Abstract group 156800.a: $C_{280}.D_{280}$

• Label: 156800.a
• Order: \( 2^{7} \cdot 5^{2} \cdot 7^{2} \)
• Exponent: \( 2^{4} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 5 \cdot 7 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 5 \cdot 7 \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{2} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 3^{2} \)
• Perm deg.: $40$
• Trans deg.: $560$
• Rank: $2$

Group information

Description:	$C_{280}.D_{280}$
Order:	\(156800\)\(\medspace = 2^{7} \cdot 5^{2} \cdot 7^{2} \)
Exponent:	\(560\)\(\medspace = 2^{4} \cdot 5 \cdot 7 \)
Automorphism group:	Group of order \(5160960\)\(\medspace = 2^{14} \cdot 3^{2} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 7, $C_5$ x 2, $C_7$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	4	5	7	8	10	14	16	20	28	35	40	56	70	80	112	140	280	560
Elements	1	283	292	24	48	608	1192	1824	1120	1408	2256	1152	3392	5664	10176	4480	6720	20544	68736	26880	156800
Conjugacy classes	1	3	8	14	27	28	42	81	4	152	300	588	592	1176	1764	16	24	6960	27744	96	39620
Divisions	1	3	5	4	5	9	11	14	1	21	27	26	41	53	75	1	1	149	297	1	745

Minimal presentations

Permutation degree:	$40$
Transitive degree:	$560$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Groups of Lie type:	$\COPlus(2,281)$
Presentation:	$\langle a, b, c \mid a^{2}=b^{280}=c^{280}=[a,b]=[b,c]=1, c^{a}=b^{139}c^{279} \rangle$
Permutation group:	Degree $40$ $\langle(1,2,5,11,4,7,13,15)(3,8,14,9,10,16,12,6)(17,18,19,21,20)(34,35,37,36,39,40,38) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 94 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{281})$
Direct product:	$C_5$ $\, \times\, $ $C_7$ $\, \times\, $ $(D_{280}:C_8)$
Semidirect product:	$C_{35}^2$ $\,\rtimes\,$ $(C_8\wr C_2)$	$C_7^2$ $\,\rtimes\,$ $(C_{40}\wr C_2)$	$C_5^2$ $\,\rtimes\,$ $(C_{56}\wr C_2)$		more information
Trans. wreath product:	not computed
Possibly split product:	$C_{140}^2$ . $D_4$	$D_{280}$ . $C_{280}$	$C_{280}$ . $D_{280}$	$C_{280}^2$ . $C_2$	all 249

Elements of the group are displayed as matrices in $\GL_{2}(\F_{281})$.

Homology

Abelianization:	$C_{2} \times C_{280} \simeq C_{2} \times C_{8} \times C_{5} \times C_{7}$
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 15056 subgroups in 1984 conjugacy classes, 268 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{280}$	$G/Z \simeq$ $D_{280}$
Commutator:	$G' \simeq$ $C_{280}$	$G/G' \simeq$ $C_2\times C_{280}$
Frattini:	$\Phi \simeq$ $C_4\times C_8$	$G/\Phi \simeq$ $C_{35}\times D_{70}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{280}^2$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{280}.D_{280}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{35}\times C_{70}$	$G/\operatorname{soc} \simeq$ $D_4:C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\wr C_2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

Order 156800: $C_{280}.D_{280}$

Order 78400: $C_{280}.C_{280}$, $C_{280}^2$, $C_{280}.D_{140}$

Order 39200: $C_{140}\times C_{280}$ x 2, $C_{140}.C_{280}$, $C_{280}.C_{140}$, $C_{140}.D_{140}$, $C_{280}.D_{70}$, $C_{140}\wr C_2$

Order 31360: $D_{56}.C_{280}$, $C_{56}.D_{280}$

Order 22400: $D_{40}.C_{280}$, $C_{40}.D_{280}$

Order 19600: $C_{70}\times C_{280}$ x 4, $C_{35}:C_{560}$, $C_{140}^2$, $C_{140}.C_{140}$, $C_{35}^2:Q_{16}$, $C_{35}\times D_{280}$, $D_{35}\times C_{280}$, $C_{280}:C_{70}$, $C_{280}:C_{70}$, $C_{140}.D_{70}$

Order 15680: $C_{56}\times C_{280}$ x 4, $C_{56}.C_{280}$, $C_{280}.C_{56}$, $C_{56}.D_{140}$, $D_{56}.C_{140}$

Order 11200: $C_{40}\times C_{280}$ x 5, $C_{40}.C_{280}$, $D_{40}.C_{140}$, $C_{280}.C_{40}$, $C_{40}.D_{140}$

Order 9800: $C_{35}\times C_{280}$ x 8, $C_{70}\times C_{140}$ x 2, $C_{35}^2:Q_8$, $C_{35}:C_{280}$, $C_{35}\times D_{140}$, $D_{35}\times C_{140}$, $C_{70}\wr C_2$

Order 7840: $C_{28}\times C_{280}$ x 10, $C_{28}.C_{280}$, $C_{140}.C_{56}$, $C_{56}.C_{140}$, $D_{280}:C_{14}$, $D_{140}:C_{28}$, $C_{280}.C_{28}$, $D_{28}:C_{140}$, $D_{28}.C_{140}$, $C_{56}.D_{70}$, $D_{56}:C_{70}$

Order 6272: $C_{56}\wr C_2$

Order 5600: $C_{20}\times C_{280}$ x 13, $C_{40}.C_{140}$, $D_{40}:C_{70}$, $C_{20}.C_{280}$, $C_{140}.C_{40}$, $C_{280}.C_{20}$, $D_{280}:C_{10}$, $D_{20}:C_{140}$, $D_{20}.C_{140}$, $C_{40}.D_{70}$, $D_{140}:C_{20}$

Order 4900: $C_{35}\times C_{140}$ x 4, $C_{35}\times D_{70}$, $C_{35}:C_{140}$, $C_{70}^2$

Order 4480: $D_{40}:C_{56}$, $D_8:C_{280}$, $D_{280}:C_8$, $D_{56}:C_{40}$

Order 3920: $C_{14}\times C_{280}$ x 20, $C_{28}\times C_{140}$ x 4, $C_{56}:C_{70}$, $C_7:C_{560}$, $C_{56}.C_{70}$, $C_{28}.C_{140}$, $C_{280}:C_{14}$, $C_{280}.C_{14}$, $C_7\times D_{280}$, $C_{35}:C_{112}$, $C_{140}.C_{28}$, $C_{56}:C_{70}$, $D_7\times C_{280}$, $C_{35}\times D_{56}$, $D_{28}:C_{70}$, $C_{280}:C_{14}$, $D_{140}:C_{14}$, $C_{56}\times D_{35}$

Order 3200: $C_{40}\wr C_2$

Order 3136: $D_{56}:C_{28}$, $C_{56}.C_{56}$, $C_{56}^2$

Order 2800: $C_{10}\times C_{280}$ x 26, $C_{20}\times C_{140}$ x 5, $C_{40}.C_{70}$, $C_{35}\times D_{40}$, $C_{40}:C_{70}$, $C_5:C_{560}$, $D_5\times C_{280}$, $C_{35}:C_{80}$, $C_{40}\times D_{35}$, $C_{140}.C_{20}$, $C_{20}.C_{140}$, $C_{280}.C_{10}$, $C_5\times D_{280}$, $C_{280}:C_{10}$, $C_{40}:C_{70}$, $C_{280}:C_{10}$, $D_{140}:C_{10}$, $D_{20}:C_{70}$

Order 2450: $C_{35}\times C_{70}$ x 2, $C_{35}\times D_{35}$

Order 2240: $C_8\times C_{280}$ x 26, $C_{40}.C_{56}$, $D_{40}:C_{28}$, $C_8.C_{280}$, $D_8:C_{140}$, $C_{280}.C_8$, $C_{280}.C_8$, $D_{280}:C_4$, $D_{56}:C_{20}$

Order 1960: $C_7\times C_{280}$ x 40, $C_{14}\times C_{140}$ x 10, $C_{140}.C_{14}$, $D_{14}:C_{70}$, $C_{35}\times D_{28}$, $D_7\times C_{140}$, $C_{28}.C_{70}$, $C_{35}:C_{56}$, $C_7:C_{280}$, $D_{70}:C_{14}$, $C_7\times D_{140}$, $C_{28}\times D_{35}$

Order 1600: $D_{40}:C_{20}$, $C_{40}^2$, $C_{40}.C_{40}$

Order 1568: $C_{28}\times C_{56}$ x 2, $D_{56}:C_{14}$, $C_{56}.D_{14}$, $C_{56}.C_{28}$, $C_{28}.C_{56}$, $C_{28}\wr C_2$

Order 1400: $C_5\times C_{280}$ x 52, $C_{10}\times C_{140}$ x 13, $C_5\times D_{140}$, $C_{20}\times D_{35}$, $C_{140}.C_{10}$, $D_{10}:C_{70}$, $C_{35}\times D_{20}$, $D_5\times C_{140}$, $C_{20}.C_{70}$, $C_{35}:C_{40}$, $C_5:C_{280}$, $D_{70}:C_{10}$

Order 1225: $C_{35}^2$

Order 1120: $C_4\times C_{280}$ x 74, $C_{280}.C_4$, $D_8:C_{70}$, $\OD_{16}:C_{70}$, $C_{28}.C_{40}$, $D_{280}:C_2$, $C_8.D_{70}$, $D_{28}:C_{20}$, $D_{40}:C_{14}$, $C_{56}.D_{10}$, $D_{56}:C_{10}$, $C_{40}.D_{14}$, $\OD_{32}\times C_{35}$, $C_8.C_{140}$, $D_4:C_{140}$, $C_{280}.C_4$, $C_{140}.C_8$, $D_{140}:C_4$, $C_{280}.C_4$, $C_{20}.C_{56}$, $D_{20}:C_{28}$

Order 980: $C_7\times C_{140}$ x 20, $C_{14}\times C_{70}$ x 4, $C_7\times D_{70}$, $D_7\times C_{70}$, $C_{35}:C_{28}$, $C_7:C_{140}$

Order 896: $D_8:C_{56}$, $D_{56}:C_8$

Order 800: $C_{20}\times C_{40}$ x 2, $D_{40}:C_{10}$, $C_{40}.D_{10}$, $C_{40}.C_{20}$, $C_{20}.C_{40}$, $C_{20}\wr C_2$

Order 784: $C_{14}\times C_{56}$ x 4, $C_{28}^2$, $C_{28}.C_{28}$, $C_7^2:Q_{16}$, $C_7\times D_{56}$, $C_{56}:C_{14}$, $C_{56}:C_{14}$, $D_7\times C_{56}$, $C_7:C_{112}$, $D_{28}:C_{14}$

Order 700: $C_5\times C_{140}$ x 26, $C_{10}\times C_{70}$ x 5, $C_5\times D_{70}$, $D_5\times C_{70}$, $C_{35}:C_{20}$, $C_5:C_{140}$

Order 640: $D_8:C_{40}$, $D_{40}:C_8$

Order 560: $C_2\times C_{280}$ x 149, $C_4\times C_{140}$ x 26, $\OD_{16}\times C_{35}$ x 2, $Q_{16}\times C_{35}$, $\SD_{16}\times C_{35}$, $D_8\times C_{35}$, $C_{140}.C_4$, $C_{35}:Q_{16}$, $D_{280}$, $C_{280}:C_2$, $C_{280}:C_2$, $C_8\times D_{35}$, $C_{20}.C_{28}$, $C_{35}:Q_{16}$, $C_7\times D_{40}$, $C_{280}:C_2$, $C_{280}:C_2$, $D_5\times C_{56}$, $C_{560}$, $C_{140}.C_4$, $C_{35}:Q_{16}$, $C_5\times D_{56}$, $C_{280}:C_2$, $C_{280}:C_2$, $D_7\times C_{40}$, $C_{35}:C_{16}$, $C_7:C_{80}$, $D_4:C_{70}$, $D_{140}:C_2$, $D_{20}:C_{14}$, $D_{28}:C_{10}$, $C_5:C_{112}$

Order 490: $C_7\times C_{70}$ x 10, $C_7\times D_{35}$, $D_7\times C_{35}$

Order 448: $C_8\times C_{56}$ x 5, $D_8:C_{28}$, $D_{56}:C_4$, $C_{56}.C_8$, $C_8.C_{56}$

Order 400: $C_{10}\times C_{40}$ x 4, $C_{20}.C_{20}$, $C_5^2:Q_{16}$, $C_5\times D_{40}$, $C_{40}:C_{10}$, $C_{40}:C_{10}$, $D_5\times C_{40}$, $C_5:C_{80}$, $D_{20}:C_{10}$, $C_{20}^2$

Order 392: $C_7\times C_{56}$ x 8, $C_{14}\times C_{28}$ x 2, $C_{14}\wr C_2$, $C_7\times D_{28}$, $D_7\times C_{28}$, $C_7^2:Q_8$, $C_7:C_{56}$

Order 350: $C_5\times C_{70}$ x 13, $C_5\times D_{35}$, $D_5\times C_{35}$

Order 320: $C_8\times C_{40}$ x 4, $D_8:C_{20}$, $D_{40}:C_4$, $C_{40}.C_8$, $C_8.C_{40}$

Order 280: $C_{280}$ x 297, $C_2\times C_{140}$ x 75, $D_4\times C_{35}$ x 2, $Q_8\times C_{35}$, $C_{35}:C_8$, $C_{35}:D_4$, $D_{140}$, $C_4\times D_{35}$, $C_{35}:Q_8$, $C_{35}:D_4$, $C_7\times D_{20}$, $D_5\times C_{28}$, $C_7:C_{40}$, $C_{35}:Q_8$, $C_{35}:D_4$, $C_5\times D_{28}$, $D_7\times C_{20}$, $C_{35}:Q_8$, $C_5:C_{56}$

Order 245: $C_7\times C_{35}$ x 4

Order 224: $C_4\times C_{56}$ x 13, $D_{56}:C_2$, $C_8.D_{14}$, $C_7\times \OD_{32}$, $C_8.C_{28}$, $D_4:C_{28}$, $C_{56}.C_4$, $C_7:\OD_{32}$, $D_8:C_{14}$, $\OD_{16}:C_{14}$, $D_{28}:C_4$

Order 200: $C_5\times C_{40}$ x 8, $C_{10}\times C_{20}$ x 2, $C_{10}\wr C_2$, $C_5\times D_{20}$, $D_5\times C_{20}$, $C_5^2:Q_8$, $C_5:C_{40}$

Order 196: $C_7\times C_{28}$ x 4, $C_7:C_{28}$, $C_{14}^2$, $C_7\times D_{14}$

Order 175: $C_5\times C_{35}$ x 5

Order 160: $C_4\times C_{40}$ x 10, $C_5\times \OD_{32}$, $C_8.C_{20}$, $D_4:C_{20}$, $C_{40}.C_4$, $D_8:C_{10}$, $\OD_{16}:C_{10}$, $C_5:\OD_{32}$, $D_{40}:C_2$, $C_8.D_{10}$, $D_{20}:C_4$

Order 140: $C_{140}$ x 149, $C_2\times C_{70}$ x 27, $C_7\times D_{10}$, $C_5\times D_{14}$, $C_{35}:C_4$, $C_7:C_{20}$, $D_{70}$, $C_5:C_{28}$

Order 128: $C_8\wr C_2$

Order 112: $C_2\times C_{56}$ x 27, $C_4\times C_{28}$ x 5, $C_7\times \OD_{16}$ x 2, $C_7:\OD_{16}$, $C_7:Q_{16}$, $D_{56}$, $C_{56}:C_2$, $D_4:C_{14}$, $C_{56}:C_2$, $D_{28}:C_2$, $C_8\times D_7$, $C_7\times Q_{16}$, $C_7\times \SD_{16}$, $C_7\times D_8$, $C_{112}$, $C_7:C_{16}$

Order 100: $C_5\times C_{20}$ x 4, $C_5:C_{20}$, $C_{10}^2$, $C_5\times D_{10}$

Order 98: $C_7\times C_{14}$ x 2, $C_7\times D_7$

Order 80: $C_2\times C_{40}$ x 21, $C_4\times C_{20}$ x 4, $C_5\times \OD_{16}$ x 2, $C_5:Q_{16}$, $D_{40}$, $C_{40}:C_2$, $C_{40}:C_2$, $D_4:C_{10}$, $C_8\times D_5$, $D_{20}:C_2$, $C_5\times Q_{16}$, $C_5\times \SD_{16}$, $C_5\times D_8$, $C_{80}$, $C_5:\OD_{16}$, $C_5:C_{16}$

Order 70: $C_{70}$ x 75, $D_{35}$, $C_5\times D_7$, $C_7\times D_5$

Order 64: $C_8.C_8$, $C_8^2$, $D_8:C_4$

Order 56: $C_{56}$ x 53, $C_2\times C_{28}$ x 14, $C_7\times D_4$ x 2, $C_7:D_4$, $D_{28}$, $C_4\times D_7$, $C_7:Q_8$, $C_7\times Q_8$, $C_7:C_8$

Order 50: $C_5\times C_{10}$ x 2, $C_5\times D_5$

Order 49: $C_7^2$

Order 40: $C_{40}$ x 41, $C_2\times C_{20}$ x 11, $C_5\times D_4$ x 2, $C_5:D_4$, $D_{20}$, $C_4\times D_5$, $C_5:Q_8$, $C_5\times Q_8$, $C_5:C_8$

Order 35: $C_{35}$ x 26

Order 32: $C_4\times C_8$ x 2, $D_8:C_2$, $\OD_{16}:C_2$, $\OD_{32}$, $C_8.C_4$, $C_4\wr C_2$

Order 28: $C_{28}$ x 27, $C_2\times C_{14}$ x 6, $D_{14}$, $C_7:C_4$

Order 25: $C_5^2$

Order 20: $C_{20}$ x 21, $C_2\times C_{10}$ x 5, $D_{10}$, $C_5:C_4$

Order 16: $C_2\times C_8$ x 5, $\OD_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$, $C_4^2$, $D_4:C_2$, $C_{16}$

Order 14: $C_{14}$ x 14, $D_7$

Order 10: $C_{10}$ x 11, $D_5$

Order 8: $C_8$ x 9, $C_2\times C_4$ x 3, $D_4$ x 2, $Q_8$

Order 7: $C_7$ x 5

Order 5: $C_5$ x 4

Order 4: $C_4$ x 5, $C_2^2$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 156800: $C_{280}.D_{280}$

Order 78400: $C_{280}.C_{280}$, $C_{280}^2$, $C_{280}.D_{140}$

Order 39200: $C_{140}\times C_{280}$ x 2, $C_{140}.C_{280}$, $C_{280}.C_{140}$, $C_{140}.D_{140}$, $C_{280}.D_{70}$, $C_{140}\wr C_2$

Order 31360: $D_{56}.C_{280}$, $C_{56}.D_{280}$

Order 22400: $D_{40}.C_{280}$, $C_{40}.D_{280}$

Order 19600: $C_{70}\times C_{280}$ x 3, $C_{35}:C_{560}$, $C_{140}^2$, $C_{140}.C_{140}$, $C_{35}^2:Q_{16}$, $C_{35}\times D_{280}$, $D_{35}\times C_{280}$, $C_{280}:C_{70}$, $C_{280}:C_{70}$, $C_{140}.D_{70}$

Order 15680: $C_{56}\times C_{280}$ x 3, $C_{56}.C_{280}$, $C_{280}.C_{56}$, $C_{56}.D_{140}$, $D_{56}.C_{140}$

Order 11200: $C_{40}\times C_{280}$ x 3, $C_{40}.C_{280}$, $D_{40}.C_{140}$, $C_{280}.C_{40}$, $C_{40}.D_{140}$

Order 9800: $C_{35}\times C_{280}$ x 5, $C_{70}\times C_{140}$ x 2, $C_{35}^2:Q_8$, $C_{35}:C_{280}$, $C_{35}\times D_{140}$, $D_{35}\times C_{140}$, $C_{70}\wr C_2$

Order 7840: $C_{28}\times C_{280}$ x 6, $C_{28}.C_{280}$, $C_{140}.C_{56}$, $C_{56}.C_{140}$, $D_{280}:C_{14}$, $D_{140}:C_{28}$, $C_{280}.C_{28}$, $D_{28}:C_{140}$, $D_{28}.C_{140}$, $C_{56}.D_{70}$, $D_{56}:C_{70}$

Order 6272: $C_{56}\wr C_2$

Order 5600: $C_{20}\times C_{280}$ x 6, $C_{40}.C_{140}$, $D_{40}:C_{70}$, $C_{20}.C_{280}$, $C_{140}.C_{40}$, $C_{280}.C_{20}$, $D_{280}:C_{10}$, $D_{20}:C_{140}$, $D_{20}.C_{140}$, $C_{40}.D_{70}$, $D_{140}:C_{20}$

Order 4900: $C_{35}\times C_{140}$ x 3, $C_{35}\times D_{70}$, $C_{35}:C_{140}$, $C_{70}^2$

Order 4480: $D_{40}:C_{56}$, $D_8:C_{280}$, $D_{280}:C_8$, $D_{56}:C_{40}$

Order 3920: $C_{14}\times C_{280}$ x 9, $C_{28}\times C_{140}$ x 3, $C_{56}:C_{70}$, $C_7:C_{560}$, $C_{56}.C_{70}$, $C_{28}.C_{140}$, $C_{280}:C_{14}$, $C_{280}.C_{14}$, $C_7\times D_{280}$, $C_{35}:C_{112}$, $C_{140}.C_{28}$, $C_{56}:C_{70}$, $D_7\times C_{280}$, $C_{35}\times D_{56}$, $D_{28}:C_{70}$, $C_{280}:C_{14}$, $D_{140}:C_{14}$, $C_{56}\times D_{35}$

Order 3200: $C_{40}\wr C_2$

Order 3136: $D_{56}:C_{28}$, $C_{56}.C_{56}$, $C_{56}^2$

Order 2800: $C_{10}\times C_{280}$ x 9, $C_{20}\times C_{140}$ x 3, $C_{40}.C_{70}$, $C_{35}\times D_{40}$, $C_{40}:C_{70}$, $C_5:C_{560}$, $D_5\times C_{280}$, $C_{35}:C_{80}$, $C_{40}\times D_{35}$, $C_{140}.C_{20}$, $C_{20}.C_{140}$, $C_{280}.C_{10}$, $C_5\times D_{280}$, $C_{280}:C_{10}$, $C_{40}:C_{70}$, $C_{280}:C_{10}$, $D_{140}:C_{10}$, $D_{20}:C_{70}$

Order 2450: $C_{35}\times C_{70}$ x 2, $C_{35}\times D_{35}$

Order 2240: $C_8\times C_{280}$ x 9, $C_{40}.C_{56}$, $D_{40}:C_{28}$, $C_8.C_{280}$, $D_8:C_{140}$, $C_{280}.C_8$, $C_{280}.C_8$, $D_{280}:C_4$, $D_{56}:C_{20}$

Order 1960: $C_7\times C_{280}$ x 15, $C_{14}\times C_{140}$ x 6, $C_{140}.C_{14}$, $D_{14}:C_{70}$, $C_{35}\times D_{28}$, $D_7\times C_{140}$, $C_{28}.C_{70}$, $C_{35}:C_{56}$, $C_7:C_{280}$, $D_{70}:C_{14}$, $C_7\times D_{140}$, $C_{28}\times D_{35}$

Order 1600: $D_{40}:C_{20}$, $C_{40}^2$, $C_{40}.C_{40}$

Order 1568: $C_{28}\times C_{56}$ x 2, $D_{56}:C_{14}$, $C_{56}.D_{14}$, $C_{56}.C_{28}$, $C_{28}.C_{56}$, $C_{28}\wr C_2$

Order 1400: $C_5\times C_{280}$ x 15, $C_{10}\times C_{140}$ x 6, $C_5\times D_{140}$, $C_{20}\times D_{35}$, $C_{140}.C_{10}$, $D_{10}:C_{70}$, $C_{35}\times D_{20}$, $D_5\times C_{140}$, $C_{20}.C_{70}$, $C_{35}:C_{40}$, $C_5:C_{280}$, $D_{70}:C_{10}$

Order 1225: $C_{35}^2$

Order 1120: $C_4\times C_{280}$ x 18, $C_{280}.C_4$, $D_8:C_{70}$, $\OD_{16}:C_{70}$, $C_{28}.C_{40}$, $D_{280}:C_2$, $C_8.D_{70}$, $D_{28}:C_{20}$, $D_{40}:C_{14}$, $C_{56}.D_{10}$, $D_{56}:C_{10}$, $C_{40}.D_{14}$, $\OD_{32}\times C_{35}$, $C_8.C_{140}$, $D_4:C_{140}$, $C_{280}.C_4$, $C_{140}.C_8$, $D_{140}:C_4$, $C_{280}.C_4$, $C_{20}.C_{56}$, $D_{20}:C_{28}$

Order 980: $C_7\times C_{140}$ x 9, $C_{14}\times C_{70}$ x 3, $C_7\times D_{70}$, $D_7\times C_{70}$, $C_{35}:C_{28}$, $C_7:C_{140}$

Order 896: $D_8:C_{56}$, $D_{56}:C_8$

Order 800: $C_{20}\times C_{40}$ x 2, $D_{40}:C_{10}$, $C_{40}.D_{10}$, $C_{40}.C_{20}$, $C_{20}.C_{40}$, $C_{20}\wr C_2$

Order 784: $C_{14}\times C_{56}$ x 3, $C_{28}^2$, $C_{28}.C_{28}$, $C_7^2:Q_{16}$, $C_7\times D_{56}$, $C_{56}:C_{14}$, $C_{56}:C_{14}$, $D_7\times C_{56}$, $C_7:C_{112}$, $D_{28}:C_{14}$

Order 700: $C_5\times C_{140}$ x 9, $C_{10}\times C_{70}$ x 3, $C_5\times D_{70}$, $D_5\times C_{70}$, $C_{35}:C_{20}$, $C_5:C_{140}$

Order 640: $D_8:C_{40}$, $D_{40}:C_8$

Order 560: $C_2\times C_{280}$ x 28, $C_4\times C_{140}$ x 9, $\OD_{16}\times C_{35}$ x 2, $Q_{16}\times C_{35}$, $\SD_{16}\times C_{35}$, $D_8\times C_{35}$, $C_{140}.C_4$, $C_{35}:Q_{16}$, $D_{280}$, $C_{280}:C_2$, $C_{280}:C_2$, $C_8\times D_{35}$, $C_{20}.C_{28}$, $C_{35}:Q_{16}$, $C_7\times D_{40}$, $C_{280}:C_2$, $C_{280}:C_2$, $D_5\times C_{56}$, $C_{560}$, $C_{140}.C_4$, $C_{35}:Q_{16}$, $C_5\times D_{56}$, $C_{280}:C_2$, $C_{280}:C_2$, $D_7\times C_{40}$, $C_{35}:C_{16}$, $C_7:C_{80}$, $D_4:C_{70}$, $D_{140}:C_2$, $D_{20}:C_{14}$, $D_{28}:C_{10}$, $C_5:C_{112}$

Order 490: $C_7\times C_{70}$ x 6, $C_7\times D_{35}$, $D_7\times C_{35}$

Order 448: $C_8\times C_{56}$ x 3, $D_8:C_{28}$, $D_{56}:C_4$, $C_{56}.C_8$, $C_8.C_{56}$

Order 400: $C_{10}\times C_{40}$ x 3, $C_{20}.C_{20}$, $C_5^2:Q_{16}$, $C_5\times D_{40}$, $C_{40}:C_{10}$, $C_{40}:C_{10}$, $D_5\times C_{40}$, $C_5:C_{80}$, $D_{20}:C_{10}$, $C_{20}^2$

Order 392: $C_7\times C_{56}$ x 5, $C_{14}\times C_{28}$ x 2, $C_{14}\wr C_2$, $C_7\times D_{28}$, $D_7\times C_{28}$, $C_7^2:Q_8$, $C_7:C_{56}$

Order 350: $C_5\times C_{70}$ x 6, $C_5\times D_{35}$, $D_5\times C_{35}$

Order 320: $C_8\times C_{40}$ x 3, $D_8:C_{20}$, $D_{40}:C_4$, $C_{40}.C_8$, $C_8.C_{40}$

Order 280: $C_{280}$ x 46, $C_2\times C_{140}$ x 19, $D_4\times C_{35}$ x 2, $Q_8\times C_{35}$, $C_{35}:C_8$, $C_{35}:D_4$, $D_{140}$, $C_4\times D_{35}$, $C_{35}:Q_8$, $C_{35}:D_4$, $C_7\times D_{20}$, $D_5\times C_{28}$, $C_7:C_{40}$, $C_{35}:Q_8$, $C_{35}:D_4$, $C_5\times D_{28}$, $D_7\times C_{20}$, $C_{35}:Q_8$, $C_5:C_{56}$

Order 245: $C_7\times C_{35}$ x 3

Order 224: $C_4\times C_{56}$ x 6, $D_{56}:C_2$, $C_8.D_{14}$, $C_7\times \OD_{32}$, $C_8.C_{28}$, $D_4:C_{28}$, $C_{56}.C_4$, $C_7:\OD_{32}$, $D_8:C_{14}$, $\OD_{16}:C_{14}$, $D_{28}:C_4$

Order 200: $C_5\times C_{40}$ x 5, $C_{10}\times C_{20}$ x 2, $C_{10}\wr C_2$, $C_5\times D_{20}$, $D_5\times C_{20}$, $C_5^2:Q_8$, $C_5:C_{40}$

Order 196: $C_7\times C_{28}$ x 3, $C_7:C_{28}$, $C_{14}^2$, $C_7\times D_{14}$

Order 175: $C_5\times C_{35}$ x 3

Order 160: $C_4\times C_{40}$ x 6, $C_5\times \OD_{32}$, $C_8.C_{20}$, $D_4:C_{20}$, $C_{40}.C_4$, $D_8:C_{10}$, $\OD_{16}:C_{10}$, $C_5:\OD_{32}$, $D_{40}:C_2$, $C_8.D_{10}$, $D_{20}:C_4$

Order 140: $C_{140}$ x 28, $C_2\times C_{70}$ x 10, $C_7\times D_{10}$, $C_5\times D_{14}$, $C_{35}:C_4$, $C_7:C_{20}$, $D_{70}$, $C_5:C_{28}$

Order 128: $C_8\wr C_2$

Order 112: $C_2\times C_{56}$ x 10, $C_4\times C_{28}$ x 3, $C_7\times \OD_{16}$ x 2, $C_7:\OD_{16}$, $C_7:Q_{16}$, $D_{56}$, $C_{56}:C_2$, $D_4:C_{14}$, $C_{56}:C_2$, $D_{28}:C_2$, $C_8\times D_7$, $C_7\times Q_{16}$, $C_7\times \SD_{16}$, $C_7\times D_8$, $C_{112}$, $C_7:C_{16}$

Order 100: $C_5\times C_{20}$ x 3, $C_5:C_{20}$, $C_{10}^2$, $C_5\times D_{10}$

Order 98: $C_7\times C_{14}$ x 2, $C_7\times D_7$

Order 80: $C_2\times C_{40}$ x 10, $C_4\times C_{20}$ x 3, $C_5\times \OD_{16}$ x 2, $C_5:Q_{16}$, $D_{40}$, $C_{40}:C_2$, $C_{40}:C_2$, $D_4:C_{10}$, $C_8\times D_5$, $D_{20}:C_2$, $C_5\times Q_{16}$, $C_5\times \SD_{16}$, $C_5\times D_8$, $C_{80}$, $C_5:\OD_{16}$, $C_5:C_{16}$

Order 70: $C_{70}$ x 19, $D_{35}$, $C_5\times D_7$, $C_7\times D_5$

Order 64: $C_8.C_8$, $C_8^2$, $D_8:C_4$

Order 56: $C_{56}$ x 16, $C_2\times C_{28}$ x 7, $C_7\times D_4$ x 2, $C_7:D_4$, $D_{28}$, $C_4\times D_7$, $C_7:Q_8$, $C_7\times Q_8$, $C_7:C_8$

Order 50: $C_5\times C_{10}$ x 2, $C_5\times D_5$

Order 49: $C_7^2$

Order 40: $C_{40}$ x 16, $C_2\times C_{20}$ x 7, $C_5\times D_4$ x 2, $C_5:D_4$, $D_{20}$, $C_4\times D_5$, $C_5:Q_8$, $C_5\times Q_8$, $C_5:C_8$

Order 35: $C_{35}$ x 9

Order 32: $C_4\times C_8$ x 2, $D_8:C_2$, $\OD_{16}:C_2$, $\OD_{32}$, $C_8.C_4$, $C_4\wr C_2$

Order 28: $C_{28}$ x 10, $C_2\times C_{14}$ x 4, $D_{14}$, $C_7:C_4$

Order 25: $C_5^2$

Order 20: $C_{20}$ x 10, $C_2\times C_{10}$ x 4, $D_{10}$, $C_5:C_4$

Order 16: $C_2\times C_8$ x 4, $\OD_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$, $C_4^2$, $D_4:C_2$, $C_{16}$

Order 14: $C_{14}$ x 7, $D_7$

Order 10: $C_{10}$ x 7, $D_5$

Order 8: $C_8$ x 6, $C_2\times C_4$ x 3, $D_4$ x 2, $Q_8$

Order 7: $C_7$ x 3

Order 5: $C_5$ x 3

Order 4: $C_4$ x 4, $C_2^2$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 156800: $C_{280}.D_{280}$ ($C_1$)

Order 78400: $C_{280}.C_{280}$ ($C_2$), $C_{280}^2$ ($C_2$), $C_{280}.D_{140}$ ($C_2$)

Order 39200: $C_{140}\times C_{280}$ ($C_2^2$), $C_{280}.C_{140}$ ($C_4$), $C_{140}.D_{140}$ ($C_4$)

Order 31360: $C_{56}.D_{280}$ ($C_5$)

Order 22400: $C_{40}.D_{280}$ ($C_7$)

Order 19600: $C_{140}^2$ ($D_4$), $C_{35}^2:Q_{16}$ ($C_8$), $C_{35}\times D_{280}$ ($C_8$), $C_{70}\times C_{280}$ ($D_4$), $C_{70}\times C_{280}$ ($C_2\times C_4$)

Order 15680: $C_{280}.C_{56}$ ($C_{10}$), $C_{56}.D_{140}$ ($C_{10}$), $C_{56}\times C_{280}$ ($C_{10}$), $C_{56}\times C_{280}$ ($D_5$)

Order 11200: $C_{40}\times C_{280}$ ($C_{14}$), $C_{40}\times C_{280}$ ($D_7$), $C_{280}.C_{40}$ ($C_{14}$), $C_{40}.D_{140}$ ($C_{14}$)

Order 9800: $C_{70}\times C_{140}$ ($C_2^2:C_4$), $C_{35}\times C_{280}$ ($\SD_{16}$), $C_{35}\times C_{280}$ ($D_8$), $C_{35}\times C_{280}$ ($\OD_{16}$), $C_{35}\times C_{280}$ ($C_2\times C_8$)

Order 7840: $D_{280}:C_{14}$ ($C_{20}$), $C_{28}\times C_{280}$ ($C_2\times C_{10}$), $C_{28}\times C_{280}$ ($D_{10}$), $C_{280}.C_{28}$ ($C_{20}$)

Order 5600: $C_{280}.C_{20}$ ($C_{28}$), $D_{280}:C_{10}$ ($C_{28}$), $C_{20}\times C_{280}$ ($C_2\times C_{14}$), $C_{20}\times C_{280}$ ($D_{14}$)

Order 4900: $C_{35}\times C_{140}$ ($D_4:C_4$), $C_{35}\times C_{140}$ ($C_2^2:C_8$), $C_{70}^2$ ($C_4\wr C_2$)

Order 4480: $D_{280}:C_8$ ($C_{35}$)

Order 3920: $C_{280}.C_{14}$ ($C_{40}$), $C_7\times D_{280}$ ($C_{40}$), $C_{14}\times C_{280}$ ($C_2\times C_{20}$), $C_{14}\times C_{280}$ ($D_{20}$), $C_{14}\times C_{280}$ ($C_4\times D_5$), $C_{14}\times C_{280}$ ($C_5\times D_4$), $C_{28}\times C_{140}$ ($C_5:D_4$), $C_{28}\times C_{140}$ ($C_5\times D_4$)

Order 3136: $C_{56}^2$ ($C_5\times D_5$)

Order 2800: $C_{20}\times C_{140}$ ($C_7\times D_4$), $C_{20}\times C_{140}$ ($C_7:D_4$), $C_{10}\times C_{280}$ ($C_7\times D_4$), $C_{10}\times C_{280}$ ($C_2\times C_{28}$), $C_{10}\times C_{280}$ ($D_{28}$), $C_{10}\times C_{280}$ ($C_4\times D_7$), $C_{280}.C_{10}$ ($C_{56}$), $C_5\times D_{280}$ ($C_{56}$)

Order 2450: $C_{35}\times C_{70}$ ($D_4:C_8$)

Order 2240: $C_8\times C_{280}$ ($C_{70}$), $C_8\times C_{280}$ ($D_{35}$), $C_8\times C_{280}$ ($C_5\times D_7$), $C_8\times C_{280}$ ($C_7\times D_5$), $C_{280}.C_8$ ($C_{70}$), $D_{280}:C_4$ ($C_{70}$)

Order 1960: $C_7\times C_{280}$ ($D_{40}$), $C_7\times C_{280}$ ($C_{40}:C_2$), $C_7\times C_{280}$ ($C_{40}:C_2$), $C_7\times C_{280}$ ($C_8\times D_5$), $C_7\times C_{280}$ ($C_5\times \SD_{16}$), $C_7\times C_{280}$ ($C_5\times D_8$), $C_7\times C_{280}$ ($C_5\times \OD_{16}$), $C_7\times C_{280}$ ($C_2\times C_{40}$), $C_{14}\times C_{140}$ ($C_2^2:C_{20}$), $C_{14}\times C_{140}$ ($D_{10}:C_4$)

Order 1600: $C_{40}^2$ ($C_7\times D_7$)

Order 1568: $C_{28}\times C_{56}$ ($C_5\times D_{10}$)

Order 1400: $C_5\times C_{280}$ ($D_{56}$), $C_5\times C_{280}$ ($C_{56}:C_2$), $C_5\times C_{280}$ ($C_{56}:C_2$), $C_5\times C_{280}$ ($C_8\times D_7$), $C_5\times C_{280}$ ($C_7\times \SD_{16}$), $C_5\times C_{280}$ ($C_7\times D_8$), $C_5\times C_{280}$ ($C_7\times \OD_{16}$), $C_5\times C_{280}$ ($C_2\times C_{56}$), $C_{10}\times C_{140}$ ($C_2^2:C_{28}$), $C_{10}\times C_{140}$ ($D_{14}:C_4$)

Order 1225: $C_{35}^2$ ($C_8\wr C_2$)

Order 1120: $D_{280}:C_2$ ($C_{140}$), $C_4\times C_{280}$ ($C_7\times D_{10}$), $C_4\times C_{280}$ ($C_5\times D_{14}$), $C_4\times C_{280}$ ($C_2\times C_{70}$), $C_4\times C_{280}$ ($D_{70}$), $C_{280}.C_4$ ($C_{140}$)

Order 980: $C_{14}\times C_{70}$ ($D_4:C_{20}$), $C_{14}\times C_{70}$ ($D_{20}:C_4$), $C_7\times C_{140}$ ($D_4:C_{20}$), $C_7\times C_{140}$ ($C_2^2:C_{40}$), $C_7\times C_{140}$ ($D_{20}:C_4$), $C_7\times C_{140}$ ($D_{10}:C_8$)

Order 800: $C_{20}\times C_{40}$ ($C_7\times D_{14}$)

Order 784: $C_{14}\times C_{56}$ ($C_5\times D_{20}$), $C_{14}\times C_{56}$ ($D_5\times C_{20}$), $C_{28}^2$ ($C_{10}\wr C_2$)

Order 700: $C_{10}\times C_{70}$ ($D_4:C_{28}$), $C_{10}\times C_{70}$ ($D_{28}:C_4$), $C_5\times C_{140}$ ($D_4:C_{28}$), $C_5\times C_{140}$ ($C_2^2:C_{56}$), $C_5\times C_{140}$ ($D_{28}:C_4$), $C_5\times C_{140}$ ($D_{14}:C_8$)

Order 560: $C_2\times C_{280}$ ($D_4\times C_{35}$), $C_2\times C_{280}$ ($C_2\times C_{140}$), $C_2\times C_{280}$ ($D_{140}$), $C_2\times C_{280}$ ($C_4\times D_{35}$), $C_2\times C_{280}$ ($C_7\times D_{20}$), $C_2\times C_{280}$ ($D_5\times C_{28}$), $C_2\times C_{280}$ ($C_5\times D_{28}$), $C_2\times C_{280}$ ($D_7\times C_{20}$), $C_4\times C_{140}$ ($D_4\times C_{35}$), $C_4\times C_{140}$ ($C_{35}:D_4$), $C_4\times C_{140}$ ($C_{35}:D_4$), $C_4\times C_{140}$ ($C_{35}:D_4$), $C_{35}:Q_{16}$ ($C_{280}$), $D_{280}$ ($C_{280}$)

Order 490: $C_7\times C_{70}$ ($D_{20}:C_8$), $C_7\times C_{70}$ ($D_4:C_{40}$)

Order 448: $C_8\times C_{56}$ ($C_5\times D_{35}$), $C_8\times C_{56}$ ($D_5\times C_{35}$)

Order 400: $C_{10}\times C_{40}$ ($C_7\times D_{28}$), $C_{10}\times C_{40}$ ($D_7\times C_{28}$), $C_{20}^2$ ($C_{14}\wr C_2$)

Order 392: $C_{14}\times C_{28}$ ($D_{10}:C_{20}$), $C_7\times C_{56}$ ($C_5\times D_{40}$), $C_7\times C_{56}$ ($C_{40}:C_{10}$), $C_7\times C_{56}$ ($C_{40}:C_{10}$), $C_7\times C_{56}$ ($D_5\times C_{40}$)

Order 350: $C_5\times C_{70}$ ($D_{28}:C_8$), $C_5\times C_{70}$ ($D_4:C_{56}$)

Order 320: $C_8\times C_{40}$ ($C_7\times D_{35}$), $C_8\times C_{40}$ ($D_7\times C_{35}$)

Order 280: $C_{280}$ ($\SD_{16}\times C_{35}$), $C_{280}$ ($D_8\times C_{35}$), $C_{280}$ ($\OD_{16}\times C_{35}$), $C_{280}$ ($C_2\times C_{280}$), $C_{280}$ ($D_{280}$), $C_{280}$ ($C_{280}:C_2$), $C_{280}$ ($C_{280}:C_2$), $C_{280}$ ($C_8\times D_{35}$), $C_{280}$ ($C_7\times D_{40}$), $C_{280}$ ($C_{280}:C_2$), $C_{280}$ ($C_{280}:C_2$), $C_{280}$ ($D_5\times C_{56}$), $C_{280}$ ($C_5\times D_{56}$), $C_{280}$ ($C_{280}:C_2$), $C_{280}$ ($C_{280}:C_2$), $C_{280}$ ($D_7\times C_{40}$), $C_2\times C_{140}$ ($C_2^2:C_{140}$), $C_2\times C_{140}$ ($D_{70}:C_4$), $C_2\times C_{140}$ ($D_{10}:C_{28}$), $C_2\times C_{140}$ ($D_{14}:C_{20}$)

Order 245: $C_7\times C_{35}$ ($D_8:C_{40}$), $C_7\times C_{35}$ ($D_{40}:C_8$)

Order 224: $C_4\times C_{56}$ ($C_5\times D_{70}$), $C_4\times C_{56}$ ($D_5\times C_{70}$)

Order 200: $C_{10}\times C_{20}$ ($D_{14}:C_{28}$), $C_5\times C_{40}$ ($C_7\times D_{56}$), $C_5\times C_{40}$ ($C_{56}:C_{14}$), $C_5\times C_{40}$ ($C_{56}:C_{14}$), $C_5\times C_{40}$ ($D_7\times C_{56}$)

Order 196: $C_7\times C_{28}$ ($D_{20}:C_{20}$), $C_7\times C_{28}$ ($D_{10}:C_{40}$), $C_{14}^2$ ($C_{20}\wr C_2$)

Order 175: $C_5\times C_{35}$ ($D_8:C_{56}$), $C_5\times C_{35}$ ($D_{56}:C_8$)

Order 160: $C_4\times C_{40}$ ($C_7\times D_{70}$), $C_4\times C_{40}$ ($D_7\times C_{70}$)

Order 140: $C_{140}$ ($D_{28}:C_{20}$), $C_{140}$ ($D_{14}:C_{40}$), $C_{140}$ ($D_4:C_{140}$), $C_{140}$ ($C_2^2:C_{280}$), $C_{140}$ ($D_{140}:C_4$), $C_{140}$ ($D_{70}:C_8$), $C_{140}$ ($D_{20}:C_{28}$), $C_{140}$ ($D_{10}:C_{56}$), $C_2\times C_{70}$ ($D_{28}:C_{20}$), $C_2\times C_{70}$ ($D_4:C_{140}$), $C_2\times C_{70}$ ($D_{140}:C_4$), $C_2\times C_{70}$ ($D_{20}:C_{28}$)

Order 112: $C_2\times C_{56}$ ($C_5\times D_{140}$), $C_2\times C_{56}$ ($C_{20}\times D_{35}$), $C_2\times C_{56}$ ($C_{35}\times D_{20}$), $C_2\times C_{56}$ ($D_5\times C_{140}$), $C_4\times C_{28}$ ($D_{10}:C_{70}$), $C_4\times C_{28}$ ($D_{70}:C_{10}$)

Order 100: $C_5\times C_{20}$ ($D_{28}:C_{28}$), $C_5\times C_{20}$ ($D_{14}:C_{56}$), $C_{10}^2$ ($C_{28}\wr C_2$)

Order 98: $C_7\times C_{14}$ ($D_{20}:C_{40}$)

Order 80: $C_2\times C_{40}$ ($C_{35}\times D_{28}$), $C_2\times C_{40}$ ($D_7\times C_{140}$), $C_2\times C_{40}$ ($C_7\times D_{140}$), $C_2\times C_{40}$ ($C_{28}\times D_{35}$), $C_4\times C_{20}$ ($D_{14}:C_{70}$), $C_4\times C_{20}$ ($D_{70}:C_{14}$)

Order 70: $C_{70}$ ($C_{35}:(D_4:C_8)$), $C_{70}$ ($C_{35}:(D_4:C_8)$), $C_{70}$ ($C_{35}:(D_4:C_8)$), $C_{70}$ ($C_{35}:(D_4:C_8)$)

Order 64: $C_8^2$ ($C_{35}\times D_{35}$)

Order 56: $C_2\times C_{28}$ ($C_7:(C_5\times D_{10}:C_4)$), $C_2\times C_{28}$ ($C_7:(C_5\times D_{10}:C_4)$), $C_{56}$ ($C_{35}\times D_{40}$), $C_{56}$ ($C_{40}:C_{70}$), $C_{56}$ ($D_5\times C_{280}$), $C_{56}$ ($C_{40}\times D_{35}$), $C_{56}$ ($C_5\times D_{280}$), $C_{56}$ ($C_{280}:C_{10}$), $C_{56}$ ($C_{40}:C_{70}$), $C_{56}$ ($C_{280}:C_{10}$)

Order 50: $C_5\times C_{10}$ ($C_{28}^2.C_2^2$)

Order 49: $C_7^2$ ($C_{40}\wr C_2$)

Order 40: $C_2\times C_{20}$ ($C_7^2:(C_5\times C_2^2:C_4)$), $C_2\times C_{20}$ ($C_7^2:(D_{10}:C_4)$), $C_{40}$ ($C_{56}:C_{70}$), $C_{40}$ ($C_{280}:C_{14}$), $C_{40}$ ($C_7\times D_{280}$), $C_{40}$ ($C_{56}:C_{70}$), $C_{40}$ ($D_7\times C_{280}$), $C_{40}$ ($C_{35}\times D_{56}$), $C_{40}$ ($C_{280}:C_{14}$), $C_{40}$ ($C_{56}\times D_{35}$)

Order 35: $C_{35}$ ($D_{40}:C_{56}$), $C_{35}$ ($D_8:C_{280}$), $C_{35}$ ($D_{280}:C_8$), $C_{35}$ ($D_{56}:C_{40}$)

Order 32: $C_4\times C_8$ ($C_{35}\times D_{70}$)

Order 28: $C_2\times C_{14}$ ($D_{20}:C_{140}$), $C_2\times C_{14}$ ($D_{140}:C_{20}$), $C_{28}$ ($C_{140}.(C_2\times C_{20})$), $C_{28}$ ($(C_{10}\times C_{280}).C_2$), $C_{28}$ ($C_{70}.(C_2\times C_{40})$), $C_{28}$ ($C_{70}.(C_2\times C_{40})$)

Order 25: $C_5^2$ ($C_{56}\wr C_2$)

Order 20: $C_2\times C_{10}$ ($D_{140}:C_{28}$), $C_2\times C_{10}$ ($D_{28}:C_{140}$), $C_{20}$ ($(C_{14}\times C_{280}).C_2$), $C_{20}$ ($C_{70}.(C_2\times C_{56})$), $C_{20}$ ($(C_{14}\times C_{280}).C_2$), $C_{20}$ ($(C_{14}\times C_{280}).C_2$)

Order 16: $C_2\times C_8$ ($C_{35}\times D_{140}$), $C_2\times C_8$ ($D_{35}\times C_{140}$), $C_4^2$ ($C_{70}\wr C_2$)

Order 14: $C_{14}$ ($(C_{20}\times C_{280}).C_2$), $C_{14}$ ($C_{140}.(C_2\times C_{40})$)

Order 10: $C_{10}$ ($(C_{28}\times C_{280}).C_2$), $C_{10}$ ($(C_{28}\times C_{280}).C_2$)

Order 8: $C_2\times C_4$ ($C_{70}^2.C_2^2$), $C_8$ ($C_{35}\times D_{280}$), $C_8$ ($D_{35}\times C_{280}$), $C_8$ ($C_{280}:C_{70}$), $C_8$ ($C_{280}:C_{70}$)

Order 7: $C_7$ ($C_{280}.(C_2\times C_{40})$), $C_7$ ($C_{40}.D_{280}$)

Order 5: $C_5$ ($D_{56}.C_{280}$), $C_5$ ($C_{56}.D_{280}$)

Order 4: $C_2^2$ ($C_{140}\wr C_2$), $C_4$ ($(C_{70}\times C_{280}).C_2$), $C_4$ ($(C_{70}\times C_{280}).C_2$)

Order 2: $C_2$ ($C_{140}^2.C_2^2$)

Order 1: $C_1$ ($C_{280}.D_{280}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 156800: $C_{280}.D_{280}$ ($C_1$)

Order 78400: $C_{280}.C_{280}$ ($C_2$), $C_{280}^2$ ($C_2$), $C_{280}.D_{140}$ ($C_2$)

Order 39200: $C_{140}\times C_{280}$ ($C_2^2$), $C_{280}.C_{140}$ ($C_4$), $C_{140}.D_{140}$ ($C_4$)

Order 31360: $C_{56}.D_{280}$ ($C_5$)

Order 22400: $C_{40}.D_{280}$ ($C_7$)

Order 19600: $C_{140}^2$ ($D_4$), $C_{35}^2:Q_{16}$ ($C_8$), $C_{35}\times D_{280}$ ($C_8$), $C_{70}\times C_{280}$ ($D_4$), $C_{70}\times C_{280}$ ($C_2\times C_4$)

Order 15680: $C_{280}.C_{56}$ ($C_{10}$), $C_{56}.D_{140}$ ($C_{10}$), $C_{56}\times C_{280}$ ($C_{10}$), $C_{56}\times C_{280}$ ($D_5$)

Order 11200: $C_{40}\times C_{280}$ ($C_{14}$), $C_{40}\times C_{280}$ ($D_7$), $C_{280}.C_{40}$ ($C_{14}$), $C_{40}.D_{140}$ ($C_{14}$)

Order 9800: $C_{70}\times C_{140}$ ($C_2^2:C_4$), $C_{35}\times C_{280}$ ($\SD_{16}$), $C_{35}\times C_{280}$ ($D_8$), $C_{35}\times C_{280}$ ($\OD_{16}$), $C_{35}\times C_{280}$ ($C_2\times C_8$)

Order 7840: $D_{280}:C_{14}$ ($C_{20}$), $C_{28}\times C_{280}$ ($C_2\times C_{10}$), $C_{28}\times C_{280}$ ($D_{10}$), $C_{280}.C_{28}$ ($C_{20}$)

Order 5600: $C_{280}.C_{20}$ ($C_{28}$), $D_{280}:C_{10}$ ($C_{28}$), $C_{20}\times C_{280}$ ($C_2\times C_{14}$), $C_{20}\times C_{280}$ ($D_{14}$)

Order 4900: $C_{35}\times C_{140}$ ($D_4:C_4$), $C_{35}\times C_{140}$ ($C_2^2:C_8$), $C_{70}^2$ ($C_4\wr C_2$)

Order 4480: $D_{280}:C_8$ ($C_{35}$)

Order 3920: $C_{280}.C_{14}$ ($C_{40}$), $C_7\times D_{280}$ ($C_{40}$), $C_{14}\times C_{280}$ ($C_2\times C_{20}$), $C_{14}\times C_{280}$ ($D_{20}$), $C_{14}\times C_{280}$ ($C_4\times D_5$), $C_{14}\times C_{280}$ ($C_5\times D_4$), $C_{28}\times C_{140}$ ($C_5:D_4$), $C_{28}\times C_{140}$ ($C_5\times D_4$)

Order 3136: $C_{56}^2$ ($C_5\times D_5$)

Order 2800: $C_{20}\times C_{140}$ ($C_7\times D_4$), $C_{20}\times C_{140}$ ($C_7:D_4$), $C_{10}\times C_{280}$ ($C_7\times D_4$), $C_{10}\times C_{280}$ ($C_2\times C_{28}$), $C_{10}\times C_{280}$ ($D_{28}$), $C_{10}\times C_{280}$ ($C_4\times D_7$), $C_{280}.C_{10}$ ($C_{56}$), $C_5\times D_{280}$ ($C_{56}$)

Order 2450: $C_{35}\times C_{70}$ ($D_4:C_8$)

Order 2240: $C_8\times C_{280}$ ($C_{70}$), $C_8\times C_{280}$ ($D_{35}$), $C_8\times C_{280}$ ($C_5\times D_7$), $C_8\times C_{280}$ ($C_7\times D_5$), $C_{280}.C_8$ ($C_{70}$), $D_{280}:C_4$ ($C_{70}$)

Order 1960: $C_7\times C_{280}$ ($D_{40}$), $C_7\times C_{280}$ ($C_{40}:C_2$), $C_7\times C_{280}$ ($C_{40}:C_2$), $C_7\times C_{280}$ ($C_8\times D_5$), $C_7\times C_{280}$ ($C_5\times \SD_{16}$), $C_7\times C_{280}$ ($C_5\times D_8$), $C_7\times C_{280}$ ($C_5\times \OD_{16}$), $C_7\times C_{280}$ ($C_2\times C_{40}$), $C_{14}\times C_{140}$ ($C_2^2:C_{20}$), $C_{14}\times C_{140}$ ($D_{10}:C_4$)

Order 1600: $C_{40}^2$ ($C_7\times D_7$)

Order 1568: $C_{28}\times C_{56}$ ($C_5\times D_{10}$)

Order 1400: $C_5\times C_{280}$ ($D_{56}$), $C_5\times C_{280}$ ($C_{56}:C_2$), $C_5\times C_{280}$ ($C_{56}:C_2$), $C_5\times C_{280}$ ($C_8\times D_7$), $C_5\times C_{280}$ ($C_7\times \SD_{16}$), $C_5\times C_{280}$ ($C_7\times D_8$), $C_5\times C_{280}$ ($C_7\times \OD_{16}$), $C_5\times C_{280}$ ($C_2\times C_{56}$), $C_{10}\times C_{140}$ ($C_2^2:C_{28}$), $C_{10}\times C_{140}$ ($D_{14}:C_4$)

Order 1225: $C_{35}^2$ ($C_8\wr C_2$)

Order 1120: $D_{280}:C_2$ ($C_{140}$), $C_4\times C_{280}$ ($C_7\times D_{10}$), $C_4\times C_{280}$ ($C_5\times D_{14}$), $C_4\times C_{280}$ ($C_2\times C_{70}$), $C_4\times C_{280}$ ($D_{70}$), $C_{280}.C_4$ ($C_{140}$)

Order 980: $C_{14}\times C_{70}$ ($D_4:C_{20}$), $C_{14}\times C_{70}$ ($D_{20}:C_4$), $C_7\times C_{140}$ ($D_4:C_{20}$), $C_7\times C_{140}$ ($C_2^2:C_{40}$), $C_7\times C_{140}$ ($D_{20}:C_4$), $C_7\times C_{140}$ ($D_{10}:C_8$)

Order 800: $C_{20}\times C_{40}$ ($C_7\times D_{14}$)

Order 784: $C_{14}\times C_{56}$ ($C_5\times D_{20}$), $C_{14}\times C_{56}$ ($D_5\times C_{20}$), $C_{28}^2$ ($C_{10}\wr C_2$)

Order 700: $C_{10}\times C_{70}$ ($D_4:C_{28}$), $C_{10}\times C_{70}$ ($D_{28}:C_4$), $C_5\times C_{140}$ ($D_4:C_{28}$), $C_5\times C_{140}$ ($C_2^2:C_{56}$), $C_5\times C_{140}$ ($D_{28}:C_4$), $C_5\times C_{140}$ ($D_{14}:C_8$)

Order 560: $C_2\times C_{280}$ ($D_4\times C_{35}$), $C_2\times C_{280}$ ($C_2\times C_{140}$), $C_2\times C_{280}$ ($D_{140}$), $C_2\times C_{280}$ ($C_4\times D_{35}$), $C_2\times C_{280}$ ($C_7\times D_{20}$), $C_2\times C_{280}$ ($D_5\times C_{28}$), $C_2\times C_{280}$ ($C_5\times D_{28}$), $C_2\times C_{280}$ ($D_7\times C_{20}$), $C_4\times C_{140}$ ($D_4\times C_{35}$), $C_4\times C_{140}$ ($C_{35}:D_4$), $C_4\times C_{140}$ ($C_{35}:D_4$), $C_4\times C_{140}$ ($C_{35}:D_4$), $C_{35}:Q_{16}$ ($C_{280}$), $D_{280}$ ($C_{280}$)

Order 490: $C_7\times C_{70}$ ($D_{20}:C_8$), $C_7\times C_{70}$ ($D_4:C_{40}$)

Order 448: $C_8\times C_{56}$ ($C_5\times D_{35}$), $C_8\times C_{56}$ ($D_5\times C_{35}$)

Order 400: $C_{10}\times C_{40}$ ($C_7\times D_{28}$), $C_{10}\times C_{40}$ ($D_7\times C_{28}$), $C_{20}^2$ ($C_{14}\wr C_2$)

Order 392: $C_{14}\times C_{28}$ ($D_{10}:C_{20}$), $C_7\times C_{56}$ ($C_5\times D_{40}$), $C_7\times C_{56}$ ($C_{40}:C_{10}$), $C_7\times C_{56}$ ($C_{40}:C_{10}$), $C_7\times C_{56}$ ($D_5\times C_{40}$)

Order 350: $C_5\times C_{70}$ ($D_{28}:C_8$), $C_5\times C_{70}$ ($D_4:C_{56}$)

Order 320: $C_8\times C_{40}$ ($C_7\times D_{35}$), $C_8\times C_{40}$ ($D_7\times C_{35}$)

Order 280: $C_{280}$ ($\SD_{16}\times C_{35}$), $C_{280}$ ($D_8\times C_{35}$), $C_{280}$ ($\OD_{16}\times C_{35}$), $C_{280}$ ($C_2\times C_{280}$), $C_{280}$ ($D_{280}$), $C_{280}$ ($C_{280}:C_2$), $C_{280}$ ($C_{280}:C_2$), $C_{280}$ ($C_8\times D_{35}$), $C_{280}$ ($C_7\times D_{40}$), $C_{280}$ ($C_{280}:C_2$), $C_{280}$ ($C_{280}:C_2$), $C_{280}$ ($D_5\times C_{56}$), $C_{280}$ ($C_5\times D_{56}$), $C_{280}$ ($C_{280}:C_2$), $C_{280}$ ($C_{280}:C_2$), $C_{280}$ ($D_7\times C_{40}$), $C_2\times C_{140}$ ($C_2^2:C_{140}$), $C_2\times C_{140}$ ($D_{70}:C_4$), $C_2\times C_{140}$ ($D_{10}:C_{28}$), $C_2\times C_{140}$ ($D_{14}:C_{20}$)

Order 245: $C_7\times C_{35}$ ($D_8:C_{40}$), $C_7\times C_{35}$ ($D_{40}:C_8$)

Order 224: $C_4\times C_{56}$ ($C_5\times D_{70}$), $C_4\times C_{56}$ ($D_5\times C_{70}$)

Order 200: $C_{10}\times C_{20}$ ($D_{14}:C_{28}$), $C_5\times C_{40}$ ($C_7\times D_{56}$), $C_5\times C_{40}$ ($C_{56}:C_{14}$), $C_5\times C_{40}$ ($C_{56}:C_{14}$), $C_5\times C_{40}$ ($D_7\times C_{56}$)

Order 196: $C_7\times C_{28}$ ($D_{20}:C_{20}$), $C_7\times C_{28}$ ($D_{10}:C_{40}$), $C_{14}^2$ ($C_{20}\wr C_2$)

Order 175: $C_5\times C_{35}$ ($D_8:C_{56}$), $C_5\times C_{35}$ ($D_{56}:C_8$)

Order 160: $C_4\times C_{40}$ ($C_7\times D_{70}$), $C_4\times C_{40}$ ($D_7\times C_{70}$)

Order 140: $C_{140}$ ($D_{28}:C_{20}$), $C_{140}$ ($D_{14}:C_{40}$), $C_{140}$ ($D_4:C_{140}$), $C_{140}$ ($C_2^2:C_{280}$), $C_{140}$ ($D_{140}:C_4$), $C_{140}$ ($D_{70}:C_8$), $C_{140}$ ($D_{20}:C_{28}$), $C_{140}$ ($D_{10}:C_{56}$), $C_2\times C_{70}$ ($D_{28}:C_{20}$), $C_2\times C_{70}$ ($D_4:C_{140}$), $C_2\times C_{70}$ ($D_{140}:C_4$), $C_2\times C_{70}$ ($D_{20}:C_{28}$)

Order 112: $C_2\times C_{56}$ ($C_5\times D_{140}$), $C_2\times C_{56}$ ($C_{20}\times D_{35}$), $C_2\times C_{56}$ ($C_{35}\times D_{20}$), $C_2\times C_{56}$ ($D_5\times C_{140}$), $C_4\times C_{28}$ ($D_{10}:C_{70}$), $C_4\times C_{28}$ ($D_{70}:C_{10}$)

Order 100: $C_5\times C_{20}$ ($D_{28}:C_{28}$), $C_5\times C_{20}$ ($D_{14}:C_{56}$), $C_{10}^2$ ($C_{28}\wr C_2$)

Order 98: $C_7\times C_{14}$ ($D_{20}:C_{40}$)

Order 80: $C_2\times C_{40}$ ($C_{35}\times D_{28}$), $C_2\times C_{40}$ ($D_7\times C_{140}$), $C_2\times C_{40}$ ($C_7\times D_{140}$), $C_2\times C_{40}$ ($C_{28}\times D_{35}$), $C_4\times C_{20}$ ($D_{14}:C_{70}$), $C_4\times C_{20}$ ($D_{70}:C_{14}$)

Order 70: $C_{70}$ ($C_{35}:(D_4:C_8)$), $C_{70}$ ($C_{35}:(D_4:C_8)$), $C_{70}$ ($C_{35}:(D_4:C_8)$), $C_{70}$ ($C_{35}:(D_4:C_8)$)

Order 64: $C_8^2$ ($C_{35}\times D_{35}$)

Order 56: $C_2\times C_{28}$ ($C_7:(C_5\times D_{10}:C_4)$), $C_2\times C_{28}$ ($C_7:(C_5\times D_{10}:C_4)$), $C_{56}$ ($C_{35}\times D_{40}$), $C_{56}$ ($C_{40}:C_{70}$), $C_{56}$ ($D_5\times C_{280}$), $C_{56}$ ($C_{40}\times D_{35}$), $C_{56}$ ($C_5\times D_{280}$), $C_{56}$ ($C_{280}:C_{10}$), $C_{56}$ ($C_{40}:C_{70}$), $C_{56}$ ($C_{280}:C_{10}$)

Order 50: $C_5\times C_{10}$ ($C_{28}^2.C_2^2$)

Order 49: $C_7^2$ ($C_{40}\wr C_2$)

Order 40: $C_2\times C_{20}$ ($C_7^2:(C_5\times C_2^2:C_4)$), $C_2\times C_{20}$ ($C_7^2:(D_{10}:C_4)$), $C_{40}$ ($C_{56}:C_{70}$), $C_{40}$ ($C_{280}:C_{14}$), $C_{40}$ ($C_7\times D_{280}$), $C_{40}$ ($C_{56}:C_{70}$), $C_{40}$ ($D_7\times C_{280}$), $C_{40}$ ($C_{35}\times D_{56}$), $C_{40}$ ($C_{280}:C_{14}$), $C_{40}$ ($C_{56}\times D_{35}$)

Order 35: $C_{35}$ ($D_{40}:C_{56}$), $C_{35}$ ($D_8:C_{280}$), $C_{35}$ ($D_{280}:C_8$), $C_{35}$ ($D_{56}:C_{40}$)

Order 32: $C_4\times C_8$ ($C_{35}\times D_{70}$)

Order 28: $C_2\times C_{14}$ ($D_{20}:C_{140}$), $C_2\times C_{14}$ ($D_{140}:C_{20}$), $C_{28}$ ($C_{140}.(C_2\times C_{20})$), $C_{28}$ ($(C_{10}\times C_{280}).C_2$), $C_{28}$ ($C_{70}.(C_2\times C_{40})$), $C_{28}$ ($C_{70}.(C_2\times C_{40})$)

Order 25: $C_5^2$ ($C_{56}\wr C_2$)

Order 20: $C_2\times C_{10}$ ($D_{140}:C_{28}$), $C_2\times C_{10}$ ($D_{28}:C_{140}$), $C_{20}$ ($(C_{14}\times C_{280}).C_2$), $C_{20}$ ($C_{70}.(C_2\times C_{56})$), $C_{20}$ ($(C_{14}\times C_{280}).C_2$), $C_{20}$ ($(C_{14}\times C_{280}).C_2$)

Order 16: $C_2\times C_8$ ($C_{35}\times D_{140}$), $C_2\times C_8$ ($D_{35}\times C_{140}$), $C_4^2$ ($C_{70}\wr C_2$)

Order 14: $C_{14}$ ($(C_{20}\times C_{280}).C_2$), $C_{14}$ ($C_{140}.(C_2\times C_{40})$)

Order 10: $C_{10}$ ($(C_{28}\times C_{280}).C_2$), $C_{10}$ ($(C_{28}\times C_{280}).C_2$)

Order 8: $C_2\times C_4$ ($C_{70}^2.C_2^2$), $C_8$ ($C_{35}\times D_{280}$), $C_8$ ($D_{35}\times C_{280}$), $C_8$ ($C_{280}:C_{70}$), $C_8$ ($C_{280}:C_{70}$)

Order 7: $C_7$ ($C_{280}.(C_2\times C_{40})$), $C_7$ ($C_{40}.D_{280}$)

Order 5: $C_5$ ($D_{56}.C_{280}$), $C_5$ ($C_{56}.D_{280}$)

Order 4: $C_2^2$ ($C_{140}\wr C_2$), $C_4$ ($(C_{70}\times C_{280}).C_2$), $C_4$ ($(C_{70}\times C_{280}).C_2$)

Order 2: $C_2$ ($C_{140}^2.C_2^2$)

Order 1: $C_1$ ($C_{280}.D_{280}$)

Series

Derived series

$C_{280}.D_{280}$

$\rhd$

$C_{280}$

$\rhd$

$C_1$

Chief series

$C_{280}.D_{280}$

$\rhd$

$C_{280}^2$

$\rhd$

$C_{140}\times C_{280}$

$\rhd$

$C_{70}\times C_{280}$

$\rhd$

$C_{35}\times C_{280}$

$\rhd$

$C_7\times C_{280}$

$\rhd$

$C_{280}$

$\rhd$

$C_{140}$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{280}.D_{280}$

$\rhd$

$C_{280}$

$\rhd$

$C_{140}$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

Upper central series

$C_1$

$\lhd$

$C_{280}$

$\lhd$

$C_2\times C_{280}$

$\lhd$

$C_4\times C_{280}$

$\lhd$

$C_8\times C_{280}$

Character theory

Complex character table

The $39620 \times 39620$ character table is not available for this group.

Rational character table

The $745 \times 745$ rational character table is not available for this group.