Abstract group 78400.a: $C_{280}.D_{140}$

• Label: 78400.a
• Order: \( 2^{6} \cdot 5^{2} \cdot 7^{2} \)
• Exponent: \( 2^{3} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 5 \cdot 7 \)
• $\card{Z(G)}$: 280
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{2} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 3^{2} \)
• Perm deg.: $40$
• Trans deg.: not computed
• Rank: $3$

Group information

Description:	$C_{280}.D_{140}$
Order:	\(78400\)\(\medspace = 2^{6} \cdot 5^{2} \cdot 7^{2} \)
Exponent:	\(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \)
Automorphism group:	Group of order \(2580480\)\(\medspace = 2^{13} \cdot 3^{2} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_5$ 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	20	28	35	40	56	70	140	280
Elements	1	283	292	24	48	576	1192	1824	1408	2256	1152	2624	4128	10176	20544	31872	78400
Conjugacy classes	1	4	9	14	27	14	46	87	156	306	588	216	420	1788	6984	9360	20020
Divisions	1	4	6	4	5	6	12	15	22	28	26	18	22	76	150	106	501

Minimal presentations

Permutation degree:	$40$
Transitive degree:	not computed
Rank:	$3$
Inequivalent generating triples:	not computed

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

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{140}=c^{280}=[b,c]=1, b^{a}=bc^{238}, c^{a}=c^{279} \rangle$
Permutation group:	Degree $40$ $\langle(1,2,4,8,13,16,15,11)(3,6,10,14,12,9,7,5)(24,25,27,29,32)(26,30,33,31,28) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 94 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{281})$
Direct product:	not computed
Semidirect product:	$C_{35}^2$ $\,\rtimes\,$ $(D_8:C_4)$	$C_7^2$ $\,\rtimes\,$ $(D_{40}:C_{20})$	$C_5^2$ $\,\rtimes\,$ $(D_{56}:C_{28})$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$D_{280}$ . $C_{140}$	$C_{280}$ . $D_{140}$ (2)	$C_{140}^2$ . $C_2^2$	$(C_8.D_{70})$ . $C_{70}$ (2)	all 210

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

Homology

Abelianization:	$C_{2}^{2} \times C_{140} \simeq C_{2}^{2} \times C_{4} \times C_{5} \times C_{7}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	not computed

Subgroups

There are 300 normal subgroups (244 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_{280}$
Commutator:	a subgroup isomorphic to $C_{140}$
Frattini:	a subgroup isomorphic to $C_2\times C_4$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
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

All subgroups of index up to 490 (order at least 160) are shown, as well as all normal subgroups of any index.

Order 78400: $C_{280}.D_{140}$

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

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

Order 15680: $C_{56}.D_{140}$, $D_{56}.C_{140}$

Order 11200: $D_{40}.C_{140}$, $C_{40}.D_{140}$

Order 9800: $C_{35}\times C_{280}$ x 4, $C_{35}^2:Q_8$ x 2, $C_{70}\times C_{140}$ x 2, $C_{35}:C_{280}$ x 2, $C_{35}\times D_{140}$ x 2, $D_{35}\times C_{140}$ x 2, $C_{70}\wr C_2$ x 2

Order 7840: $C_{28}\times C_{280}$ x 4, $D_{140}:C_{28}$ x 2, $D_{28}:C_{140}$ x 2, $D_{28}.C_{140}$ x 2, $C_{56}.D_{70}$ x 2, $C_{56}.C_{140}$, $D_{280}:C_{14}$, $C_{280}.C_{28}$, $D_{56}:C_{70}$

Order 5600: $C_{20}\times C_{280}$ x 5, $D_{20}:C_{140}$ x 2, $D_{20}.C_{140}$ x 2, $C_{40}.D_{70}$ x 2, $D_{140}:C_{20}$ x 2, $C_{40}.C_{140}$, $D_{40}:C_{70}$, $C_{280}.C_{20}$, $D_{280}:C_{10}$

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

Order 3920: $C_{14}\times C_{280}$ x 8, $C_{28}\times C_{140}$ x 4, $C_{56}:C_{70}$ x 2, $C_{28}.C_{140}$ x 2, $C_{280}:C_{14}$ x 2, $C_{140}.C_{28}$ x 2, $C_{56}:C_{70}$ x 2, $D_7\times C_{280}$ x 2, $D_{28}:C_{70}$ x 2, $C_{280}:C_{14}$ x 2, $D_{140}:C_{14}$ x 2, $C_{56}\times D_{35}$ x 2, $C_{56}.C_{70}$, $C_{280}.C_{14}$, $C_7\times D_{280}$, $C_{35}\times D_{56}$

Order 3136: $D_{56}:C_{28}$

Order 2800: $C_{10}\times C_{280}$ x 10, $C_{20}\times C_{140}$ x 5, $C_{40}:C_{70}$ x 2, $D_5\times C_{280}$ x 2, $C_{40}\times D_{35}$ x 2, $C_{140}.C_{20}$ x 2, $C_{20}.C_{140}$ x 2, $C_{280}:C_{10}$ x 2, $C_{40}:C_{70}$ x 2, $C_{280}:C_{10}$ x 2, $D_{140}:C_{10}$ x 2, $D_{20}:C_{70}$ x 2, $C_{40}.C_{70}$, $C_{35}\times D_{40}$, $C_{280}.C_{10}$, $C_5\times D_{280}$

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

Order 2240: $D_{40}:C_{28}$, $D_8:C_{140}$, $D_{280}:C_4$, $D_{56}:C_{20}$

Order 1960: $C_7\times C_{280}$ x 16, $C_{14}\times C_{140}$ x 10, $C_{140}.C_{14}$ x 2, $D_{14}:C_{70}$ x 2, $C_{35}\times D_{28}$ x 2, $D_7\times C_{140}$ x 2, $C_{28}.C_{70}$ x 2, $C_{35}:C_{56}$ x 2, $C_7:C_{280}$ x 2, $D_{70}:C_{14}$ x 2, $C_7\times D_{140}$ x 2, $C_{28}\times D_{35}$ x 2

Order 1600: $D_{40}:C_{20}$

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

Order 1400: $C_5\times C_{280}$ x 20, $C_{10}\times C_{140}$ x 13, $C_5\times D_{140}$ x 2, $C_{20}\times D_{35}$ x 2, $C_{140}.C_{10}$ x 2, $D_{10}:C_{70}$ x 2, $C_{35}\times D_{20}$ x 2, $D_5\times C_{140}$ x 2, $C_{20}.C_{70}$ x 2, $C_{35}:C_{40}$ x 2, $C_5:C_{280}$ x 2, $D_{70}:C_{10}$ x 2

Order 1225: $C_{35}^2$

Order 1120: $C_4\times C_{280}$ x 26, $\OD_{16}:C_{70}$ x 2, $C_8.D_{70}$ x 2, $D_{28}:C_{20}$ x 2, $C_{56}.D_{10}$ x 2, $C_{40}.D_{14}$ x 2, $D_4:C_{140}$ x 2, $D_{140}:C_4$ x 2, $D_{20}:C_{28}$ x 2, $C_{280}.C_4$, $D_8:C_{70}$, $D_{280}:C_2$, $D_{40}:C_{14}$, $D_{56}:C_{10}$, $C_8.C_{140}$, $C_{280}.C_4$, $C_{280}.C_4$

Order 980: $C_7\times C_{140}$ x 20, $C_{14}\times C_{70}$ x 4, $C_7\times D_{70}$ x 2, $D_7\times C_{70}$ x 2, $C_{35}:C_{28}$ x 2, $C_7:C_{140}$ x 2

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

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

Order 700: $C_5\times C_{140}$ x 26, $C_{10}\times C_{70}$ x 5, $C_5\times D_{70}$ x 2, $D_5\times C_{70}$ x 2, $C_{35}:C_{20}$ x 2, $C_5:C_{140}$ x 2

Order 560: $C_2\times C_{280}$ x 54, $C_4\times C_{140}$ x 26, $\OD_{16}\times C_{35}$ x 4, $\SD_{16}\times C_{35}$ x 2, $C_{140}.C_4$ x 2, $C_{280}:C_2$ x 2, $C_{280}:C_2$ x 2, $C_8\times D_{35}$ x 2, $C_{20}.C_{28}$ x 2, $C_{280}:C_2$ x 2, $C_{280}:C_2$ x 2, $D_5\times C_{56}$ x 2, $C_{140}.C_4$ x 2, $C_{280}:C_2$ x 2, $C_{280}:C_2$ x 2, $D_7\times C_{40}$ x 2, $D_4:C_{70}$ x 2, $D_{140}:C_2$ x 2, $D_{20}:C_{14}$ x 2, $D_{28}:C_{10}$ x 2, $Q_{16}\times C_{35}$, $D_8\times C_{35}$, $C_{35}:Q_{16}$, $D_{280}$, $C_{35}:Q_{16}$, $C_7\times D_{40}$, $C_{35}:Q_{16}$, $C_5\times D_{56}$

Order 490: $C_7\times C_{70}$ x 10, $C_7\times D_{35}$ x 2, $D_7\times C_{35}$ x 2

Order 448: $D_8:C_{28}$, $D_{56}:C_4$

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

Order 392: $C_7\times C_{56}$ x 4, $C_{14}\times C_{28}$ x 2, $C_{14}\wr C_2$ x 2, $C_7\times D_{28}$ x 2, $D_7\times C_{28}$ x 2, $C_7^2:Q_8$ x 2, $C_7:C_{56}$ x 2

Order 350: $C_5\times C_{70}$ x 13, $C_5\times D_{35}$ x 2, $D_5\times C_{35}$ x 2

Order 320: $D_8:C_{20}$, $D_{40}:C_4$

Order 280: $C_{280}$ x 106, $C_2\times C_{140}$ x 76, $D_4\times C_{35}$ x 4, $Q_8\times C_{35}$ x 2, $C_{35}:C_8$ x 2, $C_{35}:D_4$ x 2, $D_{140}$ x 2, $C_4\times D_{35}$ x 2, $C_{35}:Q_8$ x 2, $C_{35}:D_4$ x 2, $C_7\times D_{20}$ x 2, $D_5\times C_{28}$ x 2, $C_7:C_{40}$ x 2, $C_{35}:Q_8$ x 2, $C_{35}:D_4$ x 2, $C_5\times D_{28}$ x 2, $D_7\times C_{20}$ x 2, $C_{35}:Q_8$ x 2, $C_5:C_{56}$ x 2

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

Order 224: $C_4\times C_{56}$ x 5, $C_8.D_{14}$ x 2, $D_4:C_{28}$ x 2, $\OD_{16}:C_{14}$ x 2, $D_{28}:C_4$ x 2, $D_{56}:C_2$, $C_8.C_{28}$, $C_{56}.C_4$, $D_8:C_{14}$

Order 200: $C_5\times C_{40}$ x 4, $C_{10}\times C_{20}$ x 2, $C_{10}\wr C_2$ x 2, $C_5\times D_{20}$ x 2, $D_5\times C_{20}$ x 2, $C_5^2:Q_8$ x 2, $C_5:C_{40}$ x 2

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

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

Order 160: $C_4\times C_{40}$ x 4, $D_4:C_{20}$ x 2, $\OD_{16}:C_{10}$ x 2, $C_8.D_{10}$ x 2, $D_{20}:C_4$ x 2, $C_8.C_{20}$, $C_{40}.C_4$, $D_8:C_{10}$, $D_{40}:C_2$

Order 140: $C_{140}$ x 8, $C_2\times C_{70}$ x 4

Order 112: $C_2\times C_{56}$ x 4, $C_4\times C_{28}$ x 2

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

Order 98: $C_7\times C_{14}$

Order 80: $C_2\times C_{40}$ x 4, $C_4\times C_{20}$ x 2

Order 70: $C_{70}$ x 4

Order 56: $C_{56}$ x 8, $C_2\times C_{28}$ x 2

Order 50: $C_5\times C_{10}$

Order 49: $C_7^2$

Order 40: $C_{40}$ x 8, $C_2\times C_{20}$ x 2

Order 35: $C_{35}$ x 4

Order 32: $C_4\times C_8$

Order 28: $C_{28}$ x 4, $C_2\times C_{14}$ x 2

Order 25: $C_5^2$

Order 20: $C_{20}$ x 4, $C_2\times C_{10}$ x 2

Order 16: $C_2\times C_8$ x 2, $C_4^2$

Order 14: $C_{14}$ x 2

Order 10: $C_{10}$ x 2

Order 8: $C_8$ x 4, $C_2\times C_4$

Order 7: $C_7$ x 2

Order 5: $C_5$ x 2

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

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 78400: $C_{280}.D_{140}$ ($C_1$)

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

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

Order 15680: $C_{56}.D_{140}$ ($C_5$)

Order 11200: $C_{40}.D_{140}$ ($C_7$)

Order 9800: $C_{35}^2:Q_8$ ($C_2\times C_4$) x 2, $C_{35}\times C_{280}$ ($D_4$) x 2, $C_{35}\times C_{280}$ ($C_2\times C_4$) x 2, $C_{35}\times D_{140}$ ($C_2\times C_4$) x 2, $C_{70}\times C_{140}$ ($C_2^3$)

Order 7840: $D_{140}:C_{28}$ ($C_{10}$) x 2, $C_{56}.D_{70}$ ($C_{10}$) x 2, $D_{280}:C_{14}$ ($C_{10}$), $C_{28}\times C_{280}$ ($C_{10}$), $C_{28}\times C_{280}$ ($D_5$), $C_{280}.C_{28}$ ($C_{10}$)

Order 5600: $C_{40}.D_{70}$ ($C_{14}$) x 2, $D_{140}:C_{20}$ ($C_{14}$) x 2, $C_{280}.C_{20}$ ($C_{14}$), $D_{280}:C_{10}$ ($C_{14}$), $C_{20}\times C_{280}$ ($C_{14}$), $C_{20}\times C_{280}$ ($D_7$)

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

Order 3920: $C_{280}:C_{14}$ ($C_{20}$) x 2, $C_{14}\times C_{280}$ ($C_2\times C_{10}$) x 2, $C_{14}\times C_{280}$ ($D_{10}$) x 2, $C_{140}.C_{28}$ ($C_2\times C_{10}$) x 2, $D_{140}:C_{14}$ ($C_2\times C_{10}$) x 2, $C_{280}.C_{14}$ ($C_{20}$), $C_7\times D_{280}$ ($C_{20}$), $C_{28}\times C_{140}$ ($C_2\times C_{10}$), $C_{28}\times C_{140}$ ($D_{10}$)

Order 2800: $C_{140}.C_{20}$ ($C_2\times C_{14}$) x 2, $C_{10}\times C_{280}$ ($C_2\times C_{14}$) x 2, $C_{10}\times C_{280}$ ($D_{14}$) x 2, $C_{280}:C_{10}$ ($C_{28}$) x 2, $D_{140}:C_{10}$ ($C_2\times C_{14}$) x 2, $C_{20}\times C_{140}$ ($C_2\times C_{14}$), $C_{20}\times C_{140}$ ($D_{14}$), $C_{280}.C_{10}$ ($C_{28}$), $C_5\times D_{280}$ ($C_{28}$)

Order 2450: $C_{35}\times C_{70}$ ($C_4\times D_4$)

Order 2240: $D_{280}:C_4$ ($C_{35}$)

Order 1960: $C_{140}.C_{14}$ ($C_2\times C_{20}$) x 2, $C_7\times C_{280}$ ($C_2\times C_{20}$) x 2, $C_7\times C_{280}$ ($D_{20}$) x 2, $C_7\times C_{280}$ ($C_4\times D_5$) x 2, $C_7\times C_{280}$ ($C_5\times D_4$) x 2, $C_7\times D_{140}$ ($C_2\times C_{20}$) x 2, $C_{14}\times C_{140}$ ($C_2^2\times C_{10}$), $C_{14}\times C_{140}$ ($C_2\times D_{10}$)

Order 1568: $C_{28}\times C_{56}$ ($C_5\times D_5$)

Order 1400: $C_5\times D_{140}$ ($C_2\times C_{28}$) x 2, $C_{140}.C_{10}$ ($C_2\times C_{28}$) x 2, $C_5\times C_{280}$ ($C_7\times D_4$) x 2, $C_5\times C_{280}$ ($C_2\times C_{28}$) x 2, $C_5\times C_{280}$ ($D_{28}$) x 2, $C_5\times C_{280}$ ($C_4\times D_7$) x 2, $C_{10}\times C_{140}$ ($C_2^2\times C_{14}$), $C_{10}\times C_{140}$ ($C_2\times D_{14}$)

Order 1225: $C_{35}^2$ ($D_8:C_4$)

Order 1120: $C_8.D_{70}$ ($C_{70}$) x 2, $D_{140}:C_4$ ($C_{70}$) x 2, $D_{280}:C_2$ ($C_{70}$), $C_4\times C_{280}$ ($C_{70}$), $C_4\times C_{280}$ ($D_{35}$), $C_4\times C_{280}$ ($C_5\times D_7$), $C_4\times C_{280}$ ($C_7\times D_5$), $C_{280}.C_4$ ($C_{70}$)

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

Order 800: $C_{20}\times C_{40}$ ($C_7\times D_7$)

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

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

Order 560: $C_2\times C_{280}$ ($C_7\times D_{10}$) x 2, $C_2\times C_{280}$ ($C_5\times D_{14}$) x 2, $C_2\times C_{280}$ ($C_2\times C_{70}$) x 2, $C_2\times C_{280}$ ($D_{70}$) x 2, $C_{140}.C_4$ ($C_2\times C_{70}$) x 2, $C_{280}:C_2$ ($C_{140}$) x 2, $D_{140}:C_2$ ($C_2\times C_{70}$) x 2, $C_4\times C_{140}$ ($C_7\times D_{10}$), $C_4\times C_{140}$ ($C_5\times D_{14}$), $C_4\times C_{140}$ ($C_2\times C_{70}$), $C_4\times C_{140}$ ($D_{70}$), $C_{35}:Q_{16}$ ($C_{140}$), $D_{280}$ ($C_{140}$)

Order 490: $C_7\times C_{70}$ ($C_4\times D_{20}$), $C_7\times C_{70}$ ($D_4\times C_{20}$)

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

Order 392: $C_7\times C_{56}$ ($C_5\times D_{20}$) x 2, $C_7\times C_{56}$ ($D_5\times C_{20}$) x 2, $C_{14}\times C_{28}$ ($C_{10}\times D_{10}$)

Order 350: $C_5\times C_{70}$ ($C_4\times D_{28}$), $C_5\times C_{70}$ ($D_4\times C_{28}$)

Order 280: $C_{280}$ ($D_4\times C_{35}$) x 2, $C_{280}$ ($C_2\times C_{140}$) x 2, $C_{280}$ ($D_{140}$) x 2, $C_{280}$ ($C_4\times D_{35}$) x 2, $C_{280}$ ($C_7\times D_{20}$) x 2, $C_{280}$ ($D_5\times C_{28}$) x 2, $C_{280}$ ($C_5\times D_{28}$) x 2, $C_{280}$ ($D_7\times C_{20}$) x 2, $D_{140}$ ($C_2\times C_{140}$) x 2, $C_{35}:Q_8$ ($C_2\times C_{140}$) x 2, $C_2\times C_{140}$ ($C_2^2\times C_{70}$), $C_2\times C_{140}$ ($C_2\times D_{70}$), $C_2\times C_{140}$ ($C_{14}\times D_{10}$), $C_2\times C_{140}$ ($C_{10}\times D_{14}$)

Order 245: $C_7\times C_{35}$ ($D_8:C_{20}$), $C_7\times C_{35}$ ($D_{40}:C_4$)

Order 224: $C_4\times C_{56}$ ($C_5\times D_{35}$), $C_4\times C_{56}$ ($D_5\times C_{35}$)

Order 200: $C_5\times C_{40}$ ($C_7\times D_{28}$) x 2, $C_5\times C_{40}$ ($D_7\times C_{28}$) x 2, $C_{10}\times C_{20}$ ($C_{14}\times D_{14}$)

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

Order 175: $C_5\times C_{35}$ ($D_8:C_{28}$), $C_5\times C_{35}$ ($D_{56}:C_4$)

Order 160: $C_4\times C_{40}$ ($C_7\times D_{35}$), $C_4\times C_{40}$ ($D_7\times C_{35}$)

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

Order 112: $C_2\times C_{56}$ ($C_5\times D_{70}$) x 2, $C_2\times C_{56}$ ($D_5\times C_{70}$) x 2, $C_4\times C_{28}$ ($C_5\times D_{70}$), $C_4\times C_{28}$ ($D_5\times C_{70}$)

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

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

Order 80: $C_2\times C_{40}$ ($C_7\times D_{70}$) x 2, $C_2\times C_{40}$ ($D_7\times C_{70}$) x 2, $C_4\times C_{20}$ ($C_7\times D_{70}$), $C_4\times C_{20}$ ($D_7\times C_{70}$)

Order 70: $C_{70}$ ($D_4\times C_{140}$), $C_{70}$ ($C_4\times D_{140}$), $C_{70}$ ($C_{28}\times D_{20}$), $C_{70}$ ($C_{20}\times D_{28}$)

Order 56: $C_{56}$ ($C_5\times D_{140}$) x 2, $C_{56}$ ($C_{20}\times D_{35}$) x 2, $C_{56}$ ($C_{35}\times D_{20}$) x 2, $C_{56}$ ($D_5\times C_{140}$) x 2, $C_2\times C_{28}$ ($C_{10}\times D_{70}$), $C_2\times C_{28}$ ($D_{10}\times C_{70}$)

Order 50: $C_5\times C_{10}$ ($C_{28}\times D_{28}$)

Order 49: $C_7^2$ ($D_{40}:C_{20}$)

Order 40: $C_{40}$ ($C_{35}\times D_{28}$) x 2, $C_{40}$ ($D_7\times C_{140}$) x 2, $C_{40}$ ($C_7\times D_{140}$) x 2, $C_{40}$ ($C_{28}\times D_{35}$) x 2, $C_2\times C_{20}$ ($C_{14}\times D_{70}$), $C_2\times C_{20}$ ($D_{14}\times C_{70}$)

Order 35: $C_{35}$ ($D_{40}:C_{28}$), $C_{35}$ ($D_8:C_{140}$), $C_{35}$ ($D_{280}:C_4$), $C_{35}$ ($D_{56}:C_{20}$)

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

Order 28: $C_{28}$ ($C_7:(C_{10}\times D_{20})$) x 2, $C_{28}$ ($C_7:(C_2\times C_{20}\times D_5)$) x 2, $C_2\times C_{14}$ ($D_{140}:C_{10}$), $C_2\times C_{14}$ ($D_{20}:C_{70}$)

Order 25: $C_5^2$ ($D_{56}:C_{28}$)

Order 20: $C_2\times C_{10}$ ($D_{28}:C_{70}$), $C_2\times C_{10}$ ($D_{140}:C_{14}$), $C_{20}$ ($C_7^2:(C_{10}\times D_4)$), $C_{20}$ ($C_7^2:(C_2^2\times C_{20})$), $C_{20}$ ($C_7^2:(C_2\times D_{20})$), $C_{20}$ ($C_7^2:(C_2\times C_4\times D_5)$)

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

Order 14: $C_{14}$ ($C_{35}.(C_{20}\times D_4)$), $C_{14}$ ($(C_{10}\times C_{70}).C_2^3$)

Order 10: $C_{10}$ ($C_{35}.(C_{28}\times D_4)$), $C_{10}$ ($(C_{28}\times C_{140}).C_2$)

Order 8: $C_8$ ($C_{35}\times D_{140}$) x 2, $C_8$ ($D_{35}\times C_{140}$) x 2, $C_2\times C_4$ ($C_7^2:(C_2\times C_{10}\times D_5)$)

Order 7: $C_7$ ($D_{40}.C_{140}$), $C_7$ ($C_{40}.D_{140}$)

Order 5: $C_5$ ($C_{56}.D_{140}$), $C_5$ ($D_{56}.C_{140}$)

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

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

Order 1: $C_1$ ($C_{280}.D_{140}$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $20020 \times 20020$ character table is not available for this group.

Rational character table

The $501 \times 501$ rational character table is not available for this group.