Abstract group 88200.a: $C_{105}^2:D_4$

• Label: 88200.a
• Order: \( 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \cdot 5 \cdot 7 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{3} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3^{2} \)
• Perm deg.: $34$
• Trans deg.: not computed
• Rank: $2$

Group information

Description:	$C_{105}^2:D_4$
Order:	\(88200\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	Group of order \(967680\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_3$ x 2, $C_5$ 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	3	4	5	6	7	10	12	14	15	20	21	28	30	35	42	60	70	84	105	140	210	420
Elements	1	213	8	210	24	444	48	912	420	1404	192	840	384	1260	2256	1152	3672	1680	8496	2520	9216	5040	37728	10080	88200
Conjugacy classes	1	3	5	1	14	15	27	42	2	81	100	4	198	6	300	588	594	8	1764	12	4632	24	13896	48	22365
Divisions	1	3	3	1	4	8	5	11	1	14	14	1	18	1	39	26	51	1	75	1	100	1	293	1	673

Minimal presentations

Permutation degree:	$34$
Transitive degree:	not computed
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Groups of Lie type:	$\COPlus(2,211)$
Presentation:	$\langle a, b, c \mid a^{2}=b^{210}=c^{210}=[a,c]=[b,c]=1, b^{a}=b^{209}c^{57} \rangle$
Permutation group:	Degree $34$ $\langle(1,2,4,3,5)(6,7,8)(9,10,11,12,13)(14,15,16)(17,18,20,22,23,19,21)(24,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 2 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 2 & 0 \\ 0 & 106 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{211})$
Direct product:	$C_3$ $\, \times\, $ $C_5$ $\, \times\, $ $C_7$ $\, \times\, $ $(C_{105}:D_4)$
Semidirect product:	$C_{105}^2$ $\,\rtimes\,$ $D_4$	$C_{35}^2$ $\,\rtimes\,$ $(C_6\wr C_2)$	$C_7^2$ $\,\rtimes\,$ $(C_{30}\wr C_2)$	$C_5^2$ $\,\rtimes\,$ $(C_{42}\wr C_2)$	all 7
Trans. wreath product:	not computed
Possibly split product:	$D_{210}$ . $C_{210}$	$C_{210}$ . $D_{210}$	$C_{210}^2$ . $C_2$	$(C_{70}\times C_{210})$ . $S_3$	all 207

Elements of the group are displayed as matrices in $\GL_{2}(\F_{211})$.

Homology

Abelianization:	$C_{2} \times C_{210} \simeq C_{2}^{2} \times C_{3} \times C_{5} \times C_{7}$
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 216 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{210}$	$G/Z \simeq$ $D_{210}$
Commutator:	$G' \simeq$ $C_{210}$	$G/G' \simeq$ $C_2\times C_{210}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{210}.C_{210}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{210}^2$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{105}^2:D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{105}\times C_{210}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
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

All subgroups of index up to 120 (order at least 735) are shown, as well as all normal subgroups of any index.

Order 88200: $C_{105}^2:D_4$

Order 44100: $C_{105}:C_{420}$, $C_{210}.C_{210}$, $C_{210}^2$

Order 29400: $D_{70}.C_{210}$, $C_{70}.D_{210}$

Order 22050: $C_{105}\times C_{210}$ x 2, $C_{105}\times D_{105}$

Order 17640: $D_{42}.C_{210}$, $C_{42}.D_{210}$

Order 14700: $C_{70}\times C_{210}$ x 3, $C_{105}:C_{140}$, $C_{105}\times D_{70}$, $C_{105}:C_{140}$, $C_{35}\times D_{210}$

Order 12600: $D_{30}.C_{210}$, $C_{30}.D_{210}$

Order 11025: $C_{105}^2$

Order 9800: $C_{70}\wr C_2$

Order 8820: $C_{42}\times C_{210}$ x 4, $C_{21}:C_{420}$, $D_{21}\times C_{210}$, $C_{105}:C_{84}$, $C_{21}\times D_{210}$

Order 7350: $C_{35}\times C_{210}$ x 7, $D_{35}\times C_{105}$, $C_{35}\times D_{105}$

Order 6300: $C_{30}\times C_{210}$ x 5, $D_{15}\times C_{210}$, $C_{15}:C_{420}$, $C_{105}:C_{60}$, $C_{15}\times D_{210}$

Order 5880: $D_{14}:C_{210}$, $D_{42}:C_{70}$, $D_{70}:C_{42}$, $D_{210}:C_{14}$

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

Order 4410: $C_{21}\times C_{210}$ x 10, $D_{21}\times C_{105}$, $C_{21}\times D_{105}$

Order 4200: $D_{10}:C_{210}$, $D_{30}:C_{70}$, $D_{70}:C_{30}$, $D_{210}:C_{10}$

Order 3675: $C_{35}\times C_{105}$ x 3

Order 3528: $C_{42}\wr C_2$

Order 3150: $C_{15}\times C_{210}$ x 13, $D_{15}\times C_{105}$, $C_{15}\times D_{105}$

Order 2940: $C_{14}\times C_{210}$ x 14, $C_{35}:C_{84}$, $C_{21}\times D_{70}$, $D_7\times C_{210}$, $C_7:C_{420}$, $C_{105}:C_{28}$, $C_{21}:C_{140}$, $C_{35}\times D_{42}$, $C_7\times D_{210}$

Order 2520: $D_6:C_{210}$, $D_{30}:C_{42}$, $D_{42}:C_{30}$, $D_{210}:C_6$

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

Order 2205: $C_{21}\times C_{105}$ x 4

Order 2100: $C_{10}\times C_{210}$ x 18, $C_{35}\times D_{30}$, $C_5:C_{420}$, $D_5\times C_{210}$, $C_{15}\times D_{70}$, $C_{35}:C_{60}$, $C_{105}:C_{20}$, $C_{15}:C_{140}$, $C_5\times D_{210}$

Order 1960: $D_{14}:C_{70}$, $D_{70}:C_{14}$

Order 1800: $C_{30}\wr C_2$

Order 1764: $C_{42}^2$, $C_{21}\times D_{42}$, $C_{21}:C_{84}$

Order 1575: $C_{15}\times C_{105}$ x 5

Order 1470: $C_7\times C_{210}$ x 38, $C_7\times D_{105}$, $C_{35}\times D_{21}$, $C_{21}\times D_{35}$, $D_7\times C_{105}$

Order 1400: $D_{10}:C_{70}$, $D_{70}:C_{10}$

Order 1260: $C_6\times C_{210}$ x 26, $C_{105}:C_{12}$, $C_{15}:C_{84}$, $C_{21}:C_{60}$, $C_3:C_{420}$, $C_3\times D_{210}$, $C_{21}\times D_{30}$, $C_{15}\times D_{42}$, $S_3\times C_{210}$

Order 1225: $C_{35}^2$

Order 1176: $D_{42}:C_{14}$, $D_{14}:C_{42}$

Order 1050: $C_5\times C_{210}$ x 50, $C_5\times D_{105}$, $D_{15}\times C_{35}$, $C_{15}\times D_{35}$, $D_5\times C_{105}$

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

Order 900: $C_{15}:C_{60}$, $C_{30}^2$, $C_{15}\times D_{30}$

Order 882: $C_{21}\times C_{42}$ x 2, $C_{21}\times D_{21}$

Order 840: $D_4\times C_{105}$, $C_{105}:D_4$, $C_{105}:D_4$, $C_{105}:D_4$, $D_6:C_{70}$, $C_{105}:D_4$, $D_{10}:C_{42}$, $D_{14}:C_{30}$

Order 735: $C_7\times C_{105}$ x 14

Order 700: $C_{10}\times C_{70}$ x 2

Order 630: $C_3\times C_{210}$ x 4

Order 588: $C_{14}\times C_{42}$ x 2

Order 525: $C_5\times C_{105}$ x 4

Order 490: $C_7\times C_{70}$ x 2

Order 450: $C_{15}\times C_{30}$

Order 441: $C_{21}^2$

Order 420: $C_2\times C_{210}$ x 8, $D_{210}$, $C_{105}:C_4$

Order 350: $C_5\times C_{70}$ x 2

Order 315: $C_3\times C_{105}$ x 4

Order 300: $C_{10}\times C_{30}$ x 2

Order 294: $C_7\times C_{42}$ x 2

Order 252: $C_6\times C_{42}$ x 2

Order 245: $C_7\times C_{35}$ x 2

Order 225: $C_{15}^2$

Order 210: $C_{210}$ x 8

Order 196: $C_{14}^2$

Order 180: $C_6\times C_{30}$ x 2

Order 175: $C_5\times C_{35}$ x 2

Order 150: $C_5\times C_{30}$ x 2

Order 147: $C_7\times C_{21}$ x 2

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

Order 126: $C_3\times C_{42}$ x 2

Order 105: $C_{105}$ x 8

Order 100: $C_{10}^2$

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

Order 90: $C_3\times C_{30}$ x 2

Order 84: $C_2\times C_{42}$ x 4

Order 75: $C_5\times C_{15}$ x 2

Order 70: $C_{70}$ x 4

Order 63: $C_3\times C_{21}$ x 2

Order 60: $C_2\times C_{30}$ x 4

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

Order 49: $C_7^2$

Order 45: $C_3\times C_{15}$ x 2

Order 42: $C_{42}$ x 4

Order 36: $C_6^2$

Order 35: $C_{35}$ x 4

Order 30: $C_{30}$ x 4

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

Order 25: $C_5^2$

Order 21: $C_{21}$ x 4

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

Order 18: $C_3\times C_6$

Order 15: $C_{15}$ x 4

Order 14: $C_{14}$ x 2

Order 12: $C_2\times C_6$ x 2

Order 10: $C_{10}$ x 2

Order 9: $C_3^2$

Order 8: $D_4$

Order 7: $C_7$ x 2

Order 6: $C_6$ x 2

Order 5: $C_5$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 120 (order at least 735) are shown, as well as all normal subgroups of any index.

Order 88200: $C_{105}^2:D_4$

Order 44100: $C_{105}:C_{420}$, $C_{210}.C_{210}$, $C_{210}^2$

Order 29400: $D_{70}.C_{210}$, $C_{70}.D_{210}$

Order 22050: $C_{105}\times C_{210}$ x 2, $C_{105}\times D_{105}$

Order 17640: $D_{42}.C_{210}$, $C_{42}.D_{210}$

Order 14700: $C_{70}\times C_{210}$ x 3, $C_{105}:C_{140}$, $C_{105}\times D_{70}$, $C_{105}:C_{140}$, $C_{35}\times D_{210}$

Order 12600: $D_{30}.C_{210}$, $C_{30}.D_{210}$

Order 11025: $C_{105}^2$

Order 9800: $C_{70}\wr C_2$

Order 8820: $C_{42}\times C_{210}$ x 3, $C_{21}:C_{420}$, $D_{21}\times C_{210}$, $C_{105}:C_{84}$, $C_{21}\times D_{210}$

Order 7350: $C_{35}\times C_{210}$ x 6, $D_{35}\times C_{105}$, $C_{35}\times D_{105}$

Order 6300: $C_{30}\times C_{210}$ x 3, $D_{15}\times C_{210}$, $C_{15}:C_{420}$, $C_{105}:C_{60}$, $C_{15}\times D_{210}$

Order 5880: $D_{14}:C_{210}$, $D_{42}:C_{70}$, $D_{70}:C_{42}$, $D_{210}:C_{14}$

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

Order 4410: $C_{21}\times C_{210}$ x 6, $D_{21}\times C_{105}$, $C_{21}\times D_{105}$

Order 4200: $D_{10}:C_{210}$, $D_{30}:C_{70}$, $D_{70}:C_{30}$, $D_{210}:C_{10}$

Order 3675: $C_{35}\times C_{105}$ x 3

Order 3528: $C_{42}\wr C_2$

Order 3150: $C_{15}\times C_{210}$ x 6, $D_{15}\times C_{105}$, $C_{15}\times D_{105}$

Order 2940: $C_{14}\times C_{210}$ x 9, $C_{35}:C_{84}$, $C_{21}\times D_{70}$, $D_7\times C_{210}$, $C_7:C_{420}$, $C_{105}:C_{28}$, $C_{21}:C_{140}$, $C_{35}\times D_{42}$, $C_7\times D_{210}$

Order 2520: $D_6:C_{210}$, $D_{30}:C_{42}$, $D_{42}:C_{30}$, $D_{210}:C_6$

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

Order 2205: $C_{21}\times C_{105}$ x 3

Order 2100: $C_{10}\times C_{210}$ x 9, $C_{35}\times D_{30}$, $C_5:C_{420}$, $D_5\times C_{210}$, $C_{15}\times D_{70}$, $C_{35}:C_{60}$, $C_{105}:C_{20}$, $C_{15}:C_{140}$, $C_5\times D_{210}$

Order 1960: $D_{14}:C_{70}$, $D_{70}:C_{14}$

Order 1800: $C_{30}\wr C_2$

Order 1764: $C_{42}^2$, $C_{21}\times D_{42}$, $C_{21}:C_{84}$

Order 1575: $C_{15}\times C_{105}$ x 3

Order 1470: $C_7\times C_{210}$ x 18, $C_7\times D_{105}$, $C_{35}\times D_{21}$, $C_{21}\times D_{35}$, $D_7\times C_{105}$

Order 1400: $D_{10}:C_{70}$, $D_{70}:C_{10}$

Order 1260: $C_6\times C_{210}$ x 9, $C_{105}:C_{12}$, $C_{15}:C_{84}$, $C_{21}:C_{60}$, $C_3:C_{420}$, $C_3\times D_{210}$, $C_{21}\times D_{30}$, $C_{15}\times D_{42}$, $S_3\times C_{210}$

Order 1225: $C_{35}^2$

Order 1176: $D_{42}:C_{14}$, $D_{14}:C_{42}$

Order 1050: $C_5\times C_{210}$ x 18, $C_5\times D_{105}$, $D_{15}\times C_{35}$, $C_{15}\times D_{35}$, $D_5\times C_{105}$

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

Order 900: $C_{15}:C_{60}$, $C_{30}^2$, $C_{15}\times D_{30}$

Order 882: $C_{21}\times C_{42}$ x 2, $C_{21}\times D_{21}$

Order 840: $D_4\times C_{105}$, $C_{105}:D_4$, $C_{105}:D_4$, $C_{105}:D_4$, $D_6:C_{70}$, $C_{105}:D_4$, $D_{10}:C_{42}$, $D_{14}:C_{30}$

Order 735: $C_7\times C_{105}$ x 9

Order 700: $C_{10}\times C_{70}$ x 2

Order 630: $C_3\times C_{210}$ x 4

Order 588: $C_{14}\times C_{42}$ x 2

Order 525: $C_5\times C_{105}$ x 4

Order 490: $C_7\times C_{70}$ x 2

Order 450: $C_{15}\times C_{30}$

Order 441: $C_{21}^2$

Order 420: $C_2\times C_{210}$ x 8, $D_{210}$, $C_{105}:C_4$

Order 350: $C_5\times C_{70}$ x 2

Order 315: $C_3\times C_{105}$ x 4

Order 300: $C_{10}\times C_{30}$ x 2

Order 294: $C_7\times C_{42}$ x 2

Order 252: $C_6\times C_{42}$ x 2

Order 245: $C_7\times C_{35}$ x 2

Order 225: $C_{15}^2$

Order 210: $C_{210}$ x 8

Order 196: $C_{14}^2$

Order 180: $C_6\times C_{30}$ x 2

Order 175: $C_5\times C_{35}$ x 2

Order 150: $C_5\times C_{30}$ x 2

Order 147: $C_7\times C_{21}$ x 2

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

Order 126: $C_3\times C_{42}$ x 2

Order 105: $C_{105}$ x 8

Order 100: $C_{10}^2$

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

Order 90: $C_3\times C_{30}$ x 2

Order 84: $C_2\times C_{42}$ x 4

Order 75: $C_5\times C_{15}$ x 2

Order 70: $C_{70}$ x 4

Order 63: $C_3\times C_{21}$ x 2

Order 60: $C_2\times C_{30}$ x 4

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

Order 49: $C_7^2$

Order 45: $C_3\times C_{15}$ x 2

Order 42: $C_{42}$ x 4

Order 36: $C_6^2$

Order 35: $C_{35}$ x 4

Order 30: $C_{30}$ x 4

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

Order 25: $C_5^2$

Order 21: $C_{21}$ x 4

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

Order 18: $C_3\times C_6$

Order 15: $C_{15}$ x 4

Order 14: $C_{14}$ x 2

Order 12: $C_2\times C_6$ x 2

Order 10: $C_{10}$ x 2

Order 9: $C_3^2$

Order 8: $D_4$

Order 7: $C_7$ x 2

Order 6: $C_6$ x 2

Order 5: $C_5$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 88200: $C_{105}^2:D_4$ ($C_1$)

Order 44100: $C_{105}:C_{420}$ ($C_2$), $C_{210}.C_{210}$ ($C_2$), $C_{210}^2$ ($C_2$)

Order 29400: $C_{70}.D_{210}$ ($C_3$)

Order 22050: $C_{105}\times C_{210}$ ($C_2^2$)

Order 17640: $C_{42}.D_{210}$ ($C_5$)

Order 14700: $C_{70}\times C_{210}$ ($C_6$), $C_{70}\times C_{210}$ ($S_3$), $C_{105}:C_{140}$ ($C_6$), $C_{35}\times D_{210}$ ($C_6$)

Order 12600: $C_{30}.D_{210}$ ($C_7$)

Order 11025: $C_{105}^2$ ($D_4$)

Order 8820: $C_{105}:C_{84}$ ($C_{10}$), $C_{21}\times D_{210}$ ($C_{10}$), $C_{42}\times C_{210}$ ($C_{10}$), $C_{42}\times C_{210}$ ($D_5$)

Order 7350: $C_{35}\times C_{210}$ ($C_2\times C_6$), $C_{35}\times C_{210}$ ($D_6$)

Order 6300: $C_{105}:C_{60}$ ($C_{14}$), $C_{15}\times D_{210}$ ($C_{14}$), $C_{30}\times C_{210}$ ($C_{14}$), $C_{30}\times C_{210}$ ($D_7$)

Order 5880: $D_{210}:C_{14}$ ($C_{15}$)

Order 4900: $C_{70}^2$ ($C_3\times S_3$)

Order 4410: $C_{21}\times C_{210}$ ($C_2\times C_{10}$), $C_{21}\times C_{210}$ ($D_{10}$)

Order 4200: $D_{210}:C_{10}$ ($C_{21}$)

Order 3675: $C_{35}\times C_{105}$ ($C_3:D_4$), $C_{35}\times C_{105}$ ($C_3\times D_4$)

Order 3150: $C_{15}\times C_{210}$ ($C_2\times C_{14}$), $C_{15}\times C_{210}$ ($D_{14}$)

Order 2940: $C_{14}\times C_{210}$ ($C_{30}$), $C_{14}\times C_{210}$ ($D_{15}$), $C_{14}\times C_{210}$ ($C_3\times D_5$), $C_{14}\times C_{210}$ ($C_5\times S_3$), $C_{105}:C_{28}$ ($C_{30}$), $C_7\times D_{210}$ ($C_{30}$)

Order 2520: $D_{210}:C_6$ ($C_{35}$)

Order 2450: $C_{35}\times C_{70}$ ($C_6\times S_3$)

Order 2205: $C_{21}\times C_{105}$ ($C_5:D_4$), $C_{21}\times C_{105}$ ($C_5\times D_4$)

Order 2100: $C_{10}\times C_{210}$ ($C_{42}$), $C_{10}\times C_{210}$ ($D_{21}$), $C_{10}\times C_{210}$ ($C_3\times D_7$), $C_{10}\times C_{210}$ ($S_3\times C_7$), $C_{105}:C_{20}$ ($C_{42}$), $C_5\times D_{210}$ ($C_{42}$)

Order 1764: $C_{42}^2$ ($C_5\times D_5$)

Order 1575: $C_{15}\times C_{105}$ ($C_7\times D_4$), $C_{15}\times C_{105}$ ($C_7:D_4$)

Order 1470: $C_7\times C_{210}$ ($C_2\times C_{30}$), $C_7\times C_{210}$ ($D_{30}$), $C_7\times C_{210}$ ($S_3\times C_{10}$), $C_7\times C_{210}$ ($C_3\times D_{10}$)

Order 1260: $C_{105}:C_{12}$ ($C_{70}$), $C_6\times C_{210}$ ($C_{70}$), $C_6\times C_{210}$ ($D_{35}$), $C_6\times C_{210}$ ($C_5\times D_7$), $C_6\times C_{210}$ ($C_7\times D_5$), $C_3\times D_{210}$ ($C_{70}$)

Order 1225: $C_{35}^2$ ($C_6\wr C_2$)

Order 1050: $C_5\times C_{210}$ ($C_2\times C_{42}$), $C_5\times C_{210}$ ($D_{42}$), $C_5\times C_{210}$ ($S_3\times C_{14}$), $C_5\times C_{210}$ ($C_3\times D_{14}$)

Order 980: $C_{14}\times C_{70}$ ($C_3\times D_{15}$), $C_{14}\times C_{70}$ ($S_3\times C_{15}$)

Order 900: $C_{30}^2$ ($C_7\times D_7$)

Order 882: $C_{21}\times C_{42}$ ($C_5\times D_{10}$)

Order 840: $C_{105}:D_4$ ($C_{105}$)

Order 735: $C_7\times C_{105}$ ($D_4\times C_{15}$), $C_7\times C_{105}$ ($C_{15}:D_4$), $C_7\times C_{105}$ ($C_{15}:D_4$), $C_7\times C_{105}$ ($C_{15}:D_4$)

Order 700: $C_{10}\times C_{70}$ ($C_3\times D_{21}$), $C_{10}\times C_{70}$ ($S_3\times C_{21}$)

Order 630: $C_3\times C_{210}$ ($C_7\times D_{10}$), $C_3\times C_{210}$ ($C_5\times D_{14}$), $C_3\times C_{210}$ ($C_2\times C_{70}$), $C_3\times C_{210}$ ($D_{70}$)

Order 588: $C_{14}\times C_{42}$ ($D_5\times C_{15}$), $C_{14}\times C_{42}$ ($C_5\times D_{15}$)

Order 525: $C_5\times C_{105}$ ($D_4\times C_{21}$), $C_5\times C_{105}$ ($C_{21}:D_4$), $C_5\times C_{105}$ ($D_6:C_{14}$), $C_5\times C_{105}$ ($C_{21}:D_4$)

Order 490: $C_7\times C_{70}$ ($C_3\times D_{30}$), $C_7\times C_{70}$ ($S_3\times C_{30}$)

Order 450: $C_{15}\times C_{30}$ ($C_7\times D_{14}$)

Order 441: $C_{21}^2$ ($C_{10}\wr C_2$)

Order 420: $C_2\times C_{210}$ ($C_5\times D_{21}$), $C_2\times C_{210}$ ($S_3\times C_{35}$), $C_2\times C_{210}$ ($C_3\times D_{35}$), $C_2\times C_{210}$ ($D_5\times C_{21}$), $C_2\times C_{210}$ ($D_7\times C_{15}$), $C_2\times C_{210}$ ($C_{210}$), $C_2\times C_{210}$ ($D_{105}$), $C_2\times C_{210}$ ($C_7\times D_{15}$), $D_{210}$ ($C_{210}$), $C_{105}:C_4$ ($C_{210}$)

Order 350: $C_5\times C_{70}$ ($C_3\times D_{42}$), $C_5\times C_{70}$ ($S_3\times C_{42}$)

Order 315: $C_3\times C_{105}$ ($D_4\times C_{35}$), $C_3\times C_{105}$ ($C_{35}:D_4$), $C_3\times C_{105}$ ($C_{35}:D_4$), $C_3\times C_{105}$ ($C_{35}:D_4$)

Order 300: $C_{10}\times C_{30}$ ($C_7\times D_{21}$), $C_{10}\times C_{30}$ ($D_7\times C_{21}$)

Order 294: $C_7\times C_{42}$ ($C_5\times D_{30}$), $C_7\times C_{42}$ ($C_{15}\times D_{10}$)

Order 252: $C_6\times C_{42}$ ($C_5\times D_{35}$), $C_6\times C_{42}$ ($D_5\times C_{35}$)

Order 245: $C_7\times C_{35}$ ($D_6:C_{30}$), $C_7\times C_{35}$ ($D_{30}:C_6$)

Order 225: $C_{15}^2$ ($C_{14}\wr C_2$)

Order 210: $C_{210}$ ($C_2\times C_{210}$), $C_{210}$ ($D_{210}$), $C_{210}$ ($C_7\times D_{30}$), $C_{210}$ ($C_5\times D_{42}$), $C_{210}$ ($S_3\times C_{70}$), $C_{210}$ ($C_3\times D_{70}$), $C_{210}$ ($D_5\times C_{42}$), $C_{210}$ ($C_{15}\times D_{14}$)

Order 196: $C_{14}^2$ ($C_{15}\times D_{15}$)

Order 180: $C_6\times C_{30}$ ($C_7\times D_{35}$), $C_6\times C_{30}$ ($D_7\times C_{35}$)

Order 175: $C_5\times C_{35}$ ($D_{42}:C_6$), $C_5\times C_{35}$ ($D_6:C_{42}$)

Order 150: $C_5\times C_{30}$ ($C_7\times D_{42}$), $C_5\times C_{30}$ ($C_{21}\times D_{14}$)

Order 147: $C_7\times C_{21}$ ($D_{30}:C_{10}$), $C_7\times C_{21}$ ($D_{10}:C_{30}$)

Order 140: $C_2\times C_{70}$ ($C_3\times D_{105}$), $C_2\times C_{70}$ ($C_{21}\times D_{15}$), $C_2\times C_{70}$ ($C_{15}\times D_{21}$), $C_2\times C_{70}$ ($S_3\times C_{105}$)

Order 126: $C_3\times C_{42}$ ($C_5\times D_{70}$), $C_3\times C_{42}$ ($D_5\times C_{70}$)

Order 105: $C_{105}$ ($D_4\times C_{105}$), $C_{105}$ ($C_{105}:D_4$), $C_{105}$ ($C_{105}:D_4$), $C_{105}$ ($C_{105}:D_4$), $C_{105}$ ($D_6:C_{70}$), $C_{105}$ ($C_{105}:D_4$), $C_{105}$ ($D_{10}:C_{42}$), $C_{105}$ ($D_{14}:C_{30}$)

Order 100: $C_{10}^2$ ($C_{21}\times D_{21}$)

Order 98: $C_7\times C_{14}$ ($C_{15}\times D_{30}$)

Order 90: $C_3\times C_{30}$ ($C_7\times D_{70}$), $C_3\times C_{30}$ ($D_7\times C_{70}$)

Order 84: $C_2\times C_{42}$ ($C_5\times D_{105}$), $C_2\times C_{42}$ ($D_{15}\times C_{35}$), $C_2\times C_{42}$ ($C_{15}\times D_{35}$), $C_2\times C_{42}$ ($D_5\times C_{105}$)

Order 75: $C_5\times C_{15}$ ($D_{42}:C_{14}$), $C_5\times C_{15}$ ($D_{14}:C_{42}$)

Order 70: $C_{70}$ ($C_3\times D_{210}$), $C_{70}$ ($C_{21}\times D_{30}$), $C_{70}$ ($C_{15}\times D_{42}$), $C_{70}$ ($S_3\times C_{210}$)

Order 63: $C_3\times C_{21}$ ($D_{10}:C_{70}$), $C_3\times C_{21}$ ($D_{70}:C_{10}$)

Order 60: $C_2\times C_{30}$ ($C_7\times D_{105}$), $C_2\times C_{30}$ ($C_{35}\times D_{21}$), $C_2\times C_{30}$ ($C_{21}\times D_{35}$), $C_2\times C_{30}$ ($D_7\times C_{105}$)

Order 50: $C_5\times C_{10}$ ($C_{21}\times D_{42}$)

Order 49: $C_7^2$ ($C_{30}\wr C_2$)

Order 45: $C_3\times C_{15}$ ($D_{14}:C_{70}$), $C_3\times C_{15}$ ($D_{70}:C_{14}$)

Order 42: $C_{42}$ ($C_{35}\times D_{30}$), $C_{42}$ ($D_5\times C_{210}$), $C_{42}$ ($C_{15}\times D_{70}$), $C_{42}$ ($C_5\times D_{210}$)

Order 36: $C_6^2$ ($C_{35}\times D_{35}$)

Order 35: $C_{35}$ ($D_6:C_{210}$), $C_{35}$ ($D_{30}:C_{42}$), $C_{35}$ ($D_{42}:C_{30}$), $C_{35}$ ($D_{210}:C_6$)

Order 30: $C_{30}$ ($C_{21}\times D_{70}$), $C_{30}$ ($D_7\times C_{210}$), $C_{30}$ ($C_{35}\times D_{42}$), $C_{30}$ ($C_7\times D_{210}$)

Order 28: $C_2\times C_{14}$ ($D_{15}\times C_{105}$), $C_2\times C_{14}$ ($C_{15}\times D_{105}$)

Order 25: $C_5^2$ ($C_{42}\wr C_2$)

Order 21: $C_{21}$ ($D_{10}:C_{210}$), $C_{21}$ ($D_{30}:C_{70}$), $C_{21}$ ($D_{70}:C_{30}$), $C_{21}$ ($D_{210}:C_{10}$)

Order 20: $C_2\times C_{10}$ ($D_{21}\times C_{105}$), $C_2\times C_{10}$ ($C_{21}\times D_{105}$)

Order 18: $C_3\times C_6$ ($C_{35}\times D_{70}$)

Order 15: $C_{15}$ ($D_{14}:C_{210}$), $C_{15}$ ($D_{42}:C_{70}$), $C_{15}$ ($D_{70}:C_{42}$), $C_{15}$ ($D_{210}:C_{14}$)

Order 14: $C_{14}$ ($D_{15}\times C_{210}$), $C_{14}$ ($C_{15}\times D_{210}$)

Order 12: $C_2\times C_6$ ($D_{35}\times C_{105}$), $C_2\times C_6$ ($C_{35}\times D_{105}$)

Order 10: $C_{10}$ ($D_{21}\times C_{210}$), $C_{10}$ ($C_{21}\times D_{210}$)

Order 9: $C_3^2$ ($C_{70}\wr C_2$)

Order 7: $C_7$ ($D_{30}.C_{210}$), $C_7$ ($C_{30}.D_{210}$)

Order 6: $C_6$ ($C_{105}\times D_{70}$), $C_6$ ($C_{35}\times D_{210}$)

Order 5: $C_5$ ($D_{42}.C_{210}$), $C_5$ ($C_{42}.D_{210}$)

Order 4: $C_2^2$ ($C_{105}\times D_{105}$)

Order 3: $C_3$ ($D_{70}.C_{210}$), $C_3$ ($C_{70}.D_{210}$)

Order 2: $C_2$ ($C_{210}.C_{210}$)

Order 1: $C_1$ ($C_{105}^2:D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 88200: $C_{105}^2:D_4$ ($C_1$)

Order 44100: $C_{105}:C_{420}$ ($C_2$), $C_{210}.C_{210}$ ($C_2$), $C_{210}^2$ ($C_2$)

Order 29400: $C_{70}.D_{210}$ ($C_3$)

Order 22050: $C_{105}\times C_{210}$ ($C_2^2$)

Order 17640: $C_{42}.D_{210}$ ($C_5$)

Order 14700: $C_{70}\times C_{210}$ ($C_6$), $C_{70}\times C_{210}$ ($S_3$), $C_{105}:C_{140}$ ($C_6$), $C_{35}\times D_{210}$ ($C_6$)

Order 12600: $C_{30}.D_{210}$ ($C_7$)

Order 11025: $C_{105}^2$ ($D_4$)

Order 8820: $C_{105}:C_{84}$ ($C_{10}$), $C_{21}\times D_{210}$ ($C_{10}$), $C_{42}\times C_{210}$ ($C_{10}$), $C_{42}\times C_{210}$ ($D_5$)

Order 7350: $C_{35}\times C_{210}$ ($C_2\times C_6$), $C_{35}\times C_{210}$ ($D_6$)

Order 6300: $C_{105}:C_{60}$ ($C_{14}$), $C_{15}\times D_{210}$ ($C_{14}$), $C_{30}\times C_{210}$ ($C_{14}$), $C_{30}\times C_{210}$ ($D_7$)

Order 5880: $D_{210}:C_{14}$ ($C_{15}$)

Order 4900: $C_{70}^2$ ($C_3\times S_3$)

Order 4410: $C_{21}\times C_{210}$ ($C_2\times C_{10}$), $C_{21}\times C_{210}$ ($D_{10}$)

Order 4200: $D_{210}:C_{10}$ ($C_{21}$)

Order 3675: $C_{35}\times C_{105}$ ($C_3:D_4$), $C_{35}\times C_{105}$ ($C_3\times D_4$)

Order 3150: $C_{15}\times C_{210}$ ($C_2\times C_{14}$), $C_{15}\times C_{210}$ ($D_{14}$)

Order 2940: $C_{14}\times C_{210}$ ($C_{30}$), $C_{14}\times C_{210}$ ($D_{15}$), $C_{14}\times C_{210}$ ($C_3\times D_5$), $C_{14}\times C_{210}$ ($C_5\times S_3$), $C_{105}:C_{28}$ ($C_{30}$), $C_7\times D_{210}$ ($C_{30}$)

Order 2520: $D_{210}:C_6$ ($C_{35}$)

Order 2450: $C_{35}\times C_{70}$ ($C_6\times S_3$)

Order 2205: $C_{21}\times C_{105}$ ($C_5:D_4$), $C_{21}\times C_{105}$ ($C_5\times D_4$)

Order 2100: $C_{10}\times C_{210}$ ($C_{42}$), $C_{10}\times C_{210}$ ($D_{21}$), $C_{10}\times C_{210}$ ($C_3\times D_7$), $C_{10}\times C_{210}$ ($S_3\times C_7$), $C_{105}:C_{20}$ ($C_{42}$), $C_5\times D_{210}$ ($C_{42}$)

Order 1764: $C_{42}^2$ ($C_5\times D_5$)

Order 1575: $C_{15}\times C_{105}$ ($C_7\times D_4$), $C_{15}\times C_{105}$ ($C_7:D_4$)

Order 1470: $C_7\times C_{210}$ ($C_2\times C_{30}$), $C_7\times C_{210}$ ($D_{30}$), $C_7\times C_{210}$ ($S_3\times C_{10}$), $C_7\times C_{210}$ ($C_3\times D_{10}$)

Order 1260: $C_{105}:C_{12}$ ($C_{70}$), $C_6\times C_{210}$ ($C_{70}$), $C_6\times C_{210}$ ($D_{35}$), $C_6\times C_{210}$ ($C_5\times D_7$), $C_6\times C_{210}$ ($C_7\times D_5$), $C_3\times D_{210}$ ($C_{70}$)

Order 1225: $C_{35}^2$ ($C_6\wr C_2$)

Order 1050: $C_5\times C_{210}$ ($C_2\times C_{42}$), $C_5\times C_{210}$ ($D_{42}$), $C_5\times C_{210}$ ($S_3\times C_{14}$), $C_5\times C_{210}$ ($C_3\times D_{14}$)

Order 980: $C_{14}\times C_{70}$ ($C_3\times D_{15}$), $C_{14}\times C_{70}$ ($S_3\times C_{15}$)

Order 900: $C_{30}^2$ ($C_7\times D_7$)

Order 882: $C_{21}\times C_{42}$ ($C_5\times D_{10}$)

Order 840: $C_{105}:D_4$ ($C_{105}$)

Order 735: $C_7\times C_{105}$ ($D_4\times C_{15}$), $C_7\times C_{105}$ ($C_{15}:D_4$), $C_7\times C_{105}$ ($C_{15}:D_4$), $C_7\times C_{105}$ ($C_{15}:D_4$)

Order 700: $C_{10}\times C_{70}$ ($C_3\times D_{21}$), $C_{10}\times C_{70}$ ($S_3\times C_{21}$)

Order 630: $C_3\times C_{210}$ ($C_7\times D_{10}$), $C_3\times C_{210}$ ($C_5\times D_{14}$), $C_3\times C_{210}$ ($C_2\times C_{70}$), $C_3\times C_{210}$ ($D_{70}$)

Order 588: $C_{14}\times C_{42}$ ($D_5\times C_{15}$), $C_{14}\times C_{42}$ ($C_5\times D_{15}$)

Order 525: $C_5\times C_{105}$ ($D_4\times C_{21}$), $C_5\times C_{105}$ ($C_{21}:D_4$), $C_5\times C_{105}$ ($D_6:C_{14}$), $C_5\times C_{105}$ ($C_{21}:D_4$)

Order 490: $C_7\times C_{70}$ ($C_3\times D_{30}$), $C_7\times C_{70}$ ($S_3\times C_{30}$)

Order 450: $C_{15}\times C_{30}$ ($C_7\times D_{14}$)

Order 441: $C_{21}^2$ ($C_{10}\wr C_2$)

Order 420: $C_2\times C_{210}$ ($C_5\times D_{21}$), $C_2\times C_{210}$ ($S_3\times C_{35}$), $C_2\times C_{210}$ ($C_3\times D_{35}$), $C_2\times C_{210}$ ($D_5\times C_{21}$), $C_2\times C_{210}$ ($D_7\times C_{15}$), $C_2\times C_{210}$ ($C_{210}$), $C_2\times C_{210}$ ($D_{105}$), $C_2\times C_{210}$ ($C_7\times D_{15}$), $D_{210}$ ($C_{210}$), $C_{105}:C_4$ ($C_{210}$)

Order 350: $C_5\times C_{70}$ ($C_3\times D_{42}$), $C_5\times C_{70}$ ($S_3\times C_{42}$)

Order 315: $C_3\times C_{105}$ ($D_4\times C_{35}$), $C_3\times C_{105}$ ($C_{35}:D_4$), $C_3\times C_{105}$ ($C_{35}:D_4$), $C_3\times C_{105}$ ($C_{35}:D_4$)

Order 300: $C_{10}\times C_{30}$ ($C_7\times D_{21}$), $C_{10}\times C_{30}$ ($D_7\times C_{21}$)

Order 294: $C_7\times C_{42}$ ($C_5\times D_{30}$), $C_7\times C_{42}$ ($C_{15}\times D_{10}$)

Order 252: $C_6\times C_{42}$ ($C_5\times D_{35}$), $C_6\times C_{42}$ ($D_5\times C_{35}$)

Order 245: $C_7\times C_{35}$ ($D_6:C_{30}$), $C_7\times C_{35}$ ($D_{30}:C_6$)

Order 225: $C_{15}^2$ ($C_{14}\wr C_2$)

Order 210: $C_{210}$ ($C_2\times C_{210}$), $C_{210}$ ($D_{210}$), $C_{210}$ ($C_7\times D_{30}$), $C_{210}$ ($C_5\times D_{42}$), $C_{210}$ ($S_3\times C_{70}$), $C_{210}$ ($C_3\times D_{70}$), $C_{210}$ ($D_5\times C_{42}$), $C_{210}$ ($C_{15}\times D_{14}$)

Order 196: $C_{14}^2$ ($C_{15}\times D_{15}$)

Order 180: $C_6\times C_{30}$ ($C_7\times D_{35}$), $C_6\times C_{30}$ ($D_7\times C_{35}$)

Order 175: $C_5\times C_{35}$ ($D_{42}:C_6$), $C_5\times C_{35}$ ($D_6:C_{42}$)

Order 150: $C_5\times C_{30}$ ($C_7\times D_{42}$), $C_5\times C_{30}$ ($C_{21}\times D_{14}$)

Order 147: $C_7\times C_{21}$ ($D_{30}:C_{10}$), $C_7\times C_{21}$ ($D_{10}:C_{30}$)

Order 140: $C_2\times C_{70}$ ($C_3\times D_{105}$), $C_2\times C_{70}$ ($C_{21}\times D_{15}$), $C_2\times C_{70}$ ($C_{15}\times D_{21}$), $C_2\times C_{70}$ ($S_3\times C_{105}$)

Order 126: $C_3\times C_{42}$ ($C_5\times D_{70}$), $C_3\times C_{42}$ ($D_5\times C_{70}$)

Order 105: $C_{105}$ ($D_4\times C_{105}$), $C_{105}$ ($C_{105}:D_4$), $C_{105}$ ($C_{105}:D_4$), $C_{105}$ ($C_{105}:D_4$), $C_{105}$ ($D_6:C_{70}$), $C_{105}$ ($C_{105}:D_4$), $C_{105}$ ($D_{10}:C_{42}$), $C_{105}$ ($D_{14}:C_{30}$)

Order 100: $C_{10}^2$ ($C_{21}\times D_{21}$)

Order 98: $C_7\times C_{14}$ ($C_{15}\times D_{30}$)

Order 90: $C_3\times C_{30}$ ($C_7\times D_{70}$), $C_3\times C_{30}$ ($D_7\times C_{70}$)

Order 84: $C_2\times C_{42}$ ($C_5\times D_{105}$), $C_2\times C_{42}$ ($D_{15}\times C_{35}$), $C_2\times C_{42}$ ($C_{15}\times D_{35}$), $C_2\times C_{42}$ ($D_5\times C_{105}$)

Order 75: $C_5\times C_{15}$ ($D_{42}:C_{14}$), $C_5\times C_{15}$ ($D_{14}:C_{42}$)

Order 70: $C_{70}$ ($C_3\times D_{210}$), $C_{70}$ ($C_{21}\times D_{30}$), $C_{70}$ ($C_{15}\times D_{42}$), $C_{70}$ ($S_3\times C_{210}$)

Order 63: $C_3\times C_{21}$ ($D_{10}:C_{70}$), $C_3\times C_{21}$ ($D_{70}:C_{10}$)

Order 60: $C_2\times C_{30}$ ($C_7\times D_{105}$), $C_2\times C_{30}$ ($C_{35}\times D_{21}$), $C_2\times C_{30}$ ($C_{21}\times D_{35}$), $C_2\times C_{30}$ ($D_7\times C_{105}$)

Order 50: $C_5\times C_{10}$ ($C_{21}\times D_{42}$)

Order 49: $C_7^2$ ($C_{30}\wr C_2$)

Order 45: $C_3\times C_{15}$ ($D_{14}:C_{70}$), $C_3\times C_{15}$ ($D_{70}:C_{14}$)

Order 42: $C_{42}$ ($C_{35}\times D_{30}$), $C_{42}$ ($D_5\times C_{210}$), $C_{42}$ ($C_{15}\times D_{70}$), $C_{42}$ ($C_5\times D_{210}$)

Order 36: $C_6^2$ ($C_{35}\times D_{35}$)

Order 35: $C_{35}$ ($D_6:C_{210}$), $C_{35}$ ($D_{30}:C_{42}$), $C_{35}$ ($D_{42}:C_{30}$), $C_{35}$ ($D_{210}:C_6$)

Order 30: $C_{30}$ ($C_{21}\times D_{70}$), $C_{30}$ ($D_7\times C_{210}$), $C_{30}$ ($C_{35}\times D_{42}$), $C_{30}$ ($C_7\times D_{210}$)

Order 28: $C_2\times C_{14}$ ($D_{15}\times C_{105}$), $C_2\times C_{14}$ ($C_{15}\times D_{105}$)

Order 25: $C_5^2$ ($C_{42}\wr C_2$)

Order 21: $C_{21}$ ($D_{10}:C_{210}$), $C_{21}$ ($D_{30}:C_{70}$), $C_{21}$ ($D_{70}:C_{30}$), $C_{21}$ ($D_{210}:C_{10}$)

Order 20: $C_2\times C_{10}$ ($D_{21}\times C_{105}$), $C_2\times C_{10}$ ($C_{21}\times D_{105}$)

Order 18: $C_3\times C_6$ ($C_{35}\times D_{70}$)

Order 15: $C_{15}$ ($D_{14}:C_{210}$), $C_{15}$ ($D_{42}:C_{70}$), $C_{15}$ ($D_{70}:C_{42}$), $C_{15}$ ($D_{210}:C_{14}$)

Order 14: $C_{14}$ ($D_{15}\times C_{210}$), $C_{14}$ ($C_{15}\times D_{210}$)

Order 12: $C_2\times C_6$ ($D_{35}\times C_{105}$), $C_2\times C_6$ ($C_{35}\times D_{105}$)

Order 10: $C_{10}$ ($D_{21}\times C_{210}$), $C_{10}$ ($C_{21}\times D_{210}$)

Order 9: $C_3^2$ ($C_{70}\wr C_2$)

Order 7: $C_7$ ($D_{30}.C_{210}$), $C_7$ ($C_{30}.D_{210}$)

Order 6: $C_6$ ($C_{105}\times D_{70}$), $C_6$ ($C_{35}\times D_{210}$)

Order 5: $C_5$ ($D_{42}.C_{210}$), $C_5$ ($C_{42}.D_{210}$)

Order 4: $C_2^2$ ($C_{105}\times D_{105}$)

Order 3: $C_3$ ($D_{70}.C_{210}$), $C_3$ ($C_{70}.D_{210}$)

Order 2: $C_2$ ($C_{210}.C_{210}$)

Order 1: $C_1$ ($C_{105}^2:D_4$)

Series

Derived series

$C_{105}^2:D_4$

$\rhd$

$C_{210}$

$\rhd$

$C_1$

Chief series

$C_{105}^2:D_4$

$\rhd$

$C_{210}^2$

$\rhd$

$C_{105}\times C_{210}$

$\rhd$

$C_{35}\times C_{210}$

$\rhd$

$C_7\times C_{210}$

$\rhd$

$C_{210}$

$\rhd$

$C_{105}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{105}^2:D_4$

$\rhd$

$C_{210}$

$\rhd$

$C_{105}$

Upper central series

$C_1$

$\lhd$

$C_{210}$

$\lhd$

$C_2\times C_{210}$

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 $22365 \times 22365$ character table is not available for this group.

Rational character table

The $673 \times 673$ rational character table is not available for this group.