Abstract group 87780.b: $C_{209}\times D_{210}$

• Label: 87780.b
• Order: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 11 \cdot 19 \)
• $\card{Z(G)}$: \( 2 \cdot 11 \cdot 19 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{3} \cdot 5 \)
• Perm deg.: $47$
• Trans deg.: $43890$
• Rank: $2$

Group information

Description:	$C_{209}\times D_{210}$
Order:	\(87780\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \)
Exponent:	\(43890\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \)
Automorphism group:	$C_2^2\times C_{90}\times F_5\times S_3\times F_7$, of order \(1814400\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 7 \)
Composition factors:	$C_2$ x 2, $C_3$, $C_5$, $C_7$, $C_{11}$, $C_{19}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.

Group statistics

Order	1	2	3	5	6	7	10	11	14	15	19	21	22	30	33	35	38	42	55	57	66	70	77	95	105	110	114	133	154	165	190	209	210	231	266	285	330	385	399	418	462	570	627	665	770	798	1045	1155	1254	1330	1463	1995	2090	2310	2926	3135	3990	4389	6270	7315	8778	14630	21945	43890
Elements	1	211	2	4	2	6	4	10	6	8	18	12	2110	8	20	24	3798	12	40	36	20	24	60	72	48	40	36	108	60	80	72	180	48	120	108	144	80	240	216	37980	120	144	360	432	240	216	720	480	360	432	1080	864	720	480	1080	1440	864	2160	1440	4320	2160	4320	8640	8640	87780
Conjugacy classes	1	3	1	2	1	3	2	10	3	4	18	6	30	4	10	12	54	6	20	18	10	12	30	36	24	20	18	54	30	40	36	180	24	60	54	72	40	120	108	540	60	72	180	216	120	108	360	240	180	216	540	432	360	240	540	720	432	1080	720	2160	1080	2160	4320	4320	22572
Divisions	1	3	1	1	1	1	1	1	1	1	1	1	3	1	1	1	3	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	3	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	72
Autjugacy classes	1	2	1	1	1	1	1	1	1	1	1	1	2	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	68

Minimal presentations

Permutation degree:	$47$
Transitive degree:	$43890$
Rank:	$2$
Inequivalent generating pairs:	$720$

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

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{43890}=1, b^{a}=b^{419} \rangle$
Permutation group:	Degree $47$ $\langle(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 227 & 233 \\ 326 & 227 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 418 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{419})$
Direct product:	$C_2$ $\, \times\, $ $C_{11}$ $\, \times\, $ $C_{19}$ $\, \times\, $ $D_{105}$
Semidirect product:	$C_{8778}$ $\,\rtimes\,$ $D_5$	$C_{7315}$ $\,\rtimes\,$ $D_6$	$C_{6270}$ $\,\rtimes\,$ $D_7$	$C_{627}$ $\,\rtimes\,$ $D_{70}$	all 56
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2} \times C_{418} \simeq C_{2}^{2} \times C_{11} \times C_{19}$
Schur multiplier:	$C_{2}$
Commutator length:	not computed

Subgroups

There are 2368 subgroups in 160 conjugacy classes, 76 normal (68 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{418}$	$G/Z \simeq$ $D_{105}$
Commutator:	$G' \simeq$ $C_{105}$	$G/G' \simeq$ $C_2\times C_{418}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{209}\times D_{210}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{43890}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{209}\times D_{210}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{43890}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $C_{19}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 87780: $C_{209}\times D_{210}$

Order 43890: $C_{209}\times D_{105}$ x 2, $C_{43890}$

Order 29260: $D_{35}\times C_{418}$

Order 21945: $C_{21945}$

Order 17556: $D_{21}\times C_{418}$

Order 14630: $D_{35}\times C_{209}$ x 2, $C_{14630}$

Order 12540: $D_{15}\times C_{418}$

Order 8778: $D_{21}\times C_{209}$ x 2, $C_{8778}$

Order 7980: $C_{19}\times D_{210}$

Order 7315: $C_{7315}$

Order 6270: $D_{15}\times C_{209}$ x 2, $C_{6270}$

Order 5852: $D_7\times C_{418}$

Order 4620: $C_{11}\times D_{210}$

Order 4389: $C_{4389}$

Order 4180: $D_5\times C_{418}$

Order 3990: $C_{19}\times D_{105}$ x 2, $C_{3990}$

Order 3135: $C_{3135}$

Order 2926: $D_7\times C_{209}$ x 2, $C_{2926}$

Order 2660: $C_{19}\times D_{70}$

Order 2508: $S_3\times C_{418}$

Order 2310: $C_{11}\times D_{105}$ x 2, $C_{2310}$

Order 2090: $D_5\times C_{209}$ x 2, $C_{2090}$

Order 1995: $C_{1995}$

Order 1596: $C_{19}\times D_{42}$

Order 1540: $C_{11}\times D_{70}$

Order 1463: $C_{1463}$

Order 1330: $C_{19}\times D_{35}$ x 2, $C_{1330}$

Order 1254: $S_3\times C_{209}$ x 2, $C_{1254}$

Order 1155: $C_{1155}$

Order 1140: $C_{19}\times D_{30}$

Order 1045: $C_{1045}$

Order 924: $C_{11}\times D_{42}$

Order 836: $C_2\times C_{418}$

Order 798: $C_{19}\times D_{21}$ x 2, $C_{798}$

Order 770: $C_{11}\times D_{35}$ x 2, $C_{770}$

Order 665: $C_{665}$

Order 660: $C_{11}\times D_{30}$

Order 627: $C_{627}$

Order 570: $C_{19}\times D_{15}$ x 2, $C_{570}$

Order 532: $C_{19}\times D_{14}$

Order 462: $C_{11}\times D_{21}$ x 2, $C_{462}$

Order 420: $D_{210}$

Order 418: $C_{418}$ x 3

Order 399: $C_{399}$

Order 385: $C_{385}$

Order 380: $C_{19}\times D_{10}$

Order 330: $C_{11}\times D_{15}$ x 2, $C_{330}$

Order 308: $C_{11}\times D_{14}$

Order 285: $C_{285}$

Order 266: $D_7\times C_{19}$ x 2, $C_{266}$

Order 231: $C_{231}$

Order 228: $S_3\times C_{38}$

Order 220: $C_{11}\times D_{10}$

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

Order 209: $C_{209}$

Order 190: $D_5\times C_{19}$ x 2, $C_{190}$

Order 165: $C_{165}$

Order 154: $C_{11}\times D_7$ x 2, $C_{154}$

Order 140: $D_{70}$

Order 133: $C_{133}$

Order 132: $S_3\times C_{22}$

Order 114: $S_3\times C_{19}$ x 2, $C_{114}$

Order 110: $D_5\times C_{11}$ x 2, $C_{110}$

Order 105: $C_{105}$

Order 95: $C_{95}$

Order 84: $D_{42}$

Order 77: $C_{77}$

Order 76: $C_2\times C_{38}$

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

Order 66: $S_3\times C_{11}$ x 2, $C_{66}$

Order 60: $D_{30}$

Order 57: $C_{57}$

Order 55: $C_{55}$

Order 44: $C_2\times C_{22}$

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

Order 38: $C_{38}$ x 3

Order 35: $C_{35}$

Order 33: $C_{33}$

Order 30: $D_{15}$ x 2, $C_{30}$

Order 28: $D_{14}$

Order 22: $C_{22}$ x 3

Order 21: $C_{21}$

Order 20: $D_{10}$

Order 19: $C_{19}$

Order 15: $C_{15}$

Order 14: $D_7$ x 2, $C_{14}$

Order 12: $D_6$

Order 11: $C_{11}$

Order 10: $D_5$ x 2, $C_{10}$

Order 7: $C_7$

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 87780: $C_{209}\times D_{210}$

Order 43890: $C_{209}\times D_{105}$, $C_{43890}$

Order 29260: $D_{35}\times C_{418}$

Order 21945: $C_{21945}$

Order 17556: $D_{21}\times C_{418}$

Order 14630: $D_{35}\times C_{209}$, $C_{14630}$

Order 12540: $D_{15}\times C_{418}$

Order 8778: $D_{21}\times C_{209}$, $C_{8778}$

Order 7980: $C_{19}\times D_{210}$

Order 7315: $C_{7315}$

Order 6270: $D_{15}\times C_{209}$, $C_{6270}$

Order 5852: $D_7\times C_{418}$

Order 4620: $C_{11}\times D_{210}$

Order 4389: $C_{4389}$

Order 4180: $D_5\times C_{418}$

Order 3990: $C_{19}\times D_{105}$, $C_{3990}$

Order 3135: $C_{3135}$

Order 2926: $D_7\times C_{209}$, $C_{2926}$

Order 2660: $C_{19}\times D_{70}$

Order 2508: $S_3\times C_{418}$

Order 2310: $C_{11}\times D_{105}$, $C_{2310}$

Order 2090: $D_5\times C_{209}$, $C_{2090}$

Order 1995: $C_{1995}$

Order 1596: $C_{19}\times D_{42}$

Order 1540: $C_{11}\times D_{70}$

Order 1463: $C_{1463}$

Order 1330: $C_{1330}$, $C_{19}\times D_{35}$

Order 1254: $S_3\times C_{209}$, $C_{1254}$

Order 1155: $C_{1155}$

Order 1140: $C_{19}\times D_{30}$

Order 1045: $C_{1045}$

Order 924: $C_{11}\times D_{42}$

Order 836: $C_2\times C_{418}$

Order 798: $C_{798}$, $C_{19}\times D_{21}$

Order 770: $C_{770}$, $C_{11}\times D_{35}$

Order 665: $C_{665}$

Order 660: $C_{11}\times D_{30}$

Order 627: $C_{627}$

Order 570: $C_{570}$, $C_{19}\times D_{15}$

Order 532: $C_{19}\times D_{14}$

Order 462: $C_{462}$, $C_{11}\times D_{21}$

Order 420: $D_{210}$

Order 418: $C_{418}$ x 2

Order 399: $C_{399}$

Order 385: $C_{385}$

Order 380: $C_{19}\times D_{10}$

Order 330: $C_{330}$, $C_{11}\times D_{15}$

Order 308: $C_{11}\times D_{14}$

Order 285: $C_{285}$

Order 266: $C_{266}$, $D_7\times C_{19}$

Order 231: $C_{231}$

Order 228: $S_3\times C_{38}$

Order 220: $C_{11}\times D_{10}$

Order 210: $C_{210}$, $D_{105}$

Order 209: $C_{209}$

Order 190: $C_{190}$, $D_5\times C_{19}$

Order 165: $C_{165}$

Order 154: $C_{154}$, $C_{11}\times D_7$

Order 140: $D_{70}$

Order 133: $C_{133}$

Order 132: $S_3\times C_{22}$

Order 114: $C_{114}$, $S_3\times C_{19}$

Order 110: $C_{110}$, $D_5\times C_{11}$

Order 105: $C_{105}$

Order 95: $C_{95}$

Order 84: $D_{42}$

Order 77: $C_{77}$

Order 76: $C_2\times C_{38}$

Order 70: $C_{70}$, $D_{35}$

Order 66: $C_{66}$, $S_3\times C_{11}$

Order 60: $D_{30}$

Order 57: $C_{57}$

Order 55: $C_{55}$

Order 44: $C_2\times C_{22}$

Order 42: $C_{42}$, $D_{21}$

Order 38: $C_{38}$ x 2

Order 35: $C_{35}$

Order 33: $C_{33}$

Order 30: $C_{30}$, $D_{15}$

Order 28: $D_{14}$

Order 22: $C_{22}$ x 2

Order 21: $C_{21}$

Order 20: $D_{10}$

Order 19: $C_{19}$

Order 15: $C_{15}$

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

Order 12: $D_6$

Order 11: $C_{11}$

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

Order 7: $C_7$

Order 6: $C_6$, $S_3$

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 87780: $C_{209}\times D_{210}$ ($C_1$)

Order 43890: $C_{209}\times D_{105}$ ($C_2$) x 2, $C_{43890}$ ($C_2$)

Order 21945: $C_{21945}$ ($C_2^2$)

Order 14630: $C_{14630}$ ($S_3$)

Order 8778: $C_{8778}$ ($D_5$)

Order 7980: $C_{19}\times D_{210}$ ($C_{11}$)

Order 7315: $C_{7315}$ ($D_6$)

Order 6270: $C_{6270}$ ($D_7$)

Order 4620: $C_{11}\times D_{210}$ ($C_{19}$)

Order 4389: $C_{4389}$ ($D_{10}$)

Order 3990: $C_{19}\times D_{105}$ ($C_{22}$) x 2, $C_{3990}$ ($C_{22}$)

Order 3135: $C_{3135}$ ($D_{14}$)

Order 2926: $C_{2926}$ ($D_{15}$)

Order 2310: $C_{11}\times D_{105}$ ($C_{38}$) x 2, $C_{2310}$ ($C_{38}$)

Order 2090: $C_{2090}$ ($D_{21}$)

Order 1995: $C_{1995}$ ($C_2\times C_{22}$)

Order 1463: $C_{1463}$ ($D_{30}$)

Order 1330: $C_{1330}$ ($S_3\times C_{11}$)

Order 1254: $C_{1254}$ ($D_{35}$)

Order 1155: $C_{1155}$ ($C_2\times C_{38}$)

Order 1045: $C_{1045}$ ($D_{42}$)

Order 798: $C_{798}$ ($D_5\times C_{11}$)

Order 770: $C_{770}$ ($S_3\times C_{19}$)

Order 665: $C_{665}$ ($S_3\times C_{22}$)

Order 627: $C_{627}$ ($D_{70}$)

Order 570: $C_{570}$ ($C_{11}\times D_7$)

Order 462: $C_{462}$ ($D_5\times C_{19}$)

Order 420: $D_{210}$ ($C_{209}$)

Order 418: $C_{418}$ ($D_{105}$)

Order 399: $C_{399}$ ($C_{11}\times D_{10}$)

Order 385: $C_{385}$ ($S_3\times C_{38}$)

Order 330: $C_{330}$ ($D_7\times C_{19}$)

Order 285: $C_{285}$ ($C_{11}\times D_{14}$)

Order 266: $C_{266}$ ($C_{11}\times D_{15}$)

Order 231: $C_{231}$ ($C_{19}\times D_{10}$)

Order 210: $D_{105}$ ($C_{418}$) x 2, $C_{210}$ ($C_{418}$)

Order 209: $C_{209}$ ($D_{210}$)

Order 190: $C_{190}$ ($C_{11}\times D_{21}$)

Order 165: $C_{165}$ ($C_{19}\times D_{14}$)

Order 154: $C_{154}$ ($C_{19}\times D_{15}$)

Order 133: $C_{133}$ ($C_{11}\times D_{30}$)

Order 114: $C_{114}$ ($C_{11}\times D_{35}$)

Order 110: $C_{110}$ ($C_{19}\times D_{21}$)

Order 105: $C_{105}$ ($C_2\times C_{418}$)

Order 95: $C_{95}$ ($C_{11}\times D_{42}$)

Order 77: $C_{77}$ ($C_{19}\times D_{30}$)

Order 70: $C_{70}$ ($S_3\times C_{209}$)

Order 66: $C_{66}$ ($C_{19}\times D_{35}$)

Order 57: $C_{57}$ ($C_{11}\times D_{70}$)

Order 55: $C_{55}$ ($C_{19}\times D_{42}$)

Order 42: $C_{42}$ ($D_5\times C_{209}$)

Order 38: $C_{38}$ ($C_{11}\times D_{105}$)

Order 35: $C_{35}$ ($S_3\times C_{418}$)

Order 33: $C_{33}$ ($C_{19}\times D_{70}$)

Order 30: $C_{30}$ ($D_7\times C_{209}$)

Order 22: $C_{22}$ ($C_{19}\times D_{105}$)

Order 21: $C_{21}$ ($D_5\times C_{418}$)

Order 19: $C_{19}$ ($C_{11}\times D_{210}$)

Order 15: $C_{15}$ ($D_7\times C_{418}$)

Order 14: $C_{14}$ ($D_{15}\times C_{209}$)

Order 11: $C_{11}$ ($C_{19}\times D_{210}$)

Order 10: $C_{10}$ ($D_{21}\times C_{209}$)

Order 7: $C_7$ ($D_{15}\times C_{418}$)

Order 6: $C_6$ ($D_{35}\times C_{209}$)

Order 5: $C_5$ ($D_{21}\times C_{418}$)

Order 3: $C_3$ ($D_{35}\times C_{418}$)

Order 2: $C_2$ ($C_{209}\times D_{105}$)

Order 1: $C_1$ ($C_{209}\times D_{210}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 87780: $C_{209}\times D_{210}$ ($C_1$)

Order 43890: $C_{209}\times D_{105}$ ($C_2$), $C_{43890}$ ($C_2$)

Order 21945: $C_{21945}$ ($C_2^2$)

Order 14630: $C_{14630}$ ($S_3$)

Order 8778: $C_{8778}$ ($D_5$)

Order 7980: $C_{19}\times D_{210}$ ($C_{11}$)

Order 7315: $C_{7315}$ ($D_6$)

Order 6270: $C_{6270}$ ($D_7$)

Order 4620: $C_{11}\times D_{210}$ ($C_{19}$)

Order 4389: $C_{4389}$ ($D_{10}$)

Order 3990: $C_{19}\times D_{105}$ ($C_{22}$), $C_{3990}$ ($C_{22}$)

Order 3135: $C_{3135}$ ($D_{14}$)

Order 2926: $C_{2926}$ ($D_{15}$)

Order 2310: $C_{11}\times D_{105}$ ($C_{38}$), $C_{2310}$ ($C_{38}$)

Order 2090: $C_{2090}$ ($D_{21}$)

Order 1995: $C_{1995}$ ($C_2\times C_{22}$)

Order 1463: $C_{1463}$ ($D_{30}$)

Order 1330: $C_{1330}$ ($S_3\times C_{11}$)

Order 1254: $C_{1254}$ ($D_{35}$)

Order 1155: $C_{1155}$ ($C_2\times C_{38}$)

Order 1045: $C_{1045}$ ($D_{42}$)

Order 798: $C_{798}$ ($D_5\times C_{11}$)

Order 770: $C_{770}$ ($S_3\times C_{19}$)

Order 665: $C_{665}$ ($S_3\times C_{22}$)

Order 627: $C_{627}$ ($D_{70}$)

Order 570: $C_{570}$ ($C_{11}\times D_7$)

Order 462: $C_{462}$ ($D_5\times C_{19}$)

Order 420: $D_{210}$ ($C_{209}$)

Order 418: $C_{418}$ ($D_{105}$)

Order 399: $C_{399}$ ($C_{11}\times D_{10}$)

Order 385: $C_{385}$ ($S_3\times C_{38}$)

Order 330: $C_{330}$ ($D_7\times C_{19}$)

Order 285: $C_{285}$ ($C_{11}\times D_{14}$)

Order 266: $C_{266}$ ($C_{11}\times D_{15}$)

Order 231: $C_{231}$ ($C_{19}\times D_{10}$)

Order 210: $C_{210}$ ($C_{418}$), $D_{105}$ ($C_{418}$)

Order 209: $C_{209}$ ($D_{210}$)

Order 190: $C_{190}$ ($C_{11}\times D_{21}$)

Order 165: $C_{165}$ ($C_{19}\times D_{14}$)

Order 154: $C_{154}$ ($C_{19}\times D_{15}$)

Order 133: $C_{133}$ ($C_{11}\times D_{30}$)

Order 114: $C_{114}$ ($C_{11}\times D_{35}$)

Order 110: $C_{110}$ ($C_{19}\times D_{21}$)

Order 105: $C_{105}$ ($C_2\times C_{418}$)

Order 95: $C_{95}$ ($C_{11}\times D_{42}$)

Order 77: $C_{77}$ ($C_{19}\times D_{30}$)

Order 70: $C_{70}$ ($S_3\times C_{209}$)

Order 66: $C_{66}$ ($C_{19}\times D_{35}$)

Order 57: $C_{57}$ ($C_{11}\times D_{70}$)

Order 55: $C_{55}$ ($C_{19}\times D_{42}$)

Order 42: $C_{42}$ ($D_5\times C_{209}$)

Order 38: $C_{38}$ ($C_{11}\times D_{105}$)

Order 35: $C_{35}$ ($S_3\times C_{418}$)

Order 33: $C_{33}$ ($C_{19}\times D_{70}$)

Order 30: $C_{30}$ ($D_7\times C_{209}$)

Order 22: $C_{22}$ ($C_{19}\times D_{105}$)

Order 21: $C_{21}$ ($D_5\times C_{418}$)

Order 19: $C_{19}$ ($C_{11}\times D_{210}$)

Order 15: $C_{15}$ ($D_7\times C_{418}$)

Order 14: $C_{14}$ ($D_{15}\times C_{209}$)

Order 11: $C_{11}$ ($C_{19}\times D_{210}$)

Order 10: $C_{10}$ ($D_{21}\times C_{209}$)

Order 7: $C_7$ ($D_{15}\times C_{418}$)

Order 6: $C_6$ ($D_{35}\times C_{209}$)

Order 5: $C_5$ ($D_{21}\times C_{418}$)

Order 3: $C_3$ ($D_{35}\times C_{418}$)

Order 2: $C_2$ ($C_{209}\times D_{105}$)

Order 1: $C_1$ ($C_{209}\times D_{210}$)

Series

Derived series

$C_{209}\times D_{210}$

$\rhd$

$C_{105}$

$\rhd$

$C_1$

Chief series

$C_{209}\times D_{210}$

$\rhd$

$C_{209}\times D_{105}$

$\rhd$

$C_{21945}$

$\rhd$

$C_{7315}$

$\rhd$

$C_{1463}$

$\rhd$

$C_{209}$

$\rhd$

$C_{19}$

$\rhd$

$C_1$

Lower central series

$C_{209}\times D_{210}$

$\rhd$

$C_{105}$

Upper central series

$C_1$

$\lhd$

$C_{418}$

Supergroups

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

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

The $22572 \times 22572$ character table is not available for this group.

Rational character table

The $72 \times 72$ rational character table is not available for this group.