Abstract group 342798989524992000.a: $C_2^{15}.A_{16}$

• Label: 342...000.a
• Order: \( 2^{29} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \)
• Exponent: \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{30} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $32$
• Trans deg.: $32$
• Rank: $2$

Group information

Description:	$C_2^{15}.A_{16}$
Order:	\(342798989524992000\)\(\medspace = 2^{29} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \)
Exponent:	\(720720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \)
Automorphism group:	Group of order \(685597979049984000\)\(\medspace = 2^{30} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \)
Composition factors:	$C_2$ x 15, $A_{16}$
Derived length:	$0$

This group is nonabelian and perfect (hence nonsolvable).

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	18	20	21	22	24	26	28	30	33	35	36	39	40	42	44	45	48	55	56	60	63	66	70	72	78	80	84	90	105	110	120	126	140	168	210
Elements	1	13017721087	851185059200	126089221789440	114415115802624	1611430667745920	437244136243200	5567422284103680	562192043212800	1573205938956288	16231012761600	24236911835136000	1098714710016000	15345242744832000	22986016826572800	4939638217113600	5933304682905600	8771152344514560	104267081318400	2450882927001600	29369545261056000	7691002970112000	14343867688550400	35035201335214080	1298481020928000	25505877196800	8331919884288000	8789717680128000	10391094369976320	4003123809484800	5843164594176000	952219415347200	9522194153472000	6232708900454400	3315764035584000	24214642163712000	5441253801984000	9089367146496000	1402823245824000	4761097076736000	8789717680128000	4284987369062400	9692233334784000	6665535907430400	816188070297600	6232708900454400	5356234211328000	5441253801984000	1836423158169600	2040470175744000	5713316492083200	342798989524992000
Conjugacy classes	1	25	5	107	3	136	2	100	3	46	1	324	1	24	7	24	29	82	3	7	149	3	32	91	1	1	16	2	38	35	4	1	18	2	12	84	2	7	11	4	2	4	22	7	2	2	20	2	4	4	14	1526
Divisions	1	25	5	107	3	136	2	100	3	46	1	324	1	24	6	24	29	82	3	7	149	3	32	90	1	1	16	1	38	35	4	1	18	1	12	84	1	5	11	4	1	4	22	5	1	1	20	1	4	4	7	1506
Autjugacy classes	1	24	5	104	3	134	2	96	3	45	1	320	1	22	6	22	27	81	3	7	147	3	31	88	1	1	16	1	38	35	4	1	18	1	12	83	1	5	11	4	1	4	22	5	1	1	20	1	4	4	7	1478

Minimal presentations

Permutation degree:	$32$
Transitive degree:	$32$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Permutation group:	Degree $32$ $\langle(1,16,14,3,11,18,6,21,8,28,20,24)(2,15,13,4,12,17,5,22,7,27,19,23)(9,32,25,30) \!\cdots\! \rangle$
Transitive group:	32T2801269				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^{15}$ . $A_{16}$	$C_2$ . $(C_2^{14}.A_{16})$			more information

Elements of the group are displayed as permutations of degree 32.

Homology

Abelianization:	$C_1 $
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 4 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	a subgroup isomorphic to $C_2^{15}.A_{16}$
Frattini:	a subgroup isomorphic to $C_2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^{11}.C_2^6.C_2^6.C_2^5.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5:C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$

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 819200 (order at least 418455797760) are shown, as well as all normal subgroups of any index.

Order 342798989524992000: $C_2^{15}.A_{16}$

Order 21424936845312000: $C_2^{15}.A_{15}$

Order 10712468422656000: $C_2^{14}.A_{15}$

Order 2856658246041600: $C_2^{15}.A_{14}.C_2$

Order 1428329123020800: $C_2^{14}.A_{14}.C_2$ x 2, $C_2^{15}.A_{14}$

Order 714164561510400: $C_2^{14}.A_{14}$, $C_2^{13}.A_{14}.C_2$, $C_2^{14}.A_{14}$

Order 612141052723200: $A_4.C_2^{12}.C_2.A_{13}.C_2$

Order 357082280755200: $C_2^{13}.A_{14}$

Order 306070526361600: $A_4.C_2^{12}.A_{13}.C_2$ x 2, $A_4.C_2^{12}.C_2.A_{13}$

Order 204047017574400: $C_2^{15}.A_{13}.C_2$

Order 188351093145600: $(C_2\times C_2^3:A_4).C_2^{10}.C_2.A_{12}.C_2$

Order 153035263180800: $A_4.C_2^{12}.A_{13}$, $(C_2^{12}\times C_6).A_{13}.C_2$

Order 102023508787200: $C_2^{14}.A_{13}.C_2$ x 6, $C_2^{15}.A_{13}$

Order 94175546572800: $(C_2\times C_2^3:A_4).C_2^{10}.A_{12}.C_2$ x 2, $(C_2\times C_2^3:A_4).C_2^{10}.C_2.A_{12}$

Order 78479622144000: $C_2^{15}.A_{11}.S_5$

Order 76517631590400: $(C_2^{11}\times C_6).A_{13}.C_2$ x 2, $(C_2^{12}\times C_6).A_{13}$

Order 62783697715200: $(C_2\times Q_8:C_2^2).C_2^{10}.C_2.A_{12}.C_2$

Order 53271016243200: $C_2^{15}.A_8^2.C_2^2$

Order 51011754393600: $C_2^{12}.(C_2\times S_{13})$ x 4, $C_2^{14}.A_{13}$ x 3, $C_2^{13}.A_{13}.C_2$ x 2

Order 47087773286400: $(C_2\times C_2^3:A_4).C_2^{10}.A_{12}$, $(C_2\times A_4).C_2^{10}.C_2^2.A_{12}.C_2$

Order 42807066624000: $C_2^{15}.A_{10}.A_6.C_2$

Order 39239811072000: $C_2^{14}.A_{11}.S_5$ x 2, $C_2^{15}.A_{11}.A_5$

Order 38258815795200: $(C_2^{11}\times C_6).A_{13}$

Order 31391848857600: $C_2^{14}.C_2^2.A_{12}.C_2$ x 4, $C_2^{15}.C_2.A_{12}.C_2$, $(C_2^2\times D_4).C_2^{10}.C_2^2.A_{12}$, $(C_2^2\times D_4).C_2^{10}.C_2.A_{12}.C_2$

Order 29964946636800: $C_2^{15}.A_9.A_7.C_2$

Order 26635508121600: $C_2^{15}.A_8^2.C_2$ x 2, $C_2^{14}.A_8^2.C_2^2$ x 2, $C_2^{14}.A_8^2.C_2^2$ x 2, $C_2^{15}.A_8.A_8.C_2$

Order 25505877196800: $C_2^{13}.A_{13}$ x 3, $C_2^{12}.S_{13}$ x 2

Order 23543886643200: $(C_2\times A_4).C_2^{10}.C_2.A_{12}.C_2$ x 5, $(C_2^2\times A_4).C_2^{10}.A_{12}.C_2$ x 2, $(C_2\times \SL(2,3)).C_2^{10}.A_{12}.C_2$, $(C_2\times A_4).C_2^{10}.C_2^2.A_{12}$

Order 21403533312000: $C_2^{14}.A_{10}.A_6.C_2$ x 2, $C_2^{15}.A_{10}.A_6$

Order 20922789888000: $C_2.A_{16}$ x 2

Order 19619905536000: $C_2^{14}.A_{11}.A_5$

Order 15695924428800: $C_2^{14}.C_2.A_{12}.C_2$ x 7, $(C_2^2\times D_4).C_2^{10}.C_2.A_{12}$ x 3, $C_2^{13}.C_2^2.A_{12}.C_2$ x 2, $(C_2\times D_4:C_2).C_2^{10}.A_{12}.C_2$ x 2, $C_2^{15}.A_{12}.C_2$, $C_2\wr A_4.C_2^{10}.C_2.A_{11}.C_2$, $(C_2^{12}\times C_4).C_2.A_{12}.C_2$, $(C_2^2\times D_4).C_2^{10}.A_{12}.C_2$

Order 14982473318400: $C_2^{14}.A_9.A_7.C_2$ x 2, $C_2^{15}.A_9.A_7$

Order 13317754060800: $C_2^{14}.A_8^2.C_2$ x 4, $C_2^{14}.A_8.A_8.C_2$ x 2, $C_2^{15}.A_8.A_8$

Order 13079937024000: $C_2^4:C_5.C_2^{10}.C_2^2.A_{11}.C_2$

Order 12752938598400: $C_2^{12}.A_{13}$

Order 11771943321600: $(C_2\times A_4).C_2^{10}.A_{12}.C_2$ x 6, $(C_2^2\times A_4).C_2^{10}.A_{12}$ x 2, $(C_2\times A_4).C_2^{10}.C_2.A_{12}$ x 2, $(C_2^{12}\times C_6).A_{12}.C_2$, $(C_2\times \SL(2,3)).C_2^{10}.A_{12}$

Order 10701766656000: $C_2^{14}.A_{10}.A_6$

Order 10461394944000: $A_{16}$ x 2

Order 7847962214400: $C_2^{14}.A_{12}.C_2$ x 15, $(C_2^{12}\times C_4).A_{12}.C_2$ x 9, $C_2^3:A_4.C_2^{10}.C_2.A_{11}.C_2$ x 6, $C_2^{13}.C_2.A_{12}.C_2$ x 4, $C_2^{14}.C_2.A_{12}$ x 3, $C_2^{13}.C_2^2.A_{12}$ x 3, $C_2^{15}.A_{12}$, $C_2\wr A_4.C_2.C_2^{10}.A_{11}$, $(C_2^{11}\times C_4).C_2.A_{12}.C_2$, $(C_2\times Q_8).C_2^{10}.A_{12}.C_2$, $(C_2\times A_4).C_2^{10}.C_2^3.A_{11}.C_2$

Order 7491236659200: $C_2^{14}.A_9.A_7$

Order 7134511104000: $C_2^{15}.A_{10}.S_5$ x 2

Order 6658877030400: $C_2^{14}.A_8.A_8$

Order 6539968512000: $C_2^4:C_5.C_2^{10}.C_2.A_{11}.C_2$ x 2, $C_2^4:C_5.C_2^{10}.C_2^2.A_{11}$

Order 5885971660800: $(C_2^{11}\times C_6).A_{12}.C_2$ x 6, $(C_2\times A_4).C_2^{10}.A_{12}$ x 2, $(C_2^{12}\times C_6).A_{12}$

Order 5231974809600: $C_2\wr C_2^2.C_2^{10}.C_2.A_{11}.C_2$

Order 4904976384000: $C_2^{11}.A_{11}.S_5$

Order 4280706662400: $C_2^{15}.A_9.A_6.C_2$, $A_4^2.C_2^2.C_2^2.C_2^8.C_2.A_{10}.C_2$

Order 3923981107200: $C_2^{13}.A_{12}.C_2$ x 29, $(C_2\times A_4).C_2^{10}.C_2^2.A_{11}.C_2$ x 7, $C_2^{14}.A_{12}$ x 6, $(C_2^{11}\times C_4).A_{12}.C_2$ x 6, $C_2^3:A_4.C_2^{10}.A_{11}.C_2$ x 4, $C_2^3:A_4.C_2^{10}.C_2.A_{11}$ x 3, $A_4.C_2^{10}.C_2^3.A_{11}.C_2$ x 3, $(C_2^{12}\times C_4).A_{12}$ x 3, $C_2^{13}.C_2.A_{12}$ x 2, $(C_2^2\times A_4).C_2^{10}.C_2.A_{11}.C_2$ x 2, $(C_2\times D_4).C_2^{10}.A_{12}$ x 2, $\GL(2,\mathbb{Z}/4).C_2^{10}.C_2.A_{11}$, $\GL(2,\mathbb{Z}/4).C_2^{10}.A_{11}.C_2$, $C_2^{12}.C_2^2.A_{12}$, $A_4:C_4.C_2^{10}.C_2.A_{11}.C_2$

Order 3567255552000: $C_2^{14}.A_{10}.S_5$ x 8, $C_2^{15}.A_{10}.A_5$ x 2

Order 3329438515200: $C_2^{15}.A_8.A_7.C_2$

Order 3269984256000: $C_2^4:C_5.C_2^{10}.C_2.A_{11}$ x 3

Order 2942985830400: $(C_2^{10}\times C_6).A_{12}.C_2$ x 4, $(C_2^{11}\times C_6).A_{12}$ x 3

Order 2853804441600: $C_2^5.S_4.C_2.C_2^8.C_2.A_{10}.C_2$, $C_2^3:A_4.C_4.C_2.C_2^8.C_2^2.A_{10}.C_2$

Order 2615987404800: $C_2^{13}.C_2^3.A_{11}.C_2$ x 7, $(C_2\times D_4).C_2^{10}.C_2^2.A_{11}.C_2$ x 2, $C_2^{14}.C_2^2.A_{11}.C_2$, $C_2^{13}.C_2^4.A_{11}$, $C_2^{12}.C_2^4.A_{11}.C_2$, $C_2^{11}.C_2^5.A_{11}.C_2$, $C_2^2\wr C_2.C_2^{10}.C_2.A_{11}.C_2$, $C_2\wr C_2^2.C_2^{10}.A_{11}.C_2$

Order 2452488192000: $C_2^{10}.A_{11}.S_5$ x 2, $C_2^{11}.A_{11}.A_5$

Order 2140353331200: $C_2^{14}.A_9.A_6.C_2$ x 6, $A_4^2.C_2^3.C_2^8.C_2.A_{10}.C_2$ x 2, $A_4^2.C_2^2.C_2^8.C_2^2.A_{10}.C_2$ x 2, $C_2^{15}.A_9.A_6$, $A_4^2.C_4.C_2^2.C_2^8.A_{10}.C_2$, $A_4^2.C_2^2.C_2^2.C_2^8.C_2.A_{10}$, $(C_2\times A_4^2).C_2^8.C_2^3.A_{10}.C_2$

Order 1961990553600: $A_4.C_2^{10}.C_2^2.A_{11}.C_2$ x 20, $C_2^{12}.A_{12}.C_2$ x 14, $(C_2\times A_4).C_2^{10}.C_2.A_{11}.C_2$ x 14, $C_2^{13}.A_{12}$ x 9, $C_2^{12}.S_{12}$ x 4, $(C_2\times A_4).C_2^{10}.C_2^2.A_{11}$ x 3, $A_4:C_4.C_2^{10}.A_{11}.C_2$ x 2, $(C_2^{11}\times C_4).A_{12}$ x 2, $\SL(2,3).C_2^{10}.C_2.A_{11}.C_2$, $\GL(2,\mathbb{Z}/4).C_2^{10}.A_{11}$, $S_4.C_2^{10}.C_2.A_{11}.C_2$, $C_2^3:A_4.C_2^{10}.A_{11}$, $A_4:C_4.C_2^{10}.C_2.A_{11}$, $A_4.C_2^{10}.C_2^3.A_{11}$, $(C_2^{11}\times C_6).C_2^2.A_{11}.C_2$, $(C_2^2\times A_4).C_2^{10}.C_2.A_{11}$

Order 1783627776000: $C_2^{14}.A_{10}.A_5$ x 4, $C_2^{13}.A_{10}.S_5$ x 4

Order 1664719257600: $C_2^{14}.A_8.A_7.C_2$ x 6, $C_2^{15}.A_8.A_7$, $C_2^{15}.A_7^2.D_4$

Order 1634992128000: $C_2^4:C_5.C_2^{10}.A_{11}$

Order 1471492915200: $(C_2^{10}\times C_6).A_{12}$

Order 1426902220800: $C_2^5.A_4.C_2.C_2^8.C_2.A_{10}.C_2$ x 5, $C_2^3:A_4.C_2^2.C_2^8.C_2^2.A_{10}.C_2$ x 5, $C_2^3:A_4.C_4.C_2.C_2^8.C_2.A_{10}.C_2$ x 4, $C_2^3:A_4.C_2^3.C_2^8.C_2.A_{10}.C_2$ x 3, $C_2^{15}.A_9.S_5.C_2$, $C_2^5.S_4.C_2.C_2^8.A_{10}.C_2$, $C_2^5.A_4.C_2.C_2^8.C_2^2.A_{10}$, $C_2^3:A_4.D_4.C_2.C_2^8.A_{10}.C_2$, $C_2^3:A_4.C_4.C_2.C_2^8.C_2^2.A_{10}$, $C_2\wr A_4.C_2.C_2^8.C_2^2.A_{10}.C_2$

Order 1337720832000: $C_2^{10}.A_{10}.A_6.C_2$

Order 1307993702400: $C_2^{13}.C_2^2.A_{11}.C_2$ x 17, $C_2^{12}.C_2^3.A_{11}.C_2$ x 7, $(C_2\times D_4).C_2^{10}.C_2.A_{11}.C_2$ x 6, $C_2^{13}.C_2^3.A_{11}$ x 5, $C_2^{14}.C_2.A_{11}.C_2$ x 4, $Q_8:C_2^2.C_2^{10}.A_{11}.C_2$ x 3, $C_2^{11}.C_2^4.A_{11}.C_2$ x 3, $(C_2^{11}\times C_4).C_2^2.A_{11}.C_2$ x 3, $C_2^2\wr C_2.C_2^{10}.A_{11}.C_2$ x 2, $C_2^2:C_4.C_2^{10}.C_2^2.A_{11}$ x 2, $C_2^2.D_4.C_2^{10}.A_{11}.C_2$ x 2, $C_2^{15}.A_{11}.C_2$

Order 1307674368000: $C_2.A_{15}$

Order 1226244096000: $C_2^{10}.A_{11}.A_5$

Order 1189085184000: $(C_2\times C_2^4:C_5).C_2^8.C_2^3.A_{10}.C_2$

Order 1070176665600: $(C_2\times A_4^2).C_2^8.C_2^2.A_{10}.C_2$ x 8, $A_4^2.C_2^2.C_2^8.C_2.A_{10}.C_2$ x 6, $C_2^{13}.A_9.A_6.C_2$ x 4, $C_2^{14}.A_9.A_6$ x 3, $A_4^2.C_4.C_2.C_2^8.A_{10}.C_2$ x 2, $A_4^2.C_2^2.C_2^8.C_2^2.A_{10}$ x 2, $(C_2\times A_4^2).C_2^8.C_2^3.A_{10}$

Order 998831554560: $C_2^{15}.A_9.\PSL(2,7)$

Order 980995276800: $A_4.C_2^{10}.C_2.A_{11}.C_2$ x 29, $(C_2\times A_4).C_2^{10}.A_{11}.C_2$ x 8, $A_4.C_2^{10}.C_2^2.A_{11}$ x 6, $(C_2^{11}\times C_6).C_2.A_{11}.C_2$ x 4, $S_4.C_2^{10}.C_2.A_{11}$ x 3, $D_6.C_2^{10}.C_2.A_{11}.C_2$ x 3, $C_2^{12}.A_{12}$ x 3, $\SL(2,3).C_2^{10}.A_{11}.C_2$ x 2, $C_3:C_4.C_2^{10}.C_2.A_{11}.C_2$ x 2, $(C_2^{12}\times C_6).A_{11}.C_2$ x 2, $(C_2^{10}\times C_6).C_2^2.A_{11}.C_2$ x 2, $(C_2\times A_4).C_2^{10}.C_2.A_{11}$ x 2, $C_2^{11}.A_{12}.C_2$ x 2, $C_2^{12}.A_{12}$ x 2, $\SL(2,3).C_2^{10}.C_2.A_{11}$, $S_4.C_2^{10}.A_{11}.C_2$, $C_3:D_4.C_2^{10}.A_{11}.C_2$, $A_4:C_4.C_2^{10}.A_{11}$, $(C_2^{11}\times C_6).C_2^2.A_{11}$, $C_2^{11}.A_{12}.C_2$

Order 951268147200: $C_2^{15}.A_8.A_6.C_2^2$, $(C_2^3\times D_4).C_2^8.C_2^4.A_{10}.C_2$

Order 891813888000: $C_2^{13}.A_{10}.A_5$

Order 887850270720: $C_2^{15}.C_2^3.A_8.\PSL(2,7)$

Order 856141332480: $C_2.C_2^6.C_3^2.C_2^8.C_2^3.A_9.C_2$

Order 832359628800: $C_2^{14}.A_7^2.D_4$ x 4, $C_2^{13}.A_8.A_7.C_2$ x 4, $C_2^{14}.A_8.A_7$ x 3, $C_2^{15}.A_7^2.C_4$, $C_2^{15}.A_7^2.C_2^2$, $C_2^{15}.A_7.A_7.C_2^2$

Order 817496064000: $(C_2^9\times C_{10}).C_2^2.A_{11}.C_2$

Order 761014517760: $C_2^6.A_4^2.C_2^2.C_2^6.C_2^3.A_8.C_2$

Order 713451110400: $C_2^3:A_4.C_2^2.C_2^8.C_2.A_{10}.C_2$ x 17, $(C_2\times C_2^3:A_4).C_2^8.C_2^2.A_{10}.C_2$ x 12, $C_2^{14}.A_9.A_5.C_2^2$ x 7, $C_2^{14}.A_9.S_5.C_2$ x 5, $C_2^5.A_4.C_2.C_2^8.A_{10}.C_2$ x 4, $C_2^3:A_4.C_2^2.C_2^8.C_2^2.A_{10}$ x 4, $C_2^{15}.A_9.S_5$ x 3, $C_2^3:A_4.C_4.C_2.C_2^8.A_{10}.C_2$ x 3, $C_2^5.A_4.C_2.C_2^8.C_2.A_{10}$ x 2, $C_2^3:S_4.C_2.C_2^8.C_2.A_{10}.C_2$ x 2, $(C_2\times C_2^3:A_4).C_2^8.C_2^3.A_{10}$ x 2, $C_2^{15}.A_9.A_5.C_2$, $C_2^3:A_4.C_2^3.C_2^8.C_2.A_{10}$, $C_2^3:A_4.C_2^3.C_2^8.A_{10}.C_2$, $C_2^3.C_2^4.C_6.C_2^8.C_2.A_{10}$, $C_2^2:S_4.C_2^2.C_2^8.C_2.A_{10}.C_2$, $C_2\wr A_4.C_2.C_2^8.C_2.A_{10}.C_2$, $(C_2\times C_4^2:C_3).C_2^8.C_2^3.A_{10}.C_2$, $(C_2\times C_2^4:C_3).C_2^8.C_2^3.A_{10}.C_2$, $(C_2\times A_4).C_2^4.C_2^8.C_2.A_{10}.C_2$

Order 668860416000: $C_2^9.A_{10}.A_6.C_2$ x 2, $C_2^{10}.A_{10}.A_6$

Order 653996851200: $C_2^{13}.C_2.A_{11}.C_2$ x 38, $C_2^{12}.C_2^2.A_{11}.C_2$ x 23, $C_2^{14}.A_{11}.C_2$ x 19, $C_2^{13}.C_2^2.A_{11}$ x 11, $(C_2^{11}\times C_4).C_2.A_{11}.C_2$ x 11, $(C_2^{12}\times C_4).A_{11}.C_2$ x 9, $(C_2\times D_4).C_2^{10}.A_{11}.C_2$ x 5, $D_4.C_2^{10}.C_2.A_{11}.C_2$ x 4, $C_2^{12}.C_2^3.A_{11}$ x 3, $D_4:C_2.C_2^{10}.C_2.A_{11}$ x 2, $C_2^{11}.C_2^3.A_{11}.C_2$ x 2, $C_2^2:C_4.C_2^{10}.A_{11}.C_2$ x 2, $(C_2^{10}\times C_4).C_2^2.A_{11}.C_2$ x 2, $(C_2\times D_4).C_2^{10}.C_2.A_{11}$ x 2, $D_4:C_2.C_2^{10}.A_{11}.C_2$, $C_2^{15}.A_{11}$, $(C_2\times C_2^2:C_4).C_2^{10}.A_{11}$

Order 653837184000: $A_{15}$

Order 594542592000: $(C_2\times C_2^4:D_5).C_2^8.C_2.A_{10}.C_2$ x 2, $(C_2\times C_2^4:C_5).C_2^8.C_2^2.A_{10}.C_2$ x 2, $C_2^4:D_5.C_2^2.C_2^8.A_{10}.C_2$, $C_2^4:C_5.C_2^3.C_2^8.A_{10}.C_2$, $(C_2^2\times C_2^4:C_5).C_2^8.C_2^2.A_{10}$

Order 535088332800: $(C_2\times A_4^2).C_2^8.C_2.A_{10}.C_2$ x 15, $A_4^2.C_2^2.C_2^8.A_{10}.C_2$ x 5, $(C_2\times A_4^2).C_2^8.C_2^2.A_{10}$ x 4, $A_4^2.C_2^2.C_2^8.C_2.A_{10}$ x 3, $C_2^{13}.A_9.A_6$, $(C_6\times A_4).C_2^8.C_2^3.A_{10}.C_2$

Order 499415777280: $C_2^{14}.A_9.\PSL(2,7)$

Order 490497638400: $(C_2^{11}\times C_6).A_{11}.C_2$ x 16, $(C_2^{10}\times C_6).C_2.A_{11}.C_2$ x 15, $A_4.C_2^{10}.A_{11}.C_2$ x 12, $A_4.C_2^{10}.C_2.A_{11}$ x 8, $S_3.C_2^{10}.C_2.A_{11}.C_2$ x 2, $(C_2^{11}\times C_6).C_2.A_{11}$ x 2, $(C_2^{10}\times C_6).C_2^2.A_{11}$ x 2, $(C_2^9\times C_6).C_2^2.A_{11}.C_2$ x 2, $(C_2\times A_4).C_2^{10}.A_{11}$ x 2, $\SL(2,3).C_2^{10}.A_{11}$, $D_6.C_2^{10}.C_2.A_{11}$, $C_3:C_4.C_2^{10}.A_{11}.C_2$, $(C_2^{12}\times C_6).A_{11}$, $(C_2^9\times C_6).C_2^3.A_{11}$, $C_2^{11}.A_{12}$

Order 475634073600: $C_2^{14}.A_8.A_6.C_2^2$ x 12, $(C_2^3\times D_4).C_2^8.C_2^3.A_{10}.C_2$ x 10, $(C_2^2\times D_4).C_2^3.C_2^8.C_2.A_{10}.C_2$ x 4, $C_2^{15}.A_8.A_6.C_2$ x 3, $(C_2^2\times D_4).C_2^8.C_2^4.A_{10}.C_2$ x 3, $C_2^{13}.C_2^4.A_{10}.C_2$ x 2, $(C_2^2\times D_4).C_2^2.C_2^8.C_2^2.A_{10}.C_2$ x 2, $C_2^{14}.C_2^3.A_{10}.C_2$, $C_2^4.C_2^4.C_2^8.C_2.A_{10}.C_2$, $C_2^4.C_2^3.C_2^8.C_2^2.A_{10}.C_2$, $C_2^4.(C_2\times C_4).C_2^8.C_2^2.A_{10}.C_2$, $C_2^3.C_2^5.C_2^8.C_2.A_{10}.C_2$, $C_2^3.C_2^4.C_2^8.C_2^3.A_{10}$, $C_2^3.C_2^4.C_2^8.C_2^2.A_{10}.C_2$, $C_2.C_2^6.C_2^8.C_2^2.A_{10}.C_2$, $(C_2^{11}\times C_4).C_2^4.A_{10}.C_2$, $(C_2\times Q_8:C_2^2).C_2^8.C_2^3.A_{10}.C_2$, $(C_2\times A_4).C_2^6.C_2^6.A_8.A_5.C_2^2$

Order 468202291200: $C_2^9.A_9.A_7.C_2$

Order 445906944000: $C_2^{11}.A_{10}.S_5$

Order 443925135360: $C_2^{14}.C_2^3.A_8.\PSL(2,7)$

Order 428070666240: $C_2^5.A_4.C_3.C_2^8.C_2^2.A_9.C_2$ x 4, $C_2^5.A_4.S_3.C_2^8.C_2.A_9.C_2$ x 2, $C_2^5.A_4.C_6.C_2^8.C_2.A_9.C_2$ x 2, $C_2^3.A_4^2.C_2^8.C_2^2.A_9.C_2$ x 2, $C_2.C_2^6.C_3.D_6.C_2^8.A_9.C_2$ x 2, $Q_8.A_4^2.C_2.C_2^8.C_2.A_9.C_2$, $C_2^5.A_4.S_3.C_2^8.C_2^2.A_9$, $C_2^5.(C_6\times A_4).C_2^8.C_2.A_9.C_2$, $A_4^2.C_2^2.C_2^8.C_2^3.A_9.C_2$

Order 169114337280: $C_2^{15}.C_2.C_2^6.A_8.C_2$

Order 130459631616: $C_2^{12}.(C_2^{10}.C_6:S_3^3.S_4)$

Order 10569646080: $C_2^{15}.C_2^4.A_8$, $C_2^{11}.C_2^4.C_2^4.A_8$

Order 536870912: $C_2^{11}.C_2^6.C_2^6.C_2^5.C_2$

Order 32768: $C_2^{15}$

Order 729: $C_3^5:C_3$

Order 125: $C_5^3$

Order 49: $C_7^2$

Order 13: $C_{13}$

Order 11: $C_{11}$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 342798989524992000: $C_2^{15}.A_{16}$ ($C_1$)

Order 32768: $C_2^{15}$ ($A_{16}$)

Order 2: $C_2$ ($C_2^{14}.A_{16}$)

Order 1: $C_1$ ($C_2^{15}.A_{16}$)

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 3 larger groups in the database.

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

Character theory

Complex character table

The $1526 \times 1526$ character table is not available for this group.

Rational character table

The $1506 \times 1506$ rational character table is not available for this group.