Abstract group 357082280755200.a: $C_2^{13}.A_{14}$

• Label: 357...200.a
• Order: \( 2^{23} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \cdot 11 \cdot 13 \)
• 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
• $\card{\Aut(G)}$: \( 2^{24} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \cdot 11 \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $28$
• Trans deg.: $28$
• Rank: $2$

Group information

Description:	$C_2^{13}.A_{14}$
Order:	\(357082280755200\)\(\medspace = 2^{23} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \cdot 11 \cdot 13 \)
Exponent:	\(720720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \)
Automorphism group:	Group of order \(714164561510400\)\(\medspace = 2^{24} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \cdot 11 \cdot 13 \)
Composition factors:	$C_2$ x 13, $A_{14}$
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	40	42	45	48	56	60	66	70	84	90	120
Elements	1	288680575	6029065952	584614622592	18598804224	3410381889824	3643854888960	9424111656960	1673823191040	5547256915200	1352584396800	35591252336640	27467867750400	4195153797120	747193294848	7439214182400	14692448010240	26720727441408	752777625600	9468090777600	41039664906240	27467867750400	7970586624000	10003848830976	10820675174400	1275293859840	7439214182400	13762546237440	7394933145600	7935161794560	14878428364800	6376469299200	14134506946560	10820675174400	8927057018880	3188234649600	7935161794560	2975685672960	357082280755200
Conjugacy classes	1	19	4	68	2	82	2	54	2	28	1	150	2	12	4	11	14	39	2	3	61	2	12	42	2	1	4	14	18	2	6	4	22	2	7	4	2	4	709
Divisions	1	19	4	68	2	82	2	52	2	28	1	150	1	12	4	10	12	39	2	3	59	1	12	40	1	1	4	13	16	1	5	4	22	1	5	4	1	4	688
Autjugacy classes	1	19	4	66	2	82	2	51	2	28	1	148	1	12	4	10	12	38	2	3	58	1	12	40	1	1	4	13	16	1	5	4	22	1	5	4	1	4	681

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$28$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	14	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $28$ $\langle(1,16,7,20,11,18,24,5,25,28,9,21,13,2,15,8,19,12,17,23,6,26,27,10,22,14) \!\cdots\! \rangle$
Transitive group:	28T1809				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:	$C_2^{13}$ . $A_{14}$	$C_2$ . $(C_2^{12}.A_{14})$			more information

Elements of the group are displayed as permutations of degree 28.

Homology

Abelianization:	$C_1 $
Schur multiplier:	not computed
Commutator length:	$1$

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 $C_2^{13}.A_{14}$
Frattini:	a subgroup isomorphic to $C_2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.C_2^6.C_2^6.C_2^5$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4:C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
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 319488 (order at least 1117670400) are shown, as well as all normal subgroups of any index.

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

Order 25505877196800: $C_2^{13}.A_{13}$

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

Order 3923981107200: $C_2^{13}.A_{12}.C_2$

Order 1961990553600: $C_2^{12}.A_{12}.C_2$ x 2, $C_2^{13}.A_{12}$

Order 980995276800: $C_2^{12}.A_{12}$, $A_4.C_2^{10}.C_2.A_{11}.C_2$, $C_2^{11}.A_{12}.C_2$, $C_2^{12}.A_{12}$

Order 490497638400: $A_4.C_2^{10}.A_{11}.C_2$ x 2, $(C_2\times A_4).C_2^{10}.A_{11}$, $C_2^{11}.A_{12}$

Order 356725555200: $(C_2\times C_2^3:A_4).C_2^8.C_2.A_{10}.C_2$

Order 326998425600: $C_2^{13}.A_{11}.C_2$

Order 245248819200: $A_4.C_2^{10}.A_{11}$, $(C_2^{10}\times C_6).A_{11}.C_2$

Order 208089907200: $C_2^{13}.A_7^2.C_4$

Order 178362777600: $(C_2\times C_2^3:A_4).C_2^8.A_{10}.C_2$ x 2, $C_2^{13}.A_9.S_5$, $(C_2\times C_2^3:A_4).C_2^8.C_2.A_{10}$

Order 163499212800: $C_2^{12}.S_{11}$ x 4, $C_2^{13}.A_{11}$, $C_2^{12}.A_{11}.C_2$, $C_2^{12}.S_{11}$

Order 122624409600: $(C_2^9\times C_6).A_{11}.C_2$ x 2, $(C_2^{10}\times C_6).A_{11}$

Order 118908518400: $C_2^{13}.A_8.A_6.C_2$, $(C_2\times Q_8:C_2^2).C_2^8.C_2.A_{10}.C_2$

Order 104044953600: $C_2^{12}.A_7^2.C_4$ x 2, $C_2^{13}.A_7.A_7.C_2$

Order 89181388800: $C_2^{12}.A_9.S_5$ x 2, $C_2^{13}.A_9.A_5$, $(C_2\times C_2^3:A_4).C_2^8.A_{10}$, $(C_2\times A_4).C_2^8.C_2^2.A_{10}.C_2$

Order 87178291200: $C_2.A_{14}$ x 2

Order 81749606400: $C_2^{10}.(C_2\times S_{11})$ x 4, $C_2^{12}.A_{11}$ x 3, $C_2^{10}.A_{11}.C_4$ x 2

Order 61312204800: $(C_2^9\times C_6).A_{11}$

Order 59454259200: $C_2^{12}.C_2^2.A_{10}.C_2$ x 2, $C_2^{12}.A_8.A_6.C_2$ x 2, $C_2^{11}.C_2^3.A_{10}.C_2$ x 2, $(C_2^2\times D_4).C_2^8.C_2.A_{10}.C_2$ x 2, $C_2^{13}.A_8.A_6$, $(C_2^2\times D_4).C_2^8.C_2^2.A_{10}$

Order 52022476800: $C_2^{12}.A_7.A_7.C_2$ x 2, $C_2^{13}.A_7.A_7$

Order 44590694400: $(C_2\times A_4).C_2^8.C_2.A_{10}.C_2$ x 6, $C_2^{12}.A_9.A_5$, $(C_2^2\times A_4).C_2^8.A_{10}.C_2$, $(C_2\times \SL(2,3)).C_2^8.A_{10}.C_2$, $(C_2\times A_4).C_2^8.C_2^2.A_{10}$

Order 43589145600: $A_{14}$ x 2

Order 40874803200: $C_2^{11}.A_{11}$ x 3, $C_2^{10}.S_{11}$ x 2

Order 35672555520: $C_2^3:A_4.C_2^8.C_2^2.A_9.C_2$

Order 29727129600: $C_2^{12}.C_2.A_{10}.C_2$ x 8, $C_2^{12}.C_2^2.A_{10}$ x 2, $C_2^{11}.C_2^2.A_{10}.C_2$ x 2, $(C_2^2\times D_4).C_2^8.A_{10}.C_2$ x 2, $C_2^{13}.A_{10}.C_2$, $C_2^{12}.A_8.A_6$, $C_2^{11}.C_2^3.A_{10}$, $C_2^4:C_5.C_2^8.C_2^2.A_9.C_2$, $(C_2\times D_4:C_2).C_2^8.A_{10}.C_2$

Order 26011238400: $C_2^{12}.A_7.A_7$

Order 22295347200: $(C_2\times A_4).C_2^8.A_{10}.C_2$ x 6, $(C_2\times A_4).C_2^8.C_2.A_{10}$ x 3, $(C_2^{10}\times C_6).A_{10}.C_2$, $(C_2^2\times A_4).C_2^8.A_{10}$, $(C_2\times \SL(2,3)).C_2^8.A_{10}$

Order 20437401600: $C_2^{10}.A_{11}$

Order 19818086400: $C_2^{13}.A_8.S_5$ x 2

Order 17836277760: $C_2^3:A_4.C_2^8.C_2.A_9.C_2$ x 5, $C_2^3:A_4.C_2^8.C_2^2.A_9$, $(C_2^2\times A_4).C_2^8.C_2^2.A_9.C_2$, $(C_2\times C_2^3:A_4).C_2^8.A_9.C_2$

Order 14863564800: $C_2^{12}.A_{10}.C_2$ x 15, $(C_2^{10}\times C_4).A_{10}.C_2$ x 9, $C_2^{11}.C_2.A_{10}.C_2$ x 4, $C_2^{12}.C_2.A_{10}$ x 3, $C_2^{11}.C_2^2.A_{10}$ x 2, $C_2^4:C_5.C_2^8.C_2.A_9.C_2$ x 2, $C_2^{13}.A_{10}$, $C_2^{13}.A_7.A_6.C_2$, $(C_2^9\times C_4).C_2.A_{10}.C_2$, $(C_2^2\times D_4).C_2^8.A_{10}$, $(C_2\times D_4).C_2^8.A_{10}.C_2$, $(C_2\times C_2^4:D_5).C_2^8.A_9$

Order 11890851840: $C_2^{12}.C_2^3.A_9.C_2$, $A_4^2.C_2^3.C_2^6.C_2^2.A_8.C_2$

Order 11147673600: $(C_2^9\times C_6).A_{10}.C_2$ x 6, $(C_2\times A_4).C_2^8.A_{10}$ x 2, $C_2^9.A_9.S_5$, $(C_2^{10}\times C_6).A_{10}$

Order 9909043200: $C_2^{12}.A_8.S_5$ x 8, $C_2^{13}.A_8.A_5$ x 2

Order 8918138880: $C_2^3:A_4.C_2^8.A_9.C_2$ x 4, $A_4.C_2^8.C_2^3.A_9.C_2$ x 4, $(C_2\times A_4).C_2^8.C_2^2.A_9.C_2$ x 4, $A_4:C_4.C_2^8.C_2.A_9.C_2$ x 3, $C_2^3:A_4.C_2^8.C_2.A_9$ x 2, $(C_2^2\times A_4).C_2^8.C_2.A_9.C_2$ x 2, $C_2\wr A_4.C_2^8.A_9$, $(C_2\times \GL(2,\mathbb{Z}/4)).C_2^8.A_9$, $(C_2\times S_4).C_2^8.C_2.A_9.C_2$

Order 8493465600: $C_2^{13}.A_6^2.D_4$

Order 7927234560: $C_2^5.A_4.C_2.C_2^6.C_2^2.A_8.C_2$, $C_2^3:A_4.C_2^2.C_2^6.C_2^3.A_8.C_2$

Order 7431782400: $C_2^{11}.A_{10}.C_2$ x 24, $C_2^{12}.A_{10}$ x 6, $C_2^{12}.A_7.A_6.C_2$ x 6, $(C_2^9\times C_4).A_{10}.C_2$ x 6, $C_2^{10}.(C_2\times S_{10})$ x 5, $C_2^{11}.C_2.A_{10}$ x 3, $(C_2^{10}\times C_4).A_{10}$ x 3, $C_2^{13}.A_7.A_6$, $C_2^{10}.C_2^2.A_{10}$, $C_2^4:D_5.C_2^8.A_9$, $C_2^4:C_5.C_2^8.C_2.A_9$, $(C_2\times D_4).C_2^8.A_{10}$, $(C_2\times C_2^4:C_5).C_2^8.A_9$

Order 6227020800: $C_2\times A_{13}$

Order 5945425920: $(C_2\times D_4).C_2^8.C_2^2.A_9.C_2$ x 4, $C_2^2.D_4.C_2^8.C_2.A_9.C_2$ x 3, $A_4^2.C_2^3.C_2^6.C_2.A_8.C_2$ x 3, $C_2^{12}.C_2^2.A_9.C_2$ x 2, $C_2^{11}.C_2^3.A_9.C_2$ x 2, $C_2^9.C_2^5.A_9.C_2$, $C_2^2\wr C_2.C_2^8.C_2.A_9.C_2$, $C_2^2:C_4.C_2^8.C_2^2.A_9.C_2$, $C_2\wr C_2^2.C_2^8.C_2.A_9$, $A_4^2.C_4.C_2.C_2^6.C_2.A_8.C_2$, $A_4^2.C_2^2.C_2^6.C_2^2.A_8.C_2$, $A_4^2.C_2^2.C_2^2.C_2^6.C_2.A_8$, $(C_2\times A_4^2).C_2^6.C_2^3.A_8.C_2$

Order 5573836800: $(C_2^8\times C_6).A_{10}.C_2$ x 4, $(C_2^9\times C_6).A_{10}$ x 3, $C_2^8.A_9.S_5$ x 2, $C_2^9.A_9.A_5$

Order 4954521600: $C_2^{12}.A_8.A_5$ x 4, $C_2^{11}.A_8.S_5$ x 4, $C_2^{13}.A_7.S_5.C_2$

Order 4459069440: $(C_2\times A_4).C_2^8.C_2.A_9.C_2$ x 15, $A_4.C_2^8.C_2^2.A_9.C_2$ x 11, $S_4.C_2^8.C_2.A_9.C_2$ x 3, $A_4:C_4.C_2^8.A_9.C_2$ x 3, $(C_2^2\times A_4).C_2^8.A_9.C_2$ x 3, $\GL(2,\mathbb{Z}/4).C_2^8.A_9$ x 2, $(C_2\times S_4).C_2^8.C_2.A_9$ x 2, $(C_2\times S_4).C_2^8.A_9.C_2$ x 2, $(C_2\times A_4).C_2^8.C_2^2.A_9$ x 2, $\SL(2,3).C_2^8.C_2.A_9.C_2$, $C_3:D_4.C_2^8.C_2.A_9.C_2$, $C_2^3:A_4.C_2^8.A_9$, $A_4:C_4.C_2^8.C_2.A_9$

Order 4246732800: $C_2^{12}.A_6^2.D_4$ x 4, $C_2^{13}.A_6^2.C_4$, $C_2^{13}.A_6^2.C_2^2$, $C_2^{13}.A_6.A_6.C_2^2$

Order 3963617280: $C_2^3:A_4.C_2^3.C_2^6.C_2.A_8.C_2$ x 5, $C_2^5.A_4.C_2.C_2^6.C_2.A_8.C_2$ x 4, $C_2^3:A_4.C_4.C_2.C_2^6.C_2.A_8.C_2$ x 3, $C_2^3:A_4.C_2^2.C_2^6.C_2^2.A_8.C_2$ x 2, $Q_8.A_4:C_4.C_2.C_2^6.C_2.A_8.C_2$, $C_2^{13}.C_2^3.\PSL(2,7).A_6$, $C_2^5.S_4.C_2.C_2^6.C_2.A_8$, $C_2^5.S_4.C_2.C_2^6.A_8.C_2$, $C_2^3:S_4.C_2.C_2^6.C_2^2.A_8.C_2$, $C_2^3:A_4.D_4.C_2.C_2^6.C_2.A_8$, $C_2^3:A_4.D_4.C_2.C_2^6.A_8.C_2$, $C_2^3.C_2^4.C_6.C_2^6.C_2.A_8.C_2$, $(C_2\times C_2^3:A_4).C_2^6.C_2^3.A_8.C_2$

Order 3715891200: $C_2^{10}.S_{10}$ x 7, $C_2^{11}.A_{10}$ x 5, $C_2^{11}.A_7.A_6.C_2$ x 4, $C_2^{10}.A_{10}.C_2$ x 4, $C_2^{11}.A_{10}$ x 4, $C_2^{10}.S_{10}$ x 4, $C_2^{12}.A_7.A_6$ x 3, $C_2^{10}.S_{10}$ x 3, $(C_2^9\times C_4).A_{10}$ x 2, $C_2^8.A_8.A_6.C_2$, $C_2^4:C_5.C_2^8.A_9$

Order 3468165120: $C_2^{13}.\PSL(2,7).A_7$

Order 3397386240: $C_2^6.A_4^2.C_2^6.C_2^3.A_6.C_2$

Order 3303014400: $(C_2\times C_2^4:D_5).C_2^6.C_2^2.A_8.C_2$

Order 3113510400: $A_{13}$

Order 2972712960: $C_2^{11}.C_2^2.A_9.C_2$ x 17, $C_2^9.C_2^4.A_9.C_2$ x 10, $A_4^2.C_2^2.C_2^6.C_2.A_8.C_2$ x 8, $C_2^{12}.C_2.A_9.C_2$ x 5, $C_2^{10}.C_2^3.A_9.C_2$ x 5, $(C_2\times A_4^2).C_2^6.C_2^2.A_8.C_2$ x 5, $(C_2^9\times C_4).C_2^2.A_9.C_2$ x 4, $C_2^{11}.C_2^3.A_9$ x 2, $C_2^2.D_4.C_2^8.A_9.C_2$ x 2, $A_4^2.C_2^3.C_2^6.A_8.C_2$ x 2, $(C_2\times D_4).C_2^8.C_2.A_9.C_2$ x 2, $D_4:C_2.C_2^8.C_2^2.A_9$, $C_2^{13}.A_9.C_2$, $C_2^{12}.C_2^2.A_9$, $C_2^{10}.C_2^4.A_9$, $C_2^5.C_3:S_4.C_2.C_2^6.C_2.A_7.C_2$, $C_2^2:C_4.C_2^8.C_2.A_9.C_2$, $C_2^2.D_4.C_2^8.C_2.A_9$, $C_2\wr C_2^2.C_2^8.A_9$, $A_4^2.C_4.C_2.C_2^6.C_2.A_8$, $A_4^2.C_4.C_2.C_2^6.A_8.C_2$, $A_4^2.C_2^3.C_2^6.C_2.A_8$, $A_4^2.C_2^2.C_2^6.C_2^2.A_8$, $(C_2^8\times C_4).C_2^3.A_9.C_2$

Order 2831155200: $C_2^3:A_4.C_2^8.C_2^2.A_5^2.D_4$

Order 2786918400: $C_2^8.A_9.A_5$, $(C_2^8\times C_6).A_{10}$

Order 2642411520: $C_2^3.C_2^4.C_2^2.C_2^6.C_2.A_8.C_2$, $C_2^{13}.C_2^6.A_7.C_2$

Order 2477260800: $C_2^{12}.A_7.S_5.C_2$ x 12, $C_2^{13}.A_7.S_5$ x 3, $C_2^{13}.A_7.A_5.C_2$, $C_2^{11}.A_8.A_5$

Order 2229534720: $A_4.C_2^8.C_2.A_9.C_2$ x 28, $(C_2\times A_4).C_2^8.A_9.C_2$ x 7, $(C_2^9\times C_6).C_2.A_9.C_2$ x 4, $(C_2^8\times C_6).C_2^2.A_9.C_2$ x 4, $(C_2\times A_4).C_2^8.C_2.A_9$ x 4, $S_4.C_2^8.A_9.C_2$ x 3, $(C_2^2\times A_4).C_2^8.A_9$ x 3, $\SL(2,3).C_2^8.A_9.C_2$ x 2, $S_4.C_2^8.C_2.A_9$ x 2, $(C_2^{10}\times C_6).A_9.C_2$ x 2, $D_6.C_2^8.C_2.A_9.C_2$, $C_3:D_4.C_2^8.A_9.C_2$, $C_2\times A_4.C_2^8.C_2.A_9$, $A_4:C_4.C_2^8.A_9$, $A_4.C_2^8.C_2^2.A_9$, $(C_2^{10}\times C_6).C_2.A_9$, $(C_2^7\times C_6).C_2^3.A_9.C_2$, $(C_2\times \SL(2,3)).C_2^8.A_9$, $(C_2\times C_3:C_4).C_2^8.A_9.C_2$

Order 2123366400: $C_2^{12}.A_6.A_6.C_2^2$ x 7, $C_2^{11}.A_6^2.D_4$ x 4, $C_2^{11}.A_6^2.D_4$ x 4, $C_2^{12}.A_6^2:C_2^2$ x 4, $C_2^{12}.A_6^2.C_2^2$ x 3, $C_2^{13}.A_6.A_6.C_2$ x 2, $C_2^{12}.A_6^2.C_4$ x 2, $C_2^{11}.A_6^2.D_4$ x 2, $C_2^{12}.A_6^2:C_4$ x 2, $C_2^{13}.A_6^2.C_2$, $A_4.C_2^5.C_2^6.A_6.A_5.C_2^2$

Order 1981808640: $C_2^3:A_4.C_2^2.C_2^6.C_2.A_8.C_2$ x 14, $(C_2\times C_2^3:A_4).C_2^6.C_2^2.A_8.C_2$ x 7, $C_2^3:A_4.C_4.C_2.C_2^6.A_8.C_2$ x 4, $C_2^3:A_4.C_2^2.C_2^6.C_2^2.A_8$ x 4, $C_2^5.A_4.C_2.C_2^6.C_2.A_8$ x 3, $C_2^5.A_4.C_2.C_2^6.A_8.C_2$ x 3, $C_2^3:S_4.C_2.C_2^6.C_2.A_8.C_2$ x 3, $C_2^3:A_4.C_2^3.C_2^6.C_2.A_8$ x 3, $C_2^3:A_4.C_2^3.C_2^6.A_8.C_2$ x 3, $C_2^4:C_3.C_2^3.C_2^6.C_2.A_8.C_2$ x 2, $C_2^3:S_4.C_2^2.C_2^6.A_8.C_2$ x 2, $C_2\wr A_4.C_2.C_2^6.C_2.A_8.C_2$ x 2, $C_2^3.C_2^4.C_6.C_2^6.A_8.C_2$, $C_2\wr A_4.C_2^2.C_2^6.A_8.C_2$, $C_2.C_2^6.C_2^4.C_2^4.\PSL(2,7).A_6$, $(C_2^2\times C_4^2:C_3).C_2^6.C_2^2.A_8.C_2$, $(C_2^2\times A_4).C_2^6.C_2^4.A_8.C_2$

Order 1916006400: $C_2^2.A_{12}.C_2$

Order 1857945600: $C_2^{10}.A_{10}$ x 3, $C_2^7.A_8.A_6.C_2$ x 2, $C_2^{10}.A_{10}$ x 2, $C_2^9.S_{10}$ x 2, $D_5.C_2^8.C_2.A_9.C_2$, $C_2^{11}.A_7.A_6$, $C_2^8.A_8.A_6$, $C_2^9.S_{10}$

Order 1734082560: $C_2^{12}.\PSL(2,7).A_7$

Order 1698693120: $C_2^6.A_4^2.C_2^6.C_2^2.A_6.C_2$ x 3, $C_2^6.A_4^2.C_2^3.C_2^5.A_6.C_2$ x 2, $C_2^6.A_4^2.C_2^3.C_2^6.A_6$, $C_2^4.C_2^6.C_3.C_6.C_2^6.C_2.A_6.C_2$

Order 1651507200: $C_2^4:D_5.C_2^2.C_2^6.A_8.C_2$ x 2, $(C_2\times C_2^4:D_5).C_2^6.C_2.A_8.C_2$ x 2, $(C_2\times C_2^4:C_5).C_2^6.C_2^2.A_8.C_2$ x 2, $C_2^4:D_5.C_2^2.C_2^6.C_2.A_8$

Order 1625702400: $C_2^7.A_7.A_7.C_2$

Order 1486356480: $C_2^{11}.C_2.A_9.C_2$ x 45, $C_2^{12}.A_9.C_2$ x 19, $(C_2^9\times C_4).C_2.A_9.C_2$ x 13, $C_2^{10}.C_2^2.A_9.C_2$ x 11, $C_2^9.C_2^3.A_9.C_2$ x 11, $A_4^2.C_2^2.C_2^6.A_8.C_2$ x 11, $(C_2^{10}\times C_4).A_9.C_2$ x 9, $(C_2\times A_4^2).C_2^6.C_2.A_8.C_2$ x 9, $C_2^{11}.C_2^2.A_9$ x 6, $D_4.C_2^8.C_2.A_9.C_2$ x 4, $C_2^{10}.C_2^3.A_9$ x 3, $C_2^9.C_2^4.A_9$ x 3, $C_2^3.A_4^2.C_2^6.C_2^2.A_7.C_2$ x 3, $C_2^3.A_4^2.C_2.C_2^6.C_2.A_7.C_2$ x 3, $A_4^2.C_2^2.C_2^6.C_2.A_8$ x 3, $(C_2\times D_4).C_2^8.C_2.A_9$ x 3, $(C_2\times A_4^2).C_2^6.C_2^2.A_8$ x 3, $Q_8.C_2^8.C_2.A_9.C_2$ x 2, $C_2^5.A_4.S_3.C_2^6.C_2.A_7.C_2$ x 2, $(C_2^9\times C_4).C_2^2.A_9$ x 2, $Q_8.A_4^2.C_2^6.C_2^2.A_7.C_2$, $C_2^{13}.A_9$, $C_2^{12}.C_2.A_9$, $C_2^5.C_3:S_4.C_2.C_2^6.A_7.C_2$, $C_2^5.A_4.D_6.C_2^6.A_7.C_2$, $C_2^5.A_4.C_3.C_2^6.C_2^2.A_7.C_2$, $C_2^2\wr C_2.C_2^8.A_9$, $C_2^2:C_4.C_2^8.A_9.C_2$, $C_2.C_2^6.C_6^2.C_2^6.C_2.A_7$, $C_2.C_2^6.C_6^2.C_2^6.A_7.C_2$, $C_2.C_2^6.C_3^2.C_2^6.C_2^2.A_7.C_2$, $A_4^2.C_4.C_2.C_2^6.A_8$, $A_4^2.C_2^3.C_2^6.C_2^2.A_7.C_2$, $(C_2^8\times C_4).C_2^2.A_9.C_2$, $(C_2\times C_6\times A_4).C_2^6.C_2^2.A_8.C_2$

Order 1415577600: $C_2^3:A_4.C_2^8.C_2.A_5^2.D_4$ x 3, $C_2^3:A_4.C_2^8.C_2^2.A_5^2.C_2^2$, $C_2^3:A_4.C_2^4.C_2^6.A_5.S_5.C_2$, $C_2^3:A_4.C_2^2.C_2^8.A_5^2.C_4$, $(C_2\times C_2^3:A_4).C_2^8.A_5^2.D_4$

Order 1321205760: $(C_2^2\times D_4).C_2^3.C_2^6.C_2.A_8.C_2$ x 9, $(C_2^3\times D_4).C_2^6.C_2^3.A_8.C_2$ x 4, $C_2^5.C_2^3.C_2^6.C_2.A_8.C_2$ x 3, $C_2.C_2^6.C_2^6.C_2^2.A_8.C_2$ x 3, $(C_2\times D_4).D_4.C_2.C_2^6.C_2.A_8.C_2$ x 3, $(C_2\times D_4).C_2^4.C_2^6.C_2.A_8.C_2$ x 3, $C_2^{11}.C_2^4.A_8.C_2$ x 2, $C_2^3.C_2^4.C_2^6.C_2^2.A_8.C_2$, $(C_2^3\times D_4).C_2^6.C_2^4.A_8$, $(C_2^2\times D_4).C_2^6.C_2^4.A_8.C_2$, $(C_2^2\times D_4).C_2^4.C_2^6.A_8.C_2$, $C_2^{13}.C_2^6.A_7$

Order 1238630400: $C_2^{12}.A_7.S_5$ x 22, $C_2^{11}.A_7.S_5.C_2$ x 19, $C_2^{12}.A_7.A_5.C_2$ x 6, $C_2^{13}.A_7.A_5$ x 2, $C_2^{11}.A_7.A_5.C_2^2$, $C_2^9.A_8.S_5$

Order 1132462080: $C_2^5.C_2^5.S_4.C_2^6.A_6.C_2$

Order 8945664: $C_2^{13}.\PSL(2,13)$

Order 8388608: $C_2^6.C_2^6.C_2^6.C_2^5$

Order 8192: $C_2^{13}$

Order 243: $C_3^4:C_3$

Order 49: $C_7^2$

Order 25: $C_5^2$

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 357082280755200: $C_2^{13}.A_{14}$ ($C_1$)

Order 8192: $C_2^{13}$ ($A_{14}$)

Order 2: $C_2$ ($C_2^{12}.A_{14}$)

Order 1: $C_1$ ($C_2^{13}.A_{14}$)

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 4 larger groups in the database.

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

Character theory

Complex character table

The $709 \times 709$ character table is not available for this group.

Rational character table

The $688 \times 688$ rational character table is not available for this group.