Abstract group 15944266600786427904000.a: $C_2^{18}.A_{19}$

• Label: 159...000.a
• Order: \( 2^{33} \cdot 3^{8} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
• Exponent: \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{34} \cdot 3^{8} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $38$
• Trans deg.: $38$
• Rank: $2$

Group information

Description:	$C_2^{18}.A_{19}$
Order:	\(159\!\cdots\!000\)\(\medspace = 2^{33} \cdot 3^{8} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Exponent:	\(465585120\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Automorphism group:	Group of order \(318\!\cdots\!000\)\(\medspace = 2^{34} \cdot 3^{8} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Composition factors:	$C_2$ x 18, $A_{19}$
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	17	18	19	20	21	22	24	26	28	30	32	33	34	35	36	39	40	42	44	45	48	52	55	56	60	63	65	66	70	72	77	78	80	84	88	90	104	105	110	112	120	126	130	132	140	144	154	165	168	180	210	240	280	330	420
Elements	1	4287531492351	1130462209111112	443596338630319104	27682734741594624	12553509355087352760	42368908042506240	55854082750292951040	50706732451046031360	43978597372324929024	280854488678400	587135040044314245120	53232727700275200	60326931784132915200	183946911686210469888	571887643323373977600	234474508835094528000	226097410098272993280	1678343852714360832000	285956731918410448896	3610580232801484800	4318137763430400000	931335438956974571520	41681225789315481600	470414236717685145600	846628660980104970240	498258331274575872000	5158735248044851200	703423526505283584000	19117556409575669760	332345227214743142400	42586182160220160000	437923597231545384960	209906997073389158400	131642115933339648000	25189726747770224640	240055942938152140800	268292947609387008000	6039494924540313600	451200599987532595200	949798353273492602880	47453174407102464000	122648204621434060800	174012947513317785600	153386030935100620800	318331711647645696000	103534198706405376000	229965383665188864000	265737776679773798400	561479800094038425600	158536741769183232000	191829457540711710720	153310255776792576000	3361266520503091200	90592423868104704000	142359523221307392000	446218016674786836480	205630422430777344000	367944613864302182400	271777271604314112000	242011189476222566400	110724073616572416000	310602596119216128000	48315959396322508800	114678504817164288000	66434444169943449600	69400267570387353600	132868888339886899200	56943809288522956800	144947878188967526400	28471904644261478400	15944266600786427904000
Conjugacy classes	1	29	6	180	3	244	2	210	5	81	1	858	1	40	11	60	1	73	2	228	5	14	514	9	92	244	2	2	3	3	84	2	136	104	21	3	84	8	1	52	379	3	2	25	36	32	2	10	22	129	8	33	2	2	7	8	128	13	6	11	27	2	6	2	36	8	38	10	8	6	8	4428
Divisions	1	29	6	180	3	244	2	210	5	81	1	858	1	40	10	60	1	73	1	228	5	14	514	9	92	241	2	2	2	3	84	2	136	103	21	3	84	8	1	52	379	2	1	24	34	32	1	9	22	129	8	29	2	2	7	8	128	10	3	11	27	2	3	1	36	8	34	10	8	3	8	4393
Autjugacy classes	1	29	6	180	3	244	2	210	5	81	1	858	1	40	10	60	1	72	1	228	5	14	514	9	92	241	2	2	2	3	84	2	136	103	21	3	84	8	1	52	379	2	1	24	34	32	1	9	22	129	8	28	2	2	7	8	128	10	3	11	27	2	3	1	36	8	34	10	8	3	8	4391

Minimal presentations

Permutation degree:	$38$
Transitive degree:	$38$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Permutation group:	Degree $38$ $\langle(1,10,4,25,35,38,19,32,15,24,29,11,28,5,7,33,17,14,22)(2,9,3,26,36,37,20,31,16,23,30,12,27,6,8,34,18,13,21) \!\cdots\! \rangle$
Transitive group:	38T66				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:	$C_2^{18}$ . $A_{19}$				more information

Elements of the group are displayed as permutations of degree 38.

Homology

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

Subgroups

There are 3 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $C_2^{18}.A_{19}$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^6.C_2^5.C_2^6.C_2^6.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4.C_3^4$
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:	$P_{ 13 } \simeq$ $C_{13}$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $C_{19}$

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 2717000 (order at least 5868335149350912) are shown, as well as all normal subgroups of any index.

Order 15944266600786427904000: $C_2^{18}.A_{19}$

Order 839171926357180416000: $C_2^{18}.A_{18}$

Order 419585963178590208000: $C_2^{17}.A_{18}$

Order 93241325150797824000: $C_2^{18}.A_{17}.C_2$

Order 46620662575398912000: $C_2^{18}.A_{17}$, $C_2^{17}.A_{17}.C_2$, $C_2^{17}.A_{17}.C_2$

Order 23310331287699456000: $C_2^{17}.A_{17}$ x 2, $C_2^{16}.A_{17}.C_2$

Order 16454351497199616000: $(C_2\times A_4).C_2^{14}.C_2.A_{16}.C_2$

Order 11655165643849728000: $C_2^{16}.A_{17}$

Order 8227175748599808000: $(C_2\times A_4).C_2^{14}.A_{16}.C_2$ x 2, $(C_2\times A_4).C_2^{14}.C_2.A_{16}$

Order 5484783832399872000: $C_2^{18}.A_{16}.C_2$

Order 4113587874299904000: $C_2^3:A_4.C_2^{14}.C_2.A_{15}.C_2$, $(C_2^{15}\times C_6).A_{16}.C_2$, $(C_2\times A_4).C_2^{14}.A_{16}$

Order 2742391916199936000: $C_2^{17}.A_{16}.C_2$ x 6, $C_2\times C_2^{17}.A_{16}$

Order 2056793937149952000: $C_2^3:A_4.C_2^{14}.A_{15}.C_2$ x 2, $(C_2^{14}\times C_6).A_{16}.C_2$ x 2, $C_2\wr A_4.C_2^{14}.A_{15}$, $(C_2^{15}\times C_6).A_{16}$

Order 1371195958099968000: $C_2^{16}.A_{16}.C_2$ x 4, $C_2^{17}.A_{16}$ x 3, $C_2^{16}.A_{16}.C_2$ x 2, $C_2^{18}.A_{14}.S_5$, $C_2^{17}.C_2^3.A_{15}.C_2$

Order 1028396968574976000: $C_2^3:A_4.C_2^{14}.A_{15}$, $A_4.C_2^{14}.C_2^2.A_{15}.C_2$, $(C_2^{14}\times C_6).A_{16}$

Order 685597979049984000: $C_2^{17}.C_2^2.A_{15}.C_2$ x 4, $C_2^{17}.A_{14}.S_5$ x 2, $(C_2\times D_4).C_2^{14}.C_2.A_{15}.C_2$ x 2, $C_2^{16}.A_{16}$ x 2, $C_2^{18}.A_{14}.A_5$, $C_2^{17}.C_2^3.A_{15}$, $C_2^{16}.A_{16}$, $C_2^{15}.A_{16}.C_2$, $C_2^{15}.A_{16}.C_2$

Order 587655410614272000: $C_2^{18}.A_{13}.A_6.C_2$

Order 514198484287488000: $A_4.C_2^{14}.C_2.A_{15}.C_2$ x 5, $(C_2\times A_4).C_2^{14}.A_{15}.C_2$ x 2, $\SL(2,3).C_2^{14}.A_{15}.C_2$, $A_4.C_2^{14}.C_2^2.A_{15}$

Order 342798989524992000: $C_2^{17}.C_2.A_{15}.C_2$ x 5, $(C_2\times D_4).C_2^{14}.A_{15}.C_2$ x 4, $C_2^{17}.C_2^2.A_{15}$ x 3, $(C_2^{15}\times C_4).C_2.A_{15}.C_2$ x 2, $D_4:C_2.C_2^{14}.A_{15}.C_2$, $C_2^{18}.A_{15}.C_2$, $C_2^{17}.A_{14}.A_5$, $C_2^{16}.C_2^2.A_{15}.C_2$, $C_2^{15}.A_{16}$

Order 316429836484608000: $C_2^{18}.A_{12}.A_7.C_2$

Order 293827705307136000: $C_2^{17}.A_{13}.A_6.C_2$ x 2, $C_2^{18}.A_{13}.A_6$

Order 274239191619993600: $C_2\wr A_4.C_2.C_2^{12}.C_2.A_{14}.C_2$

Order 257099242143744000: $A_4.C_2^{14}.A_{15}.C_2$ x 6, $A_4.C_2^{14}.C_2.A_{15}$ x 3, $\SL(2,3).C_2^{14}.A_{15}$, $(C_2^{15}\times C_6).A_{15}.C_2$, $(C_2\times A_4).C_2^{14}.A_{15}$

Order 228532659683328000: $(C_2\times C_2^4:C_5).C_2^{12}.C_2^2.A_{14}.C_2$

Order 210953224323072000: $C_2^{18}.A_{11}.A_8.C_2$

Order 172598092627968000: $C_2^{18}.A_{10}.A_9.C_2$

Order 171399494762496000: $C_2^{17}.A_{15}.C_2$ x 15, $(C_2^{15}\times C_4).A_{15}.C_2$ x 9, $C_2^{16}.C_2.A_{15}.C_2$ x 4, $C_2^{17}.C_2.A_{15}$ x 3, $C_2^{16}.C_2^2.A_{15}$ x 2, $C_2^{18}.A_{15}$, $C_2^{15}.C_2^2.A_{15}.C_2$, $(C_2^{14}\times C_4).C_2.A_{15}.C_2$, $(C_2\times D_4).C_2^{14}.A_{15}$

Order 158214918242304000: $C_2^{17}.A_{12}.A_7.C_2$ x 2, $C_2^{18}.A_{12}.A_7$

Order 146913852653568000: $C_2^{17}.A_{13}.A_6$

Order 137119595809996800: $(C_2\times C_2^3:A_4).C_2^{12}.C_2.A_{14}.C_2$ x 4, $C_2\wr A_4.C_2.C_2^{12}.A_{14}.C_2$ x 2, $C_2\wr A_4.C_2.C_2^{12}.C_2.A_{14}$, $(C_2^2\times A_4).C_2^{12}.C_2^3.A_{14}.C_2$

Order 128549621071872000: $(C_2^{14}\times C_6).A_{15}.C_2$ x 6, $A_4.C_2^{14}.A_{15}$ x 2, $(C_2^{15}\times C_6).A_{15}$

Order 114266329841664000: $(C_2\times C_2^4:D_5).C_2^{12}.A_{14}.C_2$ x 2, $(C_2\times C_2^4:C_5).C_2^{12}.C_2^2.A_{14}$

Order 105476612161536000: $C_2^{17}.A_{11}.A_8.C_2$ x 2, $C_2^{18}.A_{11}.A_8$

Order 97942568435712000: $C_2^{18}.A_{13}.S_5$ x 2

Order 91413063873331200: $C_2^{17}.C_2^3.A_{14}.C_2$

Order 86299046313984000: $C_2^{17}.A_{10}.A_9.C_2$ x 2, $C_2^{18}.A_{10}.A_9$

Order 85699747381248000: $C_2^{16}.A_{15}.C_2$ x 29, $C_2^{17}.A_{15}$ x 6, $(C_2^{14}\times C_4).A_{15}.C_2$ x 6, $C_2^{16}.C_2.A_{15}$ x 4, $(C_2^{15}\times C_4).A_{15}$ x 3, $C_2^{14}.A_{14}.S_5$, $(C_2^{14}\times C_4).C_2.A_{15}$

Order 79107459121152000: $C_2^{17}.A_{12}.A_7$

Order 68559797904998400: $(C_2^2\times A_4).C_2^{12}.C_2^2.A_{14}.C_2$ x 11, $(C_2\times C_2^3:A_4).C_2^{12}.A_{14}.C_2$ x 4, $(C_2\times C_2^3:A_4).C_2^{12}.C_2.A_{14}$ x 2, $C_2\wr A_4.C_2.C_2^{12}.A_{14}$, $(C_2^3\times A_4).C_2^{12}.C_2.A_{14}.C_2$, $(C_2^2\times A_4).C_2^{12}.C_2^3.A_{14}$, $(C_2\times A_4:C_4).C_2^{12}.C_2.A_{14}.C_2$, $(C_2\times A_4).C_2^{12}.C_2^3.A_{14}.C_2$

Order 64274810535936000: $(C_2^{13}\times C_6).A_{15}.C_2$ x 4, $(C_2^{14}\times C_6).A_{15}$ x 3

Order 60822550204416000: $A_{19}$

Order 58765541061427200: $A_4^2.C_2^2.C_2^{12}.C_2^2.A_{13}.C_2$

Order 57133164920832000: $(C_2\times C_2^4:C_5).C_2^{12}.C_2.A_{14}$ x 2, $(C_2\times C_2^4:D_5).C_2^{12}.A_{14}$

Order 52738306080768000: $C_2^{17}.A_{11}.A_8$

Order 48971284217856000: $C_2^{17}.A_{13}.S_5$ x 8, $C_2^{18}.A_{13}.A_5$ x 2

Order 45706531936665600: $C_2^{16}.C_2^3.A_{14}.C_2$ x 7, $(C_2^2\times D_4).C_2^{12}.C_2^2.A_{14}.C_2$ x 3, $C_2^{17}.C_2^2.A_{14}.C_2$, $C_2^{16}.C_2^4.A_{14}$, $C_2^3.C_2^4.C_2^{12}.A_{14}.C_2$, $(C_2\times Q_8:C_2^2).C_2^{12}.C_2.A_{14}.C_2$, $(C_2\times C_2^2\wr C_2).C_2^{12}.C_2.A_{14}.C_2$

Order 45204262354944000: $C_2^{18}.A_{12}.A_6.C_2$

Order 43149523156992000: $C_2^{17}.A_{10}.A_9$

Order 42849873690624000: $C_2^{15}.A_{15}.C_2$ x 14, $C_2^{16}.A_{15}$ x 9, $C_2^{15}.A_{15}.C_2$ x 4, $C_2^{13}.A_{14}.S_5$ x 2, $(C_2^{14}\times C_4).A_{15}$ x 2, $C_2^{14}.A_{14}.A_5$

Order 39177027374284800: $C_2.C_4^2:A_4.C_2^{12}.C_2^2.A_{13}.C_2$, $(C_2\times C_2^3:A_4).C_2^{12}.C_2^3.A_{13}.C_2$

Order 34519618525593600: $C_2^{18}.A_9^2.C_4$

Order 34279898952499200: $(C_2\times A_4).C_2^{12}.C_2^2.A_{14}.C_2$ x 18, $(C_2^2\times A_4).C_2^{12}.C_2.A_{14}.C_2$ x 14, $(C_2^2\times A_4).C_2^{12}.C_2^2.A_{14}$ x 3, $(C_2\times S_4).C_2^{12}.C_2.A_{14}.C_2$ x 3, $(C_2\times A_4:C_4).C_2^{12}.A_{14}.C_2$ x 2, $(C_2^3\times A_4).C_2^{12}.C_2.A_{14}$, $(C_2^2\times S_4).C_2^{12}.C_2.A_{14}$, $(C_2^2\times S_3).C_2^{12}.C_2^2.A_{14}.C_2$, $(C_2\times \SL(2,3)).C_2^{12}.C_2.A_{14}.C_2$, $(C_2\times C_2^3:A_4).C_2^{12}.A_{14}$, $(C_2\times A_4:C_4).C_2^{12}.C_2.A_{14}$, $(C_2\times A_4).C_2^{12}.C_2^3.A_{14}$

Order 32137405267968000: $(C_2^{13}\times C_6).A_{15}$

Order 29382770530713600: $\PSOPlusPlus(4,3).C_2^{12}.C_2^2.A_{13}.C_2$ x 2, $A_4^2.C_2^{12}.C_2^3.A_{13}.C_2$ x 2, $A_4^2.C_2^2.C_2^{12}.C_2.A_{13}.C_2$ x 2, $A_4^2.C_2^2:C_4.C_2^{12}.A_{13}$

Order 28566582460416000: $(C_2\times C_2^4:C_5).C_2^{12}.A_{14}$

Order 26369153040384000: $C_2^{18}.A_{11}.A_7.C_2$

Order 24485642108928000: $C_2^{17}.A_{13}.A_5$ x 4, $C_2^{16}.A_{13}.S_5$ x 4

Order 22853265968332800: $C_2^{16}.C_2^2.A_{14}.C_2$ x 25, $C_2^{17}.C_2.A_{14}.C_2$ x 5, $(C_2^2\times D_4).C_2^{12}.C_2.A_{14}.C_2$ x 5, $C_2^{15}.C_2^3.A_{14}.C_2$ x 3, $(C_2\times C_2^2:C_4).C_2^{12}.C_2.A_{14}.C_2$ x 3, $(C_2\times C_2^2.D_4).C_2^{12}.A_{14}.C_2$ x 3, $C_2^{16}.C_2^3.A_{14}$ x 2, $C_2^{18}.A_{14}.C_2$, $C_2^{17}.C_2^2.A_{14}$, $(C_2^{14}\times C_4).C_2^2.A_{14}.C_2$, $(C_2^2\times D_4).C_2^{12}.C_2^2.A_{14}$, $(C_2\times D_4:C_2).C_2^{12}.C_2.A_{14}.C_2$, $(C_2\times C_2^2\wr C_2).C_2^{12}.C_2.A_{14}$, $(C_2\times C_2^2\wr C_2).C_2^{12}.A_{14}.C_2$, $(C_2\times C_2^2:C_4).C_2^{12}.C_2^2.A_{14}$, $(C_2\times C_2^2.D_4).C_2^{12}.C_2.A_{14}$

Order 22602131177472000: $C_2^{17}.A_{12}.A_6.C_2$ x 6, $C_2^{18}.A_{12}.A_6$

Order 21424936845312000: $C_2^{15}.A_{15}$ x 5, $C_2^{14}.A_{15}.C_2$ x 3, $C_2^{13}.A_{14}.A_5$

Order 19588513687142400: $(C_2\times C_2^3:A_4).C_2^{12}.C_2^2.A_{13}.C_2$ x 9, $Q_8:C_2^2.D_6.C_2^{12}.C_2.A_{13}.C_2$ x 2, $C_2^5.S_4.C_2^{12}.A_{13}.C_2$ x 2, $C_2^5.A_4.C_2^{12}.C_2.A_{13}.C_2$ x 2, $C_2^5.A_4.C_2.C_2^{12}.A_{13}.C_2$ x 2, $Q_8:C_2^2.D_6.C_2^{12}.C_2^2.A_{13}$, $C_2^5.A_4.C_2^{12}.C_2^2.A_{13}$, $C_2^3:A_4.C_2^{12}.C_2^3.A_{13}.C_2$, $C_2^3:A_4.C_2^2.C_2^{12}.C_2.A_{13}.C_2$, $C_2\wr A_4.C_2.C_2^{12}.C_2.A_{13}.C_2$

Order 19177565847552000: $C_2^{18}.A_{10}.A_8.C_2$

Order 18364231581696000: $C_2^{13}.A_{13}.A_6.C_2$

Order 17259809262796800: $C_2^{17}.A_9^2.C_4$ x 2, $C_2^{18}.A_9.A_9.C_2$

Order 17139949476249600: $(C_2\times A_4).C_2^{12}.C_2.A_{14}.C_2$ x 28, $(C_2^2\times A_4).C_2^{12}.A_{14}.C_2$ x 7, $(C_2^2\times A_4).C_2^{12}.C_2.A_{14}$ x 5, $(C_2\times A_4).C_2^{12}.C_2^2.A_{14}$ x 5, $(C_2^{14}\times C_6).C_2.A_{14}.C_2$ x 4, $(C_2^{13}\times C_6).C_2^2.A_{14}.C_2$ x 4, $(C_2\times S_4).C_2^{12}.A_{14}.C_2$ x 3, $(C_2^{15}\times C_6).A_{14}.C_2$ x 2, $(C_2^2\times S_3).C_2^{12}.C_2.A_{14}.C_2$ x 2, $(C_2\times \SL(2,3)).C_2^{12}.A_{14}.C_2$ x 2, $(C_2^{14}\times C_6).C_2^2.A_{14}$, $(C_2\times \SL(2,3)).C_2^{12}.C_2.A_{14}$, $(C_2\times S_4).C_2^{12}.C_2.A_{14}$, $(C_2\times C_3:D_4).C_2^{12}.A_{14}.C_2$, $(C_2\times C_3:C_4).C_2^{12}.C_2.A_{14}.C_2$, $(C_2\times A_4:C_4).C_2^{12}.A_{14}$

Order 16323761405952000: $C_2^4:C_5.C_2^{12}.C_2^3.A_{13}.C_2$

Order 15068087451648000: $C_2^{18}.A_{12}.A_5.C_2^2$

Order 14691385265356800: $\PSOPlusPlus(4,3).C_2^{12}.C_2.A_{13}.C_2$ x 7, $A_4^2.C_2^{12}.C_2^2.A_{13}.C_2$ x 5, $(C_2\times A_4^2).C_2^{12}.C_2.A_{13}.C_2$ x 4, $\PSOPlusPlus(4,3).C_2^{12}.C_2^2.A_{13}$, $A_4^2.C_4.C_2^{12}.C_2.A_{13}$, $A_4^2.C_2^3.C_2^{12}.A_{13}$

Order 14283291230208000: $D_{10}.C_2^{12}.C_2.A_{14}.C_2$

Order 13184576520192000: $C_2^{17}.A_{11}.A_7.C_2$ x 6, $C_2^{18}.A_{11}.A_7$

Order 13059009124761600: $(C_2\times Q_8:C_2^2).C_2^{12}.C_2^3.A_{13}.C_2$

Order 12242821054464000: $C_2^{16}.A_{13}.A_5$

Order 11426632984166400: $C_2^{16}.C_2.A_{14}.C_2$ x 36, $C_2^{15}.C_2^2.A_{14}.C_2$ x 26, $C_2^{17}.A_{14}.C_2$ x 19, $(C_2^{14}\times C_4).C_2.A_{14}.C_2$ x 14, $(C_2^{15}\times C_4).A_{14}.C_2$ x 9, $C_2^{16}.C_2^2.A_{14}$ x 7, $(C_2^2\times D_4).C_2^{12}.A_{14}.C_2$ x 5, $(C_2^2\times D_4).C_2^{12}.C_2.A_{14}$ x 4, $C_2^{15}.C_2^3.A_{14}$ x 3, $(C_2\times C_2^2:C_4).C_2^{12}.A_{14}.C_2$ x 3, $C_2^{17}.C_2.A_{14}$ x 2, $(C_2\times D_4).C_2^{12}.C_2.A_{14}.C_2$ x 2, $(C_2\times C_2^2:C_4).C_2^{12}.C_2.A_{14}$ x 2, $C_2^{18}.A_{14}$, $C_2^{14}.C_2^3.A_{14}.C_2$, $(C_2\times Q_8).C_2^{12}.C_2.A_{14}.C_2$, $(C_2\times D_4).C_2^{12}.C_2^2.A_{14}$

Order 11301065588736000: $C_2^{16}.A_{12}.A_6.C_2$ x 4, $C_2^{17}.A_{12}.A_6$ x 3

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

Order 10547661216153600: $C_2^{18}.A_{12}.\PSL(2,7)$

Order 9794256843571200: $C_2^3:A_4.C_2^{12}.C_2^2.A_{13}.C_2$ x 12, $(C_2\times C_2^3:A_4).C_2^{12}.C_2.A_{13}.C_2$ x 10, $C_2^3:A_4.C_4.C_2^{12}.A_{13}.C_2$ x 5, $C_2^3:A_4.C_2^2.C_2^{12}.C_2.A_{13}$ x 3, $Q_8:C_2^2.D_6.C_2^{12}.A_{13}.C_2$ x 2, $C_2^5.A_4.C_2^{12}.A_{13}.C_2$ x 2, $C_2^3:S_4.C_2^{12}.C_2.A_{13}.C_2$ x 2, $C_2^3:A_4.C_2^{12}.C_2^3.A_{13}$ x 2, $C_2\wr A_4.C_2^{12}.C_2.A_{13}.C_2$ x 2, $C_2\wr A_4.C_2.C_2^{12}.A_{13}.C_2$ x 2, $(C_2\times C_2^3:A_4).C_2^{12}.C_2^2.A_{13}$ x 2, $C_4^2:C_3.C_2^{12}.C_2^3.A_{13}.C_2$, $C_2^5.S_4.C_2^{12}.A_{13}$, $C_2^5.A_4.C_2.C_2^{12}.A_{13}$, $C_2^4:C_3.C_2^{12}.C_2^3.A_{13}.C_2$, $C_2^3:S_4.C_2.C_2^{12}.A_{13}.C_2$, $C_2^3.C_2^4.C_3.C_2^{12}.C_2.A_{13}$, $C_2^3.C_2^4.C_3.C_2^{12}.A_{13}.C_2$, $C_2.C_4^2:A_4.C_2^{12}.A_{13}.C_2$, $(C_2^3\times A_4).C_2^{12}.C_2^2.A_{13}.C_2$, $(C_2\times C_2^2:S_4).C_2^{12}.C_2.A_{13}.C_2$

Order 9588782923776000: $C_2^{17}.A_{10}.A_8.C_2$ x 6, $C_2^{18}.A_{10}.A_8$

Order 9182115790848000: $C_2^{12}.A_{13}.A_6.C_2$ x 2, $C_2^{13}.A_{13}.A_6$

Order 9040852470988800: $C_2^3.A_4^2.C_2.C_2^{10}.C_2^3.A_{12}.C_2$

Order 8629904631398400: $C_2^{17}.A_9.A_9.C_2$ x 4, $C_2^{16}.A_9^2.C_4$ x 2, $C_2^{18}.A_9.A_9$

Order 8569974738124800: $(C_2^{14}\times C_6).A_{14}.C_2$ x 16, $(C_2\times A_4).C_2^{12}.A_{14}.C_2$ x 12, $(C_2^{13}\times C_6).C_2.A_{14}.C_2$ x 10, $(C_2\times A_4).C_2^{12}.C_2.A_{14}$ x 7, $(C_2^{12}\times C_6).C_2^2.A_{14}.C_2$ x 4, $(C_2^{14}\times C_6).C_2.A_{14}$ x 3, $(C_2\times C_3:C_4).C_2^{12}.A_{14}.C_2$ x 3, $D_6.C_2^{12}.C_2.A_{14}.C_2$ x 2, $(C_2^{13}\times C_6).C_2^2.A_{14}$ x 2, $(C_2^2\times A_4).C_2^{12}.A_{14}$ x 2, $(C_2^{15}\times C_6).A_{14}$, $(C_2^{12}\times C_6).C_2^3.A_{14}$, $(C_2^2\times S_3).C_2^{12}.A_{14}.C_2$, $(C_2\times \SL(2,3)).C_2^{12}.A_{14}$, $(C_2\times S_4).C_2^{12}.A_{14}$

Order 8161880702976000: $C_2^4:D_5.C_2^{12}.C_2.A_{13}.C_2$ x 3, $(C_2\times C_2^4:D_5).C_2^{12}.A_{13}.C_2$ x 2, $C_2^4:C_5.C_2^{12}.C_2^3.A_{13}$, $C_2^4:C_5.C_2^{12}.C_2^2.A_{13}.C_2$

Order 7534043725824000: $C_2^{17}.A_{12}.S_5.C_2$ x 12, $C_2^{18}.A_{12}.S_5$ x 3, $C_2^{18}.A_{12}.A_5.C_2$, $C_2^{18}.A_{11}.A_6.C_2^2$

Order 7345692632678400: $A_4^2.C_2^{12}.C_2.A_{13}.C_2$ x 14, $\PSOPlusPlus(4,3).C_2^{12}.A_{13}.C_2$ x 4, $A_4^2.C_2^{12}.C_2^2.A_{13}$ x 2, $A_4^2.C_2^2.C_2^{12}.A_{13}$ x 2, $(C_2\times A_4^2).C_2^{12}.A_{13}.C_2$ x 2, $\PSOPlusPlus(4,3).C_2^{12}.C_2.A_{13}$, $A_4^2.C_4.C_2^{12}.A_{13}$, $(C_6\times A_4).C_2^{12}.C_2^2.A_{13}.C_2$, $(C_2\times A_4^2).C_2^{12}.C_2.A_{13}$

Order 7141645615104000: $D_{10}.C_2^{12}.A_{14}.C_2$, $(C_2^{13}\times C_{10}).C_2.A_{14}$, $(C_2^{12}\times C_{10}).C_2.A_{14}.C_2$

Order 7031774144102400: $C_2^7.C_2^4.C_2^{10}.A_{11}.\PSL(2,7)$

Order 6592288260096000: $C_2^{16}.A_{11}.A_7.C_2$ x 4, $C_2^{17}.A_{11}.A_7$ x 3

Order 6529504562380800: $(C_2^2\times D_4).C_2^{12}.C_2^3.A_{13}.C_2$ x 10, $(C_2\times C_2^2\wr C_2).C_2^{12}.C_2^2.A_{13}.C_2$ x 4, $C_2^{17}.C_2^3.A_{13}.C_2$ x 2, $C_2^{16}.C_2^4.A_{13}.C_2$ x 2, $(C_2\times D_4).C_2^{12}.C_2^4.A_{13}.C_2$ x 2, $(C_2\times C_2^2.D_4).C_2^{12}.C_2^2.A_{13}.C_2$ x 2, $Q_8:C_2^2.C_2^{12}.C_2^3.A_{13}.C_2$, $C_2^{15}.C_2^5.A_{13}.C_2$, $C_2^4.(C_2\times C_4).C_2^{12}.C_2.A_{13}.C_2$, $C_2^3.C_4.C_2^{12}.C_2^3.A_{13}.C_2$, $C_2.(C_2^5.C_2).C_2^{12}.C_2.A_{13}.C_2$, $(C_2^{14}\times C_4).C_2^4.A_{13}.C_2$, $(C_2^2\times D_4).C_2^{12}.C_2^4.A_{13}$, $(C_2\times Q_8:C_2^2).C_2^{12}.C_2^2.A_{13}.C_2$, $(C_2\times D_4).D_4.C_2^{12}.C_2.A_{13}.C_2$

Order 6402373705728000: $C_2.A_{18}$

Order 6121410527232000: $C_2^{14}.A_{13}.S_5$

Order 6027234980659200: $C_2^6.A_4^2.C_2.C_2^{10}.C_2^3.A_{11}.C_2$

Order 8589934592: $C_2^9.C_2^6.C_2^5.C_2^6.C_2^6.C_2$

Order 44826624: $(C_2^{18}.C_{19}):C_9$

Order 262144: $C_2^{18}$

Order 6561: $C_3^4.C_3^4$

Order 125: $C_5^3$

Order 49: $C_7^2$

Order 19: $C_{19}$

Order 17: $C_{17}$

Order 13: $C_{13}$

Order 11: $C_{11}$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 15944266600786427904000: $C_2^{18}.A_{19}$ ($C_1$)

Order 262144: $C_2^{18}$ ($A_{19}$)

Order 1: $C_1$ ($C_2^{18}.A_{19}$)

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 2 larger groups in the database.

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

Character theory

Complex character table

The $4428 \times 4428$ character table is not available for this group.

Rational character table

The $4393 \times 4393$ rational character table is not available for this group.