Abstract group 208119169843200.a: $M_{23}\wr C_2$

• Label: 208...200.a
• Order: \( 2^{15} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2} \cdot 23^{2} \)
• Exponent: \( 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 23 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2} \cdot 23^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $46$
• Trans deg.: $46$
• Rank: $2$

Group information

Description:	$M_{23}\wr C_2$
Order:	\(208119169843200\)\(\medspace = 2^{15} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2} \cdot 23^{2} \)
Exponent:	\(425040\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 23 \)
Automorphism group:	Group of order \(208119169843200\)\(\medspace = 2^{15} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2} \cdot 23^{2} \)
Composition factors:	$C_2$, $M_{23}$ x 2
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	10	11	12	14	15	16	20	21	22	23	24	28	30	33	35	40	42	44	46	55	56	60	66	69	70	77	88	92	105	115	120	138	154	161	165	184	210	253	322	345
Elements	1	24610575	3211828928	142753509360	462488404224	1403980226880	2123667912960	5700439261440	6942467347200	3439989987840	9249740881920	21258774408960	3931142595072	13007448115200	433581603840	165173944320	18934001856000	786841735680	2312435220480	16723861862400	17353587525120	210221383680	1982087331840	1734326415360	5120392273920	1182495283200	9055392192000	2522656604160	7432827494400	867163207680	3153320755200	100540661760	1982087331840	5405692723200	4729981132800	565541222400	3964174663680	1206487941120	3468652830720	1508109926400	5405692723200	2585331302400	5045313208320	2262164889600	3964174663680	3290421657600	2585331302400	2412975882240	208119169843200
Conjugacy classes	1	3	2	4	2	6	5	5	2	5	3	15	10	1	1	2	4	5	2	6	7	2	2	1	6	2	4	2	4	2	2	2	2	4	2	2	4	2	2	2	4	4	4	2	4	4	4	4	170
Divisions	1	3	2	4	2	6	3	5	2	3	3	8	6	1	1	1	2	3	2	3	4	1	1	1	3	1	2	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	96
Autjugacy classes	1	3	2	4	2	6	5	5	2	5	3	15	10	1	1	2	4	5	2	6	7	2	2	1	6	2	4	2	4	2	2	2	2	4	2	2	4	2	2	2	4	4	4	2	4	4	4	4	170

Minimal presentations

Permutation degree:	$46$
Transitive degree:	$46$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	44	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $46$ $\langle(1,46,5,36,2,40,18,33,17,44,19,35,9,24)(3,42,10,30,20,25,14,43,7,29,6,37,13,34) \!\cdots\! \rangle$
Transitive group:	46T42				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$(M_{23}.M_{23})$ . $C_2$				more information

Elements of the group are displayed as permutations of degree 46.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	not computed
Commutator length:	$1$

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:	not computed
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.C_2^5.C_2^4.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^2$
23-Sylow subgroup:	$P_{ 23 } \simeq$ $C_{23}^2$

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 52039680 (order at least 3999240) are shown, as well as all normal subgroups of any index.

Order 208119169843200: $M_{23}\wr C_2$

Order 104059584921600: $M_{23}.M_{23}$

Order 4524329779200: $M_{23}\times M_{22}$

Order 411302707200: $M_{23}\times \PSL(3,4).C_2$, $M_{23}\times C_2^4.A_7$

Order 393419980800: $M_{22}\wr C_2$

Order 205651353600: $M_{23}\times \PSL(3,4)$, $A_8\times M_{23}$

Order 196709990400: $M_{22}.M_{22}$

Order 80791603200: $M_{11}\times M_{23}$

Order 58757529600: $M_{23}\times C_2^4:C_3.S_5$, $M_{23}\times C_2^4.A_6$

Order 29378764800: $C_2^4:C_3.M_{23}.A_5$

Order 27420180480: $M_{23}\times C_2^4.\PSL(2,7)$ x 2

Order 25706419200: $M_{23}\times A_7$, $A_7\times M_{23}$

Order 19585843200: $M_{23}\times C_2^4.S_5$ x 2

Order 17882726400: $M_{22}\times \PSL(3,4).C_2$, $M_{22}\times C_2^4.A_7$

Order 13710090240: $C_2^3.\PSL(2,7)\times M_{23}$ x 2

Order 11751505920: $M_{23}\times C_2^4:A_4.S_3$

Order 9792921600: $M_{23}\times C_2^4.A_5$ x 2, $M_{23}\times F_{16}.C_4$, $C_2^4.M_{23}.A_5$

Order 8941363200: $\PSL(3,4)\times M_{22}$, $A_8\times M_{22}$

Order 7344691200: $M_{23}\times S_6$, $A_6.C_2\times M_{23}$

Order 6732633600: $M_{23}\times \PSL(2,11)$

Order 5875752960: $M_{23}\times A_4^2.C_4$, $M_{23}\times A_4\wr C_2.C_2$, $C_2^2.A_4^2.M_{23}$

Order 4896460800: $M_{23}\times F_{16}:C_2$

Order 3917168640: $M_{23}\times C_2^4:C_3.D_4$ x 2, $M_{23}\times C_2^4.S_4$, $C_2^4.S_4\times M_{23}$, $C_2\wr A_4.C_2.M_{23}$

Order 3672345600: $M_{23}\times A_6$ x 3, $M_{23}\times C_3:S_5$

Order 3512678400: $M_{22}\times M_{11}$

Order 3427522560: $C_2\times \PSL(2,7)\times M_{23}$, $C_2.M_{23}.\PSL(2,7)$, $(C_2\times F_8:C_3).M_{23}$

Order 3264307200: $M_{23}\times C_2^4:C_5:C_4$, $C_2^4:C_5:C_4\times M_{23}$

Order 3251404800: $C_2^8.A_7^2.C_2$, $(\PSL(3,4).C_2)\wr C_2$

Order 2937876480: $M_{23}\times \PSOPlusPlus(4,3)$ x 2, $A_4\wr C_2.M_{23}$ x 2

Order 2580842880: $M_{23}\times C_{23}:C_{11}$

Order 2554675200: $M_{22}\times C_2^4.A_6$, $C_2^4:C_3.S_5\times M_{22}$

Order 2448230400: $M_{23}\times F_{16}$

Order 1958584320: $M_{23}\times C_2^4:C_3:C_4$ x 2, $M_{23}\times C_2^2\wr C_2:C_2:C_3$ x 2, $M_{23}\times C_2^2:S_4:C_2$ x 2, $C_4^2:A_4.M_{23}$, $C_2^4:A_4.M_{23}$, $C_2^3:S_4.M_{23}$, $C_2^3:A_4.C_2.M_{23}$, $C_2^2:S_4:C_2.M_{23}$, $C_2\wr A_4.M_{23}$

Order 1836172800: $C_3.M_{23}.A_5$

Order 1713761280: $M_{23}\times \PSL(2,7)$ x 3, $M_{23}\times F_8:C_3$, $F_8:C_3.M_{23}$

Order 1632153600: $C_2^4:D_5\times M_{23}$, $C_2^4:D_5.M_{23}$

Order 1625702400: $\PSL(3,4).\PSL(3,4).C_2^2$, $C_2^8.A_7.A_7$, $C_2^4.A_7\times \PSL(3,4).C_2$, $\PSL(3,4)^2.C_2^2$, $\PSL(3,4)^2.C_4$

Order 1468938240: $A_4^2\times M_{23}$ x 2, $\PSU(3,2):C_2\times M_{23}$

Order 1305722880: $C_2.C_2\wr C_2^2.M_{23}$

Order 1277337600: $C_2^4:C_3.M_{22}.A_5$

Order 1224115200: $M_{23}\times S_5$ x 3

Order 1192181760: $C_2^4.\PSL(2,7)\times M_{22}$, $C_2^4.M_{22}.\PSL(2,7)$

Order 1142507520: $(C_2\times F_8).M_{23}$

Order 1117670400: $M_{22}\times A_7$, $A_7\times M_{22}$

Order 979292160: $C_2^2\wr C_2:C_3.M_{23}$ x 3, $C_2^2:S_4.M_{23}$ x 3, $\GL(2,\mathbb{Z}/4).M_{23}$ x 2, $\GL(2,3):C_2.M_{23}$, $M_{23}\times C_4^2:C_3:C_2$, $M_{23}\times C_2^2:S_4$, $C_2^3:A_4.M_{23}$

Order 851558400: $M_{22}\times C_2^4.S_5$ x 2

Order 816076800: $C_2^4:C_5.M_{23}$ x 2

Order 812851200: $\PSL(3,4).\PSL(3,4).C_2$ x 2, $\PSL(3,4)\times C_2^4.A_7$, $\PSL(3,4).C_2\times A_8$, $C_2^4.A_7\times A_8$, $\PSL(3,4)\wr C_2$, $A_8\wr C_2$

Order 734469120: $\PSU(3,2)\times M_{23}$, $M_{23}\times S_3\wr C_2$, $M_{23}\times F_9$, $C_3:S_4\times M_{23}$

Order 652861440: $C_2\wr C_4.M_{23}$ x 2, $D_4.D_4.M_{23}$, $C_2^2\wr C_2:C_2.M_{23}$, $C_2^2.D_4.C_2.M_{23}$, $C_2\wr C_2^2\times M_{23}$, $C_2\wr C_2^2.M_{23}$

Order 612057600: $M_{23}\times A_5$ x 3, $M_{23}\times C_3:F_5$, $A_5\times M_{23}$

Order 596090880: $M_{22}\times C_2^3.\PSL(2,7)$ x 2

Order 571253760: $M_{23}\times F_8$, $F_8.M_{23}$

Order 561052800: $M_{23}\times C_{11}:C_5$

Order 510935040: $M_{22}\times C_2^4:A_4.S_3$

Order 489646080: $(C_2\times S_4).M_{23}$ x 3, $C_2^4:C_3.M_{23}$ x 2, $C_2.S_4.M_{23}$ x 2, $A_4:C_4.M_{23}$ x 2, $\GL(2,3).M_{23}$, $C_4^2:C_3.M_{23}$, $C_2^4:C_3\times M_{23}$, $C_2^2\times M_{23}\times A_4$, $(C_2^2\times A_4).M_{23}$, $(C_2\times \SL(2,3)).M_{23}$

Order 428440320: $(C_2\times C_7:C_3).M_{23}$

Order 425779200: $C_2^4.A_5\times M_{22}$ x 2, $M_{22}\times F_{16}.C_4$, $C_2^4.M_{22}.A_5$

Order 406425600: $\PSL(3,4)^2$, $A_8^2$, $A_8\times \PSL(3,4)$

Order 367234560: $M_{23}\times C_3:S_3.C_2$ x 2, $M_{23}\times S_3^2$, $C_3\times M_{23}\times A_4$

Order 326430720: $C_2^2\wr C_2.M_{23}$ x 3, $C_2^2.D_4.M_{23}$ x 3, $\SD_{16}:C_2.M_{23}$, $Q_8:C_2^2.M_{23}$, $C_4\wr C_2.M_{23}$, $C_4.D_4.M_{23}$, $C_2^3.C_4.M_{23}$

Order 319334400: $\PSL(3,4).C_2\times M_{11}$, $M_{22}\times S_6$, $C_2^4.A_7\times M_{11}$, $A_6.C_2\times M_{22}$

Order 306028800: $C_3\times M_{23}\times D_5$

Order 292723200: $\PSL(2,11)\times M_{22}$

Order 255467520: $M_{22}\times A_4^2.C_4$, $M_{22}\times A_4\wr C_2.C_2$, $C_2^2.A_4^2\times M_{22}$

Order 244823040: $M_{23}\times S_4$ x 3, $\SL(2,3).M_{23}$ x 2, $S_4\times M_{23}$ x 2, $(C_2\times A_4).M_{23}$ x 2, $M_{23}\times C_2\times A_4$, $C_3:D_4.M_{23}$

Order 234622080: $C_{23}\times M_{23}$

Order 232243200: $\PSL(3,4).C_2\times C_2^4:C_3.S_5$, $C_2^8.A_6.A_7$, $C_2^4.A_7\times C_2^4:C_3.S_5$, $C_2^4.A_6\times \PSL(3,4).C_2$

Order 214220160: $M_{23}\times C_7:C_3$

Order 212889600: $F_{16}:C_2.M_{22}$

Order 204019200: $M_{23}\times F_5$

Order 183617280: $M_{23}\times C_3\times S_3$, $M_{23}\times C_3:S_3$

Order 170311680: $M_{22}\times C_2^4:C_3.D_4$ x 2, $M_{22}\times C_2^4.S_4$, $M_{22}\times C_2^3:S_4.C_2$, $C_2^4.S_4.M_{22}$

Order 163215360: $C_2^2:C_4.M_{23}$ x 3, $C_2^4.M_{23}$ x 2, $\SD_{16}.M_{23}$, $\OD_{16}.M_{23}$, $Q_{16}.M_{23}$, $D_4:C_2.M_{23}$, $C_4^2.M_{23}$, $C_2\times D_4.M_{23}$, $(C_2\times Q_8).M_{23}$, $(C_2\times D_4).M_{23}$

Order 159667200: $M_{22}\times A_6$ x 3, $\PSL(3,4)\times M_{11}$, $M_{22}\times C_3:S_5$, $A_8\times M_{11}$

Order 153014400: $C_{15}.M_{23}$

Order 149022720: $M_{22}\times C_2\times F_8:C_3$, $C_2\times M_{22}\times \PSL(2,7)$, $C_2.M_{22}.\PSL(2,7)$

Order 142813440: $C_{14}\times M_{23}$

Order 141926400: $M_{22}\times C_2^4:C_5:C_4$, $C_2^4:C_5:C_4.M_{22}$

Order 127733760: $A_4\wr C_2\times M_{22}$ x 2, $\PSOPlusPlus(4,3)\times M_{22}$, $\PSOPlusPlus(4,3).M_{22}$

Order 125452800: $M_{11}\wr C_2$

Order 122411520: $A_4.M_{23}$ x 3, $C_3:C_4.M_{23}$, $C_2\times M_{23}\times S_3$, $A_4\times M_{23}$, $(C_2\times C_6).M_{23}$

Order 116121600: $C_2^4:C_3.\PSL(3,4).S_5$, $C_2^4:C_3.\PSL(3,4).A_5.C_2$, $C_2^4:C_3.S_5\times \PSL(3,4)$, $C_2^4:C_3.C_2^4.A_7.A_5$, $C_2^4.A_6\times \PSL(3,4)$, $C_2^4.A_6\times A_8$, $A_8\times C_2^4:C_3.S_5$

Order 112210560: $M_{22}\times C_{23}:C_{11}$, $C_{11}\times M_{23}$

Order 108380160: $C_2^4.\PSL(3,4).\PSL(2,7).C_2$ x 2, $C_2^8.\PSL(2,7).A_7$, $C_2^8.A_7.\PSL(2,7)$

Order 106444800: $F_{16}.M_{22}$

Order 102009600: $M_{23}\times D_5$

Order 101606400: $A_7\times \PSL(3,4).C_2$ x 2, $C_2^4.A_7\times A_7$, $C_2^4.A_7.A_7$

Order 91808640: $C_3^2\times M_{23}$

Order 85155840: $C_2^4:C_3:C_4.M_{22}$ x 2, $C_2^2\wr C_2:C_2:C_3.M_{22}$ x 2, $M_{22}\times C_2^2:S_4:C_2$, $C_4^2:A_4.M_{22}$, $C_2^4:A_4.M_{22}$, $C_2^3:S_4\times M_{22}$, $C_2^3:A_4.C_2.M_{22}$, $C_2^2:S_4:C_2\times M_{22}$, $C_2^2:S_4:C_2.M_{22}$, $C_2\wr A_4.M_{22}$

Order 81607680: $Q_8.M_{23}$ x 2, $D_4.M_{23}$ x 2, $C_2^3.M_{23}$ x 2, $(C_2\times C_4).M_{23}$ x 2, $C_8.M_{23}$, $C_2^3\times M_{23}$

Order 79833600: $C_3.M_{22}.A_5$

Order 77414400: $\PSL(3,4).C_2\times C_2^4.S_5$ x 2, $C_2^8.A_7.S_5$ x 2

Order 74511360: $\PSL(2,7)\times M_{22}$ x 2, $M_{22}\times \PSL(2,7)$, $M_{22}\times F_8:C_3$, $F_8:C_3.M_{22}$

Order 71406720: $C_7\times M_{23}$

Order 70963200: $C_2^4:D_5.M_{22}$ x 2

Order 66355200: $C_2^8.A_6^2.C_2$, $C_2^8.A_5^2:\SOPlus(4,2)$

Order 63866880: $A_4^2.M_{22}$ x 2, $M_{22}\times \PSU(3,2):C_2$

Order 62726400: $M_{11}^2$

Order 61205760: $M_{23}\times S_3$, $C_6\times M_{23}$

Order 58060800: $C_2^4:C_3.\PSL(3,4).A_5$, $C_2^4:C_3.A_8.A_5$

Order 56770560: $C_2.C_2\wr C_2^2.M_{22}$

Order 54190080: $\PSL(3,4)\times C_2^4.\PSL(2,7)$ x 2, $C_2^4.A_8.\PSL(2,7)$ x 2, $\PSL(3,4).C_2\times C_2^3.\PSL(2,7)$, $C_2^7.A_7.\PSL(2,7)$, $C_2^3.\PSL(3,4).\PSL(2,7).C_2$, $C_2^3.\PSL(2,7)\times \PSL(3,4).C_2$, $C_2^3.\PSL(2,7)\times C_2^4.A_7$

Order 53222400: $M_{22}\times S_5$ x 3

Order 51004800: $C_5\times M_{23}$

Order 50803200: $A_7\times \PSL(3,4)$ x 2, $A_7\times A_8$ x 2

Order 49674240: $(C_2\times F_8).M_{22}$

Order 46448640: $\PSL(3,4).C_2\times C_4^2:A_4.S_3$, $C_2^2.A_4^2.C_2^5.A_7$

Order 45619200: $C_2^4:C_3.S_5\times M_{11}$, $C_2^4.A_6\times M_{11}$

Order 42577920: $C_2^2\wr C_2:C_3.M_{22}$ x 3, $C_2^2:S_4.M_{22}$ x 3, $\GL(2,\mathbb{Z}/4).M_{22}$ x 2, $\GL(2,3):C_2.M_{22}$, $C_4^2:C_3:C_2\times M_{22}$, $C_2^3:A_4.M_{22}$, $C_2^2:S_4\times M_{22}$

Order 40803840: $C_2^2.M_{23}$ x 2, $C_4.M_{23}$

Order 38707200: $C_2^8.A_7.A_5$ x 3, $C_2^4.\PSL(3,4).S_5$ x 3, $C_2^4.\PSL(3,4).A_5.C_2$ x 3, $\PSL(3,4).C_2\times F_{16}.C_4$, $F_{16}.C_2^5.C_2.A_7$, $C_2^4.S_5\times \PSL(3,4)$, $C_2^4.A_5.A_8.C_2$, $A_8\times C_2^4.S_5$

Order 35481600: $M_{22}\times C_2^4:C_5$, $C_2^4:C_5.M_{22}$

Order 33177600: $C_2^8.C_3^2.A_5.S_5.C_2$, $C_2^8.A_6.A_6$, $C_2^4.A_6\times C_2^4:C_3.S_5$, $C_2^8.C_3^2:(A_5^2:C_4)$, $C_2^8.A_5^2:S_3^2$

Order 31933440: $S_3\wr C_2.M_{22}$, $M_{22}\times \PSU(3,2)$, $M_{22}\times C_3:S_4$, $F_9.M_{22}$

Order 30602880: $C_3\times M_{23}$

Order 29030400: $\PSL(3,4).A_6.C_2^2$, $C_2^4.A_7\times S_6$, $A_6.C_2\times \PSL(3,4).C_2$, $A_6.C_2\times C_2^4.A_7$

Order 28385280: $C_2\wr C_2^2.M_{22}$ x 2, $D_4.D_4.M_{22}$, $C_2^2\wr C_2:C_2.M_{22}$, $C_2^2.D_4.C_2.M_{22}$, $C_2\wr C_4\times M_{22}$, $C_2\wr C_4.M_{22}$

Order 27095040: $\PSL(3,4)\times C_2^3.\PSL(2,7)$ x 2, $C_2^3.\PSL(2,7)\times A_8$, $A_8\times C_2^3.\PSL(2,7)$

Order 26611200: $M_{22}\times A_5$ x 3, $\PSL(3,4).C_2\times \PSL(2,11)$, $C_3:F_5\times M_{22}$, $C_2^4.A_7\times \PSL(2,11)$, $A_5\times M_{22}$

Order 24837120: $F_8\times M_{22}$, $F_8.M_{22}$

Order 24393600: $C_{11}:C_5\times M_{22}$

Order 23224320: $C_2^2.A_4^2.\PSL(3,4).C_2$ x 2, $\PSL(3,4).C_2\times A_4^2.C_2^2$, $C_4^2:A_4.S_3\times \PSL(3,4)$, $C_2^4.C_3:S_4.A_8$, $C_2^2.A_4^2.C_2^4.A_7$, $A_4^2.C_4.\PSL(3,4).C_2$, $A_4^2.C_2^6.A_7$, $A_4^2.C_2^5.C_2.A_7$

Order 22809600: $C_2^4:C_3.M_{11}.A_5$

Order 21288960: $C_2^4:C_3.M_{22}$ x 3, $C_2^4.M_{11}.\PSL(2,7)$ x 2, $C_2.S_4.M_{22}$ x 2, $A_4:C_4.M_{22}$ x 2, $(C_2^2\times A_4).M_{22}$ x 2, $(C_2\times S_4).M_{22}$ x 2, $\GL(2,3).M_{22}$, $C_4^2:C_3.M_{22}$, $C_2\times S_4\times M_{22}$, $(C_2\times \SL(2,3)).M_{22}$

Order 20401920: $C_2\times M_{23}$ x 2

Order 19958400: $M_{11}\times A_7$ x 2

Order 19353600: $\PSL(3,4)\times C_2^4.A_5$ x 2, $F_{16}:C_2.\PSL(3,4).C_2$ x 2, $C_2^4.A_5.A_8$ x 2, $F_{16}.C_4\times \PSL(3,4)$, $F_{16}.C_2^5.A_7$, $C_2^4.\PSL(3,4).A_5$, $C_2^4.A_8.A_5$, $A_8\times F_{16}.C_4$

Order 18627840: $C_2\times M_{22}\times C_7:C_3$

Order 16588800: $C_2^8.C_3^2.A_5.S_5$ x 2, $C_2^4:C_3.C_2^4.A_6.A_5$, $C_2^8.\GL(2,4)\wr C_2$

Order 15966720: $C_3:S_3.C_2.M_{22}$ x 2, $S_3^2\times M_{22}$, $C_3\times A_4\times M_{22}$

Order 15482880: $C_2^4:C_3.D_4.\PSL(3,4).C_2$ x 2, $C_2^4:C_3.C_2^6.C_2.A_7$ x 2, $C_2^4:C_3.C_2^4.\PSL(2,7).S_5$ x 2, $C_2^4.S_4.\PSL(3,4).C_2$ x 2, $C_4^2:A_4.C_2^5.A_7$, $C_2^8.\PSL(2,7).A_6$, $C_2^8.A_6.\PSL(2,7)$, $C_2^4:A_4.C_2^5.A_7$, $C_2^3:A_4.C_2^6.A_7$, $C_2\wr A_4.C_2.\PSL(3,4).C_2$

Order 15206400: $C_2^4.S_5\times M_{11}$ x 2

Order 14515200: $\PSL(3,4).A_6.C_2$ x 5, $C_2^4.A_6\times A_7$ x 2, $C_2^4.A_6.A_7$ x 2, $A_7\times C_2^4:C_3.S_5$ x 2, $\PSL(3,4)\times A_6.C_2$, $\PSL(3,4).C_2\times C_3:S_5$, $\PSL(3,4).C_2\times A_6$, $C_2^4.A_7.A_6$, $A_8\times A_6.C_2$, $(C_2^3\times C_6).A_7.S_5$, $S_6\times A_8$

Order 14450688: $C_2^8.\PSL(2,7)^2.C_2$ x 2

Order 14192640: $C_2^2\wr C_2.M_{22}$ x 3, $C_2^2.D_4.M_{22}$ x 3, $\SD_{16}:C_2.M_{22}$, $Q_8:C_2^2.M_{22}$, $C_4\wr C_2.M_{22}$, $C_4.D_4.M_{22}$, $C_2^3.C_4.M_{22}$

Order 13547520: $C_2^5.A_7.\PSL(2,7)$ x 2, $F_8:C_3.C_2^5.A_7$, $C_2\times \PSL(3,4).C_2\times \PSL(2,7)$, $C_2.\PSL(3,4).\PSL(2,7).C_2$, $(C_2\times F_8:C_3).\PSL(3,4).C_2$

Order 13305600: $\PSL(2,11)\times \PSL(3,4)$, $\PSL(2,11)\times A_8$, $(C_3\times D_5).M_{22}$

Order 12902400: $C_2^4:D_5.C_2^5.A_7$ x 2, $C_2^4:C_5:C_4\times \PSL(3,4).C_2$, $C_2^4:C_5:C_4.\PSL(3,4).C_2$

Order 12700800: $A_7\wr C_2$ x 2

Order 11612160: $A_4^2.C_2^5.A_7$ x 4, $A_4\wr C_2.\PSL(3,4).C_2$ x 4, $\PSOPlusPlus(4,3)\times \PSL(3,4).C_2$ x 2, $\PSOPlusPlus(4,3).\PSL(3,4).C_2$ x 2, $\PSL(3,4)\times A_4^2.C_2^2$, $C_2^2.A_4^2.\PSL(3,4)$, $C_2^2.A_4^2.A_8$, $A_8\times A_4^2.C_2^2$, $A_4^2.C_4.\PSL(3,4)$, $A_4^2.C_4.A_8$

Order 11059200: $C_2^8.A_5.A_6.C_2$ x 2, $C_2^4:C_3.C_2^4.A_5.S_5.C_2$ x 2, $C_2^8.A_5^2:D_6$

Order 10644480: $M_{22}\times S_4$ x 5, $\SL(2,3).M_{22}$ x 2, $C_2^3.\PSL(2,7)\times M_{11}$ x 2, $(C_2\times A_4).M_{22}$ x 2, $C_3:D_4.M_{22}$, $C_2\times M_{22}\times A_4$

Order 10200960: $M_{23}$ x 2, $C_{23}\times M_{22}$, $C_{23}:C_{11}\times \PSL(3,4).C_2$, $C_2^4.A_7\times C_{23}:C_{11}$

Order 9676800: $\PSL(3,4).C_2\times F_{16}$, $F_{16}:C_2\times A_8$, $F_{16}:C_2.\PSL(3,4)$, $F_{16}.\PSL(3,4).C_2$, $F_{16}.C_2^4.A_7$

Order 9313920: $M_{22}\times C_7:C_3$

Order 9123840: $C_2^4.C_3:S_4.M_{11}$

Order 8870400: $M_{22}\times F_5$

Order 8294400: $C_2^8.C_3^2.A_5.A_5$

Order 7983360: $M_{22}\times C_3:S_3$, $(C_3\times S_3).M_{22}$

Order 7741440: $C_2^4:C_3.C_2^6.A_7$ x 3, $C_2^2:S_4:C_2.\PSL(3,4).C_2$ x 3, $C_4^2:A_4.\PSL(3,4).C_2$ x 2, $C_2^7.\PSL(2,7).A_6$ x 2, $C_2^4:C_3:C_4\times \PSL(3,4).C_2$ x 2, $C_2^4:C_3:C_4.\PSL(3,4).C_2$ x 2, $C_2^4:C_3.D_4\times \PSL(3,4)$ x 2, $C_2^4:C_3.D_4.A_8$ x 2, $C_2^4:C_3.C_2^5.C_2.A_7$ x 2, $C_2^4:C_3.C_2^4.\PSL(2,7).A_5$ x 2, $C_2^4:A_4.\PSL(3,4).C_2$ x 2, $C_2^4.S_4.\PSL(3,4)$ x 2, $C_2^4.S_4.A_8$ x 2, $C_2^3:S_4.\PSL(3,4).C_2$ x 2, $C_2^3:A_4.C_2^5.A_7$ x 2, $C_2^3:A_4.C_2.\PSL(3,4).C_2$ x 2, $C_2^2\wr C_2:C_2:C_3\times \PSL(3,4).C_2$ x 2, $C_2^2\wr C_2:C_2:C_3.\PSL(3,4).C_2$ x 2, $C_2^2:S_4:C_2\times \PSL(3,4).C_2$ x 2, $C_2^2:S_4.C_2^5.A_7$ x 2, $C_2\wr A_4.\PSL(3,4).C_2$ x 2, $C_4^2:A_4.C_2^4.A_7$, $C_2^4:C_3.S_5\times C_2^3.\PSL(2,7)$, $C_2^4:C_3.C_2^3.\PSL(2,7).S_5$, $C_2^4:A_4.C_2^4.A_7$, $C_2^3:S_4.C_2^4.A_7$, $C_2^3:S_4.C_2.A_8$, $C_2^3.\PSL(2,7)\times C_2^4:C_3.S_5$, $C_2\wr A_4.C_2.\PSL(3,4)$

Order 7603200: $C_2^4.A_5.M_{11}$ x 2, $M_{11}\times F_{16}.C_4$, $C_2^4.M_{11}.A_5$

Order 7372800: $C_2^8.A_5^2.D_4$ x 2

Order 7257600: $A_6\times A_8$ x 3, $\PSL(3,4)\times A_6$ x 2, $C_2^4:C_3.A_7.A_5$ x 2, $\PSL(3,4)\times C_3:S_5$, $C_3.\PSL(3,4).S_5$, $C_3.\PSL(3,4).A_5.C_2$, $A_8\times C_3:S_5$, $A_6\times \PSL(3,4)$, $(C_2^3\times C_6).A_7.A_5$

Order 7225344: $C_2^8.\PSL(2,7).\PSL(2,7)$ x 3, $C_2^7.\PSL(2,7)^2.C_2$ x 2

Order 7096320: $C_2^2:C_4.M_{22}$ x 3, $C_2^4.M_{22}$ x 2, $(C_2\times D_4).M_{22}$ x 2, $\SD_{16}.M_{22}$, $\OD_{16}.M_{22}$, $Q_{16}.M_{22}$, $D_4:C_2.M_{22}$, $C_4^2.M_{22}$, $(C_2\times Q_8).M_{22}$

Order 6773760: $C_2^4.A_7.\PSL(2,7)$ x 4, $F_8:C_3.\PSL(3,4).C_2$ x 3, $C_2^4.\PSL(2,7).A_7$ x 3, $\PSL(3,4).\PSL(2,7).C_2$ x 2, $\PSL(3,4).C_2\times \PSL(2,7)$ x 2, $F_8:C_3.C_2^4.A_7$ x 2, $C_2.\PSL(3,4).\PSL(2,7)$ x 2, $C_2\times A_8\times \PSL(2,7)$, $C_2.A_8.\PSL(2,7)$, $(C_2\times F_8:C_3).\PSL(3,4)$, $(C_2\times F_8:C_3).A_8$

Order 6652800: $C_{15}.M_{22}$

Order 6635520: $C_2^2.A_4^2.C_2^5.A_6$, $C_2^2.A_4^2.C_2^4.C_6.S_5$

Order 6451200: $C_2^4:D_5.\PSL(3,4).C_2$ x 4, $C_2^4:C_5:C_4.\PSL(3,4)$ x 2, $C_2^4:C_5.C_2^5.A_7$ x 2, $C_2^4:C_5:C_4.A_8$, $A_8\times C_2^4:C_5:C_4$

Order 6350400: $A_7^2$ x 3

Order 6209280: $C_{14}\times M_{22}$

Order 5806080: $A_4^2.\PSL(3,4).C_2$ x 6, $\PSOPlusPlus(4,3).\PSL(3,4)$ x 2, $A_4^2.C_2^4.A_7$ x 2, $A_4\wr C_2\times A_8$ x 2, $A_4\wr C_2.\PSL(3,4)$ x 2, $\PSU(3,2):C_2.\PSL(3,4).C_2$, $\PSOPlusPlus(4,3)\times A_8$, $\PSOPlusPlus(4,3).A_8$, $S_3\wr C_2.C_2^5.A_7$

Order 5702400: $S_6\times M_{11}$, $M_{11}\times A_6.C_2$

Order 5529600: $C_2^4:C_3.C_2^4.A_5.S_5$ x 6, $C_2^8.A_5.A_6$ x 2, $F_{16}.C_2^5.C_2.A_6$, $F_{16}.C_2^4.C_{12}.S_5$, $C_2^8.C_3.A_5.S_5$, $C_2^8.A_6.A_5$, $C_2^4:C_3.C_2^4.A_5.A_5.C_2$, $C_2^8.A_5^2:S_3$, $C_2^8.A_5^2:C_6$

Order 5322240: $A_4.M_{22}$ x 3, $M_{22}\times A_4$, $C_3:C_4.M_{22}$, $C_2\times M_{22}\times S_3$, $(C_2\times C_6).M_{22}$

Order 5227200: $M_{11}\times \PSL(2,11)$

Order 5160960: $C_2^8.\PSL(2,7).S_5$ x 2, $C_2^8.A_5.\PSL(2,7).C_2$ x 2, $C_2^3.C_2^6.C_2^2.A_7$, $C_2\wr C_2^2.C_2.\PSL(3,4).C_2$

Order 5100480: $\PSL(3,4)\times C_{23}:C_{11}$, $A_8\times C_{23}:C_{11}$

Order 4878720: $C_{11}\times M_{22}$

Order 4838400: $C_2^4.A_7.S_5$ x 4, $C_2^4.S_5\times A_7$ x 3, $\PSL(3,4).S_5.C_2$ x 2, $\PSL(3,4).A_5.C_2^2$, $F_{16}\times A_8$, $F_{16}.\PSL(3,4)$

Order 4561920: $M_{11}\times A_4\wr C_2.C_2$, $C_2^2.A_4^2.M_{11}$, $A_4^2.C_4.M_{11}$

Order 4515840: $F_8.C_2^5.A_7$, $C_2\times F_8.\PSL(3,4).C_2$

Order 4435200: $M_{22}\times D_5$

Order 4147200: $C_2^4:C_3.S_5\times A_6.C_2$, $C_2^4:C_3.A_6.A_5.C_2^2$, $C_2^4.A_6.A_6.C_2$, $A_6.C_2\times C_2^4.A_6$

Order 128018: $C_{23}^2:(C_{11}\times D_{11})$

Order 32768: $C_2^5.C_2^5.C_2^4.C_2$

Order 529: $C_{23}^2$

Order 121: $C_{11}^2$

Order 81: $C_3^4$

Order 49: $C_7^2$

Order 25: $C_5^2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 208119169843200: $M_{23}\wr C_2$ ($C_1$)

Order 104059584921600: $M_{23}.M_{23}$ ($C_2$)

Order 1: $C_1$ ($M_{23}\wr C_2$)

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 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 $170 \times 170$ character table is not available for this group.

Rational character table

The $96 \times 96$ rational character table is not available for this group.