Abstract group 161280.b: $S_5\times C_2^3:\GL(3,2)$

• Label: 161280.b
• Order: \( 2^{9} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $13$
• Trans deg.: $40$
• Rank: $2$

Group information

Description:	$S_5\times C_2^3:\GL(3,2)$
Order:	\(161280\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \cdot 7 \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_2^4.\PSL(2,7).S_5$, of order \(322560\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 4, $A_5$, $\PSL(2,7)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	10	12	14	15	20	21	28	30	35	42
Elements	1	2391	4724	26280	24	28524	384	2184	30240	9600	5376	10080	7680	11520	5376	9216	7680	161280
Conjugacy classes	1	11	3	16	1	15	2	3	8	4	1	3	2	2	1	2	2	77
Divisions	1	11	3	16	1	15	1	3	8	2	1	3	1	1	1	1	1	70
Autjugacy classes	1	8	3	11	1	13	2	2	6	4	1	2	2	2	1	2	2	63

Dimension	1	3	4	5	6	7	8	12	14	15	18	21	24	28	30	32	35	36	40	42	48	56	70	84	105	126
Irr. complex chars.	2	4	2	2	3	6	2	4	2	4	2	4	2	6	2	2	6	1	2	3	1	2	2	5	4	2	77
Irr. rational chars.	2	0	2	2	5	6	2	0	2	0	0	4	4	6	4	2	6	2	2	3	1	2	2	5	4	2	70

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$40$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	28	28	28
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $13$ $\langle(1,2)(3,4,6,7,5,8)(9,10,12,13,11), (1,2,3,5)(4,7,8,6)(9,11)(10,12,13)\rangle$
Transitive group:	40T48040				more information
Direct product:	$S_5$ $\, \times\, $ $(C_2^3:\GL(3,2))$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^3:\GL(3,2))$ . $S_5$	$S_5$ . $(C_2^3:\GL(3,2))$	$A_5$ . $(C_2^4:\GL(3,2))$	$(C_2^3\times S_5)$ . $\PSL(2,7)$	all 9
Aut. group:	$\Aut(C_2^3\times \SL(2,5))$	$\Aut(C_2^3\times S_5)$

Elements of the group are displayed as permutations of degree 13.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 4032531 subgroups in 7163 conjugacy classes, 9 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_5\times C_2^3:\GL(3,2)$
Commutator:	$G' \simeq$ $(C_2^3\times A_5).\PSL(2,7)$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_5\times C_2^3:\GL(3,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3$	$G/\operatorname{Fit} \simeq$ $A_5.\PSL(2,7).C_2$
Radical:	$R \simeq$ $C_2^3$	$G/R \simeq$ $A_5.\PSL(2,7).C_2$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times A_5$	$G/\operatorname{soc} \simeq$ $C_2\times \GL(3,2)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3:D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

All subgroups of index up to 1008 (order at least 160) are shown, as well as all normal subgroups of any index.

Order 161280: $S_5\times C_2^3:\GL(3,2)$

Order 80640: $(C_2^3\times A_5).\PSL(2,7)$

Order 32256: $(C_2^3\times S_4).\PSL(2,7)$

Order 26880: $(C_2^3\times F_5).\PSL(2,7)$

Order 23040: $C_2^2:S_4:C_2.S_5$, $C_2^3:S_4.S_5$

Order 20160: $A_5.\PSL(2,7).C_2$ x 2, $F_8:C_3.S_5$

Order 16128: $(C_2^3\times A_4).\PSL(2,7)$, $(C_2^4\times S_3).\PSL(2,7)$

Order 13440: $(C_2^3\times D_5).\PSL(2,7)$

Order 11520: $C_2^2:S_4.S_5$ x 2, $C_2^2:S_4.S_5$ x 2, $C_2^3:A_4.S_5$, $C_2^2\wr C_2:C_3.S_5$, $C_2^2:S_4:C_2.A_5$, $C_2^3:S_4.A_5$, $C_2^3:A_4.S_5$, $C_2^2\wr C_2:C_3.S_5$

Order 10752: $(C_2^3\times D_4).\PSL(2,7)$

Order 10080: $A_5\times \PSL(2,7)$ x 2, $F_8:C_3.A_5$

Order 8064: $(C_2^3\times S_3).\PSL(2,7)$ x 2, $(C_2^3\times C_6).\PSL(2,7)$

Order 7680: $C_2\wr C_2^2.S_5$

Order 6720: $(C_2^2\times C_{10}).\PSL(2,7)$, $F_8.S_5$

Order 5760: $(C_2\times S_4).S_5$ x 3, $C_2^2:S_4.A_5$ x 2, $C_2^4:C_3.S_5$ x 2, $C_2^4.(C_6\times A_5)$, $C_2^3:A_4.A_5$, $C_2^4:C_3.S_5$, $C_2^4:C_3.S_5$

Order 5376: $C_2^5.\PSL(2,7)$ x 2, $(C_2^3\times C_4).\PSL(2,7)$

Order 4608: $C_2^6.S_3^2.C_2$, $C_2^5.A_4.D_6$

Order 4032: $S_4.\PSL(2,7)$ x 2, $(C_2^2\times C_6).\PSL(2,7)$, $C_2^5.C_{21}.C_6$

Order 3840: $C_2^2\wr C_2.S_5$ x 3, $C_2^2.D_4.S_5$ x 3, $C_2^2.D_4.S_5$ x 3, $C_2^2\wr C_2.S_5$ x 3, $Q_8:C_2^2.S_5$, $C_2^4.(S_3\times F_5).C_2$, $Q_8:C_2^2.S_5$, $C_2\wr C_2^2.A_5$, $C_2^3.(F_5\times S_4)$

Order 3360: $\PSL(2,7)\times F_5$ x 2, $F_8.A_5$, $F_8.C_{15}.C_4$

Order 2880: $S_4.S_5$ x 14, $(C_2\times A_4).S_5$ x 3, $(C_2\times A_4).S_5$ x 3, $(C_2\times S_4).A_5$ x 3, $C_2^4:C_3.A_5$, $\SL(2,3).S_5$

Order 2688: $C_2^4.\PSL(2,7)$ x 2

Order 2520: $C_7:C_3.S_5$

Order 2304: $C_2^6.S_3^2$ x 2, $C_2^6.S_3^2$ x 2, $(C_2^4\times A_4).C_6.C_2$, $C_2^3.A_4^2.C_2$, $(C_2^4\times A_4).D_6$, $C_2^2.A_4^2.C_2^2$, $C_2^3:A_4.D_6.C_2$, $(C_2^3\times A_4).A_4.C_2$, $C_2^5.C_3:S_4$, $C_2^4.D_6^2$

Order 2016: $D_6.\PSL(2,7)$ x 2, $A_4.\PSL(2,7)$ x 2, $F_8.(C_6\times S_3)$, $F_8.A_4.C_3$

Order 1920: $(C_2^2\times S_5):C_4$ x 6, $C_2\times D_4\times S_5$ x 5, $C_2.(D_4\times S_5)$ x 5, $(C_2\times D_4).S_5$ x 5, $C_2^2:C_4\times S_5$ x 5, $(C_2\times D_4):S_5$ x 3, $A_5\times C_2^2\wr C_2$ x 3, $C_2^4.S_5$ x 3, $C_2^3:C_4\times A_5$ x 3, $(C_2^3\times A_5):C_4$ x 3, $F_5\times C_2^2:S_4$ x 2, $(C_2^2\times F_5):S_4$ x 2, $C_2^4\times S_5$, $D_4:C_2^2\times A_5$, $D_4:(C_2\times S_5)$, $D_4:C_2\times S_5$, $D_5\times C_2^3:S_4$, $F_5\times Q_8:A_4$, $(C_2^2\times D_{10}).S_4$, $D_5\times C_2^3:S_4$, $C_2^3:A_4\times F_5$, $(C_2^2\times D_{10}).S_4$

Order 1680: $D_5\times \GL(3,2)$ x 2, $D_5\times F_8:C_3$

Order 1536: $D_4\times C_2^3:S_4$, $C_2\wr C_2^2\times S_4$, $D_4\times C_2^3:S_4$

Order 1440: $S_4\times A_5$ x 8, $A_4:S_5$ x 8, $A_4\times S_5$ x 8, $C_2^3:\GL(2,4)$ x 3, $D_6\times S_5$, $A_5\times \SL(2,3)$

Order 1344: $D_4\times \GL(3,2)$ x 2, $S_4\times F_8$, $C_2^3:\GL(3,2)$, $D_4\times F_8:C_3$

Order 1280: $C_2\wr C_2^2\times F_5$

Order 1260: $C_7:\GL(2,4)$

Order 1152: $S_3\times C_2^3:S_4$ x 8, $S_3\times C_2^3:S_4$ x 4, $C_2\times S_4^2$ x 3, $C_2^5:S_3^2$ x 2, $(C_2^2\times A_4):S_4$ x 2, $A_4\times C_2^2:S_4$ x 2, $C_2^5:S_3^2$ x 2, $C_2^2:A_4\times S_4$, $Q_8:A_4\times D_6$, $(C_2^3\times C_6):S_4$, $C_6\times C_2^3:S_4$, $Q_8:A_4^2$, $C_2^3:A_4\times D_6$, $(C_2^3\times C_6):S_4$, $C_6\times C_2^3:S_4$, $C_2^3:A_4^2$, $(C_2^2\times S_4):A_4$

Order 1120: $F_5\times F_8$

Order 1008: $S_3\times \GL(3,2)$ x 4, $F_8:C_3\times S_3$ x 2, $C_6\times \GL(3,2)$ x 2, $C_3\times F_8:C_6$

Order 960: $D_4\times S_5$ x 17, $C_2^3:S_5$ x 10, $C_2^3\times S_5$ x 9, $(C_2\times C_4):S_5$ x 7, $C_2^2:C_4\times A_5$ x 5, $C_2^3.S_5$ x 5, $C_2\times D_4\times A_5$ x 5, $C_2\times C_4:S_5$ x 5, $C_2\times C_4\times S_5$ x 5, $C_2\times F_5\times S_4$ x 3, $D_4:S_5$ x 3, $(C_2\times C_4):S_5$ x 3, $(C_2\times D_{10}):S_4$ x 2, $D_5\times C_2^2:S_4$ x 2, $(C_2\times D_{10}).S_4$ x 2, $D_5\times C_2^3:A_4$, $C_2^4:D_{30}$, $C_5\times C_2^3:S_4$, $(C_2^3\times C_{10}):C_{12}$, $C_2^2:A_4\times F_5$, $C_2^4\times A_5$, $D_5\times Q_8:A_4$, $(C_2^2\times C_{10}):S_4$, $C_5\times C_2^3:S_4$, $D_4:C_2\times A_5$, $Q_8:S_5$, $Q_8\times S_5$

Order 840: $C_5\times \PSL(2,7)$ x 2, $F_8:C_{15}$, $C_7\times S_5$

Order 768: $C_2\wr D_6$ x 4, $C_2^6:D_6$ x 3, $C_2^5:D_{12}$ x 3, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^3:C_4\times S_4$ x 3, $C_2^5:S_4$ x 2, $C_2^6:D_6$ x 2, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:S_4$ x 2, $(C_2^2\times D_4):S_4$ x 2, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:S_4$, $\GL(2,\mathbb{Z}/4):C_2^3$, $(C_2^3\times C_4):S_4$, $C_2^5:S_4$, $C_4\times C_2^3:S_4$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $(C_2^3\times C_4):S_4$, $C_4\times C_2^3:S_4$, $D_4\times Q_8:A_4$, $A_4\times C_2\wr C_2^2$, $D_4\times C_2^3:A_4$, $C_2\wr C_2^2\times D_6$

Order 720: $A_4\times A_5$ x 5, $S_3\times S_5$ x 4, $D_6\times A_5$, $C_6:S_5$, $C_6\times S_5$

Order 672: $C_2^2\times \GL(3,2)$ x 4, $C_2\times F_8:C_6$ x 2, $C_4\times \GL(3,2)$ x 2, $D_6\times F_8$, $A_4\times F_8$, $F_8:A_4$, $F_8:C_{12}$

Order 640: $C_2^4:(C_2\times F_5)$ x 3, $C_2^3:C_4\times F_5$ x 3, $(C_4\times D_{10}).D_4$ x 3, $C_2^2\wr C_2\times F_5$ x 3, $(C_2\times D_{20}).D_4$, $D_{10}.C_2^5$, $C_2\wr C_2^2\times D_5$

Order 576: $S_4^2$ x 14, $C_2^4:S_3^2$ x 8, $C_2^4:S_3^2$ x 8, $A_4^2:C_2^2$ x 6, $S_3\times C_2^3:A_4$ x 4, $(C_2^2\times C_6):S_4$ x 4, $C_3\times C_2^3:S_4$ x 4, $C_2^2:D_6^2$ x 3, $A_4^2:C_2^2$ x 3, $(C_2^2\times C_6):S_4$ x 2, $C_2^2:S_4\times C_6$ x 2, $S_3\times Q_8:A_4$ x 2, $(C_2^2\times C_6):S_4$ x 2, $C_3\times C_2^3:S_4$ x 2, $C_2^2:A_4\times D_6$, $C_2^2:A_4^2$, $C_6\times Q_8:A_4$, $S_4\times \SL(2,3)$, $C_2^4:C_6^2$, $C_2^4:(C_6\times S_3)$

Order 560: $D_5\times F_8$

Order 512: $C_2^3:D_4^2$

Order 504: $C_3\times \GL(3,2)$ x 2, $C_7:C_3\times S_4$, $F_8:C_3^2$

Order 480: $C_2^2\times S_5$ x 29, $C_2^2:S_5$ x 14, $F_5\times S_4$ x 14, $C_4\times S_5$ x 8, $D_4\times A_5$ x 7, $C_4:S_5$ x 7, $C_2^3\times A_5$ x 6, $C_2\times C_4\times A_5$ x 5, $C_2^2.S_5$ x 5, $D_{10}\times S_4$ x 3, $C_2\times A_4\times F_5$ x 3, $D_{10}.S_4$ x 3, $C_2^4:D_{15}$ x 2, $C_5\times C_2^2:S_4$ x 2, $F_5\times \SL(2,3)$, $Q_8\times A_5$, $A_5:Q_8$, $C_2^4:C_{30}$, $(C_2\times D_{10}):A_4$, $D_5\times C_2^2:A_4$, $C_5\times Q_8:A_4$

Order 448: $D_4\times F_8$

Order 420: $C_{35}:C_{12}$, $C_7\times A_5$

Order 384: $C_2\wr C_2^2\times S_3$ x 16, $C_2^4:S_4$ x 12, $\GL(2,\mathbb{Z}/4):C_2^2$ x 8, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^4:S_4$ x 6, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 5, $C_2^4:D_{12}$ x 5, $C_2^5.D_6$ x 5, $C_2^4:S_4$ x 4, $C_2^2\wr S_3$ x 4, $C_2\wr S_3$ x 4, $C_2^5:C_{12}$ x 3, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 3, $\GL(2,\mathbb{Z}/4):C_2^2$ x 3, $C_2^5:D_6$ x 3, $C_2^6:C_6$ x 3, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^5:D_6$ x 3, $C_2^3:C_4\times D_6$ x 3, $C_2^4:D_{12}$ x 3, $C_2\wr C_6$ x 2, $C_2^4:D_{12}$ x 2, $C_2^5.D_6$ x 2, $C_2^4:S_4$ x 2, $C_2^5:A_4$ x 2, $C_2^4:D_{12}$ x 2, $C_2^5.D_6$ x 2, $C_2^5:A_4$ x 2, $(C_2^2\times D_4):A_4$, $C_4\times C_2^3:A_4$, $C_2^4.S_4$, $C_2^4\times S_4$, $D_{12}:C_2^4$, $C_2^5:A_4$, $C_2^6:C_6$, $A_4\times D_4:C_2^2$, $\GL(2,\mathbb{Z}/4):C_2^2$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_2^5:A_4$, $C_4\times Q_8:A_4$, $C_2^4.S_4$, $C_2\wr C_2^2\times C_6$, $(C_6\times D_4):D_4$

Order 360: $S_3\times A_5$ x 2, $C_3:S_5$ x 2, $C_3\times S_5$ x 2, $C_6\times A_5$

Order 336: $C_2\times \GL(3,2)$ x 4, $F_8:C_6$ x 3, $S_3\times F_8$ x 2, $C_6\times F_8$

Order 320: $(C_2^2\times F_5):C_4$ x 6, $D_{10}.C_2^4$ x 5, $(C_2\times D_{20}):C_4$ x 5, $C_{10}.(C_4\times D_4)$ x 5, $D_{10}:C_4^2$ x 5, $C_2^4:D_{10}$ x 3, $C_2^3:C_4\times D_5$ x 3, $C_2^3:D_{20}$ x 3, $D_{10}.C_2^4$ x 3, $C_2^4:D_{10}$ x 3, $C_2^4:F_5$ x 3, $(C_4\times D_{10}):C_4$ x 3, $C_2\wr C_2^2\times C_5$, $(C_2\times C_{20}):D_4$, $C_2^4\times F_5$, $D_{20}:C_2^3$, $D_{10}.C_2^4$, $D_{10}.C_2^4$

Order 288: $D_6\times S_4$ x 39, $A_4\times S_4$ x 16, $\PSOPlus(4,3)$ x 8, $(C_2\times C_6):S_4$ x 4, $C_3\times C_2^2:S_4$ x 4, $C_2\times A_4\times D_6$ x 3, $C_2\times C_6:S_4$ x 3, $C_2\times C_6\times S_4$ x 3, $C_2\times A_4^2$ x 3, $C_3\times C_2^3:A_4$ x 2, $(C_2\times D_6):A_4$ x 2, $S_3\times C_2^2:A_4$ x 2, $C_3\times Q_8:A_4$, $D_6\times \SL(2,3)$, $A_4\times \SL(2,3)$, $C_2^5:C_3^2$

Order 280: $C_5\times F_8$

Order 256: $C_2^5:D_4$ x 6, $C_2^5:D_4$ x 6, $C_2^2:D_4^2$ x 3, $C_2^2.D_4^2$ x 3, $C_2^2.D_4^2$ x 3, $C_2^2.D_4^2$ x 3, $C_2^5:D_4$ x 2, $C_2^5:D_4$ x 2, $D_4^2:C_2^2$, $C_2^4.C_2^4$, $C_2^4.C_2^4$

Order 252: $C_{42}:C_6$, $A_4\times C_7:C_3$

Order 240: $C_2\times S_5$ x 19, $D_5\times S_4$ x 14, $C_2^2\times A_5$ x 10, $A_4\times F_5$ x 8, $A_4:F_5$ x 8, $C_4\times A_5$ x 3, $A_5:C_4$ x 3, $A_4\times D_{10}$ x 3, $C_{10}:S_4$ x 3, $C_{10}\times S_4$ x 3, $C_2^4:C_{15}$, $D_6\times F_5$, $D_5\times \SL(2,3)$

Order 224: $C_2^2\times F_8$ x 2, $C_4\times F_8$

Order 210: $C_{35}:C_6$

Order 192: $D_4\times S_4$ x 43, $C_2^4:D_6$ x 32, $C_2^4:D_6$ x 24, $C_2^3:C_4\times S_3$ x 24, $C_2^3:D_{12}$ x 24, $C_2^3:S_4$ x 16, $C_2\times \GL(2,\mathbb{Z}/4)$ x 16, $C_2^3:S_4$ x 15, $C_2^3\times S_4$ x 15, $D_{12}:C_2^3$ x 10, $C_3\times C_2\wr C_2^2$ x 8, $(C_2\times C_{12}):D_4$ x 8, $C_2^5:C_6$ x 8, $C_2^3:S_4$ x 8, $C_2^3:D_{12}$ x 8, $C_2^4.D_6$ x 8, $C_2.\GL(2,\mathbb{Z}/4)$ x 7, $C_2^3.D_{12}$ x 6, $C_2^4:A_4$ x 6, $C_2^4:C_{12}$ x 5, $C_2.\GL(2,\mathbb{Z}/4)$ x 5, $C_{12}:C_2^4$ x 5, $C_2^4:D_6$ x 5, $C_2^3:D_{12}$ x 5, $C_2^4.D_6$ x 5, $C_2^4:C_{12}$ x 3, $C_2^4.D_6$ x 3, $D_{12}:C_2^3$ x 3, $C_2^4:A_4$ x 3, $D_4:S_4$ x 3, $\GL(2,\mathbb{Z}/4):C_2$ x 3, $C_2^5:C_6$ x 3, $C_2^4:D_6$ x 3, $C_2^2\wr C_3$ x 2, $C_2^3.S_4$ x 2, $C_2^4:C_{12}$, $D_6\times C_2^4$, $C_2^4:A_4$, $A_4\times C_2^4$, $C_{12}.C_2^4$, $D_{12}:C_2^3$, $D_{12}:C_2^3$, $C_2^4:C_{12}$, $A_4\times D_4:C_2$, $Q_8:S_4$, $Q_8\times S_4$, $D_4\times \SL(2,3)$

Order 180: $\GL(2,4)$

Order 168: $F_8:C_3$ x 2, $\PSL(2,7)$ x 2, $C_7\times S_4$, $C_3\times F_8$, $C_{28}:C_6$

Order 160: $D_4\times F_5$ x 20, $C_2^3:F_5$ x 15, $C_2^3\times F_5$ x 9, $C_{10}.C_4^2$ x 7, $C_2^2.D_{20}$ x 6, $D_4\times D_{10}$ x 5, $D_{10}.D_4$ x 5, $C_{10}:C_4^2$ x 5, $D_{10}:D_4$ x 5, $C_2^2:D_{20}$ x 5, $D_{10}.D_4$ x 5, $C_2^3:F_5$ x 3, $D_{10}.D_4$ x 3, $C_2^3:C_{20}$ x 3, $C_2^3.D_{10}$ x 3, $D_4:D_{10}$ x 3, $D_{10}.C_2^3$ x 3, $C_2^4:C_{10}$ x 3, $C_2^4:D_5$ x 3, $Q_8\times F_5$ x 2, $C_2^3\times D_{10}$, $C_{10}.C_2^4$, $D_4:D_{10}$, $D_4:D_{10}$

Order 120: $S_5$

Order 60: $A_5$

Order 9: $C_3^2$

Order 8: $C_2^3$

Order 7: $C_7$

Order 5: $C_5$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 1008 (order at least 160) are shown, as well as all normal subgroups of any index.

Order 161280: $S_5\times C_2^3:\GL(3,2)$

Order 80640: $(C_2^3\times A_5).\PSL(2,7)$

Order 32256: $(C_2^3\times S_4).\PSL(2,7)$

Order 26880: $(C_2^3\times F_5).\PSL(2,7)$

Order 23040: $C_2^2:S_4:C_2.S_5$, $C_2^3:S_4.S_5$

Order 20160: $A_5.\PSL(2,7).C_2$, $F_8:C_3.S_5$

Order 16128: $(C_2^3\times A_4).\PSL(2,7)$, $(C_2^4\times S_3).\PSL(2,7)$

Order 13440: $(C_2^3\times D_5).\PSL(2,7)$

Order 11520: $C_2^3:A_4.S_5$, $C_2^2\wr C_2:C_3.S_5$, $C_2^2:S_4.S_5$, $C_2^2:S_4:C_2.A_5$, $C_2^3:S_4.A_5$, $C_2^2:S_4.S_5$, $C_2^3:A_4.S_5$, $C_2^2\wr C_2:C_3.S_5$

Order 10752: $(C_2^3\times D_4).\PSL(2,7)$

Order 10080: $A_5\times \PSL(2,7)$, $F_8:C_3.A_5$

Order 8064: $(C_2^3\times S_3).\PSL(2,7)$ x 2, $(C_2^3\times C_6).\PSL(2,7)$

Order 7680: $C_2\wr C_2^2.S_5$

Order 6720: $(C_2^2\times C_{10}).\PSL(2,7)$, $F_8.S_5$

Order 5760: $(C_2\times S_4).S_5$ x 2, $C_2^2:S_4.A_5$, $C_2^4.(C_6\times A_5)$, $C_2^4:C_3.S_5$, $C_2^3:A_4.A_5$, $C_2^4:C_3.S_5$, $C_2^4:C_3.S_5$

Order 5376: $C_2^5.\PSL(2,7)$ x 2, $(C_2^3\times C_4).\PSL(2,7)$

Order 4608: $C_2^6.S_3^2.C_2$, $C_2^5.A_4.D_6$

Order 4032: $(C_2^2\times C_6).\PSL(2,7)$, $S_4.\PSL(2,7)$, $C_2^5.C_{21}.C_6$

Order 3840: $C_2^2\wr C_2.S_5$ x 2, $C_2^2.D_4.S_5$ x 2, $C_2^2.D_4.S_5$ x 2, $C_2^2\wr C_2.S_5$ x 2, $Q_8:C_2^2.S_5$, $C_2^4.(S_3\times F_5).C_2$, $Q_8:C_2^2.S_5$, $C_2\wr C_2^2.A_5$, $C_2^3.(F_5\times S_4)$

Order 3360: $F_8.A_5$, $\PSL(2,7)\times F_5$, $F_8.C_{15}.C_4$

Order 2880: $S_4.S_5$ x 7, $(C_2\times A_4).S_5$ x 2, $(C_2\times A_4).S_5$ x 2, $(C_2\times S_4).A_5$ x 2, $C_2^4:C_3.A_5$, $\SL(2,3).S_5$

Order 2688: $C_2^4.\PSL(2,7)$ x 2

Order 2520: $C_7:C_3.S_5$

Order 2304: $(C_2^4\times A_4).C_6.C_2$, $C_2^3.A_4^2.C_2$, $C_2^6.S_3^2$, $(C_2^4\times A_4).D_6$, $C_2^2.A_4^2.C_2^2$, $C_2^6.S_3^2$, $C_2^3:A_4.D_6.C_2$, $(C_2^3\times A_4).A_4.C_2$, $C_2^5.C_3:S_4$, $C_2^4.D_6^2$

Order 2016: $D_6.\PSL(2,7)$, $A_4.\PSL(2,7)$, $F_8.(C_6\times S_3)$, $F_8.A_4.C_3$

Order 1920: $C_2\times D_4\times S_5$ x 3, $C_2.(D_4\times S_5)$ x 3, $(C_2\times D_4).S_5$ x 3, $C_2^2:C_4\times S_5$ x 3, $(C_2^2\times S_5):C_4$ x 3, $(C_2\times D_4):S_5$ x 2, $A_5\times C_2^2\wr C_2$ x 2, $C_2^4.S_5$ x 2, $C_2^3:C_4\times A_5$ x 2, $(C_2^3\times A_5):C_4$ x 2, $C_2^4\times S_5$, $D_4:C_2^2\times A_5$, $D_4:(C_2\times S_5)$, $D_4:C_2\times S_5$, $F_5\times C_2^2:S_4$, $D_5\times C_2^3:S_4$, $F_5\times Q_8:A_4$, $(C_2^2\times D_{10}).S_4$, $D_5\times C_2^3:S_4$, $C_2^3:A_4\times F_5$, $(C_2^2\times D_{10}).S_4$, $(C_2^2\times F_5):S_4$

Order 1680: $D_5\times F_8:C_3$, $D_5\times \GL(3,2)$

Order 1536: $D_4\times C_2^3:S_4$, $C_2\wr C_2^2\times S_4$, $D_4\times C_2^3:S_4$

Order 1440: $A_4\times S_5$ x 5, $S_4\times A_5$ x 4, $A_4:S_5$ x 4, $C_2^3:\GL(2,4)$ x 2, $D_6\times S_5$, $A_5\times \SL(2,3)$

Order 1344: $S_4\times F_8$, $C_2^3:\GL(3,2)$, $D_4\times F_8:C_3$, $D_4\times \GL(3,2)$

Order 1280: $C_2\wr C_2^2\times F_5$

Order 1260: $C_7:\GL(2,4)$

Order 1152: $S_3\times C_2^3:S_4$ x 6, $S_3\times C_2^3:S_4$ x 4, $C_2\times S_4^2$ x 2, $C_2^5:S_3^2$, $(C_2^2\times A_4):S_4$, $C_2^2:A_4\times S_4$, $A_4\times C_2^2:S_4$, $Q_8:A_4\times D_6$, $(C_2^3\times C_6):S_4$, $C_6\times C_2^3:S_4$, $Q_8:A_4^2$, $C_2^3:A_4\times D_6$, $(C_2^3\times C_6):S_4$, $C_6\times C_2^3:S_4$, $C_2^3:A_4^2$, $C_2^5:S_3^2$, $(C_2^2\times S_4):A_4$

Order 1120: $F_5\times F_8$

Order 1008: $F_8:C_3\times S_3$ x 2, $S_3\times \GL(3,2)$ x 2, $C_3\times F_8:C_6$, $C_6\times \GL(3,2)$

Order 960: $D_4\times S_5$ x 9, $C_2^3\times S_5$ x 6, $C_2^3:S_5$ x 6, $(C_2\times C_4):S_5$ x 4, $C_2^2:C_4\times A_5$ x 3, $C_2^3.S_5$ x 3, $C_2\times D_4\times A_5$ x 3, $C_2\times C_4:S_5$ x 3, $C_2\times C_4\times S_5$ x 3, $C_2\times F_5\times S_4$ x 2, $D_4:S_5$ x 2, $(C_2\times C_4):S_5$ x 2, $D_5\times C_2^3:A_4$, $C_2^4:D_{30}$, $C_5\times C_2^3:S_4$, $(C_2\times D_{10}):S_4$, $(C_2^3\times C_{10}):C_{12}$, $D_5\times C_2^2:S_4$, $C_2^2:A_4\times F_5$, $(C_2\times D_{10}).S_4$, $C_2^4\times A_5$, $D_5\times Q_8:A_4$, $(C_2^2\times C_{10}):S_4$, $C_5\times C_2^3:S_4$, $D_4:C_2\times A_5$, $Q_8:S_5$, $Q_8\times S_5$

Order 840: $F_8:C_{15}$, $C_7\times S_5$, $C_5\times \PSL(2,7)$

Order 768: $C_2^5:S_4$ x 2, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:S_4$ x 2, $C_2^6:D_6$ x 2, $C_2^5:D_{12}$ x 2, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $C_2\wr D_6$ x 2, $C_2^3:C_4\times S_4$ x 2, $C_2^5:S_4$, $C_2^6:D_6$, $\GL(2,\mathbb{Z}/4):C_2^3$, $(C_2^3\times C_4):S_4$, $C_2^5:S_4$, $C_4\times C_2^3:S_4$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $(C_2^3\times C_4):S_4$, $(C_2^2\times D_4):S_4$, $C_4\times C_2^3:S_4$, $D_4\times Q_8:A_4$, $A_4\times C_2\wr C_2^2$, $D_4\times C_2^3:A_4$, $C_2\wr C_2^2\times D_6$

Order 720: $A_4\times A_5$ x 3, $S_3\times S_5$ x 2, $D_6\times A_5$, $C_6:S_5$, $C_6\times S_5$

Order 672: $C_2\times F_8:C_6$ x 2, $C_2^2\times \GL(3,2)$ x 2, $D_6\times F_8$, $A_4\times F_8$, $F_8:A_4$, $F_8:C_{12}$, $C_4\times \GL(3,2)$

Order 640: $C_2^4:(C_2\times F_5)$ x 2, $C_2^3:C_4\times F_5$ x 2, $(C_4\times D_{10}).D_4$ x 2, $C_2^2\wr C_2\times F_5$ x 2, $(C_2\times D_{20}).D_4$, $D_{10}.C_2^5$, $C_2\wr C_2^2\times D_5$

Order 576: $S_4^2$ x 7, $C_2^4:S_3^2$ x 4, $A_4^2:C_2^2$ x 4, $S_3\times C_2^3:A_4$ x 4, $C_2^4:S_3^2$ x 4, $(C_2^2\times C_6):S_4$ x 3, $C_3\times C_2^3:S_4$ x 3, $C_2^2:D_6^2$ x 2, $A_4^2:C_2^2$ x 2, $S_3\times Q_8:A_4$ x 2, $(C_2^2\times C_6):S_4$ x 2, $C_3\times C_2^3:S_4$ x 2, $C_2^2:A_4\times D_6$, $(C_2^2\times C_6):S_4$, $C_2^2:S_4\times C_6$, $C_2^2:A_4^2$, $C_6\times Q_8:A_4$, $S_4\times \SL(2,3)$, $C_2^4:C_6^2$, $C_2^4:(C_6\times S_3)$

Order 560: $D_5\times F_8$

Order 512: $C_2^3:D_4^2$

Order 504: $C_7:C_3\times S_4$, $F_8:C_3^2$, $C_3\times \GL(3,2)$

Order 480: $C_2^2\times S_5$ x 17, $C_2^2:S_5$ x 7, $F_5\times S_4$ x 7, $C_4\times S_5$ x 5, $D_4\times A_5$ x 4, $C_4:S_5$ x 4, $C_2^3\times A_5$ x 4, $C_2\times C_4\times A_5$ x 3, $C_2^2.S_5$ x 3, $D_{10}\times S_4$ x 2, $C_2\times A_4\times F_5$ x 2, $D_{10}.S_4$ x 2, $F_5\times \SL(2,3)$, $Q_8\times A_5$, $A_5:Q_8$, $C_2^4:C_{30}$, $(C_2\times D_{10}):A_4$, $D_5\times C_2^2:A_4$, $C_2^4:D_{15}$, $C_5\times C_2^2:S_4$, $C_5\times Q_8:A_4$

Order 448: $D_4\times F_8$

Order 420: $C_{35}:C_{12}$, $C_7\times A_5$

Order 384: $C_2\wr C_2^2\times S_3$ x 12, $C_2^4:S_4$ x 9, $C_2^4:S_4$ x 6, $\GL(2,\mathbb{Z}/4):C_2^2$ x 5, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4:D_{12}$ x 3, $C_2^5.D_6$ x 3, $C_2^5:C_{12}$ x 2, $C_2\wr C_6$ x 2, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:S_4$ x 2, $C_2^2\wr S_3$ x 2, $C_2^5:A_4$ x 2, $C_2\wr S_3$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_2^5:D_6$ x 2, $C_2^6:C_6$ x 2, $C_2^5:A_4$ x 2, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:D_6$ x 2, $C_2^3:C_4\times D_6$ x 2, $C_2^4:D_{12}$ x 2, $(C_2^2\times D_4):A_4$, $C_4\times C_2^3:A_4$, $C_2^4.S_4$, $C_2^4:D_{12}$, $C_2^5.D_6$, $C_2^4:S_4$, $C_2^4\times S_4$, $D_{12}:C_2^4$, $C_2^5:A_4$, $C_2^6:C_6$, $A_4\times D_4:C_2^2$, $C_2^4:D_{12}$, $C_2^5.D_6$, $\GL(2,\mathbb{Z}/4):C_2^2$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_2^5:A_4$, $C_4\times Q_8:A_4$, $C_2^4.S_4$, $C_2\wr C_2^2\times C_6$, $(C_6\times D_4):D_4$

Order 360: $C_3\times S_5$ x 2, $C_6\times A_5$, $S_3\times A_5$, $C_3:S_5$

Order 336: $F_8:C_6$ x 3, $S_3\times F_8$ x 2, $C_2\times \GL(3,2)$ x 2, $C_6\times F_8$

Order 320: $(C_2^2\times F_5):C_4$ x 3, $D_{10}.C_2^4$ x 3, $(C_2\times D_{20}):C_4$ x 3, $C_{10}.(C_4\times D_4)$ x 3, $D_{10}:C_4^2$ x 3, $C_2^4:D_{10}$ x 2, $C_2^3:C_4\times D_5$ x 2, $C_2^3:D_{20}$ x 2, $D_{10}.C_2^4$ x 2, $C_2^4:D_{10}$ x 2, $C_2^4:F_5$ x 2, $(C_4\times D_{10}):C_4$ x 2, $C_2\wr C_2^2\times C_5$, $(C_2\times C_{20}):D_4$, $C_2^4\times F_5$, $D_{20}:C_2^3$, $D_{10}.C_2^4$, $D_{10}.C_2^4$

Order 288: $D_6\times S_4$ x 22, $A_4\times S_4$ x 9, $\PSOPlus(4,3)$ x 4, $C_3\times C_2^3:A_4$ x 2, $(C_2\times D_6):A_4$ x 2, $S_3\times C_2^2:A_4$ x 2, $C_2\times A_4\times D_6$ x 2, $(C_2\times C_6):S_4$ x 2, $C_3\times C_2^2:S_4$ x 2, $C_2\times C_6:S_4$ x 2, $C_2\times C_6\times S_4$ x 2, $C_2\times A_4^2$ x 2, $C_3\times Q_8:A_4$, $D_6\times \SL(2,3)$, $A_4\times \SL(2,3)$, $C_2^5:C_3^2$

Order 280: $C_5\times F_8$

Order 256: $C_2^5:D_4$ x 4, $C_2^5:D_4$ x 4, $C_2^5:D_4$ x 2, $C_2^2:D_4^2$ x 2, $C_2^5:D_4$ x 2, $C_2^2.D_4^2$ x 2, $C_2^2.D_4^2$ x 2, $C_2^2.D_4^2$ x 2, $D_4^2:C_2^2$, $C_2^4.C_2^4$, $C_2^4.C_2^4$

Order 252: $C_{42}:C_6$, $A_4\times C_7:C_3$

Order 240: $C_2\times S_5$ x 11, $D_5\times S_4$ x 7, $C_2^2\times A_5$ x 6, $A_4\times F_5$ x 5, $A_4:F_5$ x 4, $C_4\times A_5$ x 2, $A_5:C_4$ x 2, $A_4\times D_{10}$ x 2, $C_{10}:S_4$ x 2, $C_{10}\times S_4$ x 2, $C_2^4:C_{15}$, $D_6\times F_5$, $D_5\times \SL(2,3)$

Order 224: $C_2^2\times F_8$ x 2, $C_4\times F_8$

Order 210: $C_{35}:C_6$

Order 192: $D_4\times S_4$ x 22, $C_2^4:D_6$ x 20, $C_2^4:D_6$ x 14, $C_2^3:C_4\times S_3$ x 14, $C_2^3:D_{12}$ x 14, $C_2^3:S_4$ x 12, $C_2^3\times S_4$ x 10, $C_2\times \GL(2,\mathbb{Z}/4)$ x 10, $C_2^3:S_4$ x 8, $D_{12}:C_2^3$ x 8, $C_2^3:S_4$ x 7, $C_3\times C_2\wr C_2^2$ x 6, $(C_2\times C_{12}):D_4$ x 6, $C_2^4:A_4$ x 6, $C_2^5:C_6$ x 5, $C_2^3:D_{12}$ x 5, $C_2^4.D_6$ x 5, $C_2.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^4:C_{12}$ x 3, $C_2.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^3.D_{12}$ x 3, $C_{12}:C_2^4$ x 3, $C_2^4:A_4$ x 3, $C_2^4:D_6$ x 3, $C_2^3:D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^4:C_{12}$ x 2, $C_2^4.D_6$ x 2, $C_2^2\wr C_3$ x 2, $D_{12}:C_2^3$ x 2, $D_4:S_4$ x 2, $\GL(2,\mathbb{Z}/4):C_2$ x 2, $C_2^5:C_6$ x 2, $C_2^4:D_6$ x 2, $C_2^4:C_{12}$, $D_6\times C_2^4$, $C_2^4:A_4$, $A_4\times C_2^4$, $C_{12}.C_2^4$, $D_{12}:C_2^3$, $D_{12}:C_2^3$, $C_2^4:C_{12}$, $A_4\times D_4:C_2$, $C_2^3.S_4$, $Q_8:S_4$, $Q_8\times S_4$, $D_4\times \SL(2,3)$

Order 180: $\GL(2,4)$

Order 168: $F_8:C_3$ x 2, $C_7\times S_4$, $C_3\times F_8$, $\PSL(2,7)$, $C_{28}:C_6$

Order 160: $D_4\times F_5$ x 11, $C_2^3:F_5$ x 9, $C_2^3\times F_5$ x 6, $C_{10}.C_4^2$ x 4, $D_4\times D_{10}$ x 3, $D_{10}.D_4$ x 3, $C_{10}:C_4^2$ x 3, $D_{10}:D_4$ x 3, $C_2^2.D_{20}$ x 3, $C_2^2:D_{20}$ x 3, $D_{10}.D_4$ x 3, $C_2^3:F_5$ x 2, $D_{10}.D_4$ x 2, $C_2^3:C_{20}$ x 2, $C_2^3.D_{10}$ x 2, $D_4:D_{10}$ x 2, $Q_8\times F_5$ x 2, $D_{10}.C_2^3$ x 2, $C_2^4:C_{10}$ x 2, $C_2^4:D_5$ x 2, $C_2^3\times D_{10}$, $C_{10}.C_2^4$, $D_4:D_{10}$, $D_4:D_{10}$

Order 120: $S_5$

Order 60: $A_5$

Order 9: $C_3^2$

Order 8: $C_2^3$

Order 7: $C_7$

Order 5: $C_5$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 161280: $S_5\times C_2^3:\GL(3,2)$ ($C_1$)

Order 80640: $(C_2^3\times A_5).\PSL(2,7)$ ($C_2$)

Order 1344: $C_2^3:\GL(3,2)$ ($S_5$)

Order 960: $C_2^3\times S_5$ ($\PSL(2,7)$)

Order 480: $C_2^3\times A_5$ ($C_2\times \GL(3,2)$)

Order 120: $S_5$ ($C_2^3:\GL(3,2)$)

Order 60: $A_5$ ($C_2^4:\GL(3,2)$)

Order 8: $C_2^3$ ($A_5.\PSL(2,7).C_2$)

Order 1: $C_1$ ($S_5\times C_2^3:\GL(3,2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 161280: $S_5\times C_2^3:\GL(3,2)$ ($C_1$)

Order 80640: $(C_2^3\times A_5).\PSL(2,7)$ ($C_2$)

Order 1344: $C_2^3:\GL(3,2)$ ($S_5$)

Order 960: $C_2^3\times S_5$ ($\PSL(2,7)$)

Order 480: $C_2^3\times A_5$ ($C_2\times \GL(3,2)$)

Order 120: $S_5$ ($C_2^3:\GL(3,2)$)

Order 60: $A_5$ ($C_2^4:\GL(3,2)$)

Order 8: $C_2^3$ ($A_5.\PSL(2,7).C_2$)

Order 1: $C_1$ ($S_5\times C_2^3:\GL(3,2)$)

Series

Derived series	$S_5\times C_2^3:\GL(3,2)$	$\rhd$	$(C_2^3\times A_5).\PSL(2,7)$
Chief series	$S_5\times C_2^3:\GL(3,2)$	$\rhd$	$(C_2^3\times A_5).\PSL(2,7)$	$\rhd$	$C_2^3\times A_5$	$\rhd$	$C_2^3$	$\rhd$	$C_1$
Lower central series	$S_5\times C_2^3:\GL(3,2)$	$\rhd$	$(C_2^3\times A_5).\PSL(2,7)$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

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

Character theory

Complex character table

See the $77 \times 77$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $70 \times 70$ rational character table.