Abstract group 53760.a: $C_2^4:\GL(3,2)\times F_5$

• Label: 53760.a
• Order: \( 2^{9} \cdot 3 \cdot 5 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $15$
• Trans deg.: $70$
• Rank: $2$

Group information

Description:	$C_2^4:\GL(3,2)\times F_5$
Order:	\(53760\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \cdot 7 \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_2\times (C_2^4\times F_5).\PSL(2,7)$, of order \(107520\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_5$, $\PSL(2,7)$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	10	12	14	15	20	28	30	35	70
Elements	1	1103	224	15280	4	5152	384	732	8960	4224	896	3360	7680	2688	1536	1536	53760
Conjugacy classes	1	15	1	40	1	7	2	7	8	6	1	6	8	3	2	2	110
Divisions	1	15	1	26	1	7	1	7	4	3	1	6	2	3	1	1	80
Autjugacy classes	1	11	1	18	1	7	2	5	4	6	1	4	4	3	2	2	72

Dimension	1	2	3	4	6	7	8	12	14	16	21	24	28	32	42	56	84
Irr. complex chars.	8	0	16	2	8	24	8	4	8	0	16	2	6	2	0	2	4	110
Irr. rational chars.	4	2	0	2	8	12	4	4	10	2	8	4	8	2	4	2	4	80

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$70$
Rank:	$2$
Inequivalent generating pairs:	$4104$

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 $15$ $\langle(4,6,5,7), (1,2)(4,5)(6,7), (3,6,4,5)(8,14)(9,13)(10,12)(11,15), (1,2)(3,6,4,5) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $F_5$ $\, \times\, $ $(C_2^3:\GL(3,2))$
Semidirect product:	$C_2^4$ $\,\rtimes\,$ $(F_5\times \GL(3,2))$	$(C_2^4\times F_5)$ $\,\rtimes\,$ $\PSL(2,7)$	$(D_5\times C_2^3:\GL(3,2))$ $\,\rtimes\,$ $C_4$	$D_5$ $\,\rtimes\,$ $(C_4\times C_2^3:\GL(3,2))$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_5$ . $(C_2^5:\GL(3,2))$	$D_{10}$ . $(C_2^4:\GL(3,2))$	$(C_2^3:\GL(3,2)\times D_{10})$ . $C_2$	$(D_5\times C_2^3:\GL(3,2))$ . $C_2^2$	all 6
Aut. group:	$\Aut(C_{15}:C_2^4)$	$\Aut(C_2^2\times C_6\times F_5)$

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 1005994 subgroups in 8062 conjugacy classes, 30 normal (24 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $F_5\times C_2^3:\GL(3,2)$
Commutator:	$G' \simeq$ $C_5\times C_2^3:\GL(3,2)$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^4:\GL(3,2)\times F_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_{10}$	$G/\operatorname{Fit} \simeq$ $C_4\times \GL(3,2)$
Radical:	$R \simeq$ $C_2^4\times F_5$	$G/R \simeq$ $\PSL(2,7)$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_4\times \GL(3,2)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.C_2^5$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 53760: $C_2^4:\GL(3,2)\times F_5$

Order 26880: $F_5\times C_2^3:\GL(3,2)$ x 2, $C_2^3:\GL(3,2)\times D_{10}$

Order 13440: $D_5\times C_2^3:\GL(3,2)$ x 2, $C_2^3:\GL(3,2)\times C_{10}$

Order 10752: $C_4\times C_2^4:\GL(3,2)$

Order 7680: $C_2^4:S_4\times F_5$, $C_2^4:S_4\times F_5$

Order 6720: $C_2\times F_5\times \GL(3,2)$ x 2, $F_5\times F_8:C_6$, $C_5\times C_2^3:\GL(3,2)$

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

Order 3840: $F_5\times C_2^3:S_4$ x 8, $F_5\times C_2^3:S_4$ x 4, $(C_2^3\times F_5):S_4$ x 2, $C_2^3:S_4\times F_5$ x 2, $(C_2^3\times D_{10}).S_4$, $C_2^3:S_4\times D_{10}$, $C_2^4:A_4\times F_5$, $(C_2^3\times D_{10}).S_4$, $C_2^4:A_4\times F_5$, $C_2^3:S_4\times D_{10}$

Order 3360: $F_5\times \GL(3,2)$ x 4, $F_5\times F_8:C_3$ x 2, $D_{10}\times \PSL(2,7)$ x 2, $D_5\times F_8:C_6$

Order 2688: $C_2^4:\GL(3,2)$ x 3

Order 2560: $C_2^4:D_4\times F_5$

Order 2240: $C_2\times F_5\times F_8$

Order 1920: $F_5\times C_2^2:S_4$ x 8, $D_5\times C_2^3:S_4$ x 8, $(C_2^2\times F_5):S_4$ x 8, $D_5\times C_2^3:S_4$ x 4, $C_2^3:A_4\times F_5$ x 4, $(C_2^2\times D_{10}).S_4$ x 4, $C_2^2\times F_5\times S_4$ x 3, $C_2^2:S_4\times D_{10}$ x 2, $(C_2^2\times D_{10}).S_4$ x 2, $F_5\times Q_8:A_4$ x 2, $(C_2^2\times D_{10}).S_4$ x 2, $(C_2^2\times D_{10}):S_4$ x 2, $C_2^3:A_4\times F_5$, $Q_8:A_4\times D_{10}$, $(C_2^3\times C_{10}):S_4$, $C_2^3:A_4\times D_{10}$, $C_2^5:D_{30}$, $(C_2^3\times D_{10}):C_{12}$, $C_2^3:S_4\times C_{10}$, $C_2^3:S_4\times C_{10}$

Order 1680: $D_5\times \GL(3,2)$ x 4, $D_5\times F_8:C_3$ x 2, $C_{10}\times \GL(3,2)$ x 2, $F_8:C_{30}$

Order 1536: $C_4\times C_2^4:S_4$, $C_4\times C_2^4:S_4$

Order 1344: $C_2\times C_4\times \GL(3,2)$ x 2, $C_2^3:\GL(3,2)$, $C_2\times F_8:C_{12}$

Order 1280: $C_2\wr C_2^2\times F_5$ x 16, $C_2^3:D_4\times F_5$ x 3, $C_2^5:(C_2\times F_5)$ x 3, $(D_4\times D_{10}).D_4$ x 3, $C_2^4:C_4\times F_5$ x 3, $D_{10}.C_2^6$, $(C_2^3\times F_5):C_2^3$, $C_2\wr C_2^2\times D_{10}$

Order 1120: $F_5\times F_8$ x 2, $D_{10}\times F_8$

Order 960: $C_2\times F_5\times S_4$ x 38, $(C_2\times D_{10}):S_4$ x 8, $D_5\times C_2^2:S_4$ x 8, $D_5\times C_2^3:A_4$ x 4, $C_2^4:D_{30}$ x 4, $C_5\times C_2^3:S_4$ x 4, $(C_2\times D_{10}).S_4$ x 4, $C_2\times D_{10}\times S_4$ x 3, $C_2^2\times A_4\times F_5$ x 3, $C_2^2\times A_4:F_5$ x 3, $(C_2^3\times C_{10}):C_{12}$ x 2, $C_2^4:D_{30}$ x 2, $C_2^2:S_4\times C_{10}$ x 2, $C_2^2:A_4\times F_5$ x 2, $D_5\times Q_8:A_4$ x 2, $(C_2^2\times C_{10}):S_4$ x 2, $C_5\times C_2^3:S_4$ x 2, $C_2^5:C_{30}$, $(C_2^2\times D_{10}):A_4$, $C_2^2:A_4\times D_{10}$, $C_{10}\times Q_8:A_4$, $C_2\times F_5\times \SL(2,3)$

Order 840: $C_5\times \PSL(2,7)$ x 2, $C_{70}:C_{12}$, $F_8:C_{15}$

Order 768: $C_4\times C_2^3:S_4$ x 8, $C_4\times C_2^3:S_4$ x 4, $C_2^6.D_6$ x 2, $C_2^6.D_6$ x 2, $C_2^5:S_4$, $C_2^5.S_4$, $C_2^5:S_4$, $C_2^5.S_4$, $C_4\times C_2^4:A_4$, $C_4\times C_2^4:A_4$

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

Order 640: $C_2^2\wr C_2\times F_5$ x 32, $C_2^4:(C_2\times F_5)$ x 24, $C_2^3:C_4\times F_5$ x 24, $(C_4\times D_{10}).D_4$ x 24, $C_2\wr C_2^2\times D_5$ x 16, $D_{10}.C_2^5$ x 10, $(C_2\times D_{20}).D_4$ x 8, $(C_2^3\times F_5):C_4$ x 6, $D_{10}.C_2^5$ x 5, $(D_4\times D_{10}):C_4$ x 5, $D_{10}.(C_4\times D_4)$ x 5, $C_2\times D_{10}:C_4^2$ x 5, $D_{10}.C_2^5$ x 3, $C_2^5:D_{10}$ x 3, $C_2^5:F_5$ x 3, $(D_4\times D_{10}):C_4$ x 3, $C_2^5:D_{10}$ x 3, $C_2^3:C_4\times D_{10}$ x 3, $C_2^4:D_{20}$ x 3, $F_5\times C_2^5$, $D_{20}:C_2^4$, $D_{10}.C_2^5$, $D_{10}.C_2^5$, $C_2\wr C_2^2\times C_{10}$, $(D_4\times C_{10}):D_4$

Order 560: $D_5\times F_8$ x 2, $C_{10}\times F_8$

Order 512: $C_2^4.C_2^5$

Order 480: $F_5\times S_4$ x 56, $D_{10}\times S_4$ x 38, $C_2\times A_4\times F_5$ x 20, $D_{10}.S_4$ x 20, $C_2^4:D_{15}$ x 4, $C_5\times C_2^2:S_4$ x 4, $C_2\times A_4\times D_{10}$ x 3, $C_2^3:D_{30}$ x 3, $C_2\times C_{10}\times S_4$ x 3, $F_5\times \SL(2,3)$ x 2, $C_2^4:C_{30}$ x 2, $(C_2\times D_{10}):A_4$ x 2, $D_5\times C_2^2:A_4$ x 2, $C_2^4:C_{30}$, $C_2\times D_6\times F_5$, $C_5\times Q_8:A_4$, $D_{10}\times \SL(2,3)$

Order 448: $C_2^4:C_{28}$

Order 420: $C_{35}:C_{12}$ x 2, $C_{70}:C_6$

Order 384: $C_2^4:S_4$ x 12, $C_2^5.D_6$ x 8, $C_2^5.D_6$ x 8, $C_2^4:S_4$ x 6, $C_4\times C_2^3:A_4$ x 4, $C_2^4.S_4$ x 4, $C_2^5.D_6$ x 3, $C_2^2\wr S_3$ x 2, $C_2^4.S_4$ x 2, $C_4\times Q_8:A_4$ x 2, $C_2^4.S_4$ x 2, $C_2^5:C_{12}$, $C_2^5:A_4$, $C_2^5:C_{12}$, $C_2^5:A_4$

Order 336: $C_2\times \GL(3,2)$ x 6, $F_8:C_6$ x 3

Order 320: $D_{10}.C_2^4$ x 80, $(C_2\times D_{20}):C_4$ x 34, $C_{10}.(C_4\times D_4)$ x 34, $D_{10}:C_4^2$ x 34, $C_2^4:D_{10}$ x 32, $D_{10}.C_2^4$ x 30, $C_2^4:D_{10}$ x 24, $C_2^3:C_4\times D_5$ x 24, $C_2^3:D_{20}$ x 24, $(C_2^2\times F_5):C_4$ x 24, $(C_4\times D_{10}):C_4$ x 18, $C_2^4\times F_5$ x 17, $D_{10}.C_2^4$ x 16, $D_{10}.C_2^4$ x 16, $C_2^4:F_5$ x 16, $C_2^4:F_5$ x 15, $D_{20}:C_2^3$ x 10, $C_2\wr C_2^2\times C_5$ x 8, $(C_2\times C_{20}):D_4$ x 8, $D_{10}.C_4^2$ x 7, $C_2^3.D_{20}$ x 6, $C_{20}:C_2^4$ x 5, $D_{10}.C_2^4$ x 5, $D_{10}:C_4^2$ x 5, $C_2^4:D_{10}$ x 5, $C_2^3:D_{20}$ x 5, $C_2^4.D_{10}$ x 5, $C_2^4:C_{20}$ x 3, $C_2^4.D_{10}$ x 3, $C_2^4:D_{10}$ x 3, $D_{10}.C_2^4$ x 3, $C_2^5:C_{10}$ x 3, $C_2^4:D_{10}$ x 3, $C_2^4:F_5$ x 3, $(C_2\times D_{20}):C_4$ x 3, $D_{10}.C_2^4$ x 2, $C_2^4\times D_{10}$, $C_{10}.C_2^5$, $D_{20}:C_2^3$, $D_{20}:C_2^3$

Order 280: $C_{14}\times F_5$, $C_5\times F_8$

Order 256: $C_2^4.C_2^4$ x 16, $C_2^3.C_2^5$ x 3, $C_2^5.D_4$ x 3, $C_2^5.D_4$ x 3, $C_2^4:C_4^2$ x 3, $C_4^2.C_2^4$, $C_2^5:D_4$, $C_2^5.D_4$

Order 240: $D_5\times S_4$ x 56, $A_4\times D_{10}$ x 20, $C_{10}:S_4$ x 20, $C_{10}\times S_4$ x 20, $A_4\times F_5$ x 16, $A_4:F_5$ x 16, $D_6\times F_5$ x 12, $C_2^3:C_{30}$ x 3, $D_5\times \SL(2,3)$ x 2, $C_2^4:C_{15}$, $D_6\times D_{10}$, $D_{10}.D_6$, $D_{10}:C_{12}$, $C_{10}\times \SL(2,3)$

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

Order 210: $C_{35}:C_6$ x 2, $C_7:C_{30}$

Order 192: $C_2^4.D_6$ x 38, $C_2^3:S_4$ x 16, $C_2^3:S_4$ x 12, $C_2^4:A_4$ x 6, $C_2^3.S_4$ x 4, $C_2^3:S_4$ x 4, $C_2^3\times S_4$ x 3, $C_2^4:A_4$ x 3, $C_2^4:C_{12}$ x 3, $C_2^3.S_4$ x 3, $C_2^4:C_{12}$ x 2, $C_2^4:C_{12}$ x 2, $C_2\times C_4\times \SL(2,3)$, $C_2^2\wr C_3$

Order 168: $\PSL(2,7)$ x 2, $F_8:C_3$, $C_{14}:C_{12}$

Order 160: $D_4\times F_5$ x 160, $C_2^3\times F_5$ x 111, $C_2^3:F_5$ x 111, $D_4\times D_{10}$ x 77, $D_{10}.D_4$ x 38, $C_{10}:C_4^2$ x 38, $D_{10}:D_4$ x 34, $C_2^2:D_{20}$ x 34, $D_{10}.D_4$ x 34, $D_4:D_{10}$ x 30, $C_{10}.C_4^2$ x 28, $D_{10}.C_2^3$ x 24, $C_2^2.D_{20}$ x 24, $C_2^3\times D_{10}$ x 17, $D_4:D_{10}$ x 16, $D_4:D_{10}$ x 16, $C_2^4:C_{10}$ x 16, $C_2^4:D_5$ x 16, $C_2^3:F_5$ x 12, $D_{10}.D_4$ x 12, $C_2^3:C_{20}$ x 12, $C_2^3.D_{10}$ x 12, $Q_8\times F_5$ x 12, $C_2^3:D_{10}$ x 10, $C_2^2.D_{20}$ x 7, $C_{10}.C_2^4$ x 5, $C_{20}:C_2^3$ x 5, $C_2^2\times D_{20}$ x 5, $C_{20}:C_2^3$ x 5, $C_2^3:C_{20}$ x 5, $C_2^3.D_{10}$ x 5, $D_4:D_{10}$ x 3, $D_{20}:C_2^2$ x 3, $C_2^4\times C_{10}$, $C_{10}.C_2^4$, $Q_8:D_{10}$, $Q_8\times D_{10}$

Order 140: $C_7\times F_5$ x 2, $C_7\times D_{10}$

Order 128: $C_2^3.C_2^4$ x 32, $C_2^4.D_4$ x 24, $C_2^4.D_4$ x 24, $C_2^3:C_4^2$ x 24, $C_2^4:D_4$ x 24, $C_4^2.C_2^3$ x 10, $C_2^4.D_4$ x 8, $C_2^3.C_4^2$ x 6, $C_2^5:C_4$ x 6, $C_4^2:C_2^3$ x 5, $C_2^4.D_4$ x 5, $C_2^4.D_4$ x 5, $C_2^3:C_4^2$ x 5, $C_2^4:D_4$ x 3, $C_4^2:C_2^3$ x 3, $C_2^5:C_4$ x 3, $D_4:C_2^4$, $C_2^5\times C_4$, $C_4^2.C_2^3$, $C_4^2.C_2^3$

Order 120: $D_5\times A_4$ x 16, $C_5:S_4$ x 16, $C_5\times S_4$ x 16, $S_3\times F_5$ x 16, $S_3\times D_{10}$ x 12, $C_{10}\times A_4$ x 11, $C_{30}:C_4$ x 6, $C_6\times F_5$ x 6, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $C_5\times \SL(2,3)$

Order 112: $C_2\times F_8$ x 3

Order 105: $C_7:C_{15}$

Order 96: $C_4\times S_4$ x 56, $C_2^2\times S_4$ x 44, $C_2^3:C_{12}$ x 20, $C_2^2.S_4$ x 20, $C_2^2:S_4$ x 8, $C_2^3:A_4$ x 4, $C_2^3:A_4$ x 3, $C_2^3\times A_4$ x 3, $C_4\times \SL(2,3)$ x 2, $C_{12}:C_2^3$, $Q_8:A_4$, $C_2^2\times \SL(2,3)$

Order 84: $C_7:C_{12}$ x 2, $C_{14}:C_6$

Order 80: $C_2^2\times F_5$ x 213, $D_4\times D_5$ x 136, $C_2^2\times D_{10}$ x 106, $D_{10}:C_4$ x 90, $C_{10}:D_4$ x 74, $C_4\times D_{10}$ x 38, $D_4\times C_{10}$ x 37, $C_2\times D_{20}$ x 37, $C_{20}:C_4$ x 32, $C_4\times F_5$ x 32, $D_{10}:C_4$ x 28, $D_4:D_5$ x 24, $D_{20}:C_2$ x 24, $C_2^2:C_{20}$ x 17, $C_{10}.D_4$ x 17, $C_2^3\times C_{10}$ x 10, $D_4:C_{10}$ x 8, $D_{20}:C_2$ x 8, $C_2^2\times C_{20}$ x 5, $C_2^2.D_{10}$ x 5, $Q_8\times D_5$ x 4, $Q_8\times C_{10}$, $C_{10}:Q_8$

Order 70: $C_7\times D_5$ x 2, $C_{70}$

Order 64: $C_4^2:C_2^2$ x 80, $C_2\wr C_2^2$ x 64, $C_2^4:C_4$ x 60, $C_2^3:D_4$ x 48, $C_2^3.D_4$ x 34, $C_2^3.D_4$ x 34, $C_2^2:C_4^2$ x 34, $C_4^2:C_2^2$ x 30, $C_2^2.C_4^2$ x 24, $C_2^4:C_4$ x 20, $C_2^4\times C_4$ x 17, $C_2^4:C_4$ x 16, $C_4^2.C_2^2$ x 16, $C_4^2.C_2^2$ x 16, $D_4:C_2^3$ x 15, $C_2^2.C_4^2$ x 7, $D_4\times C_2^3$ x 5, $C_2^3.D_4$ x 5, $C_2^2\times C_4^2$ x 5, $C_4^2:C_2^2$ x 3, $C_4^2.C_2^2$ x 2, $C_2^6$, $D_4:C_2^3$

Order 60: $S_3\times D_5$ x 16, $D_{30}$ x 6, $S_3\times C_{10}$ x 6, $C_3\times D_{10}$ x 6, $C_5\times A_4$ x 5, $C_{15}:C_4$ x 4, $C_3\times F_5$ x 4, $C_2\times C_{30}$

Order 56: $C_2\times C_{28}$, $F_8$

Order 48: $C_2\times S_4$ x 96, $C_2^2\times A_4$ x 23, $C_4\times A_4$ x 16, $A_4:C_4$ x 16, $C_4\times D_6$ x 12, $C_2\times \SL(2,3)$ x 3, $C_2^2\times D_6$, $C_2^2:A_4$, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 42: $C_7:C_6$ x 3

Order 40: $C_2\times D_{10}$ x 181, $C_2\times F_5$ x 104, $C_5:D_4$ x 56, $C_2^2\times C_{10}$ x 35, $C_4\times D_5$ x 32, $D_{20}$ x 28, $C_5\times D_4$ x 28, $C_2\times C_{20}$ x 18, $C_{10}:C_4$ x 18, $C_5:Q_8$ x 4, $C_5\times Q_8$ x 2

Order 35: $C_{35}$

Order 32: $C_2^3:C_4$ x 162, $C_4\times D_4$ x 160, $C_2^3\times C_4$ x 116, $C_2^2\wr C_2$ x 100, $C_2^2\times D_4$ x 97, $C_2^3:C_4$ x 72, $D_4:C_2^2$ x 51, $C_2^2.D_4$ x 38, $C_2\times C_4^2$ x 38, $C_2.C_4^2$ x 28, $D_4:C_2^2$ x 24, $C_4^2:C_2$ x 24, $C_2^5$ x 18, $C_4\times Q_8$ x 12, $C_2^2\times Q_8$

Order 30: $D_{15}$ x 4, $C_3\times D_5$ x 4, $C_5\times S_3$ x 4, $C_{30}$ x 3

Order 28: $C_{28}$ x 2, $C_2\times C_{14}$

Order 24: $S_4$ x 32, $C_2\times A_4$ x 27, $C_4\times S_3$ x 16, $C_2\times D_6$ x 14, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6, $\SL(2,3)$, $C_2^2\times C_6$

Order 21: $C_7:C_3$

Order 20: $D_{10}$ x 71, $C_2\times C_{10}$ x 29, $F_5$ x 14, $C_{20}$ x 6, $C_5:C_4$ x 6

Order 16: $C_2\times D_4$ x 284, $C_2^2\times C_4$ x 261, $C_2^2:C_4$ x 152, $C_2^4$ x 116, $D_4:C_2$ x 64, $C_4:C_4$ x 32, $C_4^2$ x 32, $C_2\times Q_8$ x 6

Order 15: $C_{15}$

Order 14: $C_{14}$ x 3

Order 12: $D_6$ x 28, $C_2\times C_6$ x 7, $A_4$ x 5, $C_{12}$ x 4, $C_3:C_4$ x 4

Order 10: $D_5$ x 8, $C_{10}$ x 7

Order 8: $C_2^3$ x 216, $C_2\times C_4$ x 172, $D_4$ x 112, $Q_8$ x 6

Order 7: $C_7$

Order 6: $S_3$ x 8, $C_6$ x 7

Order 5: $C_5$

Order 4: $C_2^2$ x 100, $C_4$ x 26

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 53760: $C_2^4:\GL(3,2)\times F_5$

Order 26880: $C_2^3:\GL(3,2)\times D_{10}$, $F_5\times C_2^3:\GL(3,2)$

Order 13440: $D_5\times C_2^3:\GL(3,2)$ x 2, $C_2^3:\GL(3,2)\times C_{10}$

Order 10752: $C_4\times C_2^4:\GL(3,2)$

Order 7680: $C_2^4:S_4\times F_5$, $C_2^4:S_4\times F_5$

Order 6720: $C_2\times F_5\times \GL(3,2)$, $F_5\times F_8:C_6$, $C_5\times C_2^3:\GL(3,2)$

Order 5376: $C_2^5:\GL(3,2)$, $C_4\times C_2^3:\GL(3,2)$

Order 3840: $F_5\times C_2^3:S_4$ x 3, $F_5\times C_2^3:S_4$ x 2, $(C_2^3\times D_{10}).S_4$, $C_2^3:S_4\times D_{10}$, $C_2^4:A_4\times F_5$, $(C_2^3\times D_{10}).S_4$, $(C_2^3\times F_5):S_4$, $C_2^4:A_4\times F_5$, $C_2^3:S_4\times F_5$, $C_2^3:S_4\times D_{10}$

Order 3360: $F_5\times F_8:C_3$, $F_5\times \GL(3,2)$, $D_5\times F_8:C_6$, $D_{10}\times \PSL(2,7)$

Order 2688: $C_2^4:\GL(3,2)$ x 3

Order 2560: $C_2^4:D_4\times F_5$

Order 2240: $C_2\times F_5\times F_8$

Order 1920: $D_5\times C_2^3:S_4$ x 6, $D_5\times C_2^3:S_4$ x 4, $F_5\times C_2^2:S_4$ x 2, $C_2^2\times F_5\times S_4$ x 2, $C_2^3:A_4\times F_5$ x 2, $(C_2^2\times D_{10}).S_4$ x 2, $(C_2^2\times F_5):S_4$ x 2, $C_2^2:S_4\times D_{10}$, $C_2^3:A_4\times F_5$, $(C_2^2\times D_{10}).S_4$, $Q_8:A_4\times D_{10}$, $(C_2^3\times C_{10}):S_4$, $F_5\times Q_8:A_4$, $(C_2^2\times D_{10}).S_4$, $C_2^3:A_4\times D_{10}$, $C_2^5:D_{30}$, $(C_2^2\times D_{10}):S_4$, $(C_2^3\times D_{10}):C_{12}$, $C_2^3:S_4\times C_{10}$, $C_2^3:S_4\times C_{10}$

Order 1680: $D_5\times F_8:C_3$ x 2, $D_5\times \GL(3,2)$ x 2, $F_8:C_{30}$, $C_{10}\times \GL(3,2)$

Order 1536: $C_4\times C_2^4:S_4$, $C_4\times C_2^4:S_4$

Order 1344: $C_2^3:\GL(3,2)$, $C_2\times F_8:C_{12}$, $C_2\times C_4\times \GL(3,2)$

Order 1280: $C_2\wr C_2^2\times F_5$ x 6, $C_2^3:D_4\times F_5$ x 2, $C_2^5:(C_2\times F_5)$ x 2, $(D_4\times D_{10}).D_4$ x 2, $C_2^4:C_4\times F_5$ x 2, $D_{10}.C_2^6$, $(C_2^3\times F_5):C_2^3$, $C_2\wr C_2^2\times D_{10}$

Order 1120: $D_{10}\times F_8$, $F_5\times F_8$

Order 960: $C_2\times F_5\times S_4$ x 14, $D_5\times C_2^3:A_4$ x 4, $(C_2\times D_{10}):S_4$ x 4, $D_5\times C_2^2:S_4$ x 4, $C_2^4:D_{30}$ x 3, $C_5\times C_2^3:S_4$ x 3, $C_2\times D_{10}\times S_4$ x 2, $C_2^2\times A_4\times F_5$ x 2, $C_2^2\times A_4:F_5$ x 2, $D_5\times Q_8:A_4$ x 2, $(C_2^2\times C_{10}):S_4$ x 2, $C_5\times C_2^3:S_4$ x 2, $C_2^5:C_{30}$, $(C_2^2\times D_{10}):A_4$, $(C_2^3\times C_{10}):C_{12}$, $C_2^2:A_4\times D_{10}$, $C_2^4:D_{30}$, $C_2^2:S_4\times C_{10}$, $C_2^2:A_4\times F_5$, $(C_2\times D_{10}).S_4$, $C_{10}\times Q_8:A_4$, $C_2\times F_5\times \SL(2,3)$

Order 840: $C_{70}:C_{12}$, $F_8:C_{15}$, $C_5\times \PSL(2,7)$

Order 768: $C_4\times C_2^3:S_4$ x 3, $C_4\times C_2^3:S_4$ x 2, $C_2^5:S_4$, $C_2^6.D_6$, $C_2^5.S_4$, $C_2^5:S_4$, $C_2^5.S_4$, $C_2^6.D_6$, $C_4\times C_2^4:A_4$, $C_4\times C_2^4:A_4$

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

Order 640: $C_2\wr C_2^2\times D_5$ x 12, $C_2^2\wr C_2\times F_5$ x 10, $C_2^4:(C_2\times F_5)$ x 8, $C_2^3:C_4\times F_5$ x 7, $(C_4\times D_{10}).D_4$ x 7, $D_{10}.C_2^5$ x 4, $(C_2\times D_{20}).D_4$ x 3, $(C_2^3\times F_5):C_4$ x 3, $D_{10}.C_2^5$ x 3, $(D_4\times D_{10}):C_4$ x 3, $D_{10}.(C_4\times D_4)$ x 3, $C_2\times D_{10}:C_4^2$ x 3, $D_{10}.C_2^5$ x 2, $C_2^5:D_{10}$ x 2, $C_2^5:F_5$ x 2, $(D_4\times D_{10}):C_4$ x 2, $C_2^5:D_{10}$ x 2, $C_2^3:C_4\times D_{10}$ x 2, $C_2^4:D_{20}$ x 2, $F_5\times C_2^5$, $D_{20}:C_2^4$, $D_{10}.C_2^5$, $D_{10}.C_2^5$, $C_2\wr C_2^2\times C_{10}$, $(D_4\times C_{10}):D_4$

Order 560: $D_5\times F_8$ x 2, $C_{10}\times F_8$

Order 512: $C_2^4.C_2^5$

Order 480: $D_{10}\times S_4$ x 21, $F_5\times S_4$ x 14, $C_2\times A_4\times F_5$ x 9, $D_{10}.S_4$ x 8, $C_2^4:C_{30}$ x 2, $(C_2\times D_{10}):A_4$ x 2, $D_5\times C_2^2:A_4$ x 2, $C_2\times A_4\times D_{10}$ x 2, $C_2^4:D_{15}$ x 2, $C_5\times C_2^2:S_4$ x 2, $C_2^3:D_{30}$ x 2, $C_2\times C_{10}\times S_4$ x 2, $F_5\times \SL(2,3)$, $C_2^4:C_{30}$, $C_2\times D_6\times F_5$, $C_5\times Q_8:A_4$, $D_{10}\times \SL(2,3)$

Order 448: $C_2^4:C_{28}$

Order 420: $C_{70}:C_6$, $C_{35}:C_{12}$

Order 384: $C_2^4:S_4$ x 9, $C_2^4:S_4$ x 6, $C_4\times C_2^3:A_4$ x 2, $C_2^4.S_4$ x 2, $C_2^5.D_6$ x 2, $C_2^5.D_6$ x 2, $C_2^5.D_6$ x 2, $C_2^5:C_{12}$, $C_2^2\wr S_3$, $C_2^5:A_4$, $C_2^5:C_{12}$, $C_2^4.S_4$, $C_4\times Q_8:A_4$, $C_2^5:A_4$, $C_2^4.S_4$

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

Order 320: $D_{10}.C_2^4$ x 28, $C_2^4:D_{10}$ x 20, $C_2^4:D_{10}$ x 14, $C_2^3:C_4\times D_5$ x 14, $C_2^3:D_{20}$ x 14, $(C_2\times D_{20}):C_4$ x 12, $D_{10}.C_2^4$ x 11, $C_{10}.(C_4\times D_4)$ x 10, $D_{10}:C_4^2$ x 10, $C_2^4\times F_5$ x 9, $C_2^4:F_5$ x 9, $D_{20}:C_2^3$ x 8, $(C_4\times D_{10}):C_4$ x 8, $C_2\wr C_2^2\times C_5$ x 6, $(C_2\times C_{20}):D_4$ x 6, $(C_2^2\times F_5):C_4$ x 6, $D_{10}.C_2^4$ x 6, $D_{10}.C_2^4$ x 6, $C_2^4:F_5$ x 5, $D_{10}.C_4^2$ x 4, $C_2^3.D_{20}$ x 3, $C_{20}:C_2^4$ x 3, $D_{10}.C_2^4$ x 3, $D_{10}:C_4^2$ x 3, $C_2^4:D_{10}$ x 3, $C_2^3:D_{20}$ x 3, $C_2^4.D_{10}$ x 3, $C_2^4:C_{20}$ x 2, $C_2^4.D_{10}$ x 2, $C_2^4:D_{10}$ x 2, $D_{10}.C_2^4$ x 2, $D_{10}.C_2^4$ x 2, $C_2^5:C_{10}$ x 2, $C_2^4:D_{10}$ x 2, $C_2^4:F_5$ x 2, $(C_2\times D_{20}):C_4$ x 2, $C_2^4\times D_{10}$, $C_{10}.C_2^5$, $D_{20}:C_2^3$, $D_{20}:C_2^3$

Order 280: $C_{14}\times F_5$, $C_5\times F_8$

Order 256: $C_2^4.C_2^4$ x 6, $C_2^3.C_2^5$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^4:C_4^2$ x 2, $C_4^2.C_2^4$, $C_2^5:D_4$, $C_2^5.D_4$

Order 240: $D_5\times S_4$ x 28, $A_4\times D_{10}$ x 13, $C_{10}:S_4$ x 11, $C_{10}\times S_4$ x 11, $D_6\times F_5$ x 5, $A_4\times F_5$ x 5, $A_4:F_5$ x 4, $C_2^3:C_{30}$ x 2, $D_5\times \SL(2,3)$ x 2, $C_2^4:C_{15}$, $D_6\times D_{10}$, $D_{10}.D_6$, $D_{10}:C_{12}$, $C_{10}\times \SL(2,3)$

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

Order 210: $C_{35}:C_6$ x 2, $C_7:C_{30}$

Order 192: $C_2^4.D_6$ x 14, $C_2^3:S_4$ x 10, $C_2^3:S_4$ x 6, $C_2^4:A_4$ x 6, $C_2^3:S_4$ x 4, $C_2^4:A_4$ x 3, $C_2^3\times S_4$ x 2, $C_2^4:C_{12}$ x 2, $C_2^3.S_4$ x 2, $C_2\times C_4\times \SL(2,3)$, $C_2^4:C_{12}$, $C_2^2\wr C_3$, $C_2^4:C_{12}$, $C_2^3.S_4$

Order 168: $F_8:C_3$, $\PSL(2,7)$, $C_{14}:C_{12}$

Order 160: $C_2^3\times F_5$ x 45, $D_4\times D_{10}$ x 43, $C_2^3:F_5$ x 43, $D_4\times F_5$ x 42, $D_{10}:D_4$ x 20, $C_2^2:D_{20}$ x 20, $D_{10}.D_4$ x 20, $D_4:D_{10}$ x 17, $D_{10}.D_4$ x 14, $C_{10}:C_4^2$ x 14, $C_2^3\times D_{10}$ x 12, $D_4:D_{10}$ x 12, $C_2^2.D_{20}$ x 12, $D_4:D_{10}$ x 10, $C_2^4:C_{10}$ x 10, $C_2^4:D_5$ x 10, $C_{10}.C_4^2$ x 8, $D_{10}.C_2^3$ x 8, $C_2^3:C_{20}$ x 7, $C_2^3.D_{10}$ x 7, $C_2^3:D_{10}$ x 6, $Q_8\times F_5$ x 5, $C_2^3:F_5$ x 4, $D_{10}.D_4$ x 4, $C_{10}.C_2^4$ x 4, $C_2^2.D_{20}$ x 4, $C_{20}:C_2^3$ x 3, $C_2^2\times D_{20}$ x 3, $C_{20}:C_2^3$ x 3, $C_2^3:C_{20}$ x 3, $C_2^3.D_{10}$ x 3, $D_4:D_{10}$ x 2, $D_{20}:C_2^2$ x 2, $C_2^4\times C_{10}$, $C_{10}.C_2^4$, $Q_8:D_{10}$, $Q_8\times D_{10}$

Order 140: $C_7\times D_{10}$, $C_7\times F_5$

Order 128: $C_2^4:D_4$ x 18, $C_2^3.C_2^4$ x 10, $C_2^4.D_4$ x 8, $C_2^4.D_4$ x 7, $C_2^3:C_4^2$ x 7, $C_4^2.C_2^3$ x 4, $C_2^5:C_4$ x 4, $C_2^4.D_4$ x 3, $C_2^3.C_4^2$ x 3, $C_4^2:C_2^3$ x 3, $C_2^4.D_4$ x 3, $C_2^4.D_4$ x 3, $C_2^3:C_4^2$ x 3, $C_2^4:D_4$ x 2, $C_4^2:C_2^3$ x 2, $C_2^5:C_4$ x 2, $D_4:C_2^4$, $C_2^5\times C_4$, $C_4^2.C_2^3$, $C_4^2.C_2^3$

Order 120: $D_5\times A_4$ x 10, $S_3\times D_{10}$ x 8, $C_5:S_4$ x 8, $C_5\times S_4$ x 8, $C_{10}\times A_4$ x 7, $C_6\times F_5$ x 4, $S_3\times F_5$ x 4, $C_{30}:C_4$ x 3, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $C_5\times \SL(2,3)$

Order 112: $C_2\times F_8$ x 3

Order 105: $C_7:C_{15}$

Order 96: $C_2^2\times S_4$ x 25, $C_4\times S_4$ x 14, $C_2^3:C_{12}$ x 9, $C_2^2.S_4$ x 8, $C_2^3:A_4$ x 4, $C_2^2:S_4$ x 4, $C_2^3:A_4$ x 3, $C_2^3\times A_4$ x 2, $C_4\times \SL(2,3)$, $C_{12}:C_2^3$, $Q_8:A_4$, $C_2^2\times \SL(2,3)$

Order 84: $C_{14}:C_6$, $C_7:C_{12}$

Order 80: $C_2^2\times F_5$ x 76, $D_4\times D_5$ x 70, $C_2^2\times D_{10}$ x 62, $C_{10}:D_4$ x 41, $D_{10}:C_4$ x 24, $C_4\times D_{10}$ x 23, $D_4\times C_{10}$ x 21, $C_2\times D_{20}$ x 21, $D_{10}:C_4$ x 16, $D_4:D_5$ x 14, $D_{20}:C_2$ x 14, $C_{20}:C_4$ x 10, $C_4\times F_5$ x 10, $C_2^2:C_{20}$ x 10, $C_{10}.D_4$ x 10, $C_2^3\times C_{10}$ x 7, $D_4:C_{10}$ x 6, $D_{20}:C_2$ x 6, $Q_8\times D_5$ x 4, $C_2^2\times C_{20}$ x 3, $C_2^2.D_{10}$ x 3, $Q_8\times C_{10}$, $C_{10}:Q_8$

Order 70: $C_7\times D_5$ x 2, $C_{70}$

Order 64: $C_2\wr C_2^2$ x 40, $C_2^4:C_4$ x 33, $C_2^3:D_4$ x 30, $C_4^2:C_2^2$ x 28, $C_2^3.D_4$ x 12, $D_4:C_2^3$ x 12, $C_2^4:C_4$ x 12, $C_4^2:C_2^2$ x 11, $C_2^3.D_4$ x 10, $C_2^2:C_4^2$ x 10, $C_2^4\times C_4$ x 9, $C_2^2.C_4^2$ x 6, $C_4^2.C_2^2$ x 6, $C_4^2.C_2^2$ x 6, $C_2^4:C_4$ x 5, $C_2^2.C_4^2$ x 4, $D_4\times C_2^3$ x 3, $C_2^3.D_4$ x 3, $C_2^2\times C_4^2$ x 3, $C_4^2.C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^6$, $D_4:C_2^3$

Order 60: $S_3\times D_5$ x 8, $C_3\times D_{10}$ x 6, $D_{30}$ x 4, $S_3\times C_{10}$ x 4, $C_5\times A_4$ x 3, $C_3\times F_5$ x 2, $C_{15}:C_4$, $C_2\times C_{30}$

Order 56: $C_2\times C_{28}$, $F_8$

Order 48: $C_2\times S_4$ x 50, $C_2^2\times A_4$ x 15, $C_4\times D_6$ x 5, $C_4\times A_4$ x 5, $A_4:C_4$ x 4, $C_2\times \SL(2,3)$ x 3, $C_2^2\times D_6$, $C_2^2:A_4$, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 42: $C_7:C_6$ x 3

Order 40: $C_2\times D_{10}$ x 101, $C_2\times F_5$ x 39, $C_5:D_4$ x 28, $C_2^2\times C_{10}$ x 21, $C_4\times D_5$ x 20, $D_{20}$ x 15, $C_5\times D_4$ x 15, $C_2\times C_{20}$ x 11, $C_{10}:C_4$ x 11, $C_5:Q_8$ x 3, $C_5\times Q_8$ x 2

Order 35: $C_{35}$

Order 32: $C_2^3:C_4$ x 73, $C_2^2\wr C_2$ x 60, $C_2^2\times D_4$ x 55, $C_2^3\times C_4$ x 48, $C_4\times D_4$ x 42, $C_2^3:C_4$ x 34, $D_4:C_2^2$ x 31, $D_4:C_2^2$ x 18, $C_2^2.D_4$ x 14, $C_2\times C_4^2$ x 14, $C_2^5$ x 13, $C_4^2:C_2$ x 8, $C_2.C_4^2$ x 8, $C_4\times Q_8$ x 5, $C_2^2\times Q_8$

Order 30: $C_3\times D_5$ x 4, $C_{30}$ x 3, $D_{15}$ x 2, $C_5\times S_3$ x 2

Order 28: $C_2\times C_{14}$, $C_{28}$

Order 24: $C_2\times A_4$ x 17, $S_4$ x 16, $C_2\times D_6$ x 10, $C_2\times C_{12}$ x 4, $C_4\times S_3$ x 4, $C_6:C_4$ x 3, $\SL(2,3)$, $C_2^2\times C_6$

Order 21: $C_7:C_3$

Order 20: $D_{10}$ x 41, $C_2\times C_{10}$ x 17, $F_5$ x 5, $C_{20}$ x 4, $C_5:C_4$ x 4

Order 16: $C_2\times D_4$ x 153, $C_2^2\times C_4$ x 105, $C_2^4$ x 69, $C_2^2:C_4$ x 60, $D_4:C_2$ x 40, $C_4:C_4$ x 10, $C_4^2$ x 10, $C_2\times Q_8$ x 6

Order 15: $C_{15}$

Order 14: $C_{14}$ x 3

Order 12: $D_6$ x 16, $C_2\times C_6$ x 7, $A_4$ x 3, $C_{12}$ x 2, $C_3:C_4$

Order 10: $D_5$ x 6, $C_{10}$ x 5

Order 8: $C_2^3$ x 122, $C_2\times C_4$ x 81, $D_4$ x 58, $Q_8$ x 5

Order 7: $C_7$

Order 6: $C_6$ x 7, $S_3$ x 4

Order 5: $C_5$

Order 4: $C_2^2$ x 58, $C_4$ x 13

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 53760: $C_2^4:\GL(3,2)\times F_5$ ($C_1$)

Order 26880: $F_5\times C_2^3:\GL(3,2)$ ($C_2$) x 2, $C_2^3:\GL(3,2)\times D_{10}$ ($C_2$)

Order 13440: $C_2^3:\GL(3,2)\times C_{10}$ ($C_4$), $D_5\times C_2^3:\GL(3,2)$ ($C_2^2$), $D_5\times C_2^3:\GL(3,2)$ ($C_4$)

Order 6720: $C_5\times C_2^3:\GL(3,2)$ ($C_2\times C_4$)

Order 2688: $C_2^4:\GL(3,2)$ ($F_5$)

Order 1344: $C_2^3:\GL(3,2)$ ($C_2\times F_5$)

Order 320: $C_2^4\times F_5$ ($\PSL(2,7)$)

Order 160: $C_2^3\times F_5$ ($C_2\times \GL(3,2)$) x 2, $C_2^3\times D_{10}$ ($C_2\times \GL(3,2)$)

Order 80: $C_2^3\times C_{10}$ ($C_4\times \GL(3,2)$), $C_2^2\times D_{10}$ ($C_2^2\times \GL(3,2)$), $C_2^2\times D_{10}$ ($C_4\times \GL(3,2)$)

Order 40: $C_2^2\times C_{10}$ ($C_2\times C_4\times \GL(3,2)$), $C_2\times F_5$ ($C_2^3:\GL(3,2)$)

Order 20: $F_5$ ($C_2^4:\GL(3,2)$) x 2, $D_{10}$ ($C_2^4:\GL(3,2)$)

Order 16: $C_2^4$ ($F_5\times \GL(3,2)$)

Order 10: $C_{10}$ ($C_4\times C_2^3:\GL(3,2)$), $D_5$ ($C_2^5:\GL(3,2)$), $D_5$ ($C_4\times C_2^3:\GL(3,2)$)

Order 8: $C_2^3$ ($C_2\times F_5\times \GL(3,2)$)

Order 5: $C_5$ ($C_4\times C_2^4:\GL(3,2)$)

Order 2: $C_2$ ($F_5\times C_2^3:\GL(3,2)$)

Order 1: $C_1$ ($C_2^4:\GL(3,2)\times F_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 53760: $C_2^4:\GL(3,2)\times F_5$ ($C_1$)

Order 26880: $C_2^3:\GL(3,2)\times D_{10}$ ($C_2$), $F_5\times C_2^3:\GL(3,2)$ ($C_2$)

Order 13440: $C_2^3:\GL(3,2)\times C_{10}$ ($C_4$), $D_5\times C_2^3:\GL(3,2)$ ($C_2^2$), $D_5\times C_2^3:\GL(3,2)$ ($C_4$)

Order 6720: $C_5\times C_2^3:\GL(3,2)$ ($C_2\times C_4$)

Order 2688: $C_2^4:\GL(3,2)$ ($F_5$)

Order 1344: $C_2^3:\GL(3,2)$ ($C_2\times F_5$)

Order 320: $C_2^4\times F_5$ ($\PSL(2,7)$)

Order 160: $C_2^3\times D_{10}$ ($C_2\times \GL(3,2)$), $C_2^3\times F_5$ ($C_2\times \GL(3,2)$)

Order 80: $C_2^3\times C_{10}$ ($C_4\times \GL(3,2)$), $C_2^2\times D_{10}$ ($C_2^2\times \GL(3,2)$), $C_2^2\times D_{10}$ ($C_4\times \GL(3,2)$)

Order 40: $C_2^2\times C_{10}$ ($C_2\times C_4\times \GL(3,2)$), $C_2\times F_5$ ($C_2^3:\GL(3,2)$)

Order 20: $D_{10}$ ($C_2^4:\GL(3,2)$), $F_5$ ($C_2^4:\GL(3,2)$)

Order 16: $C_2^4$ ($F_5\times \GL(3,2)$)

Order 10: $C_{10}$ ($C_4\times C_2^3:\GL(3,2)$), $D_5$ ($C_2^5:\GL(3,2)$), $D_5$ ($C_4\times C_2^3:\GL(3,2)$)

Order 8: $C_2^3$ ($C_2\times F_5\times \GL(3,2)$)

Order 5: $C_5$ ($C_4\times C_2^4:\GL(3,2)$)

Order 2: $C_2$ ($F_5\times C_2^3:\GL(3,2)$)

Order 1: $C_1$ ($C_2^4:\GL(3,2)\times F_5$)

Series

Derived series

$C_2^4:\GL(3,2)\times F_5$

$\rhd$

$C_2^4:\GL(3,2)\times F_5$

$\rhd$

$C_5\times C_2^3:\GL(3,2)$

$\rhd$

$C_5\times C_2^3:\GL(3,2)$

$\rhd$

$C_2^3:\GL(3,2)$

$\rhd$

$C_2^3:\GL(3,2)$

Chief series

$C_2^4:\GL(3,2)\times F_5$

$\rhd$

$C_2^4:\GL(3,2)\times F_5$

$\rhd$

$C_2^4\times F_5$

$\rhd$

$C_2^4\times F_5$

$\rhd$

$C_2^3\times D_{10}$

$\rhd$

$C_2^3\times D_{10}$

$\rhd$

$C_2\times F_5$

$\rhd$

$C_2^2\times D_{10}$

$\rhd$

$C_2^2\times D_{10}$

$\rhd$

$D_{10}$

$\rhd$

$D_5$

$\rhd$

$D_5$

$\rhd$

$C_5$

$\rhd$

$C_5$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_2^4:\GL(3,2)\times F_5$

$\rhd$

$C_2^4:\GL(3,2)\times F_5$

$\rhd$

$C_5\times C_2^3:\GL(3,2)$

$\rhd$

$C_5\times C_2^3:\GL(3,2)$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

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

See the $110 \times 110$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $80 \times 80$ rational character table.