Abstract group 80640.cj: $\GL(2,5)\times \GL(3,2)$

• Label: 80640.cj
• Order: \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $31$
• Trans deg.: $168$
• Rank: $2$

Group information

Description:	$\GL(2,5)\times \GL(3,2)$
Order:	\(80640\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \)
Exponent:	\(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_2^2\times \PSL(2,7).S_5.C_2$, of order \(161280\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 3, $A_5$, $\PSL(2,7)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	10	12	14	15	20	21	24	28	30	35	42	56	60	70	84	140	168
Elements	1	703	1196	11072	24	3716	48	2560	1032	14992	1488	1344	5088	960	11840	7296	1344	1152	960	1920	2688	1152	1920	2304	3840	80640
Conjugacy classes	1	5	3	24	1	6	2	6	3	17	4	1	8	2	18	14	1	2	2	4	2	2	4	4	8	144
Divisions	1	5	3	15	1	6	1	3	3	10	2	1	5	1	5	4	1	1	1	1	1	1	1	1	1	75
Autjugacy classes	1	5	3	12	1	6	1	3	3	9	2	1	5	1	5	3	1	1	1	1	1	1	1	1	1	70

Dimension	1	2	3	4	5	6	7	8	10	12	14	15	16	18	24	28	30	32	35	36	40	42	48	56	60	64	70	72	80	84	96	112	128
Irr. complex chars.	4	0	8	10	4	10	4	4	0	20	0	8	0	12	10	10	4	10	4	6	4	6	6	0	0	0	0	0	0	0	0	0	0	144
Irr. rational chars.	2	1	0	2	2	6	2	4	1	4	1	0	2	0	4	2	4	2	2	4	2	2	6	2	2	2	1	4	1	2	4	1	1	75

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$168$
Rank:	$2$
Inequivalent generating pairs:	$12996$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	48
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $31$ $\langle(1,3,5)(2,4,7), (1,2)(4,6), (8,10,14)(9,12,17)(11,16,15)(13,19,24)(18,22,21) \!\cdots\! \rangle$
Direct product:	$\PSL(2,7)$ $\, \times\, $ $\GL(2,5)$
Semidirect product:	$\SL(2,5)$ $\,\rtimes\,$ $(C_4\times \GL(3,2))$	$(\SL(2,5)\times \PSL(2,7))$ $\,\rtimes\,$ $C_4$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4\times \GL(3,2))$ . $S_5$	$C_4$ . $(S_5\times \GL(3,2))$	$(C_4.\PSL(2,7).A_5)$ . $C_2$	$C_2$ . $(\PSL(2,7)\times A_5:C_4)$	all 6
Aut. group:	$\Aut(C_2\times C_{10}^2)$

Elements of the group are displayed as permutations of degree 31.

Homology

Abelianization:	$C_{4} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 347281 subgroups in 1643 conjugacy classes, 12 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $S_5\times \GL(3,2)$
Commutator:	$G' \simeq$ $\SL(2,5)\times \PSL(2,7)$	$G/G' \simeq$ $C_4$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $S_5\times \GL(3,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_4$	$G/\operatorname{Fit} \simeq$ $S_5\times \GL(3,2)$
Radical:	$R \simeq$ $C_4$	$G/R \simeq$ $S_5\times \GL(3,2)$
Socle:	$\operatorname{soc} \simeq$ $C_2\times \GL(3,2)$	$G/\operatorname{soc} \simeq$ $A_5:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2:C_4$
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

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 80640: $\GL(2,5)\times \GL(3,2)$

Order 40320: $C_4.\PSL(2,7).A_5$

Order 20160: $\SL(2,5)\times \PSL(2,7)$

Order 16128: $C_4.S_4.\PSL(2,7)$

Order 13440: $(C_4\times F_5).\PSL(2,7)$

Order 11520: $S_4\times \GL(2,5)$ x 2

Order 10080: $(C_4\times C_7:C_3).S_5$

Order 8064: $C_{24}:C_2.\PSL(2,7)$, $\SL(2,3):C_2.\PSL(2,7)$

Order 6720: $C_2\times F_5\times \GL(3,2)$ x 2, $(C_4\times D_5).\PSL(2,7)$

Order 5760: $A_4\times \GL(2,5)$ x 2, $A_4:\GL(2,5)$ x 2, $S_4\times \SL(2,5):C_2$ x 2

Order 5376: $C_4\wr C_2\times \PSL(2,7)$

Order 5040: $(C_4\times C_7:C_3).A_5$

Order 4032: $C_3:C_8.\PSL(2,7)$, $C_{24}.\PSL(2,7)$, $C_4\times S_3\times \GL(3,2)$, $\SL(2,3)\times \PSL(2,7)$

Order 3840: $D_4\times \GL(2,5)$

Order 3360: $F_5\times \GL(3,2)$ x 4, $C_5:C_4.\PSL(2,7)$, $C_{28}.S_5$, $C_{20}\times \GL(3,2)$, $D_{10}\times \PSL(2,7)$

Order 2880: $S_4\times \SL(2,5)$ x 2, $\SL(2,5):S_4$ x 2, $A_4\times \SL(2,5):C_2$ x 2, $S_3\times \GL(2,5)$

Order 2688: $C_4^2\times \GL(3,2)$, $\OD_{16}\times \GL(3,2)$, $D_4:C_2\times \PSL(2,7)$

Order 2520: $(C_2\times C_7:C_3).A_5$

Order 2304: $C_4.S_4^2$ x 2

Order 2016: $Q_8.(C_6\times C_7:C_3).C_2$, $C_3:C_4\times \GL(3,2)$, $C_{12}\times \GL(3,2)$, $D_6\times \GL(3,2)$

Order 1920: $C_2^2\times \GL(2,5)$ x 2, $C_2^2:\GL(2,5)$ x 2, $C_4\times F_5\times S_4$ x 2, $D_4\times \SL(2,5):C_2$, $C_4:\GL(2,5)$, $C_4\times \GL(2,5)$

Order 1680: $D_5\times \GL(3,2)$ x 2, $C_{10}\times \GL(3,2)$, $C_{140}:C_{12}$, $\SL(2,5):C_{14}$

Order 1440: $A_4\times \SL(2,5)$ x 2, $\SL(2,5):D_6$, $C_3:\GL(2,5)$, $C_3\times \GL(2,5)$

Order 1344: $C_2\times C_4\times \GL(3,2)$ x 2, $C_8\times \GL(3,2)$, $Q_8\times \PSL(2,7)$, $D_4\times \GL(3,2)$

Order 1152: $S_4\times \SL(2,3):C_2$ x 2, $C_{24}:C_2\times S_4$ x 2, $A_4:\Unitary(2,3)$ x 2, $A_4\times \Unitary(2,3)$ x 2

Order 1008: $C_6\times \GL(3,2)$, $S_3\times \GL(3,2)$, $(C_7\times \SL(2,3)):C_6$, $C_{168}:C_6$

Order 960: $C_2\times \GL(2,5)$ x 4, $C_2\times F_5\times S_4$ x 4, $A_4:C_4\times F_5$ x 4, $\SL(2,5):D_4$ x 2, $C_4\times D_5\times S_4$ x 2, $C_4\times A_4\times F_5$ x 2, $C_4\times A_4:F_5$ x 2, $\SL(2,5):C_2^3$ x 2, $D_4\times \SL(2,5)$, $\SL(2,5):D_4$, $C_4^2.A_5$

Order 840: $C_{70}:C_{12}$ x 2, $C_{140}:C_6$, $C_5\times \PSL(2,7)$, $C_7\times \SL(2,5)$

Order 768: $C_4\wr C_2\times S_4$ x 2, $D_4\times \Unitary(2,3)$

Order 720: $\SL(2,5):C_6$, $\SL(2,5):S_3$, $S_3\times \SL(2,5)$

Order 672: $C_4\times \GL(3,2)$ x 4, $\SL(2,3):C_{28}$, $C_2^2\times \GL(3,2)$, $(C_4\times C_{28}):C_6$

Order 640: $D_{20}:C_4^2$

Order 576: $C_4.A_4^2$ x 2, $C_4\times S_3\times S_4$ x 2, $S_4\times \SL(2,3)$ x 2, $\SL(2,3):S_4$ x 2, $A_4\times C_{24}:C_2$ x 2, $C_{24}:S_4$ x 2, $C_{24}\times S_4$ x 2, $C_{12}.(C_2\times S_4)$ x 2, $C_6.(C_4\times S_4)$ x 2, $C_3:C_8\times S_4$ x 2, $S_3\times \Unitary(2,3)$

Order 560: $F_5\times C_{28}$

Order 504: $C_{84}:C_6$, $C_7:C_3\times \SL(2,3)$, $C_{21}:C_{24}$, $C_{21}:C_{24}$, $C_3\times \GL(3,2)$

Order 480: $F_5\times S_4$ x 16, $C_2\times A_4\times F_5$ x 4, $D_{10}.S_4$ x 4, $\SL(2,5):C_2^2$ x 3, $\GL(2,5)$ x 3, $D_{10}.S_4$ x 2, $C_5:(C_4\times S_4)$ x 2, $C_5:C_4\times S_4$ x 2, $C_2^2\times \SL(2,5)$ x 2, $C_4\times \SL(2,5)$ x 2, $D_{10}\times S_4$ x 2, $C_4\times D_5\times A_4$ x 2, $C_{20}:S_4$ x 2, $C_{20}\times S_4$ x 2, $D_{15}:C_4^2$

Order 420: $C_{35}:C_{12}$ x 4, $C_7:C_{60}$, $C_{35}:C_{12}$, $C_{70}:C_6$

Order 384: $A_4\times C_4\wr C_2$ x 2, $C_4.\GL(2,\mathbb{Z}/4)$ x 2, $C_4.\GL(2,\mathbb{Z}/4)$ x 2, $\OD_{16}:S_4$ x 2, $C_4^2:S_4$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_2^2:\Unitary(2,3)$ x 2, $C_2^2\times \Unitary(2,3)$ x 2, $\OD_{16}\times S_4$ x 2, $C_4^2\times S_4$ x 2, $C_4^2.S_4$, $\SL(2,3):C_4^2$, $D_4\times \SL(2,3):C_2$, $C_3\times D_4\times \OD_{16}$

Order 360: $C_3\times \SL(2,5)$

Order 336: $C_2\times \GL(3,2)$ x 2, $C_{168}:C_2$, $C_{56}:C_6$, $C_{28}:C_{12}$, $C_{28}.A_4$, $\SL(2,3):C_{14}$, $(C_2\times C_{28}):C_6$

Order 320: $D_{10}:C_4^2$ x 4, $D_{10}.C_2^4$ x 2, $D_{10}:C_4^2$ x 2, $D_{10}:C_4^2$ x 2, $C_{20}:C_4^2$ x 2, $C_4^2:D_{10}$, $C_{20}:C_4^2$, $C_4^2\times F_5$

Order 288: $C_4\times S_3\times A_4$ x 2, $C_{12}:S_4$ x 2, $C_{12}\times S_4$ x 2, $A_4\times \SL(2,3)$ x 2, $D_6.S_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $A_4\times C_{24}$ x 2, $A_4\times C_3:C_8$ x 2, $C_{12}.S_4$ x 2, $A_4:C_{24}$ x 2, $D_6\times S_4$ x 2, $\SL(2,3):D_6$, $C_{24}:D_6$, $C_{12}.S_4$, $\SL(2,3):C_{12}$

Order 280: $C_{14}\times F_5$ x 2, $D_5\times C_{28}$

Order 256: $D_4^2:C_4$

Order 252: $C_{42}:C_6$, $C_{21}:C_{12}$, $C_{21}:C_{12}$

Order 240: $D_5\times S_4$ x 8, $A_4\times F_5$ x 8, $A_4:F_5$ x 8, $C_{15}:C_4^2$ x 2, $\SL(2,5):C_2$ x 2, $A_4\times D_{10}$ x 2, $C_{10}:S_4$ x 2, $C_{10}\times S_4$ x 2, $D_6\times F_5$ x 2, $A_4\times C_{20}$ x 2, $A_4\times C_5:C_4$ x 2, $C_{10}.S_4$ x 2, $A_4:C_{20}$ x 2, $C_2\times \SL(2,5)$, $C_{12}:D_{10}$, $C_{60}:C_4$, $C_{12}\times F_5$

Order 224: $D_4:C_{28}$

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

Order 192: $C_2\times \Unitary(2,3)$ x 4, $A_4:C_4^2$ x 4, $C_8:S_4$ x 4, $C_2^4.D_6$ x 4, $A_4\times C_4^2$ x 2, $A_4:\OD_{16}$ x 2, $C_8\times S_4$ x 2, $C_{24}:D_4$ x 2, $(C_2\times C_6):\OD_{16}$ x 2, $D_6:\OD_{16}$ x 2, $Q_8.(C_2\times A_4)$ x 2, $A_4\times D_4:C_2$ x 2, $\SL(2,3):C_2^3$ x 2, $Q_8:S_4$ x 2, $Q_8\times S_4$ x 2, $D_4:S_4$ x 2, $D_4\times S_4$ x 2, $\GL(2,\mathbb{Z}/4):C_2$ x 2, $C_{24}:C_2^3$ x 2, $A_4\times \OD_{16}$ x 2, $\SL(2,3):D_4$ x 2, $C_4^2.A_4$, $D_4\times C_{24}$, $C_4^2.D_6$, $C_{12}:\OD_{16}$, $C_{12}:\OD_{16}$, $C_4^2:D_6$, $C_8:D_{12}$, $C_{12}:\OD_{16}$, $C_4^2:D_6$, $\SL(2,3):D_4$, $D_4\times \SL(2,3)$

Order 168: $C_7:C_{24}$ x 2, $C_{14}:C_{12}$ x 2, $C_{168}$, $\PSL(2,7)$, $S_3\times C_{28}$, $C_3:C_{56}$, $C_7:\SL(2,3)$, $C_7\times \SL(2,3)$, $C_{28}.C_6$, $C_{28}:C_6$

Order 160: $D_4\times F_5$ x 16, $C_{10}:C_4^2$ x 12, $C_{10}.C_4^2$ x 6, $C_2^3\times F_5$ x 4, $C_2^3:F_5$ x 4, $C_{20}:C_2^3$ x 2, $D_{10}.D_4$ x 2, $C_{20}:D_4$ x 2, $C_2^3.D_{10}$ x 2, $D_{10}.D_4$ x 2, $C_4\times D_{20}$, $D_5\times C_4^2$, $D_4\times D_{10}$, $D_4\times C_{20}$, $C_2^3.D_{10}$, $D_{20}:C_4$, $D_{10}.D_4$

Order 144: $S_3\times S_4$ x 4, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $C_6\times S_4$ x 2, $C_{12}\times A_4$ x 2, $A_4\times C_3:C_4$ x 2, $C_6.S_4$ x 2, $A_4:C_{12}$ x 2, $C_{24}:S_3$, $C_{24}:C_6$, $S_3\times C_{24}$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $\SL(2,3):C_6$, $C_4\times S_3^2$, $S_3\times \SL(2,3)$, $\SL(2,3):S_3$

Order 140: $C_7\times F_5$ x 4, $C_7\times D_{10}$, $C_{140}$, $C_5:C_{28}$

Order 128: $C_4^2:D_4$ x 2, $\OD_{16}:D_4$ x 2, $C_2^4.D_4$ x 2, $C_4^2:C_2^3$ x 2, $\OD_{16}:D_4$, $C_4^3:C_2$, $C_4^2.D_4$, $D_4:C_4^2$, $D_4^2:C_2$, $D_4\times \OD_{16}$, $D_4\times C_4^2$

Order 126: $C_{21}:C_6$, $C_{21}:C_6$

Order 120: $S_3\times F_5$ x 8, $D_5\times A_4$ x 4, $C_5:S_4$ x 4, $C_5\times S_4$ x 4, $C_{10}\times A_4$ x 2, $C_{30}:C_4$ x 2, $C_6\times F_5$ x 2, $C_6.D_{10}$, $C_6.D_{10}$, $\SL(2,5)$, $S_3\times D_{10}$, $C_4\times D_{15}$, $S_3\times C_{20}$, $D_5\times C_{12}$, $D_{15}:C_4$

Order 112: $D_4:C_{14}$, $C_7\times \OD_{16}$, $C_4\times C_{28}$

Order 105: $C_7:C_{15}$

Order 96: $C_4\times S_4$ x 18, $C_8:D_6$ x 8, $\Unitary(2,3)$ x 6, $C_8\times A_4$ x 4, $A_4:C_8$ x 4, $D_6:C_8$ x 4, $C_2^3:C_{12}$ x 4, $C_2^2.S_4$ x 4, $\SL(2,3):C_2^2$ x 3, $C_2^3.D_6$ x 2, $D_6.D_4$ x 2, $S_3\times C_4^2$ x 2, $C_4\times \SL(2,3)$ x 2, $C_2^2:C_{24}$ x 2, $C_{12}.D_4$ x 2, $C_2^2\times S_4$ x 2, $C_{24}:C_4$ x 2, $C_{12}.D_4$ x 2, $C_{12}:C_2^3$ x 2, $Q_8:A_4$ x 2, $Q_8\times A_4$ x 2, $C_2^2\times \SL(2,3)$ x 2, $D_4\times A_4$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $C_4:S_4$ x 2, $A_4:Q_8$ x 2, $C_2^2\times C_{24}$ x 2, $C_{12}:D_4$ x 2, $C_{12}.C_2^3$ x 2, $D_6.D_4$, $C_{12}:C_8$, $C_4\times D_{12}$, $C_4:C_{24}$, $D_4:C_{12}$, $C_4\times C_{24}$, $C_{12}.D_4$, $D_{12}:C_4$, $D_4:D_6$, $D_4\times D_6$, $D_4\times C_{12}$, $C_2^3.D_6$, $D_{12}:C_4$, $S_3\times \OD_{16}$, $C_{12}:C_8$, $D_{12}:C_4$

Order 84: $C_7:C_{12}$ x 5, $C_{14}:C_6$, $C_{84}$, $C_3:C_{28}$, $S_3\times C_{14}$

Order 80: $C_2^2\times F_5$ x 24, $C_4\times F_5$ x 20, $D_{10}:C_4$ x 16, $C_4\times D_{10}$ x 9, $D_4\times D_5$ x 8, $C_{20}:C_4$ x 8, $D_{10}:C_4$ x 4, $C_{20}:C_4$ x 3, $C_2^2\times D_{10}$ x 2, $C_2^2\times C_{20}$ x 2, $C_{10}:D_4$ x 2, $C_2^2.D_{10}$ x 2, $C_2^2:C_{20}$ x 2, $C_{10}.D_4$ x 2, $C_{10}.D_4$ x 2, $D_4\times C_{10}$, $C_2\times D_{20}$, $C_4:C_{20}$, $C_4\times C_{20}$, $C_{20}:C_4$

Order 72: $C_3\times S_4$ x 4, $C_6\times A_4$ x 2, $S_3\times A_4$ x 2, $C_3:S_4$ x 2, $S_3\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_3:C_{24}$ x 2, $S_3\times D_6$, $C_{12}:S_3$, $C_3\times \SL(2,3)$, $C_6.D_6$, $C_3\times C_{24}$, $C_3^2:C_8$

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

Order 64: $C_4^2:C_2^2$ x 12, $C_2^2:C_4^2$ x 4, $C_4^2:C_4$ x 4, $C_2^2:\OD_{16}$ x 2, $C_4:C_4^2$ x 2, $D_4:C_2^3$ x 2, $C_2^2\times \OD_{16}$ x 2, $Q_8:D_4$ x 2, $D_4:D_4$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^2\times C_4^2$ x 2, $C_8:D_4$ x 2, $C_4\times \OD_{16}$, $C_4^3$, $Q_8:D_4$, $D_4\times Q_8$, $D_4:D_4$, $D_4^2$, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $C_8:D_4$, $C_8\times D_4$, $C_4:\OD_{16}$

Order 63: $C_{21}:C_3$

Order 60: $S_3\times D_5$ x 4, $C_{15}:C_4$ x 4, $C_3\times F_5$ x 4, $C_5\times A_4$ x 2, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $D_{30}$, $S_3\times C_{10}$, $C_3\times D_{10}$, $C_3:C_{20}$

Order 56: $C_2\times C_{28}$ x 2, $C_7\times D_4$, $C_{56}$, $C_7\times Q_8$

Order 48: $C_4\times D_6$ x 11, $C_{24}:C_2$ x 10, $C_2\times S_4$ x 10, $C_4\times A_4$ x 10, $A_4:C_4$ x 10, $S_3\times D_4$ x 5, $\SL(2,3):C_2$ x 5, $C_{12}:C_4$ x 5, $C_6:C_8$ x 4, $C_2\times C_{24}$ x 4, $D_6:C_4$ x 4, $C_2^2\times D_6$ x 2, $C_2^2\times A_4$ x 2, $C_2^2\times C_{12}$ x 2, $C_6:D_4$ x 2, $C_6.C_2^3$ x 2, $C_2^2:C_{12}$ x 2, $C_4\times C_{12}$ x 2, $C_6.D_4$ x 2, $C_6.D_4$ x 2, $D_4:C_6$, $C_6\times D_4$, $D_{12}:C_2$, $S_3\times Q_8$, $S_3\times C_8$, $D_4:S_3$, $D_{12}:C_2$, $C_2\times D_{12}$, $C_2\times \SL(2,3)$, $C_3\times \OD_{16}$, $C_4:C_{12}$, $C_{12}:C_4$, $C_3:\OD_{16}$

Order 42: $C_7:C_6$ x 3, $C_{42}$, $S_3\times C_7$

Order 40: $C_2\times F_5$ x 32, $C_4\times D_5$ x 12, $C_2\times D_{10}$ x 11, $C_5:D_4$ x 8, $C_2\times C_{20}$ x 5, $C_{10}:C_4$ x 5, $D_{20}$ x 4, $C_5\times D_4$ x 4, $C_2^2\times C_{10}$ x 2

Order 36: $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_3\times A_4$ x 2, $S_3^2$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $C_6:S_3$

Order 35: $C_{35}$

Order 32: $C_4\times D_4$ x 19, $C_4\wr C_2$ x 12, $C_2\times C_4^2$ x 9, $D_4:C_2^2$ x 7, $C_2\times \OD_{16}$ x 6, $C_4:D_4$ x 6, $C_2^2:C_8$ x 4, $C_2^3\times C_4$ x 4, $C_2^2:Q_8$ x 4, $C_2^3:C_4$ x 4, $C_2^2\times D_4$ x 3, $C_2.C_4^2$ x 3, $C_2^2\times Q_8$ x 2, $C_2^2\times C_8$ x 2, $C_4^2:C_2$ x 2, $C_2^2.D_4$ x 2, $C_2^2\wr C_2$ x 2, $C_4\times Q_8$ x 2, $C_2^2.D_4$ x 2, $C_4:C_8$ x 2, $C_8:C_4$, $C_4:Q_8$, $C_4:D_4$, $C_4\times C_8$, $C_4^2:C_2$

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

Order 28: $C_{28}$ x 4, $C_2\times C_{14}$

Order 24: $C_4\times S_3$ x 18, $C_2\times D_6$ x 8, $S_4$ x 8, $C_2\times C_{12}$ x 7, $C_6:C_4$ x 7, $C_2\times A_4$ x 6, $C_3:D_4$ x 5, $C_{24}$ x 5, $C_3\times D_4$ x 5, $C_3:C_8$ x 5, $\SL(2,3)$ x 4, $D_{12}$ x 3, $C_2^2\times C_6$ x 2, $C_3:Q_8$, $C_3\times Q_8$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 20: $D_{10}$ x 12, $F_5$ x 12, $C_2\times C_{10}$ x 5, $C_{20}$ x 5, $C_5:C_4$ x 5

Order 18: $C_3\times S_3$ x 3, $C_3\times C_6$, $C_3:S_3$

Order 16: $C_2^2\times C_4$ x 25, $C_2^2:C_4$ x 19, $C_4^2$ x 17, $C_2\times D_4$ x 14, $C_4:C_4$ x 11, $D_4:C_2$ x 10, $\OD_{16}$ x 7, $C_2\times C_8$ x 4, $C_2\times Q_8$ x 3, $C_2^4$ x 2

Order 15: $C_{15}$

Order 14: $C_{14}$ x 2

Order 12: $D_6$ x 13, $C_{12}$ x 10, $C_3:C_4$ x 9, $C_2\times C_6$ x 6, $A_4$ x 4

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 35, $D_4$ x 16, $C_2^3$ x 9, $Q_8$ x 4, $C_8$ x 3

Order 7: $C_7$

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

Order 5: $C_5$

Order 4: $C_4$ x 15, $C_2^2$ x 13

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 80640: $\GL(2,5)\times \GL(3,2)$

Order 40320: $C_4.\PSL(2,7).A_5$

Order 20160: $\SL(2,5)\times \PSL(2,7)$

Order 16128: $C_4.S_4.\PSL(2,7)$

Order 13440: $(C_4\times F_5).\PSL(2,7)$

Order 11520: $S_4\times \GL(2,5)$

Order 10080: $(C_4\times C_7:C_3).S_5$

Order 8064: $C_{24}:C_2.\PSL(2,7)$, $\SL(2,3):C_2.\PSL(2,7)$

Order 6720: $(C_4\times D_5).\PSL(2,7)$, $C_2\times F_5\times \GL(3,2)$

Order 5760: $A_4\times \GL(2,5)$, $A_4:\GL(2,5)$, $S_4\times \SL(2,5):C_2$

Order 5376: $C_4\wr C_2\times \PSL(2,7)$

Order 5040: $(C_4\times C_7:C_3).A_5$

Order 4032: $C_3:C_8.\PSL(2,7)$, $C_{24}.\PSL(2,7)$, $C_4\times S_3\times \GL(3,2)$, $\SL(2,3)\times \PSL(2,7)$

Order 3840: $D_4\times \GL(2,5)$

Order 3360: $C_5:C_4.\PSL(2,7)$, $C_{28}.S_5$, $F_5\times \GL(3,2)$, $C_{20}\times \GL(3,2)$, $D_{10}\times \PSL(2,7)$

Order 2880: $S_4\times \SL(2,5)$, $\SL(2,5):S_4$, $A_4\times \SL(2,5):C_2$, $S_3\times \GL(2,5)$

Order 2688: $C_4^2\times \GL(3,2)$, $\OD_{16}\times \GL(3,2)$, $D_4:C_2\times \PSL(2,7)$

Order 2520: $(C_2\times C_7:C_3).A_5$

Order 2304: $C_4.S_4^2$

Order 2016: $Q_8.(C_6\times C_7:C_3).C_2$, $C_3:C_4\times \GL(3,2)$, $C_{12}\times \GL(3,2)$, $D_6\times \GL(3,2)$

Order 1920: $D_4\times \SL(2,5):C_2$, $C_2^2\times \GL(2,5)$, $C_4:\GL(2,5)$, $C_2^2:\GL(2,5)$, $C_4\times \GL(2,5)$, $C_4\times F_5\times S_4$

Order 1680: $C_{10}\times \GL(3,2)$, $D_5\times \GL(3,2)$, $C_{140}:C_{12}$, $\SL(2,5):C_{14}$

Order 1440: $\SL(2,5):D_6$, $A_4\times \SL(2,5)$, $C_3:\GL(2,5)$, $C_3\times \GL(2,5)$

Order 1344: $C_2\times C_4\times \GL(3,2)$ x 2, $C_8\times \GL(3,2)$, $Q_8\times \PSL(2,7)$, $D_4\times \GL(3,2)$

Order 1152: $S_4\times \SL(2,3):C_2$, $C_{24}:C_2\times S_4$, $A_4:\Unitary(2,3)$, $A_4\times \Unitary(2,3)$

Order 1008: $C_6\times \GL(3,2)$, $S_3\times \GL(3,2)$, $(C_7\times \SL(2,3)):C_6$, $C_{168}:C_6$

Order 960: $C_2\times \GL(2,5)$ x 3, $D_4\times \SL(2,5)$, $\SL(2,5):D_4$, $\SL(2,5):D_4$, $C_4^2.A_5$, $C_2\times F_5\times S_4$, $C_4\times D_5\times S_4$, $C_4\times A_4\times F_5$, $C_4\times A_4:F_5$, $A_4:C_4\times F_5$, $\SL(2,5):C_2^3$

Order 840: $C_{140}:C_6$, $C_{70}:C_{12}$, $C_5\times \PSL(2,7)$, $C_7\times \SL(2,5)$

Order 768: $D_4\times \Unitary(2,3)$, $C_4\wr C_2\times S_4$

Order 720: $\SL(2,5):C_6$, $\SL(2,5):S_3$, $S_3\times \SL(2,5)$

Order 672: $C_4\times \GL(3,2)$ x 3, $\SL(2,3):C_{28}$, $C_2^2\times \GL(3,2)$, $(C_4\times C_{28}):C_6$

Order 640: $D_{20}:C_4^2$

Order 576: $C_4.A_4^2$, $C_4\times S_3\times S_4$, $S_4\times \SL(2,3)$, $\SL(2,3):S_4$, $A_4\times C_{24}:C_2$, $C_{24}:S_4$, $C_{24}\times S_4$, $S_3\times \Unitary(2,3)$, $C_{12}.(C_2\times S_4)$, $C_6.(C_4\times S_4)$, $C_3:C_8\times S_4$

Order 560: $F_5\times C_{28}$

Order 504: $C_{84}:C_6$, $C_7:C_3\times \SL(2,3)$, $C_{21}:C_{24}$, $C_{21}:C_{24}$, $C_3\times \GL(3,2)$

Order 480: $\GL(2,5)$ x 3, $\SL(2,5):C_2^2$ x 2, $C_4\times \SL(2,5)$ x 2, $F_5\times S_4$ x 2, $D_{15}:C_4^2$, $D_{10}.S_4$, $C_5:(C_4\times S_4)$, $C_5:C_4\times S_4$, $C_2^2\times \SL(2,5)$, $D_{10}\times S_4$, $C_2\times A_4\times F_5$, $D_{10}.S_4$, $C_4\times D_5\times A_4$, $C_{20}:S_4$, $C_{20}\times S_4$

Order 420: $C_7:C_{60}$, $C_{35}:C_{12}$, $C_{70}:C_6$, $C_{35}:C_{12}$

Order 384: $A_4\times C_4\wr C_2$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_4.\GL(2,\mathbb{Z}/4)$, $\OD_{16}:S_4$, $C_4^2.S_4$, $\SL(2,3):C_4^2$, $C_4^2:S_4$, $\GL(2,\mathbb{Z}/4):C_2^2$, $D_4\times \SL(2,3):C_2$, $C_2^2:\Unitary(2,3)$, $C_2^2\times \Unitary(2,3)$, $\OD_{16}\times S_4$, $C_4^2\times S_4$, $C_3\times D_4\times \OD_{16}$

Order 360: $C_3\times \SL(2,5)$

Order 336: $C_2\times \GL(3,2)$ x 2, $C_{168}:C_2$, $C_{56}:C_6$, $C_{28}:C_{12}$, $C_{28}.A_4$, $\SL(2,3):C_{14}$, $(C_2\times C_{28}):C_6$

Order 320: $D_{10}.C_2^4$, $D_{10}:C_4^2$, $C_4^2:D_{10}$, $D_{10}:C_4^2$, $C_{20}:C_4^2$, $D_{10}:C_4^2$, $C_{20}:C_4^2$, $C_4^2\times F_5$

Order 288: $\SL(2,3):D_6$, $C_4\times S_3\times A_4$, $C_{12}:S_4$, $C_{12}\times S_4$, $A_4\times \SL(2,3)$, $D_6.S_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $A_4\times C_{24}$, $C_{24}:D_6$, $A_4\times C_3:C_8$, $C_{12}.S_4$, $C_{12}.S_4$, $\SL(2,3):C_{12}$, $A_4:C_{24}$, $D_6\times S_4$

Order 280: $C_{14}\times F_5$, $D_5\times C_{28}$

Order 256: $D_4^2:C_4$

Order 252: $C_{42}:C_6$, $C_{21}:C_{12}$, $C_{21}:C_{12}$

Order 240: $\SL(2,5):C_2$ x 2, $D_5\times S_4$ x 2, $C_{15}:C_4^2$, $C_2\times \SL(2,5)$, $A_4\times D_{10}$, $C_{10}:S_4$, $C_{10}\times S_4$, $D_6\times F_5$, $A_4\times F_5$, $A_4:F_5$, $A_4\times C_{20}$, $C_{12}:D_{10}$, $C_{60}:C_4$, $C_{12}\times F_5$, $A_4\times C_5:C_4$, $C_{10}.S_4$, $A_4:C_{20}$

Order 224: $D_4:C_{28}$

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

Order 192: $C_2\times \Unitary(2,3)$ x 3, $A_4:C_4^2$ x 2, $C_8:S_4$ x 2, $C_2^4.D_6$ x 2, $C_4^2.A_4$, $A_4\times C_4^2$, $A_4:\OD_{16}$, $C_8\times S_4$, $D_4\times C_{24}$, $C_{24}:D_4$, $C_4^2.D_6$, $C_{12}:\OD_{16}$, $C_{12}:\OD_{16}$, $C_4^2:D_6$, $(C_2\times C_6):\OD_{16}$, $D_6:\OD_{16}$, $C_8:D_{12}$, $C_{12}:\OD_{16}$, $Q_8.(C_2\times A_4)$, $A_4\times D_4:C_2$, $\SL(2,3):C_2^3$, $Q_8:S_4$, $Q_8\times S_4$, $D_4:S_4$, $D_4\times S_4$, $\GL(2,\mathbb{Z}/4):C_2$, $C_{24}:C_2^3$, $C_4^2:D_6$, $A_4\times \OD_{16}$, $\SL(2,3):D_4$, $D_4\times \SL(2,3)$, $\SL(2,3):D_4$

Order 168: $C_7:C_{24}$ x 2, $C_{14}:C_{12}$ x 2, $C_{168}$, $\PSL(2,7)$, $S_3\times C_{28}$, $C_3:C_{56}$, $C_7:\SL(2,3)$, $C_7\times \SL(2,3)$, $C_{28}.C_6$, $C_{28}:C_6$

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

Order 144: $S_3\times S_4$ x 2, $C_{24}:S_3$, $C_{24}:C_6$, $S_3\times C_{24}$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $A_4\times D_6$, $C_6:S_4$, $C_6\times S_4$, $\SL(2,3):C_6$, $C_{12}\times A_4$, $C_4\times S_3^2$, $A_4\times C_3:C_4$, $S_3\times \SL(2,3)$, $\SL(2,3):S_3$, $C_6.S_4$, $A_4:C_{12}$

Order 140: $C_7\times D_{10}$, $C_7\times F_5$, $C_{140}$, $C_5:C_{28}$

Order 128: $\OD_{16}:D_4$, $C_4^3:C_2$, $C_4^2:D_4$, $\OD_{16}:D_4$, $C_4^2.D_4$, $C_2^4.D_4$, $D_4:C_4^2$, $D_4^2:C_2$, $D_4\times \OD_{16}$, $C_4^2:C_2^3$, $D_4\times C_4^2$

Order 126: $C_{21}:C_6$, $C_{21}:C_6$

Order 120: $C_5\times S_4$ x 2, $S_3\times F_5$ x 2, $C_6.D_{10}$, $C_6.D_{10}$, $\SL(2,5)$, $C_{10}\times A_4$, $S_3\times D_{10}$, $C_{30}:C_4$, $C_6\times F_5$, $D_5\times A_4$, $C_5:S_4$, $C_4\times D_{15}$, $S_3\times C_{20}$, $D_5\times C_{12}$, $D_{15}:C_4$

Order 112: $D_4:C_{14}$, $C_7\times \OD_{16}$, $C_4\times C_{28}$

Order 105: $C_7:C_{15}$

Order 96: $C_4\times S_4$ x 7, $\Unitary(2,3)$ x 5, $C_8:D_6$ x 5, $S_3\times C_4^2$ x 2, $C_8\times A_4$ x 2, $C_4\times \SL(2,3)$ x 2, $A_4:C_8$ x 2, $D_6:C_8$ x 2, $C_{24}:C_4$ x 2, $\SL(2,3):C_2^2$ x 2, $C_2^3:C_{12}$ x 2, $C_2^2.S_4$ x 2, $D_6.D_4$, $C_{12}:C_8$, $C_2^3.D_6$, $D_6.D_4$, $C_4\times D_{12}$, $C_4:C_{24}$, $D_4:C_{12}$, $C_2^2:C_{24}$, $C_4\times C_{24}$, $C_{12}.D_4$, $C_{12}.D_4$, $D_{12}:C_4$, $C_2^2\times S_4$, $D_4:D_6$, $C_{12}.D_4$, $D_4\times D_6$, $C_{12}:C_2^3$, $Q_8:A_4$, $Q_8\times A_4$, $C_2^2\times \SL(2,3)$, $D_4\times A_4$, $\GL(2,\mathbb{Z}/4)$, $C_4:S_4$, $A_4:Q_8$, $C_2^2\times C_{24}$, $D_4\times C_{12}$, $C_2^3.D_6$, $C_{12}:D_4$, $C_{12}.C_2^3$, $D_{12}:C_4$, $S_3\times \OD_{16}$, $C_{12}:C_8$, $D_{12}:C_4$

Order 84: $C_7:C_{12}$ x 4, $C_{14}:C_6$, $C_{84}$, $C_3:C_{28}$, $S_3\times C_{14}$

Order 80: $C_4\times F_5$ x 9, $C_4\times D_{10}$ x 6, $C_2^2\times F_5$ x 5, $D_4\times D_5$ x 3, $C_{20}:C_4$ x 3, $D_{10}:C_4$ x 2, $C_{20}:C_4$ x 2, $D_{10}:C_4$ x 2, $C_2^2\times D_{10}$, $D_4\times C_{10}$, $C_2^2\times C_{20}$, $C_{10}:D_4$, $C_2^2.D_{10}$, $C_2\times D_{20}$, $C_4:C_{20}$, $C_2^2:C_{20}$, $C_4\times C_{20}$, $C_{10}.D_4$, $C_{20}:C_4$, $C_{10}.D_4$

Order 72: $C_3\times S_4$ x 2, $S_3\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_3:C_{24}$ x 2, $C_6\times A_4$, $S_3\times D_6$, $S_3\times A_4$, $C_3:S_4$, $C_{12}:S_3$, $C_3\times \SL(2,3)$, $C_6.D_6$, $C_3\times C_{24}$, $C_3^2:C_8$

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

Order 64: $C_4^2:C_2^2$ x 7, $C_4^2:C_4$ x 3, $C_4:C_4^2$ x 2, $C_2^2:C_4^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^2:\OD_{16}$, $C_4\times \OD_{16}$, $C_4^3$, $D_4:C_2^3$, $C_2^2\times \OD_{16}$, $Q_8:D_4$, $D_4\times Q_8$, $Q_8:D_4$, $D_4:D_4$, $D_4:D_4$, $D_4^2$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_4^2.C_2^2$, $C_2^2\times C_4^2$, $C_8:D_4$, $C_8:D_4$, $C_8\times D_4$, $C_4:\OD_{16}$

Order 63: $C_{21}:C_3$

Order 60: $S_3\times D_5$ x 2, $C_5\times A_4$, $C_{15}:C_4$, $C_3\times F_5$, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $D_{30}$, $S_3\times C_{10}$, $C_3\times D_{10}$, $C_3:C_{20}$

Order 56: $C_2\times C_{28}$ x 2, $C_7\times D_4$, $C_{56}$, $C_7\times Q_8$

Order 48: $C_{24}:C_2$ x 9, $C_4\times D_6$ x 8, $C_2\times S_4$ x 5, $C_{12}:C_4$ x 5, $S_3\times D_4$ x 4, $\SL(2,3):C_2$ x 4, $C_4\times A_4$ x 4, $A_4:C_4$ x 4, $C_6:C_8$ x 3, $C_2\times C_{24}$ x 3, $C_4\times C_{12}$ x 2, $D_6:C_4$ x 2, $C_2^2\times D_6$, $C_2^2\times A_4$, $D_4:C_6$, $C_6\times D_4$, $C_2^2\times C_{12}$, $C_6:D_4$, $C_6.C_2^3$, $D_{12}:C_2$, $S_3\times Q_8$, $S_3\times C_8$, $D_4:S_3$, $D_{12}:C_2$, $C_2\times D_{12}$, $C_2\times \SL(2,3)$, $C_3\times \OD_{16}$, $C_4:C_{12}$, $C_2^2:C_{12}$, $C_6.D_4$, $C_{12}:C_4$, $C_6.D_4$, $C_3:\OD_{16}$

Order 42: $C_7:C_6$ x 3, $C_{42}$, $S_3\times C_7$

Order 40: $C_2\times F_5$ x 9, $C_4\times D_5$ x 8, $C_2\times C_{20}$ x 4, $C_{10}:C_4$ x 4, $C_2\times D_{10}$ x 4, $C_5\times D_4$ x 3, $C_5:D_4$ x 2, $D_{20}$ x 2, $C_2^2\times C_{10}$

Order 36: $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $S_3^2$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $C_6:S_3$, $C_3\times A_4$

Order 35: $C_{35}$

Order 32: $C_4\times D_4$ x 12, $C_4\wr C_2$ x 9, $C_2\times C_4^2$ x 7, $D_4:C_2^2$ x 4, $C_2\times \OD_{16}$ x 4, $C_4:D_4$ x 4, $C_2^2:C_8$ x 2, $C_2^2\times D_4$ x 2, $C_2^3\times C_4$ x 2, $C_2^2:Q_8$ x 2, $C_4\times Q_8$ x 2, $C_2^2.D_4$ x 2, $C_2^3:C_4$ x 2, $C_2.C_4^2$ x 2, $C_4:C_8$ x 2, $C_2^2\times Q_8$, $C_8:C_4$, $C_2^2\times C_8$, $C_4:Q_8$, $C_4:D_4$, $C_4^2:C_2$, $C_2^2.D_4$, $C_4\times C_8$, $C_2^2\wr C_2$, $C_4^2:C_2$

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

Order 28: $C_{28}$ x 3, $C_2\times C_{14}$

Order 24: $C_4\times S_3$ x 15, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6, $C_{24}$ x 5, $C_2\times D_6$ x 5, $C_3:C_8$ x 5, $S_4$ x 4, $C_3\times D_4$ x 4, $C_3:D_4$ x 3, $D_{12}$ x 3, $\SL(2,3)$ x 3, $C_2\times A_4$ x 3, $C_3:Q_8$, $C_2^2\times C_6$, $C_3\times Q_8$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 20: $D_{10}$ x 6, $C_{20}$ x 5, $C_5:C_4$ x 4, $C_2\times C_{10}$ x 3, $F_5$ x 3

Order 18: $C_3\times S_3$ x 3, $C_3\times C_6$, $C_3:S_3$

Order 16: $C_4^2$ x 13, $C_2^2\times C_4$ x 13, $C_2\times D_4$ x 10, $C_4:C_4$ x 8, $C_2^2:C_4$ x 8, $D_4:C_2$ x 7, $\OD_{16}$ x 6, $C_2\times C_8$ x 3, $C_2\times Q_8$ x 2, $C_2^4$

Order 15: $C_{15}$

Order 14: $C_{14}$ x 2

Order 12: $D_6$ x 11, $C_{12}$ x 9, $C_3:C_4$ x 8, $C_2\times C_6$ x 4, $A_4$ x 2

Order 10: $C_{10}$ x 3, $D_5$ x 2

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 25, $D_4$ x 12, $C_2^3$ x 5, $Q_8$ x 3, $C_8$ x 3

Order 7: $C_7$

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

Order 5: $C_5$

Order 4: $C_4$ x 12, $C_2^2$ x 9

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 80640: $\GL(2,5)\times \GL(3,2)$ ($C_1$)

Order 40320: $C_4.\PSL(2,7).A_5$ ($C_2$)

Order 20160: $\SL(2,5)\times \PSL(2,7)$ ($C_4$)

Order 672: $C_4\times \GL(3,2)$ ($S_5$)

Order 480: $\GL(2,5)$ ($\PSL(2,7)$)

Order 336: $C_2\times \GL(3,2)$ ($A_5:C_4$)

Order 240: $\SL(2,5):C_2$ ($C_2\times \GL(3,2)$)

Order 168: $\PSL(2,7)$ ($\GL(2,5)$)

Order 120: $\SL(2,5)$ ($C_4\times \GL(3,2)$)

Order 4: $C_4$ ($S_5\times \GL(3,2)$)

Order 2: $C_2$ ($\PSL(2,7)\times A_5:C_4$)

Order 1: $C_1$ ($\GL(2,5)\times \GL(3,2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 80640: $\GL(2,5)\times \GL(3,2)$ ($C_1$)

Order 40320: $C_4.\PSL(2,7).A_5$ ($C_2$)

Order 20160: $\SL(2,5)\times \PSL(2,7)$ ($C_4$)

Order 672: $C_4\times \GL(3,2)$ ($S_5$)

Order 480: $\GL(2,5)$ ($\PSL(2,7)$)

Order 336: $C_2\times \GL(3,2)$ ($A_5:C_4$)

Order 240: $\SL(2,5):C_2$ ($C_2\times \GL(3,2)$)

Order 168: $\PSL(2,7)$ ($\GL(2,5)$)

Order 120: $\SL(2,5)$ ($C_4\times \GL(3,2)$)

Order 4: $C_4$ ($S_5\times \GL(3,2)$)

Order 2: $C_2$ ($\PSL(2,7)\times A_5:C_4$)

Order 1: $C_1$ ($\GL(2,5)\times \GL(3,2)$)

Series

Derived series

$\GL(2,5)\times \GL(3,2)$

$\rhd$

$\SL(2,5)\times \PSL(2,7)$

Chief series

$\GL(2,5)\times \GL(3,2)$

$\rhd$

$C_4.\PSL(2,7).A_5$

$\rhd$

$C_4\times \GL(3,2)$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$\GL(2,5)\times \GL(3,2)$

$\rhd$

$\SL(2,5)\times \PSL(2,7)$

Upper central series

$C_1$

$\lhd$

$C_4$

Character theory

Complex character table

See the $144 \times 144$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $75 \times 75$ rational character table.