Abstract group 846720.f: $S_7\times \GL(3,2)$

• Label: 846720.f
• Order: \( 2^{7} \cdot 3^{3} \cdot 5 \cdot 7^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{3} \cdot 5 \cdot 7^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $14$
• Trans deg.: $49$
• Rank: $2$

Group information

Description:	$S_7\times \GL(3,2)$
Order:	\(846720\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 7^{2} \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$S_7\times \PGL(2,7)$, of order \(1693440\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5 \cdot 7^{2} \)
Composition factors:	$C_2$, $\PSL(2,7)$, $A_7$
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	70	84
Elements	1	5103	20006	63504	504	134946	35328	21672	173880	26208	28224	42336	57120	70560	28224	24192	70560	24192	20160	846720
Conjugacy classes	1	7	5	10	1	14	5	3	11	7	1	2	5	5	1	2	6	2	2	90
Divisions	1	7	5	10	1	14	3	3	11	4	1	2	3	3	1	1	3	1	1	75
Autjugacy classes	1	7	5	10	1	14	3	3	11	4	1	2	3	3	1	1	3	1	1	75

Dimension	1	3	6	7	8	14	15	18	20	21	35	36	42	45	48	60	63	84	90	98	105	112	120	126	140	147	160	168	210	245	280
Irr. complex chars.	2	4	4	2	2	4	2	4	1	2	2	2	10	4	2	2	4	4	2	4	6	4	3	2	1	2	1	2	2	2	2	90
Irr. rational chars.	2	0	6	2	2	4	2	0	1	2	2	4	2	0	2	0	0	8	4	4	2	4	4	4	1	2	1	2	4	2	2	75

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$49$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	36	36
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $14$ $\langle(2,3)(5,6), (8,13,14)(9,10)(11,12), (1,7,3,6)(2,4), (8,10,12), (12,13)\rangle$
Direct product:	$\PSL(2,7)$ $\, \times\, $ $S_7$
Semidirect product:	$(A_7\times \GL(3,2))$ $\,\rtimes\,$ $C_2$	$A_7$ $\,\rtimes\,$ $(C_2\times \GL(3,2))$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_2^3\times A_7)$

Elements of the group are displayed as permutations of degree 14.

Homology

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

Subgroups

There are 11257950 subgroups in 3680 conjugacy classes, 6 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_7\times \GL(3,2)$
Commutator:	$G' \simeq$ $A_7\times \GL(3,2)$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_7\times \GL(3,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $S_7\times \GL(3,2)$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $S_7\times \GL(3,2)$
Socle:	$\operatorname{soc} \simeq$ $A_7\times \GL(3,2)$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$

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 846720: $S_7\times \GL(3,2)$

Order 423360: $A_7\times \GL(3,2)$

Order 120960: $S_4\times S_7$ x 2, $S_6\times \GL(3,2)$

Order 105840: $C_7:C_3\times S_7$

Order 60480: $S_4\times A_7$ x 2, $A_4:S_7$ x 2, $A_4\times S_7$ x 2, $A_6\times \GL(3,2)$

Order 52920: $C_7:C_3\times A_7$

Order 40320: $D_4\times S_7$, $C_2\times S_5\times \GL(3,2)$

Order 35280: $C_7\times S_7$

Order 30240: $A_4\times A_7$ x 2, $S_3\times S_7$

Order 28224: $\GL(3,2)^2$

Order 24192: $S_3\times S_4\times \GL(3,2)$

Order 20160: $S_5\times \GL(3,2)$ x 3, $C_2^2\times S_7$ x 2, $C_2^2:S_7$ x 2, $C_4:S_7$, $C_4\times S_7$, $D_4\times A_7$, $C_2\times A_5\times \GL(3,2)$

Order 17640: $C_7\times A_7$

Order 17280: $S_4\times S_6$ x 2

Order 15120: $C_7:C_3\times S_6$, $C_3:S_7$, $S_3\times A_7$, $C_3\times S_7$

Order 12096: $\GL(3,2)\times \SOPlus(4,2)$, $C_3:S_4\times \GL(3,2)$, $C_3\times S_4\times \GL(3,2)$, $S_3\times A_4\times \GL(3,2)$

Order 10080: $C_2\times S_7$ x 3, $A_5\times \GL(3,2)$ x 2, $C_2^2\times A_7$ x 2, $C_4\times A_7$, $A_7:C_4$

Order 8640: $S_4\times A_6$ x 2, $A_4:S_6$ x 2, $A_4\times S_6$ x 2

Order 8064: $C_2\times S_4\times \GL(3,2)$ x 2, $S_3\times D_4\times \GL(3,2)$

Order 7560: $C_7:C_3\times A_6$, $C_3\times A_7$

Order 7056: $F_7\times \GL(3,2)$

Order 6720: $C_2\times F_5\times \GL(3,2)$

Order 6048: $S_3^2\times \GL(3,2)$ x 2, $C_3^2:C_4\times \GL(3,2)$, $C_3\times A_4\times \GL(3,2)$

Order 5760: $C_2\times S_4\times S_5$ x 2, $D_4\times S_6$

Order 5040: $S_7$ x 2, $C_7\times S_6$, $C_2\times A_7$, $C_7:C_6\times S_5$

Order 4320: $A_4\times A_6$ x 2, $S_3\times S_6$

Order 4032: $S_4\times \GL(3,2)$ x 7, $C_3:D_4\times \PSL(2,7)$ x 2, $C_2\times D_6\times \GL(3,2)$ x 2, $C_2\times A_4\times \GL(3,2)$ x 2, $C_4\times S_3\times \GL(3,2)$, $C_3\times D_4\times \GL(3,2)$, $D_{12}\times \PSL(2,7)$

Order 3528: $C_7:C_3\times \GL(3,2)$ x 2

Order 3456: $S_3\times S_4^2$ x 2

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

Order 3024: $C_3\times S_3\times \GL(3,2)$ x 2, $C_{21}:C_6\times S_4$, $C_3:S_3\times \GL(3,2)$

Order 2880: $S_4\times S_5$ x 10, $C_2\times S_4\times A_5$ x 2, $C_2\times A_4\times S_5$ x 2, $C_2\times A_4:S_5$ x 2, $C_2^2:S_6$ x 2, $C_2^2\times S_6$ x 2, $C_4:S_6$, $C_4\times S_6$, $D_4\times A_6$

Order 2688: $C_2\times D_4\times \GL(3,2)$

Order 2520: $C_7:C_3\times S_5$ x 3, $C_{14}:\GL(2,4)$, $C_7\times A_6$, $A_7$

Order 2352: $D_7\times \GL(3,2)$

Order 2160: $C_3:S_6$, $S_3\times A_6$, $C_3\times S_6$

Order 2016: $D_6\times \GL(3,2)$ x 6, $A_4\times \GL(3,2)$ x 5, $C_2\times C_6\times \GL(3,2)$ x 2, $C_3:C_4\times \GL(3,2)$, $C_{12}\times \GL(3,2)$

Order 1920: $C_2\times D_4\times S_5$

Order 1728: $A_4^2:D_6$ x 4, $C_3:S_4^2$ x 4, $A_4^2:D_6$ x 2, $C_3:S_4^2$ x 2, $C_3\times S_4^2$ x 2, $D_6^2:D_6$ x 2

Order 1680: $D_5\times \GL(3,2)$ x 2, $C_{14}\times S_5$, $C_{10}\times \GL(3,2)$

Order 1512: $A_4\times C_{21}:C_6$, $C_{21}:(C_3\times S_4)$, $C_{21}:C_3\times S_4$, $(C_7\times S_3^2):C_6$, $C_3^2\times \GL(3,2)$

Order 1440: $S_4\times A_5$ x 6, $A_4:S_5$ x 6, $A_4\times S_5$ x 6, $C_2\times S_6$ x 3, $C_2^3:\GL(2,4)$ x 2, $C_2^2\times A_6$ x 2, $D_6\times S_5$, $C_4\times A_6$, $A_6:C_4$

Order 1344: $D_4\times \GL(3,2)$ x 5, $C_2^3\times \GL(3,2)$ x 2, $C_2\times C_4\times \GL(3,2)$

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

Order 1176: $C_7\times \PSL(2,7)$ x 2

Order 1152: $C_2\times S_4^2$ x 4, $\GL(2,\mathbb{Z}/4):D_6$ x 3

Order 1080: $C_3\times A_6$

Order 1008: $S_3\times \GL(3,2)$ x 6, $C_6\times \GL(3,2)$ x 3, $C_7:C_6\times S_4$ x 2, $S_4\times F_7$ x 2, $S_3\times C_7\times S_4$, $C_{28}:C_6\times S_3$

Order 960: $D_4\times S_5$ x 9, $C_2\times F_5\times S_4$ x 2, $C_2^3\times S_5$ x 2, $C_2^3:S_5$ x 2, $C_2\times D_4\times A_5$, $C_2\times C_4:S_5$, $C_2\times C_4\times S_5$

Order 882: $C_7:C_3\times F_7$

Order 864: $S_4\times S_3^2$ x 5, $A_4^2:S_3$ x 4, $A_4^2:C_6$ x 4, $S_3^2:S_4$ x 4, $S_3\times A_4^2$ x 2, $A_4^2:S_3$ x 2, $A_4^2:C_6$ x 2, $D_6^2:C_6$ x 2, $C_6^2:D_{12}$ x 2, $C_3^2:C_4\times S_4$ x 2

Order 840: $C_7\times S_5$ x 3, $C_{70}:C_{12}$, $C_{14}\times A_5$, $C_5\times \PSL(2,7)$

Order 756: $C_{21}:C_6\times S_3$ x 2, $A_4\times C_{21}:C_3$, $(C_3\times C_{21}):C_{12}$

Order 720: $S_3\times S_5$ x 5, $A_4\times A_5$ x 4, $S_6$ x 2, $D_6\times A_5$, $C_6:S_5$, $C_6\times S_5$, $C_2\times A_6$

Order 672: $C_2^2\times \GL(3,2)$ x 7, $C_4\times \GL(3,2)$ x 3

Order 576: $S_4^2$ x 20, $A_4^2:C_2^2$ x 8, $C_2^2:D_6^2$ x 6, $\GL(2,\mathbb{Z}/4):S_3$ x 6, $C_2^4:S_3^2$ x 6, $S_3\times \GL(2,\mathbb{Z}/4)$ x 6, $A_4^2:C_2^2$ x 4, $S_3\times D_4\times A_4$ x 3, $D_4\times C_3:S_4$ x 3, $\GL(2,\mathbb{Z}/4):C_6$ x 3, $D_{12}:S_4$ x 3, $S_4\times D_{12}$ x 3, $S_3\times C_4:S_4$ x 3, $C_4\times S_3\times S_4$ x 3, $C_2^4:S_3^2$ x 2, $C_3^2:D_4^2$

Order 504: $C_7:C_3\times S_4$ x 8, $C_3\times \GL(3,2)$ x 3, $(C_2\times C_{42}):C_6$ x 2, $C_2\times C_{42}:C_6$ x 2, $C_7:C_6\times A_4$ x 2, $A_4\times F_7$ x 2, $A_4:F_7$ x 2, $C_{84}:C_6$, $C_{84}:C_6$, $S_3\times C_7:A_4$, $S_3\times C_7\times A_4$, $C_{21}:S_4$, $C_{21}\times S_4$, $S_3^2:C_{14}$, $C_{84}:C_6$

Order 480: $C_2^2\times S_5$ x 13, $C_2^2:S_5$ x 10, $F_5\times S_4$ x 8, $D_4\times A_5$ x 5, $C_4:S_5$ x 5, $C_4\times S_5$ x 5, $D_{10}\times S_4$ x 2, $C_2\times A_4\times F_5$ x 2, $D_{10}.S_4$ x 2, $C_2^3\times A_5$ x 2, $C_2\times C_4\times A_5$, $C_2^2.S_5$

Order 441: $C_7^2:C_3^2$

Order 432: $A_4:S_3^2$ x 6, $C_6^2:D_6$ x 6, $A_4\times S_3^2$ x 5, $A_4:S_3^2$ x 5, $C_6^2:D_6$ x 3, $C_3\times A_4^2$ x 2, $C_6^2:C_{12}$ x 2, $(C_3^2\times A_4):C_4$ x 2, $S_3^3:C_2$

Order 420: $C_{35}:C_{12}$ x 2, $C_7\times A_5$ x 2, $C_{70}:C_6$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$ x 4, $S_3\times D_4^2$

Order 378: $C_3\times C_{21}:C_6$ x 2, $(C_3\times C_{21}):C_6$

Order 360: $S_3\times A_5$ x 3, $C_3:S_5$ x 3, $C_3\times S_5$ x 3, $C_6\times A_5$, $A_6$

Order 336: $C_2\times \GL(3,2)$ x 4, $C_{14}\times S_4$ x 2, $D_7\times S_4$ x 2, $C_{28}:D_6$, $C_2\times C_{28}:C_6$, $D_4\times F_7$

Order 320: $D_{10}.C_2^4$

Order 294: $C_7^2:C_6$, $C_7\times F_7$, $C_7:F_7$, $C_7:F_7$

Order 288: $D_6\times S_4$ x 33, $A_4\times S_4$ x 24, $\PSOPlus(4,3)$ x 14, $A_4\times C_3:D_4$ x 6, $C_3:\GL(2,\mathbb{Z}/4)$ x 6, $C_3\times \GL(2,\mathbb{Z}/4)$ x 6, $D_6:S_4$ x 6, $D_6:S_4$ x 6, $C_2^3.S_3^2$ x 6, $C_2\times A_4\times D_6$ x 6, $C_2\times C_6:S_4$ x 6, $C_2\times C_6\times S_4$ x 6, $D_6\wr C_2$ x 4, $C_2\times A_4^2$ x 4, $C_2^3.C_6^2$ x 3, $D_4\times S_3^2$ x 3, $A_4\times D_{12}$ x 3, $C_4\times S_3\times A_4$ x 3, $C_{12}:S_4$ x 3, $C_{12}:S_4$ x 3, $C_{12}:S_4$ x 3, $C_{12}\times S_4$ x 3, $D_6.S_4$ x 3, $C_2^3.S_3^2$ x 3, $C_2^3.S_3^2$ x 3, $C_6^2:D_4$ x 2, $S_3^2:D_4$ x 2, $S_3\times C_2^2:A_4$ x 2, $(C_2\times C_6):S_4$ x 2, $C_3\times C_2^2:S_4$ x 2, $C_6^2:D_4$ x 2, $C_3^2:C_4\times D_4$, $C_4:\SOPlus(4,2)$, $C_4\times \SOPlus(4,2)$

Order 280: $C_{14}\times F_5$

Order 252: $A_4\times C_7:C_3$ x 7, $C_{42}:C_6$ x 6, $C_{42}:C_6$ x 2, $C_7\times S_3^2$ x 2, $C_{21}:A_4$, $A_4\times C_{21}$, $C_3^2:C_{28}$, $S_3\times F_7$, $C_{21}:C_{12}$, $C_{21}:C_{12}$

Order 240: $C_2\times S_5$ x 15, $D_5\times S_4$ x 8, $C_2^2\times A_5$ x 7, $A_4\times F_5$ x 4, $A_4:F_5$ x 4, $C_4\times A_5$ x 3, $A_5:C_4$ x 3, $A_4\times D_{10}$ x 2, $C_{10}:S_4$ x 2, $C_{10}\times S_4$ x 2, $D_6\times F_5$

Order 216: $C_6^2:C_6$ x 6, $C_3^2:S_4$ x 6, $C_6^2:C_6$ x 3, $C_3^2:S_4$ x 3, $C_3^2\times S_4$ x 3, $S_3^3$ x 2, $S_3^2:S_3$ x 2, $C_3^2:D_{12}$, $S_3^2:C_6$, $C_3^2:C_4\times S_3$

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

Order 192: $D_4\times S_4$ x 35, $C_2^3\times S_4$ x 8, $C_2\times \GL(2,\mathbb{Z}/4)$ x 8, $C_2^3:S_4$ x 4, $C_{12}:C_2^4$ x 4, $C_2^5:C_6$ x 4, $C_2^3:D_{12}$ x 4, $C_2^4.D_6$ x 4, $C_2^4:D_6$ x 4, $D_{12}:D_4$ x 4, $S_3\times C_4:D_4$ x 4, $C_2^4:D_6$ x 4, $C_2^4:D_6$ x 4, $D_4\times D_{12}$ x 2, $C_4^2:D_6$ x 2, $C_3\times D_4^2$, $C_3:D_4^2$, $C_4^2:D_6$

Order 189: $C_7:C_3^3$

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

Order 168: $C_7\times S_4$ x 7, $C_{28}:C_6$ x 6, $\PSL(2,7)$ x 4, $D_6\times C_{14}$ x 2, $C_{14}:A_4$ x 2, $A_4\times C_{14}$ x 2, $C_2\times C_{14}:C_6$ x 2, $D_7:A_4$ x 2, $A_4\times D_7$ x 2, $C_2^2\times F_7$ x 2, $C_7:S_4$ x 2, $D_6:C_{14}$ x 2, $D_{14}:C_6$ x 2, $C_{28}:C_6$, $C_4\times F_7$, $D_4\times C_{21}$, $C_7\times D_{12}$, $S_3\times C_{28}$, $C_{14}:C_{12}$

Order 160: $D_4\times F_5$ x 8, $C_2^3\times F_5$ x 2, $C_2^3:F_5$ x 2, $D_4\times D_{10}$, $D_{10}.D_4$, $C_{10}:C_4^2$

Order 147: $C_7:C_{21}$ x 2, $C_7^2:C_3$, $C_7^2:C_3$

Order 144: $S_3\times S_4$ x 53, $A_4\times D_6$ x 17, $C_6:S_4$ x 17, $C_6\times S_4$ x 17, $A_4^2$ x 10, $S_3^2:C_2^2$ x 9, $D_6:D_6$ x 8, $C_2^2:C_6^2$ x 6, $D_6^2$ x 6, $D_6:D_6$ x 6, $S_3^2:C_4$ x 6, $C_{12}:D_6$ x 4, $S_3\times D_{12}$ x 4, $C_{12}\times A_4$ x 3, $C_{12}:D_6$ x 3, $C_4\times S_3^2$ x 3, $A_4\times C_3:C_4$ x 3, $C_6.S_4$ x 3, $A_4:C_{12}$ x 3, $C_2^4:C_3^2$ x 2, $C_6^2:C_4$ x 2, $C_{12}:D_6$ x 2, $C_6^2:C_4$ x 2, $(C_3\times C_{12}):C_4$, $C_3^2:C_4^2$, $C_2.\SOPlus(4,2)$

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

Order 128: $C_2\times D_4^2$

Order 126: $C_{21}:C_6$ x 8, $C_{21}:C_6$ x 3, $S_3\times C_{21}$ x 2, $C_{21}:C_6$, $C_3\times F_7$, $C_{21}:S_3$

Order 120: $S_5$ x 6, $D_5\times A_4$ x 4, $C_5:S_4$ x 4, $C_5\times S_4$ x 4, $S_3\times F_5$ x 4, $C_2\times A_5$ x 4, $C_{10}\times A_4$ x 2, $S_3\times D_{10}$, $C_{30}:C_4$, $C_6\times F_5$

Order 112: $D_4\times C_{14}$, $D_4\times D_7$

Order 108: $C_3\times S_3^2$ x 4, $C_3^2\times A_4$ x 3, $C_3:S_3^2$ x 3, $C_3:S_3^2$ x 2, $C_3^3:C_4$, $C_3^2:C_{12}$

Order 105: $C_7:C_{15}$

Order 98: $C_7\times D_7$

Order 96: $C_2^2\times S_4$ x 50, $D_4\times D_6$ x 42, $\GL(2,\mathbb{Z}/4)$ x 38, $D_4\times A_4$ x 19, $C_4:S_4$ x 19, $C_4\times S_4$ x 19, $C_2^2:D_{12}$ x 12, $D_6:D_4$ x 12, $C_2^2:S_4$ x 10, $D_6:D_4$ x 8, $D_6.D_4$ x 8, $C_2^3\times A_4$ x 8, $C_2^3:D_6$ x 8, $C_2^3.D_6$ x 8, $C_{12}:D_4$ x 6, $C_2^3.D_6$ x 4, $C_2^3\times D_6$ x 4, $C_2^3:A_4$ x 4, $C_{12}:C_2^3$ x 4, $C_2^2\times D_{12}$ x 4, $C_{12}:C_2^3$ x 4, $C_2^3:C_{12}$ x 4, $C_2^2.S_4$ x 4, $C_{12}:D_4$ x 4, $C_2^4:C_6$ x 4, $C_2^4:S_3$ x 4, $C_{12}:D_4$ x 4, $C_{12}:D_4$ x 4, $C_{12}:D_4$ x 4, $C_4:D_{12}$ x 4, $D_6.D_4$ x 2, $C_4\times D_{12}$ x 2, $D_4\times C_{12}$ x 2, $C_2^3.D_6$ x 2, $D_{12}:C_4$ x 2, $C_4:D_{12}$, $S_3\times C_4^2$, $C_{12}:D_4$

Order 84: $C_{14}:C_6$ x 9, $S_3\times C_{14}$ x 6, $C_7:A_4$ x 5, $C_7\times A_4$ x 5, $C_7:C_{12}$ x 4, $C_2\times F_7$ x 3, $C_2\times C_{42}$ x 2, $S_3\times D_7$, $C_{84}$, $C_3:C_{28}$, $C_7:C_{12}$

Order 80: $C_2^2\times F_5$ x 12, $D_4\times D_5$ x 8, $D_{10}:C_4$ x 8, $C_{20}:C_4$ x 4, $C_4\times F_5$ x 4, $C_2^2\times D_{10}$ x 2, $C_{10}:D_4$ x 2, $D_4\times C_{10}$, $C_2\times D_{20}$, $C_4\times D_{10}$

Order 72: $C_3:S_4$ x 26, $S_3\times A_4$ x 25, $C_3\times S_4$ x 25, $S_3\times D_6$ x 22, $C_6\times A_4$ x 9, $C_6\times D_6$ x 8, $\SOPlus(4,2)$ x 8, $C_6\wr C_2$ x 8, $C_3:D_{12}$ x 8, $C_6:D_6$ x 4, $C_2\times C_3^2:C_4$ x 4, $C_6^2:C_2$ x 4, $C_3\times D_{12}$ x 4, $S_3\times C_{12}$ x 4, $D_6:S_3$ x 4, $C_6.D_6$ x 4, $C_6.D_6$ x 3, $D_4\times C_3^2$ x 2, $C_3:D_{12}$ x 2, $C_{12}:S_3$ x 2

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

Order 64: $D_4^2$ x 16, $D_4\times C_2^3$ x 4, $C_2^3:D_4$ x 4, $C_2^3:D_4$ x 4, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$

Order 63: $C_{21}:C_3$ x 5, $C_3\times C_{21}$

Order 60: $S_3\times D_5$ x 4, $C_5\times A_4$ x 2, $C_{15}:C_4$ x 2, $C_3\times F_5$ x 2, $A_5$ x 2, $D_{30}$, $S_3\times C_{10}$, $C_3\times D_{10}$

Order 56: $C_7\times D_4$ x 5, $C_7:D_4$ x 2, $C_2^2\times C_{14}$ x 2, $C_2\times D_{14}$ x 2, $C_2\times C_{28}$, $D_{28}$, $C_4\times D_7$

Order 54: $C_3^2:C_6$ x 3, $S_3\times C_3^2$ x 3, $C_3^2:S_3$

Order 49: $C_7^2$

Order 48: $S_3\times D_4$ x 86, $C_2\times S_4$ x 69, $C_6:D_4$ x 44, $C_2^2\times D_6$ x 32, $C_2^2\times A_4$ x 26, $C_6\times D_4$ x 22, $C_2\times D_{12}$ x 22, $C_4\times D_6$ x 21, $D_6:C_4$ x 16, $C_4\times A_4$ x 11, $A_4:C_4$ x 11, $C_2^2:C_{12}$ x 8, $C_6.D_4$ x 8, $C_2^2:A_4$ x 6, $C_2^3\times C_6$ x 4, $C_2^2\times C_{12}$ x 4, $C_6.C_2^3$ x 4, $C_6.D_4$ x 4, $C_{12}:C_4$ x 3, $C_4:C_{12}$ x 2, $C_{12}:C_4$ x 2, $C_4\times C_{12}$

Order 42: $C_7:C_6$ x 7, $S_3\times C_7$ x 6, $C_{42}$ x 3, $F_7$ x 3, $D_{21}$, $C_3\times D_7$

Order 40: $C_2\times F_5$ x 16, $C_2\times D_{10}$ x 11, $C_5:D_4$ x 8, $D_{20}$ x 4, $C_4\times D_5$ x 4, $C_5\times D_4$ x 4, $C_2^2\times C_{10}$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$

Order 36: $S_3^2$ x 27, $C_6\times S_3$ x 18, $C_3\times A_4$ x 17, $C_6:S_3$ x 9, $C_3:C_{12}$ x 4, $C_6^2$ x 4, $C_3^2:C_4$ x 3, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2

Order 35: $C_{35}$

Order 32: $C_2^2\times D_4$ x 48, $C_4:D_4$ x 32, $C_2^2\wr C_2$ x 32, $C_4\times D_4$ x 16, $C_4:D_4$ x 8, $C_2^3:C_4$ x 8, $C_2^5$ x 4, $C_2^3\times C_4$ x 4, $C_2^2.D_4$ x 2, $C_2\times C_4^2$

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

Order 28: $C_2\times C_{14}$ x 7, $D_{14}$ x 3, $C_{28}$ x 3, $C_7:C_4$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 87, $C_3:D_4$ x 56, $S_4$ x 43, $D_{12}$ x 28, $C_4\times S_3$ x 28, $C_3\times D_4$ x 28, $C_2\times A_4$ x 25, $C_2^2\times C_6$ x 16, $C_2\times C_{12}$ x 11, $C_6:C_4$ x 11

Order 21: $C_7:C_3$ x 6, $C_{21}$ x 3

Order 20: $D_{10}$ x 12, $F_5$ x 6, $C_2\times C_{10}$ x 5, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 18: $C_3\times S_3$ x 18, $C_3:S_3$ x 9, $C_3\times C_6$ x 4

Order 16: $C_2\times D_4$ x 112, $C_2^2:C_4$ x 32, $C_2^4$ x 24, $C_2^2\times C_4$ x 23, $C_4:C_4$ x 8, $C_4^2$ x 4

Order 15: $C_{15}$

Order 14: $C_{14}$ x 4, $D_7$ x 2

Order 12: $D_6$ x 67, $C_2\times C_6$ x 29, $A_4$ x 18, $C_{12}$ x 11, $C_3:C_4$ x 11

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

Order 9: $C_3^2$ x 5

Order 8: $D_4$ x 56, $C_2^3$ x 49, $C_2\times C_4$ x 28

Order 7: $C_7$ x 3

Order 6: $S_3$ x 19, $C_6$ x 14

Order 5: $C_5$

Order 4: $C_2^2$ x 35, $C_4$ x 10

Order 3: $C_3$ x 5

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 846720: $S_7\times \GL(3,2)$

Order 423360: $A_7\times \GL(3,2)$

Order 120960: $S_6\times \GL(3,2)$, $S_4\times S_7$

Order 105840: $C_7:C_3\times S_7$

Order 60480: $A_6\times \GL(3,2)$, $S_4\times A_7$, $A_4:S_7$, $A_4\times S_7$

Order 52920: $C_7:C_3\times A_7$

Order 40320: $D_4\times S_7$, $C_2\times S_5\times \GL(3,2)$

Order 35280: $C_7\times S_7$

Order 30240: $A_4\times A_7$, $S_3\times S_7$

Order 28224: $\GL(3,2)^2$

Order 24192: $S_3\times S_4\times \GL(3,2)$

Order 20160: $S_5\times \GL(3,2)$ x 3, $C_4:S_7$, $C_2^2\times S_7$, $C_4\times S_7$, $D_4\times A_7$, $C_2^2:S_7$, $C_2\times A_5\times \GL(3,2)$

Order 17640: $C_7\times A_7$

Order 17280: $S_4\times S_6$

Order 15120: $C_7:C_3\times S_6$, $C_3:S_7$, $S_3\times A_7$, $C_3\times S_7$

Order 12096: $\GL(3,2)\times \SOPlus(4,2)$, $C_3:S_4\times \GL(3,2)$, $C_3\times S_4\times \GL(3,2)$, $S_3\times A_4\times \GL(3,2)$

Order 10080: $A_5\times \GL(3,2)$ x 2, $C_2\times S_7$ x 2, $C_4\times A_7$, $C_2^2\times A_7$, $A_7:C_4$

Order 8640: $S_4\times A_6$, $A_4:S_6$, $A_4\times S_6$

Order 8064: $C_2\times S_4\times \GL(3,2)$ x 2, $S_3\times D_4\times \GL(3,2)$

Order 7560: $C_7:C_3\times A_6$, $C_3\times A_7$

Order 7056: $F_7\times \GL(3,2)$

Order 6720: $C_2\times F_5\times \GL(3,2)$

Order 6048: $S_3^2\times \GL(3,2)$ x 2, $C_3^2:C_4\times \GL(3,2)$, $C_3\times A_4\times \GL(3,2)$

Order 5760: $C_2\times S_4\times S_5$, $D_4\times S_6$

Order 5040: $S_7$ x 2, $C_7\times S_6$, $C_2\times A_7$, $C_7:C_6\times S_5$

Order 4320: $A_4\times A_6$, $S_3\times S_6$

Order 4032: $S_4\times \GL(3,2)$ x 6, $C_3:D_4\times \PSL(2,7)$ x 2, $C_2\times D_6\times \GL(3,2)$ x 2, $C_2\times A_4\times \GL(3,2)$ x 2, $C_4\times S_3\times \GL(3,2)$, $C_3\times D_4\times \GL(3,2)$, $D_{12}\times \PSL(2,7)$

Order 3528: $C_7:C_3\times \GL(3,2)$ x 2

Order 3456: $S_3\times S_4^2$

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

Order 3024: $C_3\times S_3\times \GL(3,2)$ x 2, $C_{21}:C_6\times S_4$, $C_3:S_3\times \GL(3,2)$

Order 2880: $S_4\times S_5$ x 5, $C_2\times S_4\times A_5$, $C_2\times A_4\times S_5$, $C_2\times A_4:S_5$, $C_4:S_6$, $C_4\times S_6$, $C_2^2:S_6$, $D_4\times A_6$, $C_2^2\times S_6$

Order 2688: $C_2\times D_4\times \GL(3,2)$

Order 2520: $C_7:C_3\times S_5$ x 3, $C_{14}:\GL(2,4)$, $C_7\times A_6$, $A_7$

Order 2352: $D_7\times \GL(3,2)$

Order 2160: $C_3:S_6$, $S_3\times A_6$, $C_3\times S_6$

Order 2016: $D_6\times \GL(3,2)$ x 6, $A_4\times \GL(3,2)$ x 4, $C_2\times C_6\times \GL(3,2)$ x 2, $C_3:C_4\times \GL(3,2)$, $C_{12}\times \GL(3,2)$

Order 1920: $C_2\times D_4\times S_5$

Order 1728: $A_4^2:D_6$ x 2, $C_3:S_4^2$ x 2, $A_4^2:D_6$, $C_3:S_4^2$, $C_3\times S_4^2$, $D_6^2:D_6$

Order 1680: $D_5\times \GL(3,2)$ x 2, $C_{14}\times S_5$, $C_{10}\times \GL(3,2)$

Order 1512: $A_4\times C_{21}:C_6$, $C_{21}:(C_3\times S_4)$, $C_{21}:C_3\times S_4$, $(C_7\times S_3^2):C_6$, $C_3^2\times \GL(3,2)$

Order 1440: $S_4\times A_5$ x 3, $A_4:S_5$ x 3, $A_4\times S_5$ x 3, $C_2\times S_6$ x 2, $C_2^3:\GL(2,4)$, $D_6\times S_5$, $C_2^2\times A_6$, $C_4\times A_6$, $A_6:C_4$

Order 1344: $D_4\times \GL(3,2)$ x 5, $C_2^3\times \GL(3,2)$ x 2, $C_2\times C_4\times \GL(3,2)$

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

Order 1176: $C_7\times \PSL(2,7)$ x 2

Order 1152: $C_2\times S_4^2$ x 2, $\GL(2,\mathbb{Z}/4):D_6$ x 2

Order 1080: $C_3\times A_6$

Order 1008: $S_3\times \GL(3,2)$ x 6, $C_6\times \GL(3,2)$ x 3, $C_7:C_6\times S_4$ x 2, $S_3\times C_7\times S_4$, $S_4\times F_7$, $C_{28}:C_6\times S_3$

Order 960: $D_4\times S_5$ x 7, $C_2\times F_5\times S_4$, $C_2^3\times S_5$, $C_2\times D_4\times A_5$, $C_2^3:S_5$, $C_2\times C_4:S_5$, $C_2\times C_4\times S_5$

Order 882: $C_7:C_3\times F_7$

Order 864: $S_4\times S_3^2$ x 3, $A_4^2:S_3$ x 2, $A_4^2:C_6$ x 2, $S_3^2:S_4$ x 2, $S_3\times A_4^2$, $A_4^2:S_3$, $A_4^2:C_6$, $D_6^2:C_6$, $C_6^2:D_{12}$, $C_3^2:C_4\times S_4$

Order 840: $C_7\times S_5$ x 3, $C_{70}:C_{12}$, $C_{14}\times A_5$, $C_5\times \PSL(2,7)$

Order 756: $C_{21}:C_6\times S_3$ x 2, $A_4\times C_{21}:C_3$, $(C_3\times C_{21}):C_{12}$

Order 720: $S_3\times S_5$ x 5, $A_4\times A_5$ x 2, $S_6$ x 2, $D_6\times A_5$, $C_6:S_5$, $C_6\times S_5$, $C_2\times A_6$

Order 672: $C_2^2\times \GL(3,2)$ x 6, $C_4\times \GL(3,2)$ x 3

Order 576: $S_4^2$ x 10, $A_4^2:C_2^2$ x 4, $C_2^2:D_6^2$ x 3, $\GL(2,\mathbb{Z}/4):S_3$ x 3, $C_2^4:S_3^2$ x 3, $S_3\times \GL(2,\mathbb{Z}/4)$ x 3, $A_4^2:C_2^2$ x 2, $S_3\times D_4\times A_4$ x 2, $D_4\times C_3:S_4$ x 2, $\GL(2,\mathbb{Z}/4):C_6$ x 2, $D_{12}:S_4$ x 2, $S_4\times D_{12}$ x 2, $S_3\times C_4:S_4$ x 2, $C_4\times S_3\times S_4$ x 2, $C_2^4:S_3^2$, $C_3^2:D_4^2$

Order 504: $C_7:C_3\times S_4$ x 7, $C_3\times \GL(3,2)$ x 3, $(C_2\times C_{42}):C_6$ x 2, $C_2\times C_{42}:C_6$ x 2, $C_7:C_6\times A_4$ x 2, $C_{84}:C_6$, $C_{84}:C_6$, $S_3\times C_7:A_4$, $S_3\times C_7\times A_4$, $C_{21}:S_4$, $C_{21}\times S_4$, $S_3^2:C_{14}$, $A_4\times F_7$, $A_4:F_7$, $C_{84}:C_6$

Order 480: $C_2^2\times S_5$ x 7, $C_2^2:S_5$ x 5, $C_4\times S_5$ x 5, $D_4\times A_5$ x 4, $C_4:S_5$ x 4, $F_5\times S_4$ x 4, $C_2\times C_4\times A_5$, $C_2^2.S_5$, $D_{10}\times S_4$, $C_2\times A_4\times F_5$, $D_{10}.S_4$, $C_2^3\times A_5$

Order 441: $C_7^2:C_3^2$

Order 432: $A_4:S_3^2$ x 4, $C_6^2:D_6$ x 4, $A_4\times S_3^2$ x 3, $A_4:S_3^2$ x 3, $C_6^2:D_6$ x 2, $C_3\times A_4^2$, $C_6^2:C_{12}$, $(C_3^2\times A_4):C_4$, $S_3^3:C_2$

Order 420: $C_{35}:C_{12}$ x 2, $C_7\times A_5$ x 2, $C_{70}:C_6$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$ x 3, $S_3\times D_4^2$

Order 378: $C_3\times C_{21}:C_6$ x 2, $(C_3\times C_{21}):C_6$

Order 360: $S_3\times A_5$ x 3, $C_3:S_5$ x 3, $C_3\times S_5$ x 3, $C_6\times A_5$, $A_6$

Order 336: $C_2\times \GL(3,2)$ x 4, $C_{14}\times S_4$ x 2, $D_7\times S_4$, $C_{28}:D_6$, $C_2\times C_{28}:C_6$, $D_4\times F_7$

Order 320: $D_{10}.C_2^4$

Order 294: $C_7^2:C_6$, $C_7\times F_7$, $C_7:F_7$, $C_7:F_7$

Order 288: $D_6\times S_4$ x 18, $A_4\times S_4$ x 12, $\PSOPlus(4,3)$ x 7, $D_4\times S_3^2$ x 3, $A_4\times C_3:D_4$ x 3, $C_3:\GL(2,\mathbb{Z}/4)$ x 3, $C_3\times \GL(2,\mathbb{Z}/4)$ x 3, $D_6:S_4$ x 3, $D_6:S_4$ x 3, $C_2^3.S_3^2$ x 3, $C_2\times A_4\times D_6$ x 3, $C_2\times C_6:S_4$ x 3, $C_2\times C_6\times S_4$ x 3, $C_2^3.C_6^2$ x 2, $A_4\times D_{12}$ x 2, $C_4\times S_3\times A_4$ x 2, $C_{12}:S_4$ x 2, $C_{12}:S_4$ x 2, $C_{12}:S_4$ x 2, $C_{12}\times S_4$ x 2, $D_6\wr C_2$ x 2, $S_3^2:D_4$ x 2, $D_6.S_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_2\times A_4^2$ x 2, $C_3^2:C_4\times D_4$, $C_6^2:D_4$, $C_4:\SOPlus(4,2)$, $C_4\times \SOPlus(4,2)$, $S_3\times C_2^2:A_4$, $(C_2\times C_6):S_4$, $C_3\times C_2^2:S_4$, $C_6^2:D_4$

Order 280: $C_{14}\times F_5$

Order 252: $C_{42}:C_6$ x 6, $A_4\times C_7:C_3$ x 6, $C_{42}:C_6$ x 2, $C_7\times S_3^2$ x 2, $C_{21}:A_4$, $A_4\times C_{21}$, $C_3^2:C_{28}$, $S_3\times F_7$, $C_{21}:C_{12}$, $C_{21}:C_{12}$

Order 240: $C_2\times S_5$ x 11, $D_5\times S_4$ x 4, $C_2^2\times A_5$ x 4, $C_4\times A_5$ x 3, $A_5:C_4$ x 3, $A_4\times F_5$ x 2, $A_4:F_5$ x 2, $A_4\times D_{10}$, $C_{10}:S_4$, $C_{10}\times S_4$, $D_6\times F_5$

Order 216: $C_6^2:C_6$ x 4, $C_3^2:S_4$ x 4, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2\times S_4$ x 2, $S_3^3$ x 2, $S_3^2:S_3$ x 2, $C_3^2:D_{12}$, $S_3^2:C_6$, $C_3^2:C_4\times S_3$

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

Order 192: $D_4\times S_4$ x 22, $C_2^3\times S_4$ x 4, $C_2\times \GL(2,\mathbb{Z}/4)$ x 4, $C_{12}:C_2^4$ x 3, $C_2^5:C_6$ x 3, $C_2^3:D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^4:D_6$ x 3, $D_{12}:D_4$ x 3, $S_3\times C_4:D_4$ x 3, $C_2^3:S_4$ x 2, $C_2^4:D_6$ x 2, $C_2^4:D_6$ x 2, $D_4\times D_{12}$ x 2, $C_4^2:D_6$ x 2, $C_3\times D_4^2$, $C_3:D_4^2$, $C_4^2:D_6$

Order 189: $C_7:C_3^3$

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

Order 168: $C_7\times S_4$ x 6, $C_{28}:C_6$ x 6, $\PSL(2,7)$ x 3, $D_6\times C_{14}$ x 2, $C_{14}:A_4$ x 2, $A_4\times C_{14}$ x 2, $C_2\times C_{14}:C_6$ x 2, $D_6:C_{14}$ x 2, $C_{28}:C_6$, $C_4\times F_7$, $D_7:A_4$, $A_4\times D_7$, $C_2^2\times F_7$, $C_7:S_4$, $D_4\times C_{21}$, $C_7\times D_{12}$, $S_3\times C_{28}$, $C_{14}:C_{12}$, $D_{14}:C_6$

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

Order 147: $C_7:C_{21}$ x 2, $C_7^2:C_3$, $C_7^2:C_3$

Order 144: $S_3\times S_4$ x 34, $A_4\times D_6$ x 10, $C_6:S_4$ x 10, $C_6\times S_4$ x 10, $D_6:D_6$ x 6, $S_3^2:C_2^2$ x 5, $A_4^2$ x 5, $D_6^2$ x 4, $C_{12}:D_6$ x 4, $D_6:D_6$ x 4, $S_3\times D_{12}$ x 4, $S_3^2:C_4$ x 4, $C_2^2:C_6^2$ x 3, $C_{12}:D_6$ x 3, $C_4\times S_3^2$ x 3, $C_{12}:D_6$ 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, $C_2^4:C_3^2$, $C_6^2:C_4$, $C_6^2:C_4$, $(C_3\times C_{12}):C_4$, $C_3^2:C_4^2$, $C_2.\SOPlus(4,2)$

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

Order 128: $C_2\times D_4^2$

Order 126: $C_{21}:C_6$ x 8, $C_{21}:C_6$ x 3, $S_3\times C_{21}$ x 2, $C_{21}:C_6$, $C_3\times F_7$, $C_{21}:S_3$

Order 120: $S_5$ x 6, $S_3\times F_5$ x 4, $C_2\times A_5$ x 4, $D_5\times A_4$ x 2, $C_5:S_4$ x 2, $C_5\times S_4$ x 2, $C_{10}\times A_4$, $S_3\times D_{10}$, $C_{30}:C_4$, $C_6\times F_5$

Order 112: $D_4\times C_{14}$, $D_4\times D_7$

Order 108: $C_3\times S_3^2$ x 4, $C_3:S_3^2$ x 3, $C_3^2\times A_4$ x 2, $C_3:S_3^2$ x 2, $C_3^3:C_4$, $C_3^2:C_{12}$

Order 105: $C_7:C_{15}$

Order 98: $C_7\times D_7$

Order 96: $D_4\times D_6$ x 28, $C_2^2\times S_4$ x 26, $\GL(2,\mathbb{Z}/4)$ x 19, $C_4\times S_4$ x 14, $D_4\times A_4$ x 12, $C_4:S_4$ x 12, $C_2^2:D_{12}$ x 6, $D_6:D_4$ x 6, $D_6.D_4$ x 5, $C_2^2:S_4$ x 5, $D_6:D_4$ x 4, $C_2^3\times A_4$ x 4, $C_2^3:D_6$ x 4, $C_{12}:D_4$ x 4, $C_2^3.D_6$ x 4, $C_2^3.D_6$ x 3, $C_{12}:C_2^3$ x 3, $C_2^2\times D_{12}$ x 3, $C_{12}:C_2^3$ x 3, $C_2^3:C_{12}$ x 3, $C_2^2.S_4$ x 3, $C_{12}:D_4$ x 3, $C_{12}:D_4$ x 3, $C_{12}:D_4$ x 3, $C_{12}:D_4$ x 3, $C_4:D_{12}$ x 3, $D_6.D_4$ x 2, $C_4\times D_{12}$ x 2, $C_2^3\times D_6$ x 2, $C_2^3:A_4$ x 2, $C_2^4:C_6$ x 2, $D_4\times C_{12}$ x 2, $C_2^4:S_3$ x 2, $C_2^3.D_6$ x 2, $D_{12}:C_4$ x 2, $C_4:D_{12}$, $S_3\times C_4^2$, $C_{12}:D_4$

Order 84: $C_{14}:C_6$ x 8, $S_3\times C_{14}$ x 6, $C_7:C_{12}$ x 4, $C_7:A_4$ x 4, $C_7\times A_4$ x 4, $C_2\times F_7$ x 2, $C_2\times C_{42}$ x 2, $S_3\times D_7$, $C_{84}$, $C_3:C_{28}$, $C_7:C_{12}$

Order 80: $C_2^2\times F_5$ x 7, $D_4\times D_5$ x 6, $D_{10}:C_4$ x 4, $C_4\times F_5$ x 4, $C_{20}:C_4$ x 3, $C_2^2\times D_{10}$, $D_4\times C_{10}$, $C_{10}:D_4$, $C_2\times D_{20}$, $C_4\times D_{10}$

Order 72: $S_3\times D_6$ x 18, $C_3:S_4$ x 18, $S_3\times A_4$ x 17, $C_3\times S_4$ x 17, $C_6\times D_6$ x 6, $C_6\times A_4$ x 6, $\SOPlus(4,2)$ x 6, $C_6\wr C_2$ x 6, $C_3:D_{12}$ x 6, $C_3\times D_{12}$ x 4, $S_3\times C_{12}$ x 4, $D_6:S_3$ x 4, $C_6.D_6$ x 4, $C_6:D_6$ x 3, $C_2\times C_3^2:C_4$ x 3, $C_6^2:C_2$ x 3, $C_6.D_6$ x 3, $D_4\times C_3^2$ x 2, $C_3:D_{12}$ x 2, $C_{12}:S_3$ x 2

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

Order 64: $D_4^2$ x 12, $D_4\times C_2^3$ x 3, $C_2^3:D_4$ x 3, $C_2^3:D_4$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$

Order 63: $C_{21}:C_3$ x 5, $C_3\times C_{21}$

Order 60: $S_3\times D_5$ x 4, $C_{15}:C_4$ x 2, $C_3\times F_5$ x 2, $A_5$ x 2, $C_5\times A_4$, $D_{30}$, $S_3\times C_{10}$, $C_3\times D_{10}$

Order 56: $C_7\times D_4$ x 5, $C_2^2\times C_{14}$ x 2, $C_2\times C_{28}$, $C_7:D_4$, $D_{28}$, $C_4\times D_7$, $C_2\times D_{14}$

Order 54: $C_3^2:C_6$ x 3, $S_3\times C_3^2$ x 3, $C_3^2:S_3$

Order 49: $C_7^2$

Order 48: $S_3\times D_4$ x 61, $C_2\times S_4$ x 42, $C_6:D_4$ x 24, $C_2^2\times D_6$ x 18, $C_4\times D_6$ x 18, $C_6\times D_4$ x 15, $C_2\times D_{12}$ x 15, $C_2^2\times A_4$ x 14, $D_6:C_4$ x 10, $C_4\times A_4$ x 8, $A_4:C_4$ x 8, $C_2^2:C_{12}$ x 5, $C_6.D_4$ x 5, $C_2^2:A_4$ x 3, $C_2^2\times C_{12}$ x 3, $C_6.C_2^3$ x 3, $C_6.D_4$ x 3, $C_{12}:C_4$ x 3, $C_2^3\times C_6$ x 2, $C_4:C_{12}$ x 2, $C_{12}:C_4$ x 2, $C_4\times C_{12}$

Order 42: $C_7:C_6$ x 7, $S_3\times C_7$ x 6, $C_{42}$ x 3, $F_7$ x 3, $D_{21}$, $C_3\times D_7$

Order 40: $C_2\times F_5$ x 13, $C_2\times D_{10}$ x 6, $C_5:D_4$ x 4, $C_4\times D_5$ x 4, $D_{20}$ x 3, $C_5\times D_4$ x 3, $C_2\times C_{20}$, $C_{10}:C_4$, $C_2^2\times C_{10}$

Order 36: $S_3^2$ x 25, $C_6\times S_3$ x 16, $C_3\times A_4$ x 12, $C_6:S_3$ x 8, $C_3:C_{12}$ x 4, $C_3^2:C_4$ x 3, $C_6^2$ x 3, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2

Order 35: $C_{35}$

Order 32: $C_2^2\times D_4$ x 30, $C_4:D_4$ x 20, $C_2^2\wr C_2$ x 16, $C_4\times D_4$ x 14, $C_4:D_4$ x 6, $C_2^3:C_4$ x 5, $C_2^3\times C_4$ x 3, $C_2^5$ x 2, $C_2^2.D_4$ x 2, $C_2\times C_4^2$

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

Order 28: $C_2\times C_{14}$ x 6, $C_{28}$ x 3, $D_{14}$ x 2, $C_7:C_4$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 58, $C_3:D_4$ x 37, $S_4$ x 29, $C_4\times S_3$ x 27, $D_{12}$ x 22, $C_3\times D_4$ x 22, $C_2\times A_4$ x 17, $C_2\times C_{12}$ x 10, $C_6:C_4$ x 10, $C_2^2\times C_6$ x 10

Order 21: $C_7:C_3$ x 6, $C_{21}$ x 3

Order 20: $D_{10}$ x 9, $F_5$ x 6, $C_2\times C_{10}$ x 3, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 18: $C_3\times S_3$ x 18, $C_3:S_3$ x 9, $C_3\times C_6$ x 4

Order 16: $C_2\times D_4$ x 70, $C_2^2:C_4$ x 20, $C_2^2\times C_4$ x 18, $C_2^4$ x 13, $C_4:C_4$ x 7, $C_4^2$ x 4

Order 15: $C_{15}$

Order 14: $C_{14}$ x 4, $D_7$ x 2

Order 12: $D_6$ x 56, $C_2\times C_6$ x 22, $A_4$ x 12, $C_{12}$ x 11, $C_3:C_4$ x 11

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

Order 9: $C_3^2$ x 5

Order 8: $D_4$ x 40, $C_2^3$ x 31, $C_2\times C_4$ x 25

Order 7: $C_7$ x 3

Order 6: $S_3$ x 19, $C_6$ x 14

Order 5: $C_5$

Order 4: $C_2^2$ x 26, $C_4$ x 10

Order 3: $C_3$ x 5

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 846720: $S_7\times \GL(3,2)$ ($C_1$)

Order 423360: $A_7\times \GL(3,2)$ ($C_2$)

Order 5040: $S_7$ ($\PSL(2,7)$)

Order 2520: $A_7$ ($C_2\times \GL(3,2)$)

Order 168: $\PSL(2,7)$ ($S_7$)

Order 1: $C_1$ ($S_7\times \GL(3,2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 846720: $S_7\times \GL(3,2)$ ($C_1$)

Order 423360: $A_7\times \GL(3,2)$ ($C_2$)

Order 5040: $S_7$ ($\PSL(2,7)$)

Order 2520: $A_7$ ($C_2\times \GL(3,2)$)

Order 168: $\PSL(2,7)$ ($S_7$)

Order 1: $C_1$ ($S_7\times \GL(3,2)$)

Series

Derived series	$S_7\times \GL(3,2)$	$\rhd$	$A_7\times \GL(3,2)$
Chief series	$S_7\times \GL(3,2)$	$\rhd$	$A_7\times \GL(3,2)$	$\rhd$	$A_7$	$\rhd$	$C_1$
Lower central series	$S_7\times \GL(3,2)$	$\rhd$	$A_7\times \GL(3,2)$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

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

Character theory

Complex character table

See the $90 \times 90$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

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