Abstract group 3111696.f: $C_7^4:(C_2\times C_3^3:S_4)$

• Label: 3111696.f
• Order: \( 2^{4} \cdot 3^{4} \cdot 7^{4} \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{5} \cdot 7^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 3 \)
• Perm deg.: $28$
• Trans deg.: $28$
• Rank: $2$

Group information

Description:	$C_7^4:(C_2\times C_3^3:S_4)$
Order:	\(3111696\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7^{4} \)
Exponent:	\(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \)
Automorphism group:	$C_7^4:(C_2\times C_3\wr S_4)$, of order \(9335088\)\(\medspace = 2^{4} \cdot 3^{5} \cdot 7^{4} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4, $C_7$ x 4
Derived length:	$5$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	7	9	14	18	21	28	42
Elements	1	11347	21266	111132	889742	2400	345744	207144	345744	214032	666792	296352	3111696
Conjugacy classes	1	5	5	2	17	14	2	27	2	15	6	4	100
Divisions	1	5	4	2	12	8	1	15	1	8	2	3	62
Autjugacy classes	1	5	5	2	17	8	2	15	2	9	2	4	72

Dimension	1	2	3	4	6	8	12	16	24	48	54	72	108	144	162	216	324	432	486	648	972	1296
Irr. complex chars.	4	2	4	8	8	4	4	0	2	4	6	4	11	0	18	12	3	4	0	2	0	0	100
Irr. rational chars.	4	2	4	0	4	4	6	2	2	1	0	0	8	3	2	6	1	1	6	4	1	1	62

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$28$
Rank:	$2$
Inequivalent generating pairs:	$187200$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid d^{6}=e^{6}=f^{7}=g^{21}=h^{7}=i^{7}= \!\cdots\! \rangle}$
Permutation group:	Degree $28$ $\langle(2,7)(3,6)(4,5)(8,27,13,28,11,22,9,23,14,24,12,25,10,26)(15,17)(18,21)(19,20) \!\cdots\! \rangle$
Transitive group:	28T1371				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_7^4:C_3:S_3^2)$ $\,\rtimes\,$ $D_6$	$C_7^4$ $\,\rtimes\,$ $(C_2\times C_3^3:S_4)$	$(C_7^4:(C_6:S_3^2))$ $\,\rtimes\,$ $S_3$	$(C_7^4:(C_3^3:S_4))$ $\,\rtimes\,$ $C_2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 28.

Homology

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

Subgroups

There are 10088034 subgroups in 1491 conjugacy classes, 12 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_7^4:(C_2\times C_3^3:S_4)$
Commutator:	$G' \simeq$ $C_7^4:(C_3^3:A_4)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_7^4:(C_2\times C_3^3:S_4)$
Fitting:	$\operatorname{Fit} \simeq$ $C_7^4$	$G/\operatorname{Fit} \simeq$ $C_2\times C_3^3:S_4$
Radical:	$R \simeq$ $C_7^4:(C_2\times C_3^3:S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_7^4$	$G/\operatorname{soc} \simeq$ $C_2\times C_3^3:S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^4$

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 3111696: $C_7^4:(C_2\times C_3^3:S_4)$

Order 1555848: $C_7^4:(C_2\times C_3^3:A_4)$, $C_7^4:(C_3^3:S_4)$, $C_7^4:(C_3^3:S_4)$

Order 1037232: $C_7^4:(S_3^2:D_6)$

Order 777924: $C_7^3.C_{21}.\He_3.C_2^2$, $C_7^4:(C_3^3:A_4)$

Order 518616: $C_7^4:(S_3^2:S_3)$ x 2, $C_7^4:(S_3^2:S_3)$ x 2, $C_7^4.C_3^3.C_2^3$, $C_7^4:(C_6:S_3^2)$, $C_7^4:(C_2\times C_3^3:C_4)$

Order 388962: $C_7^4.C_3^3.C_6$, $C_7^3.C_{21}.\He_3.C_2$, $C_7^3.C_{21}.\He_3.C_2$

Order 345744: $C_7^4.S_3^2.C_2^2$

Order 259308: $C_7^4:C_3:S_3^2$ x 2, $C_7^4:(C_3^3:C_4)$ x 2, $C_7^4.C_3^3.C_2^2$, $C_7^4.C_3^3.C_2^2$, $C_7^4.C_3.C_6^2$, $C_7^3.C_{21}.C_3.D_6$, $C_7^4.C_3^3.C_2^2$, $C_7^4:(C_3^2:D_6)$

Order 194481: $C_7^4.C_3.\He_3$

Order 172872: $C_7^4:\SOPlus(4,2)$ x 2, $C_7^4:\SOPlus(4,2)$ x 2, $C_7^4.C_6^2.C_2$, $C_7^4.C_3^2.C_2^3$, $C_7^4:(C_2\times C_3^2:C_4)$, $C_7^4:(S_3\times D_6)$

Order 129654: $C_7^4:(C_3^2:C_6)$ x 2, $C_7^4.C_3^3.C_2$, $C_7^4.C_3^3.C_2$, $C_7^4.C_3^3.C_2$, $C_7^3.C_{21}.C_3.S_3$, $C_7^4.C_3^2.C_6$, $C_7^4.\He_3.C_2$, $C_7\times C_7^3:(C_3^2:S_3)$

Order 115248: $C_7^4:C_6:D_4$, $C_7^4:(C_2\times S_4)$

Order 111132: $C_7^3:(C_3^3:D_6)$

Order 86436: $C_7^4:S_3^2$ x 3, $C_7^4.(C_6\times S_3)$ x 2, $C_7^4:(C_3^2:C_4)$ x 2, $C_7^4.S_3^2$, $C_7^4.(C_6\times S_3)$, $C_7^4.(C_6\times S_3)$, $C_7^4.C_6^2$, $C_7^4.(C_6\times S_3)$, $C_7^4.S_3^2$, $C_7^4.S_3^2$, $C_7^4.(C_6\times S_3)$, $C_7^4.(C_6\times S_3)$, $C_7^4.(C_6\times S_3)$, $C_7^4:(C_6\times S_3)$, $C_7^3:C_{21}:D_6$, $C_7^4:(C_6\times S_3)$

Order 64827: $C_7^4:C_3^3$, $C_7^4:(C_9:C_3)$, $C_7\times C_7^3:\He_3$

Order 57624: $C_7^2\wr C_2.D_6$ x 2, $C_7^2\wr C_2.D_6$ x 2, $C_7^4.C_6.C_2^2$, $C_7^2\wr C_2.D_6$, $C_7^4:(C_2\times A_4)$, $C_7^4:S_4$, $C_7\wr S_4$, $C_7^3:D_6\times D_7$, $C_7^4:(C_2\times D_6)$

Order 55566: $C_7^3:C_3\wr S_3$, $C_7^3:\He_3:C_6$, $C_7^3:C_3\wr S_3$

Order 43218: $C_7^4:(C_3\times S_3)$ x 2, $C_7^4:(C_3\times C_6)$ x 2, $C_7^4:(C_3\times S_3)$ x 2, $C_7^3:C_{21}:S_3$ x 2, $C_7^4:(C_3\times S_3)$ x 2, $C_7^4:(C_3\times S_3)$, $C_7\times C_7^3:(C_3\times S_3)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times C_6)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times C_6)$, $C_7^4:(C_3\times S_3)$, $C_7^4:C_{18}$, $C_7^4:(C_3\times C_6)$, $C_7^4:(C_3\times C_6)$

Order 38416: $D_7^2:D_7^2$

Order 37044: $C_7^3:C_3^2:D_6$, $C_7\wr C_3:C_6^2$

Order 28812: $C_7^4:D_6$ x 3, $C_7^4:D_6$ x 2, $C_7^2\wr C_2.S_3$ x 2, $C_7^4:(C_2\times C_6)$, $C_7^4:(C_2\times C_6)$, $C_7^4:(C_2\times C_6)$, $C_7^4:(C_2\times C_6)$, $C_7^4:(C_2\times C_6)$, $C_7^4:D_6$, $C_7^4:D_6$, $C_7^3:D_{42}$, $C_7^3:D_{42}$, $C_7^4:D_6$, $C_7^4:D_6$, $C_7^3:C_6\times D_7$, $C_7^4:(C_2\times C_6)$, $C_7\wr A_4$

Order 27783: $C_7^3:C_3\wr C_3$

Order 24696: $C_7^3.C_6^2.C_2$, $C_7^3.C_6^2.C_2$

Order 21609: $C_7^4:C_9$, $C_7^4:C_3^2$, $C_7^4:C_3^2$, $C_7^4:C_3^2$, $C_7\times C_7^3:C_3^2$

Order 19208: $C_7^3:D_{28}$ x 2, $C_7\wr D_4$ x 2, $C_7^4:C_2^3$, $C_7:D_7^3$, $C_7^4:(C_2\times C_4)$

Order 18522: $C_7^3:(C_3^2\times C_6)$, $C_7^3:(S_3\times C_3^2)$, $C_7^3:(S_3\times C_3^2)$, $C_7^3:C_9:C_6$, $C_7^3:C_3^2:S_3$, $C_7^3:C_3^2:S_3$, $C_7^3:(C_2\times \He_3)$

Order 16464: $D_7\wr S_3$

Order 14406: $C_7^4:S_3$ x 2, $C_7^4:C_6$ x 2, $C_7^3:D_{21}$ x 2, $C_7^4:S_3$ x 2, $C_7^4:C_6$ x 2, $C_7^4:C_6$, $C_7^4:C_6$, $C_7^4:C_6$, $C_7^4:S_3$, $C_7\times C_7^3:S_3$, $C_7^4:C_6$, $C_7^4:C_6$, $C_7^4:C_6$, $C_7^3:C_{42}$, $C_7^3:C_{42}$

Order 12348: $C_7^3:(C_6\times S_3)$ x 7, $C_7^3:(C_6\times S_3)$ x 2, $C_7^3:(C_6\times S_3)$ x 2, $C_7^3:(C_6\times S_3)$ x 2, $C_7^3:C_6^2$, $C_7^3:(C_6\times S_3)$, $C_7^3:(C_6\times S_3)$, $C_7^3:C_6^2$, $C_7^3:(C_6\times S_3)$, $C_7^3:(C_6\times S_3)$, $C_7^3:(C_6\times S_3)$, $C_7:F_7^2$

Order 10584: $C_7^2:(C_6\times S_3^2)$

Order 9604: $C_7\wr C_2^2$ x 2, $C_7\wr C_4$ x 2, $C_7^4:C_2^2$, $C_7^4:C_2^2$, $C_7^4:C_2^2$, $C_7^4:C_2^2$, $C_7^4:C_2^2$

Order 9261: $C_7^3:C_3^3$, $C_7^3:C_9:C_3$, $C_7^3:\He_3$

Order 8232: $C_7^3:(C_2^2\times C_6)$, $C_7^3:(C_2^2\times C_6)$, $C_2\times C_7^3:D_6$, $C_7^2:D_6\times D_7$, $C_7^3:S_4$, $C_7^3:S_4$, $D_7\wr C_3$

Order 7203: $C_7^3:C_{21}$ x 2, $C_7^4:C_3$, $C_7\times C_7^3:C_3$

Order 6174: $C_7^3:(C_3\times S_3)$ x 5, $C_7^3:(C_3\times S_3)$ x 5, $C_7^3:(C_3\times C_6)$ x 4, $C_7^3:(C_3\times C_6)$ x 2, $C_7^2:(C_3\times D_{21})$ x 2, $C_7^2:(S_3\times C_{21})$ x 2, $C_7^3:(C_3\times S_3)$ x 2, $C_7^3:(C_3\times S_3)$ x 2, $C_7^3:(C_3\times C_6)$ x 2, $C_7^3:(C_3\times C_6)$, $C_7^3:(C_3\times C_6)$, $C_7^3:(C_3\times C_6)$, $C_7^3:(C_3\times C_6)$, $C_7^3:(C_3\times C_6)$, $C_7^3:C_{18}$, $C_7^3:(C_3\times S_3)$, $C_7^3:(C_3\times S_3)$, $C_7^3:(C_3\times C_6)$, $C_7^3:(C_3\times C_6)$

Order 5488: $D_7^3:C_2$ x 2

Order 5292: $C_7^2:(C_3\times S_3^2)$ x 2, $C_7^2:(C_3\times S_3^2)$ x 2, $C_7^2:(C_3\times C_6\times S_3)$, $C_7^2:(C_3^2\times D_6)$, $C_{21}:F_7:C_6$

Order 4802: $C_7^2\wr C_2$ x 2, $C_7\times C_7^3:C_2$, $C_7^4:C_2$, $C_7^3\times D_7$

Order 4116: $C_7^3:D_6$ x 11, $C_7^3:(C_2\times C_6)$ x 7, $C_7:(D_7\times F_7)$ x 4, $C_7^3:(C_2\times C_6)$ x 2, $C_7^3:D_6$ x 2, $C_7^3:(C_2\times C_6)$, $C_7^3:(C_2\times C_6)$, $C_7^3:(C_2\times C_6)$, $C_7^3:(C_2\times C_6)$, $C_7^3:(C_2\times C_6)$, $C_7^3:(C_2\times C_6)$, $C_7^3:(C_2\times C_6)$, $C_7^2:D_{42}$, $C_7^3:D_6$, $C_7^3:D_6$, $C_7^2:D_{42}$, $D_7\times C_7:F_7$, $C_7^3:A_4$, $C_7^3:(C_2\times C_6)$, $C_2\times C_7^3:C_6$, $D_7\times C_7:F_7$

Order 3528: $S_3\times C_7^2:D_6$, $S_3\times D_7:F_7$, $C_7^2:(C_6\times D_6)$

Order 3087: $C_7^3:C_3^2$ x 3, $C_7^3:C_3^2$ x 2, $C_7^3:C_3^2$, $C_7^3:C_3^2$, $C_7\times C_7^2:C_3:C_3$, $C_7^3:C_3^2$, $C_7^3:C_9$

Order 2744: $D_7^3$ x 5, $C_7^3:D_4$ x 4, $C_7^2:C_4\times D_7$ x 2, $D_7^2:C_{14}$ x 2, $C_7^2:D_{28}$ x 2, $C_{14}:D_7^2$

Order 2646: $C_7^2:C_3^2:C_6$ x 2, $C_7^2:(S_3\times C_3^2)$ x 2, $C_7^2:(C_3^2\times S_3)$, $C_7^2:(C_3^2\times S_3)$, $C_{21}:(C_3\times F_7)$

Order 2401: $C_7^4$

Order 2352: $D_7^2:D_6$, $C_{14}^2:D_6$

Order 2268: $C_7:(C_2\times C_3\wr S_3)$

Order 2058: $C_7^2:D_{21}$ x 9, $C_7\wr S_3$ x 9, $C_7^3:C_6$ x 8, $C_7^3:C_6$ x 6, $C_7^3:C_6$ x 5, $C_7^3:C_6$ x 5, $C_7^3:C_6$ x 4, $C_7^3:C_6$ x 4, $C_7^2:C_{42}$ x 4, $C_7^3:C_6$ x 2, $C_7^3:C_6$, $C_7^3:C_6$, $C_7^3:C_6$, $C_7^3:C_6$, $C_7^3:C_6$, $C_7^3:C_6$, $C_7^2:C_{42}$, $C_7^2:C_{42}$, $C_7^2:C_{42}$, $C_7^2:C_{42}$, $C_7^3:C_6$, $C_7^3:C_6$

Order 1764: $C_7^2:(C_6\times S_3)$ x 7, $S_3\times C_7:F_7$ x 2, $S_3\times C_7:F_7$ x 2, $D_{21}:F_7$ x 2, $C_7^2:S_3^2$ x 2, $C_7^2:S_3^2$ x 2, $C_7^2:C_6^2$ x 2, $C_7^2:(C_6\times S_3)$ x 2, $D_{21}:F_7$, $S_3\times C_7:F_7$, $S_3\times C_7:F_7$, $C_7^2:C_6^2$, $(C_7\times C_{21}):D_6$, $C_3\times C_7^2:D_6$, $F_7^2$

Order 1372: $C_7:D_7^2$ x 16, $C_7\times D_7^2$ x 10, $C_7:D_7^2$ x 5, $C_7^3:C_4$ x 2, $C_7^2:C_{28}$ x 2, $C_7^2:D_{14}$, $C_7^2:D_{14}$, $D_{14}\times C_7^2$

Order 1323: $C_7^2:C_3^3$

Order 1296: $C_2\times C_3^3:S_4$

Order 1176: $D_7^2:S_3$ x 4, $C_2\times C_7^2:D_6$ x 3, $(C_7\times C_{14}).D_6$ x 2, $D_{14}:F_7$ x 2, $C_{14}^2:S_3$ x 2, $(C_7\times C_{14}).D_6$, $C_{14}^2:C_6$, $S_3\times D_7^2$, $D_{14}:F_7$, $C_7:F_7:C_4$

Order 1134: $\He_3:F_7$, $(C_7\times \He_3):C_6$, $\He_3:F_7$

Order 1029: $C_7\wr C_3$ x 7, $C_7^2:C_{21}$ x 5, $C_7^3:C_3$ x 3, $C_7^2:C_{21}$ x 2, $C_7^3:C_3$

Order 882: $C_7^2:(C_3\times S_3)$ x 10, $C_7:(C_3\times F_7)$ x 9, $C_{21}:F_7$ x 2, $C_{21}:F_7$ x 2, $C_{21}:C_{42}$ x 2, $C_{21}:F_7$ x 2, $C_{21}:F_7$ x 2, $(C_7\times C_{21}):S_3$ x 2, $C_3\times C_7^2:S_3$ x 2, $C_2\times C_7^2:C_3^2$ x 2, $C_7:C_3\times F_7$ x 2, $C_{21}:F_7$, $C_{21}:F_7$, $C_7^2:C_3\times S_3$, $C_7^2:C_3\times S_3$, $C_{21}:F_7$, $C_{21}:F_7$

Order 784: $D_7^2:C_2^2$, $D_{14}:D_{14}$

Order 756: $\He_3:D_{14}$, $C_{21}:C_6^2$

Order 686: $C_7^2:C_{14}$ x 14, $D_7\times C_7^2$ x 14, $C_7^3:C_2$ x 9, $C_7^2\times C_{14}$

Order 648: $C_3^3:S_4$ x 2, $C_2\times C_3^3:A_4$

Order 588: $C_7^2:D_6$ x 11, $C_{14}:F_7$ x 8, $D_7:F_7$ x 8, $D_7:F_7$ x 8, $C_{14}:F_7$ x 5, $C_7^2:D_6$ x 5, $C_{14}:F_7$ x 3, $D_7\times D_{21}$ x 2, $C_{21}:D_{14}$ x 2, $C_{21}:D_{14}$ x 2, $C_{14}:F_7$ x 2, $C_{14}:F_7$ x 2, $D_7\times F_7$ x 2, $C_7^2:C_3:C_4$ x 2, $(C_7\times C_{14}).S_3$ x 2, $C_{14}^2:C_3$, $C_{21}:D_{14}$, $C_3\times D_7^2$

Order 567: $C_{21}.\He_3$

Order 504: $D_6\times F_7$ x 2

Order 441: $C_7^2:C_3^2$ x 7, $C_{21}:C_{21}$ x 2, $C_7^2:C_3^2$, $C_7^2:C_3^2$

Order 432: $S_3^2:D_6$

Order 392: $D_7\times D_{14}$ x 7, $D_7\wr C_2$ x 4, $C_{14}\wr C_2$ x 2, $C_7:D_{28}$ x 2, $C_{14}:D_{14}$, $C_2\times C_7^2:C_4$, $C_{14}.D_{14}$

Order 378: $C_3^2:F_7$, $C_3\times C_{21}:C_6$, $C_3^2\times F_7$, $C_3^2:D_{21}$, $\He_3:C_{14}$, $D_7\times \He_3$, $C_3^2.F_7$

Order 343: $C_7^3$ x 8

Order 336: $D_7\times S_4$

Order 324: $C_3^3:D_6$, $C_3^3:A_4$

Order 294: $C_7:F_7$ x 17, $C_7:F_7$ x 14, $C_7^2:S_3$ x 12, $C_7:F_7$ x 12, $C_7:F_7$ x 8, $C_7:F_7$ x 8, $C_7^2:C_6$ x 6, $C_7:C_{42}$ x 5, $C_7^2:C_6$ x 2, $C_7\times F_7$ x 2, $C_7:D_{21}$ x 2, $C_7\times D_{21}$ x 2, $S_3\times C_7^2$ x 2, $C_{21}:D_7$ x 2, $D_7\times C_{21}$ x 2, $C_7^2:C_6$ x 2

Order 252: $S_3\times F_7$ x 12, $C_6\times F_7$ x 3, $C_{21}:D_6$ x 2, $C_{42}:C_6$ x 2, $C_{42}:C_6$ x 2

Order 216: $S_3^2:S_3$ x 4, $C_6:S_3^2$, $C_6\times S_3^2$, $C_2\times C_3^3:C_4$

Order 196: $D_7^2$ x 31, $C_7:D_{14}$ x 13, $C_7\times D_{14}$ x 12, $C_7^2:C_4$ x 2, $C_7:C_{28}$ x 2, $C_{14}^2$

Order 189: $C_{21}.C_3^2$, $C_7:C_3^3$, $C_7\times \He_3$

Order 168: $S_3\times D_{14}$ x 2, $C_2^2\times F_7$ x 2, $A_4\times D_7$, $C_7:S_4$, $C_7\times S_4$

Order 162: $C_3\wr S_3$ x 2, $C_3^3:C_6$

Order 147: $C_7:C_{21}$ x 12, $C_7^2:C_3$ x 8, $C_7^2:C_3$ x 6, $C_7\times C_{21}$ x 2

Order 144: $S_3^2:C_2^2$

Order 126: $C_3\times F_7$ x 10, $C_{21}:C_6$ x 8, $C_{21}:C_6$ x 8, $C_{21}:C_6$ x 3, $C_3\times D_{21}$ x 2, $S_3\times C_{21}$ x 2, $C_3^2\times D_7$ x 2, $C_7:C_{18}$

Order 112: $D_4\times D_7$ x 2

Order 108: $C_3\times S_3^2$ x 3, $C_3:S_3^2$ x 2, $C_3^3:C_4$ x 2, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2:D_6$

Order 98: $C_7\times D_7$ x 36, $C_7:D_7$ x 30, $C_7\times C_{14}$ x 11

Order 84: $C_2\times F_7$ x 19, $S_3\times D_7$ x 13, $C_3\times D_{14}$ x 3, $C_{14}:C_6$ x 2, $D_{42}$ x 2, $S_3\times C_{14}$ x 2, $C_7\times A_4$

Order 81: $C_3\wr C_3$

Order 72: $\SOPlus(4,2)$ x 4, $S_3\times D_6$ x 2, $C_6\times D_6$, $C_2\times C_3^2:C_4$

Order 63: $C_{21}:C_3$ x 7, $C_3\times C_{21}$ x 2, $C_7:C_9$

Order 56: $C_2\times D_{14}$ x 6, $C_7:D_4$ x 4, $C_7\times D_4$ x 2, $D_{28}$ x 2, $C_4\times D_7$ x 2

Order 54: $C_3^2:S_3$ x 2, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2\times C_6$, $C_9:C_6$, $C_2\times \He_3$

Order 49: $C_7^2$ x 19

Order 48: $C_2\times S_4$, $C_6:D_4$

Order 42: $F_7$ x 23, $C_7:C_6$ x 13, $C_3\times D_7$ x 11, $D_{21}$ x 9, $S_3\times C_7$ x 9, $C_{42}$ x 3

Order 36: $C_6\times S_3$ x 10, $S_3^2$ x 6, $C_3^2:C_4$ x 2, $C_6^2$, $C_6:S_3$

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

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $C_3:D_4$ x 4, $C_2\times D_6$ x 2, $S_4$ x 2, $C_6:C_4$, $C_2^2\times C_6$, $C_2\times A_4$

Order 21: $C_7:C_3$ x 10, $C_{21}$ x 8

Order 18: $C_3\times S_3$ x 12, $C_3\times C_6$ x 6, $C_3:S_3$ x 2, $C_{18}$

Order 16: $C_2\times D_4$

Order 14: $D_7$ x 23, $C_{14}$ x 15

Order 12: $D_6$ x 11, $C_2\times C_6$ x 7, $C_3:C_4$ x 2, $A_4$

Order 9: $C_3^2$ x 4, $C_9$

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 7: $C_7$ x 8

Order 6: $C_6$ x 12, $S_3$ x 8

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$ x 4

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3111696: $C_7^4:(C_2\times C_3^3:S_4)$

Order 1555848: $C_7^4:(C_2\times C_3^3:A_4)$, $C_7^4:(C_3^3:S_4)$, $C_7^4:(C_3^3:S_4)$

Order 1037232: $C_7^4:(S_3^2:D_6)$

Order 777924: $C_7^3.C_{21}.\He_3.C_2^2$, $C_7^4:(C_3^3:A_4)$

Order 518616: $C_7^4:(S_3^2:S_3)$ x 2, $C_7^4:(S_3^2:S_3)$ x 2, $C_7^4.C_3^3.C_2^3$, $C_7^4:(C_6:S_3^2)$, $C_7^4:(C_2\times C_3^3:C_4)$

Order 388962: $C_7^4.C_3^3.C_6$, $C_7^3.C_{21}.\He_3.C_2$, $C_7^3.C_{21}.\He_3.C_2$

Order 345744: $C_7^4.S_3^2.C_2^2$

Order 259308: $C_7^4:C_3:S_3^2$ x 2, $C_7^4:(C_3^3:C_4)$ x 2, $C_7^4.C_3^3.C_2^2$, $C_7^4.C_3^3.C_2^2$, $C_7^4.C_3.C_6^2$, $C_7^3.C_{21}.C_3.D_6$, $C_7^4.C_3^3.C_2^2$, $C_7^4:(C_3^2:D_6)$

Order 194481: $C_7^4.C_3.\He_3$

Order 172872: $C_7^4:\SOPlus(4,2)$ x 2, $C_7^4:\SOPlus(4,2)$ x 2, $C_7^4.C_6^2.C_2$, $C_7^4.C_3^2.C_2^3$, $C_7^4:(C_2\times C_3^2:C_4)$, $C_7^4:(S_3\times D_6)$

Order 129654: $C_7^4:(C_3^2:C_6)$ x 2, $C_7^4.C_3^3.C_2$, $C_7^4.C_3^3.C_2$, $C_7^4.C_3^3.C_2$, $C_7^3.C_{21}.C_3.S_3$, $C_7^4.C_3^2.C_6$, $C_7^4.\He_3.C_2$, $C_7\times C_7^3:(C_3^2:S_3)$

Order 115248: $C_7^4:C_6:D_4$, $C_7^4:(C_2\times S_4)$

Order 111132: $C_7^3:(C_3^3:D_6)$

Order 86436: $C_7^4:S_3^2$ x 3, $C_7^4.(C_6\times S_3)$ x 2, $C_7^4:(C_3^2:C_4)$ x 2, $C_7^4.S_3^2$, $C_7^4.(C_6\times S_3)$, $C_7^4.(C_6\times S_3)$, $C_7^4.C_6^2$, $C_7^4.(C_6\times S_3)$, $C_7^4.S_3^2$, $C_7^4.S_3^2$, $C_7^4.(C_6\times S_3)$, $C_7^4.(C_6\times S_3)$, $C_7^4.(C_6\times S_3)$, $C_7^4:(C_6\times S_3)$, $C_7^3:C_{21}:D_6$, $C_7^4:(C_6\times S_3)$

Order 64827: $C_7^4:C_3^3$, $C_7^4:(C_9:C_3)$, $C_7\times C_7^3:\He_3$

Order 57624: $C_7^2\wr C_2.D_6$ x 2, $C_7^2\wr C_2.D_6$ x 2, $C_7^4.C_6.C_2^2$, $C_7^2\wr C_2.D_6$, $C_7^4:(C_2\times A_4)$, $C_7^4:S_4$, $C_7\wr S_4$, $C_7^3:D_6\times D_7$, $C_7^4:(C_2\times D_6)$

Order 55566: $C_7^3:C_3\wr S_3$, $C_7^3:\He_3:C_6$, $C_7^3:C_3\wr S_3$

Order 43218: $C_7^4:(C_3\times S_3)$ x 2, $C_7^4:(C_3\times C_6)$ x 2, $C_7^4:(C_3\times S_3)$ x 2, $C_7^3:C_{21}:S_3$ x 2, $C_7^4:(C_3\times S_3)$ x 2, $C_7^4:(C_3\times S_3)$, $C_7\times C_7^3:(C_3\times S_3)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times C_6)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times S_3)$, $C_7^4:(C_3\times C_6)$, $C_7^4:(C_3\times S_3)$, $C_7^4:C_{18}$, $C_7^4:(C_3\times C_6)$, $C_7^4:(C_3\times C_6)$

Order 38416: $D_7^2:D_7^2$

Order 37044: $C_7^3:C_3^2:D_6$, $C_7\wr C_3:C_6^2$

Order 28812: $C_7^4:D_6$ x 3, $C_7^4:D_6$ x 2, $C_7^2\wr C_2.S_3$ x 2, $C_7^4:(C_2\times C_6)$, $C_7^4:(C_2\times C_6)$, $C_7^4:(C_2\times C_6)$, $C_7^4:(C_2\times C_6)$, $C_7^4:(C_2\times C_6)$, $C_7^4:D_6$, $C_7^4:D_6$, $C_7^3:D_{42}$, $C_7^3:D_{42}$, $C_7^4:D_6$, $C_7^4:D_6$, $C_7^3:C_6\times D_7$, $C_7^4:(C_2\times C_6)$, $C_7\wr A_4$

Order 27783: $C_7^3:C_3\wr C_3$

Order 24696: $C_7^3.C_6^2.C_2$, $C_7^3.C_6^2.C_2$

Order 21609: $C_7^4:C_9$, $C_7^4:C_3^2$, $C_7^4:C_3^2$, $C_7^4:C_3^2$, $C_7\times C_7^3:C_3^2$

Order 19208: $C_7^3:D_{28}$ x 2, $C_7\wr D_4$ x 2, $C_7^4:C_2^3$, $C_7:D_7^3$, $C_7^4:(C_2\times C_4)$

Order 18522: $C_7^3:(C_3^2\times C_6)$, $C_7^3:(S_3\times C_3^2)$, $C_7^3:(S_3\times C_3^2)$, $C_7^3:C_9:C_6$, $C_7^3:C_3^2:S_3$, $C_7^3:C_3^2:S_3$, $C_7^3:(C_2\times \He_3)$

Order 16464: $D_7\wr S_3$

Order 14406: $C_7^4:S_3$ x 2, $C_7^4:C_6$ x 2, $C_7^3:D_{21}$ x 2, $C_7^4:S_3$ x 2, $C_7^4:C_6$ x 2, $C_7^4:C_6$, $C_7^4:C_6$, $C_7^4:C_6$, $C_7^4:S_3$, $C_7\times C_7^3:S_3$, $C_7^4:C_6$, $C_7^4:C_6$, $C_7^4:C_6$, $C_7^3:C_{42}$, $C_7^3:C_{42}$

Order 12348: $C_7^3:(C_6\times S_3)$ x 7, $C_7^3:(C_6\times S_3)$ x 2, $C_7^3:(C_6\times S_3)$ x 2, $C_7^3:(C_6\times S_3)$ x 2, $C_7^3:C_6^2$, $C_7^3:(C_6\times S_3)$, $C_7^3:(C_6\times S_3)$, $C_7^3:C_6^2$, $C_7^3:(C_6\times S_3)$, $C_7^3:(C_6\times S_3)$, $C_7^3:(C_6\times S_3)$, $C_7:F_7^2$

Order 10584: $C_7^2:(C_6\times S_3^2)$

Order 9604: $C_7\wr C_2^2$ x 2, $C_7\wr C_4$ x 2, $C_7^4:C_2^2$, $C_7^4:C_2^2$, $C_7^4:C_2^2$, $C_7^4:C_2^2$, $C_7^4:C_2^2$

Order 9261: $C_7^3:C_3^3$, $C_7^3:C_9:C_3$, $C_7^3:\He_3$

Order 8232: $C_7^3:(C_2^2\times C_6)$, $C_7^3:(C_2^2\times C_6)$, $C_2\times C_7^3:D_6$, $C_7^2:D_6\times D_7$, $C_7^3:S_4$, $C_7^3:S_4$, $D_7\wr C_3$

Order 7203: $C_7^3:C_{21}$ x 2, $C_7^4:C_3$, $C_7\times C_7^3:C_3$

Order 6174: $C_7^3:(C_3\times S_3)$ x 5, $C_7^3:(C_3\times S_3)$ x 5, $C_7^3:(C_3\times C_6)$ x 4, $C_7^3:(C_3\times C_6)$ x 2, $C_7^2:(C_3\times D_{21})$ x 2, $C_7^2:(S_3\times C_{21})$ x 2, $C_7^3:(C_3\times S_3)$ x 2, $C_7^3:(C_3\times S_3)$ x 2, $C_7^3:(C_3\times C_6)$ x 2, $C_7^3:(C_3\times C_6)$, $C_7^3:(C_3\times C_6)$, $C_7^3:(C_3\times C_6)$, $C_7^3:(C_3\times C_6)$, $C_7^3:(C_3\times C_6)$, $C_7^3:C_{18}$, $C_7^3:(C_3\times S_3)$, $C_7^3:(C_3\times S_3)$, $C_7^3:(C_3\times C_6)$, $C_7^3:(C_3\times C_6)$

Order 5488: $D_7^3:C_2$ x 2

Order 5292: $C_7^2:(C_3\times S_3^2)$ x 2, $C_7^2:(C_3\times S_3^2)$ x 2, $C_7^2:(C_3\times C_6\times S_3)$, $C_7^2:(C_3^2\times D_6)$, $C_{21}:F_7:C_6$

Order 4802: $C_7^2\wr C_2$ x 2, $C_7\times C_7^3:C_2$, $C_7^4:C_2$, $C_7^3\times D_7$

Order 4116: $C_7^3:D_6$ x 11, $C_7^3:(C_2\times C_6)$ x 7, $C_7:(D_7\times F_7)$ x 4, $C_7^3:(C_2\times C_6)$ x 2, $C_7^3:D_6$ x 2, $C_7^3:(C_2\times C_6)$, $C_7^3:(C_2\times C_6)$, $C_7^3:(C_2\times C_6)$, $C_7^3:(C_2\times C_6)$, $C_7^3:(C_2\times C_6)$, $C_7^3:(C_2\times C_6)$, $C_7^3:(C_2\times C_6)$, $C_7^2:D_{42}$, $C_7^3:D_6$, $C_7^3:D_6$, $C_7^2:D_{42}$, $D_7\times C_7:F_7$, $C_7^3:A_4$, $C_7^3:(C_2\times C_6)$, $C_2\times C_7^3:C_6$, $D_7\times C_7:F_7$

Order 3528: $S_3\times C_7^2:D_6$, $S_3\times D_7:F_7$, $C_7^2:(C_6\times D_6)$

Order 3087: $C_7^3:C_3^2$ x 3, $C_7^3:C_3^2$ x 2, $C_7^3:C_3^2$, $C_7^3:C_3^2$, $C_7\times C_7^2:C_3:C_3$, $C_7^3:C_3^2$, $C_7^3:C_9$

Order 2744: $D_7^3$ x 5, $C_7^3:D_4$ x 4, $C_7^2:C_4\times D_7$ x 2, $D_7^2:C_{14}$ x 2, $C_7^2:D_{28}$ x 2, $C_{14}:D_7^2$

Order 2646: $C_7^2:C_3^2:C_6$ x 2, $C_7^2:(S_3\times C_3^2)$ x 2, $C_7^2:(C_3^2\times S_3)$, $C_7^2:(C_3^2\times S_3)$, $C_{21}:(C_3\times F_7)$

Order 2401: $C_7^4$

Order 2352: $D_7^2:D_6$, $C_{14}^2:D_6$

Order 2268: $C_7:(C_2\times C_3\wr S_3)$

Order 2058: $C_7^2:D_{21}$ x 9, $C_7\wr S_3$ x 9, $C_7^3:C_6$ x 8, $C_7^3:C_6$ x 6, $C_7^3:C_6$ x 5, $C_7^3:C_6$ x 5, $C_7^3:C_6$ x 4, $C_7^3:C_6$ x 4, $C_7^2:C_{42}$ x 4, $C_7^3:C_6$ x 2, $C_7^3:C_6$, $C_7^3:C_6$, $C_7^3:C_6$, $C_7^3:C_6$, $C_7^3:C_6$, $C_7^3:C_6$, $C_7^2:C_{42}$, $C_7^2:C_{42}$, $C_7^2:C_{42}$, $C_7^2:C_{42}$, $C_7^3:C_6$, $C_7^3:C_6$

Order 1764: $C_7^2:(C_6\times S_3)$ x 7, $S_3\times C_7:F_7$ x 2, $S_3\times C_7:F_7$ x 2, $D_{21}:F_7$ x 2, $C_7^2:S_3^2$ x 2, $C_7^2:S_3^2$ x 2, $C_7^2:C_6^2$ x 2, $C_7^2:(C_6\times S_3)$ x 2, $D_{21}:F_7$, $S_3\times C_7:F_7$, $S_3\times C_7:F_7$, $C_7^2:C_6^2$, $(C_7\times C_{21}):D_6$, $C_3\times C_7^2:D_6$, $F_7^2$

Order 1372: $C_7:D_7^2$ x 16, $C_7\times D_7^2$ x 10, $C_7:D_7^2$ x 5, $C_7^3:C_4$ x 2, $C_7^2:C_{28}$ x 2, $C_7^2:D_{14}$, $C_7^2:D_{14}$, $D_{14}\times C_7^2$

Order 1323: $C_7^2:C_3^3$

Order 1296: $C_2\times C_3^3:S_4$

Order 1176: $D_7^2:S_3$ x 4, $C_2\times C_7^2:D_6$ x 3, $(C_7\times C_{14}).D_6$ x 2, $D_{14}:F_7$ x 2, $C_{14}^2:S_3$ x 2, $(C_7\times C_{14}).D_6$, $C_{14}^2:C_6$, $S_3\times D_7^2$, $D_{14}:F_7$, $C_7:F_7:C_4$

Order 1134: $\He_3:F_7$, $(C_7\times \He_3):C_6$, $\He_3:F_7$

Order 1029: $C_7\wr C_3$ x 7, $C_7^2:C_{21}$ x 5, $C_7^3:C_3$ x 3, $C_7^2:C_{21}$ x 2, $C_7^3:C_3$

Order 882: $C_7^2:(C_3\times S_3)$ x 10, $C_7:(C_3\times F_7)$ x 9, $C_{21}:F_7$ x 2, $C_{21}:F_7$ x 2, $C_{21}:C_{42}$ x 2, $C_{21}:F_7$ x 2, $C_{21}:F_7$ x 2, $(C_7\times C_{21}):S_3$ x 2, $C_3\times C_7^2:S_3$ x 2, $C_2\times C_7^2:C_3^2$ x 2, $C_7:C_3\times F_7$ x 2, $C_{21}:F_7$, $C_{21}:F_7$, $C_7^2:C_3\times S_3$, $C_7^2:C_3\times S_3$, $C_{21}:F_7$, $C_{21}:F_7$

Order 784: $D_7^2:C_2^2$, $D_{14}:D_{14}$

Order 756: $\He_3:D_{14}$, $C_{21}:C_6^2$

Order 686: $C_7^2:C_{14}$ x 14, $D_7\times C_7^2$ x 14, $C_7^3:C_2$ x 9, $C_7^2\times C_{14}$

Order 648: $C_3^3:S_4$ x 2, $C_2\times C_3^3:A_4$

Order 588: $C_7^2:D_6$ x 11, $C_{14}:F_7$ x 8, $D_7:F_7$ x 8, $D_7:F_7$ x 8, $C_{14}:F_7$ x 5, $C_7^2:D_6$ x 5, $C_{14}:F_7$ x 3, $D_7\times D_{21}$ x 2, $C_{21}:D_{14}$ x 2, $C_{21}:D_{14}$ x 2, $C_{14}:F_7$ x 2, $C_{14}:F_7$ x 2, $D_7\times F_7$ x 2, $C_7^2:C_3:C_4$ x 2, $(C_7\times C_{14}).S_3$ x 2, $C_{14}^2:C_3$, $C_{21}:D_{14}$, $C_3\times D_7^2$

Order 567: $C_{21}.\He_3$

Order 504: $D_6\times F_7$ x 2

Order 441: $C_7^2:C_3^2$ x 7, $C_{21}:C_{21}$ x 2, $C_7^2:C_3^2$, $C_7^2:C_3^2$

Order 432: $S_3^2:D_6$

Order 392: $D_7\times D_{14}$ x 7, $D_7\wr C_2$ x 4, $C_{14}\wr C_2$ x 2, $C_7:D_{28}$ x 2, $C_{14}:D_{14}$, $C_2\times C_7^2:C_4$, $C_{14}.D_{14}$

Order 378: $C_3^2:F_7$, $C_3\times C_{21}:C_6$, $C_3^2\times F_7$, $C_3^2:D_{21}$, $\He_3:C_{14}$, $D_7\times \He_3$, $C_3^2.F_7$

Order 343: $C_7^3$ x 8

Order 336: $D_7\times S_4$

Order 324: $C_3^3:D_6$, $C_3^3:A_4$

Order 294: $C_7:F_7$ x 17, $C_7:F_7$ x 14, $C_7^2:S_3$ x 12, $C_7:F_7$ x 12, $C_7:F_7$ x 8, $C_7:F_7$ x 8, $C_7^2:C_6$ x 6, $C_7:C_{42}$ x 5, $C_7^2:C_6$ x 2, $C_7\times F_7$ x 2, $C_7:D_{21}$ x 2, $C_7\times D_{21}$ x 2, $S_3\times C_7^2$ x 2, $C_{21}:D_7$ x 2, $D_7\times C_{21}$ x 2, $C_7^2:C_6$ x 2

Order 252: $S_3\times F_7$ x 12, $C_6\times F_7$ x 3, $C_{21}:D_6$ x 2, $C_{42}:C_6$ x 2, $C_{42}:C_6$ x 2

Order 216: $S_3^2:S_3$ x 4, $C_6:S_3^2$, $C_6\times S_3^2$, $C_2\times C_3^3:C_4$

Order 196: $D_7^2$ x 31, $C_7:D_{14}$ x 13, $C_7\times D_{14}$ x 12, $C_7^2:C_4$ x 2, $C_7:C_{28}$ x 2, $C_{14}^2$

Order 189: $C_{21}.C_3^2$, $C_7:C_3^3$, $C_7\times \He_3$

Order 168: $S_3\times D_{14}$ x 2, $C_2^2\times F_7$ x 2, $A_4\times D_7$, $C_7:S_4$, $C_7\times S_4$

Order 162: $C_3\wr S_3$ x 2, $C_3^3:C_6$

Order 147: $C_7:C_{21}$ x 12, $C_7^2:C_3$ x 8, $C_7^2:C_3$ x 6, $C_7\times C_{21}$ x 2

Order 144: $S_3^2:C_2^2$

Order 126: $C_3\times F_7$ x 10, $C_{21}:C_6$ x 8, $C_{21}:C_6$ x 8, $C_{21}:C_6$ x 3, $C_3\times D_{21}$ x 2, $S_3\times C_{21}$ x 2, $C_3^2\times D_7$ x 2, $C_7:C_{18}$

Order 112: $D_4\times D_7$ x 2

Order 108: $C_3\times S_3^2$ x 3, $C_3:S_3^2$ x 2, $C_3^3:C_4$ x 2, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2:D_6$

Order 98: $C_7\times D_7$ x 36, $C_7:D_7$ x 30, $C_7\times C_{14}$ x 11

Order 84: $C_2\times F_7$ x 19, $S_3\times D_7$ x 13, $C_3\times D_{14}$ x 3, $C_{14}:C_6$ x 2, $D_{42}$ x 2, $S_3\times C_{14}$ x 2, $C_7\times A_4$

Order 81: $C_3\wr C_3$

Order 72: $\SOPlus(4,2)$ x 4, $S_3\times D_6$ x 2, $C_6\times D_6$, $C_2\times C_3^2:C_4$

Order 63: $C_{21}:C_3$ x 7, $C_3\times C_{21}$ x 2, $C_7:C_9$

Order 56: $C_2\times D_{14}$ x 6, $C_7:D_4$ x 4, $C_7\times D_4$ x 2, $D_{28}$ x 2, $C_4\times D_7$ x 2

Order 54: $C_3^2:S_3$ x 2, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2\times C_6$, $C_9:C_6$, $C_2\times \He_3$

Order 49: $C_7^2$ x 19

Order 48: $C_2\times S_4$, $C_6:D_4$

Order 42: $F_7$ x 23, $C_7:C_6$ x 13, $C_3\times D_7$ x 11, $D_{21}$ x 9, $S_3\times C_7$ x 9, $C_{42}$ x 3

Order 36: $C_6\times S_3$ x 10, $S_3^2$ x 6, $C_3^2:C_4$ x 2, $C_6^2$, $C_6:S_3$

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

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $C_3:D_4$ x 4, $C_2\times D_6$ x 2, $S_4$ x 2, $C_6:C_4$, $C_2^2\times C_6$, $C_2\times A_4$

Order 21: $C_7:C_3$ x 10, $C_{21}$ x 8

Order 18: $C_3\times S_3$ x 12, $C_3\times C_6$ x 6, $C_3:S_3$ x 2, $C_{18}$

Order 16: $C_2\times D_4$

Order 14: $D_7$ x 23, $C_{14}$ x 15

Order 12: $D_6$ x 11, $C_2\times C_6$ x 7, $C_3:C_4$ x 2, $A_4$

Order 9: $C_3^2$ x 4, $C_9$

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 7: $C_7$ x 8

Order 6: $C_6$ x 12, $S_3$ x 8

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$ x 4

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3111696: $C_7^4:(C_2\times C_3^3:S_4)$ ($C_1$)

Order 1555848: $C_7^4:(C_2\times C_3^3:A_4)$ ($C_2$), $C_7^4:(C_3^3:S_4)$ ($C_2$), $C_7^4:(C_3^3:S_4)$ ($C_2$)

Order 777924: $C_7^4:(C_3^3:A_4)$ ($C_2^2$)

Order 518616: $C_7^4:(C_6:S_3^2)$ ($S_3$)

Order 259308: $C_7^4:C_3:S_3^2$ ($D_6$)

Order 129654: $C_7^4.C_3^3.C_2$ ($S_4$)

Order 64827: $C_7^4:C_3^3$ ($C_2\times S_4$)

Order 4802: $C_7^4:C_2$ ($C_3^3:S_4$)

Order 2401: $C_7^4$ ($C_2\times C_3^3:S_4$)

Order 1: $C_1$ ($C_7^4:(C_2\times C_3^3:S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3111696: $C_7^4:(C_2\times C_3^3:S_4)$ ($C_1$)

Order 1555848: $C_7^4:(C_2\times C_3^3:A_4)$ ($C_2$), $C_7^4:(C_3^3:S_4)$ ($C_2$), $C_7^4:(C_3^3:S_4)$ ($C_2$)

Order 777924: $C_7^4:(C_3^3:A_4)$ ($C_2^2$)

Order 518616: $C_7^4:(C_6:S_3^2)$ ($S_3$)

Order 259308: $C_7^4:C_3:S_3^2$ ($D_6$)

Order 129654: $C_7^4.C_3^3.C_2$ ($S_4$)

Order 64827: $C_7^4:C_3^3$ ($C_2\times S_4$)

Order 4802: $C_7^4:C_2$ ($C_3^3:S_4$)

Order 2401: $C_7^4$ ($C_2\times C_3^3:S_4$)

Order 1: $C_1$ ($C_7^4:(C_2\times C_3^3:S_4)$)

Series

Derived series

$C_7^4:(C_2\times C_3^3:S_4)$

$\rhd$

$C_7^4:(C_2\times C_3^3:S_4)$

$\rhd$

$C_7^4:(C_3^3:A_4)$

$\rhd$

$C_7^4:(C_3^3:A_4)$

$\rhd$

$C_7^4:C_3:S_3^2$

$\rhd$

$C_7^4:C_3:S_3^2$

$\rhd$

$C_7^4:C_3^3$

$\rhd$

$C_7^4:C_3^3$

$\rhd$

$C_7^4$

$\rhd$

$C_7^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_7^4:(C_2\times C_3^3:S_4)$

$\rhd$

$C_7^4:(C_2\times C_3^3:S_4)$

$\rhd$

$C_7^4:(C_3^3:S_4)$

$\rhd$

$C_7^4:(C_3^3:S_4)$

$\rhd$

$C_7^4:(C_3^3:A_4)$

$\rhd$

$C_7^4:(C_3^3:A_4)$

$\rhd$

$C_7^4:C_3:S_3^2$

$\rhd$

$C_7^4:C_3:S_3^2$

$\rhd$

$C_7^4:C_3^3$

$\rhd$

$C_7^4:C_3^3$

$\rhd$

$C_7^4$

$\rhd$

$C_7^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_7^4:(C_2\times C_3^3:S_4)$

$\rhd$

$C_7^4:(C_2\times C_3^3:S_4)$

$\rhd$

$C_7^4:(C_3^3:A_4)$

$\rhd$

$C_7^4:(C_3^3:A_4)$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 2 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 $100 \times 100$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $62 \times 62$ rational character table.