Subgroups — C₇⋊#16810004 #37556876

Order 23660.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₅⋊(C₁₃⋊D₁₃)	C₂	1	1	yes	maximal, characteristic
2	C₅×C₇×C₁₃²	C₄	1	1	yes	characteristic, complemented, Hall
3	C₇⋊#10621729	—	5	1	no	maximal, Hall
4	#16810004	—	7	1	no	maximal, Hall
5	C₇×C₁₃⋊D₁₃	—	5	1	no	—
6	C₇⋊#63122	—	13	1	no	maximal
7	C₇⋊#63121	—	13	1	no	maximal
8	C₅⋊(C₁₃⋊D₁₃)	D₇	1	1	yes	characteristic
9	C₇×C₁₃²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
10	C₇×D₆₅	—	13	1	no	—
11	C₇×D₆₅	—	13	1	no	—
12	C₇×D₆₅	—	26	6	no	—
13	C₅×C₁₃²	Dic₇	1	1	yes	characteristic, complemented, Hall
14	#10621729	—	35	1	no	Hall
15	C₅×C₇×C₁₃	—	2	6	no	—
16	C₅×C₇×C₁₃	#261	1	1	yes	characteristic, complemented
17	C₅×C₇×C₁₃	#261	1	1	yes	characteristic, complemented
18	C₇⋊#261	—	65	1	no	—
19	C₇⋊#261	—	65	1	no	—
20	C₁₃⋊D₁₃	—	5	1	no	—
21	#63122	—	91	1	no	—
22	#63121	—	91	1	no	—
23	C₇×D₁₃	—	65	1	no	—
24	C₇×D₁₃	—	65	1	no	—
25	C₇×D₁₃	—	130	6	no	—
26	C₁₃²	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
27	C₇⋊AGL(1,𝔽₅)	—	169	1	no	Hall
28	D₆₅	—	13	1	no	—
29	D₆₅	—	13	1	no	—
30	D₆₅	—	26	6	no	—
31	C₇×C₁₃	—	2	6	no	—
32	C₇×C₁₃	#63121	1	1	yes	characteristic, complemented
33	C₇×C₁₃	#63122	1	1	yes	characteristic, complemented
34	C₇×D₅	—	169	1	no	—
35	C₅×C₁₃	—	2	6	no	—
36	C₅×C₁₃	C₇⋊#261	1	1	yes	characteristic, complemented
37	C₅×C₁₃	C₇⋊#261	1	1	yes	characteristic, complemented
38	#261	—	455	1	no	—
39	#261	—	455	1	no	—
40	C₅×C₇	#10621729	1	1	yes	characteristic, complemented, Hall
41	Dic₇	—	845	1	no	Hall
42	D₁₃	—	65	1	no	—
43	D₁₃	—	65	1	no	—
44	D₁₃	—	130	6	no	—
45	AGL(1,𝔽₅)	—	1183	1	no	Hall
46	C₂×C₇	—	845	1	no	—
47	C₁₃	—	2	6	no	—
48	C₁₃	C₇⋊#63121	1	1	yes	characteristic, complemented
49	C₁₃	C₇⋊#63122	1	1	yes	characteristic, complemented
50	D₅	—	169	1	no	—
51	C₇	#16810004	1	1	yes	characteristic, complemented, Sylow
52	C₅	C₇⋊#10621729	1	1	yes	characteristic, complemented, Sylow
53	C₄	—	5915	1	no	Sylow
54	C₂	—	845	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇×C₁₃²	C₄
prime radical, cosocle	C₇×C₅⋊(C₁₃⋊D₁₃)	C₂
LCS	G ▷ C₅×C₇×C₁₃²
UCS	C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₅×C₇×C₁₃²	C₄	semidirect
9. C₇×C₁₃²	AGL(1,𝔽₅)	semidirect
13. C₅×C₁₃²	Dic₇	semidirect
16. C₅×C₇×C₁₃	#261	semidirect
17. C₅×C₇×C₁₃	#261	semidirect
26. C₁₃²	C₇⋊AGL(1,𝔽₅)	semidirect
32. C₇×C₁₃	#63121	semidirect
33. C₇×C₁₃	#63122	semidirect
36. C₅×C₁₃	C₇⋊#261	semidirect
37. C₅×C₁₃	C₇⋊#261	semidirect
40. C₅×C₇	#10621729	semidirect
48. C₁₃	C₇⋊#63121	semidirect
49. C₁₃	C₇⋊#63122	semidirect
51. C₇	#16810004	semidirect
52. C₅	C₇⋊#10621729	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×C₅⋊(C₁₃⋊D₁₃)	2
C₇⋊#10621729	5
#16810004	7
C₇⋊#63122	13
C₇⋊#63121	13

Subgroups — C7⋊#16810004 #37556876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇⋊#16810004 #37556876