Subgroups — C₇⋊#16810006 #37556874

Order 23660.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₁₃×D₆₅	C₂	1	1	yes	maximal, characteristic
2	C₅×C₇×C₁₃²	C₄	1	1	yes	characteristic, complemented, Hall
3	C₇⋊#10621730	—	5	1	no	maximal, Hall
4	#16810006	—	7	1	no	maximal, Hall
5	C₇×C₁₃×D₁₃	—	5	1	no	—
6	C₇⋊#63120	—	13	1	no	maximal
7	C₇⋊#63121	—	13	1	no	maximal
8	C₁₃×D₆₅	D₇	1	1	yes	characteristic
9	C₇×C₁₃²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
10	C₇×D₆₅	D₁₃	1	1	yes	characteristic
11	C₇×D₅×C₁₃	—	13	1	no	—
12	C₅×C₁₃²	Dic₇	1	1	yes	characteristic, complemented, Hall
13	#10621730	—	35	1	no	Hall
14	C₅×C₇×C₁₃	—	4	3	no	—
15	C₅×C₇×C₁₃	Dic₁₃	1	1	yes	characteristic, complemented
16	C₅×C₇×C₁₃	#261	1	1	yes	characteristic, complemented
17	C₇⋊#261	—	65	1	no	—
18	Dic₉₁	—	65	1	no	—
19	C₁₃×D₁₃	—	5	1	no	—
20	#63121	—	91	1	no	—
21	#63120	—	91	1	no	—
22	C₂×C₇×C₁₃	—	65	1	no	—
23	C₇×D₁₃	—	5	1	no	—
24	C₁₃²	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
25	C₇⋊AGL(1,𝔽₅)	—	169	1	no	Hall
26	D₆₅	D₉₁	1	1	yes	characteristic
27	D₅×C₁₃	—	13	1	no	—
28	C₇×C₁₃	—	4	3	no	—
29	C₇×C₁₃	#63120	1	1	yes	characteristic, complemented
30	C₇×C₁₃	#63121	1	1	yes	characteristic, complemented
31	C₇×D₅	—	13	1	no	—
32	C₅×C₁₃	—	4	3	no	—
33	C₅×C₁₃	Dic₉₁	1	1	yes	characteristic, complemented
34	C₅×C₁₃	C₇⋊#261	1	1	yes	characteristic, complemented
35	#261	—	455	1	no	—
36	Dic₁₃	—	455	1	no	—
37	C₅×C₇	#10621730	1	1	yes	characteristic, complemented, Hall
38	Dic₇	—	845	1	no	Hall
39	C₂×C₁₃	—	65	1	no	—
40	D₁₃	—	5	1	no	—
41	AGL(1,𝔽₅)	—	1183	1	no	Hall
42	C₂×C₇	—	65	1	no	—
43	C₁₃	—	4	3	no	—
44	C₁₃	C₇⋊#63121	1	1	yes	characteristic, complemented
45	C₁₃	C₇⋊#63120	1	1	yes	characteristic, complemented
46	D₅	—	13	1	no	—
47	C₇	#16810006	1	1	yes	characteristic, complemented, Sylow
48	C₅	C₇⋊#10621730	1	1	yes	characteristic, complemented, Sylow
49	C₄	—	5915	1	no	Sylow
50	C₂	—	65	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇×C₁₃²	C₄
prime radical, cosocle	C₇×C₁₃×D₆₅	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
12. C₅×C₁₃²	Dic₇	semidirect
15. C₅×C₇×C₁₃	Dic₁₃	semidirect
16. C₅×C₇×C₁₃	#261	semidirect
24. C₁₃²	C₇⋊AGL(1,𝔽₅)	semidirect
29. C₇×C₁₃	#63120	semidirect
30. C₇×C₁₃	#63121	semidirect
33. C₅×C₁₃	Dic₉₁	semidirect
34. C₅×C₁₃	C₇⋊#261	semidirect
37. C₅×C₇	#10621730	semidirect
44. C₁₃	C₇⋊#63121	semidirect
45. C₁₃	C₇⋊#63120	semidirect
47. C₇	#16810006	semidirect
48. C₅	C₇⋊#10621730	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₁₃×D₆₅	2
C₇⋊#10621730	5
#16810006	7
C₇⋊#63120	13
C₇⋊#63121	13

Subgroups — C7⋊#16810006 #37556874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇⋊#16810006 #37556874