Subgroups — C₇⋊#15690391 #36339425

Order 21420.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₃⋊D₂₅₅	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅×C₇×C₁₇	C₄	1	1	yes	characteristic, complemented, Hall
3	C₇⋊#10597479	—	5	1	no	maximal, Hall
4	C₇×D₂₅₅	—	6	2	no	—
5	#15690391	—	7	1	no	maximal, Hall
6	C₇⋊#66776	—	9	1	no	maximal, Hall
7	C₇×C₃⋊D₅₁	—	5	1	no	—
8	C₃×C₅×C₇×C₁₇	—	2	2	no	—
9	C₃⋊D₂₅₅	D₇	1	1	yes	characteristic
10	C₇⋊#4380	—	17	1	no	maximal, Hall
11	C₇×D₈₅	—	9	1	no	—
12	C₃²×C₇×C₁₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
13	C₃²×C₅×C₁₇	Dic₇	1	1	yes	characteristic, complemented, Hall
14	C₇×D₅₁	—	30	2	no	—
15	C₇×C₃⋊D₁₅	—	17	1	no	—
16	#10597479	—	35	1	no	Hall
17	C₅×C₇×C₁₇	#157	1	1	yes	characteristic, complemented, Hall
18	D₂₅₅	—	6	2	no	—
19	C₇⋊#595	—	45	1	no	Hall
20	C₃×C₇×C₁₇	—	2	2	no	—
21	#66776	—	63	1	no	Hall
22	C₃²×C₅×C₇	#595	1	1	yes	characteristic, complemented, Hall
23	C₃⋊D₅₁	—	5	1	no	—
24	C₃×C₅×C₁₇	—	2	2	no	—
25	C₇⋊#157	—	85	1	no	Hall
26	C₇×D₁₇	—	45	1	no	—
27	C₇×D₁₅	—	102	2	no	—
28	#4380	—	119	1	no	Hall
29	D₈₅	—	9	1	no	—
30	C₃²×C₁₇	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
31	C₇⋊AGL(1,𝔽₅)	—	153	1	no	Hall
32	C₇×C₃⋊D₃	—	85	1	no	—
33	C₇×C₁₇	#4380	1	1	yes	characteristic, complemented, Hall
34	C₃×C₅×C₇	—	2	2	no	—
35	D₅₁	—	30	2	no	—
36	C₃⋊D₁₅	—	17	1	no	—
37	C₅×C₁₇	C₇⋊#157	1	1	yes	characteristic, complemented, Hall
38	C₇×D₅	—	153	1	no	—
39	#595	—	315	1	no	Hall
40	C₃²×C₇	#66776	1	1	yes	characteristic, complemented, Hall
41	C₃×C₁₇	—	2	2	no	—
42	C₃²×C₅	C₇⋊#595	1	1	yes	characteristic, complemented, Hall
43	D₃×C₇	—	510	2	no	—
44	#157	—	595	1	no	Hall
45	C₅×C₇	#10597479	1	1	yes	characteristic, complemented, Hall
46	D₁₇	—	45	1	no	—
47	D₁₅	—	102	2	no	—
48	Dic₇	—	765	1	no	Hall
49	C₃×C₇	—	2	2	no	—
50	AGL(1,𝔽₅)	—	1071	1	no	Hall
51	C₃⋊D₃	—	85	1	no	—
52	C₁₇	C₇⋊#4380	1	1	yes	characteristic, complemented, Sylow
53	C₃×C₅	—	2	2	no	—
54	C₂×C₇	—	765	1	no	—
55	D₅	—	153	1	no	—
56	C₃²	C₇⋊#66776	1	1	yes	characteristic, complemented, Sylow
57	C₇	#15690391	1	1	yes	characteristic, complemented, Sylow
58	D₃	—	510	2	no	—
59	C₅	C₇⋊#10597479	1	1	yes	characteristic, complemented, Sylow
60	C₄	—	5355	1	no	Sylow
61	C₃	—	2	2	no	—
62	C₂	—	765	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅×C₇×C₁₇	C₄
prime radical, cosocle	C₇×C₃⋊D₂₅₅	C₂
LCS	G ▷ C₃²×C₅×C₇×C₁₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅×C₇×C₁₇	C₄	semidirect
12. C₃²×C₇×C₁₇	AGL(1,𝔽₅)	semidirect
13. C₃²×C₅×C₁₇	Dic₇	semidirect
17. C₅×C₇×C₁₇	#157	semidirect
22. C₃²×C₅×C₇	#595	semidirect
30. C₃²×C₁₇	C₇⋊AGL(1,𝔽₅)	semidirect
33. C₇×C₁₇	#4380	semidirect
37. C₅×C₁₇	C₇⋊#157	semidirect
40. C₃²×C₇	#66776	semidirect
42. C₃²×C₅	C₇⋊#595	semidirect
45. C₅×C₇	#10597479	semidirect
52. C₁₇	C₇⋊#4380	semidirect
56. C₃²	C₇⋊#66776	semidirect
57. C₇	#15690391	semidirect
59. C₅	C₇⋊#10597479	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₃⋊D₂₅₅	2
C₇⋊#10597479	5
#15690391	7
C₇⋊#66776	9
C₇⋊#4380	17

Subgroups — C7⋊#15690391 #36339425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇⋊#15690391 #36339425