Subgroups — C₇×#4015 #11765113

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Dic₁₀	C₂	1	2	yes	maximal
2	C₄²×C₅×C₇	C₂	1	1	yes	maximal, characteristic
3	C₇×C₄⋊Dic₅	C₂	1	4	yes	maximal
4	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₅×C₇	C₂²	1	2	yes	—
6	C₂×C₇×Dic₅	C₂²	1	4	yes	—
7	C₇×Dic₁₀	—	2	4	no	—
8	C₇×C₄⋊Q₈	—	5	1	no	maximal, Hall
9	#4015	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
11	C₄×C₅×C₇	D₄	1	2	yes	—
12	C₄×C₅×C₇	Q₈	1	4	yes	complemented
13	C₇×Dic₅	—	2	4	no	—
14	C₂×C₇×Q₈	—	5	2	no	—
15	C₇×C₄⋊C₄	—	5	4	no	—
16	C₄²×C₇	D₅	1	1	yes	characteristic
17	C₂×Dic₁₀	C₂×C₇	1	2	yes	—
18	C₄²×C₅	C₂×C₇	1	1	yes	characteristic
19	C₄⋊Dic₅	C₂×C₇	1	4	yes	—
20	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
21	C₂×C₅×C₇	C₂×Q₈	1	2	yes	—
22	C₇×Q₈	—	10	4	no	—
23	C₂×C₄×C₇	—	5	4	no	—
24	C₂×C₄×C₇	C₂×D₅	1	2	yes	—
25	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
26	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
27	C₂×C₄×C₅	C₂²×C₇	1	2	yes	—
28	C₂×Dic₅	C₂²×C₇	1	4	yes	—
29	Dic₁₀	—	2	4	no	—
30	C₅×C₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
31	C₄⋊Q₈	—	5	1	no	Sylow
32	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
33	C₄×C₇	—	10	4	no	—
34	C₄×C₇	Dic₁₀	1	4	yes	complemented
35	C₄×C₇	D₂₀	1	2	yes	—
36	C₂²×C₅	C₂³×C₇	1	1	yes	characteristic
37	C₄×C₅	C₇×D₄	1	2	yes	—
38	C₄×C₅	C₇×Q₈	1	4	yes	complemented
39	Dic₅	—	2	4	no	—
40	C₂×Q₈	—	5	2	no	—
41	C₄⋊C₄	—	5	4	no	—
42	C₄²	C₇×D₅	1	1	yes	characteristic
43	C₂×C₇	C₂×Dic₁₀	1	2	yes	—
44	C₂×C₇	C₂×D₂₀	1	1	yes	characteristic
45	C₂×C₅	C₂×C₇×D₄	1	1	yes	characteristic
46	C₂×C₅	C₂×C₇×Q₈	1	2	yes	—
47	Q₈	—	10	4	no	—
48	C₂×C₄	—	5	4	no	—
49	C₂×C₄	C₂×C₇×D₅	1	2	yes	—
50	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
51	C₇	#4015	1	1	yes	characteristic, direct product factor, Sylow
52	C₅	C₇×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
53	C₂²	C₂²×C₇×D₅	1	1	yes	characteristic
54	C₄	—	10	4	no	—
55	C₄	C₇×Dic₁₀	1	4	yes	complemented
56	C₄	C₇×D₂₀	1	2	yes	—
57	C₂	C₂×C₇×Dic₁₀	1	2	yes	—
58	C₂	C₂×C₇×D₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂³×C₇
Frattini	C₂²	C₂²×C₇×D₅
Fitting	C₄²×C₅×C₇	C₂
socle, prime radical	C₂²×C₅×C₇	C₂³
hypercenter	C₄²×C₇	D₅
hypocenter	C₅	C₇×C₄⋊Q₈
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇

Complements

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

N	H ≅ G/N	type
9. #4015	C₇	direct (N × H)
12. C₄×C₅×C₇	Q₈	semidirect
30. C₅×C₇	C₄⋊Q₈	semidirect
34. C₄×C₇	Dic₁₀	semidirect
38. C₄×C₅	C₇×Q₈	semidirect
51. C₇	#4015	direct (N × H)
52. C₅	C₇×C₄⋊Q₈	semidirect
55. C₄	C₇×Dic₁₀	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×Dic₁₀	2
C₄²×C₅×C₇	2
C₇×C₄⋊Dic₅	2
C₇×C₄⋊Q₈	5
#4015	7

Subgroups — C7×#4015 #11765113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4015 #11765113