Subgroups — #11765199

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₀	C₂	1	2	yes	maximal
2	C₄²×C₅×C₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₃₅	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×Dic₃₅	C₂²	1	4	yes	—
7	Dic₇₀	—	2	4	no	—
8	#6432	—	5	1	no	maximal, Hall
9	#4015	—	7	1	no	maximal, 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	Dic₃₅	—	2	4	no	—
14	C₂×Dic₁₄	—	5	2	no	—
15	C₄²×C₇	D₅	1	1	yes	characteristic
16	C₄⋊Dic₇	—	5	4	no	—
17	C₂×Dic₁₀	—	7	2	no	—
18	C₄²×C₅	D₇	1	1	yes	characteristic
19	C₄⋊Dic₅	—	7	4	no	—
20	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
21	C₂×C₅×C₇	C₂×Q₈	1	2	yes	—
22	C₂×C₄×C₇	C₂×D₅	1	2	yes	—
23	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
24	C₂×Dic₇	—	5	4	no	—
25	Dic₁₄	—	10	4	no	—
26	C₂×C₄×C₅	C₂×D₇	1	2	yes	—
27	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
28	C₂×Dic₅	—	7	4	no	—
29	Dic₁₀	—	14	4	no	—
30	C₅×C₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
31	C₄⋊Q₈	—	35	1	no	Sylow
32	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
33	C₄×C₇	Dic₁₀	1	4	yes	complemented
34	C₄×C₇	D₂₀	1	2	yes	—
35	Dic₇	—	10	4	no	—
36	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
37	C₄×C₅	Dic₁₄	1	4	yes	complemented
38	C₄×C₅	D₂₈	1	2	yes	—
39	Dic₅	—	14	4	no	—
40	C₂×Q₈	—	35	2	no	—
41	C₄⋊C₄	—	35	4	no	—
42	C₄²	D₃₅	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×Dic₁₄	1	2	yes	—
46	C₂×C₅	C₂×D₂₈	1	1	yes	characteristic
47	Q₈	—	70	4	no	—
48	C₂×C₄	—	35	4	no	—
49	C₂×C₄	C₂×D₃₅	1	2	yes	—
50	C₂×C₄	C₂×D₃₅	1	1	yes	characteristic
51	C₇	#4015	1	1	yes	characteristic, complemented, Sylow
52	C₅	#6432	1	1	yes	characteristic, complemented, Sylow
53	C₂²	C₂²×D₃₅	1	1	yes	characteristic
54	C₄	—	70	4	no	—
55	C₄	Dic₇₀	1	4	yes	complemented
56	C₄	D₁₄₀	1	2	yes	—
57	C₂	C₂×Dic₇₀	1	2	yes	—
58	C₂	C₂×D₁₄₀	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₅
commutator, socle, prime radical, cosocle	C₂²×C₅×C₇	C₂³
Fitting	C₄²×C₅×C₇	C₂
hypercenter	C₄²	D₃₅
hypocenter	C₅×C₇	C₄⋊Q₈
LCS	G ▷ C₂²×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
12. C₄×C₅×C₇	Q₈	semidirect
30. C₅×C₇	C₄⋊Q₈	semidirect
33. C₄×C₇	Dic₁₀	semidirect
37. C₄×C₅	Dic₁₄	semidirect
51. C₇	#4015	semidirect
52. C₅	#6432	semidirect
55. C₄	Dic₇₀	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₀	2
C₄²×C₅×C₇	2
C₄⋊Dic₃₅	2
#6432	5
#4015	7