Subgroups — #11764779

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×Dic₁₀	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×Dic₇	C₂	1	1	yes	maximal, characteristic
4	Dic₇⋊Dic₅	C₂	1	4	yes	maximal
5	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₂×Dic₃₅	C₂²	1	2	yes	—
7	Dic₇₀	—	2	2	no	—
8	C₂×C₇×Dic₅	C₂²	1	2	yes	—
9	C₇×Dic₁₀	—	2	2	no	—
10	C₂×C₅×Dic₇	C₂²	1	2	yes	—
11	#6507	—	5	1	no	maximal, Hall
12	#4015	—	7	1	no	maximal, Hall
13	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₄×C₅×C₇	D₄	1	2	yes	—
15	Dic₃₅	—	2	2	no	—
16	C₅×Dic₇	Q₈	1	4	yes	complemented
17	C₇×Dic₅	—	2	2	no	—
18	C₂×C₇×Q₈	—	5	1	no	—
19	C₂×Dic₁₄	—	5	1	no	—
20	Dic₇⋊C₄	—	5	4	no	—
21	C₄×Dic₇	D₅	1	1	yes	characteristic
22	C₂×Dic₁₀	—	7	1	no	—
23	C₂×Dic₁₀	D₇	1	1	yes	characteristic
24	C₄²×C₅	—	7	1	no	—
25	C₄⋊Dic₅	—	7	4	no	—
26	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
27	C₂×C₅×C₇	C₂×Q₈	1	2	yes	—
28	C₇×Q₈	—	10	2	no	—
29	C₂×C₄×C₇	—	5	2	no	—
30	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
31	C₂×Dic₇	—	5	2	no	—
32	C₂×Dic₇	C₂×D₅	1	2	yes	—
33	Dic₁₄	—	10	2	no	—
34	C₂×C₄×C₅	—	7	2	no	—
35	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
36	C₂×Dic₅	—	7	2	no	—
37	C₂×Dic₅	C₂×D₇	1	2	yes	—
38	Dic₁₀	—	2	2	no	—
39	Dic₁₀	—	14	2	no	—
40	C₅×C₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
41	C₄⋊Q₈	—	35	1	no	Sylow
42	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
43	C₄×C₇	—	10	2	no	—
44	C₄×C₇	D₂₀	1	2	yes	—
45	Dic₇	—	10	2	no	—
46	Dic₇	Dic₁₀	1	4	yes	complemented
47	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
48	C₄×C₅	—	7	4	no	—
49	C₄×C₅	C₂²⋊D₇	1	2	yes	—
50	Dic₅	—	2	2	no	—
51	Dic₅	—	14	2	no	—
52	C₂×Q₈	—	5	1	no	—
53	C₂×Q₈	—	35	1	no	—
54	C₄⋊C₄	—	35	4	no	—
55	C₄²	—	7	1	no	—
56	C₂×C₇	C₂×Dic₁₀	1	2	yes	—
57	C₂×C₇	C₂×D₂₀	1	1	yes	characteristic
58	C₂×C₅	Q₈×D₇	1	2	yes	—
59	C₂×C₅	C₂×C₂²⋊D₇	1	1	yes	characteristic
60	Q₈	—	10	2	no	—
61	Q₈	—	70	2	no	—
62	C₂×C₄	—	5	2	no	—
63	C₂×C₄	—	7	2	no	—
64	C₂×C₄	—	35	2	no	—
65	C₂×C₄	D₅×D₇	1	1	yes	characteristic
66	C₇	#4015	1	1	yes	characteristic, complemented, Sylow
67	C₅	#6507	1	1	yes	characteristic, complemented, Sylow
68	C₂²	C₂×D₅×D₇	1	1	yes	characteristic
69	C₄	—	7	4	no	—
70	C₄	—	10	2	no	—
71	C₄	—	70	2	no	—
72	C₄	#63307	1	2	yes	—
73	C₂	D₇×Dic₁₀	1	2	yes	—
74	C₂	C₂×#63307	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅×D₇
commutator, socle, prime radical, cosocle	C₂²×C₅×C₇	C₂³
Fitting	C₂×C₄×C₅×C₇	C₂²
hypercenter	C₂×C₄	D₅×D₇
hypocenter	C₅×C₇	C₄⋊Q₈
LCS	G ▷ C₂²×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
16. C₅×Dic₇	Q₈	semidirect
40. C₅×C₇	C₄⋊Q₈	semidirect
46. Dic₇	Dic₁₀	semidirect
66. C₇	#4015	semidirect
67. C₅	#6507	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
C₂×C₇×Dic₁₀	2
C₄×C₅×Dic₇	2
Dic₇⋊Dic₅	2
#6507	5
#4015	7