Subgroups — #11764777

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×C₅×Dic₁₄	C₂	1	1	yes	maximal, characteristic
3	C₄×C₇×Dic₅	C₂	1	1	yes	maximal, characteristic
4	#10587859	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	C₂×C₅×Dic₇	C₂²	1	2	yes	—
10	C₅×Dic₁₄	—	2	2	no	—
11	#6432	—	5	1	no	maximal, Hall
12	#4090	—	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	—	2	2	no	—
17	C₇×Dic₅	Q₈	1	4	yes	complemented
18	C₂×Dic₁₄	—	5	1	no	—
19	C₂×Dic₁₄	D₅	1	1	yes	characteristic
20	C₄²×C₇	—	5	1	no	—
21	C₄⋊Dic₇	—	5	4	no	—
22	C₂×C₅×Q₈	—	7	1	no	—
23	C₂×Dic₁₀	—	7	1	no	—
24	#683	—	7	4	no	—
25	C₄×Dic₅	D₇	1	1	yes	characteristic
26	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
27	C₂×C₅×C₇	C₂×Q₈	1	2	yes	—
28	C₂×C₄×C₇	—	5	2	no	—
29	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
30	C₂×Dic₇	—	5	2	no	—
31	C₂×Dic₇	C₂×D₅	1	2	yes	—
32	Dic₁₄	—	2	2	no	—
33	Dic₁₄	—	10	2	no	—
34	C₅×Q₈	—	14	2	no	—
35	C₂×C₄×C₅	—	7	2	no	—
36	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
37	C₂×Dic₅	—	7	2	no	—
38	C₂×Dic₅	C₂×D₇	1	2	yes	—
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	—	5	4	no	—
44	C₄×C₇	C₂²⋊D₅	1	2	yes	—
45	Dic₇	—	2	2	no	—
46	Dic₇	—	10	2	no	—
47	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
48	C₄×C₅	—	14	2	no	—
49	C₄×C₅	D₂₈	1	2	yes	—
50	Dic₅	—	14	2	no	—
51	Dic₅	Dic₁₄	1	4	yes	complemented
52	C₂×Q₈	—	7	1	no	—
53	C₂×Q₈	—	35	1	no	—
54	C₄⋊C₄	—	35	4	no	—
55	C₄²	—	5	1	no	—
56	C₂×C₇	Q₈×D₅	1	2	yes	—
57	C₂×C₇	C₂×C₂²⋊D₅	1	1	yes	characteristic
58	C₂×C₅	C₂×Dic₁₄	1	2	yes	—
59	C₂×C₅	C₂×D₂₈	1	1	yes	characteristic
60	Q₈	—	14	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	#4090	1	1	yes	characteristic, complemented, Sylow
67	C₅	#6432	1	1	yes	characteristic, complemented, Sylow
68	C₂²	C₂×D₅×D₇	1	1	yes	characteristic
69	C₄	—	5	4	no	—
70	C₄	—	14	2	no	—
71	C₄	—	70	2	no	—
72	C₄	#63306	1	2	yes	—
73	C₂	D₅×Dic₁₄	1	2	yes	—
74	C₂	C₂×#63306	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
17. C₇×Dic₅	Q₈	semidirect
40. C₅×C₇	C₄⋊Q₈	semidirect
51. Dic₅	Dic₁₄	semidirect
66. C₇	#4090	semidirect
67. C₅	#6432	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×C₅×Dic₁₄	2
C₄×C₇×Dic₅	2
#10587859	2
#6432	5
#4090	7