Subgroups — #11764458

Order 1120.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Dic₃₅	C₂	1	1	yes	maximal, characteristic
2	C₇×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
5	C₂×Dic₃₅	—	2	1	no	—
6	C₂×C₇×Dic₅	—	2	1	no	—
7	C₅×C₇⋊C₈	C₄	1	2	yes	complemented
8	#6381	—	5	1	no	maximal, Hall
9	#3950	—	7	1	no	maximal, Hall
10	C₂²×C₅×C₇	D₄	1	1	yes	characteristic
11	C₄×C₅×C₇	C₂×C₄	1	1	yes	characteristic
12	C₄×C₅×C₇	Q₈	1	1	yes	characteristic
13	Dic₃₅	—	4	1	no	—
14	C₇×Dic₅	—	4	1	no	—
15	C₇×C₄⋊C₄	—	5	1	no	—
16	C₄⋊Dic₇	—	5	1	no	—
17	C₂×C₇⋊C₈	D₅	1	1	yes	characteristic
18	C₂×C₅×C₈	—	7	1	no	—
19	C₄⋊Dic₅	—	7	1	no	—
20	C₄⋊Dic₅	D₇	1	1	yes	characteristic
21	C₂×C₅×C₇	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₅×C₇	D₈	1	1	yes	characteristic
23	C₂×C₅×C₇	Q₁₆	1	1	yes	characteristic
24	C₂×C₄×C₇	—	10	1	no	—
25	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
26	C₂×Dic₇	—	10	1	no	—
27	C₇⋊C₈	Dic₅	1	2	yes	complemented
28	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
29	C₂×Dic₅	—	2	1	no	—
30	C₂×Dic₅	—	14	1	no	—
31	C₅×C₈	—	7	2	no	—
32	C₅×C₇	#107	1	1	yes	characteristic, complemented, Hall
33	#107	—	35	1	no	Sylow
34	C₂²×C₇	D₂₀	1	1	yes	characteristic
35	C₄×C₇	—	20	1	no	—
36	C₄×C₇	Dic₁₀	1	1	yes	characteristic
37	C₄×C₇	C₂×Dic₅	1	1	yes	characteristic
38	Dic₇	—	20	1	no	—
39	C₂²×C₅	C₂²⋊D₇	1	1	yes	characteristic
40	C₄×C₅	Dic₁₄	1	1	yes	characteristic
41	C₄×C₅	C₄×D₇	1	1	yes	characteristic
42	Dic₅	—	4	1	no	—
43	Dic₅	—	28	1	no	—
44	C₂×C₈	—	7	1	no	—
45	C₄⋊C₄	—	5	1	no	—
46	C₄⋊C₄	—	35	1	no	—
47	C₂×C₇	D₄₀	1	1	yes	characteristic
48	C₂×C₇	Dic₂₀	1	1	yes	characteristic
49	C₂×C₇	C₄⋊Dic₅	1	1	yes	characteristic
50	C₂×C₅	Dic₇⋊C₄	1	1	yes	characteristic
51	C₂×C₅	#1141	1	1	yes	characteristic
52	C₂×C₅	#1144	1	1	yes	characteristic
53	C₂×C₄	—	10	1	no	—
54	C₂×C₄	—	70	1	no	—
55	C₂×C₄	D₅×D₇	1	1	yes	characteristic
56	C₈	—	7	2	no	—
57	C₇	#3950	1	1	yes	characteristic, complemented, Sylow
58	C₅	#6381	1	1	yes	characteristic, complemented, Sylow
59	C₂²	#63307	1	1	yes	characteristic
60	C₄	—	20	1	no	—
61	C₄	—	140	1	no	—
62	C₄	D₇×Dic₅	1	1	yes	characteristic
63	C₄	#63308	1	1	yes	characteristic
64	C₂	#10587843	1	1	yes	characteristic
65	C₂	#10587852	1	1	yes	characteristic
66	C₂	Dic₇⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63307
commutator	C₄×C₅×C₇	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₅×D₇
Fitting, cosocle	C₂×C₄×C₅×C₇	C₂²
socle, prime radical	C₂²×C₅×C₇	D₄
hypocenter	C₅×C₇	#107
LCS	G ▷ C₄×C₅×C₇ ▷ C₂×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
7. C₅×C₇⋊C₈	C₄	semidirect
27. C₇⋊C₈	Dic₅	semidirect
32. C₅×C₇	#107	semidirect
57. C₇	#3950	semidirect
58. C₅	#6381	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Dic₃₅	2
C₇×C₄⋊Dic₅	2
C₂×C₅×C₇⋊C₈	2
#6381	5
#3950	7