Subgroups — #11764722

Order 1120.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10587938	C₂	1	2	yes	maximal, complemented
2	Dic₁₄₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₅₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#10587846	C₂	1	2	yes	maximal, complemented
6	Dic₇₀	C₂²	1	2	yes	—
7	C₄×C₇×D₅	C₂²	1	1	yes	characteristic
8	C₅×D₂₈	C₂²	1	2	yes	—
9	#63305	—	2	2	no	—
10	D₇×Dic₅	—	2	2	no	—
11	C₅×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×#168	C₂²	1	1	yes	characteristic
13	#6467	—	5	1	no	maximal, Hall
14	#4058	—	7	1	no	maximal, Hall
15	C₂×C₇×D₅	D₄	1	1	yes	characteristic
16	C₂×C₅×D₇	—	2	2	no	—
17	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
18	Dic₃₅	—	2	2	no	—
19	C₇×Dic₅	D₄	1	1	yes	characteristic
20	#1157	—	5	2	no	—
21	C₂×C₇×C₈	—	5	1	no	—
22	Dic₂₈	—	5	1	no	—
23	D₅₆	D₅	1	1	yes	characteristic, complemented
24	#1132	—	5	2	no	—
25	#711	—	7	2	no	—
26	C₅×D₈	—	7	1	no	—
27	#687	—	7	2	no	—
28	Dic₂₀	—	7	1	no	—
29	C₈×D₅	D₇	1	1	yes	characteristic, complemented
30	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
31	C₅×D₇	—	4	2	no	—
32	C₇×D₅	—	2	1	no	—
33	C₂×C₄×C₇	—	5	1	no	—
34	C₂²⋊D₇	—	10	2	no	—
35	D₂₈	C₂×D₅	1	2	yes	—
36	C₄×D₇	—	10	2	no	—
37	Dic₁₄	—	5	2	no	—
38	C₇×C₈	—	5	1	no	—
39	C₇×C₈	C₂×D₅	1	1	yes	characteristic
40	C₅×D₄	—	7	2	no	—
41	C₂²⋊D₅	—	14	2	no	—
42	C₂×Dic₅	—	14	2	no	—
43	C₄×D₅	C₂×D₇	1	1	yes	characteristic
44	Dic₁₀	—	7	2	no	—
45	C₅×C₈	C₂×D₇	1	1	yes	characteristic
46	#168	C₂×D₇	1	1	yes	characteristic
47	C₅×C₇	#135	1	1	yes	characteristic, complemented, Hall
48	#135	—	35	1	no	Sylow
49	C₂²×C₇	—	5	1	no	—
50	C₂×D₇	—	2	2	no	—
51	C₄×C₇	—	5	1	no	—
52	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
53	Dic₇	—	10	2	no	—
54	C₂²×C₅	—	14	2	no	—
55	C₂×D₅	D₂₈	1	1	yes	characteristic
56	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
57	Dic₅	—	14	2	no	—
58	Dic₅	D₂₈	1	1	yes	characteristic
59	#41	—	35	2	no	—
60	Q₁₆	—	35	1	no	—
61	SD₁₆	—	35	2	no	—
62	D₈	—	7	1	no	—
63	C₂×C₈	—	5	1	no	—
64	C₂×C₇	—	10	1	no	—
65	C₂×C₇	D₄×D₅	1	1	yes	characteristic
66	D₇	—	4	2	no	—
67	C₂×C₅	—	28	2	no	—
68	C₂×C₅	C₂×D₂₈	1	1	yes	characteristic
69	D₅	—	2	1	no	—
70	Q₈	—	35	2	no	—
71	D₄	—	7	2	no	—
72	D₄	—	70	2	no	—
73	C₂×C₄	—	5	1	no	—
74	C₂×C₄	—	70	2	no	—
75	C₈	—	5	1	no	—
76	C₈	D₅×D₇	1	1	yes	characteristic
77	C₇	#4058	1	1	yes	characteristic, complemented, Sylow
78	C₅	#6467	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	5	1	no	—
80	C₂²	—	14	2	no	—
81	C₄	—	5	1	no	—
82	C₄	—	70	2	no	—
83	C₄	C₂×D₅×D₇	1	1	yes	characteristic
84	C₂	—	10	1	no	—
85	C₂	—	28	2	no	—
86	C₂	D₅×D₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₅×D₂₈
commutator, cosocle	C₄×C₅×C₇	C₂³
Frattini	C₄	C₂×D₅×D₇
Fitting	C₅×C₇×C₈	C₂²
socle	C₂×C₅×C₇	C₂×D₄
hypercenter	C₈	D₅×D₇
hypocenter	C₅×C₇	#135
LCS	G ▷ C₄×C₅×C₇ ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. #10587938	C₂	semidirect
2. Dic₁₄₀	C₂	semidirect
3. C₇×C₈×D₅	C₂	semidirect
4. C₅×D₅₆	C₂	semidirect
5. #10587846	C₂	semidirect
23. D₅₆	D₅	semidirect
29. C₈×D₅	D₇	semidirect
47. C₅×C₇	#135	semidirect
77. C₇	#4058	semidirect
78. C₅	#6467	semidirect

Maximal subgroups

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

subgroup	index
#10587938	2
Dic₁₄₀	2
C₇×C₈×D₅	2
C₅×D₅₆	2
#10587846	2
#6467	5
#4058	7