Subgroups — #11764723

Order 1120.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10587942	C₂	1	2	yes	maximal, complemented
2	D₂₈₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×Dic₂₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#10587847	C₂	1	2	yes	maximal, complemented
6	D₁₄₀	C₂²	1	2	yes	—
7	C₄×C₇×D₅	C₂²	1	1	yes	characteristic
8	C₅×Dic₁₄	C₂²	1	2	yes	—
9	#63307	—	2	2	no	—
10	#63304	—	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	#4065	—	7	1	no	maximal, Hall
15	C₂×D₃₅	—	2	2	no	—
16	C₂×C₇×D₅	D₄	1	1	yes	characteristic
17	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
18	C₅×Dic₇	—	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	D₅	1	1	yes	characteristic, complemented
23	D₅₆	—	5	1	no	—
24	#1132	—	5	2	no	—
25	#713	—	7	2	no	—
26	C₅×Q₁₆	—	7	1	no	—
27	#688	—	7	2	no	—
28	D₄₀	—	7	1	no	—
29	C₈×D₅	D₇	1	1	yes	characteristic, complemented
30	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
31	D₃₅	—	4	2	no	—
32	C₇×D₅	—	2	1	no	—
33	C₂×C₄×C₇	—	5	1	no	—
34	C₂²⋊D₇	—	10	2	no	—
35	D₂₈	—	5	2	no	—
36	C₄×D₇	—	10	2	no	—
37	Dic₁₄	C₂×D₅	1	2	yes	—
38	C₇×C₈	—	5	1	no	—
39	C₇×C₈	C₂×D₅	1	1	yes	characteristic
40	C₅×Q₈	—	7	2	no	—
41	D₂₀	—	7	2	no	—
42	D₂₀	—	14	2	no	—
43	C₄×D₅	—	14	2	no	—
44	C₄×D₅	C₂×D₇	1	1	yes	characteristic
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	—	10	2	no	—
51	C₄×C₇	—	5	1	no	—
52	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
53	Dic₇	—	2	2	no	—
54	C₂×D₅	—	14	2	no	—
55	C₂×D₅	D₂₈	1	1	yes	characteristic
56	C₄×C₅	—	14	2	no	—
57	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
58	Dic₅	D₂₈	1	1	yes	characteristic
59	#41	—	35	2	no	—
60	Q₁₆	—	7	1	no	—
61	SD₁₆	—	35	2	no	—
62	D₈	—	35	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	—	20	2	no	—
67	C₂×C₅	C₂×D₂₈	1	1	yes	characteristic
68	D₅	—	2	1	no	—
69	D₅	—	28	2	no	—
70	Q₈	—	7	2	no	—
71	D₄	—	35	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	#4065	1	1	yes	characteristic, complemented, Sylow
78	C₅	#6467	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	5	1	no	—
80	C₂²	—	70	2	no	—
81	C₄	—	5	1	no	—
82	C₄	—	14	2	no	—
83	C₄	C₂×D₅×D₇	1	1	yes	characteristic
84	C₂	—	10	1	no	—
85	C₂	—	140	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. #10587942	C₂	semidirect
2. D₂₈₀	C₂	semidirect
3. C₇×C₈×D₅	C₂	semidirect
4. C₅×Dic₂₈	C₂	semidirect
5. #10587847	C₂	semidirect
22. Dic₂₈	D₅	semidirect
29. C₈×D₅	D₇	semidirect
47. C₅×C₇	#135	semidirect
77. C₇	#4065	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
#10587942	2
D₂₈₀	2
C₇×C₈×D₅	2
C₅×Dic₂₈	2
#10587847	2
#6467	5
#4065	7