Subgroups — #11765234

Order 1120.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10587987	C₂	1	2	yes	maximal, complemented
2	C₂×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₄₀	C₂	1	1	yes	maximal, characteristic, complemented
4	D₂₈₀	C₂	1	1	yes	maximal, characteristic, complemented
5	#10587894	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₂²⋊D₃₅	—	2	2	no	—
8	D₁₄₀	C₂²	1	2	yes	—
9	C₄×D₃₅	—	2	2	no	—
10	Dic₇₀	C₂²	1	2	yes	—
11	C₅×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₅×C₇×C₈	C₂²	1	1	yes	characteristic
13	#6467	—	5	1	no	maximal, Hall
14	#4050	—	7	1	no	maximal, Hall
15	C₂²×C₅×C₇	D₄	1	1	yes	characteristic
16	C₂×D₃₅	—	2	2	no	—
17	C₄×C₅×C₇	D₄	1	1	yes	characteristic
18	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
19	Dic₃₅	—	2	2	no	—
20	#1157	—	5	2	no	—
21	C₂×C₇×C₈	D₅	1	1	yes	characteristic, complemented
22	Dic₂₈	—	5	1	no	—
23	D₅₆	—	5	1	no	—
24	#1132	—	5	2	no	—
25	#709	—	7	2	no	—
26	C₂×C₅×C₈	D₇	1	1	yes	characteristic, complemented
27	Dic₂₀	—	7	1	no	—
28	D₄₀	—	7	1	no	—
29	#677	—	7	2	no	—
30	C₂×C₅×C₇	—	2	1	no	—
31	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
32	D₃₅	—	4	2	no	—
33	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
34	C₂²⋊D₇	—	10	2	no	—
35	D₂₈	—	5	2	no	—
36	C₄×D₇	—	10	2	no	—
37	Dic₁₄	—	5	2	no	—
38	C₇×C₈	C₂×D₅	1	1	yes	characteristic
39	C₇×C₈	C₂×D₅	1	1	yes	characteristic
40	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
41	C₂²⋊D₅	—	14	2	no	—
42	D₂₀	—	7	2	no	—
43	C₄×D₅	—	14	2	no	—
44	Dic₁₀	—	7	2	no	—
45	C₅×C₈	C₂×D₇	1	1	yes	characteristic
46	C₅×C₈	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	D₂₀	1	1	yes	characteristic
50	C₂×D₇	—	10	2	no	—
51	C₄×C₇	D₂₀	1	1	yes	characteristic
52	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
53	Dic₇	—	10	2	no	—
54	C₂²×C₅	D₂₈	1	1	yes	characteristic
55	C₂×D₅	—	14	2	no	—
56	C₄×C₅	D₂₈	1	1	yes	characteristic
57	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
58	Dic₅	—	14	2	no	—
59	#41	—	35	2	no	—
60	Q₁₆	—	35	1	no	—
61	SD₁₆	—	35	2	no	—
62	D₈	—	35	1	no	—
63	C₂×C₈	D₃₅	1	1	yes	characteristic, complemented
64	C₂×C₇	—	2	1	no	—
65	C₂×C₇	C₂×D₂₀	1	1	yes	characteristic
66	D₇	—	20	2	no	—
67	C₂×C₅	—	2	1	no	—
68	C₂×C₅	C₂×D₂₈	1	1	yes	characteristic
69	D₅	—	28	2	no	—
70	Q₈	—	35	2	no	—
71	D₄	—	35	2	no	—
72	D₄	—	70	2	no	—
73	C₂×C₄	—	70	2	no	—
74	C₂×C₄	C₂×D₃₅	1	1	yes	characteristic
75	C₈	C₂×D₃₅	1	1	yes	characteristic
76	C₈	C₂×D₃₅	1	1	yes	characteristic
77	C₇	#4050	1	1	yes	characteristic, complemented, Sylow
78	C₅	#6467	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	70	2	no	—
80	C₂²	D₁₄₀	1	1	yes	characteristic
81	C₄	—	70	2	no	—
82	C₄	D₁₄₀	1	1	yes	characteristic
83	C₄	C₂²×D₃₅	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	140	2	no	—
86	C₂	C₂×D₁₄₀	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₄₀
commutator, cosocle	C₄×C₅×C₇	C₂³
Frattini	C₄	C₂²×D₃₅
Fitting	C₂×C₅×C₇×C₈	C₂
socle	C₂×C₅×C₇	C₂×D₄
hypercenter	C₂×C₈	D₃₅
hypocenter	C₅×C₇	#135
LCS	G ▷ C₄×C₅×C₇ ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #10587987	C₂	semidirect
2. C₂×C₅×C₇×C₈	C₂	semidirect
3. Dic₁₄₀	C₂	semidirect
4. D₂₈₀	C₂	semidirect
5. #10587894	C₂	semidirect
21. C₂×C₇×C₈	D₅	semidirect
26. C₂×C₅×C₈	D₇	semidirect
47. C₅×C₇	#135	semidirect
63. C₂×C₈	D₃₅	semidirect
77. C₇	#4050	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
#10587987	2
C₂×C₅×C₇×C₈	2
Dic₁₄₀	2
D₂₈₀	2
#10587894	2
#6467	5
#4050	7