Subgroups — #11765460

Order 1120.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#10587991	C₂	1	4	yes	maximal, complemented
3	Q₈×D₃₅	C₂	1	4	yes	maximal, complemented
4	#10587987	C₂	1	6	yes	maximal, complemented
5	C₅×C₇×Q₈	C₂²	1	4	yes	—
6	C₂×C₄×C₅×C₇	C₂²	1	3	yes	—
7	C₂²⋊D₃₅	C₂²	1	4	yes	—
8	D₁₄₀	C₂²	1	6	yes	—
9	C₄×D₃₅	C₂²	1	12	yes	—
10	Dic₇₀	C₂²	1	6	yes	—
11	#6551	—	5	1	no	maximal, Hall
12	#4147	—	7	1	no	maximal, Hall
13	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₂×D₃₅	C₂³	1	4	yes	—
15	C₄×C₅×C₇	C₂³	1	6	yes	—
16	Dic₃₅	C₂³	1	4	yes	—
17	C₂×C₇×Q₈	D₅	1	1	yes	characteristic, complemented
18	#1161	—	5	4	no	—
19	Q₈×D₇	—	5	4	no	—
20	#1157	—	5	6	no	—
21	C₂×C₅×Q₈	D₇	1	1	yes	characteristic, complemented
22	#713	—	7	4	no	—
23	Q₈×D₅	—	7	4	no	—
24	#709	—	7	6	no	—
25	C₂×C₅×C₇	—	2	1	no	—
26	C₂×C₅×C₇	C₂⁴	1	1	yes	characteristic
27	D₃₅	—	2	4	no	—
28	C₇×Q₈	C₂×D₅	1	4	yes	—
29	C₂×C₄×C₇	C₂×D₅	1	3	yes	—
30	C₂²⋊D₇	—	5	4	no	—
31	D₂₈	—	5	6	no	—
32	C₄×D₇	—	5	12	no	—
33	Dic₁₄	—	5	6	no	—
34	C₅×Q₈	C₂×D₇	1	4	yes	—
35	C₂×C₄×C₅	C₂×D₇	1	3	yes	—
36	C₂²⋊D₅	—	7	4	no	—
37	D₂₀	—	7	6	no	—
38	C₄×D₅	—	7	12	no	—
39	Dic₁₀	—	7	6	no	—
40	C₅×C₇	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Hall
41	2₋¹⁺⁴	—	35	1	no	Sylow
42	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
43	C₂×D₇	—	5	4	no	—
44	C₄×C₇	C₂²×D₅	1	6	yes	—
45	Dic₇	—	5	4	no	—
46	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
47	C₂×D₅	—	7	4	no	—
48	C₄×C₅	C₂²×D₇	1	6	yes	—
49	Dic₅	—	7	4	no	—
50	#41	—	35	4	no	—
51	#41	—	35	6	no	—
52	C₂×Q₈	—	35	4	no	—
53	C₂×Q₈	D₃₅	1	1	yes	characteristic, complemented
54	C₂×C₇	—	2	1	no	—
55	C₂×C₇	C₂³×D₅	1	1	yes	characteristic
56	D₇	—	10	4	no	—
57	C₂×C₅	—	2	1	no	—
58	C₂×C₅	C₂³×D₇	1	1	yes	characteristic
59	D₅	—	14	4	no	—
60	Q₈	—	35	6	no	—
61	Q₈	C₂×D₃₅	1	4	yes	—
62	D₄	—	35	4	no	—
63	D₄	—	35	6	no	—
64	C₂×C₄	—	35	12	no	—
65	C₂×C₄	C₂×D₃₅	1	3	yes	—
66	C₇	#4147	1	1	yes	characteristic, complemented, Sylow
67	C₅	#6551	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	35	4	no	—
69	C₂²	C₂²×D₃₅	1	1	yes	characteristic
70	C₄	—	35	4	no	—
71	C₄	C₂²×D₃₅	1	6	yes	—
72	C₂	—	2	1	no	—
73	C₂	—	70	4	no	—
74	C₂	C₂³×D₃₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₃₅
commutator, socle, cosocle	C₂×C₅×C₇	C₂⁴
Fitting	C₂×C₅×C₇×Q₈	C₂
hypercenter	C₂×Q₈	D₃₅
hypocenter	C₅×C₇	2₋¹⁺⁴
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₇×Q₈	C₂	semidirect
2. #10587991	C₂	semidirect
3. Q₈×D₃₅	C₂	semidirect
4. #10587987	C₂	semidirect
17. C₂×C₇×Q₈	D₅	semidirect
21. C₂×C₅×Q₈	D₇	semidirect
40. C₅×C₇	2₋¹⁺⁴	semidirect
53. C₂×Q₈	D₃₅	semidirect
66. C₇	#4147	semidirect
67. C₅	#6551	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₇×Q₈	2
#10587991	2
Q₈×D₃₅	2
#10587987	2
#6551	5
#4147	7