Subgroups — #11764712

Order 1120.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₅⋊Q₈	C₂	1	2	yes	maximal
2	C₈×D₃₅	C₂	1	1	yes	maximal, characteristic
3	C₇×Dic₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×Dic₂₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#10587850	C₂	1	2	yes	maximal, complemented
6	C₄×D₃₅	C₂²	1	1	yes	characteristic
7	C₇×Dic₁₀	C₂²	1	2	yes	—
8	C₅×Dic₁₄	C₂²	1	2	yes	—
9	#63308	—	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	D₇×Q₁₆	—	5	1	no	maximal, Hall
14	D₅×Q₁₆	—	7	1	no	maximal, Hall
15	C₂×D₃₅	D₄	1	1	yes	characteristic
16	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
17	Dic₃₅	D₄	1	1	yes	characteristic
18	C₅×Dic₇	—	2	2	no	—
19	C₇×Dic₅	—	2	2	no	—
20	Q₈×D₇	—	5	2	no	—
21	C₇×Q₁₆	—	5	1	no	—
22	#1144	—	5	2	no	—
23	Dic₂₈	D₅	1	1	yes	characteristic, complemented
24	C₈×D₇	—	5	1	no	—
25	Q₈×D₅	—	7	2	no	—
26	C₅×Q₁₆	—	7	1	no	—
27	#689	—	7	2	no	—
28	Dic₂₀	D₇	1	1	yes	characteristic, complemented
29	C₈×D₅	—	7	1	no	—
30	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
31	D₃₅	Q₁₆	1	2	yes	complemented
32	C₇×Q₈	—	5	2	no	—
33	C₄×D₇	—	5	1	no	—
34	C₄×D₇	—	10	2	no	—
35	Dic₁₄	—	10	2	no	—
36	Dic₁₄	C₂×D₅	1	2	yes	—
37	C₇×C₈	C₂×D₅	1	1	yes	characteristic
38	C₇⋊C₈	—	5	1	no	—
39	C₅×Q₈	—	7	2	no	—
40	C₄×D₅	—	7	1	no	—
41	C₄×D₅	—	14	2	no	—
42	Dic₁₀	—	14	2	no	—
43	Dic₁₀	C₂×D₇	1	2	yes	—
44	C₅×C₈	C₂×D₇	1	1	yes	characteristic
45	#168	—	7	1	no	—
46	C₅×C₇	C₂×Q₁₆	1	1	yes	characteristic, complemented, Hall
47	C₂×Q₁₆	—	35	1	no	Sylow
48	C₂×D₇	—	5	1	no	—
49	C₄×C₇	—	10	2	no	—
50	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
51	Dic₇	—	2	2	no	—
52	Dic₇	—	5	1	no	—
53	C₂×D₅	—	7	1	no	—
54	C₄×C₅	—	14	2	no	—
55	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
56	Dic₅	—	2	2	no	—
57	Dic₅	—	7	1	no	—
58	C₂×Q₈	—	35	2	no	—
59	Q₁₆	—	5	1	no	—
60	Q₁₆	—	7	1	no	—
61	Q₁₆	—	35	2	no	—
62	C₂×C₈	—	35	1	no	—
63	C₂×C₇	D₄×D₅	1	1	yes	characteristic
64	D₇	—	5	2	no	—
65	C₂×C₅	D₄×D₇	1	1	yes	characteristic
66	D₅	—	7	2	no	—
67	Q₈	—	5	2	no	—
68	Q₈	—	7	2	no	—
69	Q₈	—	70	2	no	—
70	C₂×C₄	—	35	1	no	—
71	C₂×C₄	—	70	2	no	—
72	C₈	—	35	1	no	—
73	C₈	D₅×D₇	1	1	yes	characteristic
74	C₇	D₅×Q₁₆	1	1	yes	characteristic, complemented, Sylow
75	C₅	D₇×Q₁₆	1	1	yes	characteristic, complemented, Sylow
76	C₂²	—	35	1	no	—
77	C₄	—	10	2	no	—
78	C₄	—	14	2	no	—
79	C₄	—	35	1	no	—
80	C₄	C₂×D₅×D₇	1	1	yes	characteristic
81	C₂	—	35	2	no	—
82	C₂	#10587950	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10587950
commutator, cosocle	C₄×C₅×C₇	C₂³
Frattini	C₄	C₂×D₅×D₇
Fitting	C₅×C₇×C₈	C₂²
socle	C₂×C₅×C₇	C₂×D₄
prime radical	C₂×D₃₅	D₄
hypercenter	C₈	D₅×D₇
hypocenter	C₅×C₇	C₂×Q₁₆
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
3. C₇×Dic₂₀	C₂	semidirect
4. C₅×Dic₂₈	C₂	semidirect
5. #10587850	C₂	semidirect
23. Dic₂₈	D₅	semidirect
28. Dic₂₀	D₇	semidirect
31. D₃₅	Q₁₆	semidirect
46. C₅×C₇	C₂×Q₁₆	semidirect
74. C₇	D₅×Q₁₆	semidirect
75. C₅	D₇×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
D₃₅⋊Q₈	2
C₈×D₃₅	2
C₇×Dic₂₀	2
C₅×Dic₂₈	2
#10587850	2
D₇×Q₁₆	5
D₅×Q₁₆	7