Subgroups — #11765219

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₀	C₂	1	2	yes	maximal
2	C₅×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₃₅	C₂	1	1	yes	maximal, characteristic
4	#10587900	C₂	1	2	yes	maximal
5	C₄×Dic₃₅	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₅×C₇	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
8	C₂×Dic₃₅	C₂²	1	2	yes	—
9	C₂×Dic₃₅	C₂²	1	1	yes	characteristic
10	C₂×Dic₃₅	C₂²	1	1	yes	characteristic
11	Dic₇₀	—	2	4	no	—
12	#6451	—	5	1	no	maximal, Hall
13	#4034	—	7	1	no	maximal, Hall
14	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
15	C₄×C₅×C₇	—	2	2	no	—
16	C₄×C₅×C₇	Q₈	1	2	yes	complemented
17	Dic₃₅	—	2	2	no	—
18	Dic₃₅	D₄	1	2	yes	—
19	Dic₃₅	Q₈	1	2	yes	complemented
20	C₂×Dic₁₄	—	5	2	no	—
21	C₇×C₄⋊C₄	D₅	1	1	yes	characteristic
22	C₄⋊Dic₇	—	5	1	no	—
23	Dic₇⋊C₄	—	5	2	no	—
24	C₄×Dic₇	—	5	1	no	—
25	C₂×Dic₁₀	—	7	2	no	—
26	C₅×C₄⋊C₄	D₇	1	1	yes	characteristic
27	C₄⋊Dic₅	—	7	1	no	—
28	#683	—	7	2	no	—
29	C₄×Dic₅	—	7	1	no	—
30	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₅×C₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₅×C₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₇	C₂×D₅	1	2	yes	—
34	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
35	C₂×Dic₇	—	5	1	no	—
36	C₂×Dic₇	—	5	1	no	—
37	C₂×Dic₇	—	5	2	no	—
38	Dic₁₄	—	10	4	no	—
39	C₂×C₄×C₅	C₂×D₇	1	2	yes	—
40	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
41	C₂×Dic₅	—	7	1	no	—
42	C₂×Dic₅	—	7	1	no	—
43	C₂×Dic₅	—	7	2	no	—
44	Dic₁₀	—	14	4	no	—
45	C₅×C₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
46	C₄⋊Q₈	—	35	1	no	Sylow
47	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
48	C₄×C₇	—	2	2	no	—
49	C₄×C₇	Dic₁₀	1	2	yes	complemented
50	Dic₇	—	5	2	no	—
51	Dic₇	—	5	2	no	—
52	Dic₇	—	10	2	no	—
53	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
54	C₄×C₅	—	2	2	no	—
55	C₄×C₅	Dic₁₄	1	2	yes	complemented
56	Dic₅	—	7	2	no	—
57	Dic₅	—	7	2	no	—
58	Dic₅	—	14	2	no	—
59	C₂×Q₈	—	35	2	no	—
60	C₄⋊C₄	—	35	2	no	—
61	C₄⋊C₄	—	35	1	no	—
62	C₄⋊C₄	D₃₅	1	1	yes	characteristic
63	C₄²	—	35	1	no	—
64	C₂×C₇	C₂×Dic₁₀	1	1	yes	characteristic
65	C₂×C₇	D₄×D₅	1	1	yes	characteristic
66	C₂×C₇	Q₈×D₅	1	1	yes	characteristic
67	C₂×C₅	C₂×Dic₁₄	1	1	yes	characteristic
68	C₂×C₅	D₄×D₇	1	1	yes	characteristic
69	C₂×C₅	Q₈×D₇	1	1	yes	characteristic
70	Q₈	—	70	4	no	—
71	C₂×C₄	—	35	1	no	—
72	C₂×C₄	—	35	1	no	—
73	C₂×C₄	—	35	2	no	—
74	C₂×C₄	C₂×D₃₅	1	2	yes	—
75	C₂×C₄	C₂×D₃₅	1	1	yes	characteristic
76	C₇	#4034	1	1	yes	characteristic, complemented, Sylow
77	C₅	#6451	1	1	yes	characteristic, complemented, Sylow
78	C₂²	C₂²×D₃₅	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	35	2	no	—
81	C₄	—	35	2	no	—
82	C₄	—	70	2	no	—
83	C₄	Dic₇₀	1	2	yes	complemented
84	C₂	C₂×Dic₇₀	1	1	yes	characteristic
85	C₂	D₄×D₃₅	1	1	yes	characteristic
86	C₂	Q₈×D₃₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₅
commutator, socle, prime radical, cosocle	C₂²×C₅×C₇	C₂³
Fitting	C₅×C₇×C₄⋊C₄	C₂
hypercenter	C₄⋊C₄	D₃₅
hypocenter	C₅×C₇	C₄⋊Q₈
LCS	G ▷ C₂²×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
16. C₄×C₅×C₇	Q₈	semidirect
19. Dic₃₅	Q₈	semidirect
45. C₅×C₇	C₄⋊Q₈	semidirect
49. C₄×C₇	Dic₁₀	semidirect
55. C₄×C₅	Dic₁₄	semidirect
76. C₇	#4034	semidirect
77. C₅	#6451	semidirect
83. C₄	Dic₇₀	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₀	2
C₅×C₇×C₄⋊C₄	2
C₄⋊Dic₃₅	2
#10587900	2
C₄×Dic₃₅	2
#6451	5
#4034	7