Subgroups — #11764754

Order 1120.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×D₂₈	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₃₅	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₇)⋊Dic₅	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×D₇	C₂²	1	2	yes	—
6	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₃₅	C₂²	1	2	yes	—
8	C₂×C₇×Dic₅	C₂²	1	2	yes	—
9	C₇×Dic₁₀	—	2	2	no	—
10	C₅×D₂₈	—	2	2	no	—
11	#6510	—	5	1	no	maximal, Hall
12	#4082	—	7	1	no	maximal, Hall
13	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₂×C₅×D₇	—	2	2	no	—
15	C₂×C₅×D₇	—	2	4	no	—
16	C₄×C₅×C₇	D₄	1	2	yes	—
17	Dic₃₅	—	2	2	no	—
18	C₇×Dic₅	—	2	2	no	—
19	C₂×C₇×Q₈	—	5	1	no	—
20	C₂×D₂₈	D₅	1	1	yes	characteristic
21	(C₂×D₇)⋊C₄	—	5	4	no	—
22	C₄×Dic₇	—	5	1	no	—
23	C₂×C₅×D₄	—	7	1	no	—
24	C₂×Dic₁₀	D₇	1	1	yes	characteristic, complemented
25	C₂²⋊Dic₅	—	7	4	no	—
26	C₄×Dic₅	—	7	1	no	—
27	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₅×C₇	#41	1	2	yes	—
29	C₅×D₇	—	4	2	no	—
30	C₂²×D₇	C₂×D₅	1	2	yes	—
31	C₇×Q₈	—	10	2	no	—
32	C₂×C₄×C₇	—	5	2	no	—
33	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
34	C₂×Dic₇	—	5	2	no	—
35	D₂₈	—	2	2	no	—
36	C₂³×C₅	—	7	2	no	—
37	C₅×D₄	—	14	2	no	—
38	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
39	C₂×Dic₅	—	7	2	no	—
40	C₂×Dic₅	C₂×D₇	1	2	yes	—
41	Dic₁₀	—	2	2	no	—
42	C₅×C₇	#124	1	1	yes	characteristic, complemented, Hall
43	#124	—	35	1	no	Sylow
44	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
45	C₂×D₇	—	2	4	no	—
46	C₂×D₇	—	2	2	no	—
47	C₄×C₇	—	10	2	no	—
48	C₄×C₇	C₂²⋊D₅	1	2	yes	—
49	Dic₇	—	10	2	no	—
50	C₂²×C₅	—	14	4	no	—
51	C₂²×C₅	—	14	2	no	—
52	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
53	C₄×C₅	C₂²⋊D₇	1	2	yes	—
54	Dic₅	—	2	2	no	—
55	Dic₅	—	14	2	no	—
56	C₂×Q₈	—	5	1	no	—
57	C₂×D₄	—	7	1	no	—
58	C₂²⋊C₄	—	35	4	no	—
59	C₄²	—	35	1	no	—
60	C₂×C₇	#711	1	2	yes	—
61	C₂×C₇	C₂×C₂²⋊D₅	1	1	yes	characteristic
62	D₇	—	4	2	no	—
63	C₂×C₅	—	28	2	no	—
64	C₂×C₅	#1161	1	2	yes	—
65	C₂×C₅	C₂×C₂²⋊D₇	1	1	yes	characteristic
66	C₂³	—	7	2	no	—
67	Q₈	—	10	2	no	—
68	D₄	—	14	2	no	—
69	C₂×C₄	—	5	2	no	—
70	C₂×C₄	—	35	2	no	—
71	C₂×C₄	D₅×D₇	1	1	yes	characteristic
72	C₇	#4082	1	1	yes	characteristic, complemented, Sylow
73	C₅	#6510	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	14	4	no	—
75	C₂²	—	14	2	no	—
76	C₂²	C₂×D₅×D₇	1	1	yes	characteristic
77	C₄	—	10	2	no	—
78	C₄	—	70	2	no	—
79	C₄	#63305	1	2	yes	—
80	C₂	—	28	2	no	—
81	C₂	#10587939	1	2	yes	—
82	C₂	C₂×#63305	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅×D₇
commutator, socle, cosocle	C₂²×C₅×C₇	C₂³
Fitting	C₂×C₄×C₅×C₇	C₂²
prime radical	C₂×C₅×D₂₈	C₂
hypercenter	C₂×C₄	D₅×D₇
hypocenter	C₅×C₇	#124
LCS	G ▷ C₂²×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×Dic₁₀	C₂	semidirect
3. C₄×Dic₃₅	C₂	semidirect
4. (C₂×D₇)⋊Dic₅	C₂	semidirect
24. C₂×Dic₁₀	D₇	semidirect
42. C₅×C₇	#124	semidirect
72. C₇	#4082	semidirect
73. C₅	#6510	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×Dic₁₀	2
C₂×C₅×D₂₈	2
C₄×Dic₃₅	2
(C₂×D₇)⋊Dic₅	2
#6510	5
#4082	7