Subgroups — #11764772

Order 1120.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₃₅	C₂	1	1	yes	maximal, characteristic
2	C₇×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₅×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
4	#10587859	C₂	1	2	yes	maximal
5	Dic₇⋊Dic₅	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₃₅	C₂²	1	1	yes	characteristic
8	C₂×Dic₃₅	C₂²	1	1	yes	characteristic
9	C₂×C₇×Dic₅	C₂²	1	2	yes	—
10	C₂×C₅×Dic₇	C₂²	1	2	yes	—
11	#6453	—	5	1	no	maximal, Hall
12	#4036	—	7	1	no	maximal, Hall
13	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₄×C₅×C₇	Q₈	1	2	yes	—
15	Dic₃₅	—	2	1	no	—
16	Dic₃₅	—	2	1	no	—
17	C₅×Dic₇	—	2	2	no	—
18	C₇×Dic₅	—	2	2	no	—
19	C₇×C₄⋊C₄	—	5	1	no	—
20	C₄⋊Dic₇	—	5	2	no	—
21	C₄⋊Dic₇	D₅	1	1	yes	characteristic
22	Dic₇⋊C₄	—	5	2	no	—
23	C₄×Dic₇	—	5	1	no	—
24	C₅×C₄⋊C₄	—	7	1	no	—
25	C₄⋊Dic₅	—	7	2	no	—
26	C₄⋊Dic₅	D₇	1	1	yes	characteristic
27	#683	—	7	2	no	—
28	C₄×Dic₅	—	7	1	no	—
29	C₂×C₅×C₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₅×C₇	#41	1	1	yes	characteristic
31	C₂×C₅×C₇	#41	1	1	yes	characteristic
32	C₂×C₄×C₇	—	5	2	no	—
33	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
34	C₂×Dic₇	—	5	1	no	—
35	C₂×Dic₇	—	5	1	no	—
36	C₂×Dic₇	C₂×D₅	1	2	yes	—
37	C₂×C₄×C₅	—	7	2	no	—
38	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
39	C₂×Dic₅	—	7	1	no	—
40	C₂×Dic₅	—	7	1	no	—
41	C₂×Dic₅	C₂×D₇	1	2	yes	—
42	C₅×C₇	#125	1	1	yes	characteristic, complemented, Hall
43	#125	—	35	1	no	Sylow
44	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
45	C₄×C₇	—	10	2	no	—
46	C₄×C₇	Dic₁₀	1	2	yes	—
47	Dic₇	—	2	2	no	—
48	Dic₇	—	10	1	no	—
49	Dic₇	—	10	1	no	—
50	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
51	C₄×C₅	—	14	2	no	—
52	C₄×C₅	Dic₁₄	1	2	yes	—
53	Dic₅	—	2	2	no	—
54	Dic₅	—	14	1	no	—
55	Dic₅	—	14	1	no	—
56	C₄⋊C₄	—	5	1	no	—
57	C₄⋊C₄	—	7	1	no	—
58	C₄⋊C₄	—	35	2	no	—
59	C₄⋊C₄	—	35	2	no	—
60	C₄²	—	35	1	no	—
61	C₂×C₇	C₂×Dic₁₀	1	1	yes	characteristic
62	C₂×C₇	#711	1	1	yes	characteristic
63	C₂×C₇	#713	1	1	yes	characteristic
64	C₂×C₅	C₂×Dic₁₄	1	1	yes	characteristic
65	C₂×C₅	#1159	1	1	yes	characteristic
66	C₂×C₅	#1161	1	1	yes	characteristic
67	C₂×C₄	—	5	2	no	—
68	C₂×C₄	—	7	2	no	—
69	C₂×C₄	—	35	1	no	—
70	C₂×C₄	—	35	1	no	—
71	C₂×C₄	D₅×D₇	1	1	yes	characteristic
72	C₇	#4036	1	1	yes	characteristic, complemented, Sylow
73	C₅	#6453	1	1	yes	characteristic, complemented, Sylow
74	C₂²	C₂×D₅×D₇	1	1	yes	characteristic
75	C₄	—	10	2	no	—
76	C₄	—	14	2	no	—
77	C₄	—	70	1	no	—
78	C₄	—	70	1	no	—
79	C₄	#63308	1	2	yes	—
80	C₂	#10587939	1	1	yes	characteristic
81	C₂	#10587941	1	1	yes	characteristic
82	C₂	C₂×#63308	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅×D₇
commutator, socle, prime radical, cosocle	C₂²×C₅×C₇	C₂³
Fitting	C₂×C₄×C₅×C₇	C₂²
hypercenter	C₂×C₄	D₅×D₇
hypocenter	C₅×C₇	#125
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
42. C₅×C₇	#125	semidirect
72. C₇	#4036	semidirect
73. C₅	#6453	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₃₅	2
C₇×C₄⋊Dic₅	2
C₅×C₄⋊Dic₇	2
#10587859	2
Dic₇⋊Dic₅	2
#6453	5
#4036	7