Subgroups — #11765265

Order 1120.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊Dic₃₅	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₃₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₅×C₇	C₂²	1	2	yes	—
6	C₅×C₇×D₄	—	2	2	no	—
7	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
8	C₂×Dic₃₅	C₂²	1	2	yes	—
9	C₂×Dic₃₅	C₂²	1	2	yes	—
10	Dic₇₀	—	2	2	no	—
11	#6499	—	5	1	no	maximal, Hall
12	#4082	—	7	1	no	maximal, Hall
13	C₂²×C₅×C₇	—	2	2	no	—
14	C₂²×C₅×C₇	—	2	4	no	—
15	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
16	C₄×C₅×C₇	D₄	1	2	yes	—
17	Dic₃₅	—	2	2	no	—
18	Dic₃₅	—	2	2	no	—
19	C₂×C₇×D₄	D₅	1	1	yes	characteristic
20	C₂×Dic₁₄	—	5	1	no	—
21	C₂²⋊Dic₇	—	5	4	no	—
22	C₄×Dic₇	—	5	1	no	—
23	C₂×C₅×D₄	D₇	1	1	yes	characteristic
24	C₂×Dic₁₀	—	7	1	no	—
25	C₂²⋊Dic₅	—	7	4	no	—
26	C₄×Dic₅	—	7	1	no	—
27	C₂×C₅×C₇	—	4	2	no	—
28	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₅×C₇	#41	1	2	yes	—
30	C₂³×C₇	C₂×D₅	1	2	yes	—
31	C₇×D₄	—	2	2	no	—
32	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
33	C₂×Dic₇	—	5	2	no	—
34	C₂×Dic₇	—	5	2	no	—
35	Dic₁₄	—	10	2	no	—
36	C₂³×C₅	C₂×D₇	1	2	yes	—
37	C₅×D₄	—	2	2	no	—
38	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
39	C₂×Dic₅	—	7	2	no	—
40	C₂×Dic₅	—	7	2	no	—
41	Dic₁₀	—	14	2	no	—
42	C₅×C₇	#124	1	1	yes	characteristic, complemented, Hall
43	#124	—	35	1	no	Sylow
44	C₂²×C₇	—	2	4	no	—
45	C₂²×C₇	—	2	2	no	—
46	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
47	C₄×C₇	C₂²⋊D₅	1	2	yes	—
48	Dic₇	—	10	2	no	—
49	Dic₇	—	10	2	no	—
50	C₂²×C₅	—	2	4	no	—
51	C₂²×C₅	—	2	2	no	—
52	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
53	C₄×C₅	C₂²⋊D₇	1	2	yes	—
54	Dic₅	—	14	2	no	—
55	Dic₅	—	14	2	no	—
56	C₂×Q₈	—	35	1	no	—
57	C₂×D₄	D₃₅	1	1	yes	characteristic
58	C₂²⋊C₄	—	35	4	no	—
59	C₄²	—	35	1	no	—
60	C₂×C₇	—	4	2	no	—
61	C₂×C₇	#711	1	2	yes	—
62	C₂×C₇	C₂×C₂²⋊D₅	1	1	yes	characteristic
63	C₂×C₅	—	4	2	no	—
64	C₂×C₅	#1159	1	2	yes	—
65	C₂×C₅	C₂×C₂²⋊D₇	1	1	yes	characteristic
66	C₂³	C₂×D₃₅	1	2	yes	—
67	Q₈	—	70	2	no	—
68	D₄	—	2	2	no	—
69	C₂×C₄	—	35	2	no	—
70	C₂×C₄	—	35	2	no	—
71	C₂×C₄	C₂×D₃₅	1	1	yes	characteristic
72	C₇	#4082	1	1	yes	characteristic, complemented, Sylow
73	C₅	#6499	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	2	4	no	—
75	C₂²	—	2	2	no	—
76	C₂²	C₂²×D₃₅	1	1	yes	characteristic
77	C₄	—	70	2	no	—
78	C₄	—	70	2	no	—
79	C₄	C₂²⋊D₃₅	1	2	yes	—
80	C₂	—	4	2	no	—
81	C₂	#10587989	1	2	yes	—
82	C₂	C₂×C₂²⋊D₃₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₅
commutator, socle, cosocle	C₂²×C₅×C₇	C₂³
Fitting, prime radical	C₂×C₅×C₇×D₄	C₂
hypercenter	C₂×D₄	D₃₅
hypocenter	C₅×C₇	#124
LCS	G ▷ C₂²×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₇₀	C₂	semidirect
3. C₂²⋊Dic₃₅	C₂	semidirect
4. C₄×Dic₃₅	C₂	semidirect
42. C₅×C₇	#124	semidirect
72. C₇	#4082	semidirect
73. C₅	#6499	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×D₄	2
C₂×Dic₇₀	2
C₂²⋊Dic₃₅	2
C₄×Dic₃₅	2
#6499	5
#4082	7