Subgroups — #11765205

Order 1120.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₄₀	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#10587902	C₂	1	4	yes	maximal, complemented
5	C₂²×D₃₅	C₂²	1	2	yes	—
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	D₁₄₀	—	2	2	no	—
10	Dic₇₀	—	2	2	no	—
11	#6438	—	5	1	no	maximal, Hall
12	#4021	—	7	1	no	maximal, Hall
13	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₂×D₃₅	—	2	2	no	—
15	C₂×D₃₅	—	2	4	no	—
16	C₄×C₅×C₇	—	2	2	no	—
17	C₄×C₅×C₇	D₄	1	2	yes	—
18	Dic₃₅	—	2	2	no	—
19	C₂×D₂₈	—	5	1	no	—
20	C₂×Dic₁₄	—	5	1	no	—
21	C₄²×C₇	D₅	1	1	yes	characteristic, complemented
22	(C₂×D₇)⋊C₄	—	5	4	no	—
23	C₂×D₂₀	—	7	1	no	—
24	C₂×Dic₁₀	—	7	1	no	—
25	C₄²×C₅	D₇	1	1	yes	characteristic, complemented
26	#685	—	7	4	no	—
27	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₅×C₇	#41	1	2	yes	—
29	D₃₅	—	4	2	no	—
30	C₂²×D₇	—	5	2	no	—
31	C₂×C₄×C₇	C₂×D₅	1	2	yes	—
32	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
33	C₂×Dic₇	—	5	2	no	—
34	D₂₈	—	10	2	no	—
35	Dic₁₄	—	10	2	no	—
36	C₂²×D₅	—	7	2	no	—
37	C₂×C₄×C₅	C₂×D₇	1	2	yes	—
38	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
39	C₂×Dic₅	—	7	2	no	—
40	D₂₀	—	14	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	C₂²×D₅	1	1	yes	characteristic
45	C₂×D₇	—	10	2	no	—
46	C₂×D₇	—	10	4	no	—
47	C₄×C₇	—	2	2	no	—
48	C₄×C₇	D₂₀	1	2	yes	—
49	Dic₇	—	10	2	no	—
50	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
51	C₂×D₅	—	14	2	no	—
52	C₂×D₅	—	14	4	no	—
53	C₄×C₅	—	2	2	no	—
54	C₄×C₅	D₂₈	1	2	yes	—
55	Dic₅	—	14	2	no	—
56	C₂×Q₈	—	35	1	no	—
57	C₂×D₄	—	35	1	no	—
58	C₂²⋊C₄	—	35	4	no	—
59	C₄²	D₃₅	1	1	yes	characteristic, complemented
60	C₂×C₇	C₂×D₂₀	1	1	yes	characteristic
61	C₂×C₇	#709	1	2	yes	—
62	D₇	—	20	2	no	—
63	C₂×C₅	C₂×D₂₈	1	1	yes	characteristic
64	C₂×C₅	#1157	1	2	yes	—
65	D₅	—	28	2	no	—
66	C₂³	—	35	2	no	—
67	Q₈	—	70	2	no	—
68	D₄	—	70	2	no	—
69	C₂×C₄	—	35	2	no	—
70	C₂×C₄	C₂×D₃₅	1	2	yes	—
71	C₂×C₄	C₂×D₃₅	1	1	yes	characteristic
72	C₇	#4021	1	1	yes	characteristic, complemented, Sylow
73	C₅	#6438	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	70	2	no	—
75	C₂²	—	70	4	no	—
76	C₂²	C₂²×D₃₅	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	70	2	no	—
79	C₄	D₁₄₀	1	2	yes	—
80	C₂	—	140	2	no	—
81	C₂	C₂×D₁₄₀	1	1	yes	characteristic
82	C₂	#10587987	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₅
commutator, socle, cosocle	C₂²×C₅×C₇	C₂³
Fitting	C₄²×C₅×C₇	C₂
prime radical	C₂×D₁₄₀	C₂
hypercenter	C₄²	D₃₅
hypocenter	C₅×C₇	#124
LCS	G ▷ C₂²×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₇₀	C₂	semidirect
3. C₄²×C₅×C₇	C₂	semidirect
4. #10587902	C₂	semidirect
21. C₄²×C₇	D₅	semidirect
25. C₄²×C₅	D₇	semidirect
42. C₅×C₇	#124	semidirect
59. C₄²	D₃₅	semidirect
72. C₇	#4021	semidirect
73. C₅	#6438	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₄₀	2
C₂×Dic₇₀	2
C₄²×C₅×C₇	2
#10587902	2
#6438	5
#4021	7