Subgroups — #11764733

Order 1120.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₀	C₂	1	1	yes	maximal, characteristic
2	C₇×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×Dic₇	C₂	1	1	yes	maximal, characteristic
4	#10587858	C₂	1	2	yes	maximal
5	Dic₅×Dic₇	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₃₅	C₂²	1	2	yes	—
8	Dic₇₀	C₄	1	4	yes	complemented
9	C₂×C₇×Dic₅	C₂²	1	2	yes	—
10	C₂×C₅×Dic₇	C₂²	1	1	yes	characteristic
11	C₂×C₅×Dic₇	C₂²	1	1	yes	characteristic
12	#6450	—	5	1	no	maximal, Hall
13	C₄×Dic₁₀	—	7	1	no	maximal, Hall
14	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
15	C₄×C₅×C₇	C₂×C₄	1	2	yes	—
16	Dic₃₅	C₂×C₄	1	4	yes	—
17	C₅×Dic₇	—	2	1	no	—
18	C₅×Dic₇	Q₈	1	2	yes	complemented
19	C₇×Dic₅	—	2	2	no	—
20	C₂×Dic₁₄	—	5	1	no	—
21	C₇×C₄⋊C₄	—	5	1	no	—
22	Dic₇⋊C₄	—	5	2	no	—
23	C₄×Dic₇	—	5	2	no	—
24	C₄×Dic₇	D₅	1	1	yes	characteristic
25	C₂×Dic₁₀	—	7	1	no	—
26	C₄²×C₅	—	7	1	no	—
27	C₄⋊Dic₅	D₇	1	1	yes	characteristic
28	#683	—	7	2	no	—
29	C₄×Dic₅	—	7	2	no	—
30	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₅×C₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₅×C₇	#41	1	1	yes	characteristic
33	C₂×C₄×C₇	—	5	2	no	—
34	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
35	C₂×Dic₇	—	5	2	no	—
36	C₂×Dic₇	C₂×D₅	1	1	yes	characteristic
37	C₂×Dic₇	C₂×D₅	1	1	yes	characteristic
38	Dic₁₄	—	5	4	no	—
39	C₂×C₄×C₅	—	7	1	no	—
40	C₂×C₄×C₅	—	7	1	no	—
41	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
42	C₂×Dic₅	—	7	2	no	—
43	C₂×Dic₅	C₂×D₇	1	2	yes	—
44	Dic₁₀	—	7	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	—	10	2	no	—
49	C₄×C₇	C₄×D₅	1	2	yes	—
50	Dic₇	—	2	1	no	—
51	Dic₇	—	5	4	no	—
52	Dic₇	Dic₁₀	1	2	yes	complemented
53	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
54	C₄×C₅	—	7	2	no	—
55	C₄×C₅	—	14	1	no	—
56	C₄×C₅	C₄×D₇	1	2	yes	—
57	Dic₅	—	2	2	no	—
58	Dic₅	—	7	4	no	—
59	C₂×Q₈	—	35	1	no	—
60	C₄⋊C₄	—	5	1	no	—
61	C₄⋊C₄	—	35	2	no	—
62	C₄²	—	7	1	no	—
63	C₄²	—	35	2	no	—
64	C₂×C₇	C₂×Dic₁₀	1	1	yes	characteristic
65	C₂×C₇	C₂×C₄×D₅	1	1	yes	characteristic
66	C₂×C₇	#709	1	1	yes	characteristic
67	C₂×C₅	C₂×C₄×D₇	1	1	yes	characteristic
68	C₂×C₅	#1159	1	1	yes	characteristic
69	C₂×C₅	Q₈×D₇	1	1	yes	characteristic
70	Q₈	—	35	4	no	—
71	C₂×C₄	—	5	2	no	—
72	C₂×C₄	—	7	1	no	—
73	C₂×C₄	—	7	1	no	—
74	C₂×C₄	—	35	2	no	—
75	C₂×C₄	D₅×D₇	1	1	yes	characteristic
76	C₇	C₄×Dic₁₀	1	1	yes	characteristic, complemented, Sylow
77	C₅	#6450	1	1	yes	characteristic, complemented, Sylow
78	C₂²	C₂×D₅×D₇	1	1	yes	characteristic
79	C₄	—	7	2	no	—
80	C₄	—	10	2	no	—
81	C₄	—	14	1	no	—
82	C₄	—	35	4	no	—
83	C₄	#63304	1	2	yes	—
84	C₂	D₇×Dic₁₀	1	1	yes	characteristic
85	C₂	#10587945	1	1	yes	characteristic
86	C₂	C₂×#63304	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅×D₇
commutator	C₂×C₅×C₇	C₂²×C₄
Fitting	C₂×C₄×C₅×C₇	C₂²
socle, prime radical, cosocle	C₂²×C₅×C₇	C₂³
hypercenter	C₂×C₄	D₅×D₇
hypocenter	C₅×C₇	C₄×Q₈
LCS	G ▷ C₂×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
8. Dic₇₀	C₄	semidirect
18. C₅×Dic₇	Q₈	semidirect
45. C₅×C₇	C₄×Q₈	semidirect
52. Dic₇	Dic₁₀	semidirect
76. C₇	C₄×Dic₁₀	semidirect
77. C₅	#6450	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
C₇×C₄⋊Dic₅	2
C₄×C₅×Dic₇	2
#10587858	2
Dic₅×Dic₇	2
#6450	5
C₄×Dic₁₀	7