Subgroups — C₇×#4090 #11765185

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×Dic₁₀	C₂	1	1	yes	maximal, characteristic
3	C₇×#683	C₂	1	4	yes	maximal
4	C₄×C₇×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₅×C₇×Q₈	—	2	2	no	—
6	C₂×C₄×C₅×C₇	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₇×Dic₅	C₂²	1	2	yes	—
9	C₂×C₇×Dic₅	C₂²	1	2	yes	—
10	C₇×Dic₁₀	—	2	2	no	—
11	C₇×C₄⋊Q₈	—	5	1	no	maximal, Hall
12	#4090	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₄×C₅×C₇	—	2	2	no	—
15	C₄×C₅×C₇	D₄	1	2	yes	—
16	C₇×Dic₅	—	2	2	no	—
17	C₇×Dic₅	Q₈	1	4	yes	complemented
18	C₂×C₇×Q₈	—	5	1	no	—
19	C₂×C₇×Q₈	D₅	1	1	yes	characteristic
20	C₇×C₄⋊C₄	—	5	4	no	—
21	C₄²×C₇	—	5	1	no	—
22	C₂×C₅×Q₈	C₂×C₇	1	1	yes	characteristic
23	C₂×Dic₁₀	C₂×C₇	1	1	yes	characteristic
24	#683	C₂×C₇	1	4	yes	—
25	C₄×Dic₅	C₂×C₇	1	1	yes	characteristic
26	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
27	C₂×C₅×C₇	C₂×Q₈	1	2	yes	—
28	C₇×Q₈	—	2	2	no	—
29	C₇×Q₈	—	10	2	no	—
30	C₂×C₄×C₇	—	5	2	no	—
31	C₂×C₄×C₇	—	5	2	no	—
32	C₂×C₄×C₇	C₂×D₅	1	2	yes	—
33	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
34	C₅×Q₈	—	2	2	no	—
35	C₂×C₄×C₅	C₂²×C₇	1	2	yes	—
36	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
37	C₂×Dic₅	C₂²×C₇	1	2	yes	—
38	C₂×Dic₅	C₂²×C₇	1	2	yes	—
39	Dic₁₀	—	2	2	no	—
40	C₅×C₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
41	C₄⋊Q₈	—	5	1	no	Sylow
42	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
43	C₄×C₇	—	2	2	no	—
44	C₄×C₇	—	5	4	no	—
45	C₄×C₇	—	10	2	no	—
46	C₄×C₇	C₂²⋊D₅	1	2	yes	—
47	C₂²×C₅	C₂³×C₇	1	1	yes	characteristic
48	C₄×C₅	—	2	2	no	—
49	C₄×C₅	C₇×D₄	1	2	yes	—
50	Dic₅	—	2	2	no	—
51	Dic₅	C₇×Q₈	1	4	yes	complemented
52	C₂×Q₈	—	5	1	no	—
53	C₂×Q₈	C₇×D₅	1	1	yes	characteristic
54	C₄⋊C₄	—	5	4	no	—
55	C₄²	—	5	1	no	—
56	C₂×C₇	Q₈×D₅	1	2	yes	—
57	C₂×C₇	C₂×C₂²⋊D₅	1	1	yes	characteristic
58	C₂×C₅	C₂×C₇×D₄	1	1	yes	characteristic
59	C₂×C₅	C₂×C₇×Q₈	1	2	yes	—
60	Q₈	—	2	2	no	—
61	Q₈	—	10	2	no	—
62	C₂×C₄	—	5	2	no	—
63	C₂×C₄	—	5	2	no	—
64	C₂×C₄	C₂×C₇×D₅	1	2	yes	—
65	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
66	C₇	#4090	1	1	yes	characteristic, direct product factor, Sylow
67	C₅	C₇×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
68	C₂²	C₂²×C₇×D₅	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	5	4	no	—
71	C₄	—	10	2	no	—
72	C₄	C₇×C₂²⋊D₅	1	2	yes	—
73	C₂	C₇×Q₈×D₅	1	2	yes	—
74	C₂	C₂×C₇×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂³×C₇
Frattini	C₂²	C₂²×C₇×D₅
Fitting	C₂×C₅×C₇×Q₈	C₂
socle, prime radical	C₂²×C₅×C₇	C₂³
hypercenter	C₂×C₇×Q₈	D₅
hypocenter	C₅	C₇×C₄⋊Q₈
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₇×Q₈

Complements

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

N	H ≅ G/N	type
12. #4090	C₇	direct (N × H)
17. C₇×Dic₅	Q₈	semidirect
40. C₅×C₇	C₄⋊Q₈	semidirect
51. Dic₅	C₇×Q₈	semidirect
66. C₇	#4090	direct (N × H)
67. C₅	C₇×C₄⋊Q₈	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×Q₈	2
C₂×C₇×Dic₁₀	2
C₇×#683	2
C₄×C₇×Dic₅	2
C₇×C₄⋊Q₈	5
#4090	7

Subgroups — C7×#4090 #11765185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4090 #11765185