Subgroups — C₇×#4034 #11765133

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Dic₁₀	C₂	1	2	yes	maximal
2	C₅×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₇×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₇×#683	C₂	1	2	yes	maximal
5	C₄×C₇×Dic₅	C₂	1	1	yes	maximal, characteristic
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	1	yes	characteristic
10	C₂×C₇×Dic₅	C₂²	1	1	yes	characteristic
11	C₇×Dic₁₀	—	2	4	no	—
12	C₇×C₄⋊Q₈	—	5	1	no	maximal, Hall
13	#4034	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
15	C₄×C₅×C₇	—	2	2	no	—
16	C₄×C₅×C₇	Q₈	1	2	yes	complemented
17	C₇×Dic₅	—	2	2	no	—
18	C₇×Dic₅	D₄	1	2	yes	—
19	C₇×Dic₅	Q₈	1	2	yes	complemented
20	C₂×C₇×Q₈	—	5	2	no	—
21	C₇×C₄⋊C₄	—	5	2	no	—
22	C₇×C₄⋊C₄	—	5	1	no	—
23	C₇×C₄⋊C₄	D₅	1	1	yes	characteristic
24	C₄²×C₇	—	5	1	no	—
25	C₂×Dic₁₀	C₂×C₇	1	2	yes	—
26	C₅×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
27	C₄⋊Dic₅	C₂×C₇	1	1	yes	characteristic
28	#683	C₂×C₇	1	2	yes	—
29	C₄×Dic₅	C₂×C₇	1	1	yes	characteristic
30	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₅×C₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₅×C₇	C₂×Q₈	1	1	yes	characteristic
33	C₇×Q₈	—	10	4	no	—
34	C₂×C₄×C₇	—	5	1	no	—
35	C₂×C₄×C₇	—	5	1	no	—
36	C₂×C₄×C₇	—	5	2	no	—
37	C₂×C₄×C₇	C₂×D₅	1	2	yes	—
38	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
39	C₂×C₄×C₅	C₂²×C₇	1	2	yes	—
40	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
41	C₂×Dic₅	C₂²×C₇	1	2	yes	—
42	C₂×Dic₅	C₂²×C₇	1	1	yes	characteristic
43	C₂×Dic₅	C₂²×C₇	1	1	yes	characteristic
44	Dic₁₀	—	2	4	no	—
45	C₅×C₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
46	C₄⋊Q₈	—	5	1	no	Sylow
47	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
48	C₄×C₇	—	2	2	no	—
49	C₄×C₇	—	5	2	no	—
50	C₄×C₇	—	5	2	no	—
51	C₄×C₇	—	10	2	no	—
52	C₄×C₇	Dic₁₀	1	2	yes	complemented
53	C₂²×C₅	C₂³×C₇	1	1	yes	characteristic
54	C₄×C₅	—	2	2	no	—
55	C₄×C₅	C₇×Q₈	1	2	yes	complemented
56	Dic₅	—	2	2	no	—
57	Dic₅	C₇×D₄	1	2	yes	—
58	Dic₅	C₇×Q₈	1	2	yes	complemented
59	C₂×Q₈	—	5	2	no	—
60	C₄⋊C₄	—	5	2	no	—
61	C₄⋊C₄	—	5	1	no	—
62	C₄⋊C₄	C₇×D₅	1	1	yes	characteristic
63	C₄²	—	5	1	no	—
64	C₂×C₇	C₂×Dic₁₀	1	1	yes	characteristic
65	C₂×C₇	D₄×D₅	1	1	yes	characteristic
66	C₂×C₇	Q₈×D₅	1	1	yes	characteristic
67	C₂×C₅	C₂×C₇×D₄	1	1	yes	characteristic
68	C₂×C₅	C₂×C₇×Q₈	1	1	yes	characteristic
69	C₂×C₅	C₂×C₇×Q₈	1	1	yes	characteristic
70	Q₈	—	10	4	no	—
71	C₂×C₄	—	5	1	no	—
72	C₂×C₄	—	5	1	no	—
73	C₂×C₄	—	5	2	no	—
74	C₂×C₄	C₂×C₇×D₅	1	2	yes	—
75	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
76	C₇	#4034	1	1	yes	characteristic, direct product factor, Sylow
77	C₅	C₇×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
78	C₂²	C₂²×C₇×D₅	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	5	2	no	—
81	C₄	—	5	2	no	—
82	C₄	—	10	2	no	—
83	C₄	C₇×Dic₁₀	1	2	yes	complemented
84	C₂	C₂×C₇×Dic₁₀	1	1	yes	characteristic
85	C₂	C₇×D₄×D₅	1	1	yes	characteristic
86	C₂	C₇×Q₈×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	C₅×C₇×C₄⋊C₄	C₂
socle, prime radical	C₂²×C₅×C₇	C₂³
hypercenter	C₇×C₄⋊C₄	D₅
hypocenter	C₅	C₇×C₄⋊Q₈
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₇ ◁ C₇×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
13. #4034	C₇	direct (N × H)
16. C₄×C₅×C₇	Q₈	semidirect
19. C₇×Dic₅	Q₈	semidirect
45. C₅×C₇	C₄⋊Q₈	semidirect
52. C₄×C₇	Dic₁₀	semidirect
55. C₄×C₅	C₇×Q₈	semidirect
58. Dic₅	C₇×Q₈	semidirect
76. C₇	#4034	direct (N × H)
77. C₅	C₇×C₄⋊Q₈	semidirect
83. C₄	C₇×Dic₁₀	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×Dic₁₀	2
C₅×C₇×C₄⋊C₄	2
C₇×C₄⋊Dic₅	2
C₇×#683	2
C₄×C₇×Dic₅	2
C₇×C₄⋊Q₈	5
#4034	7

Subgroups — C7×#4034 #11765133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4034 #11765133