Subgroups — C₇×#4086 #11765177

Order 1120.

216 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, complemented
2	C₂×C₇×C₂²⋊D₅	C₂	1	4	yes	maximal, complemented
3	C₂×C₇×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₇×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₅×C₇	C₂²	1	2	yes	—
6	C₂²×C₇×D₅	C₂²	1	2	yes	—
7	C₅×C₇×D₄	—	2	2	no	—
8	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
9	C₇×C₂²⋊D₅	—	2	8	no	—
10	C₂×C₇×Dic₅	C₂²	1	2	yes	—
11	C₇×D₂₀	—	2	2	no	—
12	C₇×#127	—	5	1	no	maximal, Hall
13	#4086	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₅×C₇	—	2	4	no	—
15	C₂²×C₅×C₇	—	2	2	no	—
16	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
17	C₂×C₇×D₅	—	2	2	no	—
18	C₂×C₇×D₅	—	2	4	no	—
19	C₄×C₅×C₇	D₄	1	2	yes	complemented
20	C₇×Dic₅	D₄	1	4	yes	complemented
21	C₂×C₇×D₄	—	5	4	no	—
22	C₂×C₇×D₄	—	5	1	no	—
23	C₂×C₇×D₄	D₅	1	1	yes	characteristic, complemented
24	C₄²×C₇	—	5	1	no	—
25	C₂×C₅×D₄	C₂×C₇	1	1	yes	characteristic, complemented
26	C₂×C₂²⋊D₅	C₂×C₇	1	4	yes	complemented
27	C₂×D₂₀	C₂×C₇	1	1	yes	characteristic, complemented
28	C₄×Dic₅	C₂×C₇	1	1	yes	characteristic, complemented
29	C₂×C₅×C₇	—	4	2	no	—
30	C₂×C₅×C₇	C₂×D₄	1	2	yes	—
31	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
32	C₇×D₅	—	4	2	no	—
33	C₂³×C₇	—	5	2	no	—
34	C₂³×C₇	C₂×D₅	1	2	yes	—
35	C₇×D₄	—	2	2	no	—
36	C₇×D₄	—	10	8	no	—
37	C₇×D₄	—	10	2	no	—
38	C₂×C₄×C₇	—	5	2	no	—
39	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
40	C₂³×C₅	C₂²×C₇	1	2	yes	—
41	C₂²×D₅	C₂²×C₇	1	2	yes	—
42	C₅×D₄	—	2	2	no	—
43	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
44	C₂²⋊D₅	—	2	8	no	—
45	C₂×Dic₅	C₂²×C₇	1	2	yes	—
46	D₂₀	—	2	2	no	—
47	C₅×C₇	#127	1	1	yes	characteristic, complemented, Hall
48	#127	—	5	1	no	Sylow
49	C₂²×C₇	—	2	4	no	—
50	C₂²×C₇	—	2	2	no	—
51	C₂²×C₇	—	10	4	no	—
52	C₂²×C₇	—	10	2	no	—
53	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
54	C₄×C₇	—	5	4	no	—
55	C₄×C₇	C₂²⋊D₅	1	2	yes	complemented
56	C₂²×C₅	—	2	4	no	—
57	C₂²×C₅	—	2	2	no	—
58	C₂²×C₅	C₂³×C₇	1	1	yes	characteristic
59	C₂×D₅	—	2	2	no	—
60	C₂×D₅	—	2	4	no	—
61	C₄×C₅	C₇×D₄	1	2	yes	complemented
62	Dic₅	C₇×D₄	1	4	yes	complemented
63	C₂×D₄	—	5	4	no	—
64	C₂×D₄	—	5	1	no	—
65	C₂×D₄	C₇×D₅	1	1	yes	characteristic, complemented
66	C₄²	—	5	1	no	—
67	C₂×C₇	—	4	2	no	—
68	C₂×C₇	—	20	2	no	—
69	C₂×C₇	D₄×D₅	1	2	yes	—
70	C₂×C₇	C₂×C₂²⋊D₅	1	1	yes	characteristic
71	C₂×C₅	—	4	2	no	—
72	C₂×C₅	C₂×C₇×D₄	1	2	yes	—
73	C₂×C₅	C₂×C₇×D₄	1	1	yes	characteristic
74	D₅	—	4	2	no	—
75	C₂³	—	5	2	no	—
76	C₂³	C₂×C₇×D₅	1	2	yes	—
77	D₄	—	2	2	no	—
78	D₄	—	10	8	no	—
79	D₄	—	10	2	no	—
80	C₂×C₄	—	5	2	no	—
81	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
82	C₇	#4086	1	1	yes	characteristic, direct product factor, Sylow
83	C₅	C₇×#127	1	1	yes	characteristic, complemented, Sylow
84	C₂²	—	2	4	no	—
85	C₂²	—	2	2	no	—
86	C₂²	—	10	4	no	—
87	C₂²	—	10	2	no	—
88	C₂²	C₂²×C₇×D₅	1	1	yes	characteristic
89	C₄	—	5	4	no	—
90	C₄	C₇×C₂²⋊D₅	1	2	yes	complemented
91	C₂	—	4	2	no	—
92	C₂	—	20	2	no	—
93	C₂	C₇×D₄×D₅	1	2	yes	—
94	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×D₄	C₂
socle	C₂²×C₅×C₇	C₂³
hypercenter	C₂×C₇×D₄	D₅
hypocenter	C₅	C₇×#127
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₇×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₇×D₄	C₂	semidirect
2. C₂×C₇×C₂²⋊D₅	C₂	semidirect
3. C₂×C₇×D₂₀	C₂	semidirect
4. C₄×C₇×Dic₅	C₂	semidirect
13. #4086	C₇	direct (N × H)
19. C₄×C₅×C₇	D₄	semidirect
20. C₇×Dic₅	D₄	semidirect
23. C₂×C₇×D₄	D₅	semidirect
25. C₂×C₅×D₄	C₂×C₇	semidirect
26. C₂×C₂²⋊D₅	C₂×C₇	semidirect
27. C₂×D₂₀	C₂×C₇	semidirect
28. C₄×Dic₅	C₂×C₇	semidirect
47. C₅×C₇	#127	semidirect
55. C₄×C₇	C₂²⋊D₅	semidirect
61. C₄×C₅	C₇×D₄	semidirect
62. Dic₅	C₇×D₄	semidirect
65. C₂×D₄	C₇×D₅	semidirect
82. C₇	#4086	direct (N × H)
83. C₅	C₇×#127	semidirect
90. C₄	C₇×C₂²⋊D₅	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×C₇×C₂²⋊D₅	2
C₂×C₇×D₂₀	2
C₄×C₇×Dic₅	2
C₇×#127	5
#4086	7

Subgroups — C7×#4086 #11765177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4086 #11765177