Subgroups — C₇×#4021 #11765119

Order 1120.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₂₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#685	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×D₅	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×C₇×Dic₅	C₂²	1	2	yes	—
9	C₇×D₂₀	—	2	2	no	—
10	C₇×Dic₁₀	—	2	2	no	—
11	C₇×#124	—	5	1	no	maximal, Hall
12	#4021	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₂×C₇×D₅	—	2	2	no	—
15	C₂×C₇×D₅	—	2	4	no	—
16	C₄×C₅×C₇	—	2	2	no	—
17	C₄×C₅×C₇	D₄	1	2	yes	—
18	C₇×Dic₅	—	2	2	no	—
19	C₂×C₇×Q₈	—	5	1	no	—
20	C₂×C₇×D₄	—	5	1	no	—
21	C₇×C₂²⋊C₄	—	5	4	no	—
22	C₄²×C₇	D₅	1	1	yes	characteristic, complemented
23	C₂×D₂₀	C₂×C₇	1	1	yes	characteristic
24	C₂×Dic₁₀	C₂×C₇	1	1	yes	characteristic, complemented
25	C₄²×C₅	C₂×C₇	1	1	yes	characteristic, complemented
26	#685	C₂×C₇	1	4	yes	complemented
27	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₅×C₇	#41	1	2	yes	—
29	C₇×D₅	—	4	2	no	—
30	C₂³×C₇	—	5	2	no	—
31	C₇×Q₈	—	10	2	no	—
32	C₇×D₄	—	10	2	no	—
33	C₂×C₄×C₇	—	5	2	no	—
34	C₂×C₄×C₇	C₂×D₅	1	2	yes	—
35	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
36	C₂²×D₅	C₂²×C₇	1	2	yes	—
37	C₂×C₄×C₅	C₂²×C₇	1	2	yes	—
38	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
39	C₂×Dic₅	C₂²×C₇	1	2	yes	—
40	D₂₀	—	2	2	no	—
41	Dic₁₀	—	2	2	no	—
42	C₅×C₇	#124	1	1	yes	characteristic, complemented, Hall
43	#124	—	5	1	no	Sylow
44	C₂²×C₇	—	10	2	no	—
45	C₂²×C₇	—	10	4	no	—
46	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
47	C₄×C₇	—	2	2	no	—
48	C₄×C₇	—	10	2	no	—
49	C₄×C₇	D₂₀	1	2	yes	—
50	C₂²×C₅	C₂³×C₇	1	1	yes	characteristic
51	C₂×D₅	—	2	2	no	—
52	C₂×D₅	—	2	4	no	—
53	C₄×C₅	—	2	2	no	—
54	C₄×C₅	C₇×D₄	1	2	yes	—
55	Dic₅	—	2	2	no	—
56	C₂×Q₈	—	5	1	no	—
57	C₂×D₄	—	5	1	no	—
58	C₂²⋊C₄	—	5	4	no	—
59	C₄²	C₇×D₅	1	1	yes	characteristic, complemented
60	C₂×C₇	—	20	2	no	—
61	C₂×C₇	C₂×D₂₀	1	1	yes	characteristic
62	C₂×C₇	#709	1	2	yes	—
63	C₂×C₅	C₂×C₇×D₄	1	1	yes	characteristic
64	C₂×C₅	C₇×#41	1	2	yes	—
65	D₅	—	4	2	no	—
66	C₂³	—	5	2	no	—
67	Q₈	—	10	2	no	—
68	D₄	—	10	2	no	—
69	C₂×C₄	—	5	2	no	—
70	C₂×C₄	C₂×C₇×D₅	1	2	yes	—
71	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
72	C₇	#4021	1	1	yes	characteristic, direct product factor, Sylow
73	C₅	C₇×#124	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	10	2	no	—
75	C₂²	—	10	4	no	—
76	C₂²	C₂²×C₇×D₅	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	10	2	no	—
79	C₄	C₇×D₂₀	1	2	yes	—
80	C₂	—	20	2	no	—
81	C₂	C₂×C₇×D₂₀	1	1	yes	characteristic
82	C₂	C₇×#709	1	2	yes	—

	subgroup	G/H
center	C₂²×C₇	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂³×C₇
Frattini	C₂²	C₂²×C₇×D₅
Fitting	C₄²×C₅×C₇	C₂
socle	C₂²×C₅×C₇	C₂³
prime radical	C₂×C₇×D₂₀	C₂
hypercenter	C₄²×C₇	D₅
hypocenter	C₅	C₇×#124
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×Dic₁₀	C₂	semidirect
3. C₄²×C₅×C₇	C₂	semidirect
4. C₇×#685	C₂	semidirect
12. #4021	C₇	direct (N × H)
22. C₄²×C₇	D₅	semidirect
24. C₂×Dic₁₀	C₂×C₇	semidirect
25. C₄²×C₅	C₂×C₇	semidirect
26. #685	C₂×C₇	semidirect
42. C₅×C₇	#124	semidirect
59. C₄²	C₇×D₅	semidirect
72. C₇	#4021	direct (N × H)
73. C₅	C₇×#124	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×D₂₀	2
C₂×C₇×Dic₁₀	2
C₄²×C₅×C₇	2
C₇×#685	2
C₇×#124	5
#4021	7

Subgroups — C7×#4021 #11765119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4021 #11765119