Subgroups — C₇×#4065 #11765163

Order 1120.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#713	C₂	1	2	yes	maximal, complemented
2	C₅×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#688	C₂	1	2	yes	maximal, complemented
4	C₇×D₄₀	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×C₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×C₇×Q₈	C₂²	1	2	yes	—
7	C₇×D₂₀	—	2	2	no	—
8	C₇×D₂₀	C₂²	1	2	yes	—
9	C₄×C₇×D₅	—	2	2	no	—
10	C₄×C₇×D₅	C₂²	1	1	yes	characteristic
11	C₅×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×#168	C₂²	1	1	yes	characteristic
13	C₇×#135	—	5	1	no	maximal, Hall
14	#4065	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₇×D₅	—	2	2	no	—
16	C₂×C₇×D₅	D₄	1	1	yes	characteristic
17	C₄×C₅×C₇	—	2	2	no	—
18	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
19	C₇×Dic₅	D₄	1	1	yes	characteristic
20	C₇×#41	—	5	2	no	—
21	C₇×Q₁₆	D₅	1	1	yes	characteristic, complemented
22	C₇×SD₁₆	—	5	2	no	—
23	C₇×D₈	—	5	1	no	—
24	C₂×C₇×C₈	—	5	1	no	—
25	#713	C₂×C₇	1	2	yes	complemented
26	C₅×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
27	#688	C₂×C₇	1	2	yes	complemented
28	D₄₀	C₂×C₇	1	1	yes	characteristic, complemented
29	C₈×D₅	C₂×C₇	1	1	yes	characteristic, complemented
30	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
31	C₇×D₅	—	2	1	no	—
32	C₇×D₅	—	4	2	no	—
33	C₇×Q₈	C₂×D₅	1	2	yes	—
34	C₇×D₄	—	5	2	no	—
35	C₇×D₄	—	10	2	no	—
36	C₂×C₄×C₇	—	5	1	no	—
37	C₂×C₄×C₇	—	10	2	no	—
38	C₇×C₈	—	5	1	no	—
39	C₇×C₈	C₂×D₅	1	1	yes	characteristic
40	C₅×Q₈	C₂²×C₇	1	2	yes	—
41	D₂₀	—	2	2	no	—
42	D₂₀	C₂²×C₇	1	2	yes	—
43	C₄×D₅	—	2	2	no	—
44	C₄×D₅	C₂²×C₇	1	1	yes	characteristic
45	C₅×C₈	C₂²×C₇	1	1	yes	characteristic
46	#168	C₂²×C₇	1	1	yes	characteristic
47	C₅×C₇	#135	1	1	yes	characteristic, complemented, Hall
48	#135	—	5	1	no	Sylow
49	C₂²×C₇	—	5	1	no	—
50	C₂²×C₇	—	10	2	no	—
51	C₄×C₇	—	2	2	no	—
52	C₄×C₇	—	5	1	no	—
53	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
54	C₂×D₅	—	2	2	no	—
55	C₂×D₅	C₇×D₄	1	1	yes	characteristic
56	C₄×C₅	—	2	2	no	—
57	C₄×C₅	C₂³×C₇	1	1	yes	characteristic
58	Dic₅	C₇×D₄	1	1	yes	characteristic
59	#41	—	5	2	no	—
60	Q₁₆	C₇×D₅	1	1	yes	characteristic, complemented
61	SD₁₆	—	5	2	no	—
62	D₈	—	5	1	no	—
63	C₂×C₈	—	5	1	no	—
64	C₂×C₇	—	10	1	no	—
65	C₂×C₇	—	20	2	no	—
66	C₂×C₇	D₄×D₅	1	1	yes	characteristic
67	C₂×C₅	C₂×C₇×D₄	1	1	yes	characteristic
68	D₅	—	2	1	no	—
69	D₅	—	4	2	no	—
70	Q₈	C₂×C₇×D₅	1	2	yes	—
71	D₄	—	5	2	no	—
72	D₄	—	10	2	no	—
73	C₂×C₄	—	5	1	no	—
74	C₂×C₄	—	10	2	no	—
75	C₈	—	5	1	no	—
76	C₈	C₂×C₇×D₅	1	1	yes	characteristic
77	C₇	#4065	1	1	yes	characteristic, direct product factor, Sylow
78	C₅	C₇×#135	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	5	1	no	—
80	C₂²	—	10	2	no	—
81	C₄	—	2	2	no	—
82	C₄	—	5	1	no	—
83	C₄	C₂²×C₇×D₅	1	1	yes	characteristic
84	C₂	—	10	1	no	—
85	C₂	—	20	2	no	—
86	C₂	C₇×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	D₄×D₅
commutator, cosocle	C₄×C₅	C₂³×C₇
Frattini	C₄	C₂²×C₇×D₅
Fitting	C₅×C₇×Q₁₆	C₂
socle	C₂×C₅×C₇	C₂×D₄
hypercenter	C₇×Q₁₆	D₅
hypocenter	C₅	C₇×#135
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₇×#713	C₂	semidirect
2. C₅×C₇×Q₁₆	C₂	semidirect
3. C₇×#688	C₂	semidirect
4. C₇×D₄₀	C₂	semidirect
5. C₇×C₈×D₅	C₂	semidirect
14. #4065	C₇	direct (N × H)
21. C₇×Q₁₆	D₅	semidirect
25. #713	C₂×C₇	semidirect
26. C₅×Q₁₆	C₂×C₇	semidirect
27. #688	C₂×C₇	semidirect
28. D₄₀	C₂×C₇	semidirect
29. C₈×D₅	C₂×C₇	semidirect
47. C₅×C₇	#135	semidirect
60. Q₁₆	C₇×D₅	semidirect
77. C₇	#4065	direct (N × H)
78. C₅	C₇×#135	semidirect

Maximal subgroups

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

subgroup	index
C₇×#713	2
C₅×C₇×Q₁₆	2
C₇×#688	2
C₇×D₄₀	2
C₇×C₈×D₅	2
C₇×#135	5
#4065	7

Subgroups — C7×#4065 #11765163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4065 #11765163