Subgroups — C₇×#4047 #11765145

Order 1120.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#709	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#681	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#676	C₂	1	2	yes	maximal, complemented
5	C₇×C₈×D₅	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₇×C₂²⋊D₅	C₄	1	2	yes	—
8	C₇×D₂₀	C₄	1	1	yes	characteristic
9	C₄×C₇×D₅	C₂²	1	2	yes	—
10	C₇×Dic₁₀	C₄	1	1	yes	characteristic
11	C₅×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₅×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₇×#168	C₂²	1	2	yes	—
14	C₇×#131	—	5	1	no	maximal, Hall
15	#4047	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₇×D₅	C₂×C₄	1	2	yes	—
18	C₄×C₅×C₇	C₂×C₄	1	1	yes	characteristic
19	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
20	C₇×Dic₅	C₂×C₄	1	2	yes	—
21	C₇×#41	—	5	1	no	—
22	C₇×M₁₆	—	5	1	no	—
23	C₇×M₁₆	—	5	2	no	—
24	C₂×C₇×C₈	—	5	2	no	—
25	C₂×C₇×C₈	D₅	1	1	yes	characteristic, complemented
26	#709	C₂×C₇	1	1	yes	characteristic
27	C₂×C₅×C₈	C₂×C₇	1	1	yes	characteristic, complemented
28	#681	C₂×C₇	1	1	yes	characteristic, complemented
29	#676	C₂×C₇	1	2	yes	complemented
30	C₈×D₅	C₂×C₇	1	2	yes	complemented
31	C₂×C₅×C₇	—	2	1	no	—
32	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₇×D₅	—	2	2	no	—
34	C₇×Q₈	—	5	1	no	—
35	C₇×D₄	—	5	2	no	—
36	C₇×D₄	—	5	1	no	—
37	C₂×C₄×C₇	—	5	2	no	—
38	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
39	C₇×C₈	—	5	2	no	—
40	C₇×C₈	C₂×D₅	1	1	yes	characteristic
41	C₇×C₈	C₂×D₅	1	1	yes	characteristic
42	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
43	C₂²⋊D₅	C₄×C₇	1	2	yes	—
44	D₂₀	C₄×C₇	1	1	yes	characteristic
45	C₄×D₅	C₂²×C₇	1	2	yes	—
46	Dic₁₀	C₄×C₇	1	1	yes	characteristic
47	C₅×C₈	C₂²×C₇	1	1	yes	characteristic
48	C₅×C₈	C₂²×C₇	1	1	yes	characteristic
49	#168	C₂²×C₇	1	2	yes	—
50	C₅×C₇	#131	1	1	yes	characteristic, complemented, Hall
51	#131	—	5	1	no	Sylow
52	C₂²×C₇	—	5	2	no	—
53	C₂²×C₇	C₄×D₅	1	1	yes	characteristic
54	C₄×C₇	—	5	2	no	—
55	C₄×C₇	C₄×D₅	1	1	yes	characteristic
56	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
57	C₂²×C₅	C₂×C₄×C₇	1	1	yes	characteristic
58	C₂×D₅	C₂×C₄×C₇	1	2	yes	—
59	C₄×C₅	C₂×C₄×C₇	1	1	yes	characteristic
60	C₄×C₅	C₂³×C₇	1	1	yes	characteristic
61	Dic₅	C₂×C₄×C₇	1	2	yes	—
62	#41	—	5	1	no	—
63	M₁₆	—	5	1	no	—
64	M₁₆	—	5	2	no	—
65	C₂×C₈	—	5	2	no	—
66	C₂×C₈	C₇×D₅	1	1	yes	characteristic, complemented
67	C₂×C₇	—	2	1	no	—
68	C₂×C₇	—	10	2	no	—
69	C₂×C₇	C₂×C₄×D₅	1	1	yes	characteristic
70	C₂×C₅	—	2	1	no	—
71	C₂×C₅	C₂²×C₄×C₇	1	1	yes	characteristic
72	D₅	—	2	2	no	—
73	Q₈	—	5	1	no	—
74	D₄	—	5	2	no	—
75	D₄	—	5	1	no	—
76	C₂×C₄	—	5	2	no	—
77	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
78	C₈	—	5	2	no	—
79	C₈	C₂×C₇×D₅	1	1	yes	characteristic
80	C₈	C₂×C₇×D₅	1	1	yes	characteristic
81	C₇	#4047	1	1	yes	characteristic, direct product factor, Sylow
82	C₅	C₇×#131	1	1	yes	characteristic, complemented, Sylow
83	C₂²	—	5	2	no	—
84	C₂²	C₄×C₇×D₅	1	1	yes	characteristic
85	C₄	—	5	2	no	—
86	C₄	C₄×C₇×D₅	1	1	yes	characteristic
87	C₄	C₂²×C₇×D₅	1	1	yes	characteristic
88	C₂	—	2	1	no	—
89	C₂	—	10	2	no	—
90	C₂	C₂×C₄×C₇×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₇×C₈	C₂×D₅
commutator	C₂×C₅	C₂²×C₄×C₇
Frattini	C₄	C₂²×C₇×D₅
Fitting	C₂×C₅×C₇×C₈	C₂
socle	C₂×C₅×C₇	C₂²×C₄
prime radical	C₇×#709	C₂
cosocle	C₄×C₅	C₂³×C₇
hypercenter	C₂×C₇×C₈	D₅
hypocenter	C₅	C₇×#131
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₇×C₈ ◁ C₂×C₇×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₇×C₈	C₂	semidirect
3. C₇×#681	C₂	semidirect
4. C₇×#676	C₂	semidirect
5. C₇×C₈×D₅	C₂	semidirect
15. #4047	C₇	direct (N × H)
25. C₂×C₇×C₈	D₅	semidirect
27. C₂×C₅×C₈	C₂×C₇	semidirect
28. #681	C₂×C₇	semidirect
29. #676	C₂×C₇	semidirect
30. C₈×D₅	C₂×C₇	semidirect
50. C₅×C₇	#131	semidirect
66. C₂×C₈	C₇×D₅	semidirect
81. C₇	#4047	direct (N × H)
82. C₅	C₇×#131	semidirect

Maximal subgroups

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

subgroup	index
C₇×#709	2
C₂×C₅×C₇×C₈	2
C₇×#681	2
C₇×#676	2
C₇×C₈×D₅	2
C₇×#131	5
#4047	7

Subgroups — C7×#4047 #11765145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4047 #11765145