Subgroups — C₇×#6400 #13096103

Order 1568.

89 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×D₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇²×D₄	—	2	1	no	—
5	C₇×D₂₈	—	2	1	no	—
6	C₇²×C₈	C₂²	1	1	yes	characteristic
7	C₇×D₁₆	—	7	1	no	maximal
8	#6400	C₇	1	1	yes	maximal, characteristic, direct product factor
9	C₂²×C₇²	—	4	1	no	—
10	C₂×C₇×D₇	—	4	1	no	—
11	C₄×C₇²	D₄	1	1	yes	characteristic
12	C₇×D₈	—	2	3	no	—
13	C₇×D₈	—	7	1	no	—
14	C₇×D₈	D₇	1	1	yes	characteristic, complemented
15	C₇×D₈	C₂×C₇	1	1	yes	characteristic, complemented
16	D₅₆	C₂×C₇	1	1	yes	characteristic, complemented
17	C₇×C₁₆	—	7	1	no	—
18	C₇⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
19	C₂×C₇²	—	8	1	no	—
20	C₂×C₇²	D₈	1	1	yes	characteristic
21	C₇×D₇	—	8	1	no	—
22	C₇×D₄	—	2	6	no	—
23	C₇×D₄	—	2	1	no	—
24	C₇×D₄	—	2	1	no	—
25	C₇×D₄	—	14	1	no	—
26	D₂₈	—	2	1	no	—
27	C₇×C₈	—	2	3	no	—
28	C₇×C₈	C₂×D₇	1	1	yes	characteristic
29	C₇×C₈	C₂²×C₇	1	1	yes	characteristic
30	C₇²	D₁₆	1	1	yes	characteristic, complemented, Sylow
31	D₁₆	—	7	1	no	Sylow
32	C₂²×C₇	—	4	6	no	—
33	C₂²×C₇	—	4	1	no	—
34	C₂²×C₇	—	4	1	no	—
35	C₂²×C₇	—	28	1	no	—
36	C₂×D₇	—	4	1	no	—
37	C₄×C₇	—	2	3	no	—
38	C₄×C₇	C₂²⋊D₇	1	1	yes	characteristic
39	C₄×C₇	C₇×D₄	1	1	yes	characteristic
40	D₈	—	7	1	no	—
41	D₈	C₇×D₇	1	1	yes	characteristic, complemented
42	C₁₆	—	7	1	no	—
43	C₂×C₇	—	2	3	no	—
44	C₂×C₇	—	8	6	no	—
45	C₂×C₇	—	8	1	no	—
46	C₂×C₇	—	8	1	no	—
47	C₂×C₇	—	56	1	no	—
48	C₂×C₇	#1141	1	1	yes	characteristic
49	C₂×C₇	C₇×D₈	1	1	yes	characteristic
50	D₇	—	8	1	no	—
51	D₄	—	2	1	no	—
52	D₄	—	14	1	no	—
53	C₈	C₂×C₇×D₇	1	1	yes	characteristic
54	C₇	—	2	3	no	—
55	C₇	#6400	1	1	yes	characteristic, direct product factor
56	C₇	C₇×D₁₆	1	1	yes	characteristic, complemented
57	C₂²	—	4	1	no	—
58	C₂²	—	28	1	no	—
59	C₄	C₇×C₂²⋊D₇	1	1	yes	characteristic
60	C₂	—	8	1	no	—
61	C₂	—	56	1	no	—
62	C₂	C₇×#1141	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	#1141
commutator, cosocle	C₇×C₈	C₂²×C₇
Frattini	C₈	C₂×C₇×D₇
Fitting	C₇²×D₈	C₂
socle	C₂×C₇²	D₈
hypercenter	C₇×D₈	D₇
hypocenter	C₇	C₇×D₁₆
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈ ◁ C₇×D₈

Complements

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

N	H ≅ G/N	type
1. C₇²×D₈	C₂	semidirect
2. C₇×D₅₆	C₂	semidirect
3. C₇×C₇⋊C₁₆	C₂	semidirect
8. #6400	C₇	direct (N × H)
14. C₇×D₈	D₇	semidirect
15. C₇×D₈	C₂×C₇	semidirect
16. D₅₆	C₂×C₇	semidirect
18. C₇⋊C₁₆	C₂×C₇	semidirect
30. C₇²	D₁₆	semidirect
41. D₈	C₇×D₇	semidirect
55. C₇	#6400	direct (N × H)
56. C₇	C₇×D₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₇²×D₈	2
C₇×D₅₆	2
C₇×C₇⋊C₁₆	2
C₇×D₁₆	7
#6400	7

Subgroups — C7×#6400 #13096103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#6400 #13096103