Subgroups — C₇×#6401 #13096104

Order 1568.

85 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
2	C₇×Dic₂₈	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×Dic₁₄	—	2	1	no	—
6	C₇²×C₈	C₂²	1	1	yes	characteristic
7	C₇×SD₃₂	—	7	1	no	maximal
8	#6401	C₇	1	1	yes	maximal, characteristic, direct product factor
9	C₂²×C₇²	—	4	1	no	—
10	C₄×C₇²	D₄	1	1	yes	characteristic
11	C₇×Dic₇	—	4	1	no	—
12	C₇×Q₁₆	—	7	1	no	—
13	C₇×D₈	—	2	3	no	—
14	C₇×D₈	D₇	1	1	yes	characteristic
15	C₇×D₈	C₂×C₇	1	1	yes	characteristic
16	Dic₂₈	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×Q₈	—	14	1	no	—
22	C₇×D₄	—	2	6	no	—
23	C₇×D₄	—	2	1	no	—
24	C₇×D₄	—	2	1	no	—
25	Dic₁₄	—	2	1	no	—
26	C₇×C₈	—	2	3	no	—
27	C₇×C₈	C₂×D₇	1	1	yes	characteristic
28	C₇×C₈	C₂²×C₇	1	1	yes	characteristic
29	C₇²	SD₃₂	1	1	yes	characteristic, complemented, Sylow
30	SD₃₂	—	7	1	no	Sylow
31	C₂²×C₇	—	4	6	no	—
32	C₂²×C₇	—	4	1	no	—
33	C₂²×C₇	—	4	1	no	—
34	C₄×C₇	—	2	3	no	—
35	C₄×C₇	—	28	1	no	—
36	C₄×C₇	C₂²⋊D₇	1	1	yes	characteristic
37	C₄×C₇	C₇×D₄	1	1	yes	characteristic
38	Dic₇	—	4	1	no	—
39	Q₁₆	—	7	1	no	—
40	D₈	C₇×D₇	1	1	yes	characteristic
41	C₁₆	—	7	1	no	—
42	C₂×C₇	—	2	3	no	—
43	C₂×C₇	—	8	6	no	—
44	C₂×C₇	—	8	1	no	—
45	C₂×C₇	—	8	1	no	—
46	C₂×C₇	#1141	1	1	yes	characteristic
47	C₂×C₇	C₇×D₈	1	1	yes	characteristic
48	Q₈	—	14	1	no	—
49	D₄	—	2	1	no	—
50	C₈	C₂×C₇×D₇	1	1	yes	characteristic
51	C₇	—	2	3	no	—
52	C₇	#6401	1	1	yes	characteristic, direct product factor
53	C₇	C₇×SD₃₂	1	1	yes	characteristic, complemented
54	C₂²	—	4	1	no	—
55	C₄	—	28	1	no	—
56	C₄	C₇×C₂²⋊D₇	1	1	yes	characteristic
57	C₂	—	8	1	no	—
58	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, prime radical	C₇²×D₈	C₂
socle	C₂×C₇²	D₈
hypercenter	C₇×D₈	D₇
hypocenter	C₇	C₇×SD₃₂
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
2. C₇×Dic₂₈	C₂	semidirect
3. C₇×C₇⋊C₁₆	C₂	semidirect
8. #6401	C₇	direct (N × H)
16. Dic₂₈	C₂×C₇	semidirect
18. C₇⋊C₁₆	C₂×C₇	semidirect
29. C₇²	SD₃₂	semidirect
52. C₇	#6401	direct (N × H)
53. C₇	C₇×SD₃₂	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×Dic₂₈	2
C₇×C₇⋊C₁₆	2
C₇×SD₃₂	7
#6401	7

Subgroups — C7×#6401 #13096104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#6401 #13096104