Subgroups — C₇×#6129 #13089887

Order 1456.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×C₂²⋊D₁₃	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×Dic₁₃	C₂	1	2	yes	maximal, complemented
4	C₄×C₇×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×C₁₃	C₂²	1	2	yes	—
7	C₂×C₇×D₁₃	C₂²	1	1	yes	characteristic
8	C₄×C₇×C₁₃	C₂²	1	1	yes	characteristic
9	C₇×Dic₁₃	C₂²	1	2	yes	—
10	C₇×Dic₁₃	C₂²	1	1	yes	characteristic
11	#6129	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₇×C₁₃	—	2	2	no	—
13	C₂×C₇×C₁₃	C₂³	1	1	yes	characteristic
14	C₇×D₁₃	—	2	1	no	—
15	C₇×#41	—	13	1	no	maximal, Hall
16	D₄×C₁₃	C₂×C₇	1	1	yes	characteristic, complemented
17	C₂²⋊D₁₃	C₂×C₇	1	2	yes	complemented
18	C₂×Dic₁₃	C₂×C₇	1	2	yes	complemented
19	C₄×D₁₃	C₂×C₇	1	1	yes	characteristic, complemented
20	Dic₂₆	C₂×C₇	1	1	yes	characteristic, complemented
21	C₇×C₁₃	#41	1	1	yes	characteristic, complemented, Hall
22	C₇×Q₈	—	13	1	no	—
23	C₇×D₄	—	13	2	no	—
24	C₇×D₄	D₁₃	1	1	yes	characteristic, complemented
25	C₂×C₄×C₇	—	13	2	no	—
26	C₂×C₄×C₇	—	13	1	no	—
27	C₂²×C₁₃	C₂²×C₇	1	2	yes	—
28	C₂×D₁₃	C₂²×C₇	1	1	yes	characteristic
29	C₄×C₁₃	C₂²×C₇	1	1	yes	characteristic
30	Dic₁₃	C₂²×C₇	1	2	yes	—
31	Dic₁₃	C₂²×C₇	1	1	yes	characteristic
32	C₂²×C₇	—	13	1	no	—
33	C₂²×C₇	C₂×D₁₃	1	2	yes	—
34	C₄×C₇	—	13	1	no	—
35	C₄×C₇	—	13	2	no	—
36	C₄×C₇	C₂×D₁₃	1	1	yes	characteristic
37	C₂×C₁₃	—	2	2	no	—
38	C₂×C₁₃	C₂³×C₇	1	1	yes	characteristic
39	D₁₃	—	2	1	no	—
40	#41	—	13	1	no	Sylow
41	C₂×C₇	—	2	2	no	—
42	C₂×C₇	—	26	1	no	—
43	C₂×C₇	C₂²×D₁₃	1	1	yes	characteristic
44	C₁₃	C₇×#41	1	1	yes	characteristic, complemented, Sylow
45	Q₈	—	13	1	no	—
46	D₄	—	13	2	no	—
47	D₄	C₇×D₁₃	1	1	yes	characteristic, complemented
48	C₂×C₄	—	13	2	no	—
49	C₂×C₄	—	13	1	no	—
50	C₇	#6129	1	1	yes	characteristic, direct product factor, Sylow
51	C₂²	—	13	1	no	—
52	C₂²	C₂×C₇×D₁₃	1	2	yes	—
53	C₄	—	13	1	no	—
54	C₄	—	13	2	no	—
55	C₄	C₂×C₇×D₁₃	1	1	yes	characteristic
56	C₂	—	2	2	no	—
57	C₂	—	26	1	no	—
58	C₂	C₂²×C₇×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	C₂²×D₁₃
commutator, cosocle	C₂×C₁₃	C₂³×C₇
Frattini	C₂	C₂²×C₇×D₁₃
Fitting	C₇×D₄×C₁₃	C₂
socle	C₂×C₇×C₁₃	C₂³
hypercenter	C₇×D₄	D₁₃
hypocenter	C₁₃	C₇×#41
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₇ ◁ C₇×D₄

Complements

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

N	H ≅ G/N	type
1. C₇×D₄×C₁₃	C₂	semidirect
2. C₇×C₂²⋊D₁₃	C₂	semidirect
3. C₂×C₇×Dic₁₃	C₂	semidirect
4. C₄×C₇×D₁₃	C₂	semidirect
5. C₇×Dic₂₆	C₂	semidirect
11. #6129	C₇	direct (N × H)
16. D₄×C₁₃	C₂×C₇	semidirect
17. C₂²⋊D₁₃	C₂×C₇	semidirect
18. C₂×Dic₁₃	C₂×C₇	semidirect
19. C₄×D₁₃	C₂×C₇	semidirect
20. Dic₂₆	C₂×C₇	semidirect
21. C₇×C₁₃	#41	semidirect
24. C₇×D₄	D₁₃	semidirect
44. C₁₃	C₇×#41	semidirect
47. D₄	C₇×D₁₃	semidirect
50. C₇	#6129	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×D₄×C₁₃	2
C₇×C₂²⋊D₁₃	2
C₂×C₇×Dic₁₃	2
C₄×C₇×D₁₃	2
C₇×Dic₂₆	2
#6129	7
C₇×#41	13

Subgroups — C7×#6129 #13089887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#6129 #13089887