Subgroups — C₇×#711 #10587979

Order 560.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×Dic₅	C₂	1	2	yes	maximal, complemented
4	C₄×C₇×D₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×C₇	C₂²	1	2	yes	—
7	C₂×C₇×D₅	C₂²	1	1	yes	characteristic
8	C₄×C₅×C₇	C₂²	1	1	yes	characteristic
9	C₇×Dic₅	C₂²	1	2	yes	—
10	C₇×Dic₅	C₂²	1	1	yes	characteristic
11	C₇×#41	—	5	1	no	maximal, Hall
12	#711	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₅×C₇	—	2	2	no	—
14	C₂×C₅×C₇	C₂³	1	1	yes	characteristic
15	C₇×D₅	—	2	1	no	—
16	C₇×Q₈	—	5	1	no	—
17	C₇×D₄	—	5	2	no	—
18	C₇×D₄	D₅	1	1	yes	characteristic, complemented
19	C₂×C₄×C₇	—	5	2	no	—
20	C₂×C₄×C₇	—	5	1	no	—
21	C₅×D₄	C₂×C₇	1	1	yes	characteristic, complemented
22	C₂²⋊D₅	C₂×C₇	1	2	yes	complemented
23	C₂×Dic₅	C₂×C₇	1	2	yes	complemented
24	C₄×D₅	C₂×C₇	1	1	yes	characteristic, complemented
25	Dic₁₀	C₂×C₇	1	1	yes	characteristic, complemented
26	C₅×C₇	#41	1	1	yes	characteristic, complemented, Hall
27	C₂²×C₇	—	5	1	no	—
28	C₂²×C₇	C₂×D₅	1	2	yes	—
29	C₄×C₇	—	5	1	no	—
30	C₄×C₇	—	5	2	no	—
31	C₄×C₇	C₂×D₅	1	1	yes	characteristic
32	C₂²×C₅	C₂²×C₇	1	2	yes	—
33	C₂×D₅	C₂²×C₇	1	1	yes	characteristic
34	C₄×C₅	C₂²×C₇	1	1	yes	characteristic
35	Dic₅	C₂²×C₇	1	2	yes	—
36	Dic₅	C₂²×C₇	1	1	yes	characteristic
37	#41	—	5	1	no	Sylow
38	C₂×C₇	—	2	2	no	—
39	C₂×C₇	—	10	1	no	—
40	C₂×C₇	C₂²×D₅	1	1	yes	characteristic
41	C₂×C₅	—	2	2	no	—
42	C₂×C₅	C₂³×C₇	1	1	yes	characteristic
43	D₅	—	2	1	no	—
44	Q₈	—	5	1	no	—
45	D₄	—	5	2	no	—
46	D₄	C₇×D₅	1	1	yes	characteristic, complemented
47	C₂×C₄	—	5	2	no	—
48	C₂×C₄	—	5	1	no	—
49	C₇	#711	1	1	yes	characteristic, direct product factor, Sylow
50	C₅	C₇×#41	1	1	yes	characteristic, complemented, Sylow
51	C₂²	—	5	1	no	—
52	C₂²	C₂×C₇×D₅	1	2	yes	—
53	C₄	—	5	1	no	—
54	C₄	—	5	2	no	—
55	C₄	C₂×C₇×D₅	1	1	yes	characteristic
56	C₂	—	2	2	no	—
57	C₂	—	10	1	no	—
58	C₂	C₂²×C₇×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	C₂²×D₅
commutator, cosocle	C₂×C₅	C₂³×C₇
Frattini	C₂	C₂²×C₇×D₅
Fitting	C₅×C₇×D₄	C₂
socle	C₂×C₅×C₇	C₂³
hypercenter	C₇×D₄	D₅
hypocenter	C₅	C₇×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₇ ◁ C₇×D₄

Complements

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

N	H ≅ G/N	type
1. C₅×C₇×D₄	C₂	semidirect
2. C₇×C₂²⋊D₅	C₂	semidirect
3. C₂×C₇×Dic₅	C₂	semidirect
4. C₄×C₇×D₅	C₂	semidirect
5. C₇×Dic₁₀	C₂	semidirect
12. #711	C₇	direct (N × H)
18. C₇×D₄	D₅	semidirect
21. C₅×D₄	C₂×C₇	semidirect
22. C₂²⋊D₅	C₂×C₇	semidirect
23. C₂×Dic₅	C₂×C₇	semidirect
24. C₄×D₅	C₂×C₇	semidirect
25. Dic₁₀	C₂×C₇	semidirect
26. C₅×C₇	#41	semidirect
46. D₄	C₇×D₅	semidirect
49. C₇	#711	direct (N × H)
50. C₅	C₇×#41	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₇×D₄	2
C₇×C₂²⋊D₅	2
C₂×C₇×Dic₅	2
C₄×C₇×D₅	2
C₇×Dic₁₀	2
C₇×#41	5
#711	7

Subgroups — C7×#711 #10587979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#711 #10587979