Subgroups — C₇×#4016 #11765114

Order 1120.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅×C₇	C₂	1	1	yes	maximal, characteristic
2	C₇×C₄⋊Dic₅	C₂	1	2	yes	maximal
3	C₇×#683	C₂	1	4	yes	maximal
4	C₂×C₄×C₅×C₇	C₂²	1	2	yes	—
5	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₇×Dic₅	C₂²	1	4	yes	—
7	C₇×#125	—	5	1	no	maximal, Hall
8	#4016	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
10	C₄×C₅×C₇	—	2	2	no	—
11	C₄×C₅×C₇	Q₈	1	2	yes	—
12	C₇×Dic₅	—	2	4	no	—
13	C₇×C₄⋊C₄	—	5	4	no	—
14	C₇×C₄⋊C₄	—	5	2	no	—
15	C₄²×C₇	D₅	1	1	yes	characteristic
16	C₄²×C₅	C₂×C₇	1	1	yes	characteristic
17	C₄⋊Dic₅	C₂×C₇	1	2	yes	—
18	#683	C₂×C₇	1	4	yes	—
19	C₂×C₅×C₇	C₂×Q₈	1	1	yes	characteristic
20	C₂×C₅×C₇	#41	1	2	yes	—
21	C₂×C₄×C₇	—	5	4	no	—
22	C₂×C₄×C₇	C₂×D₅	1	2	yes	—
23	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
24	C₂×C₄×C₅	C₂²×C₇	1	2	yes	—
25	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
26	C₂×Dic₅	C₂²×C₇	1	4	yes	—
27	C₅×C₇	#125	1	1	yes	characteristic, complemented, Hall
28	#125	—	5	1	no	Sylow
29	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
30	C₄×C₇	—	2	2	no	—
31	C₄×C₇	—	10	4	no	—
32	C₄×C₇	Dic₁₀	1	2	yes	—
33	C₂²×C₅	C₂³×C₇	1	1	yes	characteristic
34	C₄×C₅	—	2	2	no	—
35	C₄×C₅	C₇×Q₈	1	2	yes	—
36	Dic₅	—	2	4	no	—
37	C₄⋊C₄	—	5	4	no	—
38	C₄⋊C₄	—	5	2	no	—
39	C₄²	C₇×D₅	1	1	yes	characteristic
40	C₂×C₇	C₂×Dic₁₀	1	1	yes	characteristic
41	C₂×C₇	#709	1	2	yes	—
42	C₂×C₅	C₂×C₇×Q₈	1	1	yes	characteristic
43	C₂×C₅	C₇×#41	1	2	yes	—
44	C₂×C₄	—	5	4	no	—
45	C₂×C₄	C₂×C₇×D₅	1	2	yes	—
46	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
47	C₇	#4016	1	1	yes	characteristic, direct product factor, Sylow
48	C₅	C₇×#125	1	1	yes	characteristic, complemented, Sylow
49	C₂²	C₂²×C₇×D₅	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	—	10	4	no	—
52	C₄	C₇×Dic₁₀	1	2	yes	—
53	C₂	C₂×C₇×Dic₁₀	1	1	yes	characteristic
54	C₂	C₇×#709	1	2	yes	—

	subgroup	G/H
center	C₂²×C₇	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂³×C₇
Frattini	C₂²	C₂²×C₇×D₅
Fitting	C₄²×C₅×C₇	C₂
socle, prime radical	C₂²×C₅×C₇	C₂³
hypercenter	C₄²×C₇	D₅
hypocenter	C₅	C₇×#125
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇

Complements

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

N	H ≅ G/N	type
8. #4016	C₇	direct (N × H)
27. C₅×C₇	#125	semidirect
47. C₇	#4016	direct (N × H)
48. C₅	C₇×#125	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₅×C₇	2
C₇×C₄⋊Dic₅	2
C₇×#683	2
C₇×#125	5
#4016	7

Subgroups — C7×#4016 #11765114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4016 #11765114