Subgroups — C₇×#4022 #11765120

Order 1120.

120 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, complemented
2	C₇×#685	C₂	1	3	yes	maximal
3	C₇×#683	C₂	1	3	yes	maximal, complemented
4	C₂²×C₇×D₅	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₅×C₇	C₂²	1	3	yes	—
6	C₂×C₇×Dic₅	C₂²	1	3	yes	—
7	C₇×#126	—	5	1	no	maximal, Hall
8	#4022	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
10	C₂×C₇×D₅	—	2	3	no	—
11	C₄×C₅×C₇	—	2	3	no	—
12	C₇×Dic₅	—	2	3	no	—
13	C₇×C₄⋊C₄	—	5	3	no	—
14	C₇×C₂²⋊C₄	—	5	3	no	—
15	C₄²×C₇	D₅	1	1	yes	characteristic, complemented
16	C₄²×C₅	C₂×C₇	1	1	yes	characteristic, complemented
17	#685	C₂×C₇	1	3	yes	—
18	#683	C₂×C₇	1	3	yes	complemented
19	C₂×C₅×C₇	#41	1	3	yes	—
20	C₇×D₅	—	4	1	no	—
21	C₂³×C₇	—	5	1	no	—
22	C₂×C₄×C₇	—	5	3	no	—
23	C₂×C₄×C₇	C₂×D₅	1	3	yes	—
24	C₂²×D₅	C₂²×C₇	1	1	yes	characteristic
25	C₂×C₄×C₅	C₂²×C₇	1	3	yes	—
26	C₂×Dic₅	C₂²×C₇	1	3	yes	—
27	C₅×C₇	#126	1	1	yes	characteristic, complemented, Hall
28	#126	—	5	1	no	Sylow
29	C₂²×C₇	—	10	3	no	—
30	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
31	C₄×C₇	—	2	3	no	—
32	C₄×C₇	—	10	3	no	—
33	C₂²×C₅	C₂³×C₇	1	1	yes	characteristic
34	C₂×D₅	—	2	3	no	—
35	C₄×C₅	—	2	3	no	—
36	Dic₅	—	2	3	no	—
37	C₄⋊C₄	—	5	3	no	—
38	C₂²⋊C₄	—	5	3	no	—
39	C₄²	C₇×D₅	1	1	yes	characteristic, complemented
40	C₂×C₇	—	20	1	no	—
41	C₂×C₇	#709	1	3	yes	—
42	C₂×C₅	C₇×#41	1	3	yes	—
43	D₅	—	4	1	no	—
44	C₂³	—	5	1	no	—
45	C₂×C₄	—	5	3	no	—
46	C₂×C₄	C₂×C₇×D₅	1	3	yes	—
47	C₇	#4022	1	1	yes	characteristic, direct product factor, Sylow
48	C₅	C₇×#126	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	10	3	no	—
50	C₂²	C₂²×C₇×D₅	1	1	yes	characteristic
51	C₄	—	2	3	no	—
52	C₄	—	10	3	no	—
53	C₂	—	20	1	no	—
54	C₂	C₇×#709	1	3	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	C₂²×C₅×C₇	C₂³
prime radical	C₂²×C₇×D₅	C₂²
hypercenter	C₄²×C₇	D₅
hypocenter	C₅	C₇×#126
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
1. C₄²×C₅×C₇	C₂	semidirect
3. C₇×#683	C₂	semidirect
8. #4022	C₇	direct (N × H)
15. C₄²×C₇	D₅	semidirect
16. C₄²×C₅	C₂×C₇	semidirect
18. #683	C₂×C₇	semidirect
27. C₅×C₇	#126	semidirect
39. C₄²	C₇×D₅	semidirect
47. C₇	#4022	direct (N × H)
48. C₅	C₇×#126	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×#685	2
C₇×#683	2
C₇×#126	5
#4022	7

Subgroups — C7×#4022 #11765120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4022 #11765120