Subgroups — C₇×#4099 #11765197

Order 1120.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×C₂²⋊D₅	C₂	1	3	yes	maximal, complemented
3	C₇×C₂²⋊Dic₅	C₂	1	3	yes	maximal, complemented
4	C₂³×C₅×C₇	—	2	6	no	—
5	C₂³×C₅×C₇	C₂²	1	3	yes	—
6	C₂²×C₇×D₅	C₂²	1	1	yes	characteristic, complemented
7	C₇×C₂²⋊D₅	—	2	6	no	—
8	C₂×C₇×Dic₅	C₂²	1	3	yes	complemented
9	C₇×#120	—	5	1	no	maximal, Hall
10	#4099	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₅×C₇	—	2	8	no	—
12	C₂²×C₅×C₇	—	2	6	no	—
13	C₂²×C₅×C₇	D₄	1	6	yes	complemented
14	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
15	C₂×C₇×D₅	—	2	3	no	—
16	C₇×Dic₅	—	2	3	no	—
17	C₂⁴×C₇	D₅	1	1	yes	characteristic, complemented
18	C₂×C₇×D₄	—	5	3	no	—
19	C₇×C₂²⋊C₄	—	5	3	no	—
20	C₂⁴×C₅	C₂×C₇	1	1	yes	characteristic, complemented
21	C₂×C₂²⋊D₅	C₂×C₇	1	3	yes	complemented
22	C₂²⋊Dic₅	C₂×C₇	1	3	yes	complemented
23	C₂×C₅×C₇	—	2	6	no	—
24	C₂×C₅×C₇	C₂×D₄	1	3	yes	—
25	C₇×D₅	—	4	1	no	—
26	C₂³×C₇	—	2	6	no	—
27	C₂³×C₇	—	5	1	no	—
28	C₂³×C₇	C₂×D₅	1	3	yes	—
29	C₇×D₄	—	10	6	no	—
30	C₂×C₄×C₇	—	5	3	no	—
31	C₂³×C₅	—	2	6	no	—
32	C₂³×C₅	C₂²×C₇	1	3	yes	—
33	C₂²×D₅	C₂²×C₇	1	1	yes	characteristic, complemented
34	C₂²⋊D₅	—	2	6	no	—
35	C₂×Dic₅	C₂²×C₇	1	3	yes	complemented
36	C₅×C₇	#120	1	1	yes	characteristic, complemented, Hall
37	#120	—	5	1	no	Sylow
38	C₂²×C₇	—	2	8	no	—
39	C₂²×C₇	—	2	6	no	—
40	C₂²×C₇	—	10	3	no	—
41	C₂²×C₇	C₂²⋊D₅	1	6	yes	complemented
42	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
43	C₄×C₇	—	10	3	no	—
44	C₂²×C₅	—	2	8	no	—
45	C₂²×C₅	—	2	6	no	—
46	C₂²×C₅	C₇×D₄	1	6	yes	complemented
47	C₂²×C₅	C₂³×C₇	1	1	yes	characteristic
48	C₂×D₅	—	2	3	no	—
49	Dic₅	—	2	3	no	—
50	C₂⁴	C₇×D₅	1	1	yes	characteristic, complemented
51	C₂×D₄	—	5	3	no	—
52	C₂²⋊C₄	—	5	3	no	—
53	C₂×C₇	—	2	6	no	—
54	C₂×C₇	—	20	1	no	—
55	C₂×C₇	C₂×C₂²⋊D₅	1	3	yes	—
56	C₂×C₅	—	2	6	no	—
57	C₂×C₅	C₂×C₇×D₄	1	3	yes	—
58	D₅	—	4	1	no	—
59	C₂³	—	2	6	no	—
60	C₂³	—	5	1	no	—
61	C₂³	C₂×C₇×D₅	1	3	yes	—
62	D₄	—	10	6	no	—
63	C₂×C₄	—	5	3	no	—
64	C₇	#4099	1	1	yes	characteristic, direct product factor, Sylow
65	C₅	C₇×#120	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	2	8	no	—
67	C₂²	—	2	6	no	—
68	C₂²	—	10	3	no	—
69	C₂²	C₇×C₂²⋊D₅	1	6	yes	complemented
70	C₂²	C₂²×C₇×D₅	1	1	yes	characteristic
71	C₄	—	10	3	no	—
72	C₂	—	2	6	no	—
73	C₂	—	20	1	no	—
74	C₂	C₂×C₇×C₂²⋊D₅	1	3	yes	—

	subgroup	G/H
center	C₂²×C₇	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂³×C₇
Frattini	C₂²	C₂²×C₇×D₅
Fitting	C₂⁴×C₅×C₇	C₂
socle	C₂²×C₅×C₇	C₂³
hypercenter	C₂⁴×C₇	D₅
hypocenter	C₅	C₇×#120
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₇ ◁ C₂⁴×C₇

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₅×C₇	C₂	semidirect
2. C₂×C₇×C₂²⋊D₅	C₂	semidirect
3. C₇×C₂²⋊Dic₅	C₂	semidirect
6. C₂²×C₇×D₅	C₂²	semidirect
8. C₂×C₇×Dic₅	C₂²	semidirect
10. #4099	C₇	direct (N × H)
13. C₂²×C₅×C₇	D₄	semidirect
17. C₂⁴×C₇	D₅	semidirect
20. C₂⁴×C₅	C₂×C₇	semidirect
21. C₂×C₂²⋊D₅	C₂×C₇	semidirect
22. C₂²⋊Dic₅	C₂×C₇	semidirect
33. C₂²×D₅	C₂²×C₇	semidirect
35. C₂×Dic₅	C₂²×C₇	semidirect
36. C₅×C₇	#120	semidirect
41. C₂²×C₇	C₂²⋊D₅	semidirect
46. C₂²×C₅	C₇×D₄	semidirect
50. C₂⁴	C₇×D₅	semidirect
64. C₇	#4099	direct (N × H)
65. C₅	C₇×#120	semidirect
69. C₂²	C₇×C₂²⋊D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₅×C₇	2
C₂×C₇×C₂²⋊D₅	2
C₇×C₂²⋊Dic₅	2
C₇×#120	5
#4099	7

Subgroups — C7×#4099 #11765197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4099 #11765197