Subgroups — C₅×#6516 #11765111

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×C₅×C₂²⋊D₇	C₂	1	3	yes	maximal, complemented
3	C₅×C₂²⋊Dic₇	C₂	1	3	yes	maximal, complemented
4	C₂³×C₅×C₇	—	2	6	no	—
5	C₂³×C₅×C₇	C₂²	1	3	yes	—
6	C₂²×C₅×D₇	C₂²	1	1	yes	characteristic, complemented
7	C₅×C₂²⋊D₇	—	2	6	no	—
8	C₂×C₅×Dic₇	C₂²	1	3	yes	complemented
9	#6516	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₅×#120	—	7	1	no	maximal, 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×C₅×D₇	—	2	3	no	—
16	C₅×Dic₇	—	2	3	no	—
17	C₂⁴×C₇	C₂×C₅	1	1	yes	characteristic, complemented
18	C₂×C₂²⋊D₇	C₂×C₅	1	3	yes	complemented
19	C₂²⋊Dic₇	C₂×C₅	1	3	yes	complemented
20	C₂⁴×C₅	D₇	1	1	yes	characteristic, complemented
21	C₂×C₅×D₄	—	7	3	no	—
22	C₅×C₂²⋊C₄	—	7	3	no	—
23	C₂×C₅×C₇	—	2	6	no	—
24	C₂×C₅×C₇	C₂×D₄	1	3	yes	—
25	C₅×D₇	—	4	1	no	—
26	C₂³×C₇	—	2	6	no	—
27	C₂³×C₇	C₂²×C₅	1	3	yes	—
28	C₂²×D₇	C₂²×C₅	1	1	yes	characteristic, complemented
29	C₂²⋊D₇	—	2	6	no	—
30	C₂×Dic₇	C₂²×C₅	1	3	yes	complemented
31	C₂³×C₅	—	2	6	no	—
32	C₂³×C₅	—	7	1	no	—
33	C₂³×C₅	C₂×D₇	1	3	yes	—
34	C₅×D₄	—	14	6	no	—
35	C₂×C₄×C₅	—	7	3	no	—
36	C₅×C₇	#120	1	1	yes	characteristic, complemented, Hall
37	#120	—	7	1	no	Sylow
38	C₂²×C₇	—	2	8	no	—
39	C₂²×C₇	—	2	6	no	—
40	C₂²×C₇	C₅×D₄	1	6	yes	complemented
41	C₂²×C₇	C₂³×C₅	1	1	yes	characteristic
42	C₂×D₇	—	2	3	no	—
43	Dic₇	—	2	3	no	—
44	C₂²×C₅	—	2	8	no	—
45	C₂²×C₅	—	2	6	no	—
46	C₂²×C₅	—	14	3	no	—
47	C₂²×C₅	C₂²⋊D₇	1	6	yes	complemented
48	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
49	C₄×C₅	—	14	3	no	—
50	C₂⁴	C₅×D₇	1	1	yes	characteristic, complemented
51	C₂×D₄	—	7	3	no	—
52	C₂²⋊C₄	—	7	3	no	—
53	C₂×C₇	—	2	6	no	—
54	C₂×C₇	C₂×C₅×D₄	1	3	yes	—
55	D₇	—	4	1	no	—
56	C₂×C₅	—	2	6	no	—
57	C₂×C₅	—	28	1	no	—
58	C₂×C₅	C₂×C₂²⋊D₇	1	3	yes	—
59	C₂³	—	2	6	no	—
60	C₂³	—	7	1	no	—
61	C₂³	C₂×C₅×D₇	1	3	yes	—
62	D₄	—	14	6	no	—
63	C₂×C₄	—	7	3	no	—
64	C₇	C₅×#120	1	1	yes	characteristic, complemented, Sylow
65	C₅	#6516	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²	—	2	8	no	—
67	C₂²	—	2	6	no	—
68	C₂²	—	14	3	no	—
69	C₂²	C₅×C₂²⋊D₇	1	6	yes	complemented
70	C₂²	C₂²×C₅×D₇	1	1	yes	characteristic
71	C₄	—	14	3	no	—
72	C₂	—	2	6	no	—
73	C₂	—	28	1	no	—
74	C₂	C₂×C₅×C₂²⋊D₇	1	3	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂²×D₇
commutator, cosocle	C₂²×C₇	C₂³×C₅
Frattini	C₂²	C₂²×C₅×D₇
Fitting	C₂⁴×C₅×C₇	C₂
socle	C₂²×C₅×C₇	C₂³
hypercenter	C₂⁴×C₅	D₇
hypocenter	C₇	C₅×#120
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₅ ◁ C₂⁴×C₅

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₅×C₇	C₂	semidirect
2. C₂×C₅×C₂²⋊D₇	C₂	semidirect
3. C₅×C₂²⋊Dic₇	C₂	semidirect
6. C₂²×C₅×D₇	C₂²	semidirect
8. C₂×C₅×Dic₇	C₂²	semidirect
9. #6516	C₅	direct (N × H)
13. C₂²×C₅×C₇	D₄	semidirect
17. C₂⁴×C₇	C₂×C₅	semidirect
18. C₂×C₂²⋊D₇	C₂×C₅	semidirect
19. C₂²⋊Dic₇	C₂×C₅	semidirect
20. C₂⁴×C₅	D₇	semidirect
28. C₂²×D₇	C₂²×C₅	semidirect
30. C₂×Dic₇	C₂²×C₅	semidirect
36. C₅×C₇	#120	semidirect
40. C₂²×C₇	C₅×D₄	semidirect
47. C₂²×C₅	C₂²⋊D₇	semidirect
50. C₂⁴	C₅×D₇	semidirect
64. C₇	C₅×#120	semidirect
65. C₅	#6516	direct (N × H)
69. C₂²	C₅×C₂²⋊D₇	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×C₅×C₂²⋊D₇	2
C₅×C₂²⋊Dic₇	2
#6516	5
C₅×#120	7

Subgroups — C5×#6516 #11765111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#6516 #11765111