Subgroups — C₂×C₅×D₅₆ #11765061

Order 1120.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₂₈	C₂	1	2	yes	maximal, complemented
2	C₂×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₅₆	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₅×D₇	—	2	2	no	—
5	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₅×D₂₈	—	2	2	no	—
7	C₅×D₂₈	C₂²	1	4	yes	complemented
8	C₅×C₇×C₈	C₂²	1	2	yes	complemented
9	C₂×D₅₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₅×D₈	—	7	1	no	maximal, Hall
11	C₂²×C₅×C₇	D₄	1	1	yes	characteristic
12	C₂×C₅×D₇	—	2	4	no	—
13	C₂×C₅×D₇	—	4	4	no	—
14	C₄×C₅×C₇	D₄	1	1	yes	characteristic
15	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
16	C₂×D₂₈	C₂×C₅	1	2	yes	complemented
17	C₂×C₇×C₈	C₂×C₅	1	1	yes	characteristic, complemented
18	D₅₆	C₂×C₅	1	4	yes	direct product factor
19	C₂×C₅×D₄	—	7	2	no	—
20	C₅×D₈	—	7	4	no	—
21	C₂×C₅×C₈	D₇	1	1	yes	characteristic, complemented
22	C₂×C₅×C₇	D₈	1	2	yes	complemented
23	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
24	C₅×D₇	—	4	4	no	—
25	C₂²×D₇	—	2	2	no	—
26	C₂×C₄×C₇	C₂²×C₅	1	1	yes	characteristic
27	D₂₈	—	2	2	no	—
28	D₂₈	C₂²×C₅	1	4	yes	complemented
29	C₇×C₈	C₂²×C₅	1	2	yes	complemented
30	C₂³×C₅	—	14	2	no	—
31	C₅×D₄	—	7	4	no	—
32	C₅×D₄	—	14	2	no	—
33	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
34	C₅×C₈	C₂×D₇	1	2	yes	complemented
35	C₅×C₇	C₂×D₈	1	1	yes	characteristic, complemented, Hall
36	C₂×D₈	—	7	1	no	Sylow
37	C₂²×C₇	C₅×D₄	1	1	yes	characteristic
38	C₂×D₇	—	2	4	no	—
39	C₂×D₇	—	4	4	no	—
40	C₄×C₇	C₅×D₄	1	1	yes	characteristic
41	C₄×C₇	C₂³×C₅	1	1	yes	characteristic
42	C₂²×C₅	—	14	4	no	—
43	C₂²×C₅	—	28	4	no	—
44	C₂²×C₅	D₂₈	1	1	yes	characteristic
45	C₄×C₅	D₂₈	1	1	yes	characteristic
46	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
47	C₂×D₄	—	7	2	no	—
48	D₈	—	7	4	no	—
49	C₂×C₈	C₅×D₇	1	1	yes	characteristic, complemented
50	C₂×C₇	C₅×D₈	1	2	yes	complemented
51	C₂×C₇	C₂×C₅×D₄	1	1	yes	characteristic
52	D₇	—	4	4	no	—
53	C₂×C₅	—	28	4	no	—
54	C₂×C₅	D₅₆	1	2	yes	direct product factor
55	C₂×C₅	C₂×D₂₈	1	1	yes	characteristic
56	C₂³	—	14	2	no	—
57	D₄	—	7	4	no	—
58	D₄	—	14	2	no	—
59	C₂×C₄	C₂×C₅×D₇	1	1	yes	characteristic
60	C₈	C₂×C₅×D₇	1	2	yes	complemented
61	C₇	C₂×C₅×D₈	1	1	yes	characteristic, complemented, Sylow
62	C₅	C₂×D₅₆	1	1	yes	characteristic, direct product factor, Sylow
63	C₂²	—	14	4	no	—
64	C₂²	—	28	4	no	—
65	C₂²	C₅×D₂₈	1	1	yes	characteristic
66	C₄	C₅×D₂₈	1	1	yes	characteristic
67	C₄	C₂²×C₅×D₇	1	1	yes	characteristic
68	C₂	—	28	4	no	—
69	C₂	C₅×D₅₆	1	2	yes	direct product factor
70	C₂	C₂×C₅×D₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	D₂₈
commutator, cosocle	C₄×C₇	C₂³×C₅
Frattini	C₄	C₂²×C₅×D₇
Fitting	C₂×C₅×C₇×C₈	C₂
socle	C₂²×C₅×C₇	D₄
hypercenter	C₂×C₅×C₈	D₇
hypocenter	C₇	C₂×C₅×D₈
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×D₂₈	C₂	semidirect
2. C₂×C₅×C₇×C₈	C₂	semidirect
3. C₅×D₅₆	C₂	direct (N × H)
7. C₅×D₂₈	C₂²	semidirect
8. C₅×C₇×C₈	C₂²	semidirect
9. C₂×D₅₆	C₅	direct (N × H)
16. C₂×D₂₈	C₂×C₅	semidirect
17. C₂×C₇×C₈	C₂×C₅	semidirect
18. D₅₆	C₂×C₅	direct (N × H)
21. C₂×C₅×C₈	D₇	semidirect
22. C₂×C₅×C₇	D₈	semidirect
28. D₂₈	C₂²×C₅	semidirect
29. C₇×C₈	C₂²×C₅	semidirect
34. C₅×C₈	C₂×D₇	semidirect
35. C₅×C₇	C₂×D₈	semidirect
49. C₂×C₈	C₅×D₇	semidirect
50. C₂×C₇	C₅×D₈	semidirect
54. C₂×C₅	D₅₆	direct (N × H)
60. C₈	C₂×C₅×D₇	semidirect
61. C₇	C₂×C₅×D₈	semidirect
62. C₅	C₂×D₅₆	direct (N × H)
69. C₂	C₅×D₅₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×D₂₈	2
C₂×C₅×C₇×C₈	2
C₅×D₅₆	2
C₂×D₅₆	5
C₂×C₅×D₈	7

Subgroups — C2×C5×D56 #11765061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×D₅₆ #11765061