Subgroups — C₂×C₅×C₇×D₈ #11765302

Order 1120.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂²×C₅×C₇	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₅×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₇ ◁ C₂×C₄×C₅×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₇×D₄	C₂	semidirect
2. C₅×C₇×D₈	C₂	direct (N × H)
3. C₂×C₅×C₇×C₈	C₂	semidirect
6. C₅×C₇×D₄	C₂²	semidirect
8. C₅×C₇×C₈	C₂²	semidirect
9. C₂×C₇×D₈	C₅	direct (N × H)
10. C₂×C₅×D₈	C₇	direct (N × H)
16. C₂×C₇×D₄	C₂×C₅	semidirect
17. C₇×D₈	C₂×C₅	direct (N × H)
18. C₂×C₇×C₈	C₂×C₅	semidirect
19. C₂×C₅×D₄	C₂×C₇	semidirect
20. C₅×D₈	C₂×C₇	direct (N × H)
21. C₂×C₅×C₈	C₂×C₇	semidirect
23. C₂×C₅×C₇	D₈	direct (N × H)
27. C₇×D₄	C₂²×C₅	semidirect
29. C₇×C₈	C₂²×C₅	semidirect
32. C₅×D₄	C₂²×C₇	semidirect
34. C₅×C₈	C₂²×C₇	semidirect
35. C₅×C₇	C₂×D₈	direct (N × H)
36. C₂×D₈	C₅×C₇	direct (N × H)
47. C₂×D₄	C₂×C₅×C₇	semidirect
48. D₈	C₂×C₅×C₇	direct (N × H)
49. C₂×C₈	C₂×C₅×C₇	semidirect
51. C₂×C₇	C₅×D₈	direct (N × H)
54. C₂×C₅	C₇×D₈	direct (N × H)
58. D₄	C₂²×C₅×C₇	semidirect
60. C₈	C₂²×C₅×C₇	semidirect
61. C₇	C₂×C₅×D₈	direct (N × H)
62. C₅	C₂×C₇×D₈	direct (N × H)
69. C₂	C₅×C₇×D₈	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×C₇×D₄	2
C₅×C₇×D₈	2
C₂×C₅×C₇×C₈	2
C₂×C₇×D₈	5
C₂×C₅×D₈	7

Subgroups — C2×C5×C7×D8 #11765302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×C₇×D₈ #11765302