Subgroups — C₇×D₁₂₀ #13132340

Order 1680.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×D₆₀	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₄₀	—	3	1	no	maximal, Hall
4	C₂×C₇×D₁₅	—	2	2	no	—
5	C₃×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₇×D₂₄	—	5	1	no	maximal, Hall
7	C₇×D₂₀	—	3	2	no	—
8	C₅×C₇×C₈	D₃	1	1	yes	characteristic, complemented
9	D₁₂₀	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×C₇	D₄	1	1	yes	characteristic
11	C₇×D₁₅	—	4	2	no	—
12	C₇×D₁₂	—	5	2	no	—
13	C₃×C₇×C₈	D₅	1	1	yes	characteristic, complemented
14	C₂×C₇×D₅	—	6	2	no	—
15	C₄×C₅×C₇	C₂×D₃	1	1	yes	characteristic
16	D₆₀	C₂×C₇	1	2	yes	complemented
17	C₃×C₅×C₈	C₂×C₇	1	1	yes	characteristic, complemented
18	C₇×D₈	—	15	1	no	Hall
19	C₃×C₅×C₇	D₈	1	1	yes	characteristic, complemented, Hall
20	C₂×D₃×C₇	—	10	2	no	—
21	C₃×C₄×C₇	C₂×D₅	1	1	yes	characteristic
22	D₄₀	—	3	1	no	Hall
23	C₂×C₅×C₇	D₁₂	1	1	yes	characteristic
24	C₇×D₅	—	12	2	no	—
25	C₂×D₁₅	—	2	2	no	—
26	C₃×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
27	C₇×D₄	—	15	2	no	—
28	C₇×C₈	D₁₅	1	1	yes	characteristic, complemented
29	D₂₄	—	5	1	no	Hall
30	C₂×C₃×C₇	D₂₀	1	1	yes	characteristic
31	D₃×C₇	—	20	2	no	—
32	D₂₀	—	3	2	no	—
33	C₅×C₈	D₃×C₇	1	1	yes	characteristic, complemented
34	C₅×C₇	D₂₄	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×C₅	C₇×D₄	1	1	yes	characteristic
36	D₁₅	—	4	2	no	—
37	C₂²×C₇	—	30	2	no	—
38	C₄×C₇	C₂×D₁₅	1	1	yes	characteristic
39	D₁₂	—	5	2	no	—
40	C₃×C₈	C₇×D₅	1	1	yes	characteristic, complemented
41	C₃×C₇	D₄₀	1	1	yes	characteristic, complemented, Hall
42	C₂×D₅	—	6	2	no	—
43	C₄×C₅	C₂×D₃×C₇	1	1	yes	characteristic
44	D₈	—	15	1	no	Sylow
45	C₃×C₅	C₇×D₈	1	1	yes	characteristic, complemented, Hall
46	C₂×C₇	—	60	2	no	—
47	C₂×C₇	D₆₀	1	1	yes	characteristic
48	C₂×D₃	—	10	2	no	—
49	C₃×C₄	C₂×C₇×D₅	1	1	yes	characteristic
50	C₂×C₅	C₇×D₁₂	1	1	yes	characteristic
51	D₅	—	12	2	no	—
52	D₄	—	15	2	no	—
53	C₈	C₇×D₁₅	1	1	yes	characteristic, complemented
54	C₇	D₁₂₀	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×C₃	C₇×D₂₀	1	1	yes	characteristic
56	D₃	—	20	2	no	—
57	C₅	C₇×D₂₄	1	1	yes	characteristic, complemented, Sylow
58	C₂²	—	30	2	no	—
59	C₄	C₂×C₇×D₁₅	1	1	yes	characteristic
60	C₃	C₇×D₄₀	1	1	yes	characteristic, complemented, Sylow
61	C₂	—	60	2	no	—
62	C₂	C₇×D₆₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	D₆₀
commutator, cosocle	C₃×C₄×C₅	C₂²×C₇
Frattini	C₄	C₂×C₇×D₁₅
Fitting	C₃×C₅×C₇×C₈	C₂
socle	C₂×C₃×C₅×C₇	D₄
hypercenter	C₇×C₈	D₁₅
hypocenter	C₃×C₅	C₇×D₈
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈

Complements

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

N	H ≅ G/N	type
1. C₇×D₆₀	C₂	semidirect
2. C₃×C₅×C₇×C₈	C₂	semidirect
8. C₅×C₇×C₈	D₃	semidirect
9. D₁₂₀	C₇	direct (N × H)
13. C₃×C₇×C₈	D₅	semidirect
16. D₆₀	C₂×C₇	semidirect
17. C₃×C₅×C₈	C₂×C₇	semidirect
19. C₃×C₅×C₇	D₈	semidirect
28. C₇×C₈	D₁₅	semidirect
33. C₅×C₈	D₃×C₇	semidirect
34. C₅×C₇	D₂₄	semidirect
40. C₃×C₈	C₇×D₅	semidirect
41. C₃×C₇	D₄₀	semidirect
45. C₃×C₅	C₇×D₈	semidirect
53. C₈	C₇×D₁₅	semidirect
54. C₇	D₁₂₀	direct (N × H)
57. C₅	C₇×D₂₄	semidirect
60. C₃	C₇×D₄₀	semidirect

Maximal subgroups

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

subgroup	index
C₇×D₆₀	2
C₃×C₅×C₇×C₈	2
C₇×D₄₀	3
C₇×D₂₄	5
D₁₂₀	7

Subgroups — C7×D120 #13132340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×D₁₂₀ #13132340