Subgroups — C₇×Dic₆₀ #13132341

Order 1680.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Dic₃₀	C₂	1	2	yes	maximal
2	C₃×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×Dic₂₀	—	3	1	no	maximal, Hall
4	C₃×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
5	C₇×Dic₁₅	—	2	2	no	—
6	C₇×Dic₁₂	—	5	1	no	maximal, Hall
7	C₇×Dic₁₀	—	3	2	no	—
8	C₅×C₇×C₈	D₃	1	1	yes	characteristic
9	Dic₆₀	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×C₇	D₄	1	1	yes	characteristic
11	C₇×Dic₆	—	5	2	no	—
12	C₃×C₇×C₈	D₅	1	1	yes	characteristic
13	C₄×C₅×C₇	C₂×D₃	1	1	yes	characteristic
14	C₇×Dic₅	—	6	2	no	—
15	Dic₃₀	C₂×C₇	1	2	yes	—
16	C₃×C₅×C₈	C₂×C₇	1	1	yes	characteristic
17	C₇×Q₁₆	—	15	1	no	Hall
18	C₃×C₅×C₇	Q₁₆	1	1	yes	characteristic, complemented, Hall
19	C₃×C₄×C₇	C₂×D₅	1	1	yes	characteristic
20	C₇×Dic₃	—	10	2	no	—
21	Dic₂₀	—	3	1	no	Hall
22	C₂×C₅×C₇	D₁₂	1	1	yes	characteristic
23	C₃×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
24	Dic₁₅	—	2	2	no	—
25	C₇×Q₈	—	15	2	no	—
26	C₇×C₈	D₁₅	1	1	yes	characteristic
27	Dic₁₂	—	5	1	no	Hall
28	C₂×C₃×C₇	D₂₀	1	1	yes	characteristic
29	Dic₁₀	—	3	2	no	—
30	C₅×C₈	D₃×C₇	1	1	yes	characteristic
31	C₅×C₇	Dic₁₂	1	1	yes	characteristic, complemented, Hall
32	C₂×C₃×C₅	C₇×D₄	1	1	yes	characteristic
33	C₄×C₇	—	30	2	no	—
34	C₄×C₇	C₂×D₁₅	1	1	yes	characteristic
35	Dic₆	—	5	2	no	—
36	C₃×C₈	C₇×D₅	1	1	yes	characteristic
37	C₃×C₇	Dic₂₀	1	1	yes	characteristic, complemented, Hall
38	C₄×C₅	C₂×D₃×C₇	1	1	yes	characteristic
39	Dic₅	—	6	2	no	—
40	Q₁₆	—	15	1	no	Sylow
41	C₃×C₅	C₇×Q₁₆	1	1	yes	characteristic, complemented, Hall
42	C₂×C₇	D₆₀	1	1	yes	characteristic
43	C₃×C₄	C₂×C₇×D₅	1	1	yes	characteristic
44	Dic₃	—	10	2	no	—
45	C₂×C₅	C₇×D₁₂	1	1	yes	characteristic
46	Q₈	—	15	2	no	—
47	C₈	C₇×D₁₅	1	1	yes	characteristic
48	C₇	Dic₆₀	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×C₃	C₇×D₂₀	1	1	yes	characteristic
50	C₅	C₇×Dic₁₂	1	1	yes	characteristic, complemented, Sylow
51	C₄	—	30	2	no	—
52	C₄	C₂×C₇×D₁₅	1	1	yes	characteristic
53	C₃	C₇×Dic₂₀	1	1	yes	characteristic, complemented, Sylow
54	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, prime radical	C₂×C₃×C₅×C₇	D₄
hypercenter	C₇×C₈	D₁₅
hypocenter	C₃×C₅	C₇×Q₁₆
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
9. Dic₆₀	C₇	direct (N × H)
18. C₃×C₅×C₇	Q₁₆	semidirect
31. C₅×C₇	Dic₁₂	semidirect
37. C₃×C₇	Dic₂₀	semidirect
41. C₃×C₅	C₇×Q₁₆	semidirect
48. C₇	Dic₆₀	direct (N × H)
50. C₅	C₇×Dic₁₂	semidirect
53. C₃	C₇×Dic₂₀	semidirect

Maximal subgroups

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

subgroup	index
C₇×Dic₃₀	2
C₃×C₅×C₇×C₈	2
C₇×Dic₂₀	3
C₇×Dic₁₂	5
Dic₆₀	7

Subgroups — C7×Dic60 #13132341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×Dic₆₀ #13132341