Subgroups — C₃×D₆₀ #67172

Order 360.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₁₅	C₂	1	2	yes	maximal, complemented
2	C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	D₆₀	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×D₂₀	—	3	1	no	maximal
5	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
6	C₃×D₁₅	—	2	2	no	—
7	C₃×D₁₂	—	5	1	no	maximal, Hall
8	C₂×D₁₅	C₂×C₃	1	2	yes	complemented
9	C₂×C₃×D₅	—	3	2	no	—
10	C₃×C₄×C₅	—	2	1	no	—
11	C₃×C₄×C₅	D₃	1	1	yes	characteristic, complemented
12	C₃×C₄×C₅	C₂×C₃	1	1	yes	characteristic, complemented
13	C₃²×C₅	D₄	1	1	yes	characteristic, complemented, Hall
14	D₂₀	—	3	1	no	Hall
15	C₂×C₃×D₃	—	5	2	no	—
16	C₃²×C₄	D₅	1	1	yes	characteristic, complemented
17	C₂×C₃×C₅	—	2	1	no	—
18	C₂×C₃×C₅	C₂×D₃	1	1	yes	characteristic
19	C₂×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
20	D₁₅	—	2	2	no	—
21	C₃×D₅	—	6	2	no	—
22	C₃×D₄	—	15	1	no	—
23	D₁₂	—	5	1	no	—
24	C₂×D₅	—	3	2	no	—
25	C₄×C₅	C₃×D₃	1	1	yes	characteristic, complemented
26	C₂×C₃²	C₂×D₅	1	1	yes	characteristic
27	C₃×D₃	—	10	2	no	—
28	C₃×C₅	—	2	1	no	—
29	C₃×C₅	D₁₂	1	1	yes	characteristic, complemented
30	C₃×C₅	C₃×D₄	1	1	yes	characteristic, complemented
31	C₂²×C₃	—	15	2	no	—
32	C₂×D₃	—	5	2	no	—
33	C₃×C₄	—	2	1	no	—
34	C₃×C₄	C₃×D₅	1	1	yes	characteristic, complemented
35	C₃×C₄	D₁₅	1	1	yes	characteristic, complemented
36	C₂×C₅	C₂×C₃×D₃	1	1	yes	characteristic
37	D₅	—	6	2	no	—
38	C₃²	D₂₀	1	1	yes	characteristic, complemented, Sylow
39	D₄	—	15	1	no	Sylow
40	C₂×C₃	—	2	1	no	—
41	C₂×C₃	—	30	2	no	—
42	C₂×C₃	C₂×C₃×D₅	1	1	yes	characteristic
43	C₂×C₃	C₂×D₁₅	1	1	yes	characteristic
44	D₃	—	10	2	no	—
45	C₅	C₃×D₁₂	1	1	yes	characteristic, complemented, Sylow
46	C₂²	—	15	2	no	—
47	C₄	C₃×D₁₅	1	1	yes	characteristic, complemented
48	C₃	—	2	1	no	—
49	C₃	C₃×D₂₀	1	1	yes	characteristic, complemented
50	C₃	D₆₀	1	1	yes	characteristic, direct product factor
51	C₂	—	30	2	no	—
52	C₂	C₂×C₃×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₁₅
commutator, cosocle	C₂×C₃×C₅	C₂²×C₃
Frattini	C₂	C₂×C₃×D₁₅
Fitting	C₃²×C₄×C₅	C₂
socle	C₂×C₃²×C₅	C₂²
hypercenter	C₃×C₄	D₁₅
hypocenter	C₃×C₅	C₃×D₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₁₅	C₂	semidirect
2. C₃²×C₄×C₅	C₂	semidirect
3. D₆₀	C₃	direct (N × H)
8. C₂×D₁₅	C₂×C₃	semidirect
11. C₃×C₄×C₅	D₃	semidirect
12. C₃×C₄×C₅	C₂×C₃	semidirect
13. C₃²×C₅	D₄	semidirect
16. C₃²×C₄	D₅	semidirect
25. C₄×C₅	C₃×D₃	semidirect
29. C₃×C₅	D₁₂	semidirect
30. C₃×C₅	C₃×D₄	semidirect
34. C₃×C₄	C₃×D₅	semidirect
35. C₃×C₄	D₁₅	semidirect
38. C₃²	D₂₀	semidirect
45. C₅	C₃×D₁₂	semidirect
47. C₄	C₃×D₁₅	semidirect
49. C₃	C₃×D₂₀	semidirect
50. C₃	D₆₀	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×C₃×D₁₅	2
C₃²×C₄×C₅	2
D₆₀	3
C₃×D₂₀	3
C₃×D₁₂	5

Subgroups — C3×D60 #67172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₆₀ #67172