Subgroups — C₃×C₅×D₁₂ #67167

Order 360.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂×C₃×C₅	C₂×D₃
commutator, cosocle	C₂×C₃	C₂²×C₃×C₅
Frattini	C₂	C₂×C₃×C₅×D₃
Fitting	C₃²×C₄×C₅	C₂
socle	C₂×C₃²×C₅	C₂²
hypercenter	C₃×C₄×C₅	D₃
hypocenter	C₃	C₃×C₅×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₃×C₄×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×D₃	C₂	semidirect
2. C₃²×C₄×C₅	C₂	semidirect
4. C₅×D₁₂	C₃	direct (N × H)
7. C₃×D₁₂	C₅	direct (N × H)
9. C₂×C₅×D₃	C₂×C₃	semidirect
11. C₃×C₄×C₅	D₃	semidirect
12. C₃×C₄×C₅	C₂×C₃	semidirect
13. C₃²×C₅	D₄	semidirect
15. C₂×C₃×D₃	C₂×C₅	semidirect
16. C₃²×C₄	C₂×C₅	semidirect
23. D₁₂	C₃×C₅	direct (N × H)
25. C₄×C₅	C₃×D₃	semidirect
29. C₃×C₅	D₁₂	direct (N × H)
30. C₃×C₅	C₃×D₄	semidirect
32. C₂×D₃	C₂×C₃×C₅	semidirect
34. C₃×C₄	C₅×D₃	semidirect
35. C₃×C₄	C₂×C₃×C₅	semidirect
38. C₃²	C₅×D₄	semidirect
45. C₅	C₃×D₁₂	direct (N × H)
47. C₄	C₃×C₅×D₃	semidirect
49. C₃	C₅×D₁₂	direct (N × H)
50. C₃	C₃×C₅×D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×D₃	2
C₃²×C₄×C₅	2
C₃×C₅×D₄	3
C₅×D₁₂	3
C₃×D₁₂	5

Subgroups — C3×C5×D12 #67167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×D₁₂ #67167