Subgroups — C₃×C₄×C₅×D₃ #67166

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	1	yes	maximal, characteristic
2	C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₄×C₅	—	3	1	no	maximal
5	C₄×C₅×D₃	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
7	C₃×C₅×D₃	C₄	1	2	yes	direct product factor
8	C₃×C₄×D₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₅	—	3	1	no	—
10	C₂×C₅×D₃	C₂×C₃	1	1	yes	characteristic
11	C₃×C₄×C₅	—	2	1	no	—
12	C₃×C₄×C₅	—	3	1	no	—
13	C₃×C₄×C₅	D₃	1	1	yes	characteristic, direct product factor
14	C₃×C₄×C₅	C₂×C₃	1	1	yes	characteristic, complemented
15	C₅×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
16	C₃²×C₅	C₂×C₄	1	1	yes	characteristic, complemented, Hall
17	C₂×C₄×C₅	—	3	1	no	Hall
18	C₂×C₃×D₃	C₂×C₅	1	1	yes	characteristic
19	C₃²×C₄	C₂×C₅	1	1	yes	characteristic, complemented
20	C₃×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
21	C₂×C₃×C₅	—	2	1	no	—
22	C₂×C₃×C₅	—	3	2	no	—
23	C₂×C₃×C₅	C₂×D₃	1	1	yes	characteristic
24	C₂×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
25	C₅×D₃	C₃×C₄	1	2	yes	direct product factor
26	C₂×C₃×C₄	—	3	1	no	—
27	C₄×D₃	C₃×C₅	1	1	yes	characteristic, direct product factor
28	C₂²×C₅	—	3	1	no	—
29	C₄×C₅	—	3	1	no	—
30	C₄×C₅	C₃×D₃	1	1	yes	characteristic, direct product factor
31	C₂×C₃²	C₂²×C₅	1	1	yes	characteristic
32	C₃×D₃	C₄×C₅	1	2	yes	direct product factor
33	C₃×C₅	—	2	1	no	—
34	C₃×C₅	C₄×D₃	1	1	yes	characteristic, direct product factor
35	C₃×C₅	C₂×C₃×C₄	1	1	yes	characteristic, complemented
36	C₂²×C₃	—	3	1	no	—
37	C₂×D₃	C₂×C₃×C₅	1	1	yes	characteristic
38	C₃×C₄	—	2	1	no	—
39	C₃×C₄	—	3	1	no	—
40	C₃×C₄	C₅×D₃	1	1	yes	characteristic, direct product factor
41	C₃×C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
42	Dic₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
43	C₂×C₅	—	3	2	no	—
44	C₂×C₅	C₂×C₃×D₃	1	1	yes	characteristic
45	C₃²	C₂×C₄×C₅	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₄	—	3	1	no	Sylow
47	C₂×C₃	—	2	1	no	—
48	C₂×C₃	—	3	2	no	—
49	C₂×C₃	C₂×C₅×D₃	1	1	yes	characteristic
50	C₂×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
51	D₃	C₃×C₄×C₅	1	2	yes	direct product factor
52	C₅	C₃×C₄×D₃	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²	—	3	1	no	—
54	C₄	—	3	1	no	—
55	C₄	C₃×C₅×D₃	1	1	yes	characteristic, direct product factor
56	C₃	—	2	1	no	—
57	C₃	C₄×C₅×D₃	1	1	yes	characteristic, direct product factor
58	C₃	C₂×C₃×C₄×C₅	1	1	yes	characteristic, complemented
59	C₂	—	3	2	no	—
60	C₂	C₂×C₃×C₅×D₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃²×C₄×C₅	C₂	semidirect
3. C₃×C₅×Dic₃	C₂	semidirect
5. C₄×C₅×D₃	C₃	direct (N × H)
7. C₃×C₅×D₃	C₄	direct (N × H)
8. C₃×C₄×D₃	C₅	direct (N × H)
13. C₃×C₄×C₅	D₃	direct (N × H)
14. C₃×C₄×C₅	C₂×C₃	semidirect
15. C₅×Dic₃	C₂×C₃	semidirect
16. C₃²×C₅	C₂×C₄	semidirect
19. C₃²×C₄	C₂×C₅	semidirect
20. C₃×Dic₃	C₂×C₅	semidirect
25. C₅×D₃	C₃×C₄	direct (N × H)
27. C₄×D₃	C₃×C₅	direct (N × H)
30. C₄×C₅	C₃×D₃	direct (N × H)
32. C₃×D₃	C₄×C₅	direct (N × H)
34. C₃×C₅	C₄×D₃	direct (N × H)
35. C₃×C₅	C₂×C₃×C₄	semidirect
40. C₃×C₄	C₅×D₃	direct (N × H)
41. C₃×C₄	C₂×C₃×C₅	semidirect
42. Dic₃	C₂×C₃×C₅	semidirect
45. C₃²	C₂×C₄×C₅	semidirect
51. D₃	C₃×C₄×C₅	direct (N × H)
52. C₅	C₃×C₄×D₃	direct (N × H)
55. C₄	C₃×C₅×D₃	direct (N × H)
57. C₃	C₄×C₅×D₃	direct (N × H)
58. C₃	C₂×C₃×C₄×C₅	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×Dic₃	2
C₂×C₃×C₄×C₅	3
C₄×C₅×D₃	3
C₃×C₄×D₃	5

Subgroups — C3×C4×C5×D3 #67166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₅×D₃ #67166