Subgroups — C₃²×C₅×D₁₅ #13062545

Order 1350.

184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₅²	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅×D₁₅	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅×D₅	—	3	1	no	maximal
4	C₃²×D₁₅	C₅	1	1	yes	maximal, characteristic, direct product factor
5	C₃²×C₅×D₃	—	5	1	no	maximal
6	C₃²×C₅²	—	2	4	no	—
7	C₃²×C₅²	D₃	1	1	yes	characteristic, complemented
8	C₃²×C₅²	C₂×C₃	1	4	yes	complemented
9	C₅×D₁₅	C₃²	1	1	yes	characteristic, direct product factor
10	C₃×C₅×D₅	—	3	4	no	—
11	C₃³×C₅	—	2	2	no	—
12	C₃³×C₅	D₅	1	1	yes	characteristic, complemented
13	C₃³×C₅	C₂×C₅	1	1	yes	characteristic, complemented
14	C₂×C₃²×C₅	—	15	1	no	—
15	C₃×D₁₅	C₃×C₅	1	4	yes	direct product factor
16	C₃×C₅×D₃	—	5	4	no	—
17	C₃²×D₅	—	3	1	no	—
18	C₃×C₅²	—	2	4	no	—
19	C₃×C₅²	C₃×D₃	1	4	yes	complemented
20	C₃×C₅²	C₂×C₃²	1	1	yes	characteristic, complemented
21	C₃²×D₃	—	5	1	no	Hall
22	C₅×D₅	—	3	1	no	Hall
23	C₃²×C₅	—	2	16	no	—
24	C₃²×C₅	—	2	4	no	—
25	C₃²×C₅	—	2	4	no	—
26	C₃²×C₅	—	2	8	no	—
27	C₃²×C₅	—	2	2	no	—
28	C₃²×C₅	C₅×D₃	1	1	yes	characteristic, complemented
29	C₃²×C₅	C₃×D₅	1	4	yes	complemented
30	C₃²×C₅	D₁₅	1	1	yes	characteristic, direct product factor
31	C₃²×C₅	C₂×C₃×C₅	1	4	yes	complemented
32	C₂×C₃×C₅	—	15	4	no	—
33	D₁₅	C₃²×C₅	1	1	yes	characteristic, direct product factor
34	C₃×D₅	—	3	4	no	—
35	C₅×D₃	—	5	1	no	—
36	C₃³	C₅×D₅	1	1	yes	characteristic, complemented, Sylow
37	C₅²	C₃²×D₃	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₃²	—	15	1	no	—
39	C₃×D₃	—	5	4	no	—
40	C₃×C₅	—	2	16	no	—
41	C₃×C₅	—	2	4	no	—
42	C₃×C₅	—	2	4	no	—
43	C₃×C₅	—	2	2	no	—
44	C₃×C₅	—	2	8	no	—
45	C₃×C₅	C₃²×D₅	1	1	yes	characteristic, complemented
46	C₃×C₅	C₃×C₅×D₃	1	4	yes	complemented
47	C₃×C₅	C₃×D₁₅	1	4	yes	direct product factor
48	C₃×C₅	C₂×C₃²×C₅	1	1	yes	characteristic, complemented
49	C₂×C₅	—	15	1	no	—
50	D₅	—	3	1	no	—
51	C₃²	—	2	4	no	—
52	C₃²	C₃×C₅×D₅	1	4	yes	complemented
53	C₃²	C₅×D₁₅	1	1	yes	characteristic, direct product factor
54	C₂×C₃	—	15	4	no	—
55	D₃	—	5	1	no	—
56	C₅	—	2	2	no	—
57	C₅	C₃²×C₅×D₃	1	1	yes	characteristic, complemented
58	C₅	C₃²×D₁₅	1	1	yes	characteristic, direct product factor
59	C₃	—	2	4	no	—
60	C₃	C₃²×C₅×D₅	1	1	yes	characteristic, complemented
61	C₃	C₃×C₅×D₁₅	1	4	yes	direct product factor
62	C₂	—	15	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃²×C₅	D₁₅
commutator, cosocle, hypocenter	C₃×C₅	C₂×C₃²×C₅
Frattini	C₁	G
Fitting, socle	C₃³×C₅²	C₂
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₃²×C₅

Complements

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

N	H ≅ G/N	type
1. C₃³×C₅²	C₂	semidirect
2. C₃×C₅×D₁₅	C₃	direct (N × H)
4. C₃²×D₁₅	C₅	direct (N × H)
7. C₃²×C₅²	D₃	semidirect
8. C₃²×C₅²	C₂×C₃	semidirect
9. C₅×D₁₅	C₃²	direct (N × H)
12. C₃³×C₅	D₅	semidirect
13. C₃³×C₅	C₂×C₅	semidirect
15. C₃×D₁₅	C₃×C₅	direct (N × H)
19. C₃×C₅²	C₃×D₃	semidirect
20. C₃×C₅²	C₂×C₃²	semidirect
28. C₃²×C₅	C₅×D₃	semidirect
29. C₃²×C₅	C₃×D₅	semidirect
30. C₃²×C₅	D₁₅	direct (N × H)
31. C₃²×C₅	C₂×C₃×C₅	semidirect
33. D₁₅	C₃²×C₅	direct (N × H)
36. C₃³	C₅×D₅	semidirect
37. C₅²	C₃²×D₃	semidirect
45. C₃×C₅	C₃²×D₅	semidirect
46. C₃×C₅	C₃×C₅×D₃	semidirect
47. C₃×C₅	C₃×D₁₅	direct (N × H)
48. C₃×C₅	C₂×C₃²×C₅	semidirect
52. C₃²	C₃×C₅×D₅	semidirect
53. C₃²	C₅×D₁₅	direct (N × H)
57. C₅	C₃²×C₅×D₃	semidirect
58. C₅	C₃²×D₁₅	direct (N × H)
60. C₃	C₃²×C₅×D₅	semidirect
61. C₃	C₃×C₅×D₁₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₅²	2
C₃×C₅×D₁₅	3
C₃²×C₅×D₅	3
C₃²×D₁₅	5
C₃²×C₅×D₃	5

Subgroups — C32×C5×D15 #13062545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×D₁₅ #13062545