Subgroups — C₂²×C₃×D₁₅ #67229

Order 360.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₁₅	C₂	1	6	yes	maximal, direct product factor
3	C₂²×D₁₅	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂²×C₃×D₅	—	3	1	no	maximal
5	C₂×C₃²×C₅	C₂²	1	3	yes	complemented
6	C₃×D₁₅	C₂²	1	4	yes	direct product factor
7	C₂²×C₃×D₃	—	5	1	no	maximal, Hall
8	C₂²×C₃×C₅	—	2	1	no	—
9	C₂²×C₃×C₅	D₃	1	1	yes	characteristic, complemented
10	C₂²×C₃×C₅	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×D₁₅	C₂×C₃	1	6	yes	direct product factor
12	C₂×C₃×D₅	—	3	6	no	—
13	C₃²×C₅	C₂³	1	1	yes	characteristic, complemented, Hall
14	C₂²×D₅	—	3	1	no	Hall
15	C₂²×C₃²	D₅	1	1	yes	characteristic, complemented
16	C₂×C₃×D₃	—	5	6	no	—
17	C₂×C₃×C₅	—	2	3	no	—
18	C₂×C₃×C₅	C₂×D₃	1	3	yes	complemented
19	C₂×C₃×C₅	C₂²×C₃	1	3	yes	complemented
20	D₁₅	C₂²×C₃	1	4	yes	direct product factor
21	C₃×D₅	—	3	4	no	—
22	C₂³×C₃	—	15	1	no	—
23	C₂²×D₃	—	5	1	no	—
24	C₂²×C₅	C₃×D₃	1	1	yes	characteristic, complemented
25	C₂×D₅	—	3	6	no	—
26	C₂×C₃²	C₂×D₅	1	3	yes	complemented
27	C₃×D₃	—	5	4	no	—
28	C₃×C₅	—	2	1	no	—
29	C₃×C₅	C₂²×D₃	1	1	yes	characteristic, complemented
30	C₃×C₅	C₂³×C₃	1	1	yes	characteristic, complemented
31	C₂²×C₃	—	2	1	no	—
32	C₂²×C₃	—	15	6	no	—
33	C₂²×C₃	C₃×D₅	1	1	yes	characteristic, complemented
34	C₂²×C₃	D₁₅	1	1	yes	characteristic, direct product factor
35	C₂×D₃	—	5	6	no	—
36	C₂×C₅	C₂×C₃×D₃	1	3	yes	complemented
37	D₅	—	3	4	no	—
38	C₃²	C₂²×D₅	1	1	yes	characteristic, complemented, Sylow
39	C₂³	—	15	1	no	Sylow
40	C₂×C₃	—	2	3	no	—
41	C₂×C₃	—	15	4	no	—
42	C₂×C₃	C₂×C₃×D₅	1	3	yes	complemented
43	C₂×C₃	C₂×D₁₅	1	3	yes	direct product factor
44	D₃	—	5	4	no	—
45	C₅	C₂²×C₃×D₃	1	1	yes	characteristic, complemented, Sylow
46	C₂²	—	15	6	no	—
47	C₂²	C₃×D₁₅	1	1	yes	characteristic, direct product factor
48	C₃	—	2	1	no	—
49	C₃	C₂²×C₃×D₅	1	1	yes	characteristic, complemented
50	C₃	C₂²×D₁₅	1	1	yes	characteristic, direct product factor
51	C₂	—	15	4	no	—
52	C₂	C₂×C₃×D₁₅	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃	D₁₅
commutator, cosocle, hypocenter	C₃×C₅	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃²×C₅	C₂
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₅	C₂	semidirect
2. C₂×C₃×D₁₅	C₂	direct (N × H)
3. C₂²×D₁₅	C₃	direct (N × H)
5. C₂×C₃²×C₅	C₂²	semidirect
6. C₃×D₁₅	C₂²	direct (N × H)
9. C₂²×C₃×C₅	D₃	semidirect
10. C₂²×C₃×C₅	C₂×C₃	semidirect
11. C₂×D₁₅	C₂×C₃	direct (N × H)
13. C₃²×C₅	C₂³	semidirect
15. C₂²×C₃²	D₅	semidirect
18. C₂×C₃×C₅	C₂×D₃	semidirect
19. C₂×C₃×C₅	C₂²×C₃	semidirect
20. D₁₅	C₂²×C₃	direct (N × H)
24. C₂²×C₅	C₃×D₃	semidirect
26. C₂×C₃²	C₂×D₅	semidirect
29. C₃×C₅	C₂²×D₃	semidirect
30. C₃×C₅	C₂³×C₃	semidirect
33. C₂²×C₃	C₃×D₅	semidirect
34. C₂²×C₃	D₁₅	direct (N × H)
36. C₂×C₅	C₂×C₃×D₃	semidirect
38. C₃²	C₂²×D₅	semidirect
42. C₂×C₃	C₂×C₃×D₅	semidirect
43. C₂×C₃	C₂×D₁₅	direct (N × H)
45. C₅	C₂²×C₃×D₃	semidirect
47. C₂²	C₃×D₁₅	direct (N × H)
49. C₃	C₂²×C₃×D₅	semidirect
50. C₃	C₂²×D₁₅	direct (N × H)
52. C₂	C₂×C₃×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
C₂²×C₃²×C₅	2
C₂×C₃×D₁₅	2
C₂²×D₁₅	3
C₂²×C₃×D₅	3
C₂²×C₃×D₃	5

Subgroups — C22×C3×D15 #67229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×D₁₅ #67229