Subgroups — C₂²×C₃×C₅×D₃ #67228

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

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₅	D₃
commutator, cosocle, hypocenter	C₃	C₂³×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₃²×C₅	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₅

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×C₅×D₃	C₂	direct (N × H)
4. C₂²×C₅×D₃	C₃	direct (N × H)
5. C₂×C₃²×C₅	C₂²	semidirect
6. C₃×C₅×D₃	C₂²	direct (N × H)
7. C₂²×C₃×D₃	C₅	direct (N × H)
10. C₂²×C₃×C₅	D₃	direct (N × H)
11. C₂²×C₃×C₅	C₂×C₃	semidirect
12. C₂×C₅×D₃	C₂×C₃	direct (N × H)
13. C₃²×C₅	C₂³	semidirect
15. C₂²×C₃²	C₂×C₅	semidirect
16. C₂×C₃×D₃	C₂×C₅	direct (N × H)
19. C₂×C₃×C₅	C₂×D₃	direct (N × H)
20. C₂×C₃×C₅	C₂²×C₃	semidirect
21. C₅×D₃	C₂²×C₃	direct (N × H)
23. C₂²×D₃	C₃×C₅	direct (N × H)
25. C₂²×C₅	C₃×D₃	direct (N × H)
26. C₂×C₃²	C₂²×C₅	semidirect
27. C₃×D₃	C₂²×C₅	direct (N × H)
29. C₃×C₅	C₂²×D₃	direct (N × H)
30. C₃×C₅	C₂³×C₃	semidirect
33. C₂²×C₃	C₅×D₃	direct (N × H)
34. C₂²×C₃	C₂×C₃×C₅	semidirect
35. C₂×D₃	C₂×C₃×C₅	direct (N × H)
37. C₂×C₅	C₂×C₃×D₃	direct (N × H)
38. C₃²	C₂³×C₅	semidirect
42. C₂×C₃	C₂×C₅×D₃	direct (N × H)
43. C₂×C₃	C₂²×C₃×C₅	semidirect
44. D₃	C₂²×C₃×C₅	direct (N × H)
45. C₅	C₂²×C₃×D₃	direct (N × H)
47. C₂²	C₃×C₅×D₃	direct (N × H)
49. C₃	C₂²×C₅×D₃	direct (N × H)
50. C₃	C₂³×C₃×C₅	semidirect
52. C₂	C₂×C₃×C₅×D₃	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×C₅×D₃	2
C₂³×C₃×C₅	3
C₂²×C₅×D₃	3
C₂²×C₃×D₃	5

Subgroups — C22×C3×C5×D3 #67228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅×D₃ #67228