Subgroups — C₂×C₃×C₅×Dic₃ #67168

Order 360.

74 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
2	C₃×C₅×Dic₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₄×C₅	—	3	1	no	maximal
4	C₂×C₅×Dic₃	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃²×C₅	C₄	1	2	yes	complemented
6	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₃×Dic₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₅	—	2	1	no	—
9	C₂²×C₃×C₅	D₃	1	1	yes	characteristic
10	C₂²×C₃×C₅	C₂×C₃	1	1	yes	characteristic
11	C₃×C₄×C₅	—	3	2	no	—
12	C₅×Dic₃	C₂×C₃	1	2	yes	direct product factor
13	C₃²×C₅	C₂×C₄	1	1	yes	characteristic, complemented, Hall
14	C₂×C₄×C₅	—	3	1	no	Hall
15	C₂²×C₃²	C₂×C₅	1	1	yes	characteristic
16	C₃×Dic₃	C₂×C₅	1	2	yes	direct product factor
17	C₂×C₃×C₅	—	2	2	no	—
18	C₂×C₃×C₅	—	2	1	no	—
19	C₂×C₃×C₅	Dic₃	1	2	yes	direct product factor
20	C₂×C₃×C₅	C₃×C₄	1	2	yes	complemented
21	C₂×C₃×C₅	C₂×D₃	1	1	yes	characteristic
22	C₂×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
23	C₂×C₃×C₄	—	3	1	no	—
24	C₂×Dic₃	C₃×C₅	1	1	yes	characteristic, direct product factor
25	C₂²×C₅	C₃×D₃	1	1	yes	characteristic
26	C₄×C₅	—	3	2	no	—
27	C₂×C₃²	C₄×C₅	1	2	yes	complemented
28	C₂×C₃²	C₂²×C₅	1	1	yes	characteristic
29	C₃×C₅	—	2	1	no	—
30	C₃×C₅	C₂×Dic₃	1	1	yes	characteristic, direct product factor
31	C₃×C₅	C₂×C₃×C₄	1	1	yes	characteristic, complemented
32	C₂²×C₃	—	2	1	no	—
33	C₂²×C₃	C₅×D₃	1	1	yes	characteristic
34	C₂²×C₃	C₂×C₃×C₅	1	1	yes	characteristic
35	C₃×C₄	—	3	2	no	—
36	Dic₃	C₂×C₃×C₅	1	2	yes	direct product factor
37	C₂×C₅	C₃×Dic₃	1	2	yes	direct product factor
38	C₂×C₅	C₂×C₃×D₃	1	1	yes	characteristic
39	C₃²	C₂×C₄×C₅	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₄	—	3	1	no	Sylow
41	C₂×C₃	—	2	2	no	—
42	C₂×C₃	—	2	1	no	—
43	C₂×C₃	C₅×Dic₃	1	2	yes	direct product factor
44	C₂×C₃	C₃×C₄×C₅	1	2	yes	complemented
45	C₂×C₃	C₂×C₅×D₃	1	1	yes	characteristic
46	C₂×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
47	C₅	C₂×C₃×Dic₃	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²	C₃×C₅×D₃	1	1	yes	characteristic
49	C₄	—	3	2	no	—
50	C₃	—	2	1	no	—
51	C₃	C₂×C₅×Dic₃	1	1	yes	characteristic, direct product factor
52	C₃	C₂×C₃×C₄×C₅	1	1	yes	characteristic, complemented
53	C₂	C₃×C₅×Dic₃	1	2	yes	direct product factor
54	C₂	C₂×C₃×C₅×D₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×Dic₃	C₂	direct (N × H)
4. C₂×C₅×Dic₃	C₃	direct (N × H)
5. C₂×C₃²×C₅	C₄	semidirect
7. C₂×C₃×Dic₃	C₅	direct (N × H)
12. C₅×Dic₃	C₂×C₃	direct (N × H)
13. C₃²×C₅	C₂×C₄	semidirect
16. C₃×Dic₃	C₂×C₅	direct (N × H)
19. C₂×C₃×C₅	Dic₃	direct (N × H)
20. C₂×C₃×C₅	C₃×C₄	semidirect
24. C₂×Dic₃	C₃×C₅	direct (N × H)
27. C₂×C₃²	C₄×C₅	semidirect
30. C₃×C₅	C₂×Dic₃	direct (N × H)
31. C₃×C₅	C₂×C₃×C₄	semidirect
36. Dic₃	C₂×C₃×C₅	direct (N × H)
37. C₂×C₅	C₃×Dic₃	direct (N × H)
39. C₃²	C₂×C₄×C₅	semidirect
43. C₂×C₃	C₅×Dic₃	direct (N × H)
44. C₂×C₃	C₃×C₄×C₅	semidirect
47. C₅	C₂×C₃×Dic₃	direct (N × H)
51. C₃	C₂×C₅×Dic₃	direct (N × H)
52. C₃	C₂×C₃×C₄×C₅	semidirect
53. C₂	C₃×C₅×Dic₃	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
C₃×C₅×Dic₃	2
C₂×C₃×C₄×C₅	3
C₂×C₅×Dic₃	3
C₂×C₃×Dic₃	5

Subgroups — C2×C3×C5×Dic3 #67168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×Dic₃ #67168