Subgroups — C₅×D₃×Dic₃ #67142

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	C₅×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×Dic₃	—	3	1	no	maximal
5	C₄×C₅×D₃	—	3	1	no	maximal
6	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
7	C₃×C₅×D₃	C₄	1	2	yes	complemented
8	D₃×Dic₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₅	—	3	1	no	—
10	C₂×C₅×D₃	D₃	1	1	yes	characteristic
11	C₃×C₄×C₅	—	3	1	no	—
12	C₅×Dic₃	—	3	1	no	—
13	C₅×Dic₃	—	3	1	no	—
14	C₅×Dic₃	—	6	1	no	—
15	C₅×Dic₃	D₃	1	1	yes	characteristic, direct product factor
16	C₃²×C₅	C₂×C₄	1	1	yes	characteristic, complemented, Hall
17	C₂×C₄×C₅	—	9	1	no	Hall
18	C₂×C₃×D₃	C₂×C₅	1	1	yes	characteristic
19	C₃⋊Dic₃	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	C₂×D₃	1	1	yes	characteristic
25	C₅×D₃	Dic₃	1	2	yes	direct product factor
26	C₂×Dic₃	—	3	1	no	—
27	C₄×D₃	—	3	1	no	—
28	C₂²×C₅	—	3	1	no	—
29	C₄×C₅	—	3	1	no	—
30	C₄×C₅	—	9	1	no	—
31	C₂×C₃²	C₂²×C₅	1	1	yes	characteristic
32	C₃×D₃	C₄×C₅	1	2	yes	complemented
33	C₃×C₅	—	2	1	no	—
34	C₃×C₅	C₄×D₃	1	1	yes	characteristic, complemented
35	C₃×C₅	C₂×Dic₃	1	1	yes	characteristic, complemented
36	C₂²×C₃	—	3	1	no	—
37	C₂×D₃	C₅×D₃	1	1	yes	characteristic
38	C₃×C₄	—	3	1	no	—
39	Dic₃	—	3	1	no	—
40	Dic₃	—	3	1	no	—
41	Dic₃	—	6	1	no	—
42	Dic₃	C₅×D₃	1	1	yes	characteristic, direct product factor
43	C₂×C₅	—	3	2	no	—
44	C₂×C₅	D₃²	1	1	yes	characteristic
45	C₃²	C₂×C₄×C₅	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₄	—	9	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	C₂×C₅×D₃	1	1	yes	characteristic
51	D₃	C₅×Dic₃	1	2	yes	direct product factor
52	C₅	D₃×Dic₃	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²	—	3	1	no	—
54	C₄	—	3	1	no	—
55	C₄	—	9	1	no	—
56	C₃	—	2	1	no	—
57	C₃	C₄×C₅×D₃	1	1	yes	characteristic, complemented
58	C₃	C₂×C₅×Dic₃	1	1	yes	characteristic, complemented
59	C₂	—	3	2	no	—
60	C₂	C₅×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	D₃²
commutator, hypocenter	C₃²	C₂×C₄×C₅
Frattini	C₂	C₅×D₃²
Fitting, socle	C₂×C₃²×C₅	C₂²
prime radical	C₂×C₃×C₅×D₃	C₂
cosocle	C₂×C₃²	C₂²×C₅
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₅

Complements

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

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

Subgroups — C5×D3×Dic3 #67142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₃×Dic₃ #67142