Subgroups — #67154

Order 360.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₅×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#1200	—	3	1	no	maximal
5	D₅×Dic₃	—	3	1	no	maximal
6	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
7	C₃×D₁₅	C₄	1	2	yes	complemented
8	D₃×Dic₃	—	5	1	no	maximal, Hall
9	C₂×D₁₅	D₃	1	1	yes	characteristic
10	C₂×C₃×D₅	—	3	1	no	—
11	Dic₁₅	D₃	1	1	yes	characteristic, complemented
12	C₃×Dic₅	—	3	1	no	—
13	C₅×Dic₃	—	3	1	no	—
14	C₅×Dic₃	—	3	1	no	—
15	C₅×Dic₃	—	6	1	no	—
16	C₃²×C₅	C₂×C₄	1	1	yes	characteristic, complemented, Hall
17	C₄×D₅	—	9	1	no	Hall
18	C₂×C₃×D₃	—	5	1	no	—
19	C₃⋊Dic₃	D₅	1	1	yes	characteristic, complemented
20	C₃×Dic₃	—	5	1	no	—
21	C₂×C₃×C₅	—	2	1	no	—
22	C₂×C₃×C₅	C₂×D₃	1	1	yes	characteristic
23	C₂×C₃×C₅	C₂×D₃	1	1	yes	characteristic
24	D₁₅	Dic₃	1	2	yes	complemented
25	C₃×D₅	—	3	2	no	—
26	C₂×Dic₃	—	15	1	no	—
27	C₄×D₃	—	15	1	no	—
28	C₂×D₅	—	3	1	no	—
29	C₄×C₅	—	9	1	no	—
30	Dic₅	—	3	1	no	—
31	C₂×C₃²	C₂×D₅	1	1	yes	characteristic
32	C₃×D₃	—	5	2	no	—
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	—	15	1	no	—
37	C₂×D₃	—	5	1	no	—
38	C₃×C₄	—	15	1	no	—
39	Dic₃	—	3	1	no	—
40	Dic₃	—	3	1	no	—
41	Dic₃	—	5	1	no	—
42	Dic₃	—	6	1	no	—
43	C₂×C₅	D₃²	1	1	yes	characteristic
44	D₅	—	3	2	no	—
45	C₃²	C₄×D₅	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₄	—	45	1	no	Sylow
47	C₂×C₃	—	2	1	no	—
48	C₂×C₃	—	15	2	no	—
49	C₂×C₃	D₃×D₅	1	1	yes	characteristic
50	C₂×C₃	D₃×D₅	1	1	yes	characteristic
51	D₃	—	5	2	no	—
52	C₅	D₃×Dic₃	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	15	1	no	—
54	C₄	—	9	1	no	—
55	C₄	—	15	1	no	—
56	C₃	—	2	1	no	—
57	C₃	D₅×Dic₃	1	1	yes	characteristic, complemented
58	C₃	#1200	1	1	yes	characteristic, complemented
59	C₂	—	15	2	no	—
60	C₂	(C₃⋊D₃)⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	(C₃⋊D₃)⋊D₅
commutator, hypocenter	C₃²×C₅	C₂×C₄
Fitting, socle, cosocle	C₂×C₃²×C₅	C₂²
prime radical	C₂×C₃×D₁₅	C₂
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₅×C₃⋊Dic₃	C₂	semidirect
3. C₃×Dic₁₅	C₂	semidirect
7. C₃×D₁₅	C₄	semidirect
11. Dic₁₅	D₃	semidirect
16. C₃²×C₅	C₂×C₄	semidirect
19. C₃⋊Dic₃	D₅	semidirect
24. D₁₅	Dic₃	semidirect
34. C₃×C₅	C₄×D₃	semidirect
35. C₃×C₅	C₂×Dic₃	semidirect
45. C₃²	C₄×D₅	semidirect
52. C₅	D₃×Dic₃	semidirect
57. C₃	D₅×Dic₃	semidirect
58. C₃	#1200	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₁₅	2
C₅×C₃⋊Dic₃	2
C₃×Dic₁₅	2
#1200	3
D₅×Dic₃	3
D₃×Dic₃	5