Subgroups — #67149

Order 360.

74 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	C₃×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₁₅	—	3	1	no	maximal
5	#1200	—	3	1	no	maximal
6	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
7	C₃⋊D₁₅	C₄	1	2	yes	complemented
8	#624	—	5	1	no	maximal, Hall
9	C₂×D₁₅	—	3	1	no	—
10	C₂×D₁₅	—	3	1	no	—
11	C₂×D₁₅	—	6	1	no	—
12	C₃×C₄×C₅	—	3	1	no	—
13	Dic₁₅	D₃	1	1	yes	characteristic, complemented
14	C₃×Dic₅	—	3	1	no	—
15	C₅×Dic₃	D₃	1	1	yes	characteristic, complemented
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	—	5	1	no	—
20	C₃×Dic₃	D₅	1	1	yes	characteristic, complemented
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	—	3	2	no	—
25	D₁₅	—	3	2	no	—
26	D₁₅	—	6	2	no	—
27	C₄×D₃	—	15	1	no	—
28	C₄×D₃	—	15	1	no	—
29	C₂×D₅	—	9	1	no	—
30	C₄×C₅	—	3	1	no	—
31	Dic₅	—	3	1	no	—
32	C₂×C₃²	C₂×D₅	1	1	yes	characteristic
33	C₃⋊D₃	—	5	2	no	—
34	C₃×C₅	—	2	1	no	—
35	C₃×C₅	C₄×D₃	1	1	yes	characteristic, complemented
36	C₃×C₅	C₄×D₃	1	1	yes	characteristic, complemented
37	C₂×D₃	—	15	1	no	—
38	C₂×D₃	—	15	1	no	—
39	C₂×D₃	—	30	1	no	—
40	C₃×C₄	—	3	1	no	—
41	C₃×C₄	—	15	1	no	—
42	Dic₃	—	5	1	no	—
43	Dic₃	D₁₅	1	1	yes	characteristic, complemented
44	C₂×C₅	D₃²	1	1	yes	characteristic
45	D₅	—	9	2	no	—
46	C₃²	C₄×D₅	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₄	—	45	1	no	Sylow
48	C₂×C₃	—	2	1	no	—
49	C₂×C₃	D₃×D₅	1	1	yes	characteristic
50	C₂×C₃	C₂×D₁₅	1	1	yes	characteristic
51	D₃	—	15	2	no	—
52	D₃	—	15	2	no	—
53	D₃	—	30	2	no	—
54	C₅	#624	1	1	yes	characteristic, complemented, Sylow
55	C₂²	—	45	1	no	—
56	C₄	—	3	1	no	—
57	C₄	—	15	1	no	—
58	C₃	—	2	1	no	—
59	C₃	#1200	1	1	yes	characteristic, complemented
60	C₃	C₄×D₁₅	1	1	yes	characteristic, complemented
61	C₂	—	45	2	no	—
62	C₂	D₃×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×D₁₅
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. C₃×Dic₁₅	C₂	semidirect
3. C₃×C₅×Dic₃	C₂	semidirect
7. C₃⋊D₁₅	C₄	semidirect
13. Dic₁₅	D₃	semidirect
15. C₅×Dic₃	D₃	semidirect
16. C₃²×C₅	C₂×C₄	semidirect
20. C₃×Dic₃	D₅	semidirect
35. C₃×C₅	C₄×D₃	semidirect
36. C₃×C₅	C₄×D₃	semidirect
43. Dic₃	D₁₅	semidirect
46. C₃²	C₄×D₅	semidirect
54. C₅	#624	semidirect
59. C₃	#1200	semidirect
60. C₃	C₄×D₁₅	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
C₃×Dic₁₅	2
C₃×C₅×Dic₃	2
C₄×D₁₅	3
#1200	3
#624	5