Subgroups — #67151

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, complemented
2	C₂×C₃×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	D₆₀	—	3	1	no	maximal
5	#1202	—	3	1	no	maximal
6	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
7	C₃⋊D₁₅	—	2	1	no	—
8	C₃×D₁₅	—	2	1	no	—
9	#626	—	5	1	no	maximal, Hall
10	C₂×D₁₅	—	3	1	no	—
11	C₂×D₁₅	—	3	1	no	—
12	C₂×D₁₅	—	6	1	no	—
13	C₂×D₁₅	D₃	1	1	yes	characteristic, complemented
14	C₂×C₃×D₅	—	3	1	no	—
15	C₃×C₄×C₅	—	3	1	no	—
16	C₅×Dic₃	D₃	1	1	yes	characteristic, complemented
17	C₃²×C₅	D₄	1	1	yes	characteristic, complemented, Hall
18	D₂₀	—	9	1	no	Hall
19	C₂×C₃⋊D₃	—	5	1	no	—
20	C₂×C₃×D₃	—	5	1	no	—
21	C₃×Dic₃	D₅	1	1	yes	characteristic, complemented
22	C₂×C₃×C₅	—	2	1	no	—
23	C₂×C₃×C₅	C₂×D₃	1	1	yes	characteristic
24	C₂×C₃×C₅	C₂×D₃	1	1	yes	characteristic
25	D₁₅	—	2	1	no	—
26	D₁₅	—	6	2	no	—
27	D₁₅	—	6	1	no	—
28	D₁₅	—	6	1	no	—
29	C₃×D₅	—	6	1	no	—
30	#67	—	15	1	no	—
31	D₁₂	—	15	1	no	—
32	C₂×D₅	—	3	1	no	—
33	C₂×D₅	—	9	1	no	—
34	C₄×C₅	—	3	1	no	—
35	C₂×C₃²	C₂×D₅	1	1	yes	characteristic
36	C₃⋊D₃	—	10	1	no	—
37	C₃×D₃	—	10	1	no	—
38	C₃×C₅	—	2	1	no	—
39	C₃×C₅	D₁₂	1	1	yes	characteristic, complemented
40	C₃×C₅	#67	1	1	yes	characteristic, complemented
41	C₂²×C₃	—	15	1	no	—
42	C₂×D₃	—	5	1	no	—
43	C₂×D₃	—	15	1	no	—
44	C₂×D₃	—	15	1	no	—
45	C₂×D₃	—	30	1	no	—
46	C₃×C₄	—	3	1	no	—
47	Dic₃	D₁₅	1	1	yes	characteristic, complemented
48	C₂×C₅	D₃²	1	1	yes	characteristic
49	D₅	—	6	1	no	—
50	D₅	—	18	1	no	—
51	C₃²	D₂₀	1	1	yes	characteristic, complemented, Sylow
52	D₄	—	45	1	no	Sylow
53	C₂×C₃	—	2	1	no	—
54	C₂×C₃	—	30	1	no	—
55	C₂×C₃	D₃×D₅	1	1	yes	characteristic
56	C₂×C₃	C₂×D₁₅	1	1	yes	characteristic
57	D₃	—	10	1	no	—
58	D₃	—	30	2	no	—
59	D₃	—	30	1	no	—
60	D₃	—	30	1	no	—
61	C₅	#626	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	15	1	no	—
63	C₂²	—	45	1	no	—
64	C₄	—	3	1	no	—
65	C₃	—	2	1	no	—
66	C₃	#1202	1	1	yes	characteristic, complemented
67	C₃	D₆₀	1	1	yes	characteristic, complemented
68	C₂	—	30	1	no	—
69	C₂	—	90	1	no	—
70	C₂	D₃×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×D₁₅
commutator, Fitting, socle, cosocle	C₂×C₃²×C₅	C₂²
hypocenter	C₃²×C₅	D₄
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₃⋊D₁₅	C₂	semidirect
2. C₂×C₃×D₁₅	C₂	semidirect
3. C₃×C₅×Dic₃	C₂	semidirect
13. C₂×D₁₅	D₃	semidirect
16. C₅×Dic₃	D₃	semidirect
17. C₃²×C₅	D₄	semidirect
21. C₃×Dic₃	D₅	semidirect
39. C₃×C₅	D₁₂	semidirect
40. C₃×C₅	#67	semidirect
47. Dic₃	D₁₅	semidirect
51. C₃²	D₂₀	semidirect
61. C₅	#626	semidirect
66. C₃	#1202	semidirect
67. C₃	D₆₀	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
C₂×C₃×D₁₅	2
C₃×C₅×Dic₃	2
D₆₀	3
#1202	3
#626	5