Subgroups — #67150

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, complemented
2	C₂×C₃×C₅×D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#1220	—	3	1	no	maximal
5	#1201	—	3	1	no	maximal
6	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
7	C₃×D₁₅	—	2	1	no	—
8	C₃×C₅×D₃	—	2	1	no	—
9	#625	—	5	1	no	maximal, Hall
10	C₂²×C₃×C₅	—	3	1	no	—
11	C₂×D₁₅	D₃	1	1	yes	characteristic, complemented
12	C₂×C₅×D₃	D₃	1	1	yes	characteristic, complemented
13	C₂×C₃×D₅	—	3	1	no	—
14	Dic₁₅	—	3	1	no	—
15	Dic₁₅	—	3	1	no	—
16	Dic₁₅	—	6	1	no	—
17	C₃²×C₅	D₄	1	1	yes	characteristic, complemented, Hall
18	C₂²⋊D₅	—	9	1	no	Hall
19	C₂×C₃×D₃	—	5	1	no	—
20	C₂×C₃×D₃	D₅	1	1	yes	characteristic, complemented
21	C₃⋊Dic₃	—	5	1	no	—
22	C₂×C₃×C₅	—	2	1	no	—
23	C₂×C₃×C₅	—	6	1	no	—
24	C₂×C₃×C₅	C₂×D₃	1	1	yes	characteristic
25	C₂×C₃×C₅	C₂×D₃	1	1	yes	characteristic
26	D₁₅	—	2	1	no	—
27	C₃×D₅	—	6	1	no	—
28	C₅×D₃	—	2	1	no	—
29	#67	—	15	1	no	—
30	#67	—	15	1	no	—
31	C₂²×C₅	—	3	1	no	—
32	C₂×D₅	—	3	1	no	—
33	Dic₅	—	9	1	no	—
34	C₂×C₃²	C₂×D₅	1	1	yes	characteristic
35	C₃×D₃	—	2	1	no	—
36	C₃×D₃	—	10	1	no	—
37	C₃×C₅	—	2	1	no	—
38	C₃×C₅	#67	1	1	yes	characteristic, complemented
39	C₃×C₅	#67	1	1	yes	characteristic, complemented
40	C₂²×C₃	—	3	1	no	—
41	C₂²×C₃	—	15	1	no	—
42	C₂×D₃	—	5	1	no	—
43	C₂×D₃	D₁₅	1	1	yes	characteristic, complemented
44	Dic₃	—	15	1	no	—
45	Dic₃	—	15	1	no	—
46	Dic₃	—	30	1	no	—
47	C₂×C₅	—	6	1	no	—
48	C₂×C₅	D₃²	1	1	yes	characteristic
49	D₅	—	6	1	no	—
50	C₃²	C₂²⋊D₅	1	1	yes	characteristic, complemented, Sylow
51	D₄	—	45	1	no	Sylow
52	C₂×C₃	—	2	1	no	—
53	C₂×C₃	—	6	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	—	2	1	no	—
58	D₃	—	10	1	no	—
59	C₅	#625	1	1	yes	characteristic, complemented, Sylow
60	C₂²	—	3	1	no	—
61	C₂²	—	15	1	no	—
62	C₄	—	45	1	no	—
63	C₃	—	2	1	no	—
64	C₃	#1201	1	1	yes	characteristic, complemented
65	C₃	#1220	1	1	yes	characteristic, complemented
66	C₂	—	6	1	no	—
67	C₂	—	30	1	no	—
68	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×C₅×D₃	C₂	semidirect
3. C₃⋊Dic₁₅	C₂	semidirect
11. C₂×D₁₅	D₃	semidirect
12. C₂×C₅×D₃	D₃	semidirect
17. C₃²×C₅	D₄	semidirect
20. C₂×C₃×D₃	D₅	semidirect
38. C₃×C₅	#67	semidirect
39. C₃×C₅	#67	semidirect
43. C₂×D₃	D₁₅	semidirect
50. C₃²	C₂²⋊D₅	semidirect
59. C₅	#625	semidirect
64. C₃	#1201	semidirect
65. C₃	#1220	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×C₅×D₃	2
C₃⋊Dic₁₅	2
#1220	3
#1201	3
#625	5