Subgroups — #67153

Order 360.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic
2	C₃×Dic₁₅	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
4	Dic₃₀	—	3	1	no	maximal
5	#1204	—	3	1	no	maximal
6	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
7	#627	—	5	1	no	maximal, Hall
8	C₃×C₄×C₅	—	3	1	no	—
9	Dic₁₅	—	3	1	no	—
10	Dic₁₅	—	3	1	no	—
11	Dic₁₅	—	6	1	no	—
12	Dic₁₅	D₃	1	1	yes	characteristic
13	C₃×Dic₅	—	3	1	no	—
14	C₅×Dic₃	D₃	1	1	yes	characteristic
15	C₃²×C₅	Q₈	1	1	yes	characteristic, complemented, Hall
16	Dic₁₀	—	9	1	no	Hall
17	C₃⋊Dic₃	—	5	1	no	—
18	C₃×Dic₃	—	5	1	no	—
19	C₃×Dic₃	D₅	1	1	yes	characteristic
20	C₂×C₃×C₅	—	2	1	no	—
21	C₂×C₃×C₅	C₂×D₃	1	1	yes	characteristic
22	C₂×C₃×C₅	C₂×D₃	1	1	yes	characteristic
23	Dic₆	—	15	1	no	—
24	Dic₆	—	15	1	no	—
25	C₄×C₅	—	3	1	no	—
26	Dic₅	—	3	1	no	—
27	Dic₅	—	9	1	no	—
28	C₂×C₃²	C₂×D₅	1	1	yes	characteristic
29	C₃×C₅	—	2	1	no	—
30	C₃×C₅	Dic₆	1	1	yes	characteristic, complemented
31	C₃×C₅	Dic₆	1	1	yes	characteristic, complemented
32	C₃×C₄	—	3	1	no	—
33	C₃×C₄	—	15	1	no	—
34	Dic₃	—	5	1	no	—
35	Dic₃	—	15	1	no	—
36	Dic₃	—	15	1	no	—
37	Dic₃	—	30	1	no	—
38	Dic₃	D₁₅	1	1	yes	characteristic
39	C₂×C₅	D₃²	1	1	yes	characteristic
40	C₃²	Dic₁₀	1	1	yes	characteristic, complemented, Sylow
41	Q₈	—	45	1	no	Sylow
42	C₂×C₃	—	2	1	no	—
43	C₂×C₃	D₃×D₅	1	1	yes	characteristic
44	C₂×C₃	C₂×D₁₅	1	1	yes	characteristic
45	C₅	#627	1	1	yes	characteristic, complemented, Sylow
46	C₄	—	3	1	no	—
47	C₄	—	15	1	no	—
48	C₄	—	45	1	no	—
49	C₃	—	2	1	no	—
50	C₃	#1204	1	1	yes	characteristic, complemented
51	C₃	Dic₃₀	1	1	yes	characteristic, complemented
52	C₂	D₃×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×D₁₅
commutator, Fitting, socle, prime radical, cosocle	C₂×C₃²×C₅	C₂²
hypocenter	C₃²×C₅	Q₈
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
15. C₃²×C₅	Q₈	semidirect
30. C₃×C₅	Dic₆	semidirect
31. C₃×C₅	Dic₆	semidirect
40. C₃²	Dic₁₀	semidirect
45. C₅	#627	semidirect
50. C₃	#1204	semidirect
51. C₃	Dic₃₀	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊Dic₁₅	2
C₃×Dic₁₅	2
C₃×C₅×Dic₃	2
Dic₃₀	3
#1204	3
#627	5