Subgroups — #67140

Order 360.

96 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×C₅×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#1203	—	3	4	no	maximal
5	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
6	C₃⋊D₁₅	—	2	1	no	—
7	C₅×C₃⋊D₃	—	2	1	no	—
8	C₄⋊(C₃⋊D₃)	—	5	1	no	maximal, Hall
9	C₂×D₁₅	—	3	4	no	—
10	C₂×C₅×D₃	—	3	4	no	—
11	C₃×Dic₅	D₃	1	4	yes	complemented
12	C₃²×C₅	D₄	1	1	yes	characteristic, complemented, Hall
13	C₂²⋊D₅	—	9	1	no	Hall
14	C₂×C₃⋊D₃	—	5	1	no	—
15	C₂×C₃⋊D₃	D₅	1	1	yes	characteristic, complemented
16	C₃²×C₄	—	5	1	no	—
17	C₂×C₃×C₅	C₂×D₃	1	4	yes	—
18	D₁₅	—	6	4	no	—
19	C₅×D₃	—	6	4	no	—
20	D₁₂	—	15	4	no	—
21	C₂²×C₅	—	9	1	no	—
22	C₂×D₅	—	9	1	no	—
23	Dic₅	C₃⋊D₃	1	1	yes	characteristic, complemented
24	C₂×C₃²	C₂×D₅	1	1	yes	characteristic
25	C₃⋊D₃	—	2	1	no	—
26	C₃⋊D₃	—	10	1	no	—
27	C₃×C₅	D₁₂	1	4	yes	complemented
28	C₂×D₃	—	3	4	no	—
29	C₂×D₃	—	15	4	no	—
30	C₃×C₄	—	5	4	no	—
31	C₂×C₅	—	18	1	no	—
32	C₂×C₅	C₂×C₃⋊D₃	1	1	yes	characteristic
33	D₅	—	18	1	no	—
34	C₃²	C₂²⋊D₅	1	1	yes	characteristic, complemented, Sylow
35	D₄	—	45	1	no	Sylow
36	C₂×C₃	D₃×D₅	1	4	yes	—
37	D₃	—	6	4	no	—
38	D₃	—	30	4	no	—
39	C₅	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented, Sylow
40	C₂²	—	9	1	no	—
41	C₂²	—	45	1	no	—
42	C₄	—	5	1	no	—
43	C₃	#1203	1	4	yes	complemented
44	C₂	—	18	1	no	—
45	C₂	—	90	1	no	—
46	C₂	D₅×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅×C₃⋊D₃
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×C₅×C₃⋊D₃	C₂	semidirect
3. C₃²×Dic₅	C₂	semidirect
11. C₃×Dic₅	D₃	semidirect
12. C₃²×C₅	D₄	semidirect
15. C₂×C₃⋊D₃	D₅	semidirect
23. Dic₅	C₃⋊D₃	semidirect
27. C₃×C₅	D₁₂	semidirect
34. C₃²	C₂²⋊D₅	semidirect
39. C₅	C₄⋊(C₃⋊D₃)	semidirect
43. C₃	#1203	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×C₅×C₃⋊D₃	2
C₃²×Dic₅	2
#1203	3
C₄⋊(C₃⋊D₃)	5