Subgroups — #67184

Order 360.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃⋊D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#1220	—	3	4	no	maximal
5	C₂×C₃²×C₅	—	2	1	no	—
6	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
7	C₃⋊D₁₅	—	2	1	no	—
8	#638	—	5	1	no	maximal, Hall
9	C₂²×C₃×C₅	D₃	1	4	yes	complemented
10	C₂×D₁₅	—	3	4	no	—
11	Dic₁₅	—	3	4	no	—
12	C₃²×C₅	D₄	1	1	yes	characteristic, complemented, Hall
13	C₂²⋊D₅	—	9	1	no	Hall
14	C₂²×C₃²	D₅	1	1	yes	characteristic, complemented
15	C₂×C₃⋊D₃	—	5	1	no	—
16	C₃⋊Dic₃	—	5	1	no	—
17	C₂×C₃×C₅	—	2	4	no	—
18	C₂×C₃×C₅	C₂×D₃	1	4	yes	—
19	D₁₅	—	6	4	no	—
20	#67	—	15	4	no	—
21	C₂²×C₅	C₃⋊D₃	1	1	yes	characteristic, complemented
22	C₂×D₅	—	9	1	no	—
23	Dic₅	—	9	1	no	—
24	C₂×C₃²	—	2	1	no	—
25	C₂×C₃²	C₂×D₅	1	1	yes	characteristic
26	C₃⋊D₃	—	10	1	no	—
27	C₃×C₅	#67	1	4	yes	complemented
28	C₂²×C₃	D₁₅	1	4	yes	complemented
29	C₂×D₃	—	15	4	no	—
30	Dic₃	—	15	4	no	—
31	C₂×C₅	—	2	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	—	2	4	no	—
37	C₂×C₃	C₂×D₁₅	1	4	yes	—
38	D₃	—	30	4	no	—
39	C₅	#638	1	1	yes	characteristic, complemented, Sylow
40	C₂²	—	45	1	no	—
41	C₂²	C₃⋊D₁₅	1	1	yes	characteristic, complemented
42	C₄	—	45	1	no	—
43	C₃	#1220	1	4	yes	complemented
44	C₂	—	2	1	no	—
45	C₂	—	90	1	no	—
46	C₂	C₂×C₃⋊D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×C₃⋊D₁₅
commutator, socle, cosocle	C₂×C₃²×C₅	C₂²
Fitting	C₂²×C₃²×C₅	C₂
hypercenter	C₂²	C₃⋊D₁₅
hypocenter	C₃²×C₅	D₄
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₅	C₂	semidirect
2. C₂×C₃⋊D₁₅	C₂	semidirect
3. C₃⋊Dic₁₅	C₂	semidirect
9. C₂²×C₃×C₅	D₃	semidirect
12. C₃²×C₅	D₄	semidirect
14. C₂²×C₃²	D₅	semidirect
21. C₂²×C₅	C₃⋊D₃	semidirect
27. C₃×C₅	#67	semidirect
28. C₂²×C₃	D₁₅	semidirect
34. C₃²	C₂²⋊D₅	semidirect
39. C₅	#638	semidirect
41. C₂²	C₃⋊D₁₅	semidirect
43. C₃	#1220	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₅	2
C₂×C₃⋊D₁₅	2
C₃⋊Dic₁₅	2
#1220	3
#638	5