Subgroups — C₅⋊((C₃⋊D₃)⋊C₃) #63196

Order 270.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃⋊D₁₅	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×D₁₅	—	3	1	no	maximal
4	(C₃⋊D₃)⋊C₃	—	5	1	no	maximal, Hall
5	C₃²×C₅	—	2	1	no	—
6	C₃²×C₅	D₃	1	1	yes	characteristic, complemented
7	C₃²×C₅	C₂×C₃	1	1	yes	characteristic, complemented
8	D₁₅	—	3	1	no	—
9	D₁₅	—	9	1	no	—
10	C₃×D₅	—	9	1	no	—
11	H(3,𝔽₃)	D₅	1	1	yes	characteristic, complemented, Sylow
12	C₃⋊D₃	—	5	1	no	—
13	C₃×D₃	—	15	1	no	—
14	C₃×C₅	—	3	1	no	—
15	C₃×C₅	—	3	1	no	—
16	C₃×C₅	—	6	1	no	—
17	C₃×C₅	C₃×D₃	1	1	yes	characteristic
18	D₅	—	9	1	no	Hall
19	C₃²	—	2	1	no	—
20	C₃²	C₃×D₅	1	1	yes	characteristic, complemented
21	C₃²	D₁₅	1	1	yes	characteristic, complemented
22	C₂×C₃	—	45	1	no	—
23	D₃	—	15	1	no	—
24	D₃	—	45	1	no	—
25	C₅	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented, Sylow
26	C₃	—	3	1	no	—
27	C₃	—	3	1	no	—
28	C₃	—	6	1	no	—
29	C₃	C₃×D₁₅	1	1	yes	characteristic
30	C₂	—	45	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×C₅	C₂×C₃
Frattini	C₃	C₃×D₁₅
Fitting	C₅×H(3,𝔽₃)	C₂
socle	C₃×C₅	C₃×D₃
LCS	G ▷ C₃²×C₅
UCS	C₁

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

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