Subgroups — C₅²⋊C₉ #6568

Order 225.

10 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²	C₃	1	1	yes	maximal, characteristic
2	C₅²	C₉	1	1	yes	characteristic, complemented, Sylow
3	C₃×C₅	—	3	2	no	—
4	C₉	—	25	1	no	maximal, Sylow
5	C₅	—	3	2	no	—
6	C₃	C₅²⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₅²⋊C₃
commutator, hypocenter	C₅²	C₉
Fitting, socle, prime radical, cosocle	C₃×C₅²	C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃

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

N	H ≅ G/N	type
2. C₅²	C₉	semidirect

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

subgroup	index
C₃×C₅²	3
C₉	25