Subgroups — D₉ #44

Order 18.

6 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	D₃	—	3	1	no	maximal
3	C₃	D₃	1	1	yes	characteristic
4	C₂	—	9	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉	C₂
Frattini, socle	C₃	D₃
LCS	G ▷ C₉
UCS	C₁

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

N	H ≅ G/N	type
1. C₉	C₂	semidirect

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

subgroup	index
C₉	2
D₃	3