Subgroups — Dic₉ #149

Order 36.

9 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉	C₂	1	1	yes	maximal, characteristic
2	Dic₃	—	3	1	no	maximal
3	C₉	C₄	1	1	yes	characteristic, complemented, Sylow
4	C₂×C₃	D₃	1	1	yes	characteristic
5	C₄	—	9	1	no	Sylow
6	C₃	Dic₃	1	1	yes	characteristic
7	C₂	D₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	D₉
commutator, hypocenter	C₉	C₄
Frattini, socle, prime radical	C₂×C₃	D₃
Fitting, cosocle	C₂×C₉	C₂
LCS	G ▷ C₉
UCS	C₁ ◁ C₂

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

N	H ≅ G/N	type
3. C₉	C₄	semidirect

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

subgroup	index
C₂×C₉	2
Dic₃	3