Subgroups — Dic₅ #50

Order 20.

6 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅	C₂	1	1	yes	maximal, characteristic
2	C₅	C₄	1	1	yes	characteristic, complemented, Sylow
3	C₄	—	5	1	no	maximal, Sylow
4	C₂	D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅
commutator, hypocenter	C₅	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₂

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

N	H ≅ G/N	type
2. C₅	C₄	semidirect

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

subgroup	index
C₂×C₅	2
C₄	5