Subgroups — #1035

Order 100.

9 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂₅	C₄	1	1	yes	characteristic, complemented, Sylow
3	AGL(1,𝔽₅)	—	5	1	no	maximal
4	D₅	—	5	1	no	—
5	C₅	AGL(1,𝔽₅)	1	1	yes	characteristic
6	C₄	—	25	1	no	Sylow
7	C₂	—	25	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, hypocenter	C₂₅	C₄
Frattini, socle	C₅	AGL(1,𝔽₅)
prime radical, cosocle	D₂₅	C₂
LCS	G ▷ C₂₅
UCS	C₁

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

N	H ≅ G/N	type
2. C₂₅	C₄	semidirect

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

subgroup	index
D₂₅	2
AGL(1,𝔽₅)	5