Subgroups — #1043

Order 100.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₅²	C₄	1	1	yes	characteristic, complemented, Sylow
3	AGL(1,𝔽₅)	—	5	6	no	maximal
4	D₅	—	5	6	no	—
5	C₅	AGL(1,𝔽₅)	1	6	yes	complemented
6	C₄	—	25	1	no	Sylow
7	C₂	—	25	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²	C₄
prime radical, cosocle	C₅⋊D₅	C₂
LCS	G ▷ C₅²
UCS	C₁

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

N	H ≅ G/N	type
2. C₅²	C₄	semidirect
5. C₅	AGL(1,𝔽₅)	semidirect

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

subgroup	index
C₅⋊D₅	2
AGL(1,𝔽₅)	5