Subgroups — #1044

Order 100.

16 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	2	no	maximal
4	D₅	—	5	2	no	—
5	D₅	—	10	2	no	—
6	C₅	—	2	2	no	—
7	C₅	AGL(1,𝔽₅)	1	2	yes	complemented
8	C₄	—	25	1	no	Sylow
9	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
7. 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