Subgroups — #92563

Order 500.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic
2	C₅³	C₄	1	1	yes	characteristic, complemented, Sylow
3	#1044	—	5	6	no	maximal
4	#1043	—	5	1	no	maximal
5	C₅⋊D₅	—	5	1	no	—
6	C₅⋊D₅	—	5	6	no	—
7	C₅⋊D₅	—	10	12	no	—
8	C₅²	—	2	12	no	—
9	C₅²	AGL(1,𝔽₅)	1	6	yes	complemented
10	C₅²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
11	AGL(1,𝔽₅)	—	25	6	no	—
12	AGL(1,𝔽₅)	—	25	1	no	—
13	D₅	—	25	1	no	—
14	D₅	—	25	6	no	—
15	D₅	—	50	12	no	—
16	C₅	—	2	12	no	—
17	C₅	#1043	1	1	yes	characteristic, complemented
18	C₅	#1044	1	6	yes	complemented
19	C₄	—	125	1	no	Sylow
20	C₂	—	125	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅³	C₄
prime radical, cosocle	C₅⋊(C₅⋊D₅)	C₂
LCS	G ▷ C₅³
UCS	C₁

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

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