Subgroups — #6034

Order 200.

22 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1039	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²	C₄	1	1	yes	characteristic
3	#170	—	5	2	no	maximal
4	C₅²	C₈	1	1	yes	characteristic, complemented, Sylow
5	Dic₅	—	5	2	no	—
6	Dic₅	—	10	2	no	—
7	C₂×C₅	—	2	2	no	—
8	C₂×C₅	AGL(1,𝔽₅)	1	2	yes	—
9	C₈	—	25	1	no	Sylow
10	C₅	—	2	2	no	—
11	C₅	#170	1	2	yes	complemented
12	C₄	—	25	1	no	—
13	C₂	#1044	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	#1044
commutator, hypocenter	C₅²	C₈
Fitting, socle, prime radical	C₂×C₅²	C₄
cosocle	#1039	C₂
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂

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

N	H ≅ G/N	type
4. C₅²	C₈	semidirect
11. C₅	#170	semidirect

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

subgroup	index
#1039	2
#170	5