Finite Group Databasebeta

← C₂²×5₋¹⁺² #92550

Subgroups — C₂²×5₋¹⁺² #92550

Order 500.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×5₋¹⁺²	C₂	1	3	yes	maximal, direct product factor
2	5₋¹⁺²	C₂²	1	1	yes	characteristic, direct product factor, Sylow
3	C₂²×C₅²	C₅	1	1	yes	maximal, characteristic
4	C₂²×C₂₅	C₅	1	5	yes	maximal, complemented
5	C₂×C₅²	C₂×C₅	1	3	yes	—
6	C₂×C₂₅	C₂×C₅	1	15	yes	complemented
7	C₅²	C₂²×C₅	1	1	yes	characteristic
8	C₂₅	C₂²×C₅	1	5	yes	complemented
9	C₂²×C₅	—	5	1	no	—
10	C₂²×C₅	C₅²	1	1	yes	characteristic
11	C₂×C₅	—	5	3	no	—
12	C₂×C₅	C₂×C₅²	1	3	yes	—
13	C₅	—	5	1	no	—
14	C₅	C₂²×C₅²	1	1	yes	characteristic
15	C₂²	5₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
16	C₂	C₂×5₋¹⁺²	1	3	yes	direct product factor

	subgroup	G/H
center, socle	C₂²×C₅	C₅²
commutator, Frattini, cosocle	C₅	C₂²×C₅²
prime radical	C₂²×C₅²	C₅
LCS	G ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×5₋¹⁺²	C₂	direct (N × H)
2. 5₋¹⁺²	C₂²	direct (N × H)
4. C₂²×C₂₅	C₅	semidirect
6. C₂×C₂₅	C₂×C₅	semidirect
8. C₂₅	C₂²×C₅	semidirect
15. C₂²	5₋¹⁺²	direct (N × H)
16. C₂	C₂×5₋¹⁺²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×5₋¹⁺²	2
C₂²×C₅²	5
C₂²×C₂₅	5