Finite Group Databasebeta

Subgroups — #24926271

Order 10900.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊D₅₄₅	C₂	1	1	yes	maximal, characteristic
2	C₅²×C₁₀₉	C₄	1	1	yes	characteristic, complemented, Hall
3	#14546127	—	5	6	no	maximal
4	D₅₄₅	—	5	6	no	—
5	C₅×C₁₀₉	AGL(1,𝔽₅)	1	6	yes	complemented
6	#89137	—	25	1	no	Hall
7	D₁₀₉	—	25	1	no	—
8	C₁₀₉	#1043	1	1	yes	characteristic, complemented, Sylow
9	#1043	—	109	1	no	maximal, Hall
10	C₅⋊D₅	—	109	1	no	—
11	C₅²	#89137	1	1	yes	characteristic, complemented, Sylow
12	AGL(1,𝔽₅)	—	545	6	no	—
13	D₅	—	545	6	no	—
14	C₅	#14546127	1	6	yes	complemented
15	C₄	—	2725	1	no	Sylow
16	C₂	—	2725	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₁₀₉	C₄
prime radical, cosocle	C₅⋊D₅₄₅	C₂
LCS	G ▷ C₅²×C₁₀₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₅²×C₁₀₉	C₄	semidirect
5. C₅×C₁₀₉	AGL(1,𝔽₅)	semidirect
8. C₁₀₉	#1043	semidirect
11. C₅²	#89137	semidirect
14. C₅	#14546127	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊D₅₄₅	2
#14546127	5
#1043	109