Finite Group Databasebeta

← (C₅⋊D₅)⋊C₃×C₂₁₁ #44539555

Subgroups — (C₅⋊D₅)⋊C₃×C₂₁₁ #44539555

Order 31650.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²⋊C₃×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅⋊D₅×C₂₁₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅²×C₂₁₁	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	D₅×C₂₁₁	—	15	2	no	—
5	C₂×C₃×C₂₁₁	—	25	1	no	maximal, Hall
6	C₅×C₂₁₁	—	3	2	no	—
7	C₃×C₂₁₁	—	25	1	no	Hall
8	C₂×C₂₁₁	—	25	1	no	Hall
9	C₂₁₁	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
10	(C₅⋊D₅)⋊C₃	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₅²⋊C₃	C₂×C₂₁₁	1	1	yes	characteristic, complemented, Hall
12	C₅⋊D₅	C₃×C₂₁₁	1	1	yes	characteristic, complemented, Hall
13	C₅²	C₂×C₃×C₂₁₁	1	1	yes	characteristic, complemented, Sylow
14	D₅	—	15	2	no	—
15	C₂×C₃	—	25	1	no	Hall
16	C₅	—	3	2	no	—
17	C₃	—	25	1	no	Sylow
18	C₂	—	25	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₂₁₁	(C₅⋊D₅)⋊C₃
commutator, cosocle, hypocenter	C₅²	C₂×C₃×C₂₁₁
Frattini	C₁	G
Fitting, socle	C₅²×C₂₁₁	C₂×C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂₁₁

Complements

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

N	H ≅ G/N	type
1. C₅²⋊C₃×C₂₁₁	C₂	semidirect
2. C₅⋊D₅×C₂₁₁	C₃	semidirect
3. C₅²×C₂₁₁	C₂×C₃	semidirect
9. C₂₁₁	(C₅⋊D₅)⋊C₃	direct (N × H)
10. (C₅⋊D₅)⋊C₃	C₂₁₁	direct (N × H)
11. C₅²⋊C₃	C₂×C₂₁₁	semidirect
12. C₅⋊D₅	C₃×C₂₁₁	semidirect
13. C₅²	C₂×C₃×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₅²⋊C₃×C₂₁₁	2
C₅⋊D₅×C₂₁₁	3
C₂×C₃×C₂₁₁	25
(C₅⋊D₅)⋊C₃	211