Finite Group Databasebeta

← C₃×C₁₁⋊C₅×C₂₂₃ #49145491

Subgroups — C₃×C₁₁⋊C₅×C₂₂₃ #49145491

Order 36795.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁⋊C₅×C₂₂₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₃×C₁₁×C₂₂₃	C₅	1	1	yes	maximal, characteristic, complemented, Hall
3	C₃×C₅×C₂₂₃	—	11	1	no	maximal, Hall
4	C₁₁×C₂₂₃	C₃×C₅	1	1	yes	characteristic, complemented, Hall
5	C₅×C₂₂₃	—	11	1	no	Hall
6	C₃×C₂₂₃	C₁₁⋊C₅	1	1	yes	characteristic, direct product factor, Hall
7	C₂₂₃	C₃×C₁₁⋊C₅	1	1	yes	characteristic, direct product factor, Sylow
8	C₃×C₁₁⋊C₅	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₁₁⋊C₅	C₃×C₂₂₃	1	1	yes	characteristic, direct product factor, Hall
10	C₃×C₁₁	C₅×C₂₂₃	1	1	yes	characteristic, complemented, Hall
11	C₃×C₅	—	11	1	no	Hall
12	C₁₁	C₃×C₅×C₂₂₃	1	1	yes	characteristic, complemented, Sylow
13	C₅	—	11	1	no	Sylow
14	C₃	C₁₁⋊C₅×C₂₂₃	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₃×C₂₂₃	C₁₁⋊C₅
commutator, cosocle, hypocenter	C₁₁	C₃×C₅×C₂₂₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₁×C₂₂₃	C₅
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₃×C₂₂₃

Complements

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

N	H ≅ G/N	type
1. C₁₁⋊C₅×C₂₂₃	C₃	direct (N × H)
2. C₃×C₁₁×C₂₂₃	C₅	semidirect
4. C₁₁×C₂₂₃	C₃×C₅	semidirect
6. C₃×C₂₂₃	C₁₁⋊C₅	direct (N × H)
7. C₂₂₃	C₃×C₁₁⋊C₅	direct (N × H)
8. C₃×C₁₁⋊C₅	C₂₂₃	direct (N × H)
9. C₁₁⋊C₅	C₃×C₂₂₃	direct (N × H)
10. C₃×C₁₁	C₅×C₂₂₃	semidirect
12. C₁₁	C₃×C₅×C₂₂₃	semidirect
14. C₃	C₁₁⋊C₅×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₁⋊C₅×C₂₂₃	3
C₃×C₁₁×C₂₂₃	5
C₃×C₅×C₂₂₃	11
C₃×C₁₁⋊C₅	223