Finite Group Databasebeta

← C₁₁×C₅₃×C₁₁⋊C₅ #44545461

Subgroups — C₁₁×C₅₃×C₁₁⋊C₅ #44545461

Order 32065.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁²×C₅₃	C₅	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×C₁₁×C₅₃	—	11	1	no	maximal
3	C₅₃×C₁₁⋊C₅	C₁₁	1	1	yes	maximal, characteristic, direct product factor
4	C₁₁×C₁₁⋊C₅	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₁₁×C₅₃	—	5	2	no	—
6	C₁₁×C₅₃	C₁₁⋊C₅	1	1	yes	characteristic, direct product factor
7	C₁₁×C₅₃	C₅×C₁₁	1	1	yes	characteristic, complemented
8	C₅×C₅₃	—	11	1	no	Hall
9	C₁₁²	C₅×C₅₃	1	1	yes	characteristic, complemented, Sylow
10	C₅×C₁₁	—	11	1	no	—
11	C₁₁⋊C₅	C₁₁×C₅₃	1	1	yes	characteristic, direct product factor
12	C₅₃	C₁₁×C₁₁⋊C₅	1	1	yes	characteristic, direct product factor, Sylow
13	C₁₁	—	5	2	no	—
14	C₁₁	C₅₃×C₁₁⋊C₅	1	1	yes	characteristic, direct product factor
15	C₁₁	C₅×C₁₁×C₅₃	1	1	yes	characteristic, complemented
16	C₅	—	11	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₁×C₅₃	C₁₁⋊C₅
commutator, cosocle, hypocenter	C₁₁	C₅×C₁₁×C₅₃
Frattini	C₁	G
Fitting, socle	C₁₁²×C₅₃	C₅
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₁₁×C₅₃

Complements

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

N	H ≅ G/N	type
1. C₁₁²×C₅₃	C₅	semidirect
3. C₅₃×C₁₁⋊C₅	C₁₁	direct (N × H)
4. C₁₁×C₁₁⋊C₅	C₅₃	direct (N × H)
6. C₁₁×C₅₃	C₁₁⋊C₅	direct (N × H)
7. C₁₁×C₅₃	C₅×C₁₁	semidirect
9. C₁₁²	C₅×C₅₃	semidirect
11. C₁₁⋊C₅	C₁₁×C₅₃	direct (N × H)
12. C₅₃	C₁₁×C₁₁⋊C₅	direct (N × H)
14. C₁₁	C₅₃×C₁₁⋊C₅	direct (N × H)
15. C₁₁	C₅×C₁₁×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₁²×C₅₃	5
C₅×C₁₁×C₅₃	11
C₅₃×C₁₁⋊C₅	11
C₁₁×C₁₁⋊C₅	53