Subgroups — C₁₁×C₂₃⋊AGL(1,𝔽₁₁) #42123712

Order 27830.

34 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×C₂₃×C₁₁⋊C₅	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₁×D₂₅₃	C₅	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₁²×C₂₃	C₂×C₅	1	1	yes	characteristic, complemented, Hall
4	C₂₃⋊AGL(1,𝔽₁₁)	C₁₁	1	1	yes	maximal, characteristic, direct product factor
5	C₅×C₁₁×D₂₃	—	11	1	no	maximal
6	C₅×C₁₁×C₂₃	—	11	1	no	—
7	C₂₃×C₁₁⋊C₅	C₂×C₁₁	1	1	yes	characteristic, complemented
8	C₁₁×AGL(1,𝔽₁₁)	—	23	1	no	maximal, Hall
9	C₁₁×C₁₁⋊C₅	D₂₃	1	1	yes	characteristic, complemented, Hall
10	D₂₅₃	C₅×C₁₁	1	1	yes	characteristic, complemented
11	C₁₁×D₂₃	—	55	1	no	—
12	C₁₁×C₂₃	—	10	1	no	—
13	C₁₁×C₂₃	AGL(1,𝔽₁₁)	1	1	yes	characteristic, complemented
14	C₁₁×C₂₃	C₂×C₅×C₁₁	1	1	yes	characteristic, complemented
15	C₁₁×D₁₁	—	23	1	no	Hall
16	C₅×D₂₃	—	11	1	no	Hall
17	C₁₁²	C₅×D₂₃	1	1	yes	characteristic, complemented, Sylow
18	C₅×C₂₃	—	11	1	no	Hall
19	C₂×C₅×C₁₁	—	253	1	no	—
20	AGL(1,𝔽₁₁)	—	23	1	no	—
21	C₅×C₁₁	—	11	1	no	—
22	C₁₁⋊C₅	C₁₁×D₂₃	1	1	yes	characteristic, complemented
23	D₂₃	—	11	1	no	Hall
24	C₂₃	C₁₁×AGL(1,𝔽₁₁)	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₁₁	—	1265	1	no	—
26	D₁₁	—	23	1	no	—
27	C₁₁	—	10	1	no	—
28	C₁₁	C₅×C₁₁×D₂₃	1	1	yes	characteristic, complemented
29	C₁₁	C₂₃⋊AGL(1,𝔽₁₁)	1	1	yes	characteristic, direct product factor
30	C₂×C₅	—	253	1	no	Hall
31	C₅	—	11	1	no	Sylow
32	C₂	—	1265	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₁	C₂₃⋊AGL(1,𝔽₁₁)
commutator, cosocle, hypocenter	C₁₁×C₂₃	C₂×C₅×C₁₁
Frattini	C₁	G
Fitting, socle	C₁₁²×C₂₃	C₂×C₅
LCS	G ▷ C₁₁×C₂₃
UCS	C₁ ◁ C₁₁

Complements

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

N	H ≅ G/N	type
1. C₁₁×C₂₃×C₁₁⋊C₅	C₂	semidirect
2. C₁₁×D₂₅₃	C₅	semidirect
3. C₁₁²×C₂₃	C₂×C₅	semidirect
4. C₂₃⋊AGL(1,𝔽₁₁)	C₁₁	direct (N × H)
7. C₂₃×C₁₁⋊C₅	C₂×C₁₁	semidirect
9. C₁₁×C₁₁⋊C₅	D₂₃	semidirect
10. D₂₅₃	C₅×C₁₁	semidirect
13. C₁₁×C₂₃	AGL(1,𝔽₁₁)	semidirect
14. C₁₁×C₂₃	C₂×C₅×C₁₁	semidirect
17. C₁₁²	C₅×D₂₃	semidirect
22. C₁₁⋊C₅	C₁₁×D₂₃	semidirect
24. C₂₃	C₁₁×AGL(1,𝔽₁₁)	semidirect
28. C₁₁	C₅×C₁₁×D₂₃	semidirect
29. C₁₁	C₂₃⋊AGL(1,𝔽₁₁)	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×C₂₃×C₁₁⋊C₅	2
C₁₁×D₂₅₃	5
C₂₃⋊AGL(1,𝔽₁₁)	11
C₅×C₁₁×D₂₃	11
C₁₁×AGL(1,𝔽₁₁)	23

Subgroups — C11×C23⋊AGL(1,𝔽11) #42123712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×C₂₃⋊AGL(1,𝔽₁₁) #42123712