Subgroups — C₅×C₁₁×AGL(1,𝔽₂₃) #42123700

Order 27830.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₁₁×C₂₃⋊C₁₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₁×AGL(1,𝔽₂₃)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₁₁×C₂₃⋊C₁₁	C₂×C₅	1	1	yes	characteristic, complemented, Hall
4	C₅×AGL(1,𝔽₂₃)	C₁₁	1	11	yes	maximal, direct product factor
5	C₅×C₁₁×D₂₃	C₁₁	1	1	yes	maximal, characteristic, complemented
6	C₅×C₁₁×C₂₃	C₂×C₁₁	1	1	yes	characteristic, complemented
7	C₅×C₂₃⋊C₁₁	C₂×C₁₁	1	11	yes	complemented
8	C₂×C₅×C₁₁²	—	23	1	no	maximal, Hall
9	C₅×C₁₁²	—	23	1	no	Hall
10	C₁₁×D₂₃	C₅×C₁₁	1	1	yes	characteristic, complemented
11	AGL(1,𝔽₂₃)	C₅×C₁₁	1	11	yes	direct product factor
12	C₁₁×C₂₃	C₂×C₅×C₁₁	1	1	yes	characteristic, complemented
13	C₂₃⋊C₁₁	C₂×C₅×C₁₁	1	11	yes	complemented
14	C₂×C₁₁²	—	23	1	no	Hall
15	C₅×D₂₃	C₁₁²	1	1	yes	characteristic, complemented, Hall
16	C₁₁²	—	23	1	no	Sylow
17	C₅×C₂₃	C₂×C₁₁²	1	1	yes	characteristic, complemented, Hall
18	C₂×C₅×C₁₁	—	23	11	no	—
19	C₂×C₅×C₁₁	—	253	1	no	—
20	C₅×C₁₁	—	23	11	no	—
21	C₅×C₁₁	AGL(1,𝔽₂₃)	1	1	yes	characteristic, direct product factor
22	D₂₃	C₅×C₁₁²	1	1	yes	characteristic, complemented, Hall
23	C₂₃	C₂×C₅×C₁₁²	1	1	yes	characteristic, complemented, Sylow
24	C₂×C₁₁	—	23	11	no	—
25	C₂×C₁₁	—	253	1	no	—
26	C₁₁	—	23	11	no	—
27	C₁₁	C₅×AGL(1,𝔽₂₃)	1	1	yes	characteristic, direct product factor
28	C₂×C₅	—	253	1	no	Hall
29	C₅	C₁₁×AGL(1,𝔽₂₃)	1	1	yes	characteristic, direct product factor, Sylow
30	C₂	—	253	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₁₁	AGL(1,𝔽₂₃)
commutator, cosocle, hypocenter	C₂₃	C₂×C₅×C₁₁²
Frattini	C₁	G
Fitting, socle	C₅×C₁₁×C₂₃	C₂×C₁₁
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₅×C₁₁

Complements

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

N	H ≅ G/N	type
1. C₅×C₁₁×C₂₃⋊C₁₁	C₂	semidirect
2. C₁₁×AGL(1,𝔽₂₃)	C₅	direct (N × H)
3. C₁₁×C₂₃⋊C₁₁	C₂×C₅	semidirect
4. C₅×AGL(1,𝔽₂₃)	C₁₁	direct (N × H)
5. C₅×C₁₁×D₂₃	C₁₁	semidirect
6. C₅×C₁₁×C₂₃	C₂×C₁₁	semidirect
7. C₅×C₂₃⋊C₁₁	C₂×C₁₁	semidirect
10. C₁₁×D₂₃	C₅×C₁₁	semidirect
11. AGL(1,𝔽₂₃)	C₅×C₁₁	direct (N × H)
12. C₁₁×C₂₃	C₂×C₅×C₁₁	semidirect
13. C₂₃⋊C₁₁	C₂×C₅×C₁₁	semidirect
15. C₅×D₂₃	C₁₁²	semidirect
17. C₅×C₂₃	C₂×C₁₁²	semidirect
21. C₅×C₁₁	AGL(1,𝔽₂₃)	direct (N × H)
22. D₂₃	C₅×C₁₁²	semidirect
23. C₂₃	C₂×C₅×C₁₁²	semidirect
27. C₁₁	C₅×AGL(1,𝔽₂₃)	direct (N × H)
29. C₅	C₁₁×AGL(1,𝔽₂₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₁₁×C₂₃⋊C₁₁	2
C₁₁×AGL(1,𝔽₂₃)	5
C₅×AGL(1,𝔽₂₃)	11
C₅×C₁₁×D₂₃	11
C₂×C₅×C₁₁²	23

Subgroups — C5×C11×AGL(1,𝔽23) #42123700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₁₁×AGL(1,𝔽₂₃) #42123700