Subgroups — C₅×C₁₁⋊AGL(1,𝔽₂₃) #42123701

Order 27830.

52 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)	—	11	1	no	maximal
5	C₅×D₂₅₃	C₁₁	1	1	yes	maximal, characteristic, complemented
6	C₅×C₁₁×C₂₃	C₂×C₁₁	1	1	yes	characteristic, complemented
7	C₅×C₂₃⋊C₁₁	—	2	5	no	—
8	C₅×C₂₃⋊C₁₁	D₁₁	1	1	yes	characteristic, complemented
9	C₅×C₁₁×D₁₁	—	23	1	no	maximal, Hall
10	C₅×C₁₁²	—	23	1	no	Hall
11	D₂₅₃	C₅×C₁₁	1	1	yes	characteristic, complemented
12	AGL(1,𝔽₂₃)	—	11	1	no	—
13	C₁₁×C₂₃	C₂×C₅×C₁₁	1	1	yes	characteristic, complemented
14	C₂₃⋊C₁₁	—	2	5	no	—
15	C₂₃⋊C₁₁	C₅×D₁₁	1	1	yes	characteristic, complemented
16	C₁₁×D₁₁	—	23	1	no	Hall
17	C₅×D₂₃	—	11	1	no	Hall
18	C₁₁²	—	23	1	no	Sylow
19	C₅×C₂₃	C₁₁×D₁₁	1	1	yes	characteristic, complemented, Hall
20	C₂×C₅×C₁₁	—	253	1	no	—
21	C₅×D₁₁	—	23	1	no	—
22	C₅×C₁₁	—	23	1	no	—
23	C₅×C₁₁	—	46	5	no	—
24	C₅×C₁₁	AGL(1,𝔽₂₃)	1	1	yes	characteristic, complemented
25	D₂₃	—	11	1	no	Hall
26	C₂₃	C₅×C₁₁×D₁₁	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₁₁	—	253	1	no	—
28	D₁₁	—	23	1	no	—
29	C₁₁	—	23	1	no	—
30	C₁₁	—	46	5	no	—
31	C₁₁	C₅×AGL(1,𝔽₂₃)	1	1	yes	characteristic, complemented
32	C₂×C₅	—	253	1	no	Hall
33	C₅	C₁₁⋊AGL(1,𝔽₂₃)	1	1	yes	characteristic, direct product factor, Sylow
34	C₂	—	253	1	no	Sylow

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

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
5. C₅×D₂₅₃	C₁₁	semidirect
6. C₅×C₁₁×C₂₃	C₂×C₁₁	semidirect
8. C₅×C₂₃⋊C₁₁	D₁₁	semidirect
11. D₂₅₃	C₅×C₁₁	semidirect
13. C₁₁×C₂₃	C₂×C₅×C₁₁	semidirect
15. C₂₃⋊C₁₁	C₅×D₁₁	semidirect
19. C₅×C₂₃	C₁₁×D₁₁	semidirect
24. C₅×C₁₁	AGL(1,𝔽₂₃)	semidirect
26. C₂₃	C₅×C₁₁×D₁₁	semidirect
31. C₁₁	C₅×AGL(1,𝔽₂₃)	semidirect
33. 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×D₂₅₃	11
C₅×C₁₁×D₁₁	23

Subgroups — C5×C11⋊AGL(1,𝔽23) #42123701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₁₁⋊AGL(1,𝔽₂₃) #42123701