Subgroups — C₂×C₃₁×AGL(1,𝔽₂₃) #44535878

Order 31372.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×AGL(1,𝔽₂₃)	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃₁×C₂₃⋊C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×C₂₃⋊C₁₁	C₂²	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃₁×D₂₃	C₁₁	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂×C₂₃×C₃₁	C₂×C₁₁	1	1	yes	characteristic, complemented
6	C₃₁×D₂₃	C₂×C₁₁	1	2	yes	complemented
7	C₂²×C₁₁×C₃₁	—	23	1	no	maximal, Hall
8	C₂×AGL(1,𝔽₂₃)	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂₃×C₃₁	C₂²×C₁₁	1	1	yes	characteristic, complemented, Hall
10	C₂×C₁₁×C₃₁	—	23	2	no	—
11	C₂×C₁₁×C₃₁	—	23	1	no	—
12	C₂×C₂₃⋊C₁₁	C₂×C₃₁	1	1	yes	characteristic, complemented
13	AGL(1,𝔽₂₃)	C₂×C₃₁	1	2	yes	direct product factor
14	C₁₁×C₃₁	—	23	1	no	Hall
15	C₂₃⋊C₁₁	C₂²×C₃₁	1	1	yes	characteristic, complemented, Hall
16	C₂²×C₃₁	—	23	1	no	Hall
17	C₂×D₂₃	C₁₁×C₃₁	1	1	yes	characteristic, complemented, Hall
18	C₂×C₃₁	—	23	2	no	—
19	C₂×C₃₁	AGL(1,𝔽₂₃)	1	1	yes	characteristic, direct product factor
20	C₂×C₂₃	C₂×C₁₁×C₃₁	1	1	yes	characteristic, complemented
21	D₂₃	C₂×C₁₁×C₃₁	1	2	yes	complemented
22	C₂²×C₁₁	—	23	1	no	Hall
23	C₃₁	C₂×AGL(1,𝔽₂₃)	1	1	yes	characteristic, direct product factor, Sylow
24	C₂₃	C₂²×C₁₁×C₃₁	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₁₁	—	23	2	no	—
26	C₂×C₁₁	—	23	1	no	—
27	C₁₁	—	23	1	no	Sylow
28	C₂²	—	23	1	no	Sylow
29	C₂	—	23	2	no	—
30	C₂	C₃₁×AGL(1,𝔽₂₃)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃₁	AGL(1,𝔽₂₃)
commutator, cosocle, hypocenter	C₂₃	C₂²×C₁₁×C₃₁
Frattini	C₁	G
Fitting, socle	C₂×C₂₃×C₃₁	C₂×C₁₁
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₂×C₃₁

Complements

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

N	H ≅ G/N	type
1. C₃₁×AGL(1,𝔽₂₃)	C₂	direct (N × H)
2. C₂×C₃₁×C₂₃⋊C₁₁	C₂	semidirect
3. C₃₁×C₂₃⋊C₁₁	C₂²	semidirect
4. C₂×C₃₁×D₂₃	C₁₁	semidirect
5. C₂×C₂₃×C₃₁	C₂×C₁₁	semidirect
6. C₃₁×D₂₃	C₂×C₁₁	semidirect
8. C₂×AGL(1,𝔽₂₃)	C₃₁	direct (N × H)
9. C₂₃×C₃₁	C₂²×C₁₁	semidirect
12. C₂×C₂₃⋊C₁₁	C₂×C₃₁	semidirect
13. AGL(1,𝔽₂₃)	C₂×C₃₁	direct (N × H)
15. C₂₃⋊C₁₁	C₂²×C₃₁	semidirect
17. C₂×D₂₃	C₁₁×C₃₁	semidirect
19. C₂×C₃₁	AGL(1,𝔽₂₃)	direct (N × H)
20. C₂×C₂₃	C₂×C₁₁×C₃₁	semidirect
21. D₂₃	C₂×C₁₁×C₃₁	semidirect
23. C₃₁	C₂×AGL(1,𝔽₂₃)	direct (N × H)
24. C₂₃	C₂²×C₁₁×C₃₁	semidirect
30. C₂	C₃₁×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
C₃₁×AGL(1,𝔽₂₃)	2
C₂×C₃₁×C₂₃⋊C₁₁	2
C₂×C₃₁×D₂₃	11
C₂²×C₁₁×C₃₁	23
C₂×AGL(1,𝔽₂₃)	31

Subgroups — C2×C31×AGL(1,𝔽23) #44535878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×AGL(1,𝔽₂₃) #44535878