Subgroups — C₃×C₅²×AGL(1,𝔽₂₃) #49182103

Order 37950.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×C₂₃⋊C₁₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅²×AGL(1,𝔽₂₃)	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅×AGL(1,𝔽₂₃)	C₅	1	6	yes	maximal, direct product factor
4	C₅²×C₂₃⋊C₁₁	C₂×C₃	1	1	yes	characteristic, complemented, Hall
5	C₃×C₅×C₂₃⋊C₁₁	C₂×C₅	1	6	yes	complemented
6	C₃×C₅²×D₂₃	C₁₁	1	1	yes	maximal, characteristic, complemented, Hall
7	C₅×AGL(1,𝔽₂₃)	C₃×C₅	1	6	yes	direct product factor
8	C₃×C₅²×C₂₃	C₂×C₁₁	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃×C₅²×C₁₁	—	23	1	no	maximal, Hall
10	C₃×AGL(1,𝔽₂₃)	C₅²	1	1	yes	characteristic, direct product factor, Hall
11	C₅×C₂₃⋊C₁₁	C₂×C₃×C₅	1	6	yes	complemented
12	C₅²×D₂₃	C₃×C₁₁	1	1	yes	characteristic, complemented, Hall
13	C₃×C₅²×C₁₁	—	23	1	no	Hall
14	C₃×C₂₃⋊C₁₁	C₂×C₅²	1	1	yes	characteristic, complemented, Hall
15	C₃×C₅×D₂₃	C₅×C₁₁	1	6	yes	complemented
16	C₅²×C₂₃	C₂×C₃×C₁₁	1	1	yes	characteristic, complemented, Hall
17	C₂×C₅²×C₁₁	—	23	1	no	Hall
18	AGL(1,𝔽₂₃)	C₃×C₅²	1	1	yes	characteristic, direct product factor, Hall
19	C₃×C₅×C₂₃	C₂×C₅×C₁₁	1	6	yes	complemented
20	C₂×C₃×C₅×C₁₁	—	23	6	no	—
21	C₅²×C₁₁	—	23	1	no	Hall
22	C₂₃⋊C₁₁	C₂×C₃×C₅²	1	1	yes	characteristic, complemented, Hall
23	C₅×D₂₃	C₃×C₅×C₁₁	1	6	yes	complemented
24	C₃×C₅×C₁₁	—	23	6	no	—
25	C₂×C₃×C₅²	—	23	1	no	Hall
26	C₃×D₂₃	C₅²×C₁₁	1	1	yes	characteristic, complemented, Hall
27	C₅×C₂₃	C₂×C₃×C₅×C₁₁	1	6	yes	complemented
28	C₂×C₅×C₁₁	—	23	6	no	—
29	C₃×C₅²	AGL(1,𝔽₂₃)	1	1	yes	characteristic, direct product factor, Hall
30	C₃×C₂₃	C₂×C₅²×C₁₁	1	1	yes	characteristic, complemented, Hall
31	C₂×C₃×C₁₁	—	23	1	no	Hall
32	C₅×C₁₁	—	23	6	no	—
33	C₂×C₅²	—	23	1	no	Hall
34	D₂₃	C₃×C₅²×C₁₁	1	1	yes	characteristic, complemented, Hall
35	C₃×C₁₁	—	23	1	no	Hall
36	C₂×C₃×C₅	—	23	6	no	—
37	C₅²	C₃×AGL(1,𝔽₂₃)	1	1	yes	characteristic, direct product factor, Sylow
38	C₂₃	C₂×C₃×C₅²×C₁₁	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₁₁	—	23	1	no	Hall
40	C₃×C₅	C₅×AGL(1,𝔽₂₃)	1	6	yes	direct product factor
41	C₁₁	—	23	1	no	Sylow
42	C₂×C₅	—	23	6	no	—
43	C₂×C₃	—	23	1	no	Hall
44	C₅	C₃×C₅×AGL(1,𝔽₂₃)	1	6	yes	direct product factor
45	C₃	C₅²×AGL(1,𝔽₂₃)	1	1	yes	characteristic, direct product factor, Sylow
46	C₂	—	23	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅²	AGL(1,𝔽₂₃)
commutator, cosocle, hypocenter	C₂₃	C₂×C₃×C₅²×C₁₁
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₂₃	C₂×C₁₁
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₃×C₅²

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×C₂₃⋊C₁₁	C₂	semidirect
2. C₅²×AGL(1,𝔽₂₃)	C₃	direct (N × H)
3. C₃×C₅×AGL(1,𝔽₂₃)	C₅	direct (N × H)
4. C₅²×C₂₃⋊C₁₁	C₂×C₃	semidirect
5. C₃×C₅×C₂₃⋊C₁₁	C₂×C₅	semidirect
6. C₃×C₅²×D₂₃	C₁₁	semidirect
7. C₅×AGL(1,𝔽₂₃)	C₃×C₅	direct (N × H)
8. C₃×C₅²×C₂₃	C₂×C₁₁	semidirect
10. C₃×AGL(1,𝔽₂₃)	C₅²	direct (N × H)
11. C₅×C₂₃⋊C₁₁	C₂×C₃×C₅	semidirect
12. C₅²×D₂₃	C₃×C₁₁	semidirect
14. C₃×C₂₃⋊C₁₁	C₂×C₅²	semidirect
15. C₃×C₅×D₂₃	C₅×C₁₁	semidirect
16. C₅²×C₂₃	C₂×C₃×C₁₁	semidirect
18. AGL(1,𝔽₂₃)	C₃×C₅²	direct (N × H)
19. C₃×C₅×C₂₃	C₂×C₅×C₁₁	semidirect
22. C₂₃⋊C₁₁	C₂×C₃×C₅²	semidirect
23. C₅×D₂₃	C₃×C₅×C₁₁	semidirect
26. C₃×D₂₃	C₅²×C₁₁	semidirect
27. C₅×C₂₃	C₂×C₃×C₅×C₁₁	semidirect
29. C₃×C₅²	AGL(1,𝔽₂₃)	direct (N × H)
30. C₃×C₂₃	C₂×C₅²×C₁₁	semidirect
34. D₂₃	C₃×C₅²×C₁₁	semidirect
37. C₅²	C₃×AGL(1,𝔽₂₃)	direct (N × H)
38. C₂₃	C₂×C₃×C₅²×C₁₁	semidirect
40. C₃×C₅	C₅×AGL(1,𝔽₂₃)	direct (N × H)
44. C₅	C₃×C₅×AGL(1,𝔽₂₃)	direct (N × H)
45. C₃	C₅²×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
C₃×C₅²×C₂₃⋊C₁₁	2
C₅²×AGL(1,𝔽₂₃)	3
C₃×C₅×AGL(1,𝔽₂₃)	5
C₃×C₅²×D₂₃	11
C₂×C₃×C₅²×C₁₁	23

Subgroups — C3×C52×AGL(1,𝔽23) #49182103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×AGL(1,𝔽₂₃) #49182103