Subgroups — C₃²×C₅⋊AGL(1,𝔽₂₃) #36384751

Order 22770.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₂₃⋊C₁₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅⋊AGL(1,𝔽₂₃)	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₅⋊AGL(1,𝔽₂₃)	C₃	1	3	yes	maximal, direct product factor
4	C₃²×AGL(1,𝔽₂₃)	—	5	1	no	maximal, Hall
5	C₃×C₅×C₂₃⋊C₁₁	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃×C₅×C₂₃⋊C₁₁	C₂×C₃	1	3	yes	complemented
7	C₅⋊AGL(1,𝔽₂₃)	C₃²	1	1	yes	characteristic, direct product factor, Hall
8	C₃²×C₂₃⋊C₁₁	D₅	1	1	yes	characteristic, complemented, Hall
9	C₃²×D₁₁₅	C₁₁	1	1	yes	maximal, characteristic, complemented, Hall
10	C₃×AGL(1,𝔽₂₃)	—	5	1	no	—
11	C₃×AGL(1,𝔽₂₃)	—	5	3	no	—
12	C₅×C₂₃⋊C₁₁	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
13	C₃²×C₅×C₂₃	C₂×C₁₁	1	1	yes	characteristic, complemented, Hall
14	C₃²×D₅×C₁₁	—	23	1	no	maximal, Hall
15	C₃×C₂₃⋊C₁₁	C₃×D₅	1	1	yes	characteristic, complemented
16	C₃×C₂₃⋊C₁₁	C₃×D₅	1	3	yes	complemented
17	C₃×D₁₁₅	C₃×C₁₁	1	1	yes	characteristic, complemented
18	C₃×D₁₁₅	C₃×C₁₁	1	3	yes	complemented
19	AGL(1,𝔽₂₃)	—	5	1	no	Hall
20	C₃²×C₅×C₁₁	—	23	1	no	Hall
21	C₃²×D₂₃	—	5	1	no	Hall
22	C₃×C₅×C₂₃	C₂×C₃×C₁₁	1	1	yes	characteristic, complemented
23	C₃×C₅×C₂₃	C₂×C₃×C₁₁	1	3	yes	complemented
24	C₃×D₅×C₁₁	—	23	1	no	—
25	C₃×D₅×C₁₁	—	23	3	no	—
26	C₂₃⋊C₁₁	C₃²×D₅	1	1	yes	characteristic, complemented, Hall
27	D₁₁₅	C₃²×C₁₁	1	1	yes	characteristic, complemented, Hall
28	C₃²×C₂₃	D₅×C₁₁	1	1	yes	characteristic, complemented, Hall
29	C₂×C₃²×C₁₁	—	115	1	no	Hall
30	C₃×C₅×C₁₁	—	23	1	no	—
31	C₃×C₅×C₁₁	—	23	3	no	—
32	C₃×D₂₃	—	5	1	no	—
33	C₃×D₂₃	—	5	3	no	—
34	C₅×C₂₃	C₂×C₃²×C₁₁	1	1	yes	characteristic, complemented, Hall
35	D₅×C₁₁	—	23	1	no	Hall
36	C₃²×C₁₁	—	23	1	no	Hall
37	C₃²×D₅	—	253	1	no	Hall
38	C₃×C₂₃	C₃×D₅×C₁₁	1	1	yes	characteristic, complemented
39	C₃×C₂₃	C₃×D₅×C₁₁	1	3	yes	complemented
40	C₂×C₃×C₁₁	—	115	1	no	—
41	C₂×C₃×C₁₁	—	115	3	no	—
42	C₅×C₁₁	—	23	1	no	Hall
43	D₂₃	—	5	1	no	Hall
44	C₃²×C₅	AGL(1,𝔽₂₃)	1	1	yes	characteristic, complemented, Hall
45	C₃×C₁₁	—	23	1	no	—
46	C₃×C₁₁	—	23	3	no	—
47	C₃×D₅	—	253	1	no	—
48	C₃×D₅	—	253	3	no	—
49	C₂₃	C₃²×D₅×C₁₁	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₁₁	—	115	1	no	Hall
51	C₂×C₃²	—	1265	1	no	Hall
52	C₃×C₅	C₃×AGL(1,𝔽₂₃)	1	1	yes	characteristic, complemented
53	C₃×C₅	C₃×AGL(1,𝔽₂₃)	1	3	yes	complemented
54	C₁₁	—	23	1	no	Sylow
55	D₅	—	253	1	no	Hall
56	C₃²	C₅⋊AGL(1,𝔽₂₃)	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×C₃	—	115	1	no	—
58	C₂×C₃	—	1265	3	no	—
59	C₅	C₃²×AGL(1,𝔽₂₃)	1	1	yes	characteristic, complemented, Sylow
60	C₃	C₃×C₅⋊AGL(1,𝔽₂₃)	1	1	yes	characteristic, direct product factor
61	C₃	C₃×C₅⋊AGL(1,𝔽₂₃)	1	3	yes	direct product factor
62	C₂	—	1265	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃²	C₅⋊AGL(1,𝔽₂₃)
commutator, cosocle, hypocenter	C₅×C₂₃	C₂×C₃²×C₁₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₂₃	C₂×C₁₁
LCS	G ▷ C₅×C₂₃
UCS	C₁ ◁ C₃²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×C₂₃⋊C₁₁	C₂	semidirect
2. C₃×C₅⋊AGL(1,𝔽₂₃)	C₃	direct (N × H)
3. C₃×C₅⋊AGL(1,𝔽₂₃)	C₃	direct (N × H)
5. C₃×C₅×C₂₃⋊C₁₁	C₂×C₃	semidirect
6. C₃×C₅×C₂₃⋊C₁₁	C₂×C₃	semidirect
7. C₅⋊AGL(1,𝔽₂₃)	C₃²	direct (N × H)
8. C₃²×C₂₃⋊C₁₁	D₅	semidirect
9. C₃²×D₁₁₅	C₁₁	semidirect
12. C₅×C₂₃⋊C₁₁	C₂×C₃²	semidirect
13. C₃²×C₅×C₂₃	C₂×C₁₁	semidirect
15. C₃×C₂₃⋊C₁₁	C₃×D₅	semidirect
16. C₃×C₂₃⋊C₁₁	C₃×D₅	semidirect
17. C₃×D₁₁₅	C₃×C₁₁	semidirect
18. C₃×D₁₁₅	C₃×C₁₁	semidirect
22. C₃×C₅×C₂₃	C₂×C₃×C₁₁	semidirect
23. C₃×C₅×C₂₃	C₂×C₃×C₁₁	semidirect
26. C₂₃⋊C₁₁	C₃²×D₅	semidirect
27. D₁₁₅	C₃²×C₁₁	semidirect
28. C₃²×C₂₃	D₅×C₁₁	semidirect
34. C₅×C₂₃	C₂×C₃²×C₁₁	semidirect
38. C₃×C₂₃	C₃×D₅×C₁₁	semidirect
39. C₃×C₂₃	C₃×D₅×C₁₁	semidirect
44. C₃²×C₅	AGL(1,𝔽₂₃)	semidirect
49. C₂₃	C₃²×D₅×C₁₁	semidirect
52. C₃×C₅	C₃×AGL(1,𝔽₂₃)	semidirect
53. C₃×C₅	C₃×AGL(1,𝔽₂₃)	semidirect
56. C₃²	C₅⋊AGL(1,𝔽₂₃)	direct (N × H)
59. C₅	C₃²×AGL(1,𝔽₂₃)	semidirect
60. C₃	C₃×C₅⋊AGL(1,𝔽₂₃)	direct (N × H)
61. C₃	C₃×C₅⋊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
C₃²×C₅×C₂₃⋊C₁₁	2
C₃×C₅⋊AGL(1,𝔽₂₃)	3
C₃²×AGL(1,𝔽₂₃)	5
C₃²×D₁₁₅	11
C₃²×D₅×C₁₁	23

Subgroups — C32×C5⋊AGL(1,𝔽23) #36384751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅⋊AGL(1,𝔽₂₃) #36384751