Subgroups — C₃²×C₅×C₁₃×D₂₃ #41002115

Order 26910.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₁₃×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅×C₁₃×D₂₃	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₁₃×D₂₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃×C₅×C₁₃×C₂₃	C₂×C₃	1	4	yes	complemented
5	C₅×C₁₃×D₂₃	C₃²	1	1	yes	characteristic, direct product factor, Hall
6	C₃²×C₁₃×C₂₃	C₂×C₅	1	1	yes	characteristic, complemented, Hall
7	C₃²×C₅×D₂₃	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃×C₁₃×D₂₃	C₃×C₅	1	4	yes	direct product factor
9	C₅×C₁₃×C₂₃	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
10	C₂×C₃²×C₅×C₁₃	—	23	1	no	maximal, Hall
11	C₃²×C₅×C₂₃	C₂×C₁₃	1	1	yes	characteristic, complemented, Hall
12	C₃×C₁₃×C₂₃	C₂×C₃×C₅	1	4	yes	complemented
13	C₃×C₅×D₂₃	C₃×C₁₃	1	4	yes	direct product factor
14	C₁₃×D₂₃	C₃²×C₅	1	1	yes	characteristic, direct product factor, Hall
15	C₃²×C₅×C₁₃	D₂₃	1	1	yes	characteristic, direct product factor, Hall
16	C₃²×D₂₃	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₃×C₅×C₁₃	—	23	4	no	—
18	C₃×C₅×C₂₃	C₂×C₃×C₁₃	1	4	yes	complemented
19	C₁₃×C₂₃	C₂×C₃²×C₅	1	1	yes	characteristic, complemented, Hall
20	C₂×C₃²×C₁₃	—	23	1	no	Hall
21	C₅×D₂₃	C₃²×C₁₃	1	1	yes	characteristic, direct product factor, Hall
22	C₃²×C₂₃	C₂×C₅×C₁₃	1	1	yes	characteristic, complemented, Hall
23	C₃×C₅×C₁₃	C₃×D₂₃	1	4	yes	direct product factor
24	C₃×D₂₃	C₃×C₅×C₁₃	1	4	yes	direct product factor
25	C₂×C₅×C₁₃	—	23	1	no	Hall
26	C₃²×C₁₃	C₅×D₂₃	1	1	yes	characteristic, direct product factor, Hall
27	C₅×C₂₃	C₂×C₃²×C₁₃	1	1	yes	characteristic, complemented, Hall
28	C₂×C₃²×C₅	—	23	1	no	Hall
29	C₂×C₃×C₁₃	—	23	4	no	—
30	C₃×C₂₃	C₂×C₃×C₅×C₁₃	1	4	yes	complemented
31	C₅×C₁₃	C₃²×D₂₃	1	1	yes	characteristic, direct product factor, Hall
32	D₂₃	C₃²×C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
33	C₃²×C₅	C₁₃×D₂₃	1	1	yes	characteristic, direct product factor, Hall
34	C₃×C₁₃	C₃×C₅×D₂₃	1	4	yes	direct product factor
35	C₂×C₃×C₅	—	23	4	no	—
36	C₂×C₁₃	—	23	1	no	Hall
37	C₂₃	C₂×C₃²×C₅×C₁₃	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₃²	—	23	1	no	Hall
39	C₃×C₅	C₃×C₁₃×D₂₃	1	4	yes	direct product factor
40	C₁₃	C₃²×C₅×D₂₃	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×C₅	—	23	1	no	Hall
42	C₃²	C₅×C₁₃×D₂₃	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₃	—	23	4	no	—
44	C₅	C₃²×C₁₃×D₂₃	1	1	yes	characteristic, direct product factor, Sylow
45	C₃	C₃×C₅×C₁₃×D₂₃	1	4	yes	direct product factor
46	C₂	—	23	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃²×C₅×C₁₃	D₂₃
commutator, cosocle, hypocenter	C₂₃	C₂×C₃²×C₅×C₁₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₁₃×C₂₃	C₂
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₃²×C₅×C₁₃

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×C₁₃×C₂₃	C₂	semidirect
2. C₃×C₅×C₁₃×D₂₃	C₃	direct (N × H)
3. C₃²×C₁₃×D₂₃	C₅	direct (N × H)
4. C₃×C₅×C₁₃×C₂₃	C₂×C₃	semidirect
5. C₅×C₁₃×D₂₃	C₃²	direct (N × H)
6. C₃²×C₁₃×C₂₃	C₂×C₅	semidirect
7. C₃²×C₅×D₂₃	C₁₃	direct (N × H)
8. C₃×C₁₃×D₂₃	C₃×C₅	direct (N × H)
9. C₅×C₁₃×C₂₃	C₂×C₃²	semidirect
11. C₃²×C₅×C₂₃	C₂×C₁₃	semidirect
12. C₃×C₁₃×C₂₃	C₂×C₃×C₅	semidirect
13. C₃×C₅×D₂₃	C₃×C₁₃	direct (N × H)
14. C₁₃×D₂₃	C₃²×C₅	direct (N × H)
15. C₃²×C₅×C₁₃	D₂₃	direct (N × H)
16. C₃²×D₂₃	C₅×C₁₃	direct (N × H)
18. C₃×C₅×C₂₃	C₂×C₃×C₁₃	semidirect
19. C₁₃×C₂₃	C₂×C₃²×C₅	semidirect
21. C₅×D₂₃	C₃²×C₁₃	direct (N × H)
22. C₃²×C₂₃	C₂×C₅×C₁₃	semidirect
23. C₃×C₅×C₁₃	C₃×D₂₃	direct (N × H)
24. C₃×D₂₃	C₃×C₅×C₁₃	direct (N × H)
26. C₃²×C₁₃	C₅×D₂₃	direct (N × H)
27. C₅×C₂₃	C₂×C₃²×C₁₃	semidirect
30. C₃×C₂₃	C₂×C₃×C₅×C₁₃	semidirect
31. C₅×C₁₃	C₃²×D₂₃	direct (N × H)
32. D₂₃	C₃²×C₅×C₁₃	direct (N × H)
33. C₃²×C₅	C₁₃×D₂₃	direct (N × H)
34. C₃×C₁₃	C₃×C₅×D₂₃	direct (N × H)
37. C₂₃	C₂×C₃²×C₅×C₁₃	semidirect
39. C₃×C₅	C₃×C₁₃×D₂₃	direct (N × H)
40. C₁₃	C₃²×C₅×D₂₃	direct (N × H)
42. C₃²	C₅×C₁₃×D₂₃	direct (N × H)
44. C₅	C₃²×C₁₃×D₂₃	direct (N × H)
45. C₃	C₃×C₅×C₁₃×D₂₃	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×C₁₃×C₂₃	2
C₃×C₅×C₁₃×D₂₃	3
C₃²×C₁₃×D₂₃	5
C₃²×C₅×D₂₃	13
C₂×C₃²×C₅×C₁₃	23

Subgroups — C32×C5×C13×D23 #41002115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×C₁₃×D₂₃ #41002115