Subgroups — C₃×A₄×C₁₉×C₂₃ #30628647

Order 15732.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	A₄×C₁₉×C₂₃	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₁₉×C₂₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₁₉×C₂₃	—	4	1	no	maximal, Hall
4	C₂×C₃×C₁₉×C₂₃	—	3	1	no	—
5	C₂²×C₁₉×C₂₃	C₃²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₁₉×C₂₃	—	4	3	no	—
7	C₃×C₁₉×C₂₃	A₄	1	1	yes	characteristic, direct product factor
8	C₂×C₁₉×C₂₃	—	3	1	no	—
9	C₃×A₄×C₂₃	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×A₄×C₁₉	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₁₉×C₂₃	C₃×A₄	1	1	yes	characteristic, direct product factor, Hall
12	C₂²×C₃×C₂₃	C₃×C₁₉	1	1	yes	characteristic, complemented
13	A₄×C₂₃	C₃×C₁₉	1	3	yes	direct product factor
14	C₂²×C₃×C₁₉	C₃×C₂₃	1	1	yes	characteristic, complemented
15	A₄×C₁₉	C₃×C₂₃	1	3	yes	direct product factor
16	C₃²×C₂₃	—	4	1	no	Hall
17	C₃²×C₁₉	—	4	1	no	Hall
18	C₂×C₃×C₂₃	—	3	1	no	—
19	C₂×C₃×C₁₉	—	3	1	no	—
20	C₂²×C₂₃	C₃²×C₁₉	1	1	yes	characteristic, complemented, Hall
21	C₂²×C₁₉	C₃²×C₂₃	1	1	yes	characteristic, complemented, Hall
22	C₃×C₂₃	—	4	3	no	—
23	C₃×C₂₃	A₄×C₁₉	1	1	yes	characteristic, direct product factor
24	C₃×C₁₉	—	4	3	no	—
25	C₃×C₁₉	A₄×C₂₃	1	1	yes	characteristic, direct product factor
26	C₂×C₂₃	—	3	1	no	—
27	C₂×C₁₉	—	3	1	no	—
28	C₃×A₄	C₁₉×C₂₃	1	1	yes	characteristic, direct product factor, Hall
29	C₂₃	C₃×A₄×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
30	C₁₉	C₃×A₄×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
31	C₂²×C₃	C₃×C₁₉×C₂₃	1	1	yes	characteristic, complemented
32	A₄	C₃×C₁₉×C₂₃	1	3	yes	direct product factor
33	C₃²	—	4	1	no	Sylow
34	C₂×C₃	—	3	1	no	—
35	C₂²	C₃²×C₁₉×C₂₃	1	1	yes	characteristic, complemented, Sylow
36	C₃	—	4	3	no	—
37	C₃	A₄×C₁₉×C₂₃	1	1	yes	characteristic, direct product factor
38	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₁₉×C₂₃	A₄
commutator, cosocle, hypocenter	C₂²	C₃²×C₁₉×C₂₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₁₉×C₂₃	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₁₉×C₂₃

Complements

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

N	H ≅ G/N	type
1. A₄×C₁₉×C₂₃	C₃	direct (N × H)
2. C₂²×C₃×C₁₉×C₂₃	C₃	semidirect
5. C₂²×C₁₉×C₂₃	C₃²	semidirect
7. C₃×C₁₉×C₂₃	A₄	direct (N × H)
9. C₃×A₄×C₂₃	C₁₉	direct (N × H)
10. C₃×A₄×C₁₉	C₂₃	direct (N × H)
11. C₁₉×C₂₃	C₃×A₄	direct (N × H)
12. C₂²×C₃×C₂₃	C₃×C₁₉	semidirect
13. A₄×C₂₃	C₃×C₁₉	direct (N × H)
14. C₂²×C₃×C₁₉	C₃×C₂₃	semidirect
15. A₄×C₁₉	C₃×C₂₃	direct (N × H)
20. C₂²×C₂₃	C₃²×C₁₉	semidirect
21. C₂²×C₁₉	C₃²×C₂₃	semidirect
23. C₃×C₂₃	A₄×C₁₉	direct (N × H)
25. C₃×C₁₉	A₄×C₂₃	direct (N × H)
28. C₃×A₄	C₁₉×C₂₃	direct (N × H)
29. C₂₃	C₃×A₄×C₁₉	direct (N × H)
30. C₁₉	C₃×A₄×C₂₃	direct (N × H)
31. C₂²×C₃	C₃×C₁₉×C₂₃	semidirect
32. A₄	C₃×C₁₉×C₂₃	direct (N × H)
35. C₂²	C₃²×C₁₉×C₂₃	semidirect
37. C₃	A₄×C₁₉×C₂₃	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
A₄×C₁₉×C₂₃	3
C₂²×C₃×C₁₉×C₂₃	3
C₃²×C₁₉×C₂₃	4
C₃×A₄×C₂₃	19
C₃×A₄×C₁₉	23

Subgroups — C3×A4×C19×C23 #30628647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×A₄×C₁₉×C₂₃ #30628647