Subgroups — C₃×C₂₃×C₂²⋊(C₃₁⋊C₃) #38787352

Order 25668.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×C₂²⋊(C₃₁⋊C₃)	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₂₃×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₂₃×C₃₁⋊C₃	—	4	1	no	maximal, Hall
4	C₂×C₃×C₂₃×C₃₁	—	3	1	no	—
5	C₂²×C₂₃×C₃₁	C₃²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₂₃×C₃₁	A₄	1	1	yes	characteristic, complemented
7	C₂₃×C₃₁⋊C₃	—	4	3	no	—
8	C₂×C₂₃×C₃₁	—	3	1	no	—
9	C₃×C₂²⋊(C₃₁⋊C₃)	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×A₄×C₂₃	—	31	1	no	maximal, Hall
11	C₂₃×C₃₁	C₃×A₄	1	1	yes	characteristic, complemented, Hall
12	C₂²×C₃×C₃₁	C₃×C₂₃	1	1	yes	characteristic, complemented
13	C₂²⋊(C₃₁⋊C₃)	C₃×C₂₃	1	3	yes	direct product factor
14	C₃×C₃₁⋊C₃	—	4	1	no	Hall
15	C₂²×C₃×C₂₃	C₃₁⋊C₃	1	1	yes	characteristic, complemented
16	A₄×C₂₃	—	31	3	no	—
17	C₃²×C₂₃	—	124	1	no	Hall
18	C₂×C₃×C₃₁	—	3	1	no	—
19	C₂×C₃×C₂₃	—	3	1	no	—
20	C₂²×C₃₁	C₃²×C₂₃	1	1	yes	characteristic, complemented, Hall
21	C₃×C₃₁	A₄×C₂₃	1	1	yes	characteristic, complemented
22	C₃₁⋊C₃	—	4	3	no	—
23	C₂²×C₂₃	C₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Hall
24	C₃×C₂₃	—	124	3	no	—
25	C₃×C₂₃	C₂²⋊(C₃₁⋊C₃)	1	1	yes	characteristic, direct product factor
26	C₂×C₃₁	—	3	1	no	—
27	C₂×C₂₃	—	3	1	no	—
28	C₃×A₄	—	31	1	no	Hall
29	C₃₁	C₃×A₄×C₂₃	1	1	yes	characteristic, complemented, Sylow
30	C₂₃	C₃×C₂²⋊(C₃₁⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
31	C₂²×C₃	C₂₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented
32	A₄	—	31	3	no	—
33	C₃²	—	124	1	no	Sylow
34	C₂×C₃	—	3	1	no	—
35	C₂²	C₃×C₂₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
36	C₃	—	124	3	no	—
37	C₃	C₂₃×C₂²⋊(C₃₁⋊C₃)	1	1	yes	characteristic, direct product factor
38	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₂₃	C₂²⋊(C₃₁⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₃₁	C₃²×C₂₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₂₃×C₃₁	C₃
LCS	G ▷ C₂²×C₃₁
UCS	C₁ ◁ C₃×C₂₃

Complements

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

N	H ≅ G/N	type
1. C₂₃×C₂²⋊(C₃₁⋊C₃)	C₃	direct (N × H)
2. C₂²×C₃×C₂₃×C₃₁	C₃	semidirect
5. C₂²×C₂₃×C₃₁	C₃²	semidirect
6. C₃×C₂₃×C₃₁	A₄	semidirect
9. C₃×C₂²⋊(C₃₁⋊C₃)	C₂₃	direct (N × H)
11. C₂₃×C₃₁	C₃×A₄	semidirect
12. C₂²×C₃×C₃₁	C₃×C₂₃	semidirect
13. C₂²⋊(C₃₁⋊C₃)	C₃×C₂₃	direct (N × H)
15. C₂²×C₃×C₂₃	C₃₁⋊C₃	semidirect
20. C₂²×C₃₁	C₃²×C₂₃	semidirect
21. C₃×C₃₁	A₄×C₂₃	semidirect
23. C₂²×C₂₃	C₃×C₃₁⋊C₃	semidirect
25. C₃×C₂₃	C₂²⋊(C₃₁⋊C₃)	direct (N × H)
29. C₃₁	C₃×A₄×C₂₃	semidirect
30. C₂₃	C₃×C₂²⋊(C₃₁⋊C₃)	direct (N × H)
31. C₂²×C₃	C₂₃×C₃₁⋊C₃	semidirect
35. C₂²	C₃×C₂₃×C₃₁⋊C₃	semidirect
37. C₃	C₂₃×C₂²⋊(C₃₁⋊C₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×C₂²⋊(C₃₁⋊C₃)	3
C₂²×C₃×C₂₃×C₃₁	3
C₃×C₂₃×C₃₁⋊C₃	4
C₃×C₂²⋊(C₃₁⋊C₃)	23
C₃×A₄×C₂₃	31

Subgroups — C3×C23×C22⋊(C31⋊C3) #38787352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂₃×C₂²⋊(C₃₁⋊C₃) #38787352