Subgroups — C₂²×C₃⋊D₃×C₂₃ #13110297

Order 1656.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃⋊D₃×C₂₃	C₂	1	6	yes	maximal, direct product factor
3	C₂²×D₃×C₂₃	—	3	4	no	maximal
4	C₂×C₃²×C₂₃	C₂²	1	3	yes	complemented
5	C₃⋊D₃×C₂₃	C₂²	1	4	yes	direct product factor
6	C₂²×C₃×C₂₃	D₃	1	4	yes	complemented
7	C₂×D₃×C₂₃	—	3	24	no	—
8	C₃²×C₂₃	C₂³	1	1	yes	characteristic, complemented, Hall
9	C₂³×C₂₃	—	9	1	no	Hall
10	C₂×C₃×C₂₃	C₂×D₃	1	12	yes	complemented
11	D₃×C₂₃	—	3	16	no	—
12	C₂²×C₂₃	—	9	6	no	—
13	C₂²×C₂₃	C₃⋊D₃	1	1	yes	characteristic, direct product factor
14	C₂²×C₃⋊D₃	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₃×C₂₃	C₂²×D₃	1	4	yes	complemented
16	C₂×C₂₃	—	9	4	no	—
17	C₂×C₂₃	C₂×C₃⋊D₃	1	3	yes	direct product factor
18	C₂²×C₃²	C₂×C₂₃	1	1	yes	characteristic, complemented
19	C₂×C₃⋊D₃	C₂×C₂₃	1	6	yes	direct product factor
20	C₂²×D₃	—	3	4	no	—
21	C₂₃	C₂²×C₃⋊D₃	1	1	yes	characteristic, direct product factor, Sylow
22	C₂×C₃²	C₂²×C₂₃	1	3	yes	complemented
23	C₃⋊D₃	C₂²×C₂₃	1	4	yes	direct product factor
24	C₂²×C₃	D₃×C₂₃	1	4	yes	complemented
25	C₂×D₃	—	3	24	no	—
26	C₃²	C₂³×C₂₃	1	1	yes	characteristic, complemented, Sylow
27	C₂³	—	9	1	no	Sylow
28	C₂×C₃	C₂×D₃×C₂₃	1	12	yes	complemented
29	D₃	—	3	16	no	—
30	C₂²	—	9	6	no	—
31	C₂²	C₃⋊D₃×C₂₃	1	1	yes	characteristic, direct product factor
32	C₃	C₂²×D₃×C₂₃	1	4	yes	complemented
33	C₂	—	9	4	no	—
34	C₂	C₂×C₃⋊D₃×C₂₃	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₂₃	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂³×C₂₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃²×C₂₃	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²×C₂₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₂₃	C₂	semidirect
2. C₂×C₃⋊D₃×C₂₃	C₂	direct (N × H)
4. C₂×C₃²×C₂₃	C₂²	semidirect
5. C₃⋊D₃×C₂₃	C₂²	direct (N × H)
6. C₂²×C₃×C₂₃	D₃	semidirect
8. C₃²×C₂₃	C₂³	semidirect
10. C₂×C₃×C₂₃	C₂×D₃	semidirect
13. C₂²×C₂₃	C₃⋊D₃	direct (N × H)
14. C₂²×C₃⋊D₃	C₂₃	direct (N × H)
15. C₃×C₂₃	C₂²×D₃	semidirect
17. C₂×C₂₃	C₂×C₃⋊D₃	direct (N × H)
18. C₂²×C₃²	C₂×C₂₃	semidirect
19. C₂×C₃⋊D₃	C₂×C₂₃	direct (N × H)
21. C₂₃	C₂²×C₃⋊D₃	direct (N × H)
22. C₂×C₃²	C₂²×C₂₃	semidirect
23. C₃⋊D₃	C₂²×C₂₃	direct (N × H)
24. C₂²×C₃	D₃×C₂₃	semidirect
26. C₃²	C₂³×C₂₃	semidirect
28. C₂×C₃	C₂×D₃×C₂₃	semidirect
31. C₂²	C₃⋊D₃×C₂₃	direct (N × H)
32. C₃	C₂²×D₃×C₂₃	semidirect
34. C₂	C₂×C₃⋊D₃×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
C₂²×C₃²×C₂₃	2
C₂×C₃⋊D₃×C₂₃	2
C₂²×D₃×C₂₃	3
C₂²×C₃⋊D₃	23

Subgroups — C22×C3⋊D3×C23 #13110297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃⋊D₃×C₂₃ #13110297