Subgroups — C₂²×C₃²×D₂₃ #13110294

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×C₃²×D₂₃	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×D₂₃	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₂₃	C₂²	1	3	yes	complemented
5	C₃²×D₂₃	C₂²	1	4	yes	direct product factor
6	C₂²×C₃×C₂₃	C₂×C₃	1	4	yes	complemented
7	C₂×C₃×D₂₃	C₂×C₃	1	24	yes	direct product factor
8	C₃²×C₂₃	C₂³	1	1	yes	characteristic, complemented, Hall
9	C₂²×D₂₃	C₃²	1	1	yes	characteristic, direct product factor, Hall
10	C₂×C₃×C₂₃	C₂²×C₃	1	12	yes	complemented
11	C₃×D₂₃	C₂²×C₃	1	16	yes	direct product factor
12	C₂²×C₂₃	C₂×C₃²	1	1	yes	characteristic, complemented
13	C₂×D₂₃	C₂×C₃²	1	6	yes	direct product factor
14	C₂³×C₃²	—	23	1	no	maximal, Hall
15	C₃×C₂₃	C₂³×C₃	1	4	yes	complemented
16	C₂×C₂₃	C₂²×C₃²	1	3	yes	complemented
17	D₂₃	C₂²×C₃²	1	4	yes	direct product factor
18	C₂²×C₃²	—	23	6	no	—
19	C₂²×C₃²	D₂₃	1	1	yes	characteristic, direct product factor
20	C₂³×C₃	—	23	4	no	—
21	C₂₃	C₂³×C₃²	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₃²	—	23	4	no	—
23	C₂×C₃²	C₂×D₂₃	1	3	yes	direct product factor
24	C₂²×C₃	—	23	24	no	—
25	C₂²×C₃	C₃×D₂₃	1	4	yes	direct product factor
26	C₃²	C₂²×D₂₃	1	1	yes	characteristic, direct product factor, Sylow
27	C₂³	—	23	1	no	Sylow
28	C₂×C₃	—	23	16	no	—
29	C₂×C₃	C₂×C₃×D₂₃	1	12	yes	direct product factor
30	C₂²	—	23	6	no	—
31	C₂²	C₃²×D₂₃	1	1	yes	characteristic, direct product factor
32	C₃	C₂²×C₃×D₂₃	1	4	yes	direct product factor
33	C₂	—	23	4	no	—
34	C₂	C₂×C₃²×D₂₃	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃²	D₂₃
commutator, cosocle, hypocenter	C₂₃	C₂³×C₃²
Frattini	C₁	G
Fitting, socle	C₂²×C₃²×C₂₃	C₂
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₂²×C₃²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₂₃	C₂	semidirect
2. C₂×C₃²×D₂₃	C₂	direct (N × H)
3. C₂²×C₃×D₂₃	C₃	direct (N × H)
4. C₂×C₃²×C₂₃	C₂²	semidirect
5. C₃²×D₂₃	C₂²	direct (N × H)
6. C₂²×C₃×C₂₃	C₂×C₃	semidirect
7. C₂×C₃×D₂₃	C₂×C₃	direct (N × H)
8. C₃²×C₂₃	C₂³	semidirect
9. C₂²×D₂₃	C₃²	direct (N × H)
10. C₂×C₃×C₂₃	C₂²×C₃	semidirect
11. C₃×D₂₃	C₂²×C₃	direct (N × H)
12. C₂²×C₂₃	C₂×C₃²	semidirect
13. C₂×D₂₃	C₂×C₃²	direct (N × H)
15. C₃×C₂₃	C₂³×C₃	semidirect
16. C₂×C₂₃	C₂²×C₃²	semidirect
17. D₂₃	C₂²×C₃²	direct (N × H)
19. C₂²×C₃²	D₂₃	direct (N × H)
21. C₂₃	C₂³×C₃²	semidirect
23. C₂×C₃²	C₂×D₂₃	direct (N × H)
25. C₂²×C₃	C₃×D₂₃	direct (N × H)
26. C₃²	C₂²×D₂₃	direct (N × H)
29. C₂×C₃	C₂×C₃×D₂₃	direct (N × H)
31. C₂²	C₃²×D₂₃	direct (N × H)
32. C₃	C₂²×C₃×D₂₃	direct (N × H)
34. C₂	C₂×C₃²×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
C₂²×C₃²×C₂₃	2
C₂×C₃²×D₂₃	2
C₂²×C₃×D₂₃	3
C₂³×C₃²	23

Subgroups — C22×C32×D23 #13110294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×D₂₃ #13110294