Subgroups — C₂²×C₃×D₃×C₂₃ #13110295

Order 1656.

138 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	C₂³×C₃×C₂₃	—	3	1	no	maximal
4	C₂²×D₃×C₂₃	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃²×C₂₃	C₂²	1	3	yes	complemented
6	C₃×D₃×C₂₃	C₂²	1	4	yes	direct product factor
7	C₂²×C₃×C₂₃	—	2	1	no	—
8	C₂²×C₃×C₂₃	—	3	6	no	—
9	C₂²×C₃×C₂₃	D₃	1	1	yes	characteristic, direct product factor
10	C₂²×C₃×C₂₃	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×D₃×C₂₃	C₂×C₃	1	6	yes	direct product factor
12	C₃²×C₂₃	C₂³	1	1	yes	characteristic, complemented, Hall
13	C₂³×C₂₃	—	3	1	no	Hall
14	C₂×C₃×C₂₃	—	2	3	no	—
15	C₂×C₃×C₂₃	—	3	4	no	—
16	C₂×C₃×C₂₃	C₂×D₃	1	3	yes	direct product factor
17	C₂×C₃×C₂₃	C₂²×C₃	1	3	yes	complemented
18	D₃×C₂₃	C₂²×C₃	1	4	yes	direct product factor
19	C₂²×C₂₃	—	3	6	no	—
20	C₂²×C₂₃	C₃×D₃	1	1	yes	characteristic, direct product factor
21	C₂²×C₃×D₃	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₃×C₂₃	—	2	1	no	—
23	C₃×C₂₃	C₂²×D₃	1	1	yes	characteristic, direct product factor
24	C₃×C₂₃	C₂³×C₃	1	1	yes	characteristic, complemented
25	C₂×C₂₃	—	3	4	no	—
26	C₂×C₂₃	C₂×C₃×D₃	1	3	yes	direct product factor
27	C₂²×C₃²	C₂×C₂₃	1	1	yes	characteristic, complemented
28	C₂×C₃×D₃	C₂×C₂₃	1	6	yes	direct product factor
29	C₂³×C₃	—	3	1	no	—
30	C₂²×D₃	C₃×C₂₃	1	1	yes	characteristic, direct product factor
31	C₂₃	C₂²×C₃×D₃	1	1	yes	characteristic, direct product factor, Sylow
32	C₂×C₃²	C₂²×C₂₃	1	3	yes	complemented
33	C₃×D₃	C₂²×C₂₃	1	4	yes	direct product factor
34	C₂²×C₃	—	2	1	no	—
35	C₂²×C₃	—	3	6	no	—
36	C₂²×C₃	D₃×C₂₃	1	1	yes	characteristic, direct product factor
37	C₂²×C₃	C₂×C₃×C₂₃	1	1	yes	characteristic, complemented
38	C₂×D₃	C₂×C₃×C₂₃	1	6	yes	direct product factor
39	C₃²	C₂³×C₂₃	1	1	yes	characteristic, complemented, Sylow
40	C₂³	—	3	1	no	Sylow
41	C₂×C₃	—	2	3	no	—
42	C₂×C₃	—	3	4	no	—
43	C₂×C₃	C₂×D₃×C₂₃	1	3	yes	direct product factor
44	C₂×C₃	C₂²×C₃×C₂₃	1	3	yes	complemented
45	D₃	C₂²×C₃×C₂₃	1	4	yes	direct product factor
46	C₂²	—	3	6	no	—
47	C₂²	C₃×D₃×C₂₃	1	1	yes	characteristic, direct product factor
48	C₃	—	2	1	no	—
49	C₃	C₂²×D₃×C₂₃	1	1	yes	characteristic, direct product factor
50	C₃	C₂³×C₃×C₂₃	1	1	yes	characteristic, complemented
51	C₂	—	3	4	no	—
52	C₂	C₂×C₃×D₃×C₂₃	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₂₃	D₃
commutator, cosocle, hypocenter	C₃	C₂³×C₃×C₂₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃²×C₂₃	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₃×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. C₂²×D₃×C₂₃	C₃	direct (N × H)
5. C₂×C₃²×C₂₃	C₂²	semidirect
6. C₃×D₃×C₂₃	C₂²	direct (N × H)
9. C₂²×C₃×C₂₃	D₃	direct (N × H)
10. C₂²×C₃×C₂₃	C₂×C₃	semidirect
11. C₂×D₃×C₂₃	C₂×C₃	direct (N × H)
12. C₃²×C₂₃	C₂³	semidirect
16. C₂×C₃×C₂₃	C₂×D₃	direct (N × H)
17. C₂×C₃×C₂₃	C₂²×C₃	semidirect
18. D₃×C₂₃	C₂²×C₃	direct (N × H)
20. C₂²×C₂₃	C₃×D₃	direct (N × H)
21. C₂²×C₃×D₃	C₂₃	direct (N × H)
23. C₃×C₂₃	C₂²×D₃	direct (N × H)
24. C₃×C₂₃	C₂³×C₃	semidirect
26. C₂×C₂₃	C₂×C₃×D₃	direct (N × H)
27. C₂²×C₃²	C₂×C₂₃	semidirect
28. C₂×C₃×D₃	C₂×C₂₃	direct (N × H)
30. C₂²×D₃	C₃×C₂₃	direct (N × H)
31. C₂₃	C₂²×C₃×D₃	direct (N × H)
32. C₂×C₃²	C₂²×C₂₃	semidirect
33. C₃×D₃	C₂²×C₂₃	direct (N × H)
36. C₂²×C₃	D₃×C₂₃	direct (N × H)
37. C₂²×C₃	C₂×C₃×C₂₃	semidirect
38. C₂×D₃	C₂×C₃×C₂₃	direct (N × H)
39. C₃²	C₂³×C₂₃	semidirect
43. C₂×C₃	C₂×D₃×C₂₃	direct (N × H)
44. C₂×C₃	C₂²×C₃×C₂₃	semidirect
45. D₃	C₂²×C₃×C₂₃	direct (N × H)
47. C₂²	C₃×D₃×C₂₃	direct (N × H)
49. C₃	C₂²×D₃×C₂₃	direct (N × H)
50. C₃	C₂³×C₃×C₂₃	semidirect
52. 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
C₂³×C₃×C₂₃	3
C₂²×D₃×C₂₃	3
C₂²×C₃×D₃	23

Subgroups — C22×C3×D3×C23 #13110295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×D₃×C₂₃ #13110295