Subgroups — C₂×C₃×A₄×C₂₃ #13110293

Order 1656.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₃×C₂₃	A₄
commutator, cosocle, hypocenter	C₂²	C₂×C₃²×C₂₃
Frattini	C₁	G
Fitting, socle	C₂³×C₃×C₂₃	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂×C₃×C₂₃

Complements

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

N	H ≅ G/N	type
1. C₃×A₄×C₂₃	C₂	direct (N × H)
2. C₂³×C₃×C₂₃	C₃	semidirect
3. C₂×A₄×C₂₃	C₃	direct (N × H)
7. C₂²×C₃×C₂₃	C₂×C₃	semidirect
8. A₄×C₂₃	C₂×C₃	direct (N × H)
10. C₂³×C₂₃	C₃²	semidirect
14. C₂×C₃×C₂₃	A₄	direct (N × H)
17. C₂²×C₂₃	C₂×C₃²	semidirect
18. C₂×C₃×A₄	C₂₃	direct (N × H)
20. C₃×C₂₃	C₂×A₄	direct (N × H)
23. C₂×C₂₃	C₃×A₄	direct (N × H)
24. C₃×A₄	C₂×C₂₃	direct (N × H)
25. C₂³×C₃	C₃×C₂₃	semidirect
26. C₂×A₄	C₃×C₂₃	direct (N × H)
27. C₂₃	C₂×C₃×A₄	direct (N × H)
31. C₂²×C₃	C₂×C₃×C₂₃	semidirect
32. A₄	C₂×C₃×C₂₃	direct (N × H)
34. C₂³	C₃²×C₂₃	semidirect
38. C₂×C₃	A₄×C₂₃	direct (N × H)
41. C₂²	C₂×C₃²×C₂₃	semidirect
43. C₃	C₂×A₄×C₂₃	direct (N × H)
46. C₂	C₃×A₄×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
C₃×A₄×C₂₃	2
C₂³×C₃×C₂₃	3
C₂×A₄×C₂₃	3
C₂×C₃²×C₂₃	4
C₂×C₃×A₄	23

Subgroups — C2×C3×A4×C23 #13110293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×A₄×C₂₃ #13110293