Subgroups — C₂²×C₃×C₉×C₂₂₃ #72201387

Order 24084.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₂₂₃	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃²×C₂₂₃	C₃	1	1	yes	maximal, characteristic
3	C₂²×C₉×C₂₂₃	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₉×C₂₂₃	C₂²	1	1	yes	characteristic, direct product factor, Hall
5	C₂×C₃²×C₂₂₃	C₂×C₃	1	3	yes	—
6	C₂×C₉×C₂₂₃	C₂×C₃	1	9	yes	direct product factor
7	C₂²×C₃×C₂₂₃	C₉	1	3	yes	direct product factor
8	C₂²×C₃×C₂₂₃	C₃²	1	1	yes	characteristic
9	C₃²×C₂₂₃	C₂²×C₃	1	1	yes	characteristic
10	C₉×C₂₂₃	C₂²×C₃	1	3	yes	direct product factor
11	C₂×C₃×C₂₂₃	C₂×C₉	1	9	yes	direct product factor
12	C₂×C₃×C₂₂₃	C₂×C₃²	1	3	yes	—
13	C₂²×C₂₂₃	C₃×C₉	1	1	yes	characteristic, direct product factor, Hall
14	C₃×C₂₂₃	C₂²×C₉	1	3	yes	direct product factor
15	C₃×C₂₂₃	C₂²×C₃²	1	1	yes	characteristic
16	C₂×C₂₂₃	C₂×C₃×C₉	1	3	yes	direct product factor
17	C₂₂₃	C₂²×C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
18	C₂²×C₃×C₉	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₃×C₉	C₂×C₂₂₃	1	3	yes	direct product factor
20	C₂²×C₃²	C₃×C₂₂₃	1	1	yes	characteristic
21	C₂²×C₉	C₃×C₂₂₃	1	3	yes	direct product factor
22	C₃×C₉	C₂²×C₂₂₃	1	1	yes	characteristic, direct product factor, Sylow
23	C₂×C₃²	C₂×C₃×C₂₂₃	1	3	yes	—
24	C₂×C₉	C₂×C₃×C₂₂₃	1	9	yes	direct product factor
25	C₂²×C₃	C₉×C₂₂₃	1	3	yes	direct product factor
26	C₂²×C₃	C₃²×C₂₂₃	1	1	yes	characteristic
27	C₃²	C₂²×C₃×C₂₂₃	1	1	yes	characteristic
28	C₉	C₂²×C₃×C₂₂₃	1	3	yes	direct product factor
29	C₂×C₃	C₂×C₉×C₂₂₃	1	9	yes	direct product factor
30	C₂×C₃	C₂×C₃²×C₂₂₃	1	3	yes	—
31	C₂²	C₃×C₉×C₂₂₃	1	1	yes	characteristic, direct product factor, Sylow
32	C₃	C₂²×C₉×C₂₂₃	1	3	yes	direct product factor
33	C₃	C₂²×C₃²×C₂₂₃	1	1	yes	characteristic
34	C₂	C₂×C₃×C₉×C₂₂₃	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₂²×C₃²×C₂₂₃
socle, prime radical	C₂²×C₃²×C₂₂₃	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₉×C₂₂₃	C₂	direct (N × H)
3. C₂²×C₉×C₂₂₃	C₃	direct (N × H)
4. C₃×C₉×C₂₂₃	C₂²	direct (N × H)
6. C₂×C₉×C₂₂₃	C₂×C₃	direct (N × H)
7. C₂²×C₃×C₂₂₃	C₉	direct (N × H)
10. C₉×C₂₂₃	C₂²×C₃	direct (N × H)
11. C₂×C₃×C₂₂₃	C₂×C₉	direct (N × H)
13. C₂²×C₂₂₃	C₃×C₉	direct (N × H)
14. C₃×C₂₂₃	C₂²×C₉	direct (N × H)
16. C₂×C₂₂₃	C₂×C₃×C₉	direct (N × H)
17. C₂₂₃	C₂²×C₃×C₉	direct (N × H)
18. C₂²×C₃×C₉	C₂₂₃	direct (N × H)
19. C₂×C₃×C₉	C₂×C₂₂₃	direct (N × H)
21. C₂²×C₉	C₃×C₂₂₃	direct (N × H)
22. C₃×C₉	C₂²×C₂₂₃	direct (N × H)
24. C₂×C₉	C₂×C₃×C₂₂₃	direct (N × H)
25. C₂²×C₃	C₉×C₂₂₃	direct (N × H)
28. C₉	C₂²×C₃×C₂₂₃	direct (N × H)
29. C₂×C₃	C₂×C₉×C₂₂₃	direct (N × H)
31. C₂²	C₃×C₉×C₂₂₃	direct (N × H)
32. C₃	C₂²×C₉×C₂₂₃	direct (N × H)
34. C₂	C₂×C₃×C₉×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×C₂₂₃	2
C₂²×C₃²×C₂₂₃	3
C₂²×C₉×C₂₂₃	3
C₂²×C₃×C₉	223

Subgroups — C22×C3×C9×C223 #72201387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₉×C₂₂₃ #72201387