Subgroups — C₃³×C₄×C₉×C₂₃ #72199780

Order 22356.

2376 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₉×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₃⁴×C₄×C₂₃	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₄×C₉×C₂₃	C₃	1	39	yes	maximal, direct product factor
4	C₂×C₃⁴×C₂₃	C₂×C₃	1	1	yes	characteristic
5	C₂×C₃²×C₉×C₂₃	C₂×C₃	1	39	yes	—
6	C₃³×C₄×C₂₃	C₉	1	27	yes	direct product factor
7	C₃³×C₄×C₂₃	C₃²	1	13	yes	—
8	C₃×C₄×C₉×C₂₃	C₃²	1	117	yes	direct product factor
9	C₃³×C₉×C₂₃	C₄	1	1	yes	characteristic, direct product factor, Hall
10	C₃⁴×C₂₃	C₃×C₄	1	1	yes	characteristic
11	C₃²×C₉×C₂₃	C₃×C₄	1	39	yes	direct product factor
12	C₂×C₃³×C₂₃	C₂×C₉	1	27	yes	—
13	C₂×C₃³×C₂₃	C₂×C₃²	1	13	yes	—
14	C₂×C₃×C₉×C₂₃	C₂×C₃²	1	117	yes	—
15	C₃³×C₄×C₉	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₃²×C₄×C₂₃	C₃×C₉	1	117	yes	direct product factor
17	C₃²×C₄×C₂₃	C₃³	1	13	yes	—
18	C₄×C₉×C₂₃	C₃³	1	27	yes	direct product factor
19	C₃³×C₂₃	C₄×C₉	1	27	yes	direct product factor
20	C₃³×C₂₃	C₃²×C₄	1	13	yes	—
21	C₃×C₉×C₂₃	C₃²×C₄	1	117	yes	direct product factor
22	C₂×C₃³×C₉	C₂×C₂₃	1	1	yes	characteristic
23	C₂×C₃²×C₂₃	C₂×C₃×C₉	1	117	yes	—
24	C₂×C₃²×C₂₃	C₂×C₃³	1	13	yes	—
25	C₂×C₉×C₂₃	C₂×C₃³	1	27	yes	—
26	C₃⁴×C₄	C₃×C₂₃	1	1	yes	characteristic
27	C₃²×C₄×C₉	C₃×C₂₃	1	39	yes	direct product factor
28	C₃×C₄×C₂₃	C₃²×C₉	1	39	yes	direct product factor
29	C₃×C₄×C₂₃	C₃⁴	1	1	yes	characteristic
30	C₃³×C₉	C₄×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
31	C₃²×C₂₃	C₃×C₄×C₉	1	117	yes	direct product factor
32	C₃²×C₂₃	C₃³×C₄	1	13	yes	—
33	C₉×C₂₃	C₃³×C₄	1	27	yes	direct product factor
34	C₂×C₃⁴	C₂×C₃×C₂₃	1	1	yes	characteristic
35	C₂×C₃²×C₉	C₂×C₃×C₂₃	1	39	yes	—
36	C₂×C₃×C₂₃	C₂×C₃²×C₉	1	39	yes	—
37	C₂×C₃×C₂₃	C₂×C₃⁴	1	1	yes	characteristic
38	C₃³×C₄	C₉×C₂₃	1	27	yes	direct product factor
39	C₃³×C₄	C₃²×C₂₃	1	13	yes	—
40	C₃×C₄×C₉	C₃²×C₂₃	1	117	yes	direct product factor
41	C₄×C₂₃	C₃³×C₉	1	1	yes	characteristic, direct product factor, Hall
42	C₃⁴	C₃×C₄×C₂₃	1	1	yes	characteristic
43	C₃²×C₉	C₃×C₄×C₂₃	1	39	yes	direct product factor
44	C₃×C₂₃	C₃²×C₄×C₉	1	39	yes	direct product factor
45	C₃×C₂₃	C₃⁴×C₄	1	1	yes	characteristic
46	C₂×C₃³	C₂×C₉×C₂₃	1	27	yes	—
47	C₂×C₃³	C₂×C₃²×C₂₃	1	13	yes	—
48	C₂×C₃×C₉	C₂×C₃²×C₂₃	1	117	yes	—
49	C₂×C₂₃	C₂×C₃³×C₉	1	1	yes	characteristic
50	C₃²×C₄	C₃×C₉×C₂₃	1	117	yes	direct product factor
51	C₃²×C₄	C₃³×C₂₃	1	13	yes	—
52	C₄×C₉	C₃³×C₂₃	1	27	yes	direct product factor
53	C₃³	C₄×C₉×C₂₃	1	27	yes	direct product factor
54	C₃³	C₃²×C₄×C₂₃	1	13	yes	—
55	C₃×C₉	C₃²×C₄×C₂₃	1	117	yes	direct product factor
56	C₂₃	C₃³×C₄×C₉	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×C₃²	C₂×C₃×C₉×C₂₃	1	117	yes	—
58	C₂×C₃²	C₂×C₃³×C₂₃	1	13	yes	—
59	C₂×C₉	C₂×C₃³×C₂₃	1	27	yes	—
60	C₃×C₄	C₃²×C₉×C₂₃	1	39	yes	direct product factor
61	C₃×C₄	C₃⁴×C₂₃	1	1	yes	characteristic
62	C₃²	C₃×C₄×C₉×C₂₃	1	117	yes	direct product factor
63	C₃²	C₃³×C₄×C₂₃	1	13	yes	—
64	C₉	C₃³×C₄×C₂₃	1	27	yes	direct product factor
65	C₂×C₃	C₂×C₃²×C₉×C₂₃	1	39	yes	—
66	C₂×C₃	C₂×C₃⁴×C₂₃	1	1	yes	characteristic
67	C₄	C₃³×C₉×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
68	C₃	C₃²×C₄×C₉×C₂₃	1	39	yes	direct product factor
69	C₃	C₃⁴×C₄×C₂₃	1	1	yes	characteristic
70	C₂	C₂×C₃³×C₉×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃	C₂×C₃⁴×C₂₃
socle, prime radical	C₂×C₃⁴×C₂₃	C₂×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×C₄×C₉×C₂₃	C₃	direct (N × H)
6. C₃³×C₄×C₂₃	C₉	direct (N × H)
8. C₃×C₄×C₉×C₂₃	C₃²	direct (N × H)
9. C₃³×C₉×C₂₃	C₄	direct (N × H)
11. C₃²×C₉×C₂₃	C₃×C₄	direct (N × H)
15. C₃³×C₄×C₉	C₂₃	direct (N × H)
16. C₃²×C₄×C₂₃	C₃×C₉	direct (N × H)
18. C₄×C₉×C₂₃	C₃³	direct (N × H)
19. C₃³×C₂₃	C₄×C₉	direct (N × H)
21. C₃×C₉×C₂₃	C₃²×C₄	direct (N × H)
27. C₃²×C₄×C₉	C₃×C₂₃	direct (N × H)
28. C₃×C₄×C₂₃	C₃²×C₉	direct (N × H)
30. C₃³×C₉	C₄×C₂₃	direct (N × H)
31. C₃²×C₂₃	C₃×C₄×C₉	direct (N × H)
33. C₉×C₂₃	C₃³×C₄	direct (N × H)
38. C₃³×C₄	C₉×C₂₃	direct (N × H)
40. C₃×C₄×C₉	C₃²×C₂₃	direct (N × H)
41. C₄×C₂₃	C₃³×C₉	direct (N × H)
43. C₃²×C₉	C₃×C₄×C₂₃	direct (N × H)
44. C₃×C₂₃	C₃²×C₄×C₉	direct (N × H)
50. C₃²×C₄	C₃×C₉×C₂₃	direct (N × H)
52. C₄×C₉	C₃³×C₂₃	direct (N × H)
53. C₃³	C₄×C₉×C₂₃	direct (N × H)
55. C₃×C₉	C₃²×C₄×C₂₃	direct (N × H)
56. C₂₃	C₃³×C₄×C₉	direct (N × H)
60. C₃×C₄	C₃²×C₉×C₂₃	direct (N × H)
62. C₃²	C₃×C₄×C₉×C₂₃	direct (N × H)
64. C₉	C₃³×C₄×C₂₃	direct (N × H)
67. C₄	C₃³×C₉×C₂₃	direct (N × H)
68. C₃	C₃²×C₄×C₉×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
C₂×C₃³×C₉×C₂₃	2
C₃⁴×C₄×C₂₃	3
C₃²×C₄×C₉×C₂₃	3
C₃³×C₄×C₉	23

Subgroups — C33×C4×C9×C23 #72199780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×C₄×C₉×C₂₃ #72199780