Subgroups — C₂²×C₃×C₉×C₁₃×C₂₃ #72209015

Order 32292.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₁₃×C₂₃	C₂	1	3	yes	maximal, direct product factor
2	C₃×C₉×C₁₃×C₂₃	C₂²	1	1	yes	characteristic, direct product factor, Hall
3	C₂²×C₃×C₉×C₂₃	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃²×C₁₃×C₂₃	C₃	1	1	yes	maximal, characteristic
5	C₂²×C₉×C₁₃×C₂₃	C₃	1	3	yes	maximal, direct product factor
6	C₂×C₃²×C₁₃×C₂₃	C₂×C₃	1	3	yes	—
7	C₂×C₉×C₁₃×C₂₃	C₂×C₃	1	9	yes	direct product factor
8	C₂²×C₃×C₁₃×C₂₃	C₉	1	3	yes	direct product factor
9	C₂²×C₃×C₁₃×C₂₃	C₃²	1	1	yes	characteristic
10	C₃²×C₁₃×C₂₃	C₂²×C₃	1	1	yes	characteristic
11	C₉×C₁₃×C₂₃	C₂²×C₃	1	3	yes	direct product factor
12	C₂×C₃×C₁₃×C₂₃	C₂×C₉	1	9	yes	direct product factor
13	C₂×C₃×C₁₃×C₂₃	C₂×C₃²	1	3	yes	—
14	C₂²×C₃×C₉×C₁₃	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₃×C₉×C₂₃	C₂×C₁₃	1	3	yes	direct product factor
16	C₂²×C₁₃×C₂₃	C₃×C₉	1	1	yes	characteristic, direct product factor, Hall
17	C₃×C₁₃×C₂₃	C₂²×C₉	1	3	yes	direct product factor
18	C₃×C₁₃×C₂₃	C₂²×C₃²	1	1	yes	characteristic
19	C₂²×C₃²×C₂₃	C₃×C₁₃	1	1	yes	characteristic
20	C₂²×C₉×C₂₃	C₃×C₁₃	1	3	yes	direct product factor
21	C₂×C₃×C₉×C₁₃	C₂×C₂₃	1	3	yes	direct product factor
22	C₃×C₉×C₂₃	C₂²×C₁₃	1	1	yes	characteristic, direct product factor, Hall
23	C₂×C₁₃×C₂₃	C₂×C₃×C₉	1	3	yes	direct product factor
24	C₂²×C₃²×C₁₃	C₃×C₂₃	1	1	yes	characteristic
25	C₂²×C₉×C₁₃	C₃×C₂₃	1	3	yes	direct product factor
26	C₂×C₃²×C₂₃	C₂×C₃×C₁₃	1	3	yes	—
27	C₂×C₉×C₂₃	C₂×C₃×C₁₃	1	9	yes	direct product factor
28	C₃×C₉×C₁₃	C₂²×C₂₃	1	1	yes	characteristic, direct product factor, Hall
29	C₁₃×C₂₃	C₂²×C₃×C₉	1	1	yes	characteristic, direct product factor, Hall
30	C₂²×C₃×C₂₃	C₉×C₁₃	1	3	yes	direct product factor
31	C₂²×C₃×C₂₃	C₃²×C₁₃	1	1	yes	characteristic
32	C₂×C₃²×C₁₃	C₂×C₃×C₂₃	1	3	yes	—
33	C₂×C₉×C₁₃	C₂×C₃×C₂₃	1	9	yes	direct product factor
34	C₃²×C₂₃	C₂²×C₃×C₁₃	1	1	yes	characteristic
35	C₉×C₂₃	C₂²×C₃×C₁₃	1	3	yes	direct product factor
36	C₂²×C₃×C₁₃	C₉×C₂₃	1	3	yes	direct product factor
37	C₂²×C₃×C₁₃	C₃²×C₂₃	1	1	yes	characteristic
38	C₂×C₃×C₂₃	C₂×C₉×C₁₃	1	9	yes	direct product factor
39	C₂×C₃×C₂₃	C₂×C₃²×C₁₃	1	3	yes	—
40	C₃²×C₁₃	C₂²×C₃×C₂₃	1	1	yes	characteristic
41	C₉×C₁₃	C₂²×C₃×C₂₃	1	3	yes	direct product factor
42	C₂²×C₃×C₉	C₁₃×C₂₃	1	1	yes	characteristic, direct product factor, Hall
43	C₂²×C₂₃	C₃×C₉×C₁₃	1	1	yes	characteristic, direct product factor, Hall
44	C₂×C₃×C₁₃	C₂×C₉×C₂₃	1	9	yes	direct product factor
45	C₂×C₃×C₁₃	C₂×C₃²×C₂₃	1	3	yes	—
46	C₃×C₂₃	C₂²×C₉×C₁₃	1	3	yes	direct product factor
47	C₃×C₂₃	C₂²×C₃²×C₁₃	1	1	yes	characteristic
48	C₂×C₃×C₉	C₂×C₁₃×C₂₃	1	3	yes	direct product factor
49	C₂²×C₁₃	C₃×C₉×C₂₃	1	1	yes	characteristic, direct product factor, Hall
50	C₂×C₂₃	C₂×C₃×C₉×C₁₃	1	3	yes	direct product factor
51	C₃×C₁₃	C₂²×C₉×C₂₃	1	3	yes	direct product factor
52	C₃×C₁₃	C₂²×C₃²×C₂₃	1	1	yes	characteristic
53	C₂²×C₃²	C₃×C₁₃×C₂₃	1	1	yes	characteristic
54	C₂²×C₉	C₃×C₁₃×C₂₃	1	3	yes	direct product factor
55	C₃×C₉	C₂²×C₁₃×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
56	C₂×C₁₃	C₂×C₃×C₉×C₂₃	1	3	yes	direct product factor
57	C₂₃	C₂²×C₃×C₉×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×C₃²	C₂×C₃×C₁₃×C₂₃	1	3	yes	—
59	C₂×C₉	C₂×C₃×C₁₃×C₂₃	1	9	yes	direct product factor
60	C₁₃	C₂²×C₃×C₉×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²×C₃	C₉×C₁₃×C₂₃	1	3	yes	direct product factor
62	C₂²×C₃	C₃²×C₁₃×C₂₃	1	1	yes	characteristic
63	C₃²	C₂²×C₃×C₁₃×C₂₃	1	1	yes	characteristic
64	C₉	C₂²×C₃×C₁₃×C₂₃	1	3	yes	direct product factor
65	C₂×C₃	C₂×C₉×C₁₃×C₂₃	1	9	yes	direct product factor
66	C₂×C₃	C₂×C₃²×C₁₃×C₂₃	1	3	yes	—
67	C₂²	C₃×C₉×C₁₃×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
68	C₃	C₂²×C₉×C₁₃×C₂₃	1	3	yes	direct product factor
69	C₃	C₂²×C₃²×C₁₃×C₂₃	1	1	yes	characteristic
70	C₂	C₂×C₃×C₉×C₁₃×C₂₃	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₂²×C₃²×C₁₃×C₂₃
socle, prime radical	C₂²×C₃²×C₁₃×C₂₃	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×C₁₃×C₂₃	C₂	direct (N × H)
2. C₃×C₉×C₁₃×C₂₃	C₂²	direct (N × H)
3. C₂²×C₃×C₉×C₂₃	C₁₃	direct (N × H)
5. C₂²×C₉×C₁₃×C₂₃	C₃	direct (N × H)
7. C₂×C₉×C₁₃×C₂₃	C₂×C₃	direct (N × H)
8. C₂²×C₃×C₁₃×C₂₃	C₉	direct (N × H)
11. C₉×C₁₃×C₂₃	C₂²×C₃	direct (N × H)
12. C₂×C₃×C₁₃×C₂₃	C₂×C₉	direct (N × H)
14. C₂²×C₃×C₉×C₁₃	C₂₃	direct (N × H)
15. C₂×C₃×C₉×C₂₃	C₂×C₁₃	direct (N × H)
16. C₂²×C₁₃×C₂₃	C₃×C₉	direct (N × H)
17. C₃×C₁₃×C₂₃	C₂²×C₉	direct (N × H)
20. C₂²×C₉×C₂₃	C₃×C₁₃	direct (N × H)
21. C₂×C₃×C₉×C₁₃	C₂×C₂₃	direct (N × H)
22. C₃×C₉×C₂₃	C₂²×C₁₃	direct (N × H)
23. C₂×C₁₃×C₂₃	C₂×C₃×C₉	direct (N × H)
25. C₂²×C₉×C₁₃	C₃×C₂₃	direct (N × H)
27. C₂×C₉×C₂₃	C₂×C₃×C₁₃	direct (N × H)
28. C₃×C₉×C₁₃	C₂²×C₂₃	direct (N × H)
29. C₁₃×C₂₃	C₂²×C₃×C₉	direct (N × H)
30. C₂²×C₃×C₂₃	C₉×C₁₃	direct (N × H)
33. C₂×C₉×C₁₃	C₂×C₃×C₂₃	direct (N × H)
35. C₉×C₂₃	C₂²×C₃×C₁₃	direct (N × H)
36. C₂²×C₃×C₁₃	C₉×C₂₃	direct (N × H)
38. C₂×C₃×C₂₃	C₂×C₉×C₁₃	direct (N × H)
41. C₉×C₁₃	C₂²×C₃×C₂₃	direct (N × H)
42. C₂²×C₃×C₉	C₁₃×C₂₃	direct (N × H)
43. C₂²×C₂₃	C₃×C₉×C₁₃	direct (N × H)
44. C₂×C₃×C₁₃	C₂×C₉×C₂₃	direct (N × H)
46. C₃×C₂₃	C₂²×C₉×C₁₃	direct (N × H)
48. C₂×C₃×C₉	C₂×C₁₃×C₂₃	direct (N × H)
49. C₂²×C₁₃	C₃×C₉×C₂₃	direct (N × H)
50. C₂×C₂₃	C₂×C₃×C₉×C₁₃	direct (N × H)
51. C₃×C₁₃	C₂²×C₉×C₂₃	direct (N × H)
54. C₂²×C₉	C₃×C₁₃×C₂₃	direct (N × H)
55. C₃×C₉	C₂²×C₁₃×C₂₃	direct (N × H)
56. C₂×C₁₃	C₂×C₃×C₉×C₂₃	direct (N × H)
57. C₂₃	C₂²×C₃×C₉×C₁₃	direct (N × H)
59. C₂×C₉	C₂×C₃×C₁₃×C₂₃	direct (N × H)
60. C₁₃	C₂²×C₃×C₉×C₂₃	direct (N × H)
61. C₂²×C₃	C₉×C₁₃×C₂₃	direct (N × H)
64. C₉	C₂²×C₃×C₁₃×C₂₃	direct (N × H)
65. C₂×C₃	C₂×C₉×C₁₃×C₂₃	direct (N × H)
67. C₂²	C₃×C₉×C₁₃×C₂₃	direct (N × H)
68. C₃	C₂²×C₉×C₁₃×C₂₃	direct (N × H)
70. C₂	C₂×C₃×C₉×C₁₃×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×C₃×C₉×C₁₃×C₂₃	2
C₂²×C₃×C₉×C₂₃	13
C₂²×C₃²×C₁₃×C₂₃	3
C₂²×C₉×C₁₃×C₂₃	3
C₂²×C₃×C₉×C₁₃	23

Subgroups — C22×C3×C9×C13×C23 #72209015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₉×C₁₃×C₂₃ #72209015