Subgroups — C₂²×C₃×C₅×C₉×C₄₁ #72199613

Order 22140.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₉×C₄₁	C₂	1	3	yes	maximal, direct product factor
2	C₃×C₅×C₉×C₄₁	C₂²	1	1	yes	characteristic, direct product factor, Hall
3	C₂²×C₃×C₉×C₄₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₃×C₉×C₄₁	C₂×C₅	1	3	yes	direct product factor
5	C₂²×C₃²×C₅×C₄₁	C₃	1	1	yes	maximal, characteristic
6	C₂²×C₅×C₉×C₄₁	C₃	1	3	yes	maximal, direct product factor
7	C₂×C₃²×C₅×C₄₁	C₂×C₃	1	3	yes	—
8	C₂×C₅×C₉×C₄₁	C₂×C₃	1	9	yes	direct product factor
9	C₂²×C₃×C₅×C₄₁	C₉	1	3	yes	direct product factor
10	C₂²×C₃×C₅×C₄₁	C₃²	1	1	yes	characteristic
11	C₃²×C₅×C₄₁	C₂²×C₃	1	1	yes	characteristic
12	C₅×C₉×C₄₁	C₂²×C₃	1	3	yes	direct product factor
13	C₂²×C₃²×C₄₁	C₃×C₅	1	1	yes	characteristic
14	C₂²×C₉×C₄₁	C₃×C₅	1	3	yes	direct product factor
15	C₂×C₃×C₅×C₄₁	C₂×C₉	1	9	yes	direct product factor
16	C₂×C₃×C₅×C₄₁	C₂×C₃²	1	3	yes	—
17	C₃×C₉×C₄₁	C₂²×C₅	1	1	yes	characteristic, direct product factor, Hall
18	C₂²×C₅×C₄₁	C₃×C₉	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₃²×C₄₁	C₂×C₃×C₅	1	3	yes	—
20	C₂×C₉×C₄₁	C₂×C₃×C₅	1	9	yes	direct product factor
21	C₃×C₅×C₄₁	C₂²×C₉	1	3	yes	direct product factor
22	C₃×C₅×C₄₁	C₂²×C₃²	1	1	yes	characteristic
23	C₂²×C₃×C₅×C₉	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂²×C₃×C₄₁	C₅×C₉	1	3	yes	direct product factor
25	C₂²×C₃×C₄₁	C₃²×C₅	1	1	yes	characteristic
26	C₂×C₅×C₄₁	C₂×C₃×C₉	1	3	yes	direct product factor
27	C₃²×C₄₁	C₂²×C₃×C₅	1	1	yes	characteristic
28	C₉×C₄₁	C₂²×C₃×C₅	1	3	yes	direct product factor
29	C₂×C₃×C₅×C₉	C₂×C₄₁	1	3	yes	direct product factor
30	C₂×C₃×C₄₁	C₂×C₅×C₉	1	9	yes	direct product factor
31	C₂×C₃×C₄₁	C₂×C₃²×C₅	1	3	yes	—
32	C₅×C₄₁	C₂²×C₃×C₉	1	1	yes	characteristic, direct product factor, Hall
33	C₂²×C₃²×C₅	C₃×C₄₁	1	1	yes	characteristic
34	C₂²×C₅×C₉	C₃×C₄₁	1	3	yes	direct product factor
35	C₂²×C₄₁	C₃×C₅×C₉	1	1	yes	characteristic, direct product factor, Hall
36	C₃×C₅×C₉	C₂²×C₄₁	1	1	yes	characteristic, direct product factor, Hall
37	C₃×C₄₁	C₂²×C₅×C₉	1	3	yes	direct product factor
38	C₃×C₄₁	C₂²×C₃²×C₅	1	1	yes	characteristic
39	C₂²×C₃×C₉	C₅×C₄₁	1	1	yes	characteristic, direct product factor, Hall
40	C₂×C₃²×C₅	C₂×C₃×C₄₁	1	3	yes	—
41	C₂×C₅×C₉	C₂×C₃×C₄₁	1	9	yes	direct product factor
42	C₂×C₄₁	C₂×C₃×C₅×C₉	1	3	yes	direct product factor
43	C₂²×C₃×C₅	C₉×C₄₁	1	3	yes	direct product factor
44	C₂²×C₃×C₅	C₃²×C₄₁	1	1	yes	characteristic
45	C₂×C₃×C₉	C₂×C₅×C₄₁	1	3	yes	direct product factor
46	C₃²×C₅	C₂²×C₃×C₄₁	1	1	yes	characteristic
47	C₅×C₉	C₂²×C₃×C₄₁	1	3	yes	direct product factor
48	C₄₁	C₂²×C₃×C₅×C₉	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²×C₃²	C₃×C₅×C₄₁	1	1	yes	characteristic
50	C₂²×C₉	C₃×C₅×C₄₁	1	3	yes	direct product factor
51	C₂×C₃×C₅	C₂×C₉×C₄₁	1	9	yes	direct product factor
52	C₂×C₃×C₅	C₂×C₃²×C₄₁	1	3	yes	—
53	C₃×C₉	C₂²×C₅×C₄₁	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²×C₅	C₃×C₉×C₄₁	1	1	yes	characteristic, direct product factor, Hall
55	C₂×C₃²	C₂×C₃×C₅×C₄₁	1	3	yes	—
56	C₂×C₉	C₂×C₃×C₅×C₄₁	1	9	yes	direct product factor
57	C₃×C₅	C₂²×C₉×C₄₁	1	3	yes	direct product factor
58	C₃×C₅	C₂²×C₃²×C₄₁	1	1	yes	characteristic
59	C₂²×C₃	C₅×C₉×C₄₁	1	3	yes	direct product factor
60	C₂²×C₃	C₃²×C₅×C₄₁	1	1	yes	characteristic
61	C₂×C₅	C₂×C₃×C₉×C₄₁	1	3	yes	direct product factor
62	C₃²	C₂²×C₃×C₅×C₄₁	1	1	yes	characteristic
63	C₉	C₂²×C₃×C₅×C₄₁	1	3	yes	direct product factor
64	C₂×C₃	C₂×C₅×C₉×C₄₁	1	9	yes	direct product factor
65	C₂×C₃	C₂×C₃²×C₅×C₄₁	1	3	yes	—
66	C₅	C₂²×C₃×C₉×C₄₁	1	1	yes	characteristic, direct product factor, Sylow
67	C₂²	C₃×C₅×C₉×C₄₁	1	1	yes	characteristic, direct product factor, Sylow
68	C₃	C₂²×C₅×C₉×C₄₁	1	3	yes	direct product factor
69	C₃	C₂²×C₃²×C₅×C₄₁	1	1	yes	characteristic
70	C₂	C₂×C₃×C₅×C₉×C₄₁	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₂²×C₃²×C₅×C₄₁
socle, prime radical	C₂²×C₃²×C₅×C₄₁	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×C₅×C₉×C₄₁	C₂	direct (N × H)
2. C₃×C₅×C₉×C₄₁	C₂²	direct (N × H)
3. C₂²×C₃×C₉×C₄₁	C₅	direct (N × H)
4. C₂×C₃×C₉×C₄₁	C₂×C₅	direct (N × H)
6. C₂²×C₅×C₉×C₄₁	C₃	direct (N × H)
8. C₂×C₅×C₉×C₄₁	C₂×C₃	direct (N × H)
9. C₂²×C₃×C₅×C₄₁	C₉	direct (N × H)
12. C₅×C₉×C₄₁	C₂²×C₃	direct (N × H)
14. C₂²×C₉×C₄₁	C₃×C₅	direct (N × H)
15. C₂×C₃×C₅×C₄₁	C₂×C₉	direct (N × H)
17. C₃×C₉×C₄₁	C₂²×C₅	direct (N × H)
18. C₂²×C₅×C₄₁	C₃×C₉	direct (N × H)
20. C₂×C₉×C₄₁	C₂×C₃×C₅	direct (N × H)
21. C₃×C₅×C₄₁	C₂²×C₉	direct (N × H)
23. C₂²×C₃×C₅×C₉	C₄₁	direct (N × H)
24. C₂²×C₃×C₄₁	C₅×C₉	direct (N × H)
26. C₂×C₅×C₄₁	C₂×C₃×C₉	direct (N × H)
28. C₉×C₄₁	C₂²×C₃×C₅	direct (N × H)
29. C₂×C₃×C₅×C₉	C₂×C₄₁	direct (N × H)
30. C₂×C₃×C₄₁	C₂×C₅×C₉	direct (N × H)
32. C₅×C₄₁	C₂²×C₃×C₉	direct (N × H)
34. C₂²×C₅×C₉	C₃×C₄₁	direct (N × H)
35. C₂²×C₄₁	C₃×C₅×C₉	direct (N × H)
36. C₃×C₅×C₉	C₂²×C₄₁	direct (N × H)
37. C₃×C₄₁	C₂²×C₅×C₉	direct (N × H)
39. C₂²×C₃×C₉	C₅×C₄₁	direct (N × H)
41. C₂×C₅×C₉	C₂×C₃×C₄₁	direct (N × H)
42. C₂×C₄₁	C₂×C₃×C₅×C₉	direct (N × H)
43. C₂²×C₃×C₅	C₉×C₄₁	direct (N × H)
45. C₂×C₃×C₉	C₂×C₅×C₄₁	direct (N × H)
47. C₅×C₉	C₂²×C₃×C₄₁	direct (N × H)
48. C₄₁	C₂²×C₃×C₅×C₉	direct (N × H)
50. C₂²×C₉	C₃×C₅×C₄₁	direct (N × H)
51. C₂×C₃×C₅	C₂×C₉×C₄₁	direct (N × H)
53. C₃×C₉	C₂²×C₅×C₄₁	direct (N × H)
54. C₂²×C₅	C₃×C₉×C₄₁	direct (N × H)
56. C₂×C₉	C₂×C₃×C₅×C₄₁	direct (N × H)
57. C₃×C₅	C₂²×C₉×C₄₁	direct (N × H)
59. C₂²×C₃	C₅×C₉×C₄₁	direct (N × H)
61. C₂×C₅	C₂×C₃×C₉×C₄₁	direct (N × H)
63. C₉	C₂²×C₃×C₅×C₄₁	direct (N × H)
64. C₂×C₃	C₂×C₅×C₉×C₄₁	direct (N × H)
66. C₅	C₂²×C₃×C₉×C₄₁	direct (N × H)
67. C₂²	C₃×C₅×C₉×C₄₁	direct (N × H)
68. C₃	C₂²×C₅×C₉×C₄₁	direct (N × H)
70. C₂	C₂×C₃×C₅×C₉×C₄₁	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×C₅×C₉×C₄₁	2
C₂²×C₃×C₉×C₄₁	5
C₂²×C₃²×C₅×C₄₁	3
C₂²×C₅×C₉×C₄₁	3
C₂²×C₃×C₅×C₉	41

Subgroups — C22×C3×C5×C9×C41 #72199613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅×C₉×C₄₁ #72199613