Subgroups — C₂×C₃⁴×C₅×C₉ #72186331

Order 7290.

20464 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₅×C₉	C₃	1	120	yes	maximal, direct product factor
2	C₂×C₃⁵×C₅	C₃	1	1	yes	maximal, characteristic
3	C₃⁴×C₅×C₉	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₃⁴×C₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃⁵×C₅	C₂×C₃	1	1	yes	characteristic
6	C₃³×C₅×C₉	C₂×C₃	1	120	yes	direct product factor
7	C₂×C₃⁴×C₅	C₉	1	81	yes	direct product factor
8	C₂×C₃⁴×C₅	C₃²	1	40	yes	—
9	C₂×C₃²×C₅×C₉	C₃²	1	1170	yes	direct product factor
10	C₃⁴×C₉	C₂×C₅	1	1	yes	characteristic, direct product factor, Sylow
11	C₂×C₃⁵	C₃×C₅	1	1	yes	characteristic
12	C₂×C₃³×C₉	C₃×C₅	1	120	yes	direct product factor
13	C₃⁴×C₅	C₂×C₉	1	81	yes	direct product factor
14	C₃⁴×C₅	C₂×C₃²	1	40	yes	—
15	C₃²×C₅×C₉	C₂×C₃²	1	1170	yes	direct product factor
16	C₂×C₃³×C₅	C₃×C₉	1	1080	yes	direct product factor
17	C₂×C₃³×C₅	C₃³	1	130	yes	—
18	C₂×C₃×C₅×C₉	C₃³	1	1080	yes	direct product factor
19	C₃⁵	C₂×C₃×C₅	1	1	yes	characteristic
20	C₃³×C₉	C₂×C₃×C₅	1	120	yes	direct product factor
21	C₂×C₃⁴	C₅×C₉	1	81	yes	direct product factor
22	C₂×C₃⁴	C₃²×C₅	1	40	yes	—
23	C₂×C₃²×C₉	C₃²×C₅	1	1170	yes	direct product factor
24	C₃³×C₅	C₂×C₃×C₉	1	1080	yes	direct product factor
25	C₃³×C₅	C₂×C₃³	1	130	yes	—
26	C₃×C₅×C₉	C₂×C₃³	1	1080	yes	direct product factor
27	C₂×C₃²×C₅	C₃²×C₉	1	1170	yes	direct product factor
28	C₂×C₃²×C₅	C₃⁴	1	40	yes	—
29	C₂×C₅×C₉	C₃⁴	1	81	yes	direct product factor
30	C₃⁴	C₂×C₅×C₉	1	81	yes	direct product factor
31	C₃⁴	C₂×C₃²×C₅	1	40	yes	—
32	C₃²×C₉	C₂×C₃²×C₅	1	1170	yes	direct product factor
33	C₂×C₃³	C₃×C₅×C₉	1	1080	yes	direct product factor
34	C₂×C₃³	C₃³×C₅	1	130	yes	—
35	C₂×C₃×C₉	C₃³×C₅	1	1080	yes	direct product factor
36	C₃²×C₅	C₂×C₃²×C₉	1	1170	yes	direct product factor
37	C₃²×C₅	C₂×C₃⁴	1	40	yes	—
38	C₅×C₉	C₂×C₃⁴	1	81	yes	direct product factor
39	C₂×C₃×C₅	C₃³×C₉	1	120	yes	direct product factor
40	C₂×C₃×C₅	C₃⁵	1	1	yes	characteristic
41	C₃³	C₂×C₃×C₅×C₉	1	1080	yes	direct product factor
42	C₃³	C₂×C₃³×C₅	1	130	yes	—
43	C₃×C₉	C₂×C₃³×C₅	1	1080	yes	direct product factor
44	C₂×C₃²	C₃²×C₅×C₉	1	1170	yes	direct product factor
45	C₂×C₃²	C₃⁴×C₅	1	40	yes	—
46	C₂×C₉	C₃⁴×C₅	1	81	yes	direct product factor
47	C₃×C₅	C₂×C₃³×C₉	1	120	yes	direct product factor
48	C₃×C₅	C₂×C₃⁵	1	1	yes	characteristic
49	C₂×C₅	C₃⁴×C₉	1	1	yes	characteristic, direct product factor, Hall
50	C₃²	C₂×C₃²×C₅×C₉	1	1170	yes	direct product factor
51	C₃²	C₂×C₃⁴×C₅	1	40	yes	—
52	C₉	C₂×C₃⁴×C₅	1	81	yes	direct product factor
53	C₂×C₃	C₃³×C₅×C₉	1	120	yes	direct product factor
54	C₂×C₃	C₃⁵×C₅	1	1	yes	characteristic
55	C₅	C₂×C₃⁴×C₉	1	1	yes	characteristic, direct product factor, Sylow
56	C₃	C₂×C₃⁵×C₅	1	1	yes	characteristic
57	C₃	C₂×C₃³×C₅×C₉	1	120	yes	direct product factor
58	C₂	C₃⁴×C₅×C₉	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₂×C₃⁵×C₅
socle, prime radical	C₂×C₃⁵×C₅	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×C₃³×C₅×C₉	C₃	direct (N × H)
3. C₃⁴×C₅×C₉	C₂	direct (N × H)
4. C₂×C₃⁴×C₉	C₅	direct (N × H)
6. C₃³×C₅×C₉	C₂×C₃	direct (N × H)
7. C₂×C₃⁴×C₅	C₉	direct (N × H)
9. C₂×C₃²×C₅×C₉	C₃²	direct (N × H)
10. C₃⁴×C₉	C₂×C₅	direct (N × H)
12. C₂×C₃³×C₉	C₃×C₅	direct (N × H)
13. C₃⁴×C₅	C₂×C₉	direct (N × H)
15. C₃²×C₅×C₉	C₂×C₃²	direct (N × H)
16. C₂×C₃³×C₅	C₃×C₉	direct (N × H)
18. C₂×C₃×C₅×C₉	C₃³	direct (N × H)
20. C₃³×C₉	C₂×C₃×C₅	direct (N × H)
21. C₂×C₃⁴	C₅×C₉	direct (N × H)
23. C₂×C₃²×C₉	C₃²×C₅	direct (N × H)
24. C₃³×C₅	C₂×C₃×C₉	direct (N × H)
26. C₃×C₅×C₉	C₂×C₃³	direct (N × H)
27. C₂×C₃²×C₅	C₃²×C₉	direct (N × H)
29. C₂×C₅×C₉	C₃⁴	direct (N × H)
30. C₃⁴	C₂×C₅×C₉	direct (N × H)
32. C₃²×C₉	C₂×C₃²×C₅	direct (N × H)
33. C₂×C₃³	C₃×C₅×C₉	direct (N × H)
35. C₂×C₃×C₉	C₃³×C₅	direct (N × H)
36. C₃²×C₅	C₂×C₃²×C₉	direct (N × H)
38. C₅×C₉	C₂×C₃⁴	direct (N × H)
39. C₂×C₃×C₅	C₃³×C₉	direct (N × H)
41. C₃³	C₂×C₃×C₅×C₉	direct (N × H)
43. C₃×C₉	C₂×C₃³×C₅	direct (N × H)
44. C₂×C₃²	C₃²×C₅×C₉	direct (N × H)
46. C₂×C₉	C₃⁴×C₅	direct (N × H)
47. C₃×C₅	C₂×C₃³×C₉	direct (N × H)
49. C₂×C₅	C₃⁴×C₉	direct (N × H)
50. C₃²	C₂×C₃²×C₅×C₉	direct (N × H)
52. C₉	C₂×C₃⁴×C₅	direct (N × H)
53. C₂×C₃	C₃³×C₅×C₉	direct (N × H)
55. C₅	C₂×C₃⁴×C₉	direct (N × H)
57. C₃	C₂×C₃³×C₅×C₉	direct (N × H)
58. C₂	C₃⁴×C₅×C₉	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×C₅×C₉	3
C₂×C₃⁵×C₅	3
C₃⁴×C₅×C₉	2
C₂×C₃⁴×C₉	5

Subgroups — C2×C34×C5×C9 #72186331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⁴×C₅×C₉ #72186331