Subgroups — C₃⁶×C₄×C₉ #72203404

Order 26244.

12146040 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⁶×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃⁷×C₄	C₃	1	1	yes	maximal, characteristic
3	C₃⁵×C₄×C₉	C₃	1	1092	yes	maximal, direct product factor
4	C₃⁶×C₉	C₄	1	1	yes	characteristic, direct product factor, Sylow
5	C₂×C₃⁷	C₂×C₃	1	1	yes	characteristic
6	C₂×C₃⁵×C₉	C₂×C₃	1	1092	yes	—
7	C₃⁴×C₄×C₉	C₃²	1	99099	yes	direct product factor
8	C₃⁶×C₄	C₉	1	729	yes	direct product factor
9	C₃⁶×C₄	C₃²	1	364	yes	—
10	C₃⁷	C₃×C₄	1	1	yes	characteristic
11	C₃⁵×C₉	C₃×C₄	1	1092	yes	direct product factor
12	C₂×C₃⁶	C₂×C₉	1	729	yes	—
13	C₂×C₃⁶	C₂×C₃²	1	364	yes	—
14	C₂×C₃⁴×C₉	C₂×C₃²	1	99099	yes	—
15	C₃⁵×C₄	C₃×C₉	1	88452	yes	direct product factor
16	C₃⁵×C₄	C₃³	1	11011	yes	—
17	C₃³×C₄×C₉	C₃³	1	914760	yes	direct product factor
18	C₃⁶	C₄×C₉	1	729	yes	direct product factor
19	C₃⁶	C₃²×C₄	1	364	yes	—
20	C₃⁴×C₉	C₃²×C₄	1	99099	yes	direct product factor
21	C₂×C₃⁵	C₂×C₃×C₉	1	88452	yes	—
22	C₂×C₃⁵	C₂×C₃³	1	11011	yes	—
23	C₂×C₃³×C₉	C₂×C₃³	1	914760	yes	—
24	C₃⁴×C₄	C₃²×C₉	1	891891	yes	direct product factor
25	C₃⁴×C₄	C₃⁴	1	33880	yes	—
26	C₃²×C₄×C₉	C₃⁴	1	891891	yes	direct product factor
27	C₃⁵	C₃×C₄×C₉	1	88452	yes	direct product factor
28	C₃⁵	C₃³×C₄	1	11011	yes	—
29	C₃³×C₉	C₃³×C₄	1	914760	yes	direct product factor
30	C₂×C₃⁴	C₂×C₃²×C₉	1	891891	yes	—
31	C₂×C₃⁴	C₂×C₃⁴	1	33880	yes	—
32	C₂×C₃²×C₉	C₂×C₃⁴	1	891891	yes	—
33	C₃³×C₄	C₃³×C₉	1	914760	yes	direct product factor
34	C₃³×C₄	C₃⁵	1	11011	yes	—
35	C₃×C₄×C₉	C₃⁵	1	88452	yes	direct product factor
36	C₃⁴	C₃²×C₄×C₉	1	891891	yes	direct product factor
37	C₃⁴	C₃⁴×C₄	1	33880	yes	—
38	C₃²×C₉	C₃⁴×C₄	1	891891	yes	direct product factor
39	C₂×C₃³	C₂×C₃³×C₉	1	914760	yes	—
40	C₂×C₃³	C₂×C₃⁵	1	11011	yes	—
41	C₂×C₃×C₉	C₂×C₃⁵	1	88452	yes	—
42	C₃²×C₄	C₃⁴×C₉	1	99099	yes	direct product factor
43	C₃²×C₄	C₃⁶	1	364	yes	—
44	C₄×C₉	C₃⁶	1	729	yes	direct product factor
45	C₃³	C₃³×C₄×C₉	1	914760	yes	direct product factor
46	C₃³	C₃⁵×C₄	1	11011	yes	—
47	C₃×C₉	C₃⁵×C₄	1	88452	yes	direct product factor
48	C₂×C₃²	C₂×C₃⁴×C₉	1	99099	yes	—
49	C₂×C₃²	C₂×C₃⁶	1	364	yes	—
50	C₂×C₉	C₂×C₃⁶	1	729	yes	—
51	C₃×C₄	C₃⁵×C₉	1	1092	yes	direct product factor
52	C₃×C₄	C₃⁷	1	1	yes	characteristic
53	C₃²	C₃⁶×C₄	1	364	yes	—
54	C₃²	C₃⁴×C₄×C₉	1	99099	yes	direct product factor
55	C₉	C₃⁶×C₄	1	729	yes	direct product factor
56	C₂×C₃	C₂×C₃⁵×C₉	1	1092	yes	—
57	C₂×C₃	C₂×C₃⁷	1	1	yes	characteristic
58	C₄	C₃⁶×C₉	1	1	yes	characteristic, direct product factor, Sylow
59	C₃	C₃⁵×C₄×C₉	1	1092	yes	direct product factor
60	C₃	C₃⁷×C₄	1	1	yes	characteristic
61	C₂	C₂×C₃⁶×C₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃	C₂×C₃⁷
socle, prime radical	C₂×C₃⁷	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. C₃⁵×C₄×C₉	C₃	direct (N × H)
4. C₃⁶×C₉	C₄	direct (N × H)
7. C₃⁴×C₄×C₉	C₃²	direct (N × H)
8. C₃⁶×C₄	C₉	direct (N × H)
11. C₃⁵×C₉	C₃×C₄	direct (N × H)
15. C₃⁵×C₄	C₃×C₉	direct (N × H)
17. C₃³×C₄×C₉	C₃³	direct (N × H)
18. C₃⁶	C₄×C₉	direct (N × H)
20. C₃⁴×C₉	C₃²×C₄	direct (N × H)
24. C₃⁴×C₄	C₃²×C₉	direct (N × H)
26. C₃²×C₄×C₉	C₃⁴	direct (N × H)
27. C₃⁵	C₃×C₄×C₉	direct (N × H)
29. C₃³×C₉	C₃³×C₄	direct (N × H)
33. C₃³×C₄	C₃³×C₉	direct (N × H)
35. C₃×C₄×C₉	C₃⁵	direct (N × H)
36. C₃⁴	C₃²×C₄×C₉	direct (N × H)
38. C₃²×C₉	C₃⁴×C₄	direct (N × H)
42. C₃²×C₄	C₃⁴×C₉	direct (N × H)
44. C₄×C₉	C₃⁶	direct (N × H)
45. C₃³	C₃³×C₄×C₉	direct (N × H)
47. C₃×C₉	C₃⁵×C₄	direct (N × H)
51. C₃×C₄	C₃⁵×C₉	direct (N × H)
54. C₃²	C₃⁴×C₄×C₉	direct (N × H)
55. C₉	C₃⁶×C₄	direct (N × H)
58. C₄	C₃⁶×C₉	direct (N × H)
59. C₃	C₃⁵×C₄×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×C₃⁶×C₉	2
C₃⁷×C₄	3
C₃⁵×C₄×C₉	3

Subgroups — C36×C4×C9 #72203404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⁶×C₄×C₉ #72203404