Subgroups — C₃³×C₄×C₅²×C₉ #72201576

Order 24300.

9504 conjugacy classes of subgroups.

The subgroup lattice has 108 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₅²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃⁴×C₄×C₅²	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₄×C₅²×C₉	C₃	1	39	yes	maximal, direct product factor
4	C₃³×C₅²×C₉	C₄	1	1	yes	characteristic, direct product factor, Hall
5	C₃³×C₄×C₅×C₉	C₅	1	6	yes	maximal, direct product factor
6	C₂×C₃⁴×C₅²	C₂×C₃	1	1	yes	characteristic
7	C₂×C₃²×C₅²×C₉	C₂×C₃	1	39	yes	—
8	C₃³×C₄×C₅²	C₉	1	27	yes	direct product factor
9	C₃³×C₄×C₅²	C₃²	1	13	yes	—
10	C₃×C₄×C₅²×C₉	C₃²	1	117	yes	direct product factor
11	C₂×C₃³×C₅×C₉	C₂×C₅	1	6	yes	—
12	C₃⁴×C₅²	C₃×C₄	1	1	yes	characteristic
13	C₃²×C₅²×C₉	C₃×C₄	1	39	yes	direct product factor
14	C₃⁴×C₄×C₅	C₃×C₅	1	6	yes	—
15	C₃²×C₄×C₅×C₉	C₃×C₅	1	234	yes	direct product factor
16	C₂×C₃³×C₅²	C₂×C₉	1	27	yes	—
17	C₂×C₃³×C₅²	C₂×C₃²	1	13	yes	—
18	C₂×C₃×C₅²×C₉	C₂×C₃²	1	117	yes	—
19	C₃³×C₅×C₉	C₄×C₅	1	6	yes	direct product factor
20	C₃³×C₄×C₉	C₅²	1	1	yes	characteristic, direct product factor, Hall
21	C₃²×C₄×C₅²	C₃×C₉	1	117	yes	direct product factor
22	C₃²×C₄×C₅²	C₃³	1	13	yes	—
23	C₄×C₅²×C₉	C₃³	1	27	yes	direct product factor
24	C₂×C₃⁴×C₅	C₂×C₃×C₅	1	6	yes	—
25	C₂×C₃²×C₅×C₉	C₂×C₃×C₅	1	234	yes	—
26	C₃³×C₅²	C₄×C₉	1	27	yes	direct product factor
27	C₃³×C₅²	C₃²×C₄	1	13	yes	—
28	C₃×C₅²×C₉	C₃²×C₄	1	117	yes	direct product factor
29	C₃³×C₄×C₅	C₅×C₉	1	162	yes	direct product factor
30	C₃³×C₄×C₅	C₃²×C₅	1	78	yes	—
31	C₃×C₄×C₅×C₉	C₃²×C₅	1	702	yes	direct product factor
32	C₂×C₃³×C₉	C₂×C₅²	1	1	yes	characteristic
33	C₂×C₃²×C₅²	C₂×C₃×C₉	1	117	yes	—
34	C₂×C₃²×C₅²	C₂×C₃³	1	13	yes	—
35	C₂×C₅²×C₉	C₂×C₃³	1	27	yes	—
36	C₃⁴×C₅	C₃×C₄×C₅	1	6	yes	—
37	C₃²×C₅×C₉	C₃×C₄×C₅	1	234	yes	direct product factor
38	C₃⁴×C₄	C₃×C₅²	1	1	yes	characteristic
39	C₃²×C₄×C₉	C₃×C₅²	1	39	yes	direct product factor
40	C₃×C₄×C₅²	C₃²×C₉	1	39	yes	direct product factor
41	C₃×C₄×C₅²	C₃⁴	1	1	yes	characteristic
42	C₂×C₃³×C₅	C₂×C₅×C₉	1	162	yes	—
43	C₂×C₃³×C₅	C₂×C₃²×C₅	1	78	yes	—
44	C₂×C₃×C₅×C₉	C₂×C₃²×C₅	1	702	yes	—
45	C₃³×C₉	C₄×C₅²	1	1	yes	characteristic, direct product factor, Sylow
46	C₃²×C₅²	C₃×C₄×C₉	1	117	yes	direct product factor
47	C₃²×C₅²	C₃³×C₄	1	13	yes	—
48	C₅²×C₉	C₃³×C₄	1	27	yes	direct product factor
49	C₃²×C₄×C₅	C₃×C₅×C₉	1	702	yes	direct product factor
50	C₃²×C₄×C₅	C₃³×C₅	1	78	yes	—
51	C₄×C₅×C₉	C₃³×C₅	1	162	yes	direct product factor
52	C₂×C₃⁴	C₂×C₃×C₅²	1	1	yes	characteristic
53	C₂×C₃²×C₉	C₂×C₃×C₅²	1	39	yes	—
54	C₂×C₃×C₅²	C₂×C₃²×C₉	1	39	yes	—
55	C₂×C₃×C₅²	C₂×C₃⁴	1	1	yes	characteristic
56	C₃³×C₅	C₄×C₅×C₉	1	162	yes	direct product factor
57	C₃³×C₅	C₃²×C₄×C₅	1	78	yes	—
58	C₃×C₅×C₉	C₃²×C₄×C₅	1	702	yes	direct product factor
59	C₃³×C₄	C₅²×C₉	1	27	yes	direct product factor
60	C₃³×C₄	C₃²×C₅²	1	13	yes	—
61	C₃×C₄×C₉	C₃²×C₅²	1	117	yes	direct product factor
62	C₄×C₅²	C₃³×C₉	1	1	yes	characteristic, direct product factor, Hall
63	C₂×C₃²×C₅	C₂×C₃×C₅×C₉	1	702	yes	—
64	C₂×C₃²×C₅	C₂×C₃³×C₅	1	78	yes	—
65	C₂×C₅×C₉	C₂×C₃³×C₅	1	162	yes	—
66	C₃⁴	C₃×C₄×C₅²	1	1	yes	characteristic
67	C₃²×C₉	C₃×C₄×C₅²	1	39	yes	direct product factor
68	C₃×C₅²	C₃²×C₄×C₉	1	39	yes	direct product factor
69	C₃×C₅²	C₃⁴×C₄	1	1	yes	characteristic
70	C₃×C₄×C₅	C₃²×C₅×C₉	1	234	yes	direct product factor
71	C₃×C₄×C₅	C₃⁴×C₅	1	6	yes	—
72	C₂×C₃³	C₂×C₅²×C₉	1	27	yes	—
73	C₂×C₃³	C₂×C₃²×C₅²	1	13	yes	—
74	C₂×C₃×C₉	C₂×C₃²×C₅²	1	117	yes	—
75	C₂×C₅²	C₂×C₃³×C₉	1	1	yes	characteristic
76	C₃²×C₅	C₃×C₄×C₅×C₉	1	702	yes	direct product factor
77	C₃²×C₅	C₃³×C₄×C₅	1	78	yes	—
78	C₅×C₉	C₃³×C₄×C₅	1	162	yes	direct product factor
79	C₃²×C₄	C₃×C₅²×C₉	1	117	yes	direct product factor
80	C₃²×C₄	C₃³×C₅²	1	13	yes	—
81	C₄×C₉	C₃³×C₅²	1	27	yes	direct product factor
82	C₂×C₃×C₅	C₂×C₃²×C₅×C₉	1	234	yes	—
83	C₂×C₃×C₅	C₂×C₃⁴×C₅	1	6	yes	—
84	C₃³	C₄×C₅²×C₉	1	27	yes	direct product factor
85	C₃³	C₃²×C₄×C₅²	1	13	yes	—
86	C₃×C₉	C₃²×C₄×C₅²	1	117	yes	direct product factor
87	C₅²	C₃³×C₄×C₉	1	1	yes	characteristic, direct product factor, Sylow
88	C₄×C₅	C₃³×C₅×C₉	1	6	yes	direct product factor
89	C₂×C₃²	C₂×C₃×C₅²×C₉	1	117	yes	—
90	C₂×C₃²	C₂×C₃³×C₅²	1	13	yes	—
91	C₂×C₉	C₂×C₃³×C₅²	1	27	yes	—
92	C₃×C₅	C₃²×C₄×C₅×C₉	1	234	yes	direct product factor
93	C₃×C₅	C₃⁴×C₄×C₅	1	6	yes	—
94	C₃×C₄	C₃²×C₅²×C₉	1	39	yes	direct product factor
95	C₃×C₄	C₃⁴×C₅²	1	1	yes	characteristic
96	C₂×C₅	C₂×C₃³×C₅×C₉	1	6	yes	—
97	C₃²	C₃×C₄×C₅²×C₉	1	117	yes	direct product factor
98	C₃²	C₃³×C₄×C₅²	1	13	yes	—
99	C₉	C₃³×C₄×C₅²	1	27	yes	direct product factor
100	C₂×C₃	C₂×C₃²×C₅²×C₉	1	39	yes	—
101	C₂×C₃	C₂×C₃⁴×C₅²	1	1	yes	characteristic
102	C₅	C₃³×C₄×C₅×C₉	1	6	yes	direct product factor
103	C₄	C₃³×C₅²×C₉	1	1	yes	characteristic, direct product factor, Sylow
104	C₃	C₃²×C₄×C₅²×C₉	1	39	yes	direct product factor
105	C₃	C₃⁴×C₄×C₅²	1	1	yes	characteristic
106	C₂	C₂×C₃³×C₅²×C₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃	C₂×C₃⁴×C₅²
socle, prime radical	C₂×C₃⁴×C₅²	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×C₅²×C₉	C₃	direct (N × H)
4. C₃³×C₅²×C₉	C₄	direct (N × H)
5. C₃³×C₄×C₅×C₉	C₅	direct (N × H)
8. C₃³×C₄×C₅²	C₉	direct (N × H)
10. C₃×C₄×C₅²×C₉	C₃²	direct (N × H)
13. C₃²×C₅²×C₉	C₃×C₄	direct (N × H)
15. C₃²×C₄×C₅×C₉	C₃×C₅	direct (N × H)
19. C₃³×C₅×C₉	C₄×C₅	direct (N × H)
20. C₃³×C₄×C₉	C₅²	direct (N × H)
21. C₃²×C₄×C₅²	C₃×C₉	direct (N × H)
23. C₄×C₅²×C₉	C₃³	direct (N × H)
26. C₃³×C₅²	C₄×C₉	direct (N × H)
28. C₃×C₅²×C₉	C₃²×C₄	direct (N × H)
29. C₃³×C₄×C₅	C₅×C₉	direct (N × H)
31. C₃×C₄×C₅×C₉	C₃²×C₅	direct (N × H)
37. C₃²×C₅×C₉	C₃×C₄×C₅	direct (N × H)
39. C₃²×C₄×C₉	C₃×C₅²	direct (N × H)
40. C₃×C₄×C₅²	C₃²×C₉	direct (N × H)
45. C₃³×C₉	C₄×C₅²	direct (N × H)
46. C₃²×C₅²	C₃×C₄×C₉	direct (N × H)
48. C₅²×C₉	C₃³×C₄	direct (N × H)
49. C₃²×C₄×C₅	C₃×C₅×C₉	direct (N × H)
51. C₄×C₅×C₉	C₃³×C₅	direct (N × H)
56. C₃³×C₅	C₄×C₅×C₉	direct (N × H)
58. C₃×C₅×C₉	C₃²×C₄×C₅	direct (N × H)
59. C₃³×C₄	C₅²×C₉	direct (N × H)
61. C₃×C₄×C₉	C₃²×C₅²	direct (N × H)
62. C₄×C₅²	C₃³×C₉	direct (N × H)
67. C₃²×C₉	C₃×C₄×C₅²	direct (N × H)
68. C₃×C₅²	C₃²×C₄×C₉	direct (N × H)
70. C₃×C₄×C₅	C₃²×C₅×C₉	direct (N × H)
76. C₃²×C₅	C₃×C₄×C₅×C₉	direct (N × H)
78. C₅×C₉	C₃³×C₄×C₅	direct (N × H)
79. C₃²×C₄	C₃×C₅²×C₉	direct (N × H)
81. C₄×C₉	C₃³×C₅²	direct (N × H)
84. C₃³	C₄×C₅²×C₉	direct (N × H)
86. C₃×C₉	C₃²×C₄×C₅²	direct (N × H)
87. C₅²	C₃³×C₄×C₉	direct (N × H)
88. C₄×C₅	C₃³×C₅×C₉	direct (N × H)
92. C₃×C₅	C₃²×C₄×C₅×C₉	direct (N × H)
94. C₃×C₄	C₃²×C₅²×C₉	direct (N × H)
97. C₃²	C₃×C₄×C₅²×C₉	direct (N × H)
99. C₉	C₃³×C₄×C₅²	direct (N × H)
102. C₅	C₃³×C₄×C₅×C₉	direct (N × H)
103. C₄	C₃³×C₅²×C₉	direct (N × H)
104. C₃	C₃²×C₄×C₅²×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×C₅²×C₉	2
C₃⁴×C₄×C₅²	3
C₃²×C₄×C₅²×C₉	3
C₃³×C₄×C₅×C₉	5

Subgroups — C33×C4×C52×C9 #72201576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×C₄×C₅²×C₉ #72201576