Subgroups — C₃×C₄×C₅²×C₉² #72201573

Order 24300.

3024 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×C₃×C₅²×C₉²	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄×C₅²×C₉	C₃	1	4	yes	maximal
3	C₄×C₅²×C₉²	C₃	1	9	yes	maximal, direct product factor
4	C₃×C₅²×C₉²	C₄	1	1	yes	characteristic, direct product factor, Hall
5	C₃×C₄×C₅×C₉²	C₅	1	6	yes	maximal, direct product factor
6	C₂×C₃²×C₅²×C₉	C₂×C₃	1	4	yes	—
7	C₂×C₅²×C₉²	C₂×C₃	1	9	yes	—
8	C₃³×C₄×C₅²	C₃²	1	1	yes	characteristic
9	C₃×C₄×C₅²×C₉	C₉	1	36	yes	direct product factor
10	C₃×C₄×C₅²×C₉	C₃²	1	12	yes	—
11	C₂×C₃×C₅×C₉²	C₂×C₅	1	6	yes	—
12	C₃²×C₅²×C₉	C₃×C₄	1	4	yes	—
13	C₅²×C₉²	C₃×C₄	1	9	yes	direct product factor
14	C₃²×C₄×C₅×C₉	C₃×C₅	1	24	yes	—
15	C₄×C₅×C₉²	C₃×C₅	1	54	yes	direct product factor
16	C₂×C₃³×C₅²	C₂×C₃²	1	1	yes	characteristic
17	C₂×C₃×C₅²×C₉	C₂×C₉	1	36	yes	—
18	C₂×C₃×C₅²×C₉	C₂×C₃²	1	12	yes	—
19	C₃×C₅×C₉²	C₄×C₅	1	6	yes	direct product factor
20	C₃×C₄×C₉²	C₅²	1	1	yes	characteristic, direct product factor, Hall
21	C₃²×C₄×C₅²	C₃×C₉	1	12	yes	—
22	C₃²×C₄×C₅²	C₃³	1	1	yes	characteristic
23	C₄×C₅²×C₉	C₃×C₉	1	36	yes	direct product factor
24	C₂×C₃²×C₅×C₉	C₂×C₃×C₅	1	24	yes	—
25	C₂×C₅×C₉²	C₂×C₃×C₅	1	54	yes	—
26	C₃³×C₅²	C₃²×C₄	1	1	yes	characteristic
27	C₃×C₅²×C₉	C₄×C₉	1	36	yes	direct product factor
28	C₃×C₅²×C₉	C₃²×C₄	1	12	yes	—
29	C₃³×C₄×C₅	C₃²×C₅	1	6	yes	—
30	C₃×C₄×C₅×C₉	C₅×C₉	1	216	yes	direct product factor
31	C₃×C₄×C₅×C₉	C₃²×C₅	1	72	yes	—
32	C₂×C₃×C₉²	C₂×C₅²	1	1	yes	characteristic
33	C₂×C₃²×C₅²	C₂×C₃×C₉	1	12	yes	—
34	C₂×C₃²×C₅²	C₂×C₃³	1	1	yes	characteristic
35	C₂×C₅²×C₉	C₂×C₃×C₉	1	36	yes	—
36	C₃²×C₅×C₉	C₃×C₄×C₅	1	24	yes	—
37	C₅×C₉²	C₃×C₄×C₅	1	54	yes	direct product factor
38	C₃²×C₄×C₉	C₃×C₅²	1	4	yes	—
39	C₄×C₉²	C₃×C₅²	1	9	yes	direct product factor
40	C₃×C₄×C₅²	C₉²	1	9	yes	direct product factor
41	C₃×C₄×C₅²	C₃²×C₉	1	4	yes	—
42	C₂×C₃³×C₅	C₂×C₃²×C₅	1	6	yes	—
43	C₂×C₃×C₅×C₉	C₂×C₅×C₉	1	216	yes	—
44	C₂×C₃×C₅×C₉	C₂×C₃²×C₅	1	72	yes	—
45	C₃×C₉²	C₄×C₅²	1	1	yes	characteristic, direct product factor, Sylow
46	C₃²×C₅²	C₃×C₄×C₉	1	12	yes	—
47	C₃²×C₅²	C₃³×C₄	1	1	yes	characteristic
48	C₅²×C₉	C₃×C₄×C₉	1	36	yes	direct product factor
49	C₃²×C₄×C₅	C₃×C₅×C₉	1	72	yes	—
50	C₃²×C₄×C₅	C₃³×C₅	1	6	yes	—
51	C₄×C₅×C₉	C₃×C₅×C₉	1	216	yes	direct product factor
52	C₂×C₃²×C₉	C₂×C₃×C₅²	1	4	yes	—
53	C₂×C₉²	C₂×C₃×C₅²	1	9	yes	—
54	C₂×C₃×C₅²	C₂×C₉²	1	9	yes	—
55	C₂×C₃×C₅²	C₂×C₃²×C₉	1	4	yes	—
56	C₃³×C₅	C₃²×C₄×C₅	1	6	yes	—
57	C₃×C₅×C₉	C₄×C₅×C₉	1	216	yes	direct product factor
58	C₃×C₅×C₉	C₃²×C₄×C₅	1	72	yes	—
59	C₃³×C₄	C₃²×C₅²	1	1	yes	characteristic
60	C₃×C₄×C₉	C₅²×C₉	1	36	yes	direct product factor
61	C₃×C₄×C₉	C₃²×C₅²	1	12	yes	—
62	C₄×C₅²	C₃×C₉²	1	1	yes	characteristic, direct product factor, Hall
63	C₂×C₃²×C₅	C₂×C₃×C₅×C₉	1	72	yes	—
64	C₂×C₃²×C₅	C₂×C₃³×C₅	1	6	yes	—
65	C₂×C₅×C₉	C₂×C₃×C₅×C₉	1	216	yes	—
66	C₃²×C₉	C₃×C₄×C₅²	1	4	yes	—
67	C₉²	C₃×C₄×C₅²	1	9	yes	direct product factor
68	C₃×C₅²	C₄×C₉²	1	9	yes	direct product factor
69	C₃×C₅²	C₃²×C₄×C₉	1	4	yes	—
70	C₃×C₄×C₅	C₅×C₉²	1	54	yes	direct product factor
71	C₃×C₄×C₅	C₃²×C₅×C₉	1	24	yes	—
72	C₂×C₃³	C₂×C₃²×C₅²	1	1	yes	characteristic
73	C₂×C₃×C₉	C₂×C₅²×C₉	1	36	yes	—
74	C₂×C₃×C₉	C₂×C₃²×C₅²	1	12	yes	—
75	C₂×C₅²	C₂×C₃×C₉²	1	1	yes	characteristic
76	C₃²×C₅	C₃×C₄×C₅×C₉	1	72	yes	—
77	C₃²×C₅	C₃³×C₄×C₅	1	6	yes	—
78	C₅×C₉	C₃×C₄×C₅×C₉	1	216	yes	direct product factor
79	C₃²×C₄	C₃×C₅²×C₉	1	12	yes	—
80	C₃²×C₄	C₃³×C₅²	1	1	yes	characteristic
81	C₄×C₉	C₃×C₅²×C₉	1	36	yes	direct product factor
82	C₂×C₃×C₅	C₂×C₅×C₉²	1	54	yes	—
83	C₂×C₃×C₅	C₂×C₃²×C₅×C₉	1	24	yes	—
84	C₃³	C₃²×C₄×C₅²	1	1	yes	characteristic
85	C₃×C₉	C₄×C₅²×C₉	1	36	yes	direct product factor
86	C₃×C₉	C₃²×C₄×C₅²	1	12	yes	—
87	C₅²	C₃×C₄×C₉²	1	1	yes	characteristic, direct product factor, Sylow
88	C₄×C₅	C₃×C₅×C₉²	1	6	yes	direct product factor
89	C₂×C₃²	C₂×C₃×C₅²×C₉	1	12	yes	—
90	C₂×C₃²	C₂×C₃³×C₅²	1	1	yes	characteristic
91	C₂×C₉	C₂×C₃×C₅²×C₉	1	36	yes	—
92	C₃×C₅	C₄×C₅×C₉²	1	54	yes	direct product factor
93	C₃×C₅	C₃²×C₄×C₅×C₉	1	24	yes	—
94	C₃×C₄	C₅²×C₉²	1	9	yes	direct product factor
95	C₃×C₄	C₃²×C₅²×C₉	1	4	yes	—
96	C₂×C₅	C₂×C₃×C₅×C₉²	1	6	yes	—
97	C₃²	C₃×C₄×C₅²×C₉	1	12	yes	—
98	C₃²	C₃³×C₄×C₅²	1	1	yes	characteristic
99	C₉	C₃×C₄×C₅²×C₉	1	36	yes	direct product factor
100	C₂×C₃	C₂×C₅²×C₉²	1	9	yes	—
101	C₂×C₃	C₂×C₃²×C₅²×C₉	1	4	yes	—
102	C₅	C₃×C₄×C₅×C₉²	1	6	yes	direct product factor
103	C₄	C₃×C₅²×C₉²	1	1	yes	characteristic, direct product factor, Sylow
104	C₃	C₄×C₅²×C₉²	1	9	yes	direct product factor
105	C₃	C₃²×C₄×C₅²×C₉	1	4	yes	—
106	C₂	C₂×C₃×C₅²×C₉²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃²	C₂×C₃³×C₅²
socle, prime radical	C₂×C₃³×C₅²	C₂×C₃²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₄×C₅²×C₉²	C₃	direct (N × H)
4. C₃×C₅²×C₉²	C₄	direct (N × H)
5. C₃×C₄×C₅×C₉²	C₅	direct (N × H)
9. C₃×C₄×C₅²×C₉	C₉	direct (N × H)
13. C₅²×C₉²	C₃×C₄	direct (N × H)
15. C₄×C₅×C₉²	C₃×C₅	direct (N × H)
19. C₃×C₅×C₉²	C₄×C₅	direct (N × H)
20. C₃×C₄×C₉²	C₅²	direct (N × H)
23. C₄×C₅²×C₉	C₃×C₉	direct (N × H)
27. C₃×C₅²×C₉	C₄×C₉	direct (N × H)
30. C₃×C₄×C₅×C₉	C₅×C₉	direct (N × H)
37. C₅×C₉²	C₃×C₄×C₅	direct (N × H)
39. C₄×C₉²	C₃×C₅²	direct (N × H)
40. C₃×C₄×C₅²	C₉²	direct (N × H)
45. C₃×C₉²	C₄×C₅²	direct (N × H)
48. C₅²×C₉	C₃×C₄×C₉	direct (N × H)
51. C₄×C₅×C₉	C₃×C₅×C₉	direct (N × H)
57. C₃×C₅×C₉	C₄×C₅×C₉	direct (N × H)
60. C₃×C₄×C₉	C₅²×C₉	direct (N × H)
62. C₄×C₅²	C₃×C₉²	direct (N × H)
67. C₉²	C₃×C₄×C₅²	direct (N × H)
68. C₃×C₅²	C₄×C₉²	direct (N × H)
70. C₃×C₄×C₅	C₅×C₉²	direct (N × H)
78. C₅×C₉	C₃×C₄×C₅×C₉	direct (N × H)
81. C₄×C₉	C₃×C₅²×C₉	direct (N × H)
85. C₃×C₉	C₄×C₅²×C₉	direct (N × H)
87. C₅²	C₃×C₄×C₉²	direct (N × H)
88. C₄×C₅	C₃×C₅×C₉²	direct (N × H)
92. C₃×C₅	C₄×C₅×C₉²	direct (N × H)
94. C₃×C₄	C₅²×C₉²	direct (N × H)
99. C₉	C₃×C₄×C₅²×C₉	direct (N × H)
102. C₅	C₃×C₄×C₅×C₉²	direct (N × H)
103. C₄	C₃×C₅²×C₉²	direct (N × H)
104. C₃	C₄×C₅²×C₉²	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×C₅²×C₉²	2
C₃²×C₄×C₅²×C₉	3
C₄×C₅²×C₉²	3
C₃×C₄×C₅×C₉²	5

Subgroups — C3×C4×C52×C92 #72201573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₅²×C₉² #72201573