Subgroups — C₃²×C₄³×C₅×C₉ #72203100

Order 25920.

12900 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄²×C₅×C₉	C₂	1	7	yes	maximal
2	C₃³×C₄³×C₅	C₃	1	1	yes	maximal, characteristic
3	C₃×C₄³×C₅×C₉	C₃	1	12	yes	maximal, direct product factor
4	C₂²×C₃²×C₄×C₅×C₉	C₂²	1	7	yes	—
5	C₃²×C₄²×C₅×C₉	C₄	1	28	yes	direct product factor
6	C₃²×C₄³×C₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃³×C₄²×C₅	C₂×C₃	1	7	yes	—
8	C₂×C₃×C₄²×C₅×C₉	C₂×C₃	1	84	yes	—
9	C₂³×C₃²×C₅×C₉	C₂³	1	1	yes	characteristic
10	C₂×C₃²×C₄×C₅×C₉	C₂×C₄	1	42	yes	—
11	C₃²×C₄³×C₅	C₉	1	9	yes	direct product factor
12	C₃²×C₄³×C₅	C₃²	1	4	yes	—
13	C₄³×C₅×C₉	C₃²	1	9	yes	direct product factor
14	C₂×C₃²×C₄²×C₉	C₂×C₅	1	7	yes	—
15	C₂²×C₃³×C₄×C₅	C₂²×C₃	1	7	yes	—
16	C₂²×C₃×C₄×C₅×C₉	C₂²×C₃	1	84	yes	—
17	C₃³×C₄²×C₅	C₃×C₄	1	28	yes	—
18	C₃×C₄²×C₅×C₉	C₃×C₄	1	336	yes	direct product factor
19	C₃³×C₄³	C₃×C₅	1	1	yes	characteristic
20	C₃×C₄³×C₉	C₃×C₅	1	12	yes	direct product factor
21	C₂²×C₃²×C₅×C₉	C₂²×C₄	1	7	yes	—
22	C₃²×C₄×C₅×C₉	C₄²	1	28	yes	direct product factor
23	C₂×C₃²×C₄²×C₅	C₂×C₉	1	63	yes	—
24	C₂×C₃²×C₄²×C₅	C₂×C₃²	1	28	yes	—
25	C₂×C₄²×C₅×C₉	C₂×C₃²	1	63	yes	—
26	C₂²×C₃²×C₄×C₉	C₂²×C₅	1	7	yes	—
27	C₃²×C₄²×C₉	C₄×C₅	1	28	yes	direct product factor
28	C₂³×C₃³×C₅	C₂³×C₃	1	1	yes	characteristic
29	C₂×C₃³×C₄×C₅	C₂×C₃×C₄	1	42	yes	—
30	C₂³×C₃×C₅×C₉	C₂³×C₃	1	12	yes	—
31	C₂×C₃×C₄×C₅×C₉	C₂×C₃×C₄	1	504	yes	—
32	C₃×C₄³×C₅	C₃×C₉	1	12	yes	direct product factor
33	C₃×C₄³×C₅	C₃³	1	1	yes	characteristic
34	C₂×C₃³×C₄²	C₂×C₃×C₅	1	7	yes	—
35	C₂×C₃×C₄²×C₉	C₂×C₃×C₅	1	84	yes	—
36	C₂×C₃²×C₅×C₉	C₂×C₄²	1	7	yes	—
37	C₂²×C₃²×C₄×C₅	C₂²×C₉	1	63	yes	—
38	C₂²×C₃²×C₄×C₅	C₂²×C₃²	1	28	yes	—
39	C₃²×C₄²×C₅	C₄×C₉	1	252	yes	direct product factor
40	C₃²×C₄²×C₅	C₃²×C₄	1	112	yes	—
41	C₂²×C₄×C₅×C₉	C₂²×C₃²	1	63	yes	—
42	C₄²×C₅×C₉	C₃²×C₄	1	252	yes	direct product factor
43	C₂³×C₃²×C₉	C₂³×C₅	1	1	yes	characteristic
44	C₂×C₃²×C₄×C₉	C₂×C₄×C₅	1	42	yes	—
45	C₃²×C₄³	C₅×C₉	1	9	yes	direct product factor
46	C₃²×C₄³	C₃²×C₅	1	4	yes	—
47	C₄³×C₉	C₃²×C₅	1	9	yes	direct product factor
48	C₂²×C₃³×C₅	C₂²×C₃×C₄	1	7	yes	—
49	C₃³×C₄×C₅	C₃×C₄²	1	28	yes	—
50	C₂²×C₃×C₅×C₉	C₂²×C₃×C₄	1	84	yes	—
51	C₃×C₄×C₅×C₉	C₃×C₄²	1	336	yes	direct product factor
52	C₂×C₃×C₄²×C₅	C₂×C₃×C₉	1	84	yes	—
53	C₂×C₃×C₄²×C₅	C₂×C₃³	1	7	yes	—
54	C₂²×C₃³×C₄	C₂²×C₃×C₅	1	7	yes	—
55	C₃³×C₄²	C₃×C₄×C₅	1	28	yes	—
56	C₂²×C₃×C₄×C₉	C₂²×C₃×C₅	1	84	yes	—
57	C₃×C₄²×C₉	C₃×C₄×C₅	1	336	yes	direct product factor
58	C₃²×C₅×C₉	C₄³	1	1	yes	characteristic, direct product factor, Hall
59	C₂³×C₃²×C₅	C₂³×C₉	1	9	yes	—
60	C₂³×C₃²×C₅	C₂³×C₃²	1	4	yes	—
61	C₂×C₃²×C₄×C₅	C₂×C₄×C₉	1	378	yes	—
62	C₂×C₃²×C₄×C₅	C₂×C₃²×C₄	1	168	yes	—
63	C₂³×C₅×C₉	C₂³×C₃²	1	9	yes	—
64	C₂×C₄×C₅×C₉	C₂×C₃²×C₄	1	378	yes	—
65	C₂²×C₃²×C₉	C₂²×C₄×C₅	1	7	yes	—
66	C₃²×C₄×C₉	C₄²×C₅	1	28	yes	direct product factor
67	C₄³×C₅	C₃²×C₉	1	1	yes	characteristic, direct product factor, Hall
68	C₂×C₃²×C₄²	C₂×C₅×C₉	1	63	yes	—
69	C₂×C₃²×C₄²	C₂×C₃²×C₅	1	28	yes	—
70	C₂×C₄²×C₉	C₂×C₃²×C₅	1	63	yes	—
71	C₂×C₃³×C₅	C₂×C₃×C₄²	1	7	yes	—
72	C₂×C₃×C₅×C₉	C₂×C₃×C₄²	1	84	yes	—
73	C₂²×C₃×C₄×C₅	C₂²×C₃×C₉	1	84	yes	—
74	C₂²×C₃×C₄×C₅	C₂²×C₃³	1	7	yes	—
75	C₃×C₄²×C₅	C₃×C₄×C₉	1	336	yes	direct product factor
76	C₃×C₄²×C₅	C₃³×C₄	1	28	yes	—
77	C₂³×C₃³	C₂³×C₃×C₅	1	1	yes	characteristic
78	C₂×C₃³×C₄	C₂×C₃×C₄×C₅	1	42	yes	—
79	C₂³×C₃×C₉	C₂³×C₃×C₅	1	12	yes	—
80	C₂×C₃×C₄×C₉	C₂×C₃×C₄×C₅	1	504	yes	—
81	C₃×C₄³	C₃×C₅×C₉	1	12	yes	direct product factor
82	C₃×C₄³	C₃³×C₅	1	1	yes	characteristic
83	C₂²×C₃²×C₅	C₂²×C₄×C₉	1	63	yes	—
84	C₂²×C₃²×C₅	C₂²×C₃²×C₄	1	28	yes	—
85	C₃²×C₄×C₅	C₄²×C₉	1	252	yes	direct product factor
86	C₃²×C₄×C₅	C₃²×C₄²	1	112	yes	—
87	C₂²×C₅×C₉	C₂²×C₃²×C₄	1	63	yes	—
88	C₄×C₅×C₉	C₃²×C₄²	1	252	yes	direct product factor
89	C₂×C₃²×C₉	C₂×C₄²×C₅	1	7	yes	—
90	C₂×C₄²×C₅	C₂×C₃²×C₉	1	7	yes	—
91	C₂²×C₃²×C₄	C₂²×C₅×C₉	1	63	yes	—
92	C₂²×C₃²×C₄	C₂²×C₃²×C₅	1	28	yes	—
93	C₃²×C₄²	C₄×C₅×C₉	1	252	yes	direct product factor
94	C₃²×C₄²	C₃²×C₄×C₅	1	112	yes	—
95	C₂²×C₄×C₉	C₂²×C₃²×C₅	1	63	yes	—
96	C₄²×C₉	C₃²×C₄×C₅	1	252	yes	direct product factor
97	C₃³×C₅	C₃×C₄³	1	1	yes	characteristic
98	C₃×C₅×C₉	C₃×C₄³	1	12	yes	direct product factor
99	C₂³×C₃×C₅	C₂³×C₃×C₉	1	12	yes	—
100	C₂³×C₃×C₅	C₂³×C₃³	1	1	yes	characteristic
101	C₂×C₃×C₄×C₅	C₂×C₃×C₄×C₉	1	504	yes	—
102	C₂×C₃×C₄×C₅	C₂×C₃³×C₄	1	42	yes	—
103	C₂²×C₃³	C₂²×C₃×C₄×C₅	1	7	yes	—
104	C₃³×C₄	C₃×C₄²×C₅	1	28	yes	—
105	C₂²×C₃×C₉	C₂²×C₃×C₄×C₅	1	84	yes	—
106	C₃×C₄×C₉	C₃×C₄²×C₅	1	336	yes	direct product factor
107	C₂×C₃×C₄²	C₂×C₃×C₅×C₉	1	84	yes	—
108	C₂×C₃×C₄²	C₂×C₃³×C₅	1	7	yes	—
109	C₂×C₃²×C₅	C₂×C₄²×C₉	1	63	yes	—
110	C₂×C₃²×C₅	C₂×C₃²×C₄²	1	28	yes	—
111	C₂×C₅×C₉	C₂×C₃²×C₄²	1	63	yes	—
112	C₃²×C₉	C₄³×C₅	1	1	yes	characteristic, direct product factor, Sylow
113	C₂²×C₄×C₅	C₂²×C₃²×C₉	1	7	yes	—
114	C₄²×C₅	C₃²×C₄×C₉	1	28	yes	direct product factor
115	C₂³×C₃²	C₂³×C₅×C₉	1	9	yes	—
116	C₂³×C₃²	C₂³×C₃²×C₅	1	4	yes	—
117	C₂×C₃²×C₄	C₂×C₄×C₅×C₉	1	378	yes	—
118	C₂×C₃²×C₄	C₂×C₃²×C₄×C₅	1	168	yes	—
119	C₂³×C₉	C₂³×C₃²×C₅	1	9	yes	—
120	C₂×C₄×C₉	C₂×C₃²×C₄×C₅	1	378	yes	—
121	C₄³	C₃²×C₅×C₉	1	1	yes	characteristic, direct product factor, Sylow
122	C₂²×C₃×C₅	C₂²×C₃×C₄×C₉	1	84	yes	—
123	C₂²×C₃×C₅	C₂²×C₃³×C₄	1	7	yes	—
124	C₃×C₄×C₅	C₃×C₄²×C₉	1	336	yes	direct product factor
125	C₃×C₄×C₅	C₃³×C₄²	1	28	yes	—
126	C₂×C₃³	C₂×C₃×C₄²×C₅	1	7	yes	—
127	C₂×C₃×C₉	C₂×C₃×C₄²×C₅	1	84	yes	—
128	C₂²×C₃×C₄	C₂²×C₃×C₅×C₉	1	84	yes	—
129	C₂²×C₃×C₄	C₂²×C₃³×C₅	1	7	yes	—
130	C₃×C₄²	C₃×C₄×C₅×C₉	1	336	yes	direct product factor
131	C₃×C₄²	C₃³×C₄×C₅	1	28	yes	—
132	C₃²×C₅	C₄³×C₉	1	9	yes	direct product factor
133	C₃²×C₅	C₃²×C₄³	1	4	yes	—
134	C₅×C₉	C₃²×C₄³	1	9	yes	direct product factor
135	C₂³×C₅	C₂³×C₃²×C₉	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₃²×C₄×C₉	1	42	yes	—
137	C₂²×C₃²	C₂²×C₄×C₅×C₉	1	63	yes	—
138	C₂²×C₃²	C₂²×C₃²×C₄×C₅	1	28	yes	—
139	C₃²×C₄	C₄²×C₅×C₉	1	252	yes	direct product factor
140	C₃²×C₄	C₃²×C₄²×C₅	1	112	yes	—
141	C₂²×C₉	C₂²×C₃²×C₄×C₅	1	63	yes	—
142	C₄×C₉	C₃²×C₄²×C₅	1	252	yes	direct product factor
143	C₂×C₄²	C₂×C₃²×C₅×C₉	1	7	yes	—
144	C₂×C₃×C₅	C₂×C₃×C₄²×C₉	1	84	yes	—
145	C₂×C₃×C₅	C₂×C₃³×C₄²	1	7	yes	—
146	C₃³	C₃×C₄³×C₅	1	1	yes	characteristic
147	C₃×C₉	C₃×C₄³×C₅	1	12	yes	direct product factor
148	C₂³×C₃	C₂³×C₃×C₅×C₉	1	12	yes	—
149	C₂³×C₃	C₂³×C₃³×C₅	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×C₃×C₄×C₅×C₉	1	504	yes	—
151	C₂×C₃×C₄	C₂×C₃³×C₄×C₅	1	42	yes	—
152	C₂²×C₅	C₂²×C₃²×C₄×C₉	1	7	yes	—
153	C₄×C₅	C₃²×C₄²×C₉	1	28	yes	direct product factor
154	C₂×C₃²	C₂×C₄²×C₅×C₉	1	63	yes	—
155	C₂×C₃²	C₂×C₃²×C₄²×C₅	1	28	yes	—
156	C₂×C₉	C₂×C₃²×C₄²×C₅	1	63	yes	—
157	C₂²×C₄	C₂²×C₃²×C₅×C₉	1	7	yes	—
158	C₄²	C₃²×C₄×C₅×C₉	1	28	yes	direct product factor
159	C₃×C₅	C₃×C₄³×C₉	1	12	yes	direct product factor
160	C₃×C₅	C₃³×C₄³	1	1	yes	characteristic
161	C₂²×C₃	C₂²×C₃×C₄×C₅×C₉	1	84	yes	—
162	C₂²×C₃	C₂²×C₃³×C₄×C₅	1	7	yes	—
163	C₃×C₄	C₃×C₄²×C₅×C₉	1	336	yes	direct product factor
164	C₃×C₄	C₃³×C₄²×C₅	1	28	yes	—
165	C₂×C₅	C₂×C₃²×C₄²×C₉	1	7	yes	—
166	C₃²	C₄³×C₅×C₉	1	9	yes	direct product factor
167	C₃²	C₃²×C₄³×C₅	1	4	yes	—
168	C₉	C₃²×C₄³×C₅	1	9	yes	direct product factor
169	C₂³	C₂³×C₃²×C₅×C₉	1	1	yes	characteristic
170	C₂×C₄	C₂×C₃²×C₄×C₅×C₉	1	42	yes	—
171	C₂×C₃	C₂×C₃×C₄²×C₅×C₉	1	84	yes	—
172	C₂×C₃	C₂×C₃³×C₄²×C₅	1	7	yes	—
173	C₅	C₃²×C₄³×C₉	1	1	yes	characteristic, direct product factor, Sylow
174	C₂²	C₂²×C₃²×C₄×C₅×C₉	1	7	yes	—
175	C₄	C₃²×C₄²×C₅×C₉	1	28	yes	direct product factor
176	C₃	C₃×C₄³×C₅×C₉	1	12	yes	direct product factor
177	C₃	C₃³×C₄³×C₅	1	1	yes	characteristic
178	C₂	C₂×C₃²×C₄²×C₅×C₉	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂³×C₃	C₂³×C₃³×C₅
socle, prime radical	C₂³×C₃³×C₅	C₂³×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄³×C₅×C₉	C₃	direct (N × H)
5. C₃²×C₄²×C₅×C₉	C₄	direct (N × H)
6. C₃²×C₄³×C₉	C₅	direct (N × H)
11. C₃²×C₄³×C₅	C₉	direct (N × H)
13. C₄³×C₅×C₉	C₃²	direct (N × H)
18. C₃×C₄²×C₅×C₉	C₃×C₄	direct (N × H)
20. C₃×C₄³×C₉	C₃×C₅	direct (N × H)
22. C₃²×C₄×C₅×C₉	C₄²	direct (N × H)
27. C₃²×C₄²×C₉	C₄×C₅	direct (N × H)
32. C₃×C₄³×C₅	C₃×C₉	direct (N × H)
39. C₃²×C₄²×C₅	C₄×C₉	direct (N × H)
42. C₄²×C₅×C₉	C₃²×C₄	direct (N × H)
45. C₃²×C₄³	C₅×C₉	direct (N × H)
47. C₄³×C₉	C₃²×C₅	direct (N × H)
51. C₃×C₄×C₅×C₉	C₃×C₄²	direct (N × H)
57. C₃×C₄²×C₉	C₃×C₄×C₅	direct (N × H)
58. C₃²×C₅×C₉	C₄³	direct (N × H)
66. C₃²×C₄×C₉	C₄²×C₅	direct (N × H)
67. C₄³×C₅	C₃²×C₉	direct (N × H)
75. C₃×C₄²×C₅	C₃×C₄×C₉	direct (N × H)
81. C₃×C₄³	C₃×C₅×C₉	direct (N × H)
85. C₃²×C₄×C₅	C₄²×C₉	direct (N × H)
88. C₄×C₅×C₉	C₃²×C₄²	direct (N × H)
93. C₃²×C₄²	C₄×C₅×C₉	direct (N × H)
96. C₄²×C₉	C₃²×C₄×C₅	direct (N × H)
98. C₃×C₅×C₉	C₃×C₄³	direct (N × H)
106. C₃×C₄×C₉	C₃×C₄²×C₅	direct (N × H)
112. C₃²×C₉	C₄³×C₅	direct (N × H)
114. C₄²×C₅	C₃²×C₄×C₉	direct (N × H)
121. C₄³	C₃²×C₅×C₉	direct (N × H)
124. C₃×C₄×C₅	C₃×C₄²×C₉	direct (N × H)
130. C₃×C₄²	C₃×C₄×C₅×C₉	direct (N × H)
132. C₃²×C₅	C₄³×C₉	direct (N × H)
134. C₅×C₉	C₃²×C₄³	direct (N × H)
139. C₃²×C₄	C₄²×C₅×C₉	direct (N × H)
142. C₄×C₉	C₃²×C₄²×C₅	direct (N × H)
147. C₃×C₉	C₃×C₄³×C₅	direct (N × H)
153. C₄×C₅	C₃²×C₄²×C₉	direct (N × H)
158. C₄²	C₃²×C₄×C₅×C₉	direct (N × H)
159. C₃×C₅	C₃×C₄³×C₉	direct (N × H)
163. C₃×C₄	C₃×C₄²×C₅×C₉	direct (N × H)
166. C₃²	C₄³×C₅×C₉	direct (N × H)
168. C₉	C₃²×C₄³×C₅	direct (N × H)
173. C₅	C₃²×C₄³×C₉	direct (N × H)
175. C₄	C₃²×C₄²×C₅×C₉	direct (N × H)
176. C₃	C₃×C₄³×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×C₃²×C₄²×C₅×C₉	2
C₃³×C₄³×C₅	3
C₃×C₄³×C₅×C₉	3
C₃²×C₄³×C₉	5

Subgroups — C32×C43×C5×C9 #72203100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄³×C₅×C₉ #72203100