Finite Group Databasebeta

← C2×C32×C4×C5×C7×C9 #72200113

Subgroups — C2×C32×C4×C5×C7×C9 #72200113

Order 22680.

1600 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C5×C7×C9C211yesmaximal, characteristic
2C32×C4×C5×C7×C9C212yesmaximal, direct product factor
3C2×C33×C4×C5×C7C311yesmaximal, characteristic
4C2×C3×C4×C5×C7×C9C3112yesmaximal, direct product factor
5C2×C32×C5×C7×C9C412yesdirect product factor
6C2×C32×C5×C7×C9C2211yescharacteristic
7C2×C32×C4×C7×C9C511yesmaximal, characteristic, direct product factor, Hall
8C22×C33×C5×C7C2×C311yescharacteristic
9C22×C3×C5×C7×C9C2×C3112yes—
10C33×C4×C5×C7C2×C312yes—
11C3×C4×C5×C7×C9C2×C3124yesdirect product factor
12C2×C32×C4×C5×C9C711yesmaximal, characteristic, direct product factor, Hall
13C32×C5×C7×C9C2×C411yescharacteristic, direct product factor, Hall
14C2×C32×C4×C5×C7C919yesdirect product factor
15C2×C32×C4×C5×C7C3214yes—
16C2×C4×C5×C7×C9C3219yesdirect product factor
17C22×C32×C7×C9C2×C511yescharacteristic
18C32×C4×C7×C9C2×C512yesdirect product factor
19C2×C33×C5×C7C3×C412yes—
20C2×C33×C5×C7C22×C311yescharacteristic
21C2×C3×C5×C7×C9C3×C4124yesdirect product factor
22C2×C3×C5×C7×C9C22×C3112yes—
23C22×C32×C5×C9C2×C711yescharacteristic
24C32×C4×C5×C9C2×C712yesdirect product factor
25C2×C33×C4×C7C3×C511yescharacteristic
26C2×C3×C4×C7×C9C3×C5112yesdirect product factor
27C22×C32×C5×C7C2×C919yes—
28C22×C32×C5×C7C2×C3214yes—
29C32×C4×C5×C7C2×C9118yesdirect product factor
30C32×C4×C5×C7C2×C3218yes—
31C22×C5×C7×C9C2×C3219yes—
32C4×C5×C7×C9C2×C32118yesdirect product factor
33C2×C32×C7×C9C4×C512yesdirect product factor
34C2×C32×C7×C9C22×C511yescharacteristic
35C2×C33×C4×C5C3×C711yescharacteristic
36C2×C3×C4×C5×C9C3×C7112yesdirect product factor
37C33×C5×C7C2×C3×C411yescharacteristic
38C3×C5×C7×C9C2×C3×C4112yesdirect product factor
39C2×C3×C4×C5×C7C3×C9112yesdirect product factor
40C2×C3×C4×C5×C7C3311yescharacteristic
41C2×C32×C5×C9C4×C712yesdirect product factor
42C2×C32×C5×C9C22×C711yescharacteristic
43C22×C33×C7C2×C3×C511yescharacteristic
44C33×C4×C7C2×C3×C512yes—
45C22×C3×C7×C9C2×C3×C5112yes—
46C3×C4×C7×C9C2×C3×C5124yesdirect product factor
47C2×C32×C4×C9C5×C711yescharacteristic, direct product factor, Hall
48C2×C32×C5×C7C4×C9118yesdirect product factor
49C2×C32×C5×C7C22×C919yes—
50C2×C32×C5×C7C32×C418yes—
51C2×C32×C5×C7C22×C3214yes—
52C2×C5×C7×C9C32×C4118yesdirect product factor
53C2×C5×C7×C9C22×C3219yes—
54C32×C7×C9C2×C4×C511yescharacteristic, direct product factor, Hall
55C22×C33×C5C2×C3×C711yescharacteristic
56C33×C4×C5C2×C3×C712yes—
57C22×C3×C5×C9C2×C3×C7112yes—
58C3×C4×C5×C9C2×C3×C7124yesdirect product factor
59C2×C32×C4×C7C5×C919yesdirect product factor
60C2×C32×C4×C7C32×C514yes—
61C2×C4×C7×C9C32×C519yesdirect product factor
62C22×C3×C5×C7C2×C3×C9112yes—
63C22×C3×C5×C7C2×C3311yescharacteristic
64C3×C4×C5×C7C2×C3×C9124yesdirect product factor
65C3×C4×C5×C7C2×C3312yes—
66C32×C5×C9C2×C4×C711yescharacteristic, direct product factor, Hall
67C2×C33×C7C3×C4×C512yes—
68C2×C33×C7C22×C3×C511yescharacteristic
69C2×C3×C7×C9C3×C4×C5124yesdirect product factor
70C2×C3×C7×C9C22×C3×C5112yes—
71C2×C32×C4×C5C7×C919yesdirect product factor
72C2×C32×C4×C5C32×C714yes—
73C2×C4×C5×C9C32×C719yesdirect product factor
74C22×C32×C9C2×C5×C711yescharacteristic
75C32×C4×C9C2×C5×C712yesdirect product factor
76C32×C5×C7C2×C4×C919yesdirect product factor
77C32×C5×C7C2×C32×C414yes—
78C5×C7×C9C2×C32×C419yesdirect product factor
79C2×C4×C5×C7C32×C911yescharacteristic, direct product factor, Hall
80C2×C33×C5C3×C4×C712yes—
81C2×C33×C5C22×C3×C711yescharacteristic
82C2×C3×C5×C9C3×C4×C7124yesdirect product factor
83C2×C3×C5×C9C22×C3×C7112yes—
84C22×C32×C7C2×C5×C919yes—
85C22×C32×C7C2×C32×C514yes—
86C32×C4×C7C2×C5×C9118yesdirect product factor
87C32×C4×C7C2×C32×C518yes—
88C22×C7×C9C2×C32×C519yes—
89C4×C7×C9C2×C32×C5118yesdirect product factor
90C2×C33×C4C3×C5×C711yescharacteristic
91C2×C3×C4×C9C3×C5×C7112yesdirect product factor
92C2×C3×C5×C7C3×C4×C9124yesdirect product factor
93C2×C3×C5×C7C22×C3×C9112yes—
94C2×C3×C5×C7C33×C412yes—
95C2×C3×C5×C7C22×C3311yescharacteristic
96C33×C7C2×C3×C4×C511yescharacteristic
97C3×C7×C9C2×C3×C4×C5112yesdirect product factor
98C22×C32×C5C2×C7×C919yes—
99C22×C32×C5C2×C32×C714yes—
100C32×C4×C5C2×C7×C9118yesdirect product factor
101C32×C4×C5C2×C32×C718yes—
102C22×C5×C9C2×C32×C719yes—
103C4×C5×C9C2×C32×C7118yesdirect product factor
104C2×C3×C4×C7C3×C5×C9112yesdirect product factor
105C2×C3×C4×C7C33×C511yescharacteristic
106C2×C32×C9C4×C5×C712yesdirect product factor
107C2×C32×C9C22×C5×C711yescharacteristic
108C22×C5×C7C2×C32×C911yescharacteristic
109C4×C5×C7C2×C32×C912yesdirect product factor
110C33×C5C2×C3×C4×C711yescharacteristic
111C3×C5×C9C2×C3×C4×C7112yesdirect product factor
112C2×C32×C7C4×C5×C9118yesdirect product factor
113C2×C32×C7C22×C5×C919yes—
114C2×C32×C7C32×C4×C518yes—
115C2×C32×C7C22×C32×C514yes—
116C2×C7×C9C32×C4×C5118yesdirect product factor
117C2×C7×C9C22×C32×C519yes—
118C2×C3×C4×C5C3×C7×C9112yesdirect product factor
119C2×C3×C4×C5C33×C711yescharacteristic
120C22×C33C2×C3×C5×C711yescharacteristic
121C33×C4C2×C3×C5×C712yes—
122C22×C3×C9C2×C3×C5×C7112yes—
123C3×C4×C9C2×C3×C5×C7124yesdirect product factor
124C3×C5×C7C2×C3×C4×C9112yesdirect product factor
125C3×C5×C7C2×C33×C411yescharacteristic
126C2×C32×C5C4×C7×C9118yesdirect product factor
127C2×C32×C5C22×C7×C919yes—
128C2×C32×C5C32×C4×C718yes—
129C2×C32×C5C22×C32×C714yes—
130C2×C5×C9C32×C4×C7118yesdirect product factor
131C2×C5×C9C22×C32×C719yes—
132C22×C3×C7C2×C3×C5×C9112yes—
133C22×C3×C7C2×C33×C511yescharacteristic
134C3×C4×C7C2×C3×C5×C9124yesdirect product factor
135C3×C4×C7C2×C33×C512yes—
136C32×C9C2×C4×C5×C711yescharacteristic, direct product factor, Sylow
137C2×C32×C4C5×C7×C919yesdirect product factor
138C2×C32×C4C32×C5×C714yes—
139C2×C4×C9C32×C5×C719yesdirect product factor
140C2×C5×C7C32×C4×C912yesdirect product factor
141C2×C5×C7C22×C32×C911yescharacteristic
142C32×C7C2×C4×C5×C919yesdirect product factor
143C32×C7C2×C32×C4×C514yes—
144C7×C9C2×C32×C4×C519yesdirect product factor
145C22×C3×C5C2×C3×C7×C9112yes—
146C22×C3×C5C2×C33×C711yescharacteristic
147C3×C4×C5C2×C3×C7×C9124yesdirect product factor
148C3×C4×C5C2×C33×C712yes—
149C2×C4×C7C32×C5×C911yescharacteristic, direct product factor, Hall
150C2×C33C3×C4×C5×C712yes—
151C2×C33C22×C3×C5×C711yescharacteristic
152C2×C3×C9C3×C4×C5×C7124yesdirect product factor
153C2×C3×C9C22×C3×C5×C7112yes—
154C32×C5C2×C4×C7×C919yesdirect product factor
155C32×C5C2×C32×C4×C714yes—
156C5×C9C2×C32×C4×C719yesdirect product factor
157C2×C3×C7C3×C4×C5×C9124yesdirect product factor
158C2×C3×C7C22×C3×C5×C9112yes—
159C2×C3×C7C33×C4×C512yes—
160C2×C3×C7C22×C33×C511yescharacteristic
161C2×C4×C5C32×C7×C911yescharacteristic, direct product factor, Hall
162C22×C32C2×C5×C7×C919yes—
163C22×C32C2×C32×C5×C714yes—
164C32×C4C2×C5×C7×C9118yesdirect product factor
165C32×C4C2×C32×C5×C718yes—
166C22×C9C2×C32×C5×C719yes—
167C4×C9C2×C32×C5×C7118yesdirect product factor
168C5×C7C2×C32×C4×C911yescharacteristic, direct product factor, Hall
169C2×C3×C5C3×C4×C7×C9124yesdirect product factor
170C2×C3×C5C22×C3×C7×C9112yes—
171C2×C3×C5C33×C4×C712yes—
172C2×C3×C5C22×C33×C711yescharacteristic
173C22×C7C2×C32×C5×C911yescharacteristic
174C4×C7C2×C32×C5×C912yesdirect product factor
175C33C2×C3×C4×C5×C711yescharacteristic
176C3×C9C2×C3×C4×C5×C7112yesdirect product factor
177C2×C3×C4C3×C5×C7×C9112yesdirect product factor
178C2×C3×C4C33×C5×C711yescharacteristic
179C3×C7C2×C3×C4×C5×C9112yesdirect product factor
180C3×C7C2×C33×C4×C511yescharacteristic
181C22×C5C2×C32×C7×C911yescharacteristic
182C4×C5C2×C32×C7×C912yesdirect product factor
183C2×C32C4×C5×C7×C9118yesdirect product factor
184C2×C32C22×C5×C7×C919yes—
185C2×C32C32×C4×C5×C718yes—
186C2×C32C22×C32×C5×C714yes—
187C2×C9C32×C4×C5×C7118yesdirect product factor
188C2×C9C22×C32×C5×C719yes—
189C3×C5C2×C3×C4×C7×C9112yesdirect product factor
190C3×C5C2×C33×C4×C711yescharacteristic
191C2×C7C32×C4×C5×C912yesdirect product factor
192C2×C7C22×C32×C5×C911yescharacteristic
193C22×C3C2×C3×C5×C7×C9112yes—
194C22×C3C2×C33×C5×C711yescharacteristic
195C3×C4C2×C3×C5×C7×C9124yesdirect product factor
196C3×C4C2×C33×C5×C712yes—
197C2×C5C32×C4×C7×C912yesdirect product factor
198C2×C5C22×C32×C7×C911yescharacteristic
199C32C2×C4×C5×C7×C919yesdirect product factor
200C32C2×C32×C4×C5×C714yes—
201C9C2×C32×C4×C5×C719yesdirect product factor
202C2×C4C32×C5×C7×C911yescharacteristic, direct product factor, Sylow
203C7C2×C32×C4×C5×C911yescharacteristic, direct product factor, Sylow
204C2×C3C3×C4×C5×C7×C9124yesdirect product factor
205C2×C3C33×C4×C5×C712yes—
206C2×C3C22×C3×C5×C7×C9112yes—
207C2×C3C22×C33×C5×C711yescharacteristic
208C5C2×C32×C4×C7×C911yescharacteristic, direct product factor, Sylow
209C22C2×C32×C5×C7×C911yescharacteristic
210C4C2×C32×C5×C7×C912yesdirect product factor
211C3C2×C3×C4×C5×C7×C9112yesdirect product factor
212C3C2×C33×C4×C5×C711yescharacteristic
213C2C32×C4×C5×C7×C912yesdirect product factor
214C2C22×C32×C5×C7×C911yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2×C3C22×C33×C5×C7
socle, prime radicalC22×C33×C5×C7C2×C3
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
2. C32×C4×C5×C7×C9C2direct (N × H)
4. C2×C3×C4×C5×C7×C9C3direct (N × H)
5. C2×C32×C5×C7×C9C4direct (N × H)
7. C2×C32×C4×C7×C9C5direct (N × H)
11. C3×C4×C5×C7×C9C2×C3direct (N × H)
12. C2×C32×C4×C5×C9C7direct (N × H)
13. C32×C5×C7×C9C2×C4direct (N × H)
14. C2×C32×C4×C5×C7C9direct (N × H)
16. C2×C4×C5×C7×C9C32direct (N × H)
18. C32×C4×C7×C9C2×C5direct (N × H)
21. C2×C3×C5×C7×C9C3×C4direct (N × H)
24. C32×C4×C5×C9C2×C7direct (N × H)
26. C2×C3×C4×C7×C9C3×C5direct (N × H)
29. C32×C4×C5×C7C2×C9direct (N × H)
32. C4×C5×C7×C9C2×C32direct (N × H)
33. C2×C32×C7×C9C4×C5direct (N × H)
36. C2×C3×C4×C5×C9C3×C7direct (N × H)
38. C3×C5×C7×C9C2×C3×C4direct (N × H)
39. C2×C3×C4×C5×C7C3×C9direct (N × H)
41. C2×C32×C5×C9C4×C7direct (N × H)
46. C3×C4×C7×C9C2×C3×C5direct (N × H)
47. C2×C32×C4×C9C5×C7direct (N × H)
48. C2×C32×C5×C7C4×C9direct (N × H)
52. C2×C5×C7×C9C32×C4direct (N × H)
54. C32×C7×C9C2×C4×C5direct (N × H)
58. C3×C4×C5×C9C2×C3×C7direct (N × H)
59. C2×C32×C4×C7C5×C9direct (N × H)
61. C2×C4×C7×C9C32×C5direct (N × H)
64. C3×C4×C5×C7C2×C3×C9direct (N × H)
66. C32×C5×C9C2×C4×C7direct (N × H)
69. C2×C3×C7×C9C3×C4×C5direct (N × H)
71. C2×C32×C4×C5C7×C9direct (N × H)
73. C2×C4×C5×C9C32×C7direct (N × H)
75. C32×C4×C9C2×C5×C7direct (N × H)
76. C32×C5×C7C2×C4×C9direct (N × H)
78. C5×C7×C9C2×C32×C4direct (N × H)
79. C2×C4×C5×C7C32×C9direct (N × H)
82. C2×C3×C5×C9C3×C4×C7direct (N × H)
86. C32×C4×C7C2×C5×C9direct (N × H)
89. C4×C7×C9C2×C32×C5direct (N × H)
91. C2×C3×C4×C9C3×C5×C7direct (N × H)
92. C2×C3×C5×C7C3×C4×C9direct (N × H)
97. C3×C7×C9C2×C3×C4×C5direct (N × H)
100. C32×C4×C5C2×C7×C9direct (N × H)
103. C4×C5×C9C2×C32×C7direct (N × H)
104. C2×C3×C4×C7C3×C5×C9direct (N × H)
106. C2×C32×C9C4×C5×C7direct (N × H)
109. C4×C5×C7C2×C32×C9direct (N × H)
111. C3×C5×C9C2×C3×C4×C7direct (N × H)
112. C2×C32×C7C4×C5×C9direct (N × H)
116. C2×C7×C9C32×C4×C5direct (N × H)
118. C2×C3×C4×C5C3×C7×C9direct (N × H)
123. C3×C4×C9C2×C3×C5×C7direct (N × H)
124. C3×C5×C7C2×C3×C4×C9direct (N × H)
126. C2×C32×C5C4×C7×C9direct (N × H)
130. C2×C5×C9C32×C4×C7direct (N × H)
134. C3×C4×C7C2×C3×C5×C9direct (N × H)
136. C32×C9C2×C4×C5×C7direct (N × H)
137. C2×C32×C4C5×C7×C9direct (N × H)
139. C2×C4×C9C32×C5×C7direct (N × H)
140. C2×C5×C7C32×C4×C9direct (N × H)
142. C32×C7C2×C4×C5×C9direct (N × H)
144. C7×C9C2×C32×C4×C5direct (N × H)
147. C3×C4×C5C2×C3×C7×C9direct (N × H)
149. C2×C4×C7C32×C5×C9direct (N × H)
152. C2×C3×C9C3×C4×C5×C7direct (N × H)
154. C32×C5C2×C4×C7×C9direct (N × H)
156. C5×C9C2×C32×C4×C7direct (N × H)
157. C2×C3×C7C3×C4×C5×C9direct (N × H)
161. C2×C4×C5C32×C7×C9direct (N × H)
164. C32×C4C2×C5×C7×C9direct (N × H)
167. C4×C9C2×C32×C5×C7direct (N × H)
168. C5×C7C2×C32×C4×C9direct (N × H)
169. C2×C3×C5C3×C4×C7×C9direct (N × H)
174. C4×C7C2×C32×C5×C9direct (N × H)
176. C3×C9C2×C3×C4×C5×C7direct (N × H)
177. C2×C3×C4C3×C5×C7×C9direct (N × H)
179. C3×C7C2×C3×C4×C5×C9direct (N × H)
182. C4×C5C2×C32×C7×C9direct (N × H)
183. C2×C32C4×C5×C7×C9direct (N × H)
187. C2×C9C32×C4×C5×C7direct (N × H)
189. C3×C5C2×C3×C4×C7×C9direct (N × H)
191. C2×C7C32×C4×C5×C9direct (N × H)
195. C3×C4C2×C3×C5×C7×C9direct (N × H)
197. C2×C5C32×C4×C7×C9direct (N × H)
199. C32C2×C4×C5×C7×C9direct (N × H)
201. C9C2×C32×C4×C5×C7direct (N × H)
202. C2×C4C32×C5×C7×C9direct (N × H)
203. C7C2×C32×C4×C5×C9direct (N × H)
204. C2×C3C3×C4×C5×C7×C9direct (N × H)
208. C5C2×C32×C4×C7×C9direct (N × H)
210. C4C2×C32×C5×C7×C9direct (N × H)
211. C3C2×C3×C4×C5×C7×C9direct (N × H)
213. C2C32×C4×C5×C7×C9direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C32×C5×C7×C92
C32×C4×C5×C7×C92
C2×C33×C4×C5×C73
C2×C3×C4×C5×C7×C93
C2×C32×C4×C7×C95
C2×C32×C4×C5×C97