Finite Group Databasebeta

← C2×C3×C43×C7×C9 #72201487

Subgroups — C2×C3×C43×C7×C9 #72201487

Order 24192.

12720 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C42×C7×C9C217yesmaximal
2C3×C43×C7×C9C218yesmaximal, direct product factor
3C2×C32×C43×C7C311yesmaximal, characteristic
4C2×C43×C7×C9C313yesmaximal, direct product factor
5C23×C3×C4×C7×C9C2217yes—
6C2×C3×C42×C7×C9C4156yesdirect product factor
7C2×C3×C42×C7×C9C22128yes—
8C22×C32×C42×C7C2×C317yes—
9C22×C42×C7×C9C2×C3121yes—
10C32×C43×C7C2×C318yes—
11C43×C7×C9C2×C3124yesdirect product factor
12C2×C3×C43×C9C711yesmaximal, characteristic, direct product factor, Hall
13C24×C3×C7×C9C2311yescharacteristic
14C22×C3×C4×C7×C9C2×C4184yes—
15C22×C3×C4×C7×C9C23114yes—
16C3×C42×C7×C9C2×C41112yesdirect product factor
17C2×C3×C43×C7C913yesdirect product factor
18C2×C3×C43×C7C3211yescharacteristic
19C23×C32×C4×C7C22×C317yes—
20C23×C4×C7×C9C22×C3121yes—
21C2×C42×C7×C9C3×C41168yesdirect product factor
22C2×C42×C7×C9C22×C3184yes—
23C2×C32×C42×C7C3×C4156yes—
24C2×C32×C42×C7C22×C3128yes—
25C22×C3×C42×C9C2×C717yes—
26C3×C43×C9C2×C718yesdirect product factor
27C23×C3×C7×C9C22×C4114yes—
28C23×C3×C7×C9C2411yescharacteristic
29C2×C3×C4×C7×C9C421112yesdirect product factor
30C2×C3×C4×C7×C9C22×C4184yes—
31C22×C3×C42×C7C2×C9121yes—
32C22×C3×C42×C7C2×C3217yes—
33C3×C43×C7C2×C9124yesdirect product factor
34C3×C43×C7C2×C3218yes—
35C2×C32×C43C3×C711yescharacteristic
36C2×C43×C9C3×C713yesdirect product factor
37C24×C32×C7C23×C311yescharacteristic
38C22×C32×C4×C7C2×C3×C4184yes—
39C22×C32×C4×C7C23×C3114yes—
40C24×C7×C9C23×C313yes—
41C32×C42×C7C2×C3×C41112yes—
42C22×C4×C7×C9C2×C3×C41252yes—
43C22×C4×C7×C9C23×C3142yes—
44C42×C7×C9C2×C3×C41336yesdirect product factor
45C2×C43×C7C3×C911yescharacteristic, direct product factor, Hall
46C23×C3×C4×C9C22×C717yes—
47C2×C3×C42×C9C4×C7156yesdirect product factor
48C2×C3×C42×C9C22×C7128yes—
49C22×C3×C7×C9C2×C42128yes—
50C22×C3×C7×C9C23×C417yes—
51C3×C4×C7×C9C2×C42156yesdirect product factor
52C23×C3×C4×C7C22×C9121yes—
53C23×C3×C4×C7C22×C3217yes—
54C2×C3×C42×C7C4×C91168yesdirect product factor
55C2×C3×C42×C7C22×C9184yes—
56C2×C3×C42×C7C32×C4156yes—
57C2×C3×C42×C7C22×C32128yes—
58C22×C32×C42C2×C3×C717yes—
59C32×C43C2×C3×C718yes—
60C22×C42×C9C2×C3×C7121yes—
61C43×C9C2×C3×C7124yesdirect product factor
62C23×C32×C7C22×C3×C4114yes—
63C23×C32×C7C24×C311yescharacteristic
64C2×C32×C4×C7C3×C421112yes—
65C2×C32×C4×C7C22×C3×C4184yes—
66C23×C7×C9C22×C3×C4142yes—
67C23×C7×C9C24×C313yes—
68C2×C4×C7×C9C3×C421336yesdirect product factor
69C2×C4×C7×C9C22×C3×C41252yes—
70C22×C42×C7C2×C3×C917yes—
71C43×C7C2×C3×C918yesdirect product factor
72C24×C3×C9C23×C711yescharacteristic
73C22×C3×C4×C9C2×C4×C7184yes—
74C22×C3×C4×C9C23×C7114yes—
75C3×C42×C9C2×C4×C71112yesdirect product factor
76C2×C3×C43C7×C913yesdirect product factor
77C2×C3×C43C32×C711yescharacteristic
78C2×C3×C7×C9C4318yesdirect product factor
79C2×C3×C7×C9C22×C4217yes—
80C24×C3×C7C23×C913yes—
81C24×C3×C7C23×C3211yescharacteristic
82C22×C3×C4×C7C2×C4×C91252yes—
83C22×C3×C4×C7C23×C9142yes—
84C22×C3×C4×C7C2×C32×C4184yes—
85C22×C3×C4×C7C23×C32114yes—
86C3×C42×C7C2×C4×C91336yesdirect product factor
87C3×C42×C7C2×C32×C41112yes—
88C23×C32×C4C22×C3×C717yes—
89C2×C32×C42C3×C4×C7156yes—
90C2×C32×C42C22×C3×C7128yes—
91C23×C4×C9C22×C3×C7121yes—
92C2×C42×C9C3×C4×C71168yesdirect product factor
93C2×C42×C9C22×C3×C7184yes—
94C22×C32×C7C2×C3×C42128yes—
95C22×C32×C7C23×C3×C417yes—
96C32×C4×C7C2×C3×C42156yes—
97C22×C7×C9C2×C3×C42184yes—
98C22×C7×C9C23×C3×C4121yes—
99C4×C7×C9C2×C3×C421168yesdirect product factor
100C23×C4×C7C22×C3×C917yes—
101C2×C42×C7C3×C4×C9156yesdirect product factor
102C2×C42×C7C22×C3×C9128yes—
103C23×C3×C9C22×C4×C7114yes—
104C23×C3×C9C24×C711yescharacteristic
105C2×C3×C4×C9C42×C71112yesdirect product factor
106C2×C3×C4×C9C22×C4×C7184yes—
107C22×C3×C42C2×C7×C9121yes—
108C22×C3×C42C2×C32×C717yes—
109C3×C43C2×C7×C9124yesdirect product factor
110C3×C43C2×C32×C718yes—
111C3×C7×C9C2×C4311yescharacteristic, direct product factor, Hall
112C23×C3×C7C22×C4×C9142yes—
113C23×C3×C7C24×C913yes—
114C23×C3×C7C22×C32×C4114yes—
115C23×C3×C7C24×C3211yescharacteristic
116C2×C3×C4×C7C42×C91336yesdirect product factor
117C2×C3×C4×C7C22×C4×C91252yes—
118C2×C3×C4×C7C32×C421112yes—
119C2×C3×C4×C7C22×C32×C4184yes—
120C24×C32C23×C3×C711yescharacteristic
121C22×C32×C4C2×C3×C4×C7184yes—
122C22×C32×C4C23×C3×C7114yes—
123C24×C9C23×C3×C713yes—
124C32×C42C2×C3×C4×C71112yes—
125C22×C4×C9C2×C3×C4×C71252yes—
126C22×C4×C9C23×C3×C7142yes—
127C42×C9C2×C3×C4×C71336yesdirect product factor
128C2×C43C3×C7×C911yescharacteristic, direct product factor, Sylow
129C2×C32×C7C3×C4318yes—
130C2×C32×C7C22×C3×C4217yes—
131C2×C7×C9C3×C43124yesdirect product factor
132C2×C7×C9C22×C3×C42121yes—
133C24×C7C23×C3×C911yescharacteristic
134C22×C4×C7C2×C3×C4×C9184yes—
135C22×C4×C7C23×C3×C9114yes—
136C42×C7C2×C3×C4×C91112yesdirect product factor
137C22×C3×C9C2×C42×C7128yes—
138C22×C3×C9C23×C4×C717yes—
139C3×C4×C9C2×C42×C7156yesdirect product factor
140C23×C3×C4C22×C7×C9121yes—
141C23×C3×C4C22×C32×C717yes—
142C2×C3×C42C4×C7×C91168yesdirect product factor
143C2×C3×C42C22×C7×C9184yes—
144C2×C3×C42C32×C4×C7156yes—
145C2×C3×C42C22×C32×C7128yes—
146C22×C3×C7C2×C42×C9184yes—
147C22×C3×C7C23×C4×C9121yes—
148C22×C3×C7C2×C32×C42128yes—
149C22×C3×C7C23×C32×C417yes—
150C3×C4×C7C2×C42×C91168yesdirect product factor
151C3×C4×C7C2×C32×C42156yes—
152C23×C32C22×C3×C4×C7114yes—
153C23×C32C24×C3×C711yescharacteristic
154C2×C32×C4C3×C42×C71112yes—
155C2×C32×C4C22×C3×C4×C7184yes—
156C23×C9C22×C3×C4×C7142yes—
157C23×C9C24×C3×C713yes—
158C2×C4×C9C3×C42×C71336yesdirect product factor
159C2×C4×C9C22×C3×C4×C71252yes—
160C22×C42C2×C3×C7×C917yes—
161C43C2×C3×C7×C918yesdirect product factor
162C32×C7C2×C3×C4311yescharacteristic
163C7×C9C2×C3×C4313yesdirect product factor
164C23×C7C22×C3×C4×C9114yes—
165C23×C7C24×C3×C911yescharacteristic
166C2×C4×C7C3×C42×C91112yesdirect product factor
167C2×C4×C7C22×C3×C4×C9184yes—
168C2×C3×C9C43×C718yesdirect product factor
169C2×C3×C9C22×C42×C717yes—
170C24×C3C23×C7×C913yes—
171C24×C3C23×C32×C711yescharacteristic
172C22×C3×C4C2×C4×C7×C91252yes—
173C22×C3×C4C23×C7×C9142yes—
174C22×C3×C4C2×C32×C4×C7184yes—
175C22×C3×C4C23×C32×C7114yes—
176C3×C42C2×C4×C7×C91336yesdirect product factor
177C3×C42C2×C32×C4×C71112yes—
178C2×C3×C7C43×C9124yesdirect product factor
179C2×C3×C7C22×C42×C9121yes—
180C2×C3×C7C32×C4318yes—
181C2×C3×C7C22×C32×C4217yes—
182C22×C32C2×C3×C42×C7128yes—
183C22×C32C23×C3×C4×C717yes—
184C32×C4C2×C3×C42×C7156yes—
185C22×C9C2×C3×C42×C7184yes—
186C22×C9C23×C3×C4×C7121yes—
187C4×C9C2×C3×C42×C71168yesdirect product factor
188C23×C4C22×C3×C7×C917yes—
189C2×C42C3×C4×C7×C9156yesdirect product factor
190C2×C42C22×C3×C7×C9128yes—
191C22×C7C2×C3×C42×C9128yes—
192C22×C7C23×C3×C4×C917yes—
193C4×C7C2×C3×C42×C9156yesdirect product factor
194C3×C9C2×C43×C711yescharacteristic, direct product factor, Sylow
195C23×C3C22×C4×C7×C9142yes—
196C23×C3C24×C7×C913yes—
197C23×C3C22×C32×C4×C7114yes—
198C23×C3C24×C32×C711yescharacteristic
199C2×C3×C4C42×C7×C91336yesdirect product factor
200C2×C3×C4C22×C4×C7×C91252yes—
201C2×C3×C4C32×C42×C71112yes—
202C2×C3×C4C22×C32×C4×C7184yes—
203C3×C7C2×C43×C913yesdirect product factor
204C3×C7C2×C32×C4311yescharacteristic
205C2×C32C3×C43×C718yes—
206C2×C32C22×C3×C42×C717yes—
207C2×C9C3×C43×C7124yesdirect product factor
208C2×C9C22×C3×C42×C7121yes—
209C24C23×C3×C7×C911yescharacteristic
210C22×C4C2×C3×C4×C7×C9184yes—
211C22×C4C23×C3×C7×C9114yes—
212C42C2×C3×C4×C7×C91112yesdirect product factor
213C2×C7C3×C43×C918yesdirect product factor
214C2×C7C22×C3×C42×C917yes—
215C22×C3C2×C32×C42×C7128yes—
216C22×C3C2×C42×C7×C9184yes—
217C22×C3C23×C4×C7×C9121yes—
218C22×C3C23×C32×C4×C717yes—
219C3×C4C2×C32×C42×C7156yes—
220C3×C4C2×C42×C7×C91168yesdirect product factor
221C32C2×C3×C43×C711yescharacteristic
222C9C2×C3×C43×C713yesdirect product factor
223C23C22×C3×C4×C7×C9114yes—
224C23C24×C3×C7×C911yescharacteristic
225C2×C4C3×C42×C7×C91112yesdirect product factor
226C2×C4C22×C3×C4×C7×C9184yes—
227C7C2×C3×C43×C911yescharacteristic, direct product factor, Sylow
228C2×C3C43×C7×C9124yesdirect product factor
229C2×C3C32×C43×C718yes—
230C2×C3C22×C42×C7×C9121yes—
231C2×C3C22×C32×C42×C717yes—
232C22C2×C3×C42×C7×C9128yes—
233C22C23×C3×C4×C7×C917yes—
234C4C2×C3×C42×C7×C9156yesdirect product factor
235C3C2×C43×C7×C913yesdirect product factor
236C3C2×C32×C43×C711yescharacteristic
237C2C3×C43×C7×C918yesdirect product factor
238C2C22×C3×C42×C7×C917yes—
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC23×C3C24×C32×C7
socle, prime radicalC24×C32×C7C23×C3
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×C43×C7×C9C2direct (N × H)
4. C2×C43×C7×C9C3direct (N × H)
6. C2×C3×C42×C7×C9C4direct (N × H)
11. C43×C7×C9C2×C3direct (N × H)
12. C2×C3×C43×C9C7direct (N × H)
16. C3×C42×C7×C9C2×C4direct (N × H)
17. C2×C3×C43×C7C9direct (N × H)
21. C2×C42×C7×C9C3×C4direct (N × H)
26. C3×C43×C9C2×C7direct (N × H)
29. C2×C3×C4×C7×C9C42direct (N × H)
33. C3×C43×C7C2×C9direct (N × H)
36. C2×C43×C9C3×C7direct (N × H)
44. C42×C7×C9C2×C3×C4direct (N × H)
45. C2×C43×C7C3×C9direct (N × H)
47. C2×C3×C42×C9C4×C7direct (N × H)
51. C3×C4×C7×C9C2×C42direct (N × H)
54. C2×C3×C42×C7C4×C9direct (N × H)
61. C43×C9C2×C3×C7direct (N × H)
68. C2×C4×C7×C9C3×C42direct (N × H)
71. C43×C7C2×C3×C9direct (N × H)
75. C3×C42×C9C2×C4×C7direct (N × H)
76. C2×C3×C43C7×C9direct (N × H)
78. C2×C3×C7×C9C43direct (N × H)
86. C3×C42×C7C2×C4×C9direct (N × H)
92. C2×C42×C9C3×C4×C7direct (N × H)
99. C4×C7×C9C2×C3×C42direct (N × H)
101. C2×C42×C7C3×C4×C9direct (N × H)
105. C2×C3×C4×C9C42×C7direct (N × H)
109. C3×C43C2×C7×C9direct (N × H)
111. C3×C7×C9C2×C43direct (N × H)
116. C2×C3×C4×C7C42×C9direct (N × H)
127. C42×C9C2×C3×C4×C7direct (N × H)
128. C2×C43C3×C7×C9direct (N × H)
131. C2×C7×C9C3×C43direct (N × H)
136. C42×C7C2×C3×C4×C9direct (N × H)
139. C3×C4×C9C2×C42×C7direct (N × H)
142. C2×C3×C42C4×C7×C9direct (N × H)
150. C3×C4×C7C2×C42×C9direct (N × H)
158. C2×C4×C9C3×C42×C7direct (N × H)
161. C43C2×C3×C7×C9direct (N × H)
163. C7×C9C2×C3×C43direct (N × H)
166. C2×C4×C7C3×C42×C9direct (N × H)
168. C2×C3×C9C43×C7direct (N × H)
176. C3×C42C2×C4×C7×C9direct (N × H)
178. C2×C3×C7C43×C9direct (N × H)
187. C4×C9C2×C3×C42×C7direct (N × H)
189. C2×C42C3×C4×C7×C9direct (N × H)
193. C4×C7C2×C3×C42×C9direct (N × H)
194. C3×C9C2×C43×C7direct (N × H)
199. C2×C3×C4C42×C7×C9direct (N × H)
203. C3×C7C2×C43×C9direct (N × H)
207. C2×C9C3×C43×C7direct (N × H)
212. C42C2×C3×C4×C7×C9direct (N × H)
213. C2×C7C3×C43×C9direct (N × H)
220. C3×C4C2×C42×C7×C9direct (N × H)
222. C9C2×C3×C43×C7direct (N × H)
225. C2×C4C3×C42×C7×C9direct (N × H)
227. C7C2×C3×C43×C9direct (N × H)
228. C2×C3C43×C7×C9direct (N × H)
234. C4C2×C3×C42×C7×C9direct (N × H)
235. C3C2×C43×C7×C9direct (N × H)
237. C2C3×C43×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×C3×C42×C7×C92
C3×C43×C7×C92
C2×C32×C43×C73
C2×C43×C7×C93
C2×C3×C43×C97