Finite Group Databasebeta

← C22×C47×#432 #26099813

Subgroups — C22×C47×#432 #26099813

Order 12032.

2830 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C42×C47C211yesmaximal, characteristic
2C2×C47×#432C2124yesmaximal, direct product factor
3C22×C4⋊C8×C47C212yesmaximal, complemented
4C22×C22⋊C8×C47C212yesmaximal
5C22×#97×C47C212yesmaximal, complemented
6C24×C4×C47C411yescharacteristic
7C24×C4×C47C411yescharacteristic
8C24×C4×C47C2211yescharacteristic
9C23×C8×C47C2214yes—
10C22×C42×C47C411yescharacteristic
11C22×C42×C47C4112yes—
12C22×C42×C47C411yescharacteristic
13C22×C42×C47C22112yes—
14C22×C42×C47C2211yescharacteristic
15C22×C42×C47C2211yescharacteristic
16C47×#432C22164yesdirect product factor
17C2×C4⋊C8×C47C22124yescomplemented
18C2×C22⋊C8×C47C22124yes—
19C2×#97×C47C22124yescomplemented
20C25×C47C2×C411yescharacteristic
21C23×C4×C47—28no—
22C23×C4×C47—28no—
23C23×C4×C47—28no—
24C23×C4×C47C2×C411yescharacteristic
25C23×C4×C47C2×C416yes—
26C23×C4×C47C2×C4112yes—
27C23×C4×C47C2×C411yescharacteristic
28C23×C4×C47C2×C411yescharacteristic
29C23×C4×C47C2×C411yescharacteristic
30C23×C4×C47C2×C4112yes—
31C23×C4×C47C2×C411yescharacteristic
32C23×C4×C47C2316yes—
33C23×C4×C47C2311yescharacteristic
34C22×C8×C47—216no—
35C22×C8×C47C23124yes—
36C2×C42×C47C2×C4112yes—
37C2×C42×C47C2×C4148yes—
38C2×C42×C47C2×C4112yes—
39C2×C42×C47C2×C416yes—
40C2×C42×C47C2×C416yes—
41C2×C42×C47C2316yes—
42C2×C42×C47C2316yes—
43C2×C42×C47C23116yes—
44C4⋊C8×C47C23132yescomplemented
45C22⋊C8×C47C23132yes—
46#97×C47C23132yescomplemented
47C24×C47—24no—
48C24×C47—24no—
49C24×C47M1618yes—
50C24×C47C22×C416yes—
51C24×C47C22×C411yescharacteristic
52C22×C4×C47—24no—
53C22×C4×C47—248no—
54C22×C4×C47—24no—
55C22×C4×C47—24no—
56C22×C4×C47—24no—
57C22×C4×C47—224no—
58C22×C4×C47—248no—
59C22×C4×C47—224no—
60C22×C4×C47M1618yes—
61C22×C4×C47C22×C416yes—
62C22×C4×C47C22×C4112yes—
63C22×C4×C47C22×C4116yes—
64C22×C4×C47C22×C416yes—
65C22×C4×C47C22×C416yes—
66C22×C4×C47C22×C416yes—
67C22×C4×C47C22×C4116yes—
68C22×C4×C47C22×C413yes—
69C22×C4×C47C22×C416yes—
70C22×C4×C47#4118yes—
71C22×C4×C47C2414yes—
72C22×C4×C47C2413yes—
73C2×C8×C47—248no—
74C2×C8×C47C24116yes—
75C42×C47C22×C4116yes—
76C42×C47C22×C4116yes—
77C42×C47C22×C4112yes—
78C42×C47C22×C4112yes—
79C42×C47C2414yes—
80C42×C47C2414yes—
81C23×C47—212no—
82C23×C47—212no—
83C23×C47—232no—
84C23×C47#11714yes—
85C23×C47C2×M16124yes—
86C23×C47C2×M1614yes—
87C23×C47C2×M1614yes—
88C23×C47C23×C414yes—
89C23×C47C23×C413yes—
90C2×C4×C47—212no—
91C2×C4×C47—232no—
92C2×C4×C47—212no—
93C2×C4×C47—212no—
94C2×C4×C47—212no—
95C2×C4×C47—224no—
96C2×C4×C47—232no—
97C2×C4×C47—28no—
98C2×C4×C47#117112yes—
99C2×C4×C47C2×M16124yes—
100C2×C4×C47C23×C414yes—
101C2×C4×C47C23×C414yes—
102C2×C4×C47C23×C414yes—
103C2×C4×C47C23×C414yes—
104C2×C4×C47C23×C413yes—
105C2×C4×C47C23×C414yes—
106C2×C4×C47C2×#41112yes—
107C2×C4×C47C2511yescharacteristic
108C8×C47—216no—
109C22×#432C4711yesmaximal, characteristic, direct product factor, Sylow
110C22×C47—24no—
111C22×C47—248no—
112C22×C47—24no—
113C22×C47#432116yesdirect product factor
114C22×C47C2×#11716yes—
115C22×C47C22×M1618yes—
116C22×C47C22×M1616yes—
117C22×C47C22×M1616yes—
118C22×C47C24×C411yescharacteristic
119C4×C47—24no—
120C4×C47—24no—
121C4×C47—24no—
122C4×C47—24no—
123C4×C47C2×#11716yes—
124C4×C47C22×M1618yes—
125C4×C47C22×#4112yes—
126C23×C42C2×C4711yescharacteristic
127C2×#432C2×C47124yesdirect product factor
128C22×C4⋊C8C2×C4712yescomplemented
129C22×C22⋊C8C2×C4712yes—
130C22×#97C2×C4712yescomplemented
131C2×C47—28no—
132C2×C47C2×#432112yesdirect product factor
133C2×C47C22×#11711yescharacteristic
134C2×C47C23×M1611yescharacteristic
135C2×C47C23×M1611yescharacteristic
136C24×C4C4×C4711yescharacteristic
137C24×C4C4×C4711yescharacteristic
138C24×C4C22×C4711yescharacteristic
139C23×C8C22×C4714yes—
140C22×C42C4×C4711yescharacteristic
141C22×C42C4×C47112yes—
142C22×C42C4×C4711yescharacteristic
143C22×C42C22×C4711yescharacteristic
144C22×C42C22×C4711yescharacteristic
145C22×C42C22×C47112yes—
146#432C22×C47164yesdirect product factor
147C2×C4⋊C8C22×C47124yescomplemented
148C2×C22⋊C8C22×C47124yes—
149C2×#97C22×C47124yescomplemented
150C47C22×#43211yescharacteristic, direct product factor, Sylow
151C25C2×C4×C4711yescharacteristic
152C23×C4—28no—
153C23×C4—28no—
154C23×C4—28no—
155C23×C4C2×C4×C47112yes—
156C23×C4C2×C4×C4711yescharacteristic
157C23×C4C2×C4×C4711yescharacteristic
158C23×C4C2×C4×C4711yescharacteristic
159C23×C4C2×C4×C47112yes—
160C23×C4C2×C4×C4711yescharacteristic
161C23×C4C2×C4×C4716yes—
162C23×C4C2×C4×C4711yescharacteristic
163C23×C4C23×C4716yes—
164C23×C4C23×C4711yescharacteristic
165C22×C8—216no—
166C22×C8C23×C47124yes—
167C2×C42C2×C4×C47112yes—
168C2×C42C2×C4×C47148yes—
169C2×C42C2×C4×C47112yes—
170C2×C42C2×C4×C4716yes—
171C2×C42C2×C4×C4716yes—
172C2×C42C23×C4716yes—
173C2×C42C23×C4716yes—
174C2×C42C23×C47116yes—
175C4⋊C8C23×C47132yescomplemented
176C22⋊C8C23×C47132yes—
177#97C23×C47132yescomplemented
178C24—24no—
179C24—24no—
180C24M16×C4718yes—
181C24C22×C4×C4716yes—
182C24C22×C4×C4711yescharacteristic
183C22×C4—24no—
184C22×C4—248no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—224no—
189C22×C4—248no—
190C22×C4—224no—
191C22×C4M16×C4718yes—
192C22×C4C22×C4×C4716yes—
193C22×C4C22×C4×C47112yes—
194C22×C4C22×C4×C47116yes—
195C22×C4C22×C4×C4716yes—
196C22×C4C22×C4×C4716yes—
197C22×C4C22×C4×C4716yes—
198C22×C4C22×C4×C47116yes—
199C22×C4C22×C4×C4713yes—
200C22×C4C22×C4×C4716yes—
201C22×C4#41×C4718yes—
202C22×C4C24×C4714yes—
203C22×C4C24×C4713yes—
204C2×C8—248no—
205C2×C8C24×C47116yes—
206C42C22×C4×C47116yes—
207C42C22×C4×C47116yes—
208C42C22×C4×C47112yes—
209C42C22×C4×C47112yes—
210C42C24×C4714yes—
211C42C24×C4714yes—
212C23—212no—
213C23—212no—
214C23—232no—
215C23#117×C4714yes—
216C23C2×M16×C47124yes—
217C23C2×M16×C4714yes—
218C23C2×M16×C4714yes—
219C23C23×C4×C4714yes—
220C23C23×C4×C4713yes—
221C2×C4—212no—
222C2×C4—232no—
223C2×C4—212no—
224C2×C4—212no—
225C2×C4—212no—
226C2×C4—224no—
227C2×C4—232no—
228C2×C4—28no—
229C2×C4#117×C47112yes—
230C2×C4C2×M16×C47124yes—
231C2×C4C23×C4×C4714yes—
232C2×C4C23×C4×C4714yes—
233C2×C4C23×C4×C4714yes—
234C2×C4C23×C4×C4714yes—
235C2×C4C23×C4×C4713yes—
236C2×C4C23×C4×C4714yes—
237C2×C4C2×#41×C47112yes—
238C2×C4C25×C4711yescharacteristic
239C8—216no—
240C22—24no—
241C22—24no—
242C22—248no—
243C22C47×#432116yesdirect product factor
244C22C2×#117×C4716yes—
245C22C22×M16×C4718yes—
246C22C22×M16×C4716yes—
247C22C22×M16×C4716yes—
248C22C24×C4×C4711yescharacteristic
249C4—24no—
250C4—24no—
251C4—24no—
252C4—24no—
253C4C2×#117×C4716yes—
254C4C22×M16×C4718yes—
255C4C22×#41×C4712yes—
256C2—28no—
257C2C2×C47×#432112yesdirect product factor
258C2C22×#117×C4711yescharacteristic
259C2C23×M16×C4711yescharacteristic
260C2C23×M16×C4711yescharacteristic
subgroupG/H
centerC23×C4×C47C23
commutatorC22C24×C4×C47
Frattini, cosocleC2×C4C25×C47
socleC24×C47C22×C4
prime radicalC25×C47C2×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C47×#432C2direct (N × H)
3. C22×C4⋊C8×C47C2semidirect
5. C22×#97×C47C2semidirect
16. C47×#432C22direct (N × H)
17. C2×C4⋊C8×C47C22semidirect
19. C2×#97×C47C22semidirect
44. C4⋊C8×C47C23semidirect
46. #97×C47C23semidirect
109. C22×#432C47direct (N × H)
113. C22×C47#432direct (N × H)
127. C2×#432C2×C47direct (N × H)
128. C22×C4⋊C8C2×C47semidirect
130. C22×#97C2×C47semidirect
132. C2×C47C2×#432direct (N × H)
146. #432C22×C47direct (N × H)
147. C2×C4⋊C8C22×C47semidirect
149. C2×#97C22×C47semidirect
150. C47C22×#432direct (N × H)
175. C4⋊C8C23×C47semidirect
177. #97C23×C47semidirect
243. C22C47×#432direct (N × H)
257. C2C2×C47×#432direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C42×C472
C2×C47×#4322
C22×C4⋊C8×C472
C22×C22⋊C8×C472
C22×#97×C472
C22×#43247