Finite Group Databasebeta

← #123×AGL(1,𝔽11) #72248837

Subgroups — #123×AGL(1,𝔽11) #72248837

Order 3520.

684 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×AGL(1,𝔽11)C211yesmaximal, characteristic
2C22×C4×AGL(1,𝔽11)C211yesmaximal, characteristic, complemented
3#123×C11⋊C5C211yesmaximal, characteristic, complemented
4#66988⋊C5C211yesmaximal, characteristic, complemented
5#66982⋊C5C211yesmaximal, characteristic, complemented
6#66947⋊C5C212yesmaximal, complemented
7C4⋊C4×AGL(1,𝔽11)C212yesmaximal, complemented
8#66939⋊C5C211yesmaximal, characteristic, complemented
9#66936⋊C5C212yesmaximal, complemented
10C22⋊C4×AGL(1,𝔽11)C211yesmaximal, characteristic, complemented
11C22⋊C4×AGL(1,𝔽11)C212yesmaximal, complemented
12C23×AGL(1,𝔽11)C2211yescharacteristic
13C23×AGL(1,𝔽11)C2211yescharacteristic
14C2×D4×C11⋊C5C2211yescharacteristic
15C22×C4×C11⋊C5C2211yescharacteristic, complemented
16C2×C22⋊AGL(1,𝔽11)C2211yescharacteristic
17C2×C22⋊AGL(1,𝔽11)C2211yescharacteristic
18C22×Dic11⋊C5C2211yescharacteristic, complemented
19D4×AGL(1,𝔽11)—24no—
20C2×C4⋊AGL(1,𝔽11)C2211yescharacteristic
21C2×C4×AGL(1,𝔽11)—24no—
22C2×C4×AGL(1,𝔽11)C2212yes—
23C2×C4×AGL(1,𝔽11)C2212yes—
24C2×C4×AGL(1,𝔽11)C2211yescharacteristic
25C2×C4×AGL(1,𝔽11)C2212yescomplemented
26C4⋊C4×C11⋊C5C2212yescomplemented
27C22⋊C4×C11⋊C5C2212yescomplemented
28C22⋊C4×C11⋊C5C2211yescharacteristic, complemented
29C22⋊(Dic11⋊C5)C2212yescomplemented
30C22⋊(Dic11⋊C5)C2211yescharacteristic, complemented
31((C2×D11)⋊C4)⋊C5C2212yescomplemented
32((C2×D11)⋊C4)⋊C5C2212yescomplemented
33((C2×D11)⋊C4)⋊C5C2212yescomplemented
34C4⋊(Dic11⋊C5)C2212yescomplemented
35(Dic11⋊C4)⋊C5C2212yescomplemented
36(Dic11⋊C4)⋊C5C2212yescomplemented
37D11×#123C511yesmaximal, characteristic, complemented, Hall
38C23×C11⋊C5C2311yescharacteristic
39C23×C11⋊C5C2311yescharacteristic
40C22×AGL(1,𝔽11)—24no—
41C22×AGL(1,𝔽11)—22no—
42C22×AGL(1,𝔽11)—24no—
43C22×AGL(1,𝔽11)D414yes—
44C22×AGL(1,𝔽11)C2311yescharacteristic
45C22×AGL(1,𝔽11)C2311yescharacteristic
46C22×AGL(1,𝔽11)C2311yescharacteristic
47D4×C11⋊C5—22no—
48C2×C4×C11⋊C5—22no—
49C2×C4×C11⋊C5C2311yescharacteristic
50C2×C4×C11⋊C5C2312yes—
51C2×C4×C11⋊C5C2312yes—
52C22⋊AGL(1,𝔽11)—22no—
53C22⋊AGL(1,𝔽11)—22no—
54C2×Dic11⋊C5—22no—
55C2×Dic11⋊C5C2311yescharacteristic
56C2×Dic11⋊C5C2312yes—
57C2×Dic11⋊C5C2312yes—
58C4⋊AGL(1,𝔽11)—22no—
59C4×AGL(1,𝔽11)—24no—
60C4×AGL(1,𝔽11)—24no—
61C4×AGL(1,𝔽11)—22no—
62C4×AGL(1,𝔽11)—24no—
63C2×D4×D11C2×C511yescharacteristic
64C22×C4×D11C2×C511yescharacteristic, complemented
65C11×#123C2×C511yescharacteristic, complemented
66#66988C2×C511yescharacteristic, complemented
67#66982C2×C511yescharacteristic, complemented
68#66947C2×C512yescomplemented
69C4⋊C4×D11C2×C512yescomplemented
70#66939C2×C511yescharacteristic, complemented
71#66936C2×C512yescomplemented
72C22⋊C4×D11C2×C511yescharacteristic, complemented
73C22⋊C4×D11C2×C512yescomplemented
74C2×C5×#123—111nomaximal, Hall
75C22×C11⋊C5—22no—
76C22×C11⋊C5—21no—
77C22×C11⋊C5—22no—
78C22×C11⋊C5C2×D412yes—
79C22×C11⋊C5C2411yescharacteristic
80C2×AGL(1,𝔽11)—22no—
81C2×AGL(1,𝔽11)—28no—
82C2×AGL(1,𝔽11)—21no—
83C2×AGL(1,𝔽11)—22no—
84C2×AGL(1,𝔽11)—44no—
85C2×AGL(1,𝔽11)C2×D412yes—
86C2×AGL(1,𝔽11)C2×D412yes—
87C2×AGL(1,𝔽11)#4114yes—
88C4×C11⋊C5—22no—
89C4×C11⋊C5—21no—
90C4×C11⋊C5—22no—
91Dic11⋊C5—22no—
92Dic11⋊C5—21no—
93Dic11⋊C5—22no—
94C23×D11C22×C511yescharacteristic
95C23×D11C22×C511yescharacteristic
96C2×D4×C11C22×C511yescharacteristic
97C22×C4×C11C22×C511yescharacteristic, complemented
98C2×C22⋊D11C22×C511yescharacteristic
99C2×C22⋊D11C22×C511yescharacteristic
100C22×Dic11C22×C511yescharacteristic, complemented
101D4×D11—24no—
102C2×D44C22×C511yescharacteristic
103C2×C4×D11—24no—
104C2×C4×D11C22×C512yes—
105C2×C4×D11C22×C512yes—
106C2×C4×D11C22×C511yescharacteristic
107C2×C4×D11C22×C512yescomplemented
108C11×C4⋊C4C22×C512yescomplemented
109C11×C22⋊C4C22×C512yescomplemented
110C11×C22⋊C4C22×C511yescharacteristic, complemented
111C22⋊Dic11C22×C512yescomplemented
112C22⋊Dic11C22×C511yescharacteristic, complemented
113(C2×D11)⋊C4C22×C512yescomplemented
114(C2×D11)⋊C4C22×C512yescomplemented
115(C2×D11)⋊C4C22×C512yescomplemented
116C4⋊Dic11C22×C512yescomplemented
117Dic11⋊C4C22×C512yescomplemented
118Dic11⋊C4C22×C512yescomplemented
119C22×C5×D4—111no—
120C23×C4×C5—111no—
121C5×#123—111no—
122C5×#123—111no—
123C5×#123—112no—
124C5×#123—112no—
125C5×#123—111no—
126C5×#123—221no—
127C2×C5×C4⋊C4—112no—
128C2×C5×C22⋊C4—111no—
129C2×C5×C22⋊C4—112no—
130C2×C11⋊C5—22no—
131C2×C11⋊C5—41no—
132C2×C11⋊C5C22×D411yescharacteristic
133C2×C11⋊C5C2×#4112yes—
134AGL(1,𝔽11)—22no—
135AGL(1,𝔽11)—41no—
136AGL(1,𝔽11)#12314yesdirect product factor
137C23×C11C23×C511yescharacteristic
138C23×C11C23×C511yescharacteristic
139C22×D11—24no—
140C22×D11—22no—
141C22×D11—24no—
142C22×D11C5×D414yes—
143C22×D11C23×C511yescharacteristic
144C22×D11C23×C511yescharacteristic
145C22×D11C23×C511yescharacteristic
146D4×C11—22no—
147C2×C4×C11—22no—
148C2×C4×C11C23×C511yescharacteristic
149C2×C4×C11C23×C512yes—
150C2×C4×C11C23×C512yes—
151C22⋊D11—22no—
152C22⋊D11—22no—
153C2×Dic11—22no—
154C2×Dic11C23×C511yescharacteristic
155C2×Dic11C23×C512yes—
156C2×Dic11C23×C512yes—
157D44—22no—
158C4×D11—24no—
159C4×D11—24no—
160C4×D11—22no—
161C4×D11—24no—
162C24×C5—111no—
163C24×C5—111no—
164C2×C5×D4—111no—
165C2×C5×D4—111no—
166C2×C5×D4—111no—
167C2×C5×D4—224no—
168C2×C5×D4—221no—
169C22×C4×C5—111no—
170C22×C4×C5—112no—
171C22×C4×C5—112no—
172C22×C4×C5—111no—
173C22×C4×C5—112no—
174C22×C4×C5—224no—
175C22×C4×C5—221no—
176C5×C4⋊C4—112no—
177C5×C4⋊C4—112no—
178C5×C4⋊C4—112no—
179C5×C4⋊C4—222no—
180C5×C22⋊C4—111no—
181C5×C22⋊C4—112no—
182C5×C22⋊C4—112no—
183C5×C22⋊C4—112no—
184C5×C22⋊C4—112no—
185C5×C22⋊C4—221no—
186C5×C22⋊C4—222no—
187C2×#123—111noSylow
188C11⋊C5C2×#12311yescharacteristic, complemented, Hall
189C22×C11—22no—
190C22×C11—21no—
191C22×C11—22no—
192C22×C11C2×C5×D412yes—
193C22×C11C24×C511yescharacteristic
194C2×D11—22no—
195C2×D11—28no—
196C2×D11—21no—
197C2×D11—22no—
198C2×D11—44no—
199C2×D11C2×C5×D412yes—
200C2×D11C2×C5×D412yes—
201C2×D11C5×#4114yes—
202C4×C11—22no—
203C4×C11—21no—
204C4×C11—22no—
205Dic11—22no—
206Dic11—21no—
207Dic11—22no—
208C23×C5—111no—
209C23×C5—114no—
210C23×C5—111no—
211C23×C5—111no—
212C23×C5—224no—
213C23×C5—222no—
214C23×C5—224no—
215C23×C5—221no—
216C23×C5—221no—
217C5×D4—222no—
218C5×D4—222no—
219C5×D4—222no—
220C5×D4—442no—
221C2×C4×C5—112no—
222C2×C4×C5—112no—
223C2×C4×C5—111no—
224C2×C4×C5—222no—
225C2×C4×C5—224no—
226C2×C4×C5—224no—
227C2×C4×C5—224no—
228C2×C4×C5—222no—
229C2×C4×C5—222no—
230C2×C4×C5—222no—
231C2×C4×C5—221no—
232C2×C4×C5—442no—
233C22×D4—111no—
234C23×C4—111no—
235#123—111no—
236#123—111no—
237#123—112no—
238#123—112no—
239#123—111no—
240#123AGL(1,𝔽11)11yescharacteristic, direct product factor
241C2×C4⋊C4—112no—
242C2×C22⋊C4—111no—
243C2×C22⋊C4—112no—
244C2×C11—22no—
245C2×C11—41no—
246C2×C11C22×C5×D411yescharacteristic
247C2×C11C2×C5×#4112yes—
248D11—22no—
249D11—41no—
250D11C5×#12314yescomplemented
251C22×C5—112no—
252C22×C5—114no—
253C22×C5—112no—
254C22×C5—228no—
255C22×C5—222no—
256C22×C5—221no—
257C22×C5—222no—
258C22×C5—222no—
259C22×C5—221no—
260C22×C5—442no—
261C22×C5—441no—
262C22×C5—442no—
263C22×C5—444no—
264C4×C5—222no—
265C4×C5—221no—
266C4×C5—222no—
267C4×C5—442no—
268C4×C5—441no—
269C4×C5—442no—
270C24—111no—
271C24—111no—
272C2×D4—111no—
273C2×D4—111no—
274C2×D4—111no—
275C2×D4—224no—
276C2×D4C2×AGL(1,𝔽11)11yescharacteristic
277C22×C4—111no—
278C22×C4—112no—
279C22×C4—112no—
280C22×C4—111no—
281C22×C4—112no—
282C22×C4—224no—
283C22×C4C2×AGL(1,𝔽11)11yescharacteristic, complemented
284C4⋊C4—112no—
285C4⋊C4—112no—
286C4⋊C4—112no—
287C4⋊C4C2×AGL(1,𝔽11)12yescomplemented
288C22⋊C4—111no—
289C22⋊C4—112no—
290C22⋊C4—112no—
291C22⋊C4—112no—
292C22⋊C4—112no—
293C22⋊C4C2×AGL(1,𝔽11)11yescharacteristic, complemented
294C22⋊C4C2×AGL(1,𝔽11)12yescomplemented
295C11C2×C5×#12311yescharacteristic, complemented, Sylow
296C2×C5—114no—
297C2×C5—222no—
298C2×C5—221no—
299C2×C5—222no—
300C2×C5—442no—
301C2×C5—441no—
302C2×C5—881no—
303C23—111no—
304C23—114no—
305C23—111no—
306C23—111no—
307C23—224no—
308C23—222no—
309C23—224no—
310C23C22×AGL(1,𝔽11)11yescharacteristic
311C23C22×AGL(1,𝔽11)11yescharacteristic
312D4—22no—
313D4—222no—
314D4—222no—
315D4—222no—
316C2×C4—22no—
317C2×C4—112no—
318C2×C4—112no—
319C2×C4—111no—
320C2×C4—224no—
321C2×C4—224no—
322C2×C4—222no—
323C2×C4—224no—
324C2×C4—222no—
325C2×C4C22×AGL(1,𝔽11)12yes—
326C2×C4C22×AGL(1,𝔽11)12yes—
327C2×C4C22×AGL(1,𝔽11)11yescharacteristic
328C5—221noSylow
329C22—22no—
330C22—21no—
331C22—22no—
332C22—112no—
333C22—114no—
334C22—112no—
335C22—228no—
336C22—222no—
337C22—221no—
338C22—222no—
339C22—444no—
340C22D4×AGL(1,𝔽11)12yes—
341C22C23×AGL(1,𝔽11)11yescharacteristic
342C4—22no—
343C4—21no—
344C4—22no—
345C4—222no—
346C4—221no—
347C4—222no—
348C2—22no—
349C2—41no—
350C2—114no—
351C2—222no—
352C2—441no—
353C2C2×D4×AGL(1,𝔽11)11yescharacteristic
354C2#41×AGL(1,𝔽11)12yes—
subgroupG/H
center, FrattiniC22C23×AGL(1,𝔽11)
commutator, socle, cosocleC22×C11C24×C5
FittingC11×#123C2×C5
prime radicalC2×D4×AGL(1,𝔽11)C2
hypercenter#123AGL(1,𝔽11)
hypocenterC11C2×C5×#123
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×AGL(1,𝔽11)C2semidirect
3. #123×C11⋊C5C2semidirect
4. #66988⋊C5C2semidirect
5. #66982⋊C5C2semidirect
6. #66947⋊C5C2semidirect
7. C4⋊C4×AGL(1,𝔽11)C2semidirect
8. #66939⋊C5C2semidirect
9. #66936⋊C5C2semidirect
10. C22⋊C4×AGL(1,𝔽11)C2semidirect
11. C22⋊C4×AGL(1,𝔽11)C2semidirect
15. C22×C4×C11⋊C5C22semidirect
18. C22×Dic11⋊C5C22semidirect
25. C2×C4×AGL(1,𝔽11)C22semidirect
26. C4⋊C4×C11⋊C5C22semidirect
27. C22⋊C4×C11⋊C5C22semidirect
28. C22⋊C4×C11⋊C5C22semidirect
29. C22⋊(Dic11⋊C5)C22semidirect
30. C22⋊(Dic11⋊C5)C22semidirect
31. ((C2×D11)⋊C4)⋊C5C22semidirect
32. ((C2×D11)⋊C4)⋊C5C22semidirect
33. ((C2×D11)⋊C4)⋊C5C22semidirect
34. C4⋊(Dic11⋊C5)C22semidirect
35. (Dic11⋊C4)⋊C5C22semidirect
36. (Dic11⋊C4)⋊C5C22semidirect
37. D11×#123C5semidirect
64. C22×C4×D11C2×C5semidirect
65. C11×#123C2×C5semidirect
66. #66988C2×C5semidirect
67. #66982C2×C5semidirect
68. #66947C2×C5semidirect
69. C4⋊C4×D11C2×C5semidirect
70. #66939C2×C5semidirect
71. #66936C2×C5semidirect
72. C22⋊C4×D11C2×C5semidirect
73. C22⋊C4×D11C2×C5semidirect
97. C22×C4×C11C22×C5semidirect
100. C22×Dic11C22×C5semidirect
107. C2×C4×D11C22×C5semidirect
108. C11×C4⋊C4C22×C5semidirect
109. C11×C22⋊C4C22×C5semidirect
110. C11×C22⋊C4C22×C5semidirect
111. C22⋊Dic11C22×C5semidirect
112. C22⋊Dic11C22×C5semidirect
113. (C2×D11)⋊C4C22×C5semidirect
114. (C2×D11)⋊C4C22×C5semidirect
115. (C2×D11)⋊C4C22×C5semidirect
116. C4⋊Dic11C22×C5semidirect
117. Dic11⋊C4C22×C5semidirect
118. Dic11⋊C4C22×C5semidirect
136. AGL(1,𝔽11)#123direct (N × H)
188. C11⋊C5C2×#123semidirect
240. #123AGL(1,𝔽11)direct (N × H)
250. D11C5×#123semidirect
283. C22×C4C2×AGL(1,𝔽11)semidirect
287. C4⋊C4C2×AGL(1,𝔽11)semidirect
293. C22⋊C4C2×AGL(1,𝔽11)semidirect
294. C22⋊C4C2×AGL(1,𝔽11)semidirect
295. C11C2×C5×#123semidirect

Maximal subgroups

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

subgroupindex
C2×D4×AGL(1,𝔽11)2
C22×C4×AGL(1,𝔽11)2
#123×C11⋊C52
#66988⋊C52
#66982⋊C52
#66947⋊C52
C4⋊C4×AGL(1,𝔽11)2
#66939⋊C52
#66936⋊C52
C22⋊C4×AGL(1,𝔽11)2
D11×#1235
C2×C5×#12311