Finite Group Databasebeta

← C2×#10623227 #13081981

Subgroups — C2×#10623227 #13081981

Order 1408.

1004 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C11×#125C211yesmaximal, characteristic, complemented
2#10623227C2116yesmaximal, direct product factor
3C2×#66951C211yesmaximal, characteristic
4C2×#66951C211yesmaximal, characteristic
5C2×#66951C211yesmaximal, characteristic
6C2×#66949C211yesmaximal, characteristic
7C2×#66949C211yesmaximal, characteristic
8C2×#66945C211yesmaximal, characteristic
9C2×#66943C211yesmaximal, characteristic, complemented
10C2×#66942C211yesmaximal, characteristic, complemented
11C2×#66942C211yesmaximal, characteristic, complemented
12C2×#66942C211yesmaximal, characteristic, complemented
13C2×#66941C211yesmaximal, characteristic, complemented
14C2×#66940C211yesmaximal, characteristic, complemented
15C2×#66929C211yesmaximal, characteristic
16C2×C4×Dic22C211yesmaximal, characteristic, complemented
17C22×C4×D11C2211yescharacteristic
18C22×Dic22C2211yescharacteristic
19C22×Dic22C2211yescharacteristic
20C11×#125C2218yescomplemented
21C2×C11×C4⋊C4C2211yescharacteristic
22C2×C11×C4⋊C4C2211yescharacteristic
23C2×C11×C4⋊C4C2211yescharacteristic
24C2×C11×C4⋊C4C2211yescharacteristic
25C2×C11×C4⋊C4C2211yescharacteristic
26C2×C11×C4⋊C4C2211yescharacteristic
27C2×C42×C11C2211yescharacteristic
28C2×(C2×D11)⋊C4C2211yescharacteristic
29C2×(C2×D11)⋊C4C2211yescharacteristic
30C2×(C2×D11)⋊C4C2211yescharacteristic
31C2×(C2×D11)⋊C4C2211yescharacteristic
32C2×(C2×D11)⋊C4C2211yescharacteristic
33C2×(C2×D11)⋊C4C2211yescharacteristic
34C2×C4⋊Dic11C2211yescharacteristic
35C2×C4⋊Dic11C2211yescharacteristic
36C2×C4⋊Dic11C2211yescharacteristic
37C2×C4⋊Dic11C2211yescharacteristic
38C2×Dic11⋊C4C2211yescharacteristic
39C2×Dic11⋊C4C2211yescharacteristic
40C2×Dic11⋊C4C2211yescharacteristic
41C2×Dic11⋊C4C2211yescharacteristic
42C2×Dic11⋊C4C2211yescharacteristic
43C2×Dic11⋊C4C2211yescharacteristic
44C2×Dic11⋊C4C2211yescharacteristic
45C2×Dic11⋊C4C2211yescharacteristic
46C2×Dic11⋊C4C2211yescharacteristic
47C2×Dic11⋊C4C2211yescharacteristic
48C2×C4×Dic11C2211yescharacteristic
49C2×C4×Dic11C2211yescharacteristic
50C2×C4×Dic11C2211yescharacteristic
51C2×C4×Dic11C2211yescharacteristic
52C2×C4×Dic11C2211yescharacteristic
53#66951C2218yes—
54#66951C2218yes—
55#66951C2218yes—
56#66949C2218yes—
57#66949C2218yes—
58#66945C2218yes—
59#66943C2218yescomplemented
60#66942C2218yescomplemented
61#66942C2218yescomplemented
62#66942C2218yescomplemented
63#66941C2218yescomplemented
64#66940C2218yescomplemented
65#66929C2218yes—
66C4×Dic22C2218yescomplemented
67C23×D11C2311yescharacteristic
68C22×C4×C11C2311yescharacteristic
69C22×C4×C11C2311yescharacteristic
70C22×C4×C11C2311yescharacteristic
71C22×C4×C11C2311yescharacteristic
72C22×C4×C11C2311yescharacteristic
73C22×C4×C11C2311yescharacteristic
74C22×C4×C11C2311yescharacteristic
75C22×Dic11C2311yescharacteristic
76C22×Dic11C2311yescharacteristic
77C22×Dic11C2311yescharacteristic
78C22×Dic11C2311yescharacteristic
79C22×Dic11C2311yescharacteristic
80C22×Dic11C2311yescharacteristic
81C22×Dic11C2311yescharacteristic
82C2×C4×D11—24no—
83C2×C4×D11C2314yes—
84C2×Dic22—24no—
85C2×Dic22—24no—
86C2×Dic22C2314yes—
87C2×Dic22C2314yes—
88C11×C4⋊C4C2314yes—
89C11×C4⋊C4C2314yes—
90C11×C4⋊C4C2314yes—
91C11×C4⋊C4C2314yes—
92C11×C4⋊C4C2314yes—
93C11×C4⋊C4C2314yes—
94C42×C11C2314yes—
95(C2×D11)⋊C4C2314yes—
96(C2×D11)⋊C4C2314yes—
97(C2×D11)⋊C4C2314yes—
98(C2×D11)⋊C4C2314yes—
99(C2×D11)⋊C4C2314yes—
100(C2×D11)⋊C4C2314yes—
101C4⋊Dic11C2314yes—
102C4⋊Dic11C2314yes—
103C4⋊Dic11C2314yes—
104C4⋊Dic11C2314yes—
105Dic11⋊C4C2314yes—
106Dic11⋊C4C2314yes—
107Dic11⋊C4C2314yes—
108Dic11⋊C4C2314yes—
109Dic11⋊C4C2314yes—
110Dic11⋊C4C2314yes—
111Dic11⋊C4C2314yes—
112Dic11⋊C4C2314yes—
113Dic11⋊C4C2314yes—
114Dic11⋊C4C2314yes—
115C4×Dic11C2314yes—
116C4×Dic11C2314yes—
117C4×Dic11C2314yes—
118C4×Dic11C2314yes—
119C4×Dic11C2314yes—
120C2×#541—111nomaximal, Sylow
121C23×C11C2411yescharacteristic
122C22×D11—22no—
123C22×D11—22no—
124C22×D11—22no—
125C22×D11C2412yes—
126C2×C4×C11—22no—
127C2×C4×C11—22no—
128C2×C4×C11—22no—
129C2×C4×C11—22no—
130C2×C4×C11—22no—
131C2×C4×C11—22no—
132C2×C4×C11—22no—
133C2×C4×C11C2412yes—
134C2×C4×C11C2412yes—
135C2×C4×C11C2412yes—
136C2×C4×C11C2412yes—
137C2×C4×C11C2412yes—
138C2×C4×C11C2412yes—
139C2×C4×C11C2412yes—
140C2×Dic11—22no—
141C2×Dic11—22no—
142C2×Dic11—22no—
143C2×Dic11—22no—
144C2×Dic11—22no—
145C2×Dic11—22no—
146C2×Dic11#4114yes—
147C2×Dic11C2412yes—
148C2×Dic11C2412yes—
149C2×Dic11C2412yes—
150C2×Dic11C2412yes—
151C2×Dic11C2412yes—
152C2×Dic11C2412yes—
153C2×Dic11C2412yes—
154C4×D11—28no—
155Dic22—28no—
156Dic22—28no—
157#541—1116no—
158C2×C4⋊Q8—111no—
159C2×#126—111no—
160C2×#126—111no—
161C2×#126—111no—
162C2×#126—111no—
163C2×#125—111no—
164C2×#125—111no—
165C2×#125—111no—
166C2×#125—111no—
167C2×#125D1111yescharacteristic, complemented
168C2×C22⋊Q8—111no—
169C2×C22⋊Q8—111no—
170C2×C4×Q8—111no—
171C2×C4×Q8—111no—
172C2×#117—111no—
173C22×C11C2×#4112yes—
174C22×C112−1+412yes—
175C22×C112−1+412yes—
176C22×C11C2511yescharacteristic
177C2×D11—22no—
178C2×D11—22no—
179C2×D11—22no—
180C2×D11—44no—
181C4×C11—22no—
182C4×C11—22no—
183C4×C11—22no—
184C4×C11—22no—
185C4×C11—22no—
186C4×C11—22no—
187C4×C11—22no—
188Dic11—22no—
189Dic11—22no—
190Dic11—22no—
191Dic11—22no—
192Dic11—22no—
193Dic11—22no—
194Dic11C2×#4114yes—
195C22×Q8—111no—
196C22×Q8—111no—
197C23×C4—111no—
198C4⋊Q8—118no—
199#126—118no—
200#126—118no—
201#126—118no—
202#126—118no—
203#125—118no—
204#125—118no—
205#125—118no—
206#125—118no—
207#125C2×D1118yescomplemented
208C22⋊Q8—118no—
209C22⋊Q8—118no—
210C4×Q8—118no—
211C4×Q8—118no—
212#117—118no—
213C2×C4⋊C4—111no—
214C2×C4⋊C4—111no—
215C2×C4⋊C4—111no—
216C2×C4⋊C4—111no—
217C2×C4⋊C4—111no—
218C2×C4⋊C4—111no—
219C2×C4⋊C4—111no—
220C2×C4⋊C4—111no—
221C2×C4⋊C4—111no—
222C2×C4⋊C4—111no—
223C2×C4⋊C4—111no—
224C2×C4⋊C4—111no—
225C2×C4⋊C4—111no—
226C2×C4⋊C4—111no—
227C2×C4⋊C4C2×D1111yescharacteristic
228C2×C4⋊C4C2×D1111yescharacteristic
229C2×C4⋊C4C2×D1111yescharacteristic
230C2×C4⋊C4C2×D1111yescharacteristic
231C2×C4⋊C4C2×D1111yescharacteristic
232C2×C4⋊C4C2×D1111yescharacteristic
233C2×C22⋊C4—111no—
234C2×C22⋊C4—111no—
235C2×C22⋊C4—111no—
236C2×C22⋊C4—111no—
237C2×C22⋊C4—111no—
238C2×C22⋊C4—111no—
239C2×C42—111no—
240C2×C42—111no—
241C2×C42—111no—
242C2×C42—111no—
243C2×C42—111no—
244C2×C42C2×D1111yescharacteristic
245C2×C11#54114yescomplemented
246C2×C11C22×#4111yescharacteristic
247C2×C11C2×2−1+411yescharacteristic
248C2×C11C2×2−1+411yescharacteristic
249D11—42no—
250C24—111no—
251C2×Q8—114no—
252C2×Q8—114no—
253C2×Q8—224no—
254C2×Q8—224no—
255C22×C4—111no—
256C22×C4—111no—
257C22×C4—111no—
258C22×C4—111no—
259C22×C4—111no—
260C22×C4—111no—
261C22×C4—111no—
262C22×C4—114no—
263C22×C4—224no—
264C22×C4C22×D1111yescharacteristic
265C22×C4C22×D1111yescharacteristic
266C22×C4C22×D1111yescharacteristic
267C22×C4C22×D1111yescharacteristic
268C22×C4C22×D1111yescharacteristic
269C22×C4C22×D1111yescharacteristic
270C22×C4C22×D1111yescharacteristic
271C4⋊C4—114no—
272C4⋊C4—114no—
273C4⋊C4—114no—
274C4⋊C4—114no—
275C4⋊C4—114no—
276C4⋊C4—114no—
277C4⋊C4—114no—
278C4⋊C4—114no—
279C4⋊C4—114no—
280C4⋊C4—114no—
281C4⋊C4—114no—
282C4⋊C4—114no—
283C4⋊C4—114no—
284C4⋊C4—114no—
285C4⋊C4C22×D1114yes—
286C4⋊C4C22×D1114yes—
287C4⋊C4C22×D1114yes—
288C4⋊C4C22×D1114yes—
289C4⋊C4C22×D1114yes—
290C4⋊C4C22×D1114yes—
291C22⋊C4—114no—
292C22⋊C4—114no—
293C22⋊C4—114no—
294C22⋊C4—114no—
295C22⋊C4—114no—
296C22⋊C4—114no—
297C42—114no—
298C42—114no—
299C42—114no—
300C42—114no—
301C42—114no—
302C42C22×D1114yes—
303C11C2×#54111yescharacteristic, complemented, Sylow
304C23—112no—
305C23—222no—
306C23—222no—
307C23—222no—
308C23C23×D1111yescharacteristic
309Q8—228no—
310Q8—228no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—114no—
319C2×C4—112no—
320C2×C4—112no—
321C2×C4—112no—
322C2×C4—112no—
323C2×C4—112no—
324C2×C4—112no—
325C2×C4—112no—
326C2×C4—228no—
327C2×C4—222no—
328C2×C4—222no—
329C2×C4—222no—
330C2×C4—222no—
331C2×C4—222no—
332C2×C4—222no—
333C2×C4C23×D1112yes—
334C2×C4C23×D1112yes—
335C2×C4C23×D1112yes—
336C2×C4C23×D1112yes—
337C2×C4C23×D1112yes—
338C2×C4C23×D1112yes—
339C2×C4C23×D1112yes—
340C22—222no—
341C22—222no—
342C22—222no—
343C22—444no—
344C22#6704012yes—
345C22#41×D1112yes—
346C22#6704312yes—
347C22C24×D1111yescharacteristic
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—114no—
356C4—222no—
357C4—222no—
358C4—222no—
359C4—222no—
360C4—222no—
361C4—222no—
362C2—442no—
363C2#1062322714yesdirect product factor
364C2C2×#6704011yescharacteristic
365C2C2×#41×D1111yescharacteristic
366C2C2×#6704311yescharacteristic
subgroupG/H
centerC23C23×D11
commutator, cosocleC22×C11C25
FrattiniC22C24×D11
FittingC2×C11×#125C2
socleC23×C11C24
prime radicalC23×D11C23
hypercenterC2×#125D11
hypocenterC11C2×#541
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. C2×C11×#125C2semidirect
2. #10623227C2direct (N × H)
9. C2×#66943C2semidirect
10. C2×#66942C2semidirect
11. C2×#66942C2semidirect
12. C2×#66942C2semidirect
13. C2×#66941C2semidirect
14. C2×#66940C2semidirect
16. C2×C4×Dic22C2semidirect
20. C11×#125C22semidirect
59. #66943C22semidirect
60. #66942C22semidirect
61. #66942C22semidirect
62. #66942C22semidirect
63. #66941C22semidirect
64. #66940C22semidirect
66. C4×Dic22C22semidirect
167. C2×#125D11semidirect
207. #125C2×D11semidirect
245. C2×C11#541semidirect
303. C11C2×#541semidirect
363. C2#10623227direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C11×#1252
#106232272
C2×#669512
C2×#669492
C2×#669452
C2×#669432
C2×#669422
C2×#669412
C2×#669402
C2×#669292
C2×C4×Dic222
C2×#54111