Finite Group Databasebeta

← C2×#20251385 #26422396

Subgroups — C2×#20251385 #26422396

Order 12032.

4438 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 4438 conjugacy classes of subgroups — too many to draw. Only its 667 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093993C212yesmaximal, complemented
2#20251385C2116yesmaximal, direct product factor
3C2×#15689048C213yesmaximal, complemented
4C2×#15689039C216yesmaximal, complemented
5C2×#15689032C213yesmaximal, complemented
6C2×C47×#379C211yesmaximal, characteristic, complemented
7C23×D188C2211yescharacteristic, complemented
8C23×C22⋊D47C2216yes—
9C2×#13093993—216no—
10C2×#13093993C22116yescomplemented
11C22×C4⋊Dic47C2213yescomplemented
12C22×C22⋊Dic47C2216yes—
13C22×(C2×D47)⋊C4C2216yescomplemented
14#15689048C22124yescomplemented
15#15689039C22148yescomplemented
16#15689032C22124yescomplemented
17C2×C47⋊#95C2213yescomplemented
18C2×C47⋊#95C2213yescomplemented
19C26×C47C2211yescharacteristic
20C22×C22⋊C4×C47C2213yes—
21C22×C22⋊C4×C47C2213yes—
22C47×#379C2218yescomplemented
23C24×D47C2312yes—
24C22×D188—46no—
25C22×D188C2314yescomplemented
26C22×C22⋊D47—224no—
27C22×C22⋊D47—412no—
28C22×C22⋊D47—412no—
29C22×C22⋊D47C23124yes—
30C23×Dic47C2316yes—
31#13093993—264no—
32C2×C4⋊Dic47—46no—
33C2×C4⋊Dic47C23112yescomplemented
34C2×C22⋊Dic47—224no—
35C2×C22⋊Dic47C23124yes—
36C2×(C2×D47)⋊C4—412no—
37C2×(C2×D47)⋊C4C23124yescomplemented
38C47⋊#95C23112yescomplemented
39C47⋊#95C23112yescomplemented
40C25×C47—46no—
41C25×C47—46no—
42C25×C47D418yescomplemented
43C25×C47C2313yes—
44C25×C47C2314yes—
45C23×C4×C47C2311yescharacteristic
46C23×C4×C47C2313yes—
47C2×C22⋊C4×C47—46no—
48C2×C22⋊C4×C47—46no—
49C2×C22⋊C4×C47C23112yes—
50C2×C22⋊C4×C47C23112yes—
51C24×C47—26no—
52C24×C47—26no—
53C24×C47—212no—
54C24×C47—212no—
55C24×C47—448no—
56C24×C47—412no—
57C24×C47—412no—
58C24×C47—448no—
59C24×C47—416no—
60C24×C47C2×D4112yes—
61C24×C47C2×D4116yescomplemented
62C24×C47C2416yes—
63C24×C47C2411yescharacteristic
64C23×D47—24no—
65C23×D47—212no—
66C23×D47—212no—
67C23×D47C2414yes—
68C22×C4×C47—212no—
69C22×C4×C47—26no—
70C22×C4×C47—26no—
71C22×C4×C47C2412yes—
72C22×C4×C47C2416yes—
73C2×C22⋊D47—248no—
74C2×C22⋊D47—424no—
75C2×C22⋊D47—424no—
76C2×C22⋊D47—496no—
77C22×Dic47—212no—
78C22×Dic47—212no—
79C22×Dic47—212no—
80C22×Dic47C24112yes—
81C2×D188—412no—
82C2×D188—416no—
83C22⋊C4×C47—412no—
84C22⋊C4×C47—412no—
85C22⋊Dic47—248no—
86(C2×D47)⋊C4—424no—
87C4⋊Dic47—412no—
88C23×C47—248no—
89C23×C47—224no—
90C23×C47—26no—
91C23×C47—26no—
92C23×C47—212no—
93C23×C47—212no—
94C23×C47—448no—
95C23×C47—448no—
96C23×C47—416no—
97C23×C47—4128no—
98C23×C47—424no—
99C23×C47—412no—
100C23×C47—412no—
101C23×C47C22×D4112yes—
102C23×C47C22×D412yes—
103C23×C47H(5,𝔽2)16yes—
104C23×C47H(5,𝔽2)16yes—
105C23×C47C2511yescharacteristic
106C22×D47—24no—
107C22×D47—212no—
108C22×D47—212no—
109C22×D47—424no—
110C22×D47—424no—
111C22×D47—48no—
112C2×C4×C47—26no—
113C2×C4×C47—212no—
114C2×C4×C47—26no—
115C2×C4×C47—412no—
116C2×C4×C47—44no—
117C22⋊D47—496no—
118C2×Dic47—212no—
119C2×Dic47—212no—
120C2×Dic47—212no—
121C2×Dic47—424no—
122D188—416no—
123C2×#2812—471nomaximal, Sylow
124C22×C47—224no—
125C22×C47—212no—
126C22×C47—464no—
127C22×C47—412no—
128C22×C47—46no—
129C22×C47—46no—
130C22×C47—448no—
131C22×C47#534112yes—
132C22×C47#560112yes—
133C22×C47H(3,𝔽4)14yes—
134C22×C47C23×D411yescharacteristic
135C22×C47C2×H(5,𝔽2)13yes—
136C22×C47C2×H(5,𝔽2)13yes—
137C2×D47—412no—
138C2×D47—412no—
139C2×D47—44no—
140C2×D47—816no—
141C4×C47—46no—
142C4×C47—42no—
143Dic47—412no—
144C22×#120—472no—
145#2812—4716no—
146C2×#397—473no—
147C2×#394—476no—
148C2×#392—473no—
149C2×#379D4711yescharacteristic, complemented
150C2×C47—412no—
151C2×C47#281218yescomplemented
152C2×C47C2×#53413yes—
153C2×C47C2×#56013yes—
154C2×C47C2×H(3,𝔽4)11yescharacteristic
155D47—84no—
156C26C2×D4711yescharacteristic
157C23×D4—476no—
158C23×D4—471no—
159C2×#120—4716no—
160C2×#120—9416no—
161C22×C4⋊C4—473no—
162C22×C22⋊C4—476no—
163C22×C22⋊C4—476no—
164C22×C22⋊C4C2×D4713yes—
165C22×C22⋊C4C2×D4713yes—
166#397—4724no—
167#394—4748no—
168#392—4724no—
169#379C2×D4718yescomplemented
170C2×#95—473no—
171C2×#95—473no—
172C47C2×#281211yescharacteristic, complemented, Sylow
173C25—46no—
174C25—46no—
175C25—472no—
176C25D18818yescomplemented
177C25C22×D4713yes—
178C25C22×D4714yes—
179C22×D4—4724no—
180C22×D4—474no—
181C22×D4—9424no—
182C22×D4—1886no—
183C22×D4—18812no—
184C22×D4—18812no—
185C23×C4—476no—
186C23×C4C22×D4713yes—
187C23×C4C22×D4711yescharacteristic
188#120—9464no—
189C2×C4⋊C4—4712no—
190C2×C4⋊C4—1886no—
191C2×C22⋊C4—46no—
192C2×C22⋊C4—46no—
193C2×C22⋊C4—4724no—
194C2×C22⋊C4—4724no—
195C2×C22⋊C4—9424no—
196C2×C22⋊C4—18812no—
197C2×C22⋊C4C22×D47112yes—
198C2×C22⋊C4C22×D47112yes—
199#95—4712no—
200#95—4712no—
201C24—26no—
202C24—26no—
203C24—212no—
204C24—212no—
205C24—448no—
206C24—412no—
207C24—412no—
208C24—448no—
209C24—416no—
210C24—474no—
211C24—944no—
212C24—9412no—
213C24—9412no—
214C24C2×D188112yes—
215C24C2×D188116yescomplemented
216C24C23×D4716yes—
217C24C23×D4711yescharacteristic
218C2×D4—9448no—
219C2×D4—18812no—
220C2×D4—18816no—
221C2×D4—18824no—
222C2×D4—18824no—
223C2×D4—18896no—
224C22×C4—212no—
225C22×C4—26no—
226C22×C4—26no—
227C22×C4—4712no—
228C22×C4—9412no—
229C22×C4—9412no—
230C22×C4—9412no—
231C22×C4C23×D4712yes—
232C22×C4C23×D4716yes—
233C4⋊C4—18812no—
234C22⋊C4—412no—
235C22⋊C4—412no—
236C22⋊C4—9448no—
237C22⋊C4—18824no—
238C23—248no—
239C23—224no—
240C23—26no—
241C23—26no—
242C23—212no—
243C23—212no—
244C23—448no—
245C23—448no—
246C23—416no—
247C23—4128no—
248C23—424no—
249C23—412no—
250C23—412no—
251C23—944no—
252C23—9412no—
253C23—9412no—
254C23—18824no—
255C23—1888no—
256C23—18824no—
257C23C22×D188112yes—
258C23C22×D18812yes—
259C23#1309406916yes—
260C23#1309406916yes—
261C23C24×D4711yescharacteristic
262D4—18816no—
263D4—18896no—
264C2×C4—212no—
265C2×C4—26no—
266C2×C4—26no—
267C2×C4—412no—
268C2×C4—44no—
269C2×C4—9412no—
270C2×C4—9412no—
271C2×C4—9412no—
272C2×C4—18824no—
273C22—224no—
274C22—212no—
275C22—464no—
276C22—412no—
277C22—46no—
278C22—46no—
279C22—448no—
280C22—37616no—
281C22—18812no—
282C22—18812no—
283C22—1884no—
284C22#15689685112yes—
285C22#15689862112yes—
286C22#1568986614yes—
287C22C23×D18811yescharacteristic
288C22C2×#1309406913yes—
289C22C2×#1309406913yes—
290C4—46no—
291C4—42no—
292C4—18812no—
293C2—412no—
294C2—3764no—
295C2#2025138518yesdirect product factor
296C2C2×#1568968513yes—
297C2C2×#1568986213yes—
298C2C2×#1568986611yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#379C2
socleC24×C47C24
hypercenterC2×#379D47
hypocenterC47C2×#2812
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
1. C22×#13093993C2semidirect
2. #20251385C2direct (N × H)
3. C2×#15689048C2semidirect
4. C2×#15689039C2semidirect
5. C2×#15689032C2semidirect
6. C2×C47×#379C2semidirect
7. C23×D188C22semidirect
10. C2×#13093993C22semidirect
11. C22×C4⋊Dic47C22semidirect
13. C22×(C2×D47)⋊C4C22semidirect
14. #15689048C22semidirect
15. #15689039C22semidirect
16. #15689032C22semidirect
17. C2×C47⋊#95C22semidirect
18. C2×C47⋊#95C22semidirect
22. C47×#379C22semidirect
25. C22×D188C23semidirect
33. C2×C4⋊Dic47C23semidirect
37. C2×(C2×D47)⋊C4C23semidirect
38. C47⋊#95C23semidirect
39. C47⋊#95C23semidirect
42. C25×C47D4semidirect
61. C24×C47C2×D4semidirect
149. C2×#379D47semidirect
151. C2×C47#2812semidirect
169. #379C2×D47semidirect
172. C47C2×#2812semidirect
176. C25D188semidirect
215. C24C2×D188semidirect
295. C2#20251385direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939932
#202513852
C2×#156890482
C2×#156890392
C2×#156890322
C2×C47×#3792
C2×#281247