Finite Group Databasebeta

← #38752628

Subgroups — #38752628

Order 25216.

420 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#27161592C211yesmaximal, characteristic
2C2×#20263284C211yesmaximal, characteristic, complemented
3#27161419C212yesmaximal, complemented
4#27161407C211yesmaximal, characteristic, complemented
5#27161406C212yesmaximal, complemented
6#27161397C211yesmaximal, characteristic, complemented
7#27161393C211yesmaximal, characteristic, complemented
8#27161116C212yesmaximal, complemented
9#27161074C212yesmaximal, complemented
10C22⋊C8×D197C211yesmaximal, characteristic, complemented
11#484×C197C211yesmaximal, characteristic, complemented
12C2×#15691359C2211yescharacteristic
13C22×C4×D197C2211yescharacteristic
14C2×C8×D197C2212yes—
15#20263311—21no—
16#20263302C2211yescharacteristic
17#20263301C2211yescharacteristic
18#20263300C2211yescharacteristic
19#20263296—21no—
20#20263288—21no—
21#20263286—21no—
22#20263284—21no—
23#20263284—21no—
24#20263284C2211yescharacteristic
25#20263284C2212yes—
26#20263284C2212yes—
27C2×C4⋊Dic197C2211yescharacteristic
28C2×#15691335—21no—
29C2×C4×Dic197—21no—
30#20263260C2212yescomplemented
31#20263257C2212yescomplemented
32#20263257C2212yescomplemented
33#20263250C2212yescomplemented
34#20263248C2212yescomplemented
35#20263246C2212yescomplemented
36C22⋊#13096802C2211yescharacteristic
37#20263238C2212yes—
38C22⋊Q8×C197C2211yescharacteristic
39C2×C4⋊C4×C197C2211yescharacteristic
40#107×C197C2212yescomplemented
41#103×C197C2212yescomplemented
42C22⋊C8×C197C2211yescharacteristic
43C23×D197D411yescharacteristic
44#15691359—24no—
45C2×D788—21no—
46C2×D788—21no—
47C2×D788C2311yescharacteristic
48C2×C22⋊D197—21no—
49C22×Dic197—21no—
50C22×Dic197D411yescharacteristic
51C2×C4×D197—24no—
52C2×C4×D197—21no—
53C2×C4×D197D411yescharacteristic
54C2×C4×D197D411yescharacteristic
55C2×C4×D197C2311yescharacteristic
56C2×C4×D197C2311yescharacteristic
57C2×#13096802C2312yes—
58C8×D197—24no—
59C4⋊Dic197—21no—
60C4⋊Dic197C2312yes—
61C4⋊Dic197C2311yescharacteristic
62#15691337—21no—
63#15691335—22no—
64#15691335—22no—
65#15691335—21no—
66#15691335—21no—
67#15691335—21no—
68#15691335—21no—
69C4×Dic197—22no—
70C4×Dic197—22no—
71C4×Dic197—21no—
72C2×Q8×C197C2311yescharacteristic
73C22×C4×C197—21no—
74C22×C4×C197C2311yescharacteristic
75C2×C8×C197C2312yes—
76C4⋊C4×C197—21no—
77C4⋊C4×C197—21no—
78C4⋊C4×C197C2312yes—
79C4⋊C4×C197C2311yescharacteristic
80C22⋊C4×C197—21no—
81C22×D197—22no—
82C22×D197—22no—
83C22×D197—21no—
84C22×D197—41no—
85C22×D197C2×D411yescharacteristic
86C22×D197C2×D411yescharacteristic
87D788—22no—
88D788—42no—
89D788—42no—
90C22⋊D197—42no—
91C2×Dic197—22no—
92C2×Dic197—22no—
93C2×Dic197—21no—
94C2×Dic197—42no—
95C2×Dic197C2×D411yescharacteristic
96C2×Dic197C2×D411yescharacteristic
97C4×D197—22no—
98C4×D197—22no—
99C4×D197—22no—
100C4×D197—42no—
101C4×D197#4114yes—
102#13096802—22no—
103C23×C197C2×D411yescharacteristic
104Q8×C197—22no—
105C2×C4×C197—22no—
106C2×C4×C197—21no—
107C2×C4×C197—21no—
108C2×C4×C197—22no—
109C2×C4×C197—42no—
110C2×C4×C197C2×D411yescharacteristic
111C2×C4×C197C2411yescharacteristic
112C8×C197—22no—
113C22×C197—21no—
114C22×C197—21no—
115C22×C197#13512yes—
116C22×C197C22×D411yescharacteristic
117C2×D197—21no—
118C2×D197—21no—
119C2×D197—21no—
120C2×D197—21no—
121C2×D197—21no—
122C2×D197—44no—
123C2×D197—41no—
124C2×D197—41no—
125C2×D197—41no—
126C2×D197#12312yes—
127C4×C197—21no—
128C4×C197—41no—
129C4×C197—42no—
130C4×C197—41no—
131C4×C197C2×#4112yes—
132Dic197—21no—
133Dic197—42no—
134Dic197—41no—
135Dic197#12312yes—
136C2×C197—22no—
137C2×C197C2×#12311yescharacteristic
138C2×C197C2×#13511yescharacteristic
139C2×C197#57511yescharacteristic
140D197—22no—
141D197—41no—
142D197—81no—
143C197#309111yescharacteristic, complemented, Sylow
144#3091—1971nomaximal, Sylow
145#525—1971no—
146C2×#117—1971no—
147#485—1972no—
148#484D19711yescharacteristic, complemented
149#483—1971no—
150#482—1972no—
151#481—1971no—
152#480—1971no—
153#427—1972no—
154#416—1972no—
155C2×C22⋊C8—1971no—
156C2×#41—1971no—
157C23×C4—1971no—
158C22×C8—1972no—
159#123—3941no—
160C22⋊Q8—1971no—
161C22⋊Q8C2×D19711yescharacteristic
162#121—1971no—
163#121—1971no—
164#120—3941no—
165C4×D4—3941no—
166C4×D4—3941no—
167#117—3941no—
168#117—3941no—
169#117—1972no—
170#117—1972no—
171#117—1971no—
172C2×C4⋊C4—1971no—
173C2×C4⋊C4C2×D19711yescharacteristic
174C2×C22⋊C4—3941no—
175C2×C42—3941no—
176#107—1972no—
177#107C2×D19712yescomplemented
178#106—1972no—
179#106—1972no—
180#103—1972no—
181#103C2×D19712yescomplemented
182#102—1972no—
183#102—1972no—
184C22⋊C8—1971no—
185C22⋊C8—1972no—
186C22⋊C8C2×D19711yescharacteristic
187C24—1971no—
188#41—3944no—
189C2×Q8C22×D19711yescharacteristic
190C2×D4—3941no—
191C2×D4—3941no—
192C2×D4—3941no—
193C2×D4—1971no—
194C22×C4—21no—
195C22×C4—3941no—
196C22×C4—3944no—
197C22×C4—3941no—
198C22×C4—1971no—
199C22×C4—1971no—
200C22×C4—1971no—
201C22×C4—1971no—
202C22×C4—1971no—
203C22×C4C22×D19711yescharacteristic
204C2×C8—3944no—
205C2×C8—1972no—
206C2×C8C22×D19712yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—3941no—
210C4⋊C4—3941no—
211C4⋊C4—1972no—
212C4⋊C4—1971no—
213C4⋊C4C22×D19712yes—
214C4⋊C4C22×D19711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—3942no—
217C22⋊C4—3942no—
218C22⋊C4—3941no—
219C22⋊C4—3941no—
220C22⋊C4—3941no—
221C22⋊C4—3941no—
222C42—3942no—
223C42—3942no—
224C42—3941no—
225C23—7881no—
226C23—3942no—
227C23—3941no—
228C23—3942no—
229C23—1971no—
230C23—1971no—
231C23D4×D19711yescharacteristic
232Q8—22no—
233D4—7882no—
234D4—7882no—
235D4—7882no—
236D4—3942no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—7882no—
243C2×C4—7882no—
244C2×C4—3942no—
245C2×C4—3942no—
246C2×C4—3942no—
247C2×C4—3941no—
248C2×C4—3942no—
249C2×C4—3942no—
250C2×C4—1971no—
251C2×C4—1971no—
252C2×C4—1974no—
253C2×C4D4×D19711yescharacteristic
254C2×C4C23×D19711yescharacteristic
255C8—22no—
256C8—3942no—
257C22—21no—
258C22—21no—
259C22—7884no—
260C22—7881no—
261C22—7881no—
262C22—7881no—
263C22—3941no—
264C22—3941no—
265C22—3941no—
266C22—3941no—
267C22—3941no—
268C22—1972no—
269C22#2026334912yes—
270C22C2×D4×D19711yescharacteristic
271C4—21no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4—7882no—
276C4—7881no—
277C4—3941no—
278C4—1972no—
279C4#41×D19712yes—
280C2—22no—
281C2—7881no—
282C2—15761no—
283C2—3942no—
284C2#123×D19711yescharacteristic
285C2C2×#2026334911yescharacteristic
286C2#2716187511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#484×C197C2
socleC22×C197C22×D4
prime radical#27161592C2
hypercenter#484D197
hypocenterC197#3091
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#20263284C2semidirect
3. #27161419C2semidirect
4. #27161407C2semidirect
5. #27161406C2semidirect
6. #27161397C2semidirect
7. #27161393C2semidirect
8. #27161116C2semidirect
9. #27161074C2semidirect
10. C22⋊C8×D197C2semidirect
11. #484×C197C2semidirect
30. #20263260C22semidirect
31. #20263257C22semidirect
32. #20263257C22semidirect
33. #20263250C22semidirect
34. #20263248C22semidirect
35. #20263246C22semidirect
40. #107×C197C22semidirect
41. #103×C197C22semidirect
143. C197#3091semidirect
148. #484D197semidirect
177. #107C2×D197semidirect
181. #103C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271615922
C2×#202632842
#271614192
#271614072
#271614062
#271613972
#271613932
#271611162
#271610742
C22⋊C8×D1972
#484×C1972
#3091197