Finite Group Databasebeta

← #38752853

Subgroups — #38752853

Order 25216.

450 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 117 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#27161615C211yesmaximal, characteristic, complemented
2C2×#20263288C211yesmaximal, characteristic, complemented
3#27161392C211yesmaximal, characteristic, complemented
4#27161378C212yesmaximal, complemented
5#27161297C212yesmaximal, complemented
6#27161229C211yesmaximal, characteristic, complemented
7#27161215C211yesmaximal, characteristic
8#27161190C212yesmaximal, complemented
9#27161182C212yesmaximal, complemented
10#27161133C211yesmaximal, characteristic, complemented
11#483×C197C211yesmaximal, characteristic, complemented
12C22×D788C2211yescharacteristic
13C22×C4×D197—21no—
14C2×#15691345C2212yes—
15C2×D1576C2212yes—
16#20263321—21no—
17#20263317C2211yescharacteristic, complemented
18#20263315C2211yescharacteristic, complemented
19#20263310—21no—
20#20263300C2211yescharacteristic
21Q8×Dic197C2211yescharacteristic, complemented
22#20263288—21no—
23#20263288—21no—
24#20263288C2212yes—
25#20263288C2212yes—
26#20263288C2211yescharacteristic, complemented
27#20263286—21no—
28#20263284—21no—
29C2×#15691335—21no—
30C2×C4×Dic197C2211yescharacteristic
31#20263259C2212yescomplemented
32#20263255C2212yescomplemented
33#20263248C2212yes—
34#20263246C2212yes—
35#20263246C2212yescomplemented
36C22⋊#13096802C2211yescharacteristic
37#20263236C2212yescomplemented
38C22⋊Q8×C197C2211yescharacteristic
39C2×C4⋊C4×C197C2211yescharacteristic
40#106×C197C2212yescomplemented
41#103×C197C2212yescomplemented
42C22⋊C8×C197C2211yescharacteristic
43C23×D197—21no—
44C2×D788—24no—
45C2×D788—21no—
46C2×D788C2311yescharacteristic
47C2×D788C2312yes—
48C2×C22⋊D197—21no—
49C22×Dic197D411yescharacteristic
50C22×Dic197D411yescharacteristic
51C2×C4×D197—21no—
52C2×C4×D197—22no—
53C2×C4×D197—22no—
54C2×C4×D197—44no—
55#15691345—24no—
56D1576—24no—
57C2×#13096802C2312yes—
58C4⋊Dic197—21no—
59C4⋊Dic197—21no—
60C4⋊Dic197C2311yescharacteristic
61#15691337—21no—
62#15691335—22no—
63#15691335—22no—
64#15691335—21no—
65#15691335—21no—
66#15691335—21no—
67#15691335—21no—
68C4×Dic197—21no—
69C4×Dic197D411yescharacteristic
70C4×Dic197D411yescharacteristic
71C4×Dic197C2311yescharacteristic
72C4×Dic197C2311yescharacteristic
73C2×Q8×C197C2311yescharacteristic
74C22×C4×C197—21no—
75C22×C4×C197C2311yescharacteristic
76C2×C8×C197C2312yes—
77C4⋊C4×C197—21no—
78C4⋊C4×C197—21no—
79C4⋊C4×C197C2312yes—
80C4⋊C4×C197C2311yescharacteristic
81C22⋊C4×C197—21no—
82C22×D197—24no—
83C22×D197—22no—
84C22×D197—21no—
85C22×D197—42no—
86C22×D197—42no—
87D788—24no—
88D788—22no—
89D788—22no—
90D788#4114yes—
91C22⋊D197—42no—
92C2×Dic197—22no—
93C2×Dic197—21no—
94C2×Dic197—41no—
95C2×Dic197C2×D411yescharacteristic
96C2×Dic197C2×D411yescharacteristic
97C2×Dic197C2×D411yescharacteristic
98C2×Dic197C2×D411yescharacteristic
99C4×D197—42no—
100C4×D197—44no—
101C4×D197—44no—
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×C197AGL(1,ℤ/8)12yes—
116C22×C197C22×D411yescharacteristic
117C2×D197—24no—
118C2×D197—48no—
119C2×D197—42no—
120C2×D197—42no—
121C2×D197—41no—
122C2×D197—41no—
123C2×D197—41no—
124C4×C197—21no—
125C4×C197—41no—
126C4×C197—42no—
127C4×C197—41no—
128C4×C197C2×#4112yes—
129Dic197—21no—
130Dic197—21no—
131Dic197—21no—
132Dic197—21no—
133Dic197—41no—
134C2×C197—22no—
135C2×C197#52511yescharacteristic
136C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C197#57511yescharacteristic
138D197—44no—
139D197—81no—
140C197#311411yescharacteristic, complemented, Sylow
141#3114—1971nomaximal, Sylow
142#529—1971no—
143C2×C4×D4—1971no—
144#483D19711yescharacteristic, complemented
145#480—1971no—
146#478—1972no—
147#463—1972no—
148#450—1971no—
149#447—1971no—
150#442—1972no—
151#440—1972no—
152#432—1971no—
153C22×D4—1971no—
154C23×C4—3941no—
155C2×SD16—1972no—
156C2×D8—1972no—
157#126—3941no—
158#125—1971no—
159#124—1971no—
160#123—3941no—
161C22⋊Q8C2×D19711yescharacteristic
162#121—1971no—
163C4×Q8—1971no—
164C4×D4—3941no—
165C4×D4—3941no—
166C4×D4—3941no—
167C4×D4—1972no—
168C4×D4—1972no—
169C4×D4—1971no—
170#117—3941no—
171C2×C4⋊C4C2×D19711yescharacteristic
172C2×C22⋊C4—3941no—
173C2×C42—1971no—
174#107—1972no—
175#106C2×D19712yescomplemented
176C4⋊C8—1972no—
177#103C2×D19712yescomplemented
178#102—1972no—
179#102—1972no—
180#102—1972no—
181C22⋊C8—1971no—
182C22⋊C8C2×D19711yescharacteristic
183#97—1972no—
184C24—3941no—
185C2×Q8C22×D19711yescharacteristic
186C2×D4—3944no—
187C2×D4—3941no—
188C2×D4—3941no—
189C2×D4—1971no—
190C2×D4—1972no—
191C22×C4—21no—
192C22×C4—7884no—
193C22×C4—3941no—
194C22×C4—3942no—
195C22×C4—3942no—
196C22×C4—1971no—
197C22×C4—1971no—
198C22×C4C22×D19711yescharacteristic
199SD16—3944no—
200D8—3944no—
201C2×C8—1972no—
202C2×C8C22×D19712yes—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—3941no—
206C4⋊C4—3941no—
207C4⋊C4—3941no—
208C4⋊C4—1971no—
209C4⋊C4C22×D19711yescharacteristic
210C4⋊C4C22×D19712yes—
211C22⋊C4—21no—
212C22⋊C4—3942no—
213C22⋊C4—3942no—
214C22⋊C4—3941no—
215C22⋊C4—3941no—
216C22⋊C4—3941no—
217C22⋊C4—3941no—
218C42—3941no—
219C42—1971no—
220C42—1971no—
221C42—1971no—
222C42—1971no—
223C23—7882no—
224C23—7882no—
225C23—3944no—
226C23—3941no—
227C23—3942no—
228C23D4×D19711yescharacteristic
229Q8—22no—
230D4—7882no—
231D4—3944no—
232D4—3942no—
233D4—3942no—
234D4—1974no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—7884no—
241C2×C4—7884no—
242C2×C4—7882no—
243C2×C4—7881no—
244C2×C4—3941no—
245C2×C4—3942no—
246C2×C4—1971no—
247C2×C4—1971no—
248C2×C4—1971no—
249C2×C4—1971no—
250C2×C4D4×D19711yescharacteristic
251C2×C4C23×D19711yescharacteristic
252C8—22no—
253C8—3942no—
254C22—21no—
255C22—21no—
256C22—7888no—
257C22—7882no—
258C22—7882no—
259C22—7881no—
260C22—7881no—
261C22—7881no—
262C22—3944no—
263C22#2026335312yes—
264C22C2×D4×D19711yescharacteristic
265C4—21no—
266C4—42no—
267C4—41no—
268C4—41no—
269C4—7881no—
270C4—3941no—
271C4—3941no—
272C4—3941no—
273C4—3941no—
274C4#41×D19712yes—
275C2—22no—
276C2—7884no—
277C2—15761no—
278C2#2716158711yescharacteristic
279C2C2×#2026335311yescharacteristic
280C2#2716187811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#483×C197C2
socleC22×C197C22×D4
prime radical#27161215C2
hypercenter#483D197
hypocenterC197#3114
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #27161615C2semidirect
2. C2×#20263288C2semidirect
3. #27161392C2semidirect
4. #27161378C2semidirect
5. #27161297C2semidirect
6. #27161229C2semidirect
8. #27161190C2semidirect
9. #27161182C2semidirect
10. #27161133C2semidirect
11. #483×C197C2semidirect
17. #20263317C22semidirect
18. #20263315C22semidirect
21. Q8×Dic197C22semidirect
26. #20263288C22semidirect
31. #20263259C22semidirect
32. #20263255C22semidirect
35. #20263246C22semidirect
37. #20263236C22semidirect
40. #106×C197C22semidirect
41. #103×C197C22semidirect
140. C197#3114semidirect
144. #483D197semidirect
175. #106C2×D197semidirect
177. #103C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616152
C2×#202632882
#271613922
#271613782
#271612972
#271612292
#271612152
#271611902
#271611822
#271611332
#483×C1972
#3114197