Finite Group Databasebeta

← #38752103

Subgroups — #38752103

Order 25216.

766 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 145 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#27161915C211yesmaximal, characteristic, complemented
2#27161590C212yesmaximal, complemented
3#27161268C214yesmaximal, complemented
4#27161265C214yesmaximal, complemented
5#27161053C211yesmaximal, characteristic, complemented
6#27161052C212yesmaximal, complemented
7#525×C197C211yesmaximal, characteristic, complemented
8#20263374—21no—
9#20263373—22no—
10#20263372—22no—
11#20263371—21no—
12#20263371C2214yes—
13C2×#15691358—22no—
14C2×#15691358C2212yes—
15#41×D197—24no—
16#20263309C2214yescomplemented
17#20263305—22no—
18#20263300—22no—
19#20263295C2214yescomplemented
20C4×C22⋊D197—24no—
21#20263285C2212yescomplemented
22#20263241C2214yescomplemented
23#20263240C2218yescomplemented
24C2×#41×C197C2211yescharacteristic
25C23×C4×C197C2211yescharacteristic, complemented
26#123×C197C2212yescomplemented
27C22⋊Q8×C197—21no—
28#121×C197—21no—
29#120×C197C2212yescomplemented
30C4×D4×C197—22no—
31#117×C197C2211yescharacteristic, complemented
32#15691359—24no—
33#15691358—24no—
34#15691358—24no—
35#15691358—22no—
36#15691358—22no—
37#15691358—24no—
38#15691358—22no—
39#15691358—22no—
40#15691357—24no—
41#15691357—28no—
42C2×Dic394—22no—
43C2×Dic394D412yescomplemented
44Q8×D197—24no—
45C2×D788—22no—
46C2×D788D412yescomplemented
47C2×C22⋊D197—24no—
48C2×C22⋊D197—24no—
49C2×C22⋊D197C2314yes—
50D4×D197—24no—
51D4×D197—28no—
52C2×C4×D197—24no—
53C2×C4×D197D414yescomplemented
54C4⋊Dic197—22no—
55#15691337—24no—
56C22⋊Dic197—24no—
57C22⋊Dic197C2314yes—
58#15691335—24no—
59C4×Dic197—22no—
60C24×C197C2311yescharacteristic
61#41×C197—22no—
62#41×C197—22no—
63C2×Q8×C197D411yescharacteristic, complemented
64C2×D4×C197—22no—
65C2×D4×C197D411yescharacteristic, complemented
66C2×D4×C197C2312yes—
67C22×C4×C197—21no—
68C22×C4×C197—21no—
69C22×C4×C197—22no—
70C22×C4×C197—22no—
71C22×C4×C197D412yescomplemented
72C22×C4×C197C2311yescharacteristic
73C22×C4×C197C2311yescharacteristic
74C4⋊C4×C197—21no—
75C4⋊C4×C197—22no—
76C22⋊C4×C197—22no—
77C22⋊C4×C197—22no—
78C22⋊C4×C197C2312yes—
79C42×C197—21no—
80C22×D197—24no—
81C22×D197C2×D414yes—
82Dic394—24no—
83Dic394—22no—
84Dic394—22no—
85D788—24no—
86D788—22no—
87D788—22no—
88C22⋊D197—28no—
89C22⋊D197—24no—
90C22⋊D197—24no—
91C22⋊D197—48no—
92C2×Dic197—24no—
93C2×Dic197—24no—
94C2×Dic197C2×D414yes—
95C4×D197—28no—
96C4×D197—24no—
97C4×D197—24no—
98C23×C197—21no—
99C23×C197—22no—
100C23×C197—42no—
101C23×C197C2×D412yes—
102C23×C197C2411yescharacteristic
103Q8×C197—22no—
104D4×C197—24no—
105D4×C197—22no—
106D4×C197—44no—
107C2×C4×C197—22no—
108C2×C4×C197—22no—
109C2×C4×C197—22no—
110C2×C4×C197—21no—
111C2×C4×C197—22no—
112C2×C4×C197—41no—
113C2×C4×C197—42no—
114C2×C4×C197—41no—
115C2×C4×C197C2×D412yes—
116C2×C4×C197C2×D412yes—
117C2×C4×C197C2×D412yes—
118C2×C4×C197C2×D411yescharacteristic
119C2×C4×C197C2×D411yescharacteristic
120C22×C197—22no—
121C22×C197—24no—
122C22×C197—22no—
123C22×C197—41no—
124C22×C197—42no—
125C22×C197—42no—
126C22×C197—41no—
127C22×C197—41no—
128C22×C197C22×D412yes—
129C22×C197C22×D411yescharacteristic
130C2×D197—24no—
131C2×D197—44no—
132C2×D197—84no—
133C4×C197—22no—
134C4×C197—42no—
135C4×C197—41no—
136C4×C197#12011yescharacteristic
137C4×C197#12012yes—
138C4×C197#12011yescharacteristic
139Dic197—24no—
140Dic197—44no—
141C2×C197—22no—
142C2×C197—21no—
143C2×C197—42no—
144C2×C197—42no—
145C2×C197C2×#12011yescharacteristic
146D197—44no—
147C197#302511yescharacteristic, complemented, Sylow
148#3025—1971nomaximal, Sylow
149#585—1971no—
150#525—1972no—
151#525D19711yescharacteristic, complemented
152#458—1974no—
153B(4,𝔽2)—1974no—
154#410—1971no—
155#410—1972no—
1562−1+4—3942no—
1572−1+4—3941no—
158H(5,𝔽2)—3942no—
159H(5,𝔽2)—3941no—
160H(5,𝔽2)—1974no—
161C2×#41—3944no—
162C2×#41—3942no—
163C2×#41—1972no—
164C2×#41C2×D19711yescharacteristic
165C23×C4C2×D19711yescharacteristic, complemented
166#123—1974no—
167#123C2×D19712yescomplemented
168C22⋊Q8—21no—
169C22⋊Q8—3942no—
170#121—21no—
171#121—3942no—
172#120—1974no—
173#120C2×D19712yescomplemented
174C4×D4—22no—
175C4×D4—3944no—
176#117—1972no—
177#117C2×D19711yescharacteristic, complemented
178#99—1974no—
179#99—1978no—
180C24C22×D19711yescharacteristic
181#41—22no—
182#41—22no—
183#41—3944no—
184#41—3944no—
185#41—3944no—
186#41—3944no—
187#41—3948no—
188#41—3942no—
189#41—3942no—
190#41—3942no—
191#41—3944no—
192#41—3942no—
193C2×Q8—3942no—
194C2×Q8—3944no—
195C2×Q8—1972no—
196C2×Q8C22⋊D19711yescharacteristic, complemented
197C2×D4—22no—
198C2×D4—3942no—
199C2×D4—3944no—
200C2×D4—3948no—
201C2×D4—3944no—
202C2×D4—3944no—
203C2×D4—1974no—
204C2×D4—1972no—
205C2×D4C22⋊D19711yescharacteristic, complemented
206C2×D4C22×D19712yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—3944no—
212C22×C4—1974no—
213C22×C4C22⋊D19712yescomplemented
214C22×C4C22×D19711yescharacteristic
215C22×C4C22×D19711yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—3944no—
219C4⋊C4—3942no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—3944no—
223C22⋊C4—3944no—
224C22⋊C4—1974no—
225C22⋊C4C22×D19712yes—
226C42—21no—
227C42—3942no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—3944no—
232C23—1974no—
233C23C2×C22⋊D19712yes—
234C23C23×D19711yescharacteristic
235Q8—22no—
236Q8—3944no—
237Q8—3942no—
238Q8—3942no—
239D4—24no—
240D4—22no—
241D4—44no—
242D4—7888no—
243D4—3944no—
244D4—3948no—
245D4—3944no—
246D4—3944no—
247D4—3942no—
248D4—3942no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—3948no—
258C2×C4—3944no—
259C2×C4—3944no—
260C2×C4—3944no—
261C2×C4—3944no—
262C2×C4—1974no—
263C2×C4D4×D19712yes—
264C2×C4D4×D19712yes—
265C2×C4C2×C22⋊D19712yes—
266C2×C4C2×C22⋊D19711yescharacteristic
267C2×C4C2×C22⋊D19711yescharacteristic
268C22—22no—
269C22—24no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—7884no—
277C22—15764no—
278C22—3944no—
279C22C2×D4×D19712yes—
280C22C22×C22⋊D19711yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—7884no—
285C4—3944no—
286C4#2026329411yescharacteristic
287C4#2026329412yes—
288C4#2026329411yescharacteristic
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—7884no—
294C2C2×#2026329411yescharacteristic
subgroupG/H
centerC4#20263294
commutator, cosocleC23×C197C24
FrattiniC23C23×D197
Fitting#525×C197C2
socleC2×C197C2×#120
hypercenter#525D197
hypocenterC197#3025
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #27161915C2semidirect
2. #27161590C2semidirect
3. #27161268C2semidirect
4. #27161265C2semidirect
5. #27161053C2semidirect
6. #27161052C2semidirect
7. #525×C197C2semidirect
16. #20263309C22semidirect
19. #20263295C22semidirect
21. #20263285C22semidirect
22. #20263241C22semidirect
23. #20263240C22semidirect
25. C23×C4×C197C22semidirect
26. #123×C197C22semidirect
29. #120×C197C22semidirect
31. #117×C197C22semidirect
43. C2×Dic394D4semidirect
46. C2×D788D4semidirect
53. C2×C4×D197D4semidirect
63. C2×Q8×C197D4semidirect
65. C2×D4×C197D4semidirect
71. C22×C4×C197D4semidirect
147. C197#3025semidirect
151. #525D197semidirect
165. C23×C4C2×D197semidirect
167. #123C2×D197semidirect
173. #120C2×D197semidirect
177. #117C2×D197semidirect
196. C2×Q8C22⋊D197semidirect
205. C2×D4C22⋊D197semidirect
213. C22×C4C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271619152
#271615902
#271612682
#271612652
#271610532
#271610522
#525×C1972
#3025197