Finite Group Databasebeta

← #38752290

Subgroups — #38752290

Order 25216.

520 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#20263346C211yesmaximal, characteristic, complemented
2C23×#13096802C211yesmaximal, characteristic, complemented
3#27161590C211yesmaximal, characteristic, complemented
4#27161318C212yesmaximal, complemented
5#27161313C212yesmaximal, complemented
6#27161310C212yesmaximal, complemented
7#27161306C212yesmaximal, complemented
8#27161115C211yesmaximal, characteristic, complemented
9#27161075C211yesmaximal, characteristic, complemented
10#27161073C211yesmaximal, characteristic, complemented
11#525×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2211yescharacteristic
13#20263346—24no—
14C2×#15691348C2211yescharacteristic, complemented
15C2×#15691346C2211yescharacteristic, complemented
16C2×#15691345C2211yescharacteristic, complemented
17C2×#15691343C2211yescharacteristic, complemented
18C22×#13096802—24no—
19C22×#13096802C2212yes—
20C22×#13096802C2212yes—
21C22×#13096802C2211yescharacteristic
22C22×#13096802C2211yescharacteristic
23#20263309—21no—
24#20263305C2212yes—
25#20263300C2212yes—
26#20263295—21no—
27C4×C22⋊D197—22no—
28#20263285C2211yescharacteristic
29#20263259C2212yescomplemented
30#20263257C2212yescomplemented
31#20263250C2212yescomplemented
32#20263249C2212yescomplemented
33#20263247C2212yescomplemented
34#20263246C2212yescomplemented
35C2×#41×C197C2211yescharacteristic
36C23×C4×C197C2211yescharacteristic
37#123×C197—21no—
38C22⋊Q8×C197C2212yes—
39#121×C197C2212yes—
40#120×C197—21no—
41C4×D4×C197—22no—
42#117×C197C2211yescharacteristic
43#15691358—22no—
44#15691358—22no—
45C2×Dic394C2311yescharacteristic
46C2×D788C2311yescharacteristic
47C2×C22⋊D197—22no—
48C2×C22⋊D197—21no—
49C2×C4×D197—21no—
50#15691348—22no—
51#15691346—22no—
52#15691345—22no—
53#15691343—22no—
54C2×#13096802—24no—
55C2×#13096802—24no—
56C2×#13096802—22no—
57C2×#13096802D414yescomplemented
58C2×#13096802C2312yes—
59C2×#13096802C2311yescharacteristic
60C2×#13096802C2311yescharacteristic
61C4⋊Dic197C2312yes—
62#15691337—22no—
63C22⋊Dic197—21no—
64C22⋊Dic197—22no—
65#15691335—22no—
66C4×Dic197—21no—
67C24×C197D411yescharacteristic
68#41×C197—22no—
69#41×C197—22no—
70C2×Q8×C197C2311yescharacteristic
71C2×D4×C197—22no—
72C2×D4×C197—21no—
73C2×D4×C197C2311yescharacteristic
74C22×C4×C197—22no—
75C22×C4×C197—22no—
76C22×C4×C197—21no—
77C22×C4×C197D412yes—
78C22×C4×C197D411yescharacteristic
79C22×C4×C197C2312yes—
80C22×C4×C197C2311yescharacteristic
81C4⋊C4×C197—22no—
82C4⋊C4×C197C2312yes—
83C22⋊C4×C197—21no—
84C22⋊C4×C197—22no—
85C22⋊C4×C197—22no—
86C42×C197—21no—
87C22×D197—21no—
88Dic394—21no—
89Dic394—21no—
90D788—21no—
91D788—21no—
92C22⋊D197—42no—
93C22⋊D197—44no—
94C2×Dic197—22no—
95C2×Dic197—21no—
96C4×D197—42no—
97#13096802—22no—
98#13096802C2×D412yes—
99#13096802C2×D412yes—
100C23×C197—22no—
101C23×C197—24no—
102C23×C197—21no—
103C23×C197C2×D411yescharacteristic
104C23×C197C2×D412yes—
105Q8×C197—21no—
106Q8×C197—21no—
107D4×C197—21no—
108D4×C197—21no—
109D4×C197—42no—
110D4×C197—44no—
111C2×C4×C197—22no—
112C2×C4×C197—24no—
113C2×C4×C197—22no—
114C2×C4×C197—22no—
115C2×C4×C197—22no—
116C2×C4×C197—21no—
117C2×C4×C197—42no—
118C2×C4×C197C2×D412yes—
119C2×C4×C197C2×D412yes—
120C2×C4×C197C2×D411yescharacteristic
121C2×C4×C197#4112yes—
122C2×C4×C197C2411yescharacteristic
123C22×C197—28no—
124C22×C197—22no—
125C22×C197—22no—
126C22×C197—22no—
127C22×C197—41no—
128C22×C197—42no—
129C22×C197#12112yes—
130C22×C197#13514yescomplemented
131C22×C197C22×D411yescharacteristic
132C2×D197—41no—
133C2×D197—42no—
134C4×C197—22no—
135C4×C197—41no—
136C4×C197—42no—
137C4×C197#12112yes—
138C4×C197C22×D411yescharacteristic
139C4×C197C2×#4111yescharacteristic
140Dic197—41no—
141Dic197—42no—
142C2×C197—22no—
143C2×C197—24no—
144C2×C197—81no—
145C2×C197C2×#12111yescharacteristic
146C2×C197C2×#13512yes—
147D197—81no—
148C197#305011yescharacteristic, complemented, Sylow
149#3050—1971nomaximal, Sylow
150C2×#135—1971no—
151C23×C8—1971no—
152#525—1971no—
153#525D19711yescharacteristic, complemented
154#467—1972no—
155#466—1972no—
156#465—1972no—
157#465—1972no—
158#427—1971no—
159#416—1971no—
160#416—1971no—
161C2×#41—1971no—
162C2×#41C2×D19711yescharacteristic
163C23×C4C2×D19711yescharacteristic
164#135—3944no—
165C2×Q16—1971no—
166C2×SD16—1971no—
167C2×SD16—1971no—
168C2×D8—1971no—
169C22×C8—3944no—
170C22×C8—1971no—
171C22×C8—1972no—
172C22×C8—1972no—
173C22×C8—1971no—
174#123—21no—
175#123—3941no—
176C22⋊Q8—1972no—
177C22⋊Q8C2×D19712yes—
178#121—1972no—
179#121C2×D19712yes—
180#120—21no—
181#120—3941no—
182C4×D4—22no—
183C4×D4—3942no—
184#117—1971no—
185#117C2×D19711yescharacteristic
186#107—1972no—
187#106—1972no—
188#103—1972no—
189#103—1972no—
190#102—1972no—
191#102—1972no—
192C24C22⋊D19711yescharacteristic
193#41—22no—
194#41—22no—
195#41—3942no—
196#41—3942no—
197C2×Q8—1971no—
198C2×Q8C22×D19711yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—3942no—
202C2×D4—3941no—
203C2×D4—1971no—
204C2×D4C22×D19711yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—3941no—
209C22×C4C22⋊D19711yescharacteristic
210C22×C4C22⋊D19712yes—
211C22×C4C22×D19712yes—
212C22×C4C22×D19711yescharacteristic
213Q16—3942no—
214SD16—3942no—
215SD16—3942no—
216D8—3942no—
217C2×C8—3942no—
218C2×C8—3944no—
219C2×C8—3944no—
220C2×C8—1972no—
221C2×C8—1974no—
222C2×C8—1971no—
223C2×C8—1971no—
224C4⋊C4—22no—
225C4⋊C4—3942no—
226C4⋊C4—1972no—
227C4⋊C4C22×D19712yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—3941no—
232C22⋊C4—3942no—
233C22⋊C4—3942no—
234C42—21no—
235C42—3941no—
236C23—22no—
237C23—24no—
238C23—21no—
239C23—3941no—
240C23C2×C22⋊D19711yescharacteristic
241C23C2×C22⋊D19712yes—
242Q8—21no—
243Q8—21no—
244Q8—3941no—
245Q8—3941no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—44no—
250D4—7882no—
251D4—7884no—
252D4—3941no—
253D4—3941no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—7882no—
262C2×C4—3942no—
263C2×C4—3941no—
264C2×C4D4×D19712yes—
265C2×C4C2×C22⋊D19712yes—
266C2×C4C2×C22⋊D19711yescharacteristic
267C2×C4#1569135712yes—
268C2×C4C23×D19711yescharacteristic
269C8—3942no—
270C8—1972no—
271C8—1972no—
272C22—28no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—41no—
278C22—7882no—
279C22—7881no—
280C22#2026329812yes—
281C22#2026334614yescomplemented
282C22C22×C22⋊D19711yescharacteristic
283C4—22no—
284C4—42no—
285C4—41no—
286C4—7882no—
287C4—7881no—
288C4#2026329812yes—
289C4C2×D4×D19711yescharacteristic
290C4C2×#1569135711yescharacteristic
291C2—22no—
292C2—24no—
293C2—81no—
294C2—15761no—
295C2C2×#2026329811yescharacteristic
296C2C2×#2026334612yes—
subgroupG/H
centerC2×C4C2×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#525×C197C2
socleC22×C197C22×D4
hypercenter#525D197
hypocenterC197#3050
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#20263346C2semidirect
2. C23×#13096802C2semidirect
3. #27161590C2semidirect
4. #27161318C2semidirect
5. #27161313C2semidirect
6. #27161310C2semidirect
7. #27161306C2semidirect
8. #27161115C2semidirect
9. #27161075C2semidirect
10. #27161073C2semidirect
11. #525×C197C2semidirect
14. C2×#15691348C22semidirect
15. C2×#15691346C22semidirect
16. C2×#15691345C22semidirect
17. C2×#15691343C22semidirect
29. #20263259C22semidirect
30. #20263257C22semidirect
31. #20263250C22semidirect
32. #20263249C22semidirect
33. #20263247C22semidirect
34. #20263246C22semidirect
57. C2×#13096802D4semidirect
130. C22×C197#135semidirect
148. C197#3050semidirect
153. #525D197semidirect
281. C22#20263346semidirect

Maximal subgroups

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

subgroupindex
C2×#202633462
C23×#130968022
#271615902
#271613182
#271613132
#271613102
#271613062
#271611152
#271610752
#271610732
#525×C1972
#3050197