Finite Group Databasebeta

← #38752061

Subgroups — #38752061

Order 25216.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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#27161877C212yesmaximal, complemented
2#27161625C211yesmaximal, characteristic, complemented
3#27161262C212yesmaximal, complemented
4#27161257C212yesmaximal, complemented
5#27161252C212yesmaximal, complemented
6#27161247C212yesmaximal, complemented
7C2×#20263253C212yesmaximal, complemented
8#27161063C211yesmaximal, characteristic, complemented
9#585×C197C211yesmaximal, characteristic, complemented
10C2×#15691358C2211yescharacteristic, complemented
11#20263357—22no—
12#20263356C2214yescomplemented
13#20263352—22no—
14#20263351C2214yescomplemented
15#20263346—24no—
16C2×#15691342C2212yescomplemented
17#20263324C2211yescharacteristic, complemented
18#20263322C2211yescharacteristic, complemented
19#20263315C2211yescharacteristic, complemented
20#20263313C2211yescharacteristic, complemented
21#20263298—22no—
22C4×C22⋊D197—22no—
23C2×C4×Dic197C2211yescharacteristic, complemented
24#20263253C2214yescomplemented
25#20263253C2214yescomplemented
26#20263245C2212yescomplemented
27#20263243C2212yescomplemented
282−1+4×C197—22no—
292−1+4×C197C2212yes—
30H(5,𝔽2)×C197—22no—
31H(5,𝔽2)×C197—22no—
32H(5,𝔽2)×C197C2212yes—
33C2×#41×C197—24no—
34C2×#41×C197—22no—
35C2×#41×C197C2212yes—
36C2×#41×C197C2211yescharacteristic
37#15691358—22no—
38#15691358—22no—
39C2×Dic394C2311yescharacteristic
40C2×D788C2311yescharacteristic
41C2×C22⋊D197—22no—
42C2×C22⋊D197—21no—
43C22×Dic197—21no—
44C2×C4×D197—21no—
45#15691348—24no—
46#15691346—24no—
47#15691345—24no—
48#15691343—24no—
49#15691342—22no—
50#15691342C2314yes—
51C2×#13096802—22no—
52#15691337—22no—
53C22⋊Dic197—22no—
54#15691335—22no—
55C4×Dic197—21no—
56C4×Dic197C2311yescharacteristic
57C4×Dic197C2311yescharacteristic
58#41×C197—24no—
59#41×C197—28no—
60#41×C197—24no—
61#41×C197—24no—
62#41×C197—24no—
63#41×C197—22no—
64#41×C197—22no—
65#41×C197—24no—
66#41×C197—22no—
67#41×C197—22no—
68#41×C197D414yescomplemented
69#41×C197C2314yes—
70C2×Q8×C197—22no—
71C2×Q8×C197—24no—
72C2×Q8×C197D412yescomplemented
73C2×Q8×C197C2311yescharacteristic
74C2×D4×C197—28no—
75C2×D4×C197—22no—
76C2×D4×C197—24no—
77C2×D4×C197—22no—
78C2×D4×C197—24no—
79C2×D4×C197D412yes—
80C2×D4×C197D412yescomplemented
81C2×D4×C197C2311yescharacteristic
82C22×C4×C197—24no—
83C22×C4×C197—22no—
84C22×C4×C197D412yes—
85C22×C4×C197C2311yescharacteristic
86C22×D197—21no—
87Dic394—22no—
88D788—22no—
89C22⋊D197—44no—
90C22⋊D197—42no—
91C2×Dic197—21no—
92C2×Dic197—21no—
93C2×Dic197—21no—
94C2×Dic197—42no—
95C4×D197—41no—
96C4×D197—41no—
97#13096802—24no—
98C23×C197—24no—
99C23×C197—22no—
100C23×C197C2×D412yes—
101C23×C197C2×D411yescharacteristic
102Q8×C197—24no—
103Q8×C197—22no—
104Q8×C197—22no—
105Q8×C197C2×D414yes—
106D4×C197—28no—
107D4×C197—24no—
108D4×C197—24no—
109D4×C197—24no—
110D4×C197—24no—
111D4×C197—22no—
112D4×C197—22no—
113D4×C197C2×D414yes—
114C2×C4×C197—24no—
115C2×C4×C197—28no—
116C2×C4×C197—22no—
117C2×C4×C197—24no—
118C2×C4×C197—22no—
119C2×C4×C197—22no—
120C2×C4×C197—24no—
121C2×C4×C197C2×D412yes—
122C2×C4×C197C2×D411yescharacteristic
123C2×C4×C197C2×D412yes—
124C2×C4×C197C2×D412yes—
125C2×C4×C197C2411yescharacteristic
126C22×C197—22no—
127C22×C197—24no—
128C22×C197—41no—
129C22×C197—42no—
130C22×C197—82no—
131C22×C197—84no—
132C22×C197#12012yes—
133C22×C197C22×D411yescharacteristic
134C2×D197—42no—
135C2×D197—41no—
136C4×C197—24no—
137C4×C197—22no—
138C4×C197#12011yescharacteristic
139C4×C197#12011yescharacteristic
140C4×C197C22×D412yes—
141Dic197—22no—
142Dic197—41no—
143Dic197—41no—
144C2×C197—21no—
145C2×C197—22no—
146C2×C197—42no—
147C2×C197—44no—
148C2×C197C2×#12011yescharacteristic
149D197—81no—
150C197#301811yescharacteristic, complemented, Sylow
151#3018—1971nomaximal, Sylow
152#585D19711yescharacteristic, complemented
153#575—1972no—
154#532—1971no—
155#456—1972no—
156#455—1972no—
157#454—1972no—
158#453—1972no—
159C2×#104—1972no—
160#413—1971no—
1612−1+4—22no—
1622−1+4C2×D19712yes—
163H(5,𝔽2)—22no—
164H(5,𝔽2)—22no—
165H(5,𝔽2)C2×D19712yes—
166C2×#41—24no—
167C2×#41—22no—
168C2×#41—1971no—
169C2×#41C2×D19712yes—
170C2×#41C2×D19711yescharacteristic
171#137—3942no—
172#137—1974no—
173AGL(1,ℤ/8)—3942no—
174AGL(1,ℤ/8)—1974no—
175#135—3944no—
176C2×M16—1972no—
177C4⋊Q8—1971no—
178#127—1971no—
179#124—1971no—
180#124—1971no—
181#121—3942no—
182C4×D4—3942no—
183C2×C42—1971no—
184#104—1974no—
185#104—1974no—
186#101—1972no—
187#100—1972no—
188#41—24no—
189#41—28no—
190#41—24no—
191#41—24no—
192#41—24no—
193#41—22no—
194#41—22no—
195#41—24no—
196#41—22no—
197#41—22no—
198#41—3942no—
199#41—3942no—
200#41C22⋊D19714yescomplemented
201#41C22×D19714yes—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—1971no—
205C2×Q8C22⋊D19712yescomplemented
206C2×Q8C22×D19711yescharacteristic
207C2×D4—28no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—3942no—
213C2×D4—3941no—
214C2×D4—1971no—
215C2×D4C22⋊D19712yes—
216C2×D4C22⋊D19712yescomplemented
217C2×D4C22×D19711yescharacteristic
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—3941no—
221C22×C4—3941no—
222C22×C4C22⋊D19712yes—
223C22×C4C22×D19711yescharacteristic
224Q16—3944no—
225SD16—3944no—
226SD16—3944no—
227D8—3944no—
228M16—3942no—
229M16—1974no—
230C2×C8—3942no—
231C4⋊C4—3942no—
232C22⋊C4—3942no—
233C22⋊C4—3942no—
234C42—3941no—
235C42—1971no—
236C42—1971no—
237C23—24no—
238C23—22no—
239C23—3941no—
240C23C2×C22⋊D19712yes—
241C23C2×C22⋊D19711yescharacteristic
242Q8—24no—
243Q8—22no—
244Q8—22no—
245Q8—3942no—
246Q8C2×C22⋊D19714yes—
247D4—28no—
248D4—24no—
249D4—24no—
250D4—24no—
251D4—24no—
252D4—22no—
253D4—22no—
254D4—7882no—
255D4—7884no—
256D4—3942no—
257D4C2×C22⋊D19714yes—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—7882no—
266C2×C4—7881no—
267C2×C4—7881no—
268C2×C4—3941no—
269C2×C4—3941no—
270C2×C4—3941no—
271C2×C4C2×C22⋊D19712yes—
272C2×C4C2×C22⋊D19711yescharacteristic
273C2×C4C2×C22⋊D19712yes—
274C2×C4C2×C22⋊D19712yes—
275C2×C4C23×D19711yescharacteristic
276C8—3944no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—42no—
281C22—82no—
282C22—84no—
283C22—7882no—
284C22—7881no—
285C22#2026329512yes—
286C22C22×C22⋊D19711yescharacteristic
287C4—24no—
288C4—22no—
289C4—7881no—
290C4—7881no—
291C4—3942no—
292C4#2026329511yescharacteristic
293C4#2026329511yescharacteristic
294C4C22×C22⋊D19712yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—44no—
299C2—15761no—
300C2C2×#2026329511yescharacteristic
subgroupG/H
centerC4#20263295
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#585×C197C2
socleC2×C197C2×#120
hypercenter#585D197
hypocenterC197#3018
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #27161877C2semidirect
2. #27161625C2semidirect
3. #27161262C2semidirect
4. #27161257C2semidirect
5. #27161252C2semidirect
6. #27161247C2semidirect
7. C2×#20263253C2semidirect
8. #27161063C2semidirect
9. #585×C197C2semidirect
10. C2×#15691358C22semidirect
12. #20263356C22semidirect
14. #20263351C22semidirect
16. C2×#15691342C22semidirect
17. #20263324C22semidirect
18. #20263322C22semidirect
19. #20263315C22semidirect
20. #20263313C22semidirect
23. C2×C4×Dic197C22semidirect
24. #20263253C22semidirect
25. #20263253C22semidirect
26. #20263245C22semidirect
27. #20263243C22semidirect
68. #41×C197D4semidirect
72. C2×Q8×C197D4semidirect
80. C2×D4×C197D4semidirect
150. C197#3018semidirect
152. #585D197semidirect
200. #41C22⋊D197semidirect
205. C2×Q8C22⋊D197semidirect
216. C2×D4C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271618772
#271616252
#271612622
#271612572
#271612522
#271612472
C2×#202632532
#271610632
#585×C1972
#3018197