Finite Group Databasebeta

← #38751996

Subgroups — #38751996

Order 25216.

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

normal subgroups only

This group has 732 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×#20263373C211yesmaximal, characteristic, complemented
2C2×#20263352C211yesmaximal, characteristic, complemented
3C22×#15691346C211yesmaximal, characteristic, complemented
4#27161222C214yesmaximal, complemented
5#27161217C214yesmaximal, complemented
6#27161087C211yesmaximal, characteristic, complemented
7#27161073C211yesmaximal, characteristic, complemented
8#27161046C211yesmaximal, characteristic, complemented
9#535×C197C211yesmaximal, characteristic, complemented
10#20263373—24no—
11#20263373—24no—
12C2×#15691358C2211yescharacteristic
13C2×#15691358C2212yescomplemented
14C2×#15691357—22no—
15C2×#15691357—21no—
16C22×Dic394C2211yescharacteristic
17C22×Dic394C2212yescomplemented
18C2×Q8×D197—21no—
19#20263352—24no—
20C2×#15691346—24no—
21C2×#15691346C2212yes—
22C2×#15691346C2214yes—
23C2×#15691343C2212yes—
24C2×#15691342C2211yescharacteristic
25C22×#13096802C2211yescharacteristic
26#20263249C2212yescomplemented
27#20263249C2214yescomplemented
28#20263246C2212yescomplemented
29C22⋊#13096802C2214yescomplemented
30C2×#41×C197C2211yescharacteristic
31C22×D4×C197C2211yescharacteristic
32#127×C197—21no—
33#124×C197—21no—
34#121×C197C2214yes—
35#120×C197—22no—
36#117×C197C2211yescharacteristic
37#15691358—24no—
38#15691358—24no—
39#15691358—22no—
40#15691358D414yescomplemented
41#15691357—216no—
42#15691357—28no—
43C2×Dic394—24no—
44C2×Dic394—24no—
45C2×Dic394—21no—
46C2×Dic394—22no—
47C2×Dic394—22no—
48C2×Dic394D414yescomplemented
49C2×Dic394C2312yes—
50C2×Dic394C2311yescharacteristic
51Q8×D197—28no—
52C2×D788C2311yescharacteristic
53C2×C22⋊D197—21no—
54C2×C22⋊D197—22no—
55C22×Dic197—21no—
56C22×Dic197—22no—
57C2×C4×D197—21no—
58C2×C4×D197—22no—
59#15691346—28no—
60#15691346—24no—
61#15691343—24no—
62#15691342—22no—
63C2×#13096802—22no—
64C2×#13096802C2312yes—
65C2×#13096802C2312yes—
66C24×C197—21no—
67#41×C197—22no—
68#41×C197—22no—
69C2×Q8×C197D411yescharacteristic
70C2×D4×C197—24no—
71C2×D4×C197—22no—
72C2×D4×C197—22no—
73C2×D4×C197—21no—
74C2×D4×C197D412yes—
75C2×D4×C197D411yescharacteristic
76C2×D4×C197C2312yes—
77C22×C4×C197C2312yes—
78C22×C4×C197C2311yescharacteristic
79C4⋊C4×C197C2312yes—
80C22⋊C4×C197—21no—
81C22⋊C4×C197—24no—
82C42×C197—21no—
83C22×D197—21no—
84Dic394—28no—
85Dic394—24no—
86Dic394—21no—
87Dic394—22no—
88Dic394C2×D412yes—
89Dic394C2×D414yes—
90D788—21no—
91D788C2×D412yes—
92C22⋊D197—28no—
93C22⋊D197—24no—
94C2×Dic197—28no—
95C2×Dic197—24no—
96C2×Dic197—21no—
97C2×Dic197—22no—
98C4×D197—28no—
99C4×D197—24no—
100#13096802—22no—
101#13096802—22no—
102C23×C197—22no—
103C23×C197—42no—
104C23×C197—82no—
105C23×C197C2×D412yes—
106C23×C197C2×D411yescharacteristic
107Q8×C197—22no—
108D4×C197—24no—
109D4×C197—22no—
110D4×C197—22no—
111D4×C197—22no—
112D4×C197—42no—
113D4×C197—84no—
114C2×C4×C197—22no—
115C2×C4×C197—22no—
116C2×C4×C197—22no—
117C2×C4×C197C2×D412yes—
118C2×C4×C197C2×D412yes—
119C2×C4×C197C2×D412yes—
120C2×C4×C197C2×D411yescharacteristic
121C2×C4×C197C2411yescharacteristic
122C22×C197—22no—
123C22×C197—22no—
124C22×C197—24no—
125C22×C197—42no—
126C22×C197—44no—
127C22×C197—84no—
128C22×C197#12012yes—
129C22×C197C22×D411yescharacteristic
130C2×D197—22no—
131C2×D197—42no—
132C4×C197—22no—
133C4×C197—42no—
134C4×C197#12012yes—
135C4×C197C22×D411yescharacteristic
136C4×C197C22×D411yescharacteristic
137Dic197—24no—
138Dic197—22no—
139C2×C197—22no—
140C2×C197—42no—
141C2×C197—82no—
142C2×C197C2×#12011yescharacteristic
143C2×C197#57712yes—
144D197—42no—
145C197#301011yescharacteristic, complemented, Sylow
146#3010—1971nomaximal, Sylow
147C2×2−1+4—1971no—
148C2×AGL(1,ℤ/8)—1971no—
149C22×SD16—1971no—
150#535D19711yescharacteristic, complemented
151#449—1974no—
152#448—1974no—
153#419—1971no—
154#416—1971no—
155#408—1971no—
1562−1+4—3944no—
1572−1+4—3944no—
158C2×#41—3942no—
159C2×#41—3941no—
160C2×#41—1972no—
161C2×#41—1971no—
162C2×#41C2×D19711yescharacteristic
163C22×Q8—3941no—
164C22×Q8—1972no—
165C22×Q8—1971no—
166C22×D4C2×D19711yescharacteristic
167AGL(1,ℤ/8)—3944no—
168C2×SD16—3944no—
169C2×SD16—1974no—
170C2×SD16—1972no—
171C2×D8—1972no—
172C2×M16—1971no—
173C22×C8—1971no—
174#127—21no—
175#124—21no—
176#121C2×D19714yes—
177#120—22no—
178#117C2×D19711yescharacteristic
179#103—1972no—
180#103—1974no—
181#102—1972no—
182C22⋊C8—1974no—
183C24—21no—
184#41—22no—
185#41—22no—
186#41—3944no—
187#41—3944no—
188#41—39416no—
189#41—3948no—
190#41—3942no—
191#41—1974no—
192C2×Q8—3944no—
193C2×Q8—3944no—
194C2×Q8—3948no—
195C2×Q8—3942no—
196C2×Q8—3942no—
197C2×Q8—3941no—
198C2×Q8—1974no—
199C2×Q8—1971no—
200C2×Q8—1972no—
201C2×Q8C22⋊D19711yescharacteristic
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—3942no—
207C2×D4—3941no—
208C2×D4—1971no—
209C2×D4C22⋊D19712yes—
210C2×D4C22⋊D19711yescharacteristic
211C2×D4C22×D19712yes—
212C22×C4—3942no—
213C22×C4—3941no—
214C22×C4—3942no—
215C22×C4—3941no—
216C22×C4C22×D19712yes—
217C22×C4C22×D19711yescharacteristic
218SD16—3948no—
219SD16—3944no—
220D8—3944no—
221M16—3942no—
222C2×C8—3942no—
223C2×C8—1972no—
224C2×C8—1972no—
225C4⋊C4C22×D19712yes—
226C22⋊C4—21no—
227C22⋊C4—24no—
228C42—21no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—3941no—
233C23C2×C22⋊D19712yes—
234C23C2×C22⋊D19711yescharacteristic
235Q8—22no—
236Q8—3948no—
237Q8—3944no—
238Q8—3941no—
239Q8—3942no—
240Q8—1972no—
241Q8—1974no—
242D4—24no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—84no—
248D4—3948no—
249D4—3944no—
250D4—3941no—
251D4—1972no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—3948no—
256C2×C4—3948no—
257C2×C4—3944no—
258C2×C4—3942no—
259C2×C4—3941no—
260C2×C4—3944no—
261C2×C4D4×D19712yes—
262C2×C4D4×D19712yes—
263C2×C4C2×C22⋊D19712yes—
264C2×C4C2×C22⋊D19711yescharacteristic
265C2×C4C23×D19711yescharacteristic
266C8—3942no—
267C8—3942no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—44no—
272C22—42no—
273C22—84no—
274C22—7882no—
275C22—3942no—
276C22#2026329412yes—
277C22C22×C22⋊D19711yescharacteristic
278C4—22no—
279C4—42no—
280C4—3944no—
281C4—3942no—
282C4#2026329412yes—
283C4C2×D4×D19711yescharacteristic
284C4C2×D4×D19711yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—7882no—
289C2C2×#2026329411yescharacteristic
290C2#2716189012yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#535×C197C2
socleC22×C197C22×D4
hypercenter#535D197
hypocenterC197#3010
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#20263373C2semidirect
2. C2×#20263352C2semidirect
3. C22×#15691346C2semidirect
4. #27161222C2semidirect
5. #27161217C2semidirect
6. #27161087C2semidirect
7. #27161073C2semidirect
8. #27161046C2semidirect
9. #535×C197C2semidirect
13. C2×#15691358C22semidirect
17. C22×Dic394C22semidirect
26. #20263249C22semidirect
27. #20263249C22semidirect
28. #20263246C22semidirect
29. C22⋊#13096802C22semidirect
40. #15691358D4semidirect
48. C2×Dic394D4semidirect
145. C197#3010semidirect
150. #535D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633732
C2×#202633522
C22×#156913462
#271612222
#271612172
#271610872
#271610732
#271610462
#535×C1972
#3010197