Finite Group Databasebeta

← #38751999

Subgroups — #38751999

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×#20263374C211yesmaximal, characteristic, complemented
2C2×#20263354C211yesmaximal, characteristic, complemented
3C22×#15691345C211yesmaximal, characteristic, complemented
4#27161648C211yesmaximal, characteristic, complemented
5#27161227C214yesmaximal, complemented
6#27161219C214yesmaximal, complemented
7#27161073C211yesmaximal, characteristic, complemented
8#27161047C211yesmaximal, characteristic, complemented
9#419×C197C211yesmaximal, characteristic, complemented
10#20263374—24no—
11#20263374—24no—
12C2×#15691359—21no—
13C2×#15691359C2212yescomplemented
14C2×#15691358—22no—
15C2×#15691358C2211yescharacteristic
16C2×#15691358C2211yescharacteristic
17C2×Q8×D197—21no—
18C2×Q8×D197C2212yescomplemented
19C22×D788C2211yescharacteristic
20#20263354—24no—
21C2×#15691347C2212yes—
22C2×#15691345—24no—
23C2×#15691345C2214yes—
24C2×D1576C2212yes—
25C22×#13096802C2211yescharacteristic
26#20263323—21no—
27#20263314—21no—
28#20263301C2214yes—
29#20263296—22no—
30#20263249C2212yescomplemented
31#20263246C2212yescomplemented
32#20263238C2214yescomplemented
33C22×Q8×C197C2211yescharacteristic
34C2×M16×C197C2211yescharacteristic
35#117×C197C2211yescharacteristic
36#103×C197C2214yescomplemented
37C23×D197—21no—
38#15691359—28no—
39#15691359—24no—
40#15691359—24no—
41#15691358—216no—
42#15691358—22no—
43#15691358—22no—
44#15691358—22no—
45#15691358D414yescomplemented
46C2×Dic394—22no—
47C2×Dic394D411yescharacteristic
48C2×Dic394C2311yescharacteristic
49Q8×D197—28no—
50Q8×D197—24no—
51Q8×D197—24no—
52C2×D788—22no—
53C2×D788—24no—
54C2×D788—22no—
55C2×D788—22no—
56C2×D788—21no—
57C2×D788D411yescharacteristic
58C2×D788C2312yes—
59C2×D788C2311yescharacteristic
60C2×C22⋊D197—21no—
61C2×C22⋊D197D412yes—
62C2×C4×D197—22no—
63C2×C4×D197—22no—
64C2×C4×D197—21no—
65C2×C4×D197C2312yes—
66#15691347—24no—
67#15691345—28no—
68D1576—24no—
69C2×#13096802—22no—
70C2×#13096802C2312yes—
71#15691335—24no—
72C2×Q8×C197—21no—
73C2×Q8×C197—22no—
74C2×Q8×C197D414yescomplemented
75C2×Q8×C197C2312yes—
76C22×C4×C197—21no—
77C22×C4×C197C2311yescharacteristic
78M16×C197—22no—
79C2×C8×C197C2312yes—
80C4⋊C4×C197C2312yes—
81C22⋊C4×C197—21no—
82C42×C197—21no—
83C22×D197—21no—
84C22×D197—22no—
85C22×D197—42no—
86C22×D197—82no—
87C22×D197C2×D412yes—
88Dic394—28no—
89Dic394—22no—
90Dic394—21no—
91Dic394C2×D412yes—
92D788—28no—
93D788—22no—
94D788—21no—
95D788—22no—
96D788—22no—
97D788—42no—
98D788—84no—
99D788C2×D412yes—
100C22⋊D197—24no—
101C22⋊D197—24no—
102C2×Dic197—21no—
103C2×Dic197C2×D412yes—
104C4×D197—28no—
105C4×D197—28no—
106C4×D197—24no—
107C4×D197—22no—
108C4×D197—22no—
109#13096802—22no—
110C23×C197C2×D411yescharacteristic
111Q8×C197—24no—
112Q8×C197—22no—
113Q8×C197C2×D414yes—
114C2×C4×C197—24no—
115C2×C4×C197—22no—
116C2×C4×C197—22no—
117C2×C4×C197C2×D411yescharacteristic
118C2×C4×C197C2×D412yes—
119C2×C4×C197C2×D412yes—
120C2×C4×C197C2411yescharacteristic
121C8×C197—22no—
122C22×C197—22no—
123C22×C197#12012yes—
124C22×C197C22×D411yescharacteristic
125C2×D197—22no—
126C2×D197—22no—
127C2×D197—24no—
128C2×D197—42no—
129C2×D197—42no—
130C2×D197—44no—
131C2×D197—84no—
132C4×C197—24no—
133C4×C197—42no—
134C4×C197#12012yes—
135C4×C197C22×D411yescharacteristic
136C4×C197C22×D411yescharacteristic
137Dic197—22no—
138Dic197—22no—
139C2×C197—22no—
140C2×C197C2×#12011yescharacteristic
141C2×C197#57712yes—
142D197—42no—
143D197—42no—
144D197—82no—
145C197#301011yescharacteristic, complemented, Sylow
146#3010—1971nomaximal, Sylow
147C2×2−1+4—1971no—
148C2×AGL(1,ℤ/8)—1971no—
149C22×SD16—1971no—
150#535—1971no—
151#449—1974no—
152#448—1974no—
153#419D19711yescharacteristic, complemented
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×#41—1971no—
163C22×Q8—3941no—
164C22×Q8—1972no—
165C22×Q8C2×D19711yescharacteristic
166C22×D4—1971no—
167AGL(1,ℤ/8)—3944no—
168C2×SD16—3944no—
169C2×SD16—1974no—
170C2×SD16—1972no—
171C2×D8—1972no—
172C2×M16C2×D19711yescharacteristic
173C22×C8—1971no—
174#127—3941no—
175#124—3941no—
176#121—1974no—
177#120—3942no—
178#117C2×D19711yescharacteristic
179#103—1972no—
180#103C2×D19714yescomplemented
181#102—1972no—
182C22⋊C8—1974no—
183C24—3941no—
184#41—39416no—
185#41—3948no—
186#41—3944no—
187#41—3944no—
188#41—3942no—
189#41—3942no—
190#41—3942no—
191#41—1974no—
192C2×Q8—21no—
193C2×Q8—22no—
194C2×Q8—3942no—
195C2×Q8—3948no—
196C2×Q8—3944no—
197C2×Q8—3944no—
198C2×Q8—1971no—
199C2×Q8—1971no—
200C2×Q8D78814yescomplemented
201C2×Q8C22×D19712yes—
202C2×D4—3944no—
203C2×D4—3942no—
204C2×D4—3941no—
205C2×D4—3941no—
206C2×D4—3942no—
207C2×D4—3942no—
208C2×D4—1972no—
209C2×D4—1971no—
210C2×D4—1972no—
211C2×D4—1971no—
212C22×C4—21no—
213C22×C4—3942no—
214C22×C4—3942no—
215C22×C4—3941no—
216C22×C4—1972no—
217C22×C4C22×D19711yescharacteristic
218SD16—3944no—
219SD16—3948no—
220D8—3944no—
221M16—22no—
222C2×C8—3942no—
223C2×C8—1972no—
224C2×C8C22×D19712yes—
225C4⋊C4C22×D19712yes—
226C22⋊C4—21no—
227C22⋊C4—3944no—
228C42—21no—
229C23—7882no—
230C23—15762no—
231C23—3942no—
232C23—3941no—
233C23—1972no—
234C23D4×D19711yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—3948no—
238Q8—3942no—
239Q8—3941no—
240Q8—1972no—
241Q8C2×D78814yes—
242D4—7882no—
243D4—15764no—
244D4—3948no—
245D4—3944no—
246D4—3944no—
247D4—3942no—
248D4—3942no—
249D4—3942no—
250D4—3941no—
251D4—1972no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—3948no—
256C2×C4—3948no—
257C2×C4—3941no—
258C2×C4—3944no—
259C2×C4—3942no—
260C2×C4—3942no—
261C2×C4—1972no—
262C2×C4D4×D19711yescharacteristic
263C2×C4D4×D19712yes—
264C2×C4C2×D78812yes—
265C2×C4C23×D19711yescharacteristic
266C8—22no—
267C8—3942no—
268C22—22no—
269C22—7882no—
270C22—7884no—
271C22—7882no—
272C22—15764no—
273C22—3944no—
274C22—3942no—
275C22—3942no—
276C22#2026329612yes—
277C22C2×D4×D19711yescharacteristic
278C4—24no—
279C4—42no—
280C4—3942no—
281C4—3942no—
282C4#2026329612yes—
283C4C2×D4×D19711yescharacteristic
284C4C22×D78811yescharacteristic
285C2—22no—
286C2—7882no—
287C2—7882no—
288C2—15762no—
289C2C2×#2026329611yescharacteristic
290C2#2716188812yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#419×C197C2
socleC22×C197C22×D4
hypercenter#419D197
hypocenterC197#3010
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#20263374C2semidirect
2. C2×#20263354C2semidirect
3. C22×#15691345C2semidirect
4. #27161648C2semidirect
5. #27161227C2semidirect
6. #27161219C2semidirect
7. #27161073C2semidirect
8. #27161047C2semidirect
9. #419×C197C2semidirect
13. C2×#15691359C22semidirect
18. C2×Q8×D197C22semidirect
30. #20263249C22semidirect
31. #20263246C22semidirect
32. #20263238C22semidirect
36. #103×C197C22semidirect
45. #15691358D4semidirect
74. C2×Q8×C197D4semidirect
145. C197#3010semidirect
153. #419D197semidirect
180. #103C2×D197semidirect
200. C2×Q8D788semidirect

Maximal subgroups

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

subgroupindex
C2×#202633742
C2×#202633542
C22×#156913452
#271616482
#271612272
#271612192
#271610732
#271610472
#419×C1972
#3010197