Finite Group Databasebeta

← #38745849

Subgroups — #38745849

Order 25216.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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×#20263294C211yesmaximal, characteristic
2#27160979C212yesmaximal, complemented
3#27160970C212yesmaximal, complemented
4#27160969C212yesmaximal, complemented
5#27160967C212yesmaximal, complemented
6#27160959C211yesmaximal, characteristic, complemented
7#27160956C212yesmaximal, complemented
8#27160934C212yesmaximal, complemented
9#390×C197C211yesmaximal, characteristic, complemented
10C24×D197C2211yescharacteristic
11C22×C22⋊D197C2212yes—
12C23×Dic197C2211yescharacteristic, complemented
13#20263294C418yescomplemented
14C2×#15691337C2212yescomplemented
15C2×C22⋊Dic197C2212yes—
16C2×C22⋊Dic197C2211yescharacteristic
17C2×C22⋊Dic197C2211yescharacteristic, complemented
18C2×#15691335C2212yes—
19C2×#15691335C2212yes—
20C2×#15691335C2212yes—
21C2×#15691335C2212yes—
22C2×#15691335C2212yes—
23C2×C4×Dic197C2212yescomplemented
24#20263233C2212yescomplemented
25#20263233C2212yescomplemented
26#20263233C2212yescomplemented
27C22×D4×C197C2211yescharacteristic
28C2×C4⋊C4×C197C2211yescharacteristic, complemented
29C2×C22⋊C4×C197C2212yes—
30C2×C22⋊C4×C197C2212yescomplemented
31C2×C42×C197C2211yescharacteristic, complemented
32C23×D197—42no—
33C23×D197—41no—
34C23×D197—41no—
35C23×D197—42no—
36C23×D197C2×C414yescomplemented
37C23×D197C2312yes—
38C2×C22⋊D197—42no—
39C2×C22⋊D197—42no—
40C2×C22⋊D197C2×C418yes—
41C22×Dic197—41no—
42C22×Dic197—42no—
43C22×Dic197C2312yes—
44C22×Dic197C2312yes—
45C22×Dic197C2312yes—
46#15691337—42no—
47C22⋊Dic197—42no—
48C22⋊Dic197—41no—
49C22⋊Dic197C2×C414yescomplemented
50#15691335—42no—
51#15691335—42no—
52#15691335—42no—
53#15691335—42no—
54#15691335C2×C418yescomplemented
55C4×Dic197—42no—
56C24×C197C2311yescharacteristic
57C24×C197C2311yescharacteristic
58C2×D4×C197—41no—
59C2×D4×C197—41no—
60C2×D4×C197C2×C414yes—
61C22×C4×C197C2312yes—
62C22×C4×C197C2312yes—
63C22×C4×C197C2311yescharacteristic
64C4⋊C4×C197—41no—
65C22⋊C4×C197—42no—
66C22⋊C4×C197—42no—
67C42×C197—41no—
68C22×D197—22no—
69C22×D197—22no—
70C22×D197—22no—
71C22×D197—22no—
72C22×D197—22no—
73C22×D197—22no—
74C22×D197—48no—
75C22×D197—44no—
76C22×D197—44no—
77C22×D197—48no—
78C22×D197—44no—
79C22×D197C22×C414yes—
80C22⋊D197—48no—
81C2×Dic197—22no—
82C2×Dic197—22no—
83C2×Dic197—22no—
84C2×Dic197—22no—
85C2×Dic197—22no—
86C2×Dic197—22no—
87C2×Dic197—22no—
88C2×Dic197—22no—
89C2×Dic197—44no—
90C2×Dic197C22×C414yes—
91C23×C197—22no—
92C23×C197—22no—
93C23×C197—21no—
94C23×C197—21no—
95C23×C197—21no—
96C23×C197—21no—
97C23×C197—22no—
98C23×C197—22no—
99C23×C197C22×C412yes—
100C23×C197C22×C412yes—
101C23×C197C2411yescharacteristic
102D4×C197—44no—
103C2×C4×C197—21no—
104C2×C4×C197—21no—
105C2×C4×C197—22no—
106C2×C4×C197—22no—
107C2×C4×C197—22no—
108C2×C4×C197—22no—
109C2×C4×C197—22no—
110C2×C4×C197—22no—
111C2×C4×C197C22×C412yes—
112C22×C197—22no—
113C22×C197—22no—
114C22×C197—24no—
115C22×C197—24no—
116C22×C197—41no—
117C22×C197—42no—
118C22×C197—41no—
119C22×C197—42no—
120C22×C197—41no—
121C22×C197—41no—
122C22×C197C23×C411yescharacteristic
123C22×C197H(5,𝔽2)12yes—
124C22×C197H(5,𝔽2)11yescharacteristic
125C22×C197H(5,𝔽2)11yescharacteristic
126C22×C197H(5,𝔽2)12yes—
127C2×D197—24no—
128C2×D197—24no—
129C2×D197—24no—
130C2×D197—416no—
131C2×D197—42no—
132C2×D197—42no—
133C2×D197—42no—
134C2×D197—42no—
135C4×C197—22no—
136C4×C197—42no—
137C4×C197—42no—
138Dic197—24no—
139Dic197—42no—
140Dic197—42no—
141C2×C197—42no—
142C2×C197—42no—
143C2×C197#51812yes—
144C2×C197#51811yescharacteristic
145C2×C197#56011yescharacteristic
146C2×C197#56012yes—
147C2×C197H(3,𝔽4)11yescharacteristic
148D197—44no—
149C197#237511yescharacteristic, complemented, Sylow
150#2375—1971nomaximal, Sylow
151C2×#120—1971no—
152#390—1972no—
153#390D19711yescharacteristic, complemented
154#388—1972no—
155#388—1972no—
156#388—1972no—
157#386—1971no—
158#386—1972no—
159#379—1972no—
160C25—1971no—
161C22×D4—1972no—
162C22×D4C2×D19711yescharacteristic
163C23×C4—1971no—
164#120—1978no—
165C2×C4⋊C4—1972no—
166C2×C4⋊C4C2×D19711yescharacteristic, complemented
167C2×C22⋊C4—1972no—
168C2×C22⋊C4—1971no—
169C2×C22⋊C4—1972no—
170C2×C22⋊C4—1972no—
171C2×C22⋊C4—1972no—
172C2×C22⋊C4—1972no—
173C2×C22⋊C4—1971no—
174C2×C22⋊C4—1972no—
175C2×C22⋊C4C2×D19712yes—
176C2×C22⋊C4C2×D19712yescomplemented
177C2×C42—1972no—
178C2×C42C2×D19711yescharacteristic, complemented
179#95—1972no—
180#95—1972no—
181#95—1972no—
182C24—7882no—
183C24—7881no—
184C24—7881no—
185C24—7882no—
186C24—1974no—
187C24—1972no—
188C24C22×D19711yescharacteristic
189C24C22×D19711yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—7882no—
193C2×D4—7882no—
194C2×D4—1978no—
195C2×D4C4×D19714yes—
196C22×C4—7881no—
197C22×C4—7882no—
198C22×C4—1972no—
199C22×C4—1972no—
200C22×C4—1972no—
201C22×C4C22×D19712yes—
202C22×C4C22×D19712yes—
203C22×C4C22×D19711yescharacteristic
204C4⋊C4—41no—
205C4⋊C4—7882no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—7882no—
209C22⋊C4—7882no—
210C22⋊C4—7882no—
211C22⋊C4—7882no—
212C22⋊C4—7881no—
213C22⋊C4—7882no—
214C22⋊C4—1974no—
215C22⋊C4—1978no—
216C42—41no—
217C42—7882no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—7888no—
227C23—7884no—
228C23—7884no—
229C23—7888no—
230C23—7884no—
231C23—3942no—
232C23—3942no—
233C23—3942no—
234C23—3942no—
235C23—3942no—
236C23—3942no—
237C23—1974no—
238C23C2×C4×D19712yes—
239C23C2×C4×D19712yes—
240C23C23×D19711yescharacteristic
241D4—44no—
242D4—7888no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—7884no—
252C2×C4—3942no—
253C2×C4—3942no—
254C2×C4—3942no—
255C2×C4—3942no—
256C2×C4—3942no—
257C2×C4—3942no—
258C2×C4—3942no—
259C2×C4—3942no—
260C2×C4—1974no—
261C2×C4C2×C4×D19712yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—78816no—
273C22—7882no—
274C22—7882no—
275C22—7882no—
276C22—7882no—
277C22—3944no—
278C22—3944no—
279C22—3944no—
280C22C22×C4×D19711yescharacteristic
281C22#2026337112yes—
282C22#2026337111yescharacteristic
283C22#2026337111yescharacteristic
284C22#2026337212yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—7882no—
289C4—7882no—
290C4—3944no—
291C2—42no—
292C2—42no—
293C2—7884no—
294C2#2716155112yes—
295C2#2716155211yescharacteristic
296C2#2716181611yescharacteristic
297C2#2716181712yes—
298C2#2716182111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#390×C197C2
socle, cosocleC23×C197C24
prime radicalC2×#20263294C2
hypercenter#390D197
hypocenterC197#2375
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #27160979C2semidirect
3. #27160970C2semidirect
4. #27160969C2semidirect
5. #27160967C2semidirect
6. #27160959C2semidirect
7. #27160956C2semidirect
8. #27160934C2semidirect
9. #390×C197C2semidirect
12. C23×Dic197C22semidirect
13. #20263294C4semidirect
14. C2×#15691337C22semidirect
17. C2×C22⋊Dic197C22semidirect
23. C2×C4×Dic197C22semidirect
24. #20263233C22semidirect
25. #20263233C22semidirect
26. #20263233C22semidirect
28. C2×C4⋊C4×C197C22semidirect
30. C2×C22⋊C4×C197C22semidirect
31. C2×C42×C197C22semidirect
36. C23×D197C2×C4semidirect
49. C22⋊Dic197C2×C4semidirect
54. #15691335C2×C4semidirect
149. C197#2375semidirect
153. #390D197semidirect
166. C2×C4⋊C4C2×D197semidirect
176. C2×C22⋊C4C2×D197semidirect
178. C2×C42C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632942
#271609792
#271609702
#271609692
#271609672
#271609592
#271609562
#271609342
#390×C1972
#2375197