Finite Group Databasebeta

← #38752647

Subgroups — #38752647

Order 25216.

462 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 121 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#121×D197C211yesmaximal, characteristic
2C2×#20263284C211yesmaximal, characteristic, complemented
3#27161406C212yesmaximal, complemented
4#27161402C212yesmaximal, complemented
5#27161398C211yesmaximal, characteristic, complemented
6#27161395C211yesmaximal, characteristic, complemented
7#27161392C211yesmaximal, characteristic, complemented
8#27161120C212yesmaximal, complemented
9#27161082C212yesmaximal, complemented
10#27161041C211yesmaximal, characteristic, complemented
11#480×C197C211yesmaximal, characteristic, complemented
12C2×D4×D197—21no—
13C2×D4×D197C2211yescharacteristic
14C22×C4×D197C2211yescharacteristic
15C2×#15691341C2212yes—
16#20263301C2211yescharacteristic
17#20263300C2211yescharacteristic
18#20263299—21no—
19#20263298—21no—
20#20263297C2211yescharacteristic
21#20263284—21no—
22#20263284—21no—
23#20263284C2212yes—
24#20263284C2212yescomplemented
25C2×C4⋊Dic197C2211yescharacteristic, complemented
26C4⋊C4×D197C2211yescharacteristic
27C2×#15691335—21no—
28C22⋊C4×D197—21no—
29C2×C4×Dic197—21no—
30#20263259C2212yescomplemented
31#20263258C2212yescomplemented
32#20263257C2212yescomplemented
33#20263248C2212yescomplemented
34#20263247C2212yescomplemented
35#20263246C2212yescomplemented
36C22⋊#13096802C2211yescharacteristic, complemented
37#20263238C2212yescomplemented
38#121×C197C2211yescharacteristic
39C2×C4⋊C4×C197C2211yescharacteristic, complemented
40#107×C197C2212yescomplemented
41#102×C197C2212yescomplemented
42C22⋊C8×C197C2211yescharacteristic, complemented
43C23×D197—21no—
44C23×D197D411yescharacteristic
45C2×D788—21no—
46C2×D788C2311yescharacteristic
47C2×C22⋊D197—21no—
48C2×C22⋊D197—21no—
49C2×C22⋊D197—21no—
50D4×D197—24no—
51D4×D197—44no—
52C22×Dic197—21no—
53C22×Dic197D411yescharacteristic
54C2×C4×D197—24no—
55C2×C4×D197—21no—
56C2×C4×D197D411yescharacteristic
57C2×C4×D197D411yescharacteristic
58C2×C4×D197C2311yescharacteristic
59C2×C4×D197C2311yescharacteristic
60#15691341—24no—
61C2×#13096802C2312yes—
62C4⋊Dic197—21no—
63C4⋊Dic197C2312yes—
64C4⋊Dic197C2311yescharacteristic
65#15691337—21no—
66C22⋊Dic197—21no—
67#15691335—22no—
68#15691335—22no—
69#15691335—21no—
70#15691335—21no—
71C4×Dic197—22no—
72C4×Dic197—22no—
73C2×D4×C197—21no—
74C2×D4×C197C2311yescharacteristic
75C22×C4×C197—21no—
76C22×C4×C197C2311yescharacteristic
77C2×C8×C197C2312yes—
78C4⋊C4×C197—21no—
79C4⋊C4×C197C2312yes—
80C4⋊C4×C197C2311yescharacteristic
81C22⋊C4×C197—21no—
82C22×D197—22no—
83C22×D197—22no—
84C22×D197—21no—
85C22×D197—41no—
86C22×D197—42no—
87C22×D197—42no—
88C22×D197—42no—
89C22×D197C2×D411yescharacteristic
90C22×D197C2×D411yescharacteristic
91D788—22no—
92D788—42no—
93C22⋊D197—42no—
94C22⋊D197—42no—
95C22⋊D197—42no—
96C2×Dic197—22no—
97C2×Dic197—22no—
98C2×Dic197—21no—
99C2×Dic197—42no—
100C2×Dic197C2×D411yescharacteristic
101C2×Dic197C2×D411yescharacteristic
102C4×D197—22no—
103C4×D197—22no—
104C4×D197—22no—
105C4×D197—42no—
106C4×D197#4114yes—
107#13096802—22no—
108C23×C197—21no—
109C23×C197C2×D411yescharacteristic
110D4×C197—22no—
111D4×C197—42no—
112C2×C4×C197—22no—
113C2×C4×C197—21no—
114C2×C4×C197—22no—
115C2×C4×C197—42no—
116C2×C4×C197C2×D411yescharacteristic
117C2×C4×C197C2411yescharacteristic
118C8×C197—22no—
119C22×C197—21no—
120C22×C197—21no—
121C22×C197—41no—
122C22×C197—41no—
123C22×C197—41no—
124C22×C197AGL(1,ℤ/8)12yes—
125C22×C197C22×D411yescharacteristic
126C2×D197—21no—
127C2×D197—21no—
128C2×D197—21no—
129C2×D197—21no—
130C2×D197—21no—
131C2×D197—44no—
132C2×D197—41no—
133C2×D197—41no—
134C2×D197—41no—
135C2×D197—84no—
136C2×D197#12312yes—
137C4×C197—21no—
138C4×C197—42no—
139C4×C197—41no—
140C4×C197C2×#4112yes—
141Dic197—21no—
142Dic197—41no—
143Dic197—42no—
144Dic197#12312yes—
145C2×C197—22no—
146C2×C197—81no—
147C2×C197C2×#12311yescharacteristic
148C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C197#57511yescharacteristic
150D197—22no—
151D197—41no—
152D197—81no—
153C197#309411yescharacteristic, complemented, Sylow
154#3094—1971nomaximal, Sylow
155C2×#121—1971no—
156C2×#117—1971no—
157#482—1972no—
158#482—1972no—
159#481—1971no—
160#481—1971no—
161#480—1971no—
162#480D19711yescharacteristic, complemented
163#428—1972no—
164#418—1972no—
165#407—1971no—
166C22×D4—3941no—
167C22×D4—1971no—
168C23×C4—1971no—
169C2×M16—1972no—
170#121—3941no—
171#121—3941no—
172#121—1971no—
173#121—1971no—
174#121—1971no—
175#121C2×D19711yescharacteristic
176#117—3941no—
177#117—3941no—
178#117—1972no—
179#117—1972no—
180C2×C4⋊C4—1971no—
181C2×C4⋊C4—1971no—
182C2×C4⋊C4C2×D19711yescharacteristic, complemented
183C2×C22⋊C4—3941no—
184C2×C22⋊C4—3941no—
185C2×C42—3941no—
186#107—1972no—
187#107C2×D19712yescomplemented
188#106—1972no—
189#106—1972no—
190#102—1972no—
191#102—1972no—
192#102—1972no—
193#102C2×D19712yescomplemented
194C22⋊C8—1971no—
195C22⋊C8—1972no—
196C22⋊C8C2×D19711yescharacteristic, complemented
197C24—3941no—
198C24—1971no—
199C2×D4—21no—
200C2×D4—7884no—
201C2×D4—3941no—
202C2×D4—3941no—
203C2×D4—3941no—
204C2×D4—3944no—
205C2×D4—3941no—
206C2×D4—1971no—
207C2×D4C22×D19711yescharacteristic
208C22×C4—21no—
209C22×C4—3941no—
210C22×C4—3944no—
211C22×C4—3941no—
212C22×C4—1971no—
213C22×C4—1971no—
214C22×C4—1971no—
215C22×C4—1971no—
216C22×C4—1971no—
217C22×C4C22×D19711yescharacteristic
218M16—3944no—
219C2×C8—1972no—
220C2×C8C22×D19712yes—
221C4⋊C4—21no—
222C4⋊C4—3941no—
223C4⋊C4—3941no—
224C4⋊C4—1972no—
225C4⋊C4—1971no—
226C4⋊C4C22×D19712yes—
227C4⋊C4C22×D19711yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—3942no—
230C22⋊C4—3942no—
231C22⋊C4—3941no—
232C22⋊C4—3941no—
233C22⋊C4—3941no—
234C42—3942no—
235C42—3942no—
236C23—21no—
237C23—7881no—
238C23—7882no—
239C23—7882no—
240C23—7882no—
241C23—3942no—
242C23—3941no—
243C23—3942no—
244C23—1971no—
245C23—1971no—
246C23D4×D19711yescharacteristic
247D4—22no—
248D4—42no—
249D4—7882no—
250D4—7882no—
251D4—7882no—
252D4—7882no—
253D4—3942no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—7882no—
259C2×C4—7882no—
260C2×C4—3942no—
261C2×C4—3942no—
262C2×C4—3942no—
263C2×C4—3941no—
264C2×C4—3942no—
265C2×C4—3942no—
266C2×C4—1971no—
267C2×C4—1971no—
268C2×C4—1974no—
269C2×C4D4×D19711yescharacteristic
270C2×C4C23×D19711yescharacteristic
271C8—22no—
272C8—3942no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—7884no—
279C22—7881no—
280C22—7881no—
281C22—7881no—
282C22—15764no—
283C22—3941no—
284C22—3941no—
285C22—3941no—
286C22—3941no—
287C22—3941no—
288C22—1972no—
289C22#2026335012yes—
290C22C2×D4×D19711yescharacteristic
291C4—21no—
292C4—42no—
293C4—41no—
294C4—7881no—
295C4—7882no—
296C4—3941no—
297C4—1972no—
298C4#41×D19712yes—
299C2—22no—
300C2—81no—
301C2—7881no—
302C2—15761no—
303C2—3942no—
304C2#123×D19711yescharacteristic
305C2C2×#2026335011yescharacteristic
306C2#2716187811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#480×C197C2
socleC22×C197C22×D4
prime radical#121×D197C2
hypercenter#480D197
hypocenterC197#3094
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#20263284C2semidirect
3. #27161406C2semidirect
4. #27161402C2semidirect
5. #27161398C2semidirect
6. #27161395C2semidirect
7. #27161392C2semidirect
8. #27161120C2semidirect
9. #27161082C2semidirect
10. #27161041C2semidirect
11. #480×C197C2semidirect
24. #20263284C22semidirect
25. C2×C4⋊Dic197C22semidirect
30. #20263259C22semidirect
31. #20263258C22semidirect
32. #20263257C22semidirect
33. #20263248C22semidirect
34. #20263247C22semidirect
35. #20263246C22semidirect
36. C22⋊#13096802C22semidirect
37. #20263238C22semidirect
39. C2×C4⋊C4×C197C22semidirect
40. #107×C197C22semidirect
41. #102×C197C22semidirect
42. C22⋊C8×C197C22semidirect
153. C197#3094semidirect
162. #480D197semidirect
182. C2×C4⋊C4C2×D197semidirect
187. #107C2×D197semidirect
193. #102C2×D197semidirect
196. C22⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#121×D1972
C2×#202632842
#271614062
#271614022
#271613982
#271613952
#271613922
#271611202
#271610822
#271610412
#480×C1972
#3094197