Finite Group Databasebeta

← #38755320

Subgroups — #38755320

Order 25216.

342 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161800C211yesmaximal, characteristic
2#27161784C211yesmaximal, characteristic
3#27161476C211yesmaximal, characteristic, complemented
4#27161423C211yesmaximal, characteristic, complemented
5#27161421C211yesmaximal, characteristic, complemented
6#27161408C211yesmaximal, characteristic, complemented
7#27161372C211yesmaximal, characteristic, complemented
8#27161363C211yesmaximal, characteristic, complemented
9#27161336C211yesmaximal, characteristic, complemented
10#27161335C211yesmaximal, characteristic, complemented
11#27161171C211yesmaximal, characteristic, complemented
12#27161154C211yesmaximal, characteristic
13#27161119C211yesmaximal, characteristic, complemented
14#27161118C211yesmaximal, characteristic, complemented
15#483×C197C211yesmaximal, characteristic, complemented
16C2×#15691348C2211yescharacteristic
17C2×#15691342C2211yescharacteristic
18C2×#15691341C2211yescharacteristic
19#20263325C2211yescharacteristic
20#20263325C2211yescharacteristic
21#20263321—21no—
22#20263319—21no—
23#20263316—21no—
24#20263312—21no—
25#20263306C2211yescharacteristic
26#20263305—21no—
27#20263305C2211yescharacteristic
28#20263304—21no—
29#20263302C2211yescharacteristic
30Q8×Dic197C2211yescharacteristic
31#20263291—21no—
32#20263291C2211yescharacteristic
33C4×C22⋊D197—21no—
34C4×C22⋊D197C2211yescharacteristic
35#20263285C2211yescharacteristic
36#20263284C2211yescharacteristic
37C4⋊C4×D197C2211yescharacteristic
38#20263260C2211yescharacteristic
39#20263259C2211yescharacteristic
40#20263259C2211yescharacteristic
41#20263258C2211yescharacteristic
42#20263257C2211yescharacteristic
43#20263257C2211yescharacteristic
44#20263257C2211yescharacteristic
45#20263255C2211yescharacteristic
46#20263251C2211yescharacteristic
47#20263251C2211yescharacteristic
48#20263250C2211yescharacteristic
49#20263249C2211yescharacteristic
50#20263238C2211yescharacteristic
51C8×Dic197C2211yescharacteristic
52C22⋊Q8×C197C2211yescharacteristic
53C2×C4⋊C4×C197C2211yescharacteristic
54#106×C197C2211yescharacteristic
55#106×C197C2211yescharacteristic
56#103×C197C2211yescharacteristic
57#103×C197C2211yescharacteristic
58C22⋊C8×C197C2211yescharacteristic
59C2×Dic394—21no—
60C2×Dic394—21no—
61C2×Dic394C2311yescharacteristic
62C2×C22⋊D197D411yescharacteristic
63C2×C4×D197—21no—
64C2×C4×D197C2311yescharacteristic
65#15691348—22no—
66#15691342—22no—
67#15691341—22no—
68C2×#13096802C2311yescharacteristic
69C2×#13096802C2311yescharacteristic
70C4⋊Dic197—21no—
71C4⋊Dic197—21no—
72C4⋊Dic197C2311yescharacteristic
73C4⋊Dic197C2311yescharacteristic
74C4⋊Dic197C2311yescharacteristic
75#15691337—21no—
76#15691337—21no—
77#15691337—21no—
78#15691337—21no—
79#15691337—21no—
80#15691337—21no—
81#15691337D411yescharacteristic
82C22⋊Dic197—21no—
83C22⋊Dic197—21no—
84C22⋊Dic197D411yescharacteristic
85#15691335—21no—
86#15691335—21no—
87#15691335—21no—
88#15691335D411yescharacteristic
89C4×Dic197—21no—
90C4×Dic197—21no—
91C4×Dic197—21no—
92C4×Dic197—21no—
93C4×Dic197C2311yescharacteristic
94C2×Q8×C197C2311yescharacteristic
95C22×C4×C197—21no—
96C22×C4×C197C2311yescharacteristic
97C2×C8×C197C2311yescharacteristic
98C2×C8×C197C2311yescharacteristic
99C4⋊C4×C197—21no—
100C4⋊C4×C197—21no—
101C4⋊C4×C197C2311yescharacteristic
102C4⋊C4×C197C2311yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C22⋊C4×C197—21no—
105C22×D197C2×D411yescharacteristic
106Dic394—21no—
107Dic394—21no—
108Dic394—42no—
109Dic394—81no—
110C22⋊D197—22no—
111C2×Dic197—21no—
112C2×Dic197—21no—
113C2×Dic197—21no—
114C2×Dic197—21no—
115C2×Dic197C2×D411yescharacteristic
116C2×Dic197C2×D411yescharacteristic
117C2×Dic197C2×D411yescharacteristic
118C4×D197—21no—
119C4×D197—21no—
120C4×D197—81no—
121#13096802—21no—
122#13096802#4112yes—
123C23×C197C2×D411yescharacteristic
124Q8×C197—21no—
125Q8×C197—21no—
126C2×C4×C197—21no—
127C2×C4×C197—21no—
128C2×C4×C197—21no—
129C2×C4×C197—21no—
130C2×C4×C197—21no—
131C2×C4×C197—21no—
132C2×C4×C197—81no—
133C2×C4×C197C2×D411yescharacteristic
134C2×C4×C197C2411yescharacteristic
135C8×C197—21no—
136C8×C197—21no—
137C22×C197—21no—
138C22×C197—21no—
139C22×C197—21no—
140C22×C197C22×D411yescharacteristic
141C2×D197—21no—
142C2×D197—21no—
143C2×D197—21no—
144C4×C197—21no—
145C4×C197—41no—
146C4×C197—41no—
147C4×C197—41no—
148C4×C197—41no—
149C4×C197C2×#4111yescharacteristic
150C4×C1972−1+411yescharacteristic
151Dic197—21no—
152Dic197—21no—
153Dic197—41no—
154Dic197—41no—
155Dic197—41no—
156Dic197—41no—
157Dic197#13712yes—
158C2×C197—41no—
159C2×C197#54711yescharacteristic
160C2×C197C2×#13711yescharacteristic
161C2×C197#57711yescharacteristic
162D197—41no—
163C197#335611yescharacteristic, complemented, Sylow
164#3356—1971nomaximal, Sylow
165#556—1971no—
166#554—1971no—
167#498—1971no—
168#485—1971no—
169#485—1971no—
170#483—1971no—
171#483D19711yescharacteristic, complemented
172#477—1971no—
173#475—1971no—
174#470—1971no—
175#470—1971no—
176C4×Q16—1971no—
177#436—1971no—
178#428—1971no—
179#428—1971no—
180C2×Q16—1971no—
181C2×M16—1971no—
182C2×M16—1971no—
183C4⋊Q8—1971no—
184C4⋊Q8—1971no—
185#126—3941no—
186#126—3941no—
187#125—3941no—
188#124—3941no—
189C22⋊Q8—3941no—
190C22⋊Q8—3941no—
191C22⋊Q8—1971no—
192C22⋊Q8—1971no—
193C22⋊Q8—1971no—
194C22⋊Q8C2×D19711yescharacteristic
195C4×Q8—3941no—
196C4×Q8—1971no—
197C4×Q8—1971no—
198C4×D4—3941no—
199C4×D4—1971no—
200#117—1971no—
201#117—1971no—
202C2×C4⋊C4—1971no—
203C2×C4⋊C4C2×D19711yescharacteristic
204#107—1971no—
205#107—1971no—
206#107—1971no—
207#106—1971no—
208#106—1971no—
209#106—1971no—
210#106—1971no—
211#106C2×D19711yescharacteristic
212#106C2×D19711yescharacteristic
213C4⋊C8—1971no—
214#103—1971no—
215#103—1971no—
216#103—1971no—
217#103—1971no—
218#103C2×D19711yescharacteristic
219#103C2×D19711yescharacteristic
220C22⋊C8—1971no—
221C22⋊C8C2×D19711yescharacteristic
222C4×C8—1971no—
223C2×Q8—3941no—
224C2×Q8—3941no—
225C2×Q8—1971no—
226C2×Q8C22×D19711yescharacteristic
227C2×D4—1971no—
228C22×C4—21no—
229C22×C4—3941no—
230C22×C4—1971no—
231C22×C4C22×D19711yescharacteristic
232Q16—3942no—
233M16—3942no—
234M16—3942no—
235C2×C8—1971no—
236C2×C8—1971no—
237C2×C8C22×D19711yescharacteristic
238C2×C8C22×D19711yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—3941no—
242C4⋊C4—3941no—
243C4⋊C4—3941no—
244C4⋊C4—3941no—
245C4⋊C4—3941no—
246C4⋊C4—3941no—
247C4⋊C4—3941no—
248C4⋊C4—3941no—
249C4⋊C4—1971no—
250C4⋊C4—1971no—
251C4⋊C4—1971no—
252C4⋊C4—1971no—
253C4⋊C4C22×D19711yescharacteristic
254C4⋊C4C22×D19711yescharacteristic
255C4⋊C4C22×D19711yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—3941no—
258C22⋊C4—3941no—
259C22⋊C4—3941no—
260C22⋊C4—3941no—
261C22⋊C4—3941no—
262C22⋊C4—1971no—
263C22⋊C4—1971no—
264C42—3941no—
265C42—3941no—
266C42—3941no—
267C42—3941no—
268C42—1971no—
269C23—1971no—
270C23D4×D19711yescharacteristic
271Q8—21no—
272Q8—21no—
273Q8—7882no—
274Q8—15761no—
275Q8—3941no—
276Q8—3941no—
277D4—3942no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—15761no—
286C2×C4—3941no—
287C2×C4—3941no—
288C2×C4—3941no—
289C2×C4—3941no—
290C2×C4—3941no—
291C2×C4—3941no—
292C2×C4—1971no—
293C2×C4—1971no—
294C2×C4—1971no—
295C2×C4D4×D19711yescharacteristic
296C2×C4C23×D19711yescharacteristic
297C8—21no—
298C8—21no—
299C8—3941no—
300C8—1972no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—3941no—
305C22—3941no—
306C22—3941no—
307C22C2×D4×D19711yescharacteristic
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—7881no—
314C4—7881no—
315C4—7881no—
316C4—7881no—
317C4—3941no—
318C4—3941no—
319C4—1972no—
320C4#41×D19711yescharacteristic
321C4#2026337311yescharacteristic
322C2—41no—
323C2—7881no—
324C2#2716172711yescharacteristic
325C2#137×D19711yescharacteristic
326C2#2716188511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#483×C197C2
socleC22×C197C22×D4
prime radicalC2×C22⋊D197D4
hypercenter#483D197
hypocenterC197#3356
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #27161476C2semidirect
4. #27161423C2semidirect
5. #27161421C2semidirect
6. #27161408C2semidirect
7. #27161372C2semidirect
8. #27161363C2semidirect
9. #27161336C2semidirect
10. #27161335C2semidirect
11. #27161171C2semidirect
13. #27161119C2semidirect
14. #27161118C2semidirect
15. #483×C197C2semidirect
163. C197#3356semidirect
171. #483D197semidirect

Maximal subgroups

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

subgroupindex
#271618002
#271617842
#271614762
#271614232
#271614212
#271614082
#271613722
#271613632
#271613362
#271613352
#271611712
#271611542
#271611192
#271611182
#483×C1972
#3356197