Finite Group Databasebeta

← #38752499

Subgroups — #38752499

Order 25216.

388 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161635C211yesmaximal, characteristic, complemented
2C4×#15691359C211yesmaximal, characteristic
3#27161386C211yesmaximal, characteristic, complemented
4#27161380C211yesmaximal, characteristic, complemented
5#27161376C211yesmaximal, characteristic, complemented
6#27161372C211yesmaximal, characteristic, complemented
7#27161368C211yesmaximal, characteristic, complemented
8#27161362C211yesmaximal, characteristic, complemented
9#27161358C211yesmaximal, characteristic, complemented
10#27161116C211yesmaximal, characteristic, complemented
11#27161114C211yesmaximal, characteristic, complemented
12C4⋊C8×D197C211yesmaximal, characteristic, complemented
13#27161074C211yesmaximal, characteristic
14#27161074C211yesmaximal, characteristic
15#479×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×C8×D197C2211yescharacteristic
18C2×C8×D197C2211yescharacteristic
19#20263325C2211yescharacteristic
20#20263325C2211yescharacteristic
21#20263318C2211yescharacteristic
22#20263304—21no—
23#20263304—21no—
24Q8×Dic197C2211yescharacteristic
25#20263291—21no—
26#20263291—21no—
27C4×D788—21no—
28C4×D788C2211yescharacteristic
29#20263288—21no—
30#20263288C2211yescharacteristic
31#20263284—21no—
32#20263284C2211yescharacteristic
33#20263284C2211yescharacteristic
34#20263284C2211yescharacteristic
35C42×D197—21no—
36C42×D197C2211yescharacteristic
37#20263260C2211yescharacteristic
38#20263259C2211yescharacteristic
39#20263259C2211yescharacteristic
40#20263258C2211yescharacteristic
41#20263257C2211yescharacteristic
42#20263257C2211yescharacteristic
43C4⋊#13096802C2211yescharacteristic
44#20263255C2211yescharacteristic
45#20263255C2211yescharacteristic
46#20263250C2211yescharacteristic
47#20263250C2211yescharacteristic
48#20263248C2211yescharacteristic
49#20263248C2211yescharacteristic
50#20263246C2211yescharacteristic
51#20263246C2211yescharacteristic
52#125×C197C2211yescharacteristic
53C4×Q8×C197C2211yescharacteristic
54#107×C197C2211yescharacteristic
55#106×C197C2211yescharacteristic
56C4⋊C8×C197C2211yescharacteristic
57#103×C197C2211yescharacteristic
58#103×C197C2211yescharacteristic
59#15691359—22no—
60#15691359Q814yescomplemented
61C2×Dic394—21no—
62C2×Dic394—21no—
63C2×D788—21no—
64C2×D788C2311yescharacteristic
65C2×C4×D197—21no—
66C2×C4×D197—21no—
67C2×C4×D197—21no—
68C2×C4×D197D411yescharacteristic
69C2×C4×D197D411yescharacteristic
70C2×C4×D197C2311yescharacteristic
71C2×#13096802C2311yescharacteristic
72C2×#13096802C2311yescharacteristic
73C8×D197—22no—
74C8×D197—22no—
75C4⋊Dic197—21no—
76C4⋊Dic197C2311yescharacteristic
77C4⋊Dic197C2311yescharacteristic
78C4⋊Dic197C2311yescharacteristic
79#15691337—21no—
80#15691337—21no—
81#15691337—21no—
82#15691337—21no—
83#15691335—21no—
84#15691335—21no—
85#15691335—21no—
86#15691335—21no—
87#15691335—21no—
88C4×Dic197—21no—
89C4×Dic197—21no—
90C4×Dic197—21no—
91C4×Dic197—21no—
92C4×Dic197D411yescharacteristic
93C4×Dic197D411yescharacteristic
94C4×Dic197C2311yescharacteristic
95C2×Q8×C197C2311yescharacteristic
96C2×C8×C197C2311yescharacteristic
97C2×C8×C197C2311yescharacteristic
98C4⋊C4×C197—21no—
99C4⋊C4×C197—21no—
100C4⋊C4×C197—21no—
101C4⋊C4×C197C2311yescharacteristic
102C4⋊C4×C197C2311yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C42×C197—21no—
105C42×C197C2311yescharacteristic
106C22×D197—21no—
107C22×D197C2×D411yescharacteristic
108Dic394—42no—
109Dic394—42no—
110D788—24no—
111D788—21no—
112D788C2×Q812yes—
113C2×Dic197—21no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197C2×D411yescharacteristic
117C2×Dic197C2×D411yescharacteristic
118C2×Dic197C2×D411yescharacteristic
119C4×D197—24no—
120C4×D197—22no—
121C4×D197—41no—
122C4×D197—42no—
123C4×D197—42no—
124C4×D197C2×Q812yes—
125C4×D197#4112yes—
126#13096802—21no—
127#13096802—21no—
128Q8×C197—21no—
129Q8×C197C2×Q812yes—
130C2×C4×C197—21no—
131C2×C4×C197—21no—
132C2×C4×C197—21no—
133C2×C4×C197—21no—
134C2×C4×C197C2×D411yescharacteristic
135C2×C4×C197C2×D411yescharacteristic
136C2×C4×C197C2411yescharacteristic
137C8×C197—21no—
138C8×C197—21no—
139C22×C197C22×D411yescharacteristic
140C2×D197—21no—
141C2×D197—21no—
142C2×D197—22no—
143C2×D197—41no—
144C2×D197—41no—
145C2×D197C22⋊Q812yes—
146C4×C197—22no—
147C4×C197—21no—
148C4×C197—21no—
149C4×C197—41no—
150C4×C197—41no—
151C4×C197—41no—
152C4×C197C22×Q811yescharacteristic
153C4×C197C2×#4111yescharacteristic
154Dic197—21no—
155Dic197—41no—
156Dic197—41no—
157Dic197—41no—
158Dic197C22⋊Q812yes—
159Dic197#13512yes—
160C2×C197C2×C22⋊Q811yescharacteristic
161C2×C197C2×#13511yescharacteristic
162C2×C197#57511yescharacteristic
163D197—22no—
164D197—42no—
165C197#307611yescharacteristic, complemented, Sylow
166#3076—1971nomaximal, Sylow
167#533—1971no—
168C4×#41—1971no—
169#479—1971no—
170#479D19711yescharacteristic, complemented
171#478—1971no—
172#478—1971no—
173#477—1971no—
174#476—1971no—
175#475—1971no—
176#474—1971no—
177#427—1971no—
178#427—1971no—
179C2×C4⋊C8—1971no—
180#416—1971no—
181#416—1971no—
182C2×#41—1971no—
183C22×C8—1971no—
184C22×C8—1971no—
185C4⋊Q8—1971no—
186C4⋊Q8—1971no—
187#125—1971no—
188#125C2×D19711yescharacteristic
189C22⋊Q8—3941no—
190C22⋊Q8—3941no—
191C4×Q8—3941no—
192C4×Q8—3941no—
193C4×Q8—1971no—
194C4×Q8C2×D19711yescharacteristic
195C4×D4—3941no—
196C4×D4—3941no—
197C4×D4—1971no—
198C4×D4—1971no—
199#117—3941no—
200#117—1971no—
201#117—1971no—
202#117—1971no—
203C2×C42—3941no—
204C2×C42—1971no—
205#107—1971no—
206#107—1971no—
207#107—1971no—
208#107C2×D19711yescharacteristic
209#106—1971no—
210#106—1971no—
211#106—1971no—
212#106C2×D19711yescharacteristic
213C4⋊C8—1971no—
214C4⋊C8—1971no—
215C4⋊C8—1971no—
216C4⋊C8C2×D19711yescharacteristic
217#103—1971no—
218#103—1971no—
219#103C2×D19711yescharacteristic
220#103C2×D19711yescharacteristic
221#102—1971no—
222#102—1971no—
223#102—1971no—
224#102—1971no—
225#41—3942no—
226#41—1974no—
227C2×Q8—3941no—
228C2×Q8—3941no—
229C2×Q8C22×D19711yescharacteristic
230C2×D4—3941no—
231C2×D4—1971no—
232C22×C4—3941no—
233C22×C4—3941no—
234C22×C4—3941no—
235C22×C4—1971no—
236C22×C4—1971no—
237C22×C4—1971no—
238C2×C8—3942no—
239C2×C8—3942no—
240C2×C8—1971no—
241C2×C8—1971no—
242C2×C8C22×D19711yescharacteristic
243C2×C8C22×D19711yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—3941no—
248C4⋊C4—3941no—
249C4⋊C4—3941no—
250C4⋊C4—3941no—
251C4⋊C4—3941no—
252C4⋊C4—1971no—
253C4⋊C4—1971no—
254C4⋊C4—1971no—
255C4⋊C4C22×D19711yescharacteristic
256C4⋊C4C22×D19711yescharacteristic
257C4⋊C4C22×D19711yescharacteristic
258C22⋊C4—3941no—
259C22⋊C4—3941no—
260C22⋊C4—3941no—
261C22⋊C4—3941no—
262C22⋊C4—3941no—
263C42—21no—
264C42—3941no—
265C42—3941no—
266C42—3941no—
267C42—3941no—
268C42—1971no—
269C42—1971no—
270C42—1971no—
271C42C22×D19711yescharacteristic
272C23—3941no—
273C23—1971no—
274Q8—21no—
275Q8—7882no—
276Q8—7882no—
277Q8Q8×D19712yes—
278D4—3944no—
279D4—3941no—
280D4—1972no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—7882no—
286C2×C4—7881no—
287C2×C4—7882no—
288C2×C4—3944no—
289C2×C4—3942no—
290C2×C4—3941no—
291C2×C4—3941no—
292C2×C4—3941no—
293C2×C4—1971no—
294C2×C4—1971no—
295C2×C4—1971no—
296C2×C4—1972no—
297C2×C4—1972no—
298C2×C4D4×D19711yescharacteristic
299C2×C4D4×D19711yescharacteristic
300C2×C4C23×D19711yescharacteristic
301C8—21no—
302C8—21no—
303C8—3941no—
304C8—3941no—
305C22—7881no—
306C22—7881no—
307C22—3942no—
308C22—3941no—
309C22—3941no—
310C22—1972no—
311C22C2×D4×D19711yescharacteristic
312C4—21no—
313C4—21no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—7881no—
319C4—7881no—
320C4—7881no—
321C4—3941no—
322C4—1972no—
323C4—1972no—
324C4C2×Q8×D19711yescharacteristic
325C4#41×D19711yescharacteristic
326C2—7882no—
327C2—3942no—
328C2C22⋊Q8×D19711yescharacteristic
329C2#135×D19711yescharacteristic
330C2#2716187811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#479×C197C2
socleC22×C197C22×D4
prime radicalC2×#15691359C22
hypercenter#479D197
hypocenterC197#3076
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. #27161635C2semidirect
3. #27161386C2semidirect
4. #27161380C2semidirect
5. #27161376C2semidirect
6. #27161372C2semidirect
7. #27161368C2semidirect
8. #27161362C2semidirect
9. #27161358C2semidirect
10. #27161116C2semidirect
11. #27161114C2semidirect
12. C4⋊C8×D197C2semidirect
15. #479×C197C2semidirect
60. #15691359Q8semidirect
165. C197#3076semidirect
170. #479D197semidirect

Maximal subgroups

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

subgroupindex
#271616352
C4×#156913592
#271613862
#271613802
#271613762
#271613722
#271613682
#271613622
#271613582
#271611162
#271611142
C4⋊C8×D1972
#271610742
#479×C1972
#3076197