Finite Group Databasebeta

← #38753067

Subgroups — #38753067

Order 25216.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161709C211yesmaximal, characteristic, complemented
2#27161642C211yesmaximal, characteristic
3#27161442C211yesmaximal, characteristic, complemented
4#27161439C211yesmaximal, characteristic, complemented
5#27161437C211yesmaximal, characteristic, complemented
6#27161436C211yesmaximal, characteristic, complemented
7#27161432C211yesmaximal, characteristic, complemented
8#27161425C211yesmaximal, characteristic, complemented
9#27161424C211yesmaximal, characteristic, complemented
10#27161082C211yesmaximal, characteristic, complemented
11#27161077C211yesmaximal, characteristic, complemented
12#27161072C211yesmaximal, characteristic, complemented
13#102×D197C211yesmaximal, characteristic, complemented
14#27161036C211yesmaximal, characteristic, complemented
15#490×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic
17C2×D4×D197C2211yescharacteristic
18C2×#15691341C2211yescharacteristic
19C2×C8×D197C2211yescharacteristic
20#20263326C2211yescharacteristic, complemented
21#20263325—21no—
22#20263325C2211yescharacteristic, complemented
23#20263322C2211yescharacteristic
24#20263317C2211yescharacteristic, complemented
25#20263316—21no—
26#20263314C2211yescharacteristic
27#20263313C2211yescharacteristic
28#20263306—21no—
29#20263304—21no—
30#20263298—21no—
31#20263298—21no—
32#20263294—21no—
33#20263294—21no—
34#20263284C2211yescharacteristic
35#20263283C2211yescharacteristic
36C4⋊C4×D197C2211yescharacteristic
37#20263251C2211yescharacteristic, complemented
38#20263251C2211yescharacteristic, complemented
39#20263249C2211yescharacteristic, complemented
40#20263249C2211yescharacteristic, complemented
41#20263248C2211yescharacteristic
42#20263248C2211yescharacteristic
43#20263247C2211yescharacteristic
44#20263247C2211yescharacteristic
45#20263247C2211yescharacteristic, complemented
46#20263247C2211yescharacteristic, complemented
47#20263246C2211yescharacteristic
48#20263246C2211yescharacteristic
49#20263236C2211yescharacteristic, complemented
50C4×#13096802C2211yescharacteristic, complemented
51C8×Dic197C2211yescharacteristic, complemented
52#127×C197C2211yescharacteristic
53#125×C197C2211yescharacteristic, complemented
54#102×C197C2211yescharacteristic
55#102×C197C2211yescharacteristic
56#102×C197C2211yescharacteristic, complemented
57#102×C197C2211yescharacteristic, complemented
58#97×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60#15691357—22no—
61#15691357—22no—
62C2×Dic394—21no—
63C2×Dic394—21no—
64C2×Dic394C2311yescharacteristic
65C2×D788C2311yescharacteristic
66C2×C22⋊D197—21no—
67C2×C22⋊D197—21no—
68C2×C22⋊D197—21no—
69C2×C22⋊D197—21no—
70D4×D197—22no—
71D4×D197—22no—
72C22×Dic197—21no—
73C2×C4×D197—21no—
74C2×C4×D197C2311yescharacteristic
75#15691341—22no—
76C2×#13096802C2311yescharacteristic
77C2×#13096802C2311yescharacteristic
78C8×D197—22no—
79C4⋊Dic197—21no—
80C4⋊Dic197—21no—
81C4⋊Dic197C2311yescharacteristic
82C4⋊Dic197C2311yescharacteristic
83#15691337—21no—
84#15691337—21no—
85#15691337—21no—
86#15691337D411yescharacteristic
87#15691337D411yescharacteristic
88C22⋊Dic197—21no—
89C22⋊Dic197—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335D411yescharacteristic
94#15691335D411yescharacteristic
95C4×Dic197—21no—
96C4×Dic197C2311yescharacteristic
97C2×D4×C197—21no—
98C2×D4×C197—21no—
99C2×D4×C197C2311yescharacteristic
100C2×D4×C197C2311yescharacteristic
101C2×C8×C197C2311yescharacteristic
102C2×C8×C197C2311yescharacteristic
103C4⋊C4×C197—21no—
104C4⋊C4×C197—21no—
105C4⋊C4×C197C2311yescharacteristic
106C4⋊C4×C197C2311yescharacteristic
107C42×C197C2311yescharacteristic
108C22×D197—21no—
109C22×D197—42no—
110C22×D197—81no—
111C22×D197—81no—
112C22×D197C2×D411yescharacteristic
113Dic394—21no—
114Dic394—21no—
115Dic394—81no—
116Dic394—81no—
117D788—21no—
118D788—21no—
119C22⋊D197—42no—
120C22⋊D197—42no—
121C22⋊D197—81no—
122C22⋊D197—81no—
123C2×Dic197—21no—
124C2×Dic197—21no—
125C2×Dic197—21no—
126C2×Dic197—42no—
127C2×Dic197C2×D411yescharacteristic
128C2×Dic197C2×D411yescharacteristic
129C2×Dic197C2×D411yescharacteristic
130C4×D197—81no—
131C4×D197#4112yes—
132C4×D197#4112yes—
133#13096802—21no—
134#13096802—21no—
135C23×C197—21no—
136C23×C197—21no—
137D4×C197—21no—
138D4×C197—21no—
139D4×C197—21no—
140D4×C197—21no—
141D4×C197—81no—
142D4×C197—81no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197C2×D411yescharacteristic
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2411yescharacteristic
148C8×C197—21no—
149C8×C197—21no—
150C22×C197—41no—
151C22×C197—41no—
152C22×C197—41no—
153C22×C197—41no—
154C22×C197—41no—
155C22×C197—41no—
156C22×C197C22×D411yescharacteristic
157C2×D197—21no—
158C2×D197—21no—
159C2×D197—41no—
160C2×D197—41no—
161C2×D197—41no—
162C2×D197—84no—
163C2×D197#12412yes—
164C4×C197—21no—
165C4×C197—21no—
166C4×C197—41no—
167C4×C197—41no—
168C4×C197C2×#4111yescharacteristic
169C4×C197C2×#4111yescharacteristic
170Dic197—21no—
171Dic197—21no—
172Dic197—41no—
173Dic197—41no—
174Dic197—41no—
175Dic197#12412yes—
176C2×C197—81no—
177C2×C197—81no—
178C2×C197C2×#12411yescharacteristic
179C2×C197#57611yescharacteristic
180C2×C197#57711yescharacteristic
181D197—22no—
182D197—81no—
183C197#313611yescharacteristic, complemented, Sylow
184#3136—1971nomaximal, Sylow
185#544—1971no—
186#535—1971no—
187#490—1971no—
188#490D19711yescharacteristic, complemented
189#489—1971no—
190#489—1971no—
191#488—1971no—
192#488—1971no—
193#486—1971no—
194#486—1971no—
195#418—1971no—
196#417—1971no—
197#416—1971no—
198C2×#102—1971no—
199#405—1971no—
200C2×#41—1971no—
201C22×D4—1971no—
202C2×M16—1971no—
203C22×C8—1971no—
204C4⋊Q8—3941no—
205C4⋊Q8—1971no—
206C4⋊Q8—1971no—
207#127—1971no—
208#127C2×D19711yescharacteristic
209#125—3941no—
210#125—1971no—
211#125C2×D19711yescharacteristic, complemented
212#124—1971no—
213#124—1971no—
214C22⋊Q8—3941no—
215C22⋊Q8—3941no—
216#121—3941no—
217#121—3941no—
218#120—3941no—
219#120—3941no—
220#117—1971no—
221#117—1971no—
222C2×C4⋊C4—1971no—
223#103—1971no—
224#103—1971no—
225#103—1971no—
226#103—1971no—
227#102—1971no—
228#102—1971no—
229#102—1971no—
230#102—1971no—
231#102—1971no—
232#102—1971no—
233#102—1971no—
234#102—1971no—
235#102C2×D19711yescharacteristic
236#102C2×D19711yescharacteristic
237#102C2×D19711yescharacteristic, complemented
238#102C2×D19711yescharacteristic, complemented
239#97—1971no—
240#97C2×D19711yescharacteristic, complemented
241C4×C8—1971no—
242C4×C8—1971no—
243C24—3941no—
244#41—3942no—
245#41—3942no—
246C2×Q8—3941no—
247C2×Q8—3941no—
248C2×Q8—1971no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—3941no—
252C2×D4—3942no—
253C2×D4—3942no—
254C2×D4—3941no—
255C2×D4—3941no—
256C2×D4—3941no—
257C2×D4—1971no—
258C2×D4C22×D19711yescharacteristic
259C2×D4C22×D19711yescharacteristic
260C22×C4—3941no—
261C22×C4—3941no—
262C22×C4—1971no—
263M16—3942no—
264C2×C8—3942no—
265C2×C8—1971no—
266C2×C8—1971no—
267C2×C8C22×D19711yescharacteristic
268C2×C8C22×D19711yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—3941no—
272C4⋊C4—3941no—
273C4⋊C4—3941no—
274C4⋊C4—3941no—
275C4⋊C4—3941no—
276C4⋊C4—1971no—
277C4⋊C4—1971no—
278C4⋊C4—1971no—
279C4⋊C4—1971no—
280C4⋊C4C22×D19711yescharacteristic
281C4⋊C4C22×D19711yescharacteristic
282C22⋊C4—3941no—
283C22⋊C4—3941no—
284C22⋊C4—3941no—
285C22⋊C4—3941no—
286C22⋊C4—3941no—
287C22⋊C4—1971no—
288C22⋊C4—1971no—
289C42—3941no—
290C42—1971no—
291C42C22×D19711yescharacteristic
292C23—21no—
293C23—21no—
294C23—7882no—
295C23—15761no—
296C23—15761no—
297C23—3941no—
298C23—1971no—
299Q8—15761no—
300Q8—15761no—
301Q8—3941no—
302Q8—3941no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—81no—
308D4—81no—
309D4—7882no—
310D4—7882no—
311D4—15761no—
312D4—15761no—
313D4—3941no—
314D4—3941no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—7882no—
318C2×C4—15761no—
319C2×C4—3941no—
320C2×C4—3941no—
321C2×C4—3941no—
322C2×C4—1971no—
323C2×C4—1971no—
324C2×C4—1971no—
325C2×C4—1972no—
326C2×C4—1972no—
327C2×C4D4×D19711yescharacteristic
328C2×C4D4×D19711yescharacteristic
329C2×C4C23×D19711yescharacteristic
330C8—21no—
331C8—21no—
332C8—3941no—
333C8—3941no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—7881no—
341C22—7881no—
342C22—7881no—
343C22—15764no—
344C22—3941no—
345C22—3941no—
346C22—1972no—
347C22C2×D4×D19711yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—7881no—
353C4—7881no—
354C4—7881no—
355C4—3941no—
356C4—3941no—
357C4—1972no—
358C4#41×D19711yescharacteristic
359C4#41×D19711yescharacteristic
360C2—81no—
361C2—81no—
362C2—15761no—
363C2—3942no—
364C2#124×D19711yescharacteristic
365C2#2716188311yescharacteristic
366C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#490×C197C2
socleC22×C197C22×D4
prime radical#27161642C2
hypercenter#490D197
hypocenterC197#3136
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #27161709C2semidirect
3. #27161442C2semidirect
4. #27161439C2semidirect
5. #27161437C2semidirect
6. #27161436C2semidirect
7. #27161432C2semidirect
8. #27161425C2semidirect
9. #27161424C2semidirect
10. #27161082C2semidirect
11. #27161077C2semidirect
12. #27161072C2semidirect
13. #102×D197C2semidirect
14. #27161036C2semidirect
15. #490×C197C2semidirect
20. #20263326C22semidirect
22. #20263325C22semidirect
24. #20263317C22semidirect
37. #20263251C22semidirect
38. #20263251C22semidirect
39. #20263249C22semidirect
40. #20263249C22semidirect
45. #20263247C22semidirect
46. #20263247C22semidirect
49. #20263236C22semidirect
50. C4×#13096802C22semidirect
51. C8×Dic197C22semidirect
53. #125×C197C22semidirect
56. #102×C197C22semidirect
57. #102×C197C22semidirect
58. #97×C197C22semidirect
183. C197#3136semidirect
188. #490D197semidirect
211. #125C2×D197semidirect
237. #102C2×D197semidirect
238. #102C2×D197semidirect
240. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617092
#271616422
#271614422
#271614392
#271614372
#271614362
#271614322
#271614252
#271614242
#271610822
#271610772
#271610722
#102×D1972
#271610362
#490×C1972
#3136197