Finite Group Databasebeta

← #38754666

Subgroups — #38754666

Order 25216.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161729C211yesmaximal, characteristic, complemented
2#27161717C211yesmaximal, characteristic, complemented
3#27161440C211yesmaximal, characteristic, complemented
4#27161408C211yesmaximal, characteristic, complemented
5#27161398C211yesmaximal, characteristic, complemented
6#27161367C211yesmaximal, characteristic, complemented
7#27161330C211yesmaximal, characteristic, complemented
8#27161305C211yesmaximal, characteristic, complemented
9#27161296C211yesmaximal, characteristic, complemented
10#27161211C211yesmaximal, characteristic, complemented
11#27161191C211yesmaximal, characteristic, complemented
12#27161153C211yesmaximal, characteristic, complemented
13#27161080C211yesmaximal, characteristic, complemented
14#102×D197C211yesmaximal, characteristic, complemented
15#449×C197C211yesmaximal, characteristic, complemented
16C2×#15691357—21no—
17C2×D4×D197C2211yescharacteristic
18C2×#15691343C2211yescharacteristic
19C2×#15691342C2211yescharacteristic, complemented
20C2×C8×D197C2211yescharacteristic
21#20263325C2211yescharacteristic, complemented
22#20263315C2211yescharacteristic
23#20263309—21no—
24#20263308—21no—
25#20263307—21no—
26#20263303—21no—
27#20263302C2211yescharacteristic
28#20263300C2211yescharacteristic
29#20263298—21no—
30#20263297C2211yescharacteristic
31#20263294—21no—
32D4×Dic197C2211yescharacteristic
33D4×Dic197C2211yescharacteristic, complemented
34C4×C22⋊D197C2211yescharacteristic
35C2×C4⋊Dic197C2211yescharacteristic, complemented
36C4⋊C4×D197C2211yescharacteristic
37C2×#15691335—21no—
38#20263258C2211yescharacteristic, complemented
39#20263258C2211yescharacteristic, complemented
40#20263257C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263250C2211yescharacteristic, complemented
43#20263250C2211yescharacteristic, complemented
44#20263248C2211yescharacteristic
45#20263248C2211yescharacteristic
46#20263247C2211yescharacteristic
47#20263247C2211yescharacteristic
48#20263247C2211yescharacteristic, complemented
49#20263246C2211yescharacteristic
50#20263238C2211yescharacteristic
51#20263236C2211yescharacteristic, complemented
52C2×#41×C197C2211yescharacteristic
53C2×SD16×C197C2211yescharacteristic, complemented
54C2×D8×C197C2211yescharacteristic
55#121×C197C2211yescharacteristic, complemented
56#103×C197C2211yescharacteristic, complemented
57#102×C197C2211yescharacteristic
58C22⋊C8×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60#15691357—44no—
61C2×Dic394—21no—
62C2×D788C2311yescharacteristic
63C2×C22⋊D197—21no—
64C2×C22⋊D197—21no—
65C2×C22⋊D197—21no—
66C2×C22⋊D197D411yescharacteristic
67D4×D197—22no—
68D4×D197—41no—
69C22×Dic197—21no—
70C22×Dic197—21no—
71C22×Dic197—21no—
72C2×C4×D197—21no—
73C2×C4×D197C2311yescharacteristic
74#15691343—41no—
75#15691342—41no—
76C2×#13096802C2311yescharacteristic
77C2×#13096802C2311yescharacteristic
78C8×D197—41no—
79C4⋊Dic197—21no—
80C4⋊Dic197C2311yescharacteristic
81C4⋊Dic197C2311yescharacteristic
82C4⋊Dic197C2311yescharacteristic
83#15691337—21no—
84#15691337—21no—
85#15691337—21no—
86#15691337D411yescharacteristic
87C22⋊Dic197—21no—
88C22⋊Dic197—21no—
89C22⋊Dic197—21no—
90C22⋊Dic197D411yescharacteristic
91#15691335—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335D411yescharacteristic
96C4×Dic197C2311yescharacteristic
97#41×C197—22no—
98#41×C197—41no—
99C2×Q8×C197C2311yescharacteristic
100C2×D4×C197—21no—
101C2×D4×C197—21no—
102C2×D4×C197C2311yescharacteristic
103C2×D4×C197C2311yescharacteristic
104C22×C4×C197—21no—
105C22×C4×C197C2311yescharacteristic
106SD16×C197—41no—
107D8×C197—22no—
108C2×C8×C197C2311yescharacteristic
109C2×C8×C197C2311yescharacteristic
110C4⋊C4×C197C2311yescharacteristic
111C22⋊C4×C197—21no—
112C22×D197—21no—
113C22×D197—42no—
114C22×D197—42no—
115C22×D197—81no—
116C22×D197C2×D411yescharacteristic
117Dic394—42no—
118D788—21no—
119D788—21no—
120C22⋊D197—22no—
121C22⋊D197—42no—
122C22⋊D197—42no—
123C22⋊D197—81no—
124C2×Dic197—21no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C2×Dic197—42no—
128C2×Dic197—42no—
129C2×Dic197—81no—
130C2×Dic197C2×D411yescharacteristic
131C2×Dic197C2×D411yescharacteristic
132C2×Dic197C2×D411yescharacteristic
133C4×D197—21no—
134C4×D197—21no—
135C4×D197—42no—
136#13096802—21no—
137#13096802—21no—
138C23×C197—21no—
139C23×C197—21no—
140C23×C197C2×D411yescharacteristic
141Q8×C197—21no—
142Q8×C197—21no—
143D4×C197—21no—
144D4×C197—21no—
145D4×C197—21no—
146D4×C197—42no—
147D4×C197—42no—
148D4×C197#4112yescomplemented
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197—42no—
154C2×C4×C197C2×D411yescharacteristic
155C2×C4×C197C2411yescharacteristic
156C8×C197—21no—
157C8×C197—21no—
158C22×C197—21no—
159C22×C197—22no—
160C22×C197—21no—
161C22×C197—21no—
162C22×C197—41no—
163C22×C197—41no—
164C22×C197—41no—
165C22×C197—41no—
166C22×C197—41no—
167C22×C197C22×D411yescharacteristic
168C2×D197—21no—
169C2×D197—21no—
170C2×D197—41no—
171C2×D197—41no—
172C2×D197—41no—
173C2×D197—84no—
174C2×D197AGL(1,ℤ/8)12yes—
175C4×C197—21no—
176C4×C197—41no—
177C4×C197—41no—
178C4×C197C2×#4111yescharacteristic
179C4×C197H(5,𝔽2)11yescharacteristic
180Dic197—21no—
181Dic197—21no—
182Dic197—21no—
183Dic197—41no—
184Dic197—41no—
185Dic197—41no—
186C2×C197—42no—
187C2×C197—41no—
188C2×C197—81no—
189C2×C197#54611yescharacteristic
190C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C197#57711yescharacteristic
192D197—22no—
193D197—81no—
194C197#330611yescharacteristic, complemented, Sylow
195#3306—1971nomaximal, Sylow
196#547—1971no—
197#546—1971no—
198#489—1971no—
199#483—1971no—
200#481—1971no—
201#476—1971no—
202#469—1971no—
203#465—1971no—
204#463—1971no—
205#449D19711yescharacteristic, complemented
206#447—1971no—
207#442—1971no—
208#435—1971no—
209#417—1971no—
210C2×#102—1971no—
211C2×#41—3941no—
212C2×#41C2×D19711yescharacteristic
213C22×D4—1971no—
214C2×SD16C2×D19711yescharacteristic, complemented
215C2×D8—1971no—
216C2×D8C2×D19711yescharacteristic
217C2×M16—1971no—
218C22×C8—1971no—
219C4⋊Q8—1971no—
220#124—1971no—
221#123—3941no—
222#123—3941no—
223#123—3941no—
224C22⋊Q8—3941no—
225C22⋊Q8—1971no—
226#121—3941no—
227#121—1971no—
228#121—1971no—
229#121C2×D19711yescharacteristic, complemented
230#120—3941no—
231C4×D4—1971no—
232C4×D4—1971no—
233C4×D4—1971no—
234C2×C4⋊C4—1971no—
235C2×C4⋊C4—1971no—
236C2×C22⋊C4—3941no—
237#106—1971no—
238#106—1971no—
239#106—1971no—
240C4⋊C8—1971no—
241#103—1971no—
242#103—1971no—
243#103C2×D19711yescharacteristic, complemented
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248#102—1971no—
249#102—1971no—
250#102C2×D19711yescharacteristic
251C22⋊C8—1971no—
252C22⋊C8C2×D19711yescharacteristic, complemented
253#97—1971no—
254C24—3941no—
255#41—22no—
256#41—41no—
257#41—7884no—
258C2×Q8—3941no—
259C2×Q8C22×D19711yescharacteristic
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—7881no—
263C2×D4—3941no—
264C2×D4—3942no—
265C2×D4—3941no—
266C2×D4—3941no—
267C2×D4—1971no—
268C2×D4—1971no—
269C2×D4C22×D19711yescharacteristic
270C2×D4C22×D19711yescharacteristic
271C22×C4—21no—
272C22×C4—3941no—
273C22×C4—3941no—
274C22×C4—3941no—
275C22×C4—3941no—
276C22×C4—1971no—
277C22×C4C22×D19711yescharacteristic
278SD16—41no—
279D8—22no—
280D8—7881no—
281M16—7881no—
282C2×C8—7881no—
283C2×C8—1971no—
284C2×C8—1971no—
285C2×C8C22×D19711yescharacteristic
286C2×C8C22×D19711yescharacteristic
287C4⋊C4—3941no—
288C4⋊C4—3941no—
289C4⋊C4—3941no—
290C4⋊C4—3941no—
291C4⋊C4—1971no—
292C4⋊C4—1971no—
293C4⋊C4—1971no—
294C4⋊C4—1971no—
295C4⋊C4C22×D19711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—3941no—
298C22⋊C4—3941no—
299C22⋊C4—3941no—
300C22⋊C4—3941no—
301C22⋊C4—3941no—
302C22⋊C4—3941no—
303C22⋊C4—3941no—
304C22⋊C4—1971no—
305C22⋊C4—1971no—
306C42—1971no—
307C23—21no—
308C23—21no—
309C23—7882no—
310C23—7882no—
311C23—15761no—
312C23—3941no—
313C23—1971no—
314C23D4×D19711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—7882no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—7882no—
324D4—7882no—
325D4—15761no—
326D4—3942no—
327D4—3941no—
328D4—3941no—
329D4#1569135712yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—7882no—
336C2×C4—7882no—
337C2×C4—7882no—
338C2×C4—15761no—
339C2×C4—3941no—
340C2×C4—3941no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4—1971no—
345C2×C4—1971no—
346C2×C4—1971no—
347C2×C4D4×D19711yescharacteristic
348C2×C4C23×D19711yescharacteristic
349C8—21no—
350C8—21no—
351C8—3941no—
352C8—3941no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—7881no—
363C22—7881no—
364C22—7881no—
365C22—15764no—
366C22—3941no—
367C22—3941no—
368C22—1972no—
369C22C2×D4×D19711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—7881no—
374C4—7881no—
375C4—7881no—
376C4—3941no—
377C4—3941no—
378C4—3941no—
379C4C2×#1569135711yescharacteristic
380C4#2026337111yescharacteristic
381C2—41no—
382C2—42no—
383C2—81no—
384C2—15761no—
385C2—3942no—
386C2#2716171611yescharacteristic
387C2AGL(1,ℤ/8)×D19711yescharacteristic
388C2#2716188611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#449×C197C2
socleC22×C197C22×D4
hypercenter#449D197
hypocenterC197#3306
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #27161729C2semidirect
2. #27161717C2semidirect
3. #27161440C2semidirect
4. #27161408C2semidirect
5. #27161398C2semidirect
6. #27161367C2semidirect
7. #27161330C2semidirect
8. #27161305C2semidirect
9. #27161296C2semidirect
10. #27161211C2semidirect
11. #27161191C2semidirect
12. #27161153C2semidirect
13. #27161080C2semidirect
14. #102×D197C2semidirect
15. #449×C197C2semidirect
19. C2×#15691342C22semidirect
21. #20263325C22semidirect
33. D4×Dic197C22semidirect
35. C2×C4⋊Dic197C22semidirect
38. #20263258C22semidirect
39. #20263258C22semidirect
40. #20263257C22semidirect
41. #20263255C22semidirect
42. #20263250C22semidirect
43. #20263250C22semidirect
48. #20263247C22semidirect
51. #20263236C22semidirect
53. C2×SD16×C197C22semidirect
55. #121×C197C22semidirect
56. #103×C197C22semidirect
58. C22⋊C8×C197C22semidirect
148. D4×C197#41semidirect
194. C197#3306semidirect
205. #449D197semidirect
214. C2×SD16C2×D197semidirect
229. #121C2×D197semidirect
243. #103C2×D197semidirect
252. C22⋊C8C2×D197semidirect
329. D4#15691357semidirect

Maximal subgroups

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

subgroupindex
#271617292
#271617172
#271614402
#271614082
#271613982
#271613672
#271613302
#271613052
#271612962
#271612112
#271611912
#271611532
#271610802
#102×D1972
#449×C1972
#3306197