Finite Group Databasebeta

← #38754551

Subgroups — #38754551

Order 25216.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161771C211yesmaximal, characteristic, complemented
2#27161723C211yesmaximal, characteristic
3#27161432C211yesmaximal, characteristic, complemented
4#27161410C211yesmaximal, characteristic, complemented
5#27161402C211yesmaximal, characteristic, complemented
6#27161381C211yesmaximal, characteristic, complemented
7#27161318C211yesmaximal, characteristic, complemented
8#27161305C211yesmaximal, characteristic, complemented
9#27161299C211yesmaximal, characteristic, complemented
10#27161238C211yesmaximal, characteristic, complemented
11SD16×Dic197C211yesmaximal, characteristic, complemented
12#27161142C211yesmaximal, characteristic, complemented
13#27161072C211yesmaximal, characteristic, complemented
14C2×#20263247C211yesmaximal, characteristic, complemented
15#450×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic
17C2×#15691348C2211yescharacteristic, complemented
18C22×#13096802C2211yescharacteristic
19C2×C8×D197C2211yescharacteristic, complemented
20#20263321—21no—
21#20263317—21no—
22#20263317C2211yescharacteristic, complemented
23#20263313C2211yescharacteristic
24#20263310—21no—
25#20263307—21no—
26#20263305C2211yescharacteristic
27#20263302C2211yescharacteristic, complemented
28#20263298—21no—
29#20263297C2211yescharacteristic
30#20263295—21no—
31Q8×Dic197C2211yescharacteristic, complemented
32D4×Dic197C2211yescharacteristic
33C4×C22⋊D197C2211yescharacteristic
34#20263284C2211yescharacteristic, complemented
35#20263282—21no—
36C2×C4⋊Dic197C2211yescharacteristic
37C2×C22⋊Dic197—21no—
38#20263259C2211yescharacteristic, complemented
39#20263258C2211yescharacteristic, complemented
40#20263258C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263251C2211yescharacteristic, complemented
43#20263251C2211yescharacteristic, complemented
44#20263250C2211yescharacteristic, complemented
45#20263249C2211yescharacteristic, complemented
46#20263247C2211yescharacteristic
47#20263247C2211yescharacteristic
48#20263247C2211yescharacteristic
49#20263247C2211yescharacteristic
50#20263238C2211yescharacteristic, complemented
51C8×Dic197C2211yescharacteristic, complemented
52C22×D4×C197C2211yescharacteristic
53C2×SD16×C197C2211yescharacteristic
54C2×SD16×C197C2211yescharacteristic
55C22⋊Q8×C197C2211yescharacteristic
56#102×C197C2211yescharacteristic
57#102×C197C2211yescharacteristic
58C22⋊C8×C197C2211yescharacteristic
59#15691357—22no—
60#15691357—41no—
61C2×Dic394C2311yescharacteristic
62C2×C22⋊D197—21no—
63C2×C22⋊D197—21no—
64C2×C22⋊D197D411yescharacteristic
65C22×Dic197—21no—
66C22×Dic197—21no—
67C22×Dic197—21no—
68C2×C4×D197C2311yescharacteristic
69#15691348—41no—
70C2×#13096802—41no—
71C2×#13096802C2311yescharacteristic
72C2×#13096802C2311yescharacteristic
73C8×D197—41no—
74C4⋊Dic197—21no—
75C4⋊Dic197—21no—
76C4⋊Dic197—21no—
77C4⋊Dic197C2311yescharacteristic
78C4⋊Dic197C2311yescharacteristic
79C4⋊Dic197C2311yescharacteristic
80#15691337—21no—
81#15691337—21no—
82#15691337—21no—
83#15691337D411yescharacteristic
84C22⋊Dic197—21no—
85C22⋊Dic197—21no—
86C22⋊Dic197—21no—
87C22⋊Dic197—21no—
88C22⋊Dic197—21no—
89C22⋊Dic197D411yescharacteristic
90#15691335—21no—
91#15691335—21no—
92#15691335D411yescharacteristic
93C4×Dic197—21no—
94C4×Dic197—21no—
95C4×Dic197C2311yescharacteristic
96C24×C197—21no—
97C2×Q8×C197C2311yescharacteristic
98C2×D4×C197—22no—
99C2×D4×C197—21no—
100C2×D4×C197—41no—
101C2×D4×C197C2311yescharacteristic
102C2×D4×C197C2311yescharacteristic
103C22×C4×C197C2311yescharacteristic
104SD16×C197—22no—
105SD16×C197—41no—
106C2×C8×C197C2311yescharacteristic
107C2×C8×C197C2311yescharacteristic
108C4⋊C4×C197—21no—
109C4⋊C4×C197C2311yescharacteristic
110C22⋊C4×C197—21no—
111C22×D197C2×D411yescharacteristic
112Dic394—21no—
113Dic394—21no—
114C22⋊D197—22no—
115C22⋊D197—42no—
116C22⋊D197—81no—
117C2×Dic197—21no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197—42no—
122C2×Dic197—42no—
123C2×Dic197—81no—
124C2×Dic197C2×D411yescharacteristic
125C2×Dic197C2×D411yescharacteristic
126C2×Dic197C2×D411yescharacteristic
127C4×D197—21no—
128C4×D197—21no—
129#13096802—21no—
130#13096802—21no—
131C23×C197—21no—
132C23×C197—21no—
133C23×C197—42no—
134C23×C197—42no—
135C23×C197—81no—
136C23×C197C2×D411yescharacteristic
137Q8×C197—21no—
138Q8×C197—21no—
139D4×C197—21no—
140D4×C197—21no—
141D4×C197—21no—
142D4×C197—42no—
143D4×C197#4112yes—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197C2×D411yescharacteristic
149C2×C4×C197C2411yescharacteristic
150C8×C197—21no—
151C8×C197—21no—
152C22×C197—21no—
153C22×C197—21no—
154C22×C197—22no—
155C22×C197—41no—
156C22×C197—41no—
157C22×C197—41no—
158C22×C197—41no—
159C22×C197—41no—
160C22×C197—84no—
161C22×C197#13512yes—
162C22×C197C22×D411yescharacteristic
163C2×D197—21no—
164C2×D197—21no—
165C2×D197—21no—
166C4×C197—21no—
167C4×C197—41no—
168C4×C197—41no—
169C4×C197C2×#4111yescharacteristic
170C4×C197H(5,𝔽2)11yescharacteristic
171Dic197—21no—
172Dic197—21no—
173Dic197—21no—
174Dic197—41no—
175Dic197—41no—
176Dic197—41no—
177Dic197—41no—
178C2×C197—22no—
179C2×C197—42no—
180C2×C197—81no—
181C2×C197#54611yescharacteristic
182C2×C197C2×#13511yescharacteristic
183C2×C197#57711yescharacteristic
184D197—41no—
185C197#329611yescharacteristic, complemented, Sylow
186#3296—1971nomaximal, Sylow
187#552—1971no—
188#546—1971no—
189#488—1971no—
190#483—1971no—
191#482—1971no—
192#478—1971no—
193#467—1971no—
194#465—1971no—
195#464—1971no—
196#452—1971no—
197#450D19711yescharacteristic, complemented
198C4×SD16—1971no—
199C8×D4—1971no—
200#416—1971no—
201C2×#102—1971no—
202C2×#41—1971no—
203C22×D4C2×D19711yescharacteristic
204C2×Q16—1971no—
205C2×SD16C2×D19711yescharacteristic
206C2×SD16C2×D19711yescharacteristic
207C22×C8—1971no—
208C22×C8—1971no—
209#126—3941no—
210#125—3941no—
211#125—1971no—
212#124—1971no—
213#123—3941no—
214#123—3941no—
215C22⋊Q8—1971no—
216C22⋊Q8—1971no—
217C22⋊Q8C2×D19711yescharacteristic
218#121—3941no—
219#121—1971no—
220#120—3941no—
221C4×Q8—1971no—
222C4×D4—1971no—
223C4×D4—1971no—
224#117—3941no—
225#117—1971no—
226C2×C4⋊C4—1971no—
227C2×C22⋊C4—3941no—
228#107—1971no—
229#106—1971no—
230#106—1971no—
231C4⋊C8—1971no—
232#103—1971no—
233#103—1971no—
234#103—1971no—
235#103—1971no—
236#102—1971no—
237#102—1971no—
238#102—1971no—
239#102—1971no—
240#102C2×D19711yescharacteristic
241#102C2×D19711yescharacteristic
242C22⋊C8—1971no—
243C22⋊C8C2×D19711yescharacteristic
244C4×C8—1971no—
245C24—21no—
246#41—7881no—
247#41—3942no—
248C2×Q8—1971no—
249C2×Q8C22×D19711yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—3941no—
254C2×D4—3941no—
255C2×D4—1971no—
256C2×D4C22×D19711yescharacteristic
257C2×D4C22×D19711yescharacteristic
258C22×C4—3941no—
259C22×C4—3941no—
260C22×C4—3941no—
261C22×C4—1971no—
262C22×C4C22×D19711yescharacteristic
263Q16—7881no—
264SD16—22no—
265SD16—41no—
266C2×C8—7881no—
267C2×C8—7881no—
268C2×C8—1971no—
269C2×C8—1971no—
270C2×C8C22×D19711yescharacteristic
271C2×C8C22×D19711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—3941no—
274C4⋊C4—3941no—
275C4⋊C4—3941no—
276C4⋊C4—3941no—
277C4⋊C4—3941no—
278C4⋊C4—3941no—
279C4⋊C4—1971no—
280C4⋊C4—1971no—
281C4⋊C4—1971no—
282C4⋊C4—1971no—
283C4⋊C4C22×D19711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—3941no—
286C22⋊C4—3941no—
287C22⋊C4—3941no—
288C22⋊C4—3941no—
289C22⋊C4—3941no—
290C22⋊C4—3941no—
291C22⋊C4—3941no—
292C22⋊C4—1971no—
293C22⋊C4—1971no—
294C42—3941no—
295C42—3941no—
296C42—1971no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—1971no—
303C23D4×D19711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3941no—
307Q8—3941no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—7882no—
313D4—15761no—
314D4—3942no—
315D4#1569135712yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—7882no—
321C2×C4—7882no—
322C2×C4—15761no—
323C2×C4—3941no—
324C2×C4—3941no—
325C2×C4—3941no—
326C2×C4—3941no—
327C2×C4—3941no—
328C2×C4—3941no—
329C2×C4—1971no—
330C2×C4—1971no—
331C2×C4—1971no—
332C2×C4D4×D19711yescharacteristic
333C2×C4C23×D19711yescharacteristic
334C8—21no—
335C8—21no—
336C8—3941no—
337C8—3941no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—3941no—
348C22—3941no—
349C22—3941no—
350C22#2026334712yes—
351C22C2×D4×D19711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—7881no—
356C4—7881no—
357C4—7881no—
358C4—7881no—
359C4—3941no—
360C4—3941no—
361C4—3941no—
362C4C2×#1569135711yescharacteristic
363C4#2026337111yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—7881no—
368C2#2716171611yescharacteristic
369C2C2×#2026334711yescharacteristic
370C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#450×C197C2
socleC22×C197C22×D4
prime radical#27161723C2
hypercenter#450D197
hypocenterC197#3296
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #27161771C2semidirect
3. #27161432C2semidirect
4. #27161410C2semidirect
5. #27161402C2semidirect
6. #27161381C2semidirect
7. #27161318C2semidirect
8. #27161305C2semidirect
9. #27161299C2semidirect
10. #27161238C2semidirect
11. SD16×Dic197C2semidirect
12. #27161142C2semidirect
13. #27161072C2semidirect
14. C2×#20263247C2semidirect
15. #450×C197C2semidirect
17. C2×#15691348C22semidirect
19. C2×C8×D197C22semidirect
22. #20263317C22semidirect
27. #20263302C22semidirect
31. Q8×Dic197C22semidirect
34. #20263284C22semidirect
38. #20263259C22semidirect
39. #20263258C22semidirect
40. #20263258C22semidirect
41. #20263255C22semidirect
42. #20263251C22semidirect
43. #20263251C22semidirect
44. #20263250C22semidirect
45. #20263249C22semidirect
50. #20263238C22semidirect
51. C8×Dic197C22semidirect
185. C197#3296semidirect
197. #450D197semidirect

Maximal subgroups

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

subgroupindex
#271617712
#271617232
#271614322
#271614102
#271614022
#271613812
#271613182
#271613052
#271612992
#271612382
SD16×Dic1972
#271611422
#271610722
C2×#202632472
#450×C1972
#3296197