Finite Group Databasebeta

← #38754553

Subgroups — #38754553

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#27161720C211yesmaximal, characteristic
3#27161435C211yesmaximal, characteristic, complemented
4#27161407C211yesmaximal, characteristic, complemented
5#27161406C211yesmaximal, characteristic, complemented
6#27161382C211yesmaximal, characteristic, complemented
7#27161311C211yesmaximal, characteristic, complemented
8#27161303C211yesmaximal, characteristic, complemented
9#27161240C211yesmaximal, characteristic, complemented
10#27161229C211yesmaximal, characteristic, complemented
11#27161164C211yesmaximal, characteristic, complemented
12C8×C22⋊D197C211yesmaximal, characteristic, complemented
13#27161074C211yesmaximal, characteristic, complemented
14C2×#20263248C211yesmaximal, characteristic, complemented
15#467×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C2×#15691345C2211yescharacteristic
19C2×#15691345C2211yescharacteristic
20C2×C8×D197C2211yescharacteristic, complemented
21#20263321—21no—
22#20263317—21no—
23#20263317C2211yescharacteristic, complemented
24#20263315C2211yescharacteristic
25#20263311—21no—
26#20263310—21no—
27#20263302C2211yescharacteristic, complemented
28#20263301—21no—
29#20263301C2211yescharacteristic
30#20263296—21no—
31Q8×Dic197C2211yescharacteristic, complemented
32#20263288C2211yescharacteristic
33C4×C22⋊D197C2211yescharacteristic
34#20263284C2211yescharacteristic, complemented
35#20263282—21no—
36C2×C4⋊Dic197C2211yescharacteristic
37C22⋊C4×D197—21no—
38#20263257C2211yescharacteristic, complemented
39#20263257C2211yescharacteristic, complemented
40#20263255C2211yescharacteristic, complemented
41#20263250C2211yescharacteristic, complemented
42#20263250C2211yescharacteristic, complemented
43#20263248C2211yescharacteristic
44#20263248C2211yescharacteristic
45#20263248C2211yescharacteristic
46#20263248C2211yescharacteristic
47#20263246C2211yescharacteristic
48#20263246C2211yescharacteristic
49C22⋊#13096802C2211yescharacteristic
50#20263238C2211yescharacteristic, complemented
51C8×Dic197C2211yescharacteristic, complemented
52C2×Q16×C197C2211yescharacteristic, complemented
53C22×C8×C197C2211yescharacteristic
54C22⋊Q8×C197C2211yescharacteristic
55C22⋊Q8×C197C2211yescharacteristic
56#107×C197C2211yescharacteristic, complemented
57#103×C197C2211yescharacteristic, complemented
58#103×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60#15691359—22no—
61#15691359—41no—
62C2×D788—22no—
63C2×D788—21no—
64C2×D788—21no—
65C2×D788—21no—
66C2×D788—41no—
67C2×D788C2311yescharacteristic
68C2×D788C2311yescharacteristic
69C2×C22⋊D197D411yescharacteristic
70C22×Dic197—21no—
71C2×C4×D197—21no—
72C2×C4×D197—21no—
73C2×C4×D197C2311yescharacteristic
74#15691345—22no—
75#15691345—41no—
76C2×#13096802C2311yescharacteristic
77C2×#13096802C2311yescharacteristic
78C8×D197—41no—
79C4⋊Dic197—21no—
80C4⋊Dic197—21no—
81C4⋊Dic197—21no—
82C4⋊Dic197C2311yescharacteristic
83C4⋊Dic197C2311yescharacteristic
84#15691337—21no—
85#15691337—21no—
86#15691337D411yescharacteristic
87C22⋊Dic197D411yescharacteristic
88#15691335—21no—
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335D411yescharacteristic
95C4×Dic197—21no—
96C4×Dic197—21no—
97C4×Dic197C2311yescharacteristic
98C2×Q8×C197C2311yescharacteristic
99C2×Q8×C197C2311yescharacteristic
100C22×C4×C197C2311yescharacteristic
101Q16×C197—41no—
102C2×C8×C197—41no—
103C2×C8×C197C2311yescharacteristic
104C2×C8×C197C2311yescharacteristic
105C4⋊C4×C197—21no—
106C4⋊C4×C197—21no—
107C4⋊C4×C197C2311yescharacteristic
108C4⋊C4×C197C2311yescharacteristic
109C22⋊C4×C197—21no—
110C22⋊C4×C197—21no—
111C22×D197—21no—
112C22×D197—21no—
113C22×D197—42no—
114C22×D197—42no—
115C22×D197—81no—
116C22×D197C2×D411yescharacteristic
117D788—21no—
118D788—21no—
119D788—21no—
120D788—42no—
121D788—42no—
122D788—81no—
123D788#4112yes—
124C22⋊D197—22no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C2×Dic197—42no—
128C2×Dic197C2×D411yescharacteristic
129C2×Dic197C2×D411yescharacteristic
130C2×Dic197C2×D411yescharacteristic
131C4×D197—21no—
132C4×D197—21no—
133C4×D197—42no—
134C4×D197—81no—
135#13096802—21no—
136#13096802—21no—
137C23×C197C2×D411yescharacteristic
138Q8×C197—21no—
139Q8×C197—21no—
140Q8×C197—21no—
141Q8×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
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#13512yes—
155C22×C197C22×D411yescharacteristic
156C2×D197—21no—
157C2×D197—22no—
158C2×D197—21no—
159C2×D197—21no—
160C2×D197—41no—
161C2×D197—41no—
162C2×D197—41no—
163C2×D197—41no—
164C2×D197—41no—
165C2×D197—84no—
166C4×C197—21no—
167C4×C197—41no—
168C4×C197—41no—
169C4×C197—41no—
170C4×C197—41no—
171C4×C197C2×#4111yescharacteristic
172C4×C197H(5,𝔽2)11yescharacteristic
173Dic197—21no—
174Dic197—21no—
175Dic197—21no—
176Dic197—41no—
177Dic197—41no—
178C2×C197—22no—
179C2×C197#54611yescharacteristic
180C2×C197C2×#13511yescharacteristic
181C2×C197#57711yescharacteristic
182D197—41no—
183D197—42no—
184D197—81no—
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#467D19711yescharacteristic, complemented
194#465—1971no—
195#464—1971no—
196#452—1971no—
197#450—1971no—
198C4×SD16—1971no—
199C8×D4—1971no—
200#416—1971no—
201C2×#102—1971no—
202C2×#41—1971no—
203C22×D4—1971no—
204C2×Q16C2×D19711yescharacteristic, complemented
205C2×SD16—1971no—
206C2×SD16—1971no—
207C22×C8—1971no—
208C22×C8C2×D19711yescharacteristic
209#126—3941no—
210#125—3941no—
211#125—1971no—
212#124—1971no—
213#123—3941no—
214#123—3941no—
215C22⋊Q8—1971no—
216C22⋊Q8C2×D19711yescharacteristic
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#107C2×D19711yescharacteristic, complemented
229#106—1971no—
230#106—1971no—
231C4⋊C8—1971no—
232#103—1971no—
233#103—1971no—
234#103C2×D19711yescharacteristic, complemented
235#103C2×D19711yescharacteristic, complemented
236#102—1971no—
237#102—1971no—
238#102—1971no—
239#102—1971no—
240#102—1971no—
241#102—1971no—
242C22⋊C8—1971no—
243C22⋊C8—1971no—
244C4×C8—1971no—
245C24—3941no—
246#41—7881no—
247#41—3942no—
248C2×Q8C22×D19711yescharacteristic
249C2×Q8C22×D19711yescharacteristic
250C2×D4—7881no—
251C2×D4—3941no—
252C2×D4—3942no—
253C2×D4—3941no—
254C2×D4—3941no—
255C2×D4—1971no—
256C2×D4—1971no—
257C2×D4—1971no—
258C22×C4—3941no—
259C22×C4—3941no—
260C22×C4—3941no—
261C22×C4—1971no—
262C22×C4C22×D19711yescharacteristic
263Q16—41no—
264SD16—7881no—
265SD16—3942no—
266C2×C8—41no—
267C2×C8—7881no—
268C2×C8—1971no—
269C2×C8—1971no—
270C2×C8C22×D19711yescharacteristic
271C2×C8C22×D19711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
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⋊C4C22×D19711yescharacteristic
283C4⋊C4C22×D19711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
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—7882no—
298C23—7882no—
299C23—15761no—
300C23—3941no—
301C23—3941no—
302C23—1971no—
303C23D4×D19711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—7882no—
309D4—7882no—
310D4—15761no—
311D4—3942no—
312D4—3941no—
313D4—3941no—
314D4—3941no—
315D4—1972no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—7882no—
323C2×C4—7882no—
324C2×C4—15761no—
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—7881no—
341C22—7881no—
342C22—7881no—
343C22—7881no—
344C22—7881no—
345C22—15764no—
346C22—3941no—
347C22—3941no—
348C22—3942no—
349C22—3941no—
350C22#2026334912yes—
351C22C2×D4×D19711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—7881no—
358C4—7881no—
359C4—3941no—
360C4—3941no—
361C4—3941no—
362C4#41×D19711yescharacteristic
363C4#2026337111yescharacteristic
364C2—22no—
365C2—7882no—
366C2—7881no—
367C2—15761no—
368C2#2716171811yescharacteristic
369C2C2×#2026334911yescharacteristic
370C2#2716188611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#467×C197C2
socleC22×C197C22×D4
prime radical#27161720C2
hypercenter#467D197
hypocenterC197#3296
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #27161771C2semidirect
3. #27161435C2semidirect
4. #27161407C2semidirect
5. #27161406C2semidirect
6. #27161382C2semidirect
7. #27161311C2semidirect
8. #27161303C2semidirect
9. #27161240C2semidirect
10. #27161229C2semidirect
11. #27161164C2semidirect
12. C8×C22⋊D197C2semidirect
13. #27161074C2semidirect
14. C2×#20263248C2semidirect
15. #467×C197C2semidirect
20. C2×C8×D197C22semidirect
23. #20263317C22semidirect
27. #20263302C22semidirect
31. Q8×Dic197C22semidirect
34. #20263284C22semidirect
38. #20263257C22semidirect
39. #20263257C22semidirect
40. #20263255C22semidirect
41. #20263250C22semidirect
42. #20263250C22semidirect
50. #20263238C22semidirect
51. C8×Dic197C22semidirect
52. C2×Q16×C197C22semidirect
56. #107×C197C22semidirect
57. #103×C197C22semidirect
58. #103×C197C22semidirect
185. C197#3296semidirect
193. #467D197semidirect
204. C2×Q16C2×D197semidirect
228. #107C2×D197semidirect
234. #103C2×D197semidirect
235. #103C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617712
#271617202
#271614352
#271614072
#271614062
#271613822
#271613112
#271613032
#271612402
#271612292
#271611642
C8×C22⋊D1972
#271610742
C2×#202632482
#467×C1972
#3296197