Finite Group Databasebeta

← #38754544

Subgroups — #38754544

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#27161767C211yesmaximal, characteristic, complemented
2#27161724C211yesmaximal, characteristic
3#27161436C211yesmaximal, characteristic, complemented
4#27161411C211yesmaximal, characteristic, complemented
5#27161406C211yesmaximal, characteristic, complemented
6#27161321C211yesmaximal, characteristic, complemented
7#27161309C211yesmaximal, characteristic, complemented
8#27161302C211yesmaximal, characteristic, complemented
9#27161242C211yesmaximal, characteristic, complemented
10#27161231C211yesmaximal, characteristic, complemented
11C4×#15691346C211yesmaximal, characteristic, complemented
12#27161144C211yesmaximal, characteristic, complemented
13#27161072C211yesmaximal, characteristic, complemented
14#102×D197C211yesmaximal, characteristic, complemented
15#478×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic
17C2×D4×D197C2211yescharacteristic
18C2×Dic788C2211yescharacteristic, complemented
19C2×#15691347C2211yescharacteristic
20C2×#15691346C2211yescharacteristic
21C2×C8×D197C2211yescharacteristic
22C2×C8×D197C2211yescharacteristic, complemented
23#20263321—21no—
24#20263317—21no—
25#20263314C2211yescharacteristic
26#20263311—21no—
27#20263310—21no—
28#20263306C2211yescharacteristic
29#20263306C2211yescharacteristic
30#20263306C2211yescharacteristic, complemented
31#20263301C2211yescharacteristic
32#20263300—21no—
33#20263296—21no—
34C4×Dic394C2211yescharacteristic, complemented
35C4×D788C2211yescharacteristic
36#20263284—21no—
37#20263284C2211yescharacteristic, complemented
38C4⋊C4×D197C2211yescharacteristic
39C2×#15691335—21no—
40#20263257C2211yescharacteristic, complemented
41#20263251C2211yescharacteristic, complemented
42#20263249C2211yescharacteristic, complemented
43#20263249C2211yescharacteristic, complemented
44#20263249C2211yescharacteristic, complemented
45#20263248C2211yescharacteristic
46#20263247C2211yescharacteristic
47#20263246C2211yescharacteristic
48#20263246C2211yescharacteristic
49#20263238C2211yescharacteristic
50#20263238C2211yescharacteristic, complemented
51C4×#13096802C2211yescharacteristic, complemented
52#125×C197C2211yescharacteristic, complemented
53C4×D4×C197C2211yescharacteristic
54#107×C197C2211yescharacteristic, complemented
55#106×C197C2211yescharacteristic, complemented
56C4⋊C8×C197C2211yescharacteristic, complemented
57#102×C197C2211yescharacteristic
58#102×C197C2211yescharacteristic
59C23×D197—21no—
60#15691357—22no—
61#15691357—41no—
62C2×Dic394C2311yescharacteristic
63C2×Dic394C2311yescharacteristic
64C2×D788—21no—
65C2×D788D411yescharacteristic
66C2×D788C2311yescharacteristic
67C2×C22⋊D197—21no—
68C2×C22⋊D197—21no—
69D4×D197—22no—
70D4×D197—41no—
71C22×Dic197—21no—
72C2×C4×D197—21no—
73C2×C4×D197C2311yescharacteristic
74C2×C4×D197C2311yescharacteristic
75Dic788—41no—
76#15691347—41no—
77#15691346—22no—
78C2×#13096802C2311yescharacteristic
79C2×#13096802C2311yescharacteristic
80C8×D197—41no—
81C8×D197—41no—
82C4⋊Dic197—21no—
83C4⋊Dic197—21no—
84C4⋊Dic197—21no—
85C4⋊Dic197D411yescharacteristic
86C4⋊Dic197C2311yescharacteristic
87#15691337—21no—
88#15691337—21no—
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335—21no—
96#15691335D411yescharacteristic
97#15691335D411yescharacteristic
98C4×Dic197—21no—
99C4×Dic197—21no—
100C2×D4×C197C2311yescharacteristic
101C22×C4×C197—21no—
102C2×C8×C197C2311yescharacteristic
103C2×C8×C197C2311yescharacteristic
104C4⋊C4×C197—21no—
105C4⋊C4×C197—21no—
106C4⋊C4×C197C2311yescharacteristic
107C4⋊C4×C197C2311yescharacteristic
108C4⋊C4×C197C2311yescharacteristic
109C22⋊C4×C197—21no—
110C42×C197C2311yescharacteristic
111C22×D197—21no—
112C22×D197—42no—
113C22×D197—42no—
114C22×D197—81no—
115C22×D197C2×D411yescharacteristic
116C22×D197C2×D411yescharacteristic
117Dic394—21no—
118Dic394—21no—
119Dic394—21no—
120Dic394—21no—
121D788—22no—
122D788—21no—
123D788—21no—
124D788—81no—
125C22⋊D197—42no—
126C22⋊D197—42no—
127C2×Dic197—21no—
128C2×Dic197—21no—
129C2×Dic197—21no—
130C2×Dic197—42no—
131C2×Dic197C2×D411yescharacteristic
132C2×Dic197C2×D411yescharacteristic
133C4×D197—21no—
134C4×D197—21no—
135C4×D197—21no—
136C4×D197—21no—
137C4×D197—42no—
138#13096802—21no—
139#13096802—21no—
140C23×C197—21no—
141D4×C197—21no—
142D4×C197#4112yes—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—81no—
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2×D411yescharacteristic
149C2×C4×C197C2411yescharacteristic
150C8×C197—21no—
151C8×C197—21no—
152C22×C197—22no—
153C22×C197—41no—
154C22×C197—41no—
155C22×C197C22×D411yescharacteristic
156C2×D197—21no—
157C2×D197—21no—
158C2×D197—21no—
159C2×D197—21no—
160C2×D197—21no—
161C2×D197—41no—
162C2×D197—41no—
163C2×D197—41no—
164C2×D197—84no—
165C2×D197#13512yes—
166C4×C197—21no—
167C4×C197—21no—
168C4×C197—41no—
169C4×C197—41no—
170C4×C197—41no—
171C4×C197C2×#4111yescharacteristic
172C4×C197H(5,𝔽2)11yescharacteristic
173Dic197—21no—
174Dic197—21no—
175Dic197—41no—
176Dic197—41no—
177Dic197—41no—
178C2×C197—42no—
179C2×C197#54611yescharacteristic
180C2×C197C2×#13511yescharacteristic
181C2×C197#57711yescharacteristic
182D197—22no—
183D197—41no—
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#478D19711yescharacteristic, complemented
193#467—1971no—
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×Q16—1971no—
205C2×SD16—1971no—
206C2×SD16—1971no—
207C22×C8—1971no—
208C22×C8—1971no—
209#126—3941no—
210#125—3941no—
211#125C2×D19711yescharacteristic, complemented
212#124—1971no—
213#123—3941no—
214#123—3941no—
215C22⋊Q8—1971no—
216C22⋊Q8—1971no—
217C22⋊Q8—1971no—
218#121—3941no—
219#121—1971no—
220#120—3941no—
221C4×Q8—1971no—
222C4×D4—1971no—
223C4×D4C2×D19711yescharacteristic
224#117—3941no—
225#117—1971no—
226C2×C4⋊C4—1971no—
227C2×C22⋊C4—3941no—
228#107C2×D19711yescharacteristic, complemented
229#106—1971no—
230#106C2×D19711yescharacteristic, complemented
231C4⋊C8C2×D19711yescharacteristic, complemented
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⋊C8—1971no—
244C4×C8—1971no—
245C24—3941no—
246#41—7881no—
247#41—3942no—
248C2×Q8—1971no—
249C2×Q8—1971no—
250C2×D4—7881no—
251C2×D4—3941no—
252C2×D4—3941no—
253C2×D4—3942no—
254C2×D4—3941no—
255C2×D4—1971no—
256C2×D4—1971no—
257C2×D4C22×D19711yescharacteristic
258C22×C4—21no—
259C22×C4—3941no—
260C22×C4—3941no—
261C22×C4—1971no—
262C22×C4—1971no—
263Q16—7881no—
264SD16—7881no—
265SD16—3942no—
266C2×C8—7881no—
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⋊C4C22×D19711yescharacteristic
282C4⋊C4C22×D19711yescharacteristic
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—
296C42C22×D19711yescharacteristic
297C23—21no—
298C23—7882no—
299C23—7882no—
300C23—15761no—
301C23—3941no—
302C23—1971no—
303C23—1971no—
304Q8—3941no—
305Q8—3941no—
306Q8—3941no—
307Q8—3941no—
308D4—21no—
309D4—7882no—
310D4—7882no—
311D4—15761no—
312D4—3942no—
313D4—3941no—
314D4—3941no—
315D4#1569135912yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
320C2×C4—7882no—
321C2×C4—7882no—
322C2×C4—3941no—
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×C4D4×D19711yescharacteristic
332C2×C4D4×D19711yescharacteristic
333C2×C4C23×D19711yescharacteristic
334C8—21no—
335C8—21no—
336C8—3941no—
337C8—3941no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—7881no—
342C22—7881no—
343C22—7881no—
344C22—15764no—
345C22—3941no—
346C22—3941no—
347C22—3941no—
348C22—3941no—
349C22—3941no—
350C22—1972no—
351C22C2×D4×D19711yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—7881no—
358C4—7881no—
359C4—7881no—
360C4—3941no—
361C4—3941no—
362C4C2×#1569135911yescharacteristic
363C4#2026337211yescharacteristic
364C2—42no—
365C2—7881no—
366C2—15761no—
367C2—3942no—
368C2#2716172011yescharacteristic
369C2#135×D19711yescharacteristic
370C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#478×C197C2
socleC22×C197C22×D4
prime radical#27161724C2
hypercenter#478D197
hypocenterC197#3296
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #27161767C2semidirect
3. #27161436C2semidirect
4. #27161411C2semidirect
5. #27161406C2semidirect
6. #27161321C2semidirect
7. #27161309C2semidirect
8. #27161302C2semidirect
9. #27161242C2semidirect
10. #27161231C2semidirect
11. C4×#15691346C2semidirect
12. #27161144C2semidirect
13. #27161072C2semidirect
14. #102×D197C2semidirect
15. #478×C197C2semidirect
18. C2×Dic788C22semidirect
22. C2×C8×D197C22semidirect
30. #20263306C22semidirect
34. C4×Dic394C22semidirect
37. #20263284C22semidirect
40. #20263257C22semidirect
41. #20263251C22semidirect
42. #20263249C22semidirect
43. #20263249C22semidirect
44. #20263249C22semidirect
50. #20263238C22semidirect
51. C4×#13096802C22semidirect
52. #125×C197C22semidirect
54. #107×C197C22semidirect
55. #106×C197C22semidirect
56. C4⋊C8×C197C22semidirect
185. C197#3296semidirect
192. #478D197semidirect
211. #125C2×D197semidirect
228. #107C2×D197semidirect
230. #106C2×D197semidirect
231. C4⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617672
#271617242
#271614362
#271614112
#271614062
#271613212
#271613092
#271613022
#271612422
#271612312
C4×#156913462
#271611442
#271610722
#102×D1972
#478×C1972
#3296197