Finite Group Databasebeta

← #38754754

Subgroups — #38754754

Order 25216.

402 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#27161800C211yesmaximal, characteristic, complemented
2#27161726C211yesmaximal, characteristic
3#27161437C211yesmaximal, characteristic, complemented
4#27161423C211yesmaximal, characteristic, complemented
5#27161400C211yesmaximal, characteristic, complemented
6#27161368C211yesmaximal, characteristic, complemented
7#27161336C211yesmaximal, characteristic, complemented
8#27161299C211yesmaximal, characteristic, complemented
9#27161239C211yesmaximal, characteristic, complemented
10#27161228C211yesmaximal, characteristic, complemented
11#27161182C211yesmaximal, characteristic, complemented
12#27161150C211yesmaximal, characteristic, complemented
13#27161082C211yesmaximal, characteristic, complemented
14#27161076C211yesmaximal, characteristic, complemented
15#465×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C2×D4×D197C2211yescharacteristic
18C2×#15691348C2211yescharacteristic, complemented
19C2×#15691345C2211yescharacteristic
20C2×#15691341C2211yescharacteristic
21#20263325C2211yescharacteristic, complemented
22#20263321—21no—
23#20263319—21no—
24#20263313C2211yescharacteristic
25#20263312—21no—
26#20263308—21no—
27#20263306C2211yescharacteristic
28#20263302C2211yescharacteristic
29#20263297—21no—
30#20263294—21no—
31Q8×Dic197C2211yescharacteristic, complemented
32#20263291—21no—
33#20263288C2211yescharacteristic
34C4×C22⋊D197C2211yescharacteristic
35#20263285C2211yescharacteristic, complemented
36#20263284C2211yescharacteristic
37C22⋊C4×D197—21no—
38#20263259C2211yescharacteristic, complemented
39#20263257C2211yescharacteristic, complemented
40#20263255C2211yescharacteristic, complemented
41#20263251C2211yescharacteristic, complemented
42#20263251C2211yescharacteristic, complemented
43#20263249C2211yescharacteristic, complemented
44#20263248C2211yescharacteristic
45#20263248C2211yescharacteristic
46#20263247C2211yescharacteristic
47#20263247C2211yescharacteristic
48#20263246C2211yescharacteristic
49C22⋊#13096802C2211yescharacteristic, complemented
50#20263238C2211yescharacteristic
51#20263236C2211yescharacteristic, complemented
52C2×SD16×C197C2211yescharacteristic
53C22×C8×C197C2211yescharacteristic, complemented
54C22⋊Q8×C197C2211yescharacteristic, complemented
55#121×C197C2211yescharacteristic
56#106×C197C2211yescharacteristic, complemented
57#103×C197C2211yescharacteristic, complemented
58#102×C197C2211yescharacteristic
59C23×D197—21no—
60#15691358—22no—
61#15691358—41no—
62C2×Dic394—21no—
63C2×Dic394C2311yescharacteristic
64C2×D788C2311yescharacteristic
65C2×C22⋊D197—21no—
66C2×C22⋊D197—21no—
67C2×C22⋊D197D411yescharacteristic
68D4×D197—22no—
69D4×D197—41no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72C2×C4×D197C2311yescharacteristic
73#15691348—41no—
74#15691345—22no—
75#15691341—41no—
76C2×#13096802C2311yescharacteristic
77C2×#13096802C2311yescharacteristic
78C4⋊Dic197—21no—
79C4⋊Dic197—21no—
80C4⋊Dic197C2311yescharacteristic
81C4⋊Dic197C2311yescharacteristic
82#15691337—21no—
83#15691337—21no—
84#15691337D411yescharacteristic
85C22⋊Dic197—21no—
86C22⋊Dic197—21no—
87C22⋊Dic197—21no—
88C22⋊Dic197D411yescharacteristic
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335D411yescharacteristic
94C4×Dic197—21no—
95C4×Dic197—21no—
96C4×Dic197—21no—
97C4×Dic197C2311yescharacteristic
98C2×Q8×C197C2311yescharacteristic
99C2×D4×C197—21no—
100C2×D4×C197C2311yescharacteristic
101C22×C4×C197C2311yescharacteristic
102SD16×C197—41no—
103C2×C8×C197—41no—
104C2×C8×C197C2311yescharacteristic
105C2×C8×C197C2311yescharacteristic
106C4⋊C4×C197—21no—
107C4⋊C4×C197C2311yescharacteristic
108C4⋊C4×C197C2311yescharacteristic
109C22⋊C4×C197—21no—
110C22⋊C4×C197—21no—
111C22×D197—21no—
112C22×D197—42no—
113C22×D197—42no—
114C22×D197—81no—
115C22×D197C2×D411yescharacteristic
116Dic394—21no—
117Dic394—21no—
118Dic394—42no—
119D788—21no—
120D788#4112yes—
121C22⋊D197—41no—
122C22⋊D197—42no—
123C22⋊D197—42no—
124C2×Dic197—21no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C2×Dic197C2×D411yescharacteristic
128C2×Dic197C2×D411yescharacteristic
129C2×Dic197C2×D411yescharacteristic
130C4×D197—21no—
131C4×D197—21no—
132C4×D197—42no—
133C4×D197—81no—
134#13096802—21no—
135#13096802—21no—
136C23×C197—21no—
137C23×C197C2×D411yescharacteristic
138Q8×C197—21no—
139Q8×C197—21no—
140D4×C197—21no—
141D4×C197—21no—
142D4×C197—81no—
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—21no—
155C22×C197—41no—
156C22×C197—41no—
157C22×C197—41no—
158C22×C197C22×D411yescharacteristic
159C2×D197—21no—
160C2×D197—22no—
161C2×D197—21no—
162C2×D197—21no—
163C2×D197—41no—
164C2×D197—41no—
165C2×D197—84no—
166C4×C197—21no—
167C4×C197—41no—
168C4×C197—41no—
169C4×C197—41no—
170C4×C197C2×#4111yescharacteristic
171C4×C197H(5,𝔽2)11yescharacteristic
172Dic197—21no—
173Dic197—21no—
174Dic197—21no—
175Dic197—41no—
176Dic197—41no—
177Dic197—41no—
178C2×C197—41no—
179C2×C197—81no—
180C2×C197#54611yescharacteristic
181C2×C197#57511yescharacteristic
182C2×C197#57711yescharacteristic
183D197—41no—
184D197—42no—
185C197#331111yescharacteristic, complemented, Sylow
186#3311—1971nomaximal, Sylow
187#556—1971no—
188#546—1971no—
189#489—1971no—
190#485—1971no—
191#482—1971no—
192#476—1971no—
193#470—1971no—
194#465D19711yescharacteristic, complemented
195#464—1971no—
196#452—1971no—
197#450—1971no—
198#440—1971no—
199#435—1971no—
200#418—1971no—
201#416—1971no—
202C2×#41—1971no—
203C22×D4—1971no—
204C2×Q16—1971no—
205C2×SD16—1971no—
206C2×SD16C2×D19711yescharacteristic
207C2×M16—1971no—
208C22×C8C2×D19711yescharacteristic, complemented
209C4⋊Q8—1971no—
210#126—3941no—
211#126—3941no—
212#124—3941no—
213#124—1971no—
214#123—3941no—
215C22⋊Q8—1971no—
216C22⋊Q8—1971no—
217C22⋊Q8C2×D19711yescharacteristic, complemented
218#121—3941no—
219#121C2×D19711yescharacteristic
220#120—3941no—
221C4×Q8—3941no—
222C4×Q8—1971no—
223C4×D4—1971no—
224C4×D4—1971no—
225#117—1971no—
226#117—1971no—
227C2×C22⋊C4—3941no—
228#107—1971no—
229#106—1971no—
230#106C2×D19711yescharacteristic, complemented
231C4⋊C8—1971no—
232#103—1971no—
233#103—1971no—
234#103—1971no—
235#103C2×D19711yescharacteristic, complemented
236#102—1971no—
237#102—1971no—
238#102—1971no—
239#102—1971no—
240#102—1971no—
241#102C2×D19711yescharacteristic
242C22⋊C8—1971no—
243C22⋊C8—1971no—
244#97—1971no—
245C24—3941no—
246#41—7881no—
247#41—3942no—
248C2×Q8—3941no—
249C2×Q8—1971no—
250C2×Q8C22×D19711yescharacteristic
251C2×D4—21no—
252C2×D4—7881no—
253C2×D4—3942no—
254C2×D4—3941no—
255C2×D4—3941no—
256C2×D4—1971no—
257C2×D4—1971no—
258C2×D4C22×D19711yescharacteristic
259C22×C4—3941no—
260C22×C4—3941no—
261C22×C4—1971no—
262C22×C4C22×D19711yescharacteristic
263Q16—7881no—
264SD16—41no—
265SD16—3942no—
266M16—7881no—
267C2×C8—41no—
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—1971no—
278C4⋊C4—1971no—
279C4⋊C4—1971no—
280C4⋊C4C22×D19711yescharacteristic
281C4⋊C4C22×D19711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—3941no—
285C22⋊C4—3941no—
286C22⋊C4—3941no—
287C22⋊C4—3941no—
288C22⋊C4—3941no—
289C22⋊C4—3941no—
290C22⋊C4—3941no—
291C22⋊C4—1971no—
292C22⋊C4—1971no—
293C42—3941no—
294C42—3941no—
295C42—3941no—
296C42—1971no—
297C23—21no—
298C23—7882no—
299C23—7882no—
300C23—15761no—
301C23—3941no—
302C23—1971no—
303C23D4×D19711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—7882no—
307Q8—3941no—
308Q8—3941no—
309D4—21no—
310D4—21no—
311D4—81no—
312D4—7882no—
313D4—7881no—
314D4—7882no—
315D4—3941no—
316D4—1972no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—7882no—
323C2×C4—15761no—
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—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—7881no—
345C22—7881no—
346C22—15764no—
347C22—3941no—
348C22—3941no—
349C22—3942no—
350C22—3941no—
351C22C2×D4×D19711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—7881no—
357C4—7881no—
358C4—7881no—
359C4—3941no—
360C4—3941no—
361C4—3941no—
362C4#41×D19711yescharacteristic
363C4#2026337111yescharacteristic
364C2—41no—
365C2—81no—
366C2—7882no—
367C2—7881no—
368C2#2716171811yescharacteristic
369C2#2716187611yescharacteristic
370C2#2716188511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#465×C197C2
socleC22×C197C22×D4
prime radical#27161726C2
hypercenter#465D197
hypocenterC197#3311
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #27161800C2semidirect
3. #27161437C2semidirect
4. #27161423C2semidirect
5. #27161400C2semidirect
6. #27161368C2semidirect
7. #27161336C2semidirect
8. #27161299C2semidirect
9. #27161239C2semidirect
10. #27161228C2semidirect
11. #27161182C2semidirect
12. #27161150C2semidirect
13. #27161082C2semidirect
14. #27161076C2semidirect
15. #465×C197C2semidirect
18. C2×#15691348C22semidirect
21. #20263325C22semidirect
31. Q8×Dic197C22semidirect
35. #20263285C22semidirect
38. #20263259C22semidirect
39. #20263257C22semidirect
40. #20263255C22semidirect
41. #20263251C22semidirect
42. #20263251C22semidirect
43. #20263249C22semidirect
49. C22⋊#13096802C22semidirect
51. #20263236C22semidirect
53. C22×C8×C197C22semidirect
54. C22⋊Q8×C197C22semidirect
56. #106×C197C22semidirect
57. #103×C197C22semidirect
185. C197#3311semidirect
194. #465D197semidirect
208. C22×C8C2×D197semidirect
217. C22⋊Q8C2×D197semidirect
230. #106C2×D197semidirect
235. #103C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618002
#271617262
#271614372
#271614232
#271614002
#271613682
#271613362
#271612992
#271612392
#271612282
#271611822
#271611502
#271610822
#271610762
#465×C1972
#3311197