Finite Group Databasebeta

← #38752854

Subgroups — #38752854

Order 25216.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161616C211yesmaximal, characteristic, complemented
2C4×D4×D197C211yesmaximal, characteristic, complemented
3#27161408C211yesmaximal, characteristic, complemented
4#27161394C211yesmaximal, characteristic, complemented
5#27161382C211yesmaximal, characteristic, complemented
6#27161381C211yesmaximal, characteristic, complemented
7#27161297C211yesmaximal, characteristic, complemented
8#27161296C211yesmaximal, characteristic, complemented
9#27161228C211yesmaximal, characteristic, complemented
10#27161214C211yesmaximal, characteristic
11#27161192C211yesmaximal, characteristic, complemented
12#27161189C211yesmaximal, characteristic, complemented
13#27161177C211yesmaximal, characteristic, complemented
14#27161134C211yesmaximal, characteristic, complemented
15#440×C197C211yesmaximal, characteristic, complemented
16C2×D4×D197C2211yescharacteristic
17C22×C4×D197—21no—
18C2×#15691345C2211yescharacteristic
19C2×D1576C2211yescharacteristic
20C2×#15691343C2211yescharacteristic
21#20263321—21no—
22#20263317C2211yescharacteristic, complemented
23#20263315C2211yescharacteristic, complemented
24#20263311—21no—
25#20263302C2211yescharacteristic
26#20263301C2211yescharacteristic
27C4×D788—21no—
28C4×D788C2211yescharacteristic
29C4×D788C2211yescharacteristic, complemented
30D4×Dic197C2211yescharacteristic
31#20263288C2211yescharacteristic
32C4×C22⋊D197—21no—
33#20263286—21no—
34#20263283—21no—
35C4⋊C4×D197C2211yescharacteristic
36C22⋊C4×D197—21no—
37C42×D197C2211yescharacteristic
38#20263259C2211yescharacteristic, complemented
39#20263258C2211yescharacteristic, complemented
40#20263257C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263255C2211yescharacteristic, complemented
43#20263250C2211yescharacteristic, complemented
44#20263248C2211yescharacteristic
45#20263248C2211yescharacteristic, complemented
46#20263247C2211yescharacteristic
47#20263246C2211yescharacteristic
48#20263246C2211yescharacteristic, complemented
49#20263238C2211yescharacteristic
50#20263238C2211yescharacteristic
51#20263237C2211yescharacteristic, complemented
52C2×SD16×C197C2211yescharacteristic
53C4×Q8×C197C2211yescharacteristic, complemented
54C4×D4×C197C2211yescharacteristic
55#107×C197C2211yescharacteristic, complemented
56#103×C197C2211yescharacteristic, complemented
57#102×C197C2211yescharacteristic
58#97×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C2×D788—21no—
61C2×D788C2311yescharacteristic
62C2×D788C2311yescharacteristic
63C2×C22⋊D197—21no—
64D4×D197—22no—
65D4×D197—22no—
66C22×Dic197—21no—
67C2×C4×D197—21no—
68C2×C4×D197—21no—
69C2×C4×D197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—44no—
72C2×C4×D197D411yescharacteristic
73C2×C4×D197D411yescharacteristic
74C2×C4×D197C2311yescharacteristic
75#15691345—22no—
76D1576—22no—
77#15691343—22no—
78C2×#13096802C2311yescharacteristic
79C2×#13096802C2311yescharacteristic
80C4⋊Dic197—21no—
81C4⋊Dic197C2311yescharacteristic
82C4⋊Dic197C2311yescharacteristic
83#15691337—21no—
84#15691337—21no—
85#15691337—21no—
86C22⋊Dic197—21no—
87#15691335—21no—
88#15691335—21no—
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335—21no—
94C4×Dic197D411yescharacteristic
95C4×Dic197D411yescharacteristic
96C4×Dic197C2311yescharacteristic
97C2×Q8×C197C2311yescharacteristic
98C2×D4×C197C2311yescharacteristic
99C22×C4×C197—21no—
100SD16×C197—22no—
101C2×C8×C197C2311yescharacteristic
102C2×C8×C197C2311yescharacteristic
103C4⋊C4×C197—21no—
104C4⋊C4×C197C2311yescharacteristic
105C4⋊C4×C197C2311yescharacteristic
106C22⋊C4×C197—21no—
107C42×C197—21no—
108C42×C197C2311yescharacteristic
109C22×D197—24no—
110C22×D197—21no—
111C22×D197—21no—
112C22×D197—42no—
113C22×D197—42no—
114C22×D197C2×D411yescharacteristic
115D788—21no—
116D788—21no—
117D788—21no—
118D788—42no—
119D788#4112yes—
120C22⋊D197—24no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197—42no—
124C2×Dic197C2×D411yescharacteristic
125C2×Dic197C2×D411yescharacteristic
126C2×Dic197C2×D411yescharacteristic
127C4×D197—22no—
128C4×D197—21no—
129C4×D197—21no—
130C4×D197—41no—
131C4×D197—42no—
132C4×D197—42no—
133C4×D197—42no—
134C4×D197—42no—
135#13096802—21no—
136#13096802—21no—
137C23×C197—21no—
138Q8×C197—21no—
139Q8×C197—21no—
140D4×C197—21no—
141D4×C197#4112yes—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—42no—
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2411yescharacteristic
149C8×C197—21no—
150C8×C197—21no—
151C22×C197—22no—
152C22×C197—41no—
153C22×C197—41no—
154C22×C197C22×D411yescharacteristic
155C2×D197—21no—
156C2×D197—21no—
157C2×D197—22no—
158C2×D197—48no—
159C2×D197—41no—
160C2×D197—41no—
161C2×D197—41no—
162C2×D197—41no—
163C2×D197—41no—
164C2×D197AGL(1,ℤ/8)12yes—
165C4×C197—21no—
166C4×C197—21no—
167C4×C197—41no—
168C4×C197—41no—
169C4×C197—41no—
170C4×C197C2×#4111yescharacteristic
171C4×C197C2×#4111yescharacteristic
172Dic197—21no—
173Dic197—21no—
174Dic197—21no—
175Dic197—41no—
176Dic197—41no—
177C2×C197—42no—
178C2×C197#52511yescharacteristic
179C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C197#57511yescharacteristic
181D197—22no—
182D197—42no—
183D197—81no—
184C197#311411yescharacteristic, complemented, Sylow
185#3114—1971nomaximal, Sylow
186#529—1971no—
187C2×C4×D4—1971no—
188#483—1971no—
189#480—1971no—
190#478—1971no—
191#478—1971no—
192#463—1971no—
193#463—1971no—
194#450—1971no—
195#447—1971no—
196#442—1971no—
197#442—1971no—
198#440—1971no—
199#440D19711yescharacteristic, complemented
200#432—1971no—
201C22×D4—1971no—
202C23×C4—3941no—
203C2×SD16—1971no—
204C2×SD16C2×D19711yescharacteristic
205C2×D8—1971no—
206C2×D8—1971no—
207#126—3941no—
208#125—1971no—
209#124—1971no—
210#123—3941no—
211C22⋊Q8—1971no—
212#121—1971no—
213C4×Q8C2×D19711yescharacteristic, complemented
214C4×D4—3941no—
215C4×D4—3941no—
216C4×D4—3941no—
217C4×D4—1971no—
218C4×D4—1971no—
219C4×D4—1971no—
220C4×D4—1971no—
221C4×D4C2×D19711yescharacteristic
222#117—3941no—
223C2×C4⋊C4—1971no—
224C2×C22⋊C4—3941no—
225C2×C42—1971no—
226#107—1971no—
227#107C2×D19711yescharacteristic, complemented
228#106—1971no—
229#106—1971no—
230C4⋊C8—1971no—
231C4⋊C8—1971no—
232#103—1971no—
233#103C2×D19711yescharacteristic, complemented
234#102—1971no—
235#102—1971no—
236#102—1971no—
237#102—1971no—
238#102—1971no—
239#102C2×D19711yescharacteristic
240C22⋊C8—1971no—
241C22⋊C8—1971no—
242#97—1971no—
243#97C2×D19711yescharacteristic, complemented
244C24—3941no—
245C2×Q8C22×D19711yescharacteristic
246C2×D4—3941no—
247C2×D4—3942no—
248C2×D4—3942no—
249C2×D4—3941no—
250C2×D4—1971no—
251C2×D4—1971no—
252C2×D4C22×D19711yescharacteristic
253C22×C4—21no—
254C22×C4—7884no—
255C22×C4—3941no—
256C22×C4—3941no—
257C22×C4—3941no—
258C22×C4—3941no—
259C22×C4—3941no—
260C22×C4—1971no—
261C22×C4—1971no—
262C22×C4—1971no—
263SD16—22no—
264SD16—3942no—
265D8—3942no—
266D8—3942no—
267C2×C8—1971no—
268C2×C8—1971no—
269C2×C8C22×D19711yescharacteristic
270C2×C8C22×D19711yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—3941no—
273C4⋊C4—3941no—
274C4⋊C4—3941no—
275C4⋊C4—3941no—
276C4⋊C4—1971no—
277C4⋊C4—1971no—
278C4⋊C4C22×D19711yescharacteristic
279C4⋊C4C22×D19711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4—3941no—
284C22⋊C4—3941no—
285C22⋊C4—3941no—
286C22⋊C4—3941no—
287C22⋊C4—3941no—
288C22⋊C4—3941no—
289C42—21no—
290C42—1971no—
291C42—1971no—
292C42—1971no—
293C42C22×D19711yescharacteristic
294C23—21no—
295C23—7882no—
296C23—7882no—
297C23—3944no—
298C23—3941no—
299C23—3941no—
300C23—1971no—
301Q8—21no—
302Q8—21no—
303D4—21no—
304D4—7882no—
305D4—3944no—
306D4—3941no—
307D4—3941no—
308D4—3941no—
309D4—1972no—
310D4#1569135812yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—7882no—
316C2×C4—7882no—
317C2×C4—7881no—
318C2×C4—7882no—
319C2×C4—7882no—
320C2×C4—7882no—
321C2×C4—3942no—
322C2×C4—3941no—
323C2×C4—3941no—
324C2×C4—3941no—
325C2×C4—3941no—
326C2×C4—1971no—
327C2×C4—1971no—
328C2×C4—1971no—
329C2×C4D4×D19711yescharacteristic
330C2×C4D4×D19711yescharacteristic
331C2×C4C23×D19711yescharacteristic
332C8—21no—
333C8—21no—
334C8—3941no—
335C8—3941no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—7888no—
340C22—7881no—
341C22—7881no—
342C22—7881no—
343C22—7881no—
344C22—7881no—
345C22—3942no—
346C22—3941no—
347C22—3941no—
348C22—1972no—
349C22C2×D4×D19711yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—7881no—
356C4—7881no—
357C4—3941no—
358C4—3941no—
359C4—3941no—
360C4#41×D19711yescharacteristic
361C4C2×#1569135811yescharacteristic
362C2—42no—
363C2—7882no—
364C2—15761no—
365C2—3942no—
366C2#2716158811yescharacteristic
367C2AGL(1,ℤ/8)×D19711yescharacteristic
368C2#2716187811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#440×C197C2
socleC22×C197C22×D4
prime radical#27161214C2
hypercenter#440D197
hypocenterC197#3114
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #27161616C2semidirect
2. C4×D4×D197C2semidirect
3. #27161408C2semidirect
4. #27161394C2semidirect
5. #27161382C2semidirect
6. #27161381C2semidirect
7. #27161297C2semidirect
8. #27161296C2semidirect
9. #27161228C2semidirect
11. #27161192C2semidirect
12. #27161189C2semidirect
13. #27161177C2semidirect
14. #27161134C2semidirect
15. #440×C197C2semidirect
22. #20263317C22semidirect
23. #20263315C22semidirect
29. C4×D788C22semidirect
38. #20263259C22semidirect
39. #20263258C22semidirect
40. #20263257C22semidirect
41. #20263255C22semidirect
42. #20263255C22semidirect
43. #20263250C22semidirect
45. #20263248C22semidirect
48. #20263246C22semidirect
51. #20263237C22semidirect
53. C4×Q8×C197C22semidirect
55. #107×C197C22semidirect
56. #103×C197C22semidirect
58. #97×C197C22semidirect
184. C197#3114semidirect
199. #440D197semidirect
213. C4×Q8C2×D197semidirect
227. #107C2×D197semidirect
233. #103C2×D197semidirect
243. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616162
C4×D4×D1972
#271614082
#271613942
#271613822
#271613812
#271612972
#271612962
#271612282
#271612142
#271611922
#271611892
#271611772
#271611342
#440×C1972
#3114197