Finite Group Databasebeta

← #38752855

Subgroups — #38752855

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#27161614C211yesmaximal, characteristic, complemented
2C4×D4×D197C211yesmaximal, characteristic, complemented
3#27161411C211yesmaximal, characteristic, complemented
4#27161394C211yesmaximal, characteristic, complemented
5#27161376C211yesmaximal, characteristic, complemented
6#27161295C211yesmaximal, characteristic, complemented
7#27161294C211yesmaximal, characteristic, complemented
8#27161231C211yesmaximal, characteristic, complemented
9#27161214C211yesmaximal, characteristic
10#27161190C211yesmaximal, characteristic, complemented
11#27161188C211yesmaximal, characteristic, complemented
12#27161178C211yesmaximal, characteristic, complemented
13#27161176C211yesmaximal, characteristic, complemented
14#27161135C211yesmaximal, characteristic, complemented
15#478×C197C211yesmaximal, characteristic, complemented
16C2×D4×D197C2211yescharacteristic
17C22×C4×D197—21no—
18C2×#15691347C2211yescharacteristic
19C2×#15691346C2211yescharacteristic
20C2×D1576C2211yescharacteristic
21C2×#15691343C2211yescharacteristic
22#20263321—21no—
23#20263314C2211yescharacteristic, complemented
24#20263311—21no—
25#20263306C2211yescharacteristic
26#20263301C2211yescharacteristic
27#20263291C2211yescharacteristic, complemented
28C4×D788C2211yescharacteristic
29D4×Dic197C2211yescharacteristic
30#20263288—21no—
31#20263288C2211yescharacteristic
32#20263288C2211yescharacteristic, complemented
33C4×C22⋊D197—21no—
34#20263286—21no—
35#20263284—21no—
36C4⋊C4×D197C2211yescharacteristic
37C22⋊C4×D197—21no—
38C42×D197C2211yescharacteristic
39#20263259C2211yescharacteristic, complemented
40#20263257C2211yescharacteristic, complemented
41C4⋊#13096802C2211yescharacteristic, complemented
42#20263251C2211yescharacteristic, complemented
43#20263249C2211yescharacteristic, complemented
44#20263248C2211yescharacteristic, complemented
45#20263246C2211yescharacteristic
46#20263246C2211yescharacteristic
47#20263246C2211yescharacteristic, complemented
48#20263238C2211yescharacteristic
49#20263238C2211yescharacteristic
50#20263236C2211yescharacteristic, complemented
51#20263236C2211yescharacteristic, 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—
60C2×Dic394C2311yescharacteristic
61C2×D788—21no—
62C2×D788C2311yescharacteristic
63C2×D788C2311yescharacteristic
64C2×C22⋊D197—21no—
65D4×D197—22no—
66D4×D197—22no—
67C22×Dic197—21no—
68C2×C4×D197—21no—
69C2×C4×D197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72C2×C4×D197—44no—
73C2×C4×D197D411yescharacteristic
74C2×C4×D197D411yescharacteristic
75C2×C4×D197C2311yescharacteristic
76#15691347—22no—
77#15691346—22no—
78D1576—22no—
79#15691343—22no—
80C2×#13096802C2311yescharacteristic
81C2×#13096802C2311yescharacteristic
82C4⋊Dic197—21no—
83C4⋊Dic197C2311yescharacteristic
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×Dic197—21no—
95C4×Dic197D411yescharacteristic
96C4×Dic197D411yescharacteristic
97C4×Dic197C2311yescharacteristic
98C2×D4×C197C2311yescharacteristic
99C22×C4×C197—21no—
100C2×C8×C197C2311yescharacteristic
101C2×C8×C197C2311yescharacteristic
102C4⋊C4×C197—21no—
103C4⋊C4×C197—21no—
104C4⋊C4×C197C2311yescharacteristic
105C4⋊C4×C197C2311yescharacteristic
106C4⋊C4×C197C2311yescharacteristic
107C22⋊C4×C197—21no—
108C42×C197C2311yescharacteristic
109C22×D197—24no—
110C22×D197—21no—
111C22×D197—21no—
112C22×D197—42no—
113C22×D197—42no—
114C22×D197C2×D411yescharacteristic
115Dic394—21no—
116Dic394—21no—
117D788—21no—
118D788—21no—
119D788—21no—
120D788—42no—
121D788#4112yes—
122C22⋊D197—24no—
123C2×Dic197—21no—
124C2×Dic197—21no—
125C2×Dic197—42no—
126C2×Dic197C2×D411yescharacteristic
127C2×Dic197C2×D411yescharacteristic
128C2×Dic197C2×D411yescharacteristic
129C4×D197—22no—
130C4×D197—21no—
131C4×D197—21no—
132C4×D197—41no—
133C4×D197—42no—
134C4×D197—42no—
135C4×D197—42no—
136C4×D197—42no—
137#13096802—21no—
138#13096802—21no—
139C23×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#478D19711yescharacteristic, complemented
192#463—1971no—
193#463—1971no—
194#450—1971no—
195#447—1971no—
196#442—1971no—
197#442—1971no—
198#440—1971no—
199#440—1971no—
200#432—1971no—
201C22×D4—1971no—
202C23×C4—3941no—
203C2×SD16—1971no—
204C2×SD16—1971no—
205C2×D8—1971no—
206C2×D8—1971no—
207#126—3941no—
208#125C2×D19711yescharacteristic, complemented
209#124—1971no—
210#123—3941no—
211C22⋊Q8—1971no—
212#121—1971no—
213C4×Q8—1971no—
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#106C2×D19711yescharacteristic, complemented
230C4⋊C8—1971no—
231C4⋊C8C2×D19711yescharacteristic, complemented
232#103—1971no—
233#103—1971no—
234#102—1971no—
235#102—1971no—
236#102—1971no—
237#102—1971no—
238#102C2×D19711yescharacteristic
239#102C2×D19711yescharacteristic
240C22⋊C8—1971no—
241C22⋊C8—1971no—
242#97—1971no—
243#97—1971no—
244C24—3941no—
245C2×Q8—1971no—
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—3942no—
264SD16—3942no—
265D8—3942no—
266D8—3942no—
267C2×C8—1971no—
268C2×C8—1971no—
269C2×C8C22×D19711yescharacteristic
270C2×C8C22×D19711yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—3941no—
274C4⋊C4—3941no—
275C4⋊C4—3941no—
276C4⋊C4—1971no—
277C4⋊C4C22×D19711yescharacteristic
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—3941no—
290C42—1971no—
291C42—1971no—
292C42—1971no—
293C42C22×D19711yescharacteristic
294C23—21no—
295C23—7882no—
296C23—7882no—
297C23—3944no—
298C23—3941no—
299C23—3941no—
300C23—1971no—
301Q8—3941no—
302Q8—3941no—
303D4—21no—
304D4—7882no—
305D4—3944no—
306D4—3941no—
307D4—3941no—
308D4—3941no—
309D4—1972no—
310D4#1569135912yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—7882no—
316C2×C4—7881no—
317C2×C4—7882no—
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×#1569135911yescharacteristic
362C2—42no—
363C2—7882no—
364C2—15761no—
365C2—3942no—
366C2#2716159211yescharacteristic
367C2AGL(1,ℤ/8)×D19711yescharacteristic
368C2#2716187611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#478×C197C2
socleC22×C197C22×D4
prime radical#27161214C2
hypercenter#478D197
hypocenterC197#3114
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. #27161614C2semidirect
2. C4×D4×D197C2semidirect
3. #27161411C2semidirect
4. #27161394C2semidirect
5. #27161376C2semidirect
6. #27161295C2semidirect
7. #27161294C2semidirect
8. #27161231C2semidirect
10. #27161190C2semidirect
11. #27161188C2semidirect
12. #27161178C2semidirect
13. #27161176C2semidirect
14. #27161135C2semidirect
15. #478×C197C2semidirect
23. #20263314C22semidirect
27. #20263291C22semidirect
32. #20263288C22semidirect
39. #20263259C22semidirect
40. #20263257C22semidirect
41. C4⋊#13096802C22semidirect
42. #20263251C22semidirect
43. #20263249C22semidirect
44. #20263248C22semidirect
47. #20263246C22semidirect
50. #20263236C22semidirect
51. #20263236C22semidirect
52. #125×C197C22semidirect
54. #107×C197C22semidirect
55. #106×C197C22semidirect
56. C4⋊C8×C197C22semidirect
184. C197#3114semidirect
191. #478D197semidirect
208. #125C2×D197semidirect
227. #107C2×D197semidirect
229. #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
#271616142
C4×D4×D1972
#271614112
#271613942
#271613762
#271612952
#271612942
#271612312
#271612142
#271611902
#271611882
#271611782
#271611762
#271611352
#478×C1972
#3114197