Finite Group Databasebeta

← #38751568

Subgroups — #38751568

Order 25216.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D197C211yesmaximal, characteristic
2#27161555C211yesmaximal, characteristic, complemented
3#27161551C211yesmaximal, characteristic
4#27161181C211yesmaximal, characteristic, complemented
5#27161180C211yesmaximal, characteristic, complemented
6#27161178C211yesmaximal, characteristic, complemented
7#27161170C211yesmaximal, characteristic, complemented
8C4×#15691345C211yesmaximal, characteristic, complemented
9C4×#15691346C211yesmaximal, characteristic, complemented
10#27161164C211yesmaximal, characteristic, complemented
11#27161116C211yesmaximal, characteristic, complemented
12#27161086C211yesmaximal, characteristic, complemented
13#27161082C211yesmaximal, characteristic
14#27161036C211yesmaximal, characteristic, complemented
15#440×C197C211yesmaximal, characteristic, complemented
16C2×Q8×D197C2211yescharacteristic
17C2×D4×D197C2211yescharacteristic
18C2×#15691347C2211yescharacteristic
19C2×#15691346C2211yescharacteristic
20C2×#15691345C2211yescharacteristic
21SD16×D197C418yescomplemented
22C2×#15691341C2211yescharacteristic
23C2×C8×D197C2211yescharacteristic
24Q8×Dic197C2211yescharacteristic, complemented
25C4×Dic394—21no—
26C4×Dic394C2211yescharacteristic, complemented
27#20263291—21no—
28#20263291C2211yescharacteristic, complemented
29C4×D788C2211yescharacteristic
30D4×Dic197C2211yescharacteristic
31#20263288C2211yescharacteristic
32C4×C22⋊D197—21no—
33#20263286—21no—
34#20263284—21no—
35#20263284C2211yescharacteristic
36#20263284C2211yescharacteristic
37#20263283—21no—
38#20263283C2211yescharacteristic
39C2×#15691335—21no—
40C22⋊C4×D197—21no—
41#20263260C2211yescharacteristic, complemented
42#20263257C2211yescharacteristic, complemented
43#20263257C2211yescharacteristic, complemented
44#20263251C2211yescharacteristic, complemented
45#20263250C2211yescharacteristic, complemented
46#20263249C2211yescharacteristic, complemented
47#20263248C2211yescharacteristic
48#20263247C2211yescharacteristic
49#20263246C2211yescharacteristic
50#20263236C2211yescharacteristic, complemented
51C4×#13096802C2211yescharacteristic, complemented
52C8×Dic197C2211yescharacteristic, complemented
53C2×SD16×C197C2211yescharacteristic
54C4×Q8×C197C2211yescharacteristic, complemented
55C4×D4×C197C2211yescharacteristic
56#107×C197C2211yescharacteristic, complemented
57#103×C197C2211yescharacteristic, complemented
58#102×C197C2211yescharacteristic
59#97×C197C2211yescharacteristic, complemented
60C23×D197—21no—
61C2×Dic394—21no—
62C2×Dic394C2311yescharacteristic
63Q8×D197—22no—
64Q8×D197C2×C414yescomplemented
65C2×D788C2311yescharacteristic
66C2×C22⋊D197—21no—
67D4×D197—22no—
68D4×D197C2×C414yes—
69C22×Dic197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72C2×C4×D197—21no—
73C2×C4×D197C2311yescharacteristic
74#15691347C2×C414yescomplemented
75#15691346C2×C414yescomplemented
76#15691345C2×C414yescomplemented
77#15691341—22no—
78C2×#13096802C2311yescharacteristic
79C2×#13096802C2311yescharacteristic
80C8×D197C2×C414yescomplemented
81C4⋊Dic197—21no—
82C4⋊Dic197C2311yescharacteristic
83C4⋊Dic197C2311yescharacteristic
84#15691337—21no—
85#15691337—21no—
86#15691337D411yescharacteristic
87#15691337D411yescharacteristic
88C22⋊Dic197—21no—
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335D411yescharacteristic
94#15691335D411yescharacteristic
95C4×Dic197—21no—
96C4×Dic197—21no—
97C4×Dic197—21no—
98C4×Dic197—21no—
99C4×Dic197—21no—
100C4×Dic197C2311yescharacteristic
101C2×Q8×C197C2311yescharacteristic
102C2×D4×C197C2311yescharacteristic
103C22×C4×C197—21no—
104SD16×C197C2×C414yescomplemented
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C4⋊C4×C197—21no—
108C4⋊C4×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C22⋊C4×C197—21no—
111C42×C197—21no—
112C42×C197C2311yescharacteristic
113C22×D197—21no—
114C22×D197—24no—
115C22×D197—81no—
116C22×D197—81no—
117C22×D197C2×D411yescharacteristic
118Dic394—21no—
119Dic394—24no—
120Dic394C22×C412yes—
121D788—21no—
122D788C22×C412yes—
123C22⋊D197—24no—
124C2×Dic197—21no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C2×Dic197—81no—
128C2×Dic197C2×D411yescharacteristic
129C2×Dic197C2×D411yescharacteristic
130C2×Dic197C2×D411yescharacteristic
131C4×D197—24no—
132C4×D197—81no—
133C4×D197—81no—
134C4×D197C22×C412yes—
135C4×D197#4112yes—
136#13096802—21no—
137#13096802C22×C412yes—
138C23×C197—21no—
139Q8×C197—21no—
140Q8×C197C22×C412yes—
141D4×C197—21no—
142D4×C197C22×C412yes—
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×C197C22×C412yes—
152C22×C197—22no—
153C22×C197—41no—
154C22×C197—41no—
155C22×C197C22×D411yescharacteristic
156C2×D197—21no—
157C2×D197—21no—
158C2×D197—22no—
159C2×D197—41no—
160C2×D197—41no—
161C2×D197—84no—
162C2×D197C4×D412yes—
163C4×C197—22no—
164C4×C197—21no—
165C4×C197—21no—
166C4×C197—41no—
167C4×C197—41no—
168C4×C197C23×C411yescharacteristic
169C4×C197C2×#4111yescharacteristic
170Dic197—22no—
171Dic197—21no—
172Dic197—21no—
173Dic197—41no—
174Dic197—41no—
175Dic197C4×D412yes—
176C2×C197—42no—
177C2×C197C2×C4×D411yescharacteristic
178C2×C197#57711yescharacteristic
179C2×C197#57711yescharacteristic
180D197—22no—
181D197—42no—
182C197#294911yescharacteristic, complemented, Sylow
183#2949—1971nomaximal, Sylow
184C22×SD16—1971no—
185#519—1971no—
186#518—1971no—
187#440—1971no—
188#440—1971no—
189#440—1971no—
190#440D19711yescharacteristic, complemented
191C4×SD16—1971no—
192C4×SD16—1971no—
193C4×SD16—1971no—
194C4×SD16—1971no—
195#427—1971no—
196#419—1971no—
197#418—1971no—
198#405—1971no—
199C22×Q8—1971no—
200C22×D4—1971no—
201C2×SD16—1971no—
202C2×SD16—1971no—
203C2×SD16—1978no—
204C2×SD16—1971no—
205C2×SD16C2×D19711yescharacteristic
206C2×M16—1971no—
207C22×C8—1971no—
208C4×Q8—3941no—
209C4×Q8—3941no—
210C4×Q8—1971no—
211C4×Q8—1971no—
212C4×Q8—1971no—
213C4×Q8C2×D19711yescharacteristic, complemented
214C4×D4—3941no—
215C4×D4—3941no—
216C4×D4—1971no—
217C4×D4—1971no—
218C4×D4—1971no—
219C4×D4C2×D19711yescharacteristic
220#117—3941no—
221#117—3941no—
222#117—1971no—
223#117—1971no—
224#117—1971no—
225C2×C22⋊C4—3941no—
226C2×C22⋊C4—3941no—
227#107—1971no—
228#107C2×D19711yescharacteristic, complemented
229#106—1971no—
230#106—1971no—
231#103—1971no—
232#103—1971no—
233#103—1971no—
234#103C2×D19711yescharacteristic, complemented
235#102—1971no—
236#102—1971no—
237#102—1971no—
238#102C2×D19711yescharacteristic
239#97—1971no—
240#97C2×D19711yescharacteristic, complemented
241C4×C8—1971no—
242C4×C8—1971no—
243C24—3941no—
244C2×Q8—3941no—
245C2×Q8—3942no—
246C2×Q8—1974no—
247C2×Q8—1971no—
248C2×Q8C22×D19711yescharacteristic
249C2×D4—3942no—
250C2×D4—3941no—
251C2×D4—1974no—
252C2×D4—1971no—
253C2×D4C22×D19711yescharacteristic
254C22×C4—21no—
255C22×C4—3941no—
256C22×C4—3941no—
257C22×C4—3941no—
258C22×C4—3941no—
259C22×C4—1971no—
260SD16—1974no—
261SD16—1974no—
262SD16—1974no—
263SD16C4×D19714yescomplemented
264M16—3942no—
265C2×C8—1971no—
266C2×C8—1971no—
267C2×C8—1974no—
268C2×C8C22×D19711yescharacteristic
269C2×C8C22×D19711yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—3941no—
272C4⋊C4—3941no—
273C4⋊C4—3941no—
274C4⋊C4—1971no—
275C4⋊C4—1971no—
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—1971no—
287C22⋊C4—1971no—
288C42—21no—
289C42—3941no—
290C42—3941no—
291C42—3941no—
292C42—3941no—
293C42—3941no—
294C42—1971no—
295C42C22×D19711yescharacteristic
296C23—21no—
297C23—15761no—
298C23—15761no—
299C23—3944no—
300C23—3941no—
301C23—1971no—
302Q8—21no—
303Q8—3944no—
304Q8—3941no—
305Q8—1972no—
306Q8C2×C4×D19712yes—
307D4—21no—
308D4—3944no—
309D4—3941no—
310D4—1972no—
311D4C2×C4×D19712yes—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—15761no—
317C2×C4—15761no—
318C2×C4—15761no—
319C2×C4—3944no—
320C2×C4—3941no—
321C2×C4—3941no—
322C2×C4—3941no—
323C2×C4—1971no—
324C2×C4—1971no—
325C2×C4—1971no—
326C2×C4—1972no—
327C2×C4—1972no—
328C2×C4D4×D19711yescharacteristic
329C2×C4D4×D19711yescharacteristic
330C2×C4C23×D19711yescharacteristic
331C8—21no—
332C8—3941no—
333C8—1972no—
334C8C2×C4×D19712yes—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—7881no—
339C22—7881no—
340C22—15764no—
341C22—3942no—
342C22—3941no—
343C22—3941no—
344C22—1972no—
345C22C2×D4×D19711yescharacteristic
346C4—21no—
347C4—21no—
348C4—22no—
349C4—41no—
350C4—41no—
351C4—7881no—
352C4—7881no—
353C4—3941no—
354C4—3941no—
355C4—3942no—
356C4—1972no—
357C4C22×C4×D19711yescharacteristic
358C4#41×D19711yescharacteristic
359C2—42no—
360C2—7882no—
361C2—3942no—
362C2C4×D4×D19711yescharacteristic
363C2#2716188811yescharacteristic
364C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutatorC4×C197C23×C4
FrattiniC2×C4C23×D197
Fitting#440×C197C2
socleC22×C197C22×D4
prime radicalC2×D4×D197C22
cosocleC2×C4×C197C24
hypercenter#440D197
hypocenterC197#2949
LCSG ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #27161555C2semidirect
4. #27161181C2semidirect
5. #27161180C2semidirect
6. #27161178C2semidirect
7. #27161170C2semidirect
8. C4×#15691345C2semidirect
9. C4×#15691346C2semidirect
10. #27161164C2semidirect
11. #27161116C2semidirect
12. #27161086C2semidirect
14. #27161036C2semidirect
15. #440×C197C2semidirect
21. SD16×D197C4semidirect
24. Q8×Dic197C22semidirect
26. C4×Dic394C22semidirect
28. #20263291C22semidirect
41. #20263260C22semidirect
42. #20263257C22semidirect
43. #20263257C22semidirect
44. #20263251C22semidirect
45. #20263250C22semidirect
46. #20263249C22semidirect
50. #20263236C22semidirect
51. C4×#13096802C22semidirect
52. C8×Dic197C22semidirect
54. C4×Q8×C197C22semidirect
56. #107×C197C22semidirect
57. #103×C197C22semidirect
59. #97×C197C22semidirect
64. Q8×D197C2×C4semidirect
74. #15691347C2×C4semidirect
75. #15691346C2×C4semidirect
76. #15691345C2×C4semidirect
80. C8×D197C2×C4semidirect
104. SD16×C197C2×C4semidirect
182. C197#2949semidirect
190. #440D197semidirect
213. C4×Q8C2×D197semidirect
228. #107C2×D197semidirect
234. #103C2×D197semidirect
240. #97C2×D197semidirect
263. SD16C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1972
#271615552
#271615512
#271611812
#271611802
#271611782
#271611702
C4×#156913452
C4×#156913462
#271611642
#271611162
#271610862
#271610822
#271610362
#440×C1972
#2949197