Finite Group Databasebeta

← #541×D197 #38756074

Subgroups — #541×D197 #38756074

Order 25216.

1004 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161692C211yesmaximal, characteristic, complemented
2#27161691C211yesmaximal, characteristic, complemented
3#27161690C211yesmaximal, characteristic, complemented
4#27161689C212yesmaximal, complemented
5#27161688C214yesmaximal, complemented
6#27161687C212yesmaximal, complemented
7#27161686C214yesmaximal, complemented
8C4⋊Q8×D197C211yesmaximal, characteristic, complemented
9#126×D197C214yesmaximal
10#125×D197C214yesmaximal, complemented
11#125×D197C211yesmaximal, characteristic, complemented
12C22⋊Q8×D197C212yesmaximal
13C4×Q8×D197C212yesmaximal, complemented
14#117×D197C211yesmaximal, characteristic
15#541×C197C211yesmaximal, characteristic, complemented
16C2×Q8×D197C2212yes—
17C22×C4×D197C2211yescharacteristic
18#20263326C2211yescharacteristic, complemented
19#20263325C2214yescomplemented
20#20263324C2212yescomplemented
21#20263321C2214yes—
22#20263321C2214yes—
23#20263321C2214yes—
24#20263320C2214yes—
25#20263319C2214yes—
26#20263319C2214yes—
27#20263319C2214yes—
28#20263318C2214yescomplemented
29#20263318C2211yescharacteristic, complemented
30#20263317C2214yescomplemented
31#20263317C2212yescomplemented
32#20263317C2214yescomplemented
33#20263316C2214yescomplemented
34#20263316C2214yescomplemented
35#20263316C2214yescomplemented
36#20263316C2214yescomplemented
37#20263316C2214yescomplemented
38#20263306C2212yes—
39#20263305C2212yes—
40#20263304C2214yes—
41#20263303C2214yes—
42#20263302C2212yes—
43Q8×Dic197C2212yescomplemented
44C4×Dic394C2214yescomplemented
45C4×Dic394C2212yescomplemented
46#20263291C2214yescomplemented
47#20263291C2212yescomplemented
48#20263285C2211yescharacteristic
49#20263284C2212yes—
50#20263283C2211yescharacteristic
51#20263283C2211yescharacteristic
52#20263282C2212yes—
53C4⋊C4×D197C2214yes—
54C4⋊C4×D197C2214yes—
55C4⋊C4×D197C2214yes—
56C4⋊C4×D197C2212yes—
57C4⋊C4×D197C2212yes—
58C4⋊C4×D197C2214yes—
59C22⋊C4×D197C2212yes—
60C22⋊C4×D197C2214yes—
61C42×D197C2214yes—
62C42×D197C2211yescharacteristic
63C42×D197C2211yescharacteristic
64C4⋊Q8×C197C2211yescharacteristic, complemented
65#126×C197C2214yes—
66#125×C197C2214yescomplemented
67#125×C197C2211yescharacteristic, complemented
68C22⋊Q8×C197C2212yes—
69C4×Q8×C197C2212yescomplemented
70#117×C197C2211yescharacteristic
71C23×D197C2311yescharacteristic
72C2×Dic394C2314yes—
73C2×Dic394C2312yes—
74Q8×D197—28no—
75C22×Dic197C2311yescharacteristic
76C2×C4×D197—24no—
77C2×C4×D197C2312yes—
78C2×C4×D197C2314yes—
79C2×C4×D197C2312yes—
80C2×C4×D197C2314yes—
81C2×C4×D197C2311yescharacteristic
82C2×C4×D197C2311yescharacteristic
83C2×C4×D197C2311yescharacteristic
84C2×C4×D197C2311yescharacteristic
85C4⋊Dic197C2312yes—
86C4⋊Dic197C2312yes—
87C4⋊Dic197C2314yes—
88C4⋊Dic197C2314yes—
89C4⋊Dic197C2314yes—
90C4⋊Dic197C2314yes—
91#15691337C2312yes—
92#15691337C2312yes—
93#15691337C2314yes—
94#15691337C2314yes—
95#15691337C2314yes—
96#15691337C2314yes—
97#15691337C2314yes—
98#15691337C2314yes—
99#15691337C2314yes—
100#15691337C2314yes—
101#15691337C2314yes—
102C22⋊Dic197C2312yes—
103C22⋊Dic197C2314yes—
104#15691335C2312yes—
105#15691335C2312yes—
106#15691335C2314yes—
107#15691335C2314yes—
108C4×Dic197C2312yes—
109C4×Dic197C2312yes—
110C4×Dic197C2311yescharacteristic
111C4×Dic197C2311yescharacteristic
112C4×Dic197C2314yes—
113C4×Dic197C2314yes—
114C4×Dic197C2314yes—
115C2×Q8×C197C2312yes—
116C22×C4×C197C2311yescharacteristic
117C4⋊C4×C197C2312yes—
118C4⋊C4×C197C2312yes—
119C4⋊C4×C197C2314yes—
120C4⋊C4×C197C2314yes—
121C4⋊C4×C197C2314yes—
122C4⋊C4×C197C2314yes—
123C22⋊C4×C197C2312yes—
124C22⋊C4×C197C2314yes—
125C42×C197C2311yescharacteristic
126C42×C197C2311yescharacteristic
127C42×C197C2314yes—
128C22×D197—22no—
129C22×D197—24no—
130C22×D197C2411yescharacteristic
131C22×D197C2411yescharacteristic
132Dic394—24no—
133Dic394—28no—
134C2×Dic197—22no—
135C2×Dic197C2412yes—
136C2×Dic197C2412yes—
137C2×Dic197C2414yes—
138C2×Dic197C2414yes—
139C2×Dic197C2411yescharacteristic
140C2×Dic197C2411yescharacteristic
141C4×D197—22no—
142C4×D197—28no—
143C4×D197—24no—
144C4×D197—28no—
145C4×D197—24no—
146C4×D197—22no—
147C4×D197—22no—
148C4×D197#4114yes—
149C23×C197C2411yescharacteristic
150Q8×C197—24no—
151C2×C4×C197—22no—
152C2×C4×C197C2412yes—
153C2×C4×C197C2412yes—
154C2×C4×C197C2414yes—
155C2×C4×C197C2414yes—
156C2×C4×C197C2411yescharacteristic
157C2×C4×C197C2411yescharacteristic
158C22×C197—21no—
159C22×C197—22no—
160C22×C197C2511yescharacteristic
161C2×D197—21no—
162C2×D197—22no—
163C2×D197—44no—
164C2×D197C2×#4112yes—
165C2×D1972−1+414yes—
166C4×C197—21no—
167C4×C197—24no—
168C4×C197—22no—
169C4×C197—22no—
170C4×C197—24no—
171C4×C197C2×#4112yes—
172Dic197—21no—
173Dic197—24no—
174Dic197—22no—
175Dic197—22no—
176Dic197—24no—
177Dic197C2×#4112yes—
178C2×C197—41no—
179C2×C197C22×#4111yescharacteristic
180C2×C197C2×2−1+412yes—
181D197—41no—
182D197#54114yesdirect product factor
183C197C2×#54111yescharacteristic, complemented, Sylow
184C2×#541—1971nomaximal, Sylow
185#541—1971no—
186#541—1974no—
187#541—1974no—
188#541—1971no—
189#541—1972no—
190#541—1972no—
191#541—1971no—
192#541D19711yescharacteristic, direct product factor
193C2×C4⋊Q8—1971no—
194C2×#126—1974no—
195C2×#125—1971no—
196C2×#125—1974no—
197C2×C22⋊Q8—1972no—
198C2×C4×Q8—1972no—
199C2×#117—1971no—
200C22×Q8—1972no—
201C23×C4—1971no—
202C4⋊Q8—1972no—
203C4⋊Q8—1974no—
204C4⋊Q8—1971no—
205C4⋊Q8C2×D19711yescharacteristic, complemented
206#126—1974no—
207#126—1974no—
208#126—1974no—
209#126—1974no—
210#126—1974no—
211#126—1974no—
212#126—1974no—
213#126C2×D19714yes—
214#125—1974no—
215#125—1974no—
216#125—1974no—
217#125—1974no—
218#125—1974no—
219#125—1972no—
220#125—1974no—
221#125—1974no—
222#125—1971no—
223#125—1974no—
224#125C2×D19711yescharacteristic, complemented
225#125C2×D19714yescomplemented
226C22⋊Q8—1972no—
227C22⋊Q8—1972no—
228C22⋊Q8—1974no—
229C22⋊Q8—1972no—
230C22⋊Q8—1974no—
231C22⋊Q8C2×D19712yes—
232C4×Q8—1972no—
233C4×Q8—1974no—
234C4×Q8—1972no—
235C4×Q8—1974no—
236C4×Q8—1972no—
237C4×Q8C2×D19712yescomplemented
238#117—1971no—
239#117—1972no—
240#117—1972no—
241#117—1971no—
242#117—1971no—
243#117C2×D19711yescharacteristic
244C2×C4⋊C4—1972no—
245C2×C4⋊C4—1972no—
246C2×C4⋊C4—1974no—
247C2×C4⋊C4—1974no—
248C2×C4⋊C4—1974no—
249C2×C4⋊C4—1974no—
250C2×C22⋊C4—1972no—
251C2×C22⋊C4—1974no—
252C2×C42—1971no—
253C2×C42—1971no—
254C2×C42—1974no—
255C24—1971no—
256C2×Q8—3948no—
257C2×Q8—1972no—
258C2×Q8—1974no—
259C2×Q8C22×D19712yes—
260C22×C4—3944no—
261C22×C4—1971no—
262C22×C4—1972no—
263C22×C4—1972no—
264C22×C4—1974no—
265C22×C4—1974no—
266C22×C4—1971no—
267C22×C4—1971no—
268C22×C4—1971no—
269C22×C4—1971no—
270C22×C4C22×D19711yescharacteristic
271C4⋊C4—1972no—
272C4⋊C4—1972no—
273C4⋊C4—1974no—
274C4⋊C4—1974no—
275C4⋊C4—1974no—
276C4⋊C4—1974no—
277C4⋊C4—1974no—
278C4⋊C4—1974no—
279C4⋊C4—1974no—
280C4⋊C4—1974no—
281C4⋊C4—1974no—
282C4⋊C4—1972no—
283C4⋊C4—1972no—
284C4⋊C4—1974no—
285C4⋊C4—1974no—
286C4⋊C4—1974no—
287C4⋊C4—1974no—
288C4⋊C4C22×D19712yes—
289C4⋊C4C22×D19712yes—
290C4⋊C4C22×D19714yes—
291C4⋊C4C22×D19714yes—
292C4⋊C4C22×D19714yes—
293C4⋊C4C22×D19714yes—
294C22⋊C4—1972no—
295C22⋊C4—1974no—
296C22⋊C4—1972no—
297C22⋊C4—1972no—
298C22⋊C4—1974no—
299C22⋊C4—1974no—
300C22⋊C4C22×D19712yes—
301C22⋊C4C22×D19714yes—
302C42—1971no—
303C42—1972no—
304C42—1971no—
305C42—1972no—
306C42—1974no—
307C42—1974no—
308C42—1974no—
309C42C22×D19711yescharacteristic
310C42C22×D19711yescharacteristic
311C42C22×D19714yes—
312C23—3942no—
313C23—3944no—
314C23—1971no—
315C23—1971no—
316C23C23×D19711yescharacteristic
317Q8—24no—
318Q8—3948no—
319Q8—3944no—
320C2×C4—22no—
321C2×C4—3942no—
322C2×C4—3948no—
323C2×C4—3944no—
324C2×C4—3944no—
325C2×C4—3948no—
326C2×C4—3942no—
327C2×C4—3942no—
328C2×C4—3942no—
329C2×C4—1972no—
330C2×C4—1972no—
331C2×C4—1974no—
332C2×C4—1974no—
333C2×C4—1971no—
334C2×C4—1971no—
335C2×C4—1974no—
336C2×C4C23×D19712yes—
337C2×C4C23×D19712yes—
338C2×C4C23×D19714yes—
339C2×C4C23×D19714yes—
340C2×C4C23×D19711yescharacteristic
341C2×C4C23×D19711yescharacteristic
342C22—21no—
343C22—22no—
344C22—7884no—
345C22—3941no—
346C22—3942no—
347C22—1972no—
348C22—1974no—
349C22C24×D19711yescharacteristic
350C4—21no—
351C4—24no—
352C4—22no—
353C4—22no—
354C4—24no—
355C4—3941no—
356C4—3944no—
357C4—3942no—
358C4—3942no—
359C4—3944no—
360C4—1972no—
361C4#41×D19712yes—
362C2—41no—
363C2—7881no—
364C2—1974no—
365C2C2×#41×D19711yescharacteristic
366C22−1+4×D19712yes—
subgroupG/H
center, FrattiniC22C24×D197
commutator, socle, cosocleC22×C197C25
Fitting#541×C197C2
prime radicalC23×D197C23
hypercenter#541D197
hypocenterC197C2×#541
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ #541

Complements

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

NH ≅ G/Ntype
1. #27161692C2semidirect
2. #27161691C2semidirect
3. #27161690C2semidirect
4. #27161689C2semidirect
5. #27161688C2semidirect
6. #27161687C2semidirect
7. #27161686C2semidirect
8. C4⋊Q8×D197C2semidirect
10. #125×D197C2semidirect
11. #125×D197C2semidirect
13. C4×Q8×D197C2semidirect
15. #541×C197C2semidirect
18. #20263326C22semidirect
19. #20263325C22semidirect
20. #20263324C22semidirect
28. #20263318C22semidirect
29. #20263318C22semidirect
30. #20263317C22semidirect
31. #20263317C22semidirect
32. #20263317C22semidirect
33. #20263316C22semidirect
34. #20263316C22semidirect
35. #20263316C22semidirect
36. #20263316C22semidirect
37. #20263316C22semidirect
43. Q8×Dic197C22semidirect
44. C4×Dic394C22semidirect
45. C4×Dic394C22semidirect
46. #20263291C22semidirect
47. #20263291C22semidirect
64. C4⋊Q8×C197C22semidirect
66. #125×C197C22semidirect
67. #125×C197C22semidirect
69. C4×Q8×C197C22semidirect
182. D197#541direct (N × H)
183. C197C2×#541semidirect
192. #541D197direct (N × H)
205. C4⋊Q8C2×D197semidirect
224. #125C2×D197semidirect
225. #125C2×D197semidirect
237. C4×Q8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616922
#271616912
#271616902
#271616892
#271616882
#271616872
#271616862
C4⋊Q8×D1972
#126×D1972
#125×D1972
C22⋊Q8×D1972
C4×Q8×D1972
#117×D1972
#541×C1972
C2×#541197