Finite Group Databasebeta

← #38756039

Subgroups — #38756039

Order 25216.

1556 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D197C212yesmaximal, complemented
2#27161656C214yesmaximal, complemented
3#27161654C212yesmaximal, complemented
4#27161649C212yesmaximal, complemented
5#27161647C212yesmaximal, complemented
6#27161642C212yesmaximal, complemented
7#27161637C214yesmaximal, complemented
8#27161629C211yesmaximal, characteristic, complemented
9#27161625C212yesmaximal, complemented
10#27161588C218yesmaximal, complemented
11C2×#20263283C211yesmaximal, characteristic, complemented
12#532×C197C211yesmaximal, characteristic, complemented
13C2×#15691359C2212yes—
14C2×#15691358C2214yes—
15C2×#15691358C2212yescomplemented
16C2×#15691357C2212yes—
17C2×#15691357C2214yescomplemented
18#41×D197—216no—
19C2×Q8×D197C2212yescomplemented
20C2×D4×D197C2214yes—
21C2×D4×D197C2212yescomplemented
22C22×C4×D197C2211yescharacteristic
23C22×C4×D197C2214yescomplemented
24#20263326C2211yescharacteristic, complemented
25#20263324C2212yescomplemented
26#20263323C2211yescharacteristic, complemented
27#20263322C2212yescomplemented
28#20263315C2212yescomplemented
29#20263314C2212yescomplemented
30#20263313C2212yescomplemented
31#20263311C2218yescomplemented
32#20263308C2218yescomplemented
33#20263304C2218yescomplemented
34#20263303C2218yescomplemented
35#20263300C2214yescomplemented
36#20263299C2218yescomplemented
37#20263298C2218yescomplemented
38#20263298C2218yescomplemented
39#20263296C2218yescomplemented
40#20263294C2218yescomplemented
41C4×D788C2214yescomplemented
42C4×C22⋊D197C2218yescomplemented
43#20263283C2212yescomplemented
44#20263283C2212yescomplemented
45#20263283C2214yescomplemented
46C2×#15691337C2212yescomplemented
47C2×#15691335C2212yes—
48C2×C4×Dic197C2211yescharacteristic, complemented
49C2×#41×C197C2212yescomplemented
50C4⋊Q8×C197C2211yescharacteristic, complemented
51#127×C197C2211yescharacteristic, complemented
52#124×C197C2212yescomplemented
53#121×C197C2214yescomplemented
54C4×D4×C197C2214yescomplemented
55C2×C42×C197C2211yescharacteristic, complemented
56C23×D197C2314yes—
57C23×D197C2311yescharacteristic
58#15691359—28no—
59#15691358—216no—
60#15691358—28no—
61#15691357—216no—
62#15691357—28no—
63C2×Dic394C2314yes—
64C2×Dic394C2312yes—
65Q8×D197—28no—
66C2×D788C2314yes—
67C2×D788C2312yes—
68C2×D788C2314yes—
69C2×C22⋊D197C2314yes—
70C2×C22⋊D197C2314yes—
71C2×C22⋊D197C2314yes—
72C2×C22⋊D197C2318yes—
73D4×D197—216no—
74D4×D197—28no—
75C22×Dic197C2314yes—
76C22×Dic197C2312yes—
77C22×Dic197C2311yescharacteristic
78C2×C4×D197—216no—
79C2×C4×D197D418yescomplemented
80C2×C4×D197C2314yes—
81C2×C4×D197C2314yes—
82C2×C4×D197C2314yes—
83C2×C4×D197C2312yes—
84C2×C4×D197C2312yes—
85C4⋊Dic197C2314yes—
86#15691337C2314yes—
87#15691337C2314yes—
88#15691337C2318yes—
89C22⋊Dic197C2318yes—
90#15691335C2314yes—
91#15691335C2314yes—
92#15691335C2318yes—
93#15691335C2318yes—
94C4×Dic197C2312yes—
95C4×Dic197C2312yes—
96#41×C197—28no—
97C2×Q8×C197C2312yes—
98C2×D4×C197C2314yes—
99C2×D4×C197C2312yes—
100C2×D4×C197C2314yes—
101C22×C4×C197C2314yes—
102C22×C4×C197C2312yes—
103C22×C4×C197C2311yescharacteristic
104C4⋊C4×C197C2314yes—
105C22⋊C4×C197C2318yes—
106C42×C197C2312yes—
107C42×C197C2311yescharacteristic
108C42×C197C2311yescharacteristic
109C22×D197—28no—
110C22×D197—42no—
111C22×D197—48no—
112C22×D197C2×D414yes—
113C22×D197C2414yes—
114C22×D197C2412yes—
115Dic394—24no—
116Dic394—28no—
117D788—28no—
118D788—24no—
119D788—44no—
120C22⋊D197—28no—
121C22⋊D197—28no—
122C22⋊D197—28no—
123C22⋊D197—48no—
124C2×Dic197—28no—
125C2×Dic197—42no—
126C2×Dic197C2×D414yes—
127C2×Dic197C2414yes—
128C2×Dic197C2414yes—
129C2×Dic197C2412yes—
130C4×D197—28no—
131C4×D197—28no—
132C4×D197—28no—
133C4×D197C2×D418yes—
134C4×D197C2×D418yes—
135C23×C197C2414yes—
136C23×C197C2411yescharacteristic
137Q8×C197—24no—
138D4×C197—28no—
139D4×C197—24no—
140D4×C197—44no—
141C2×C4×C197—28no—
142C2×C4×C197—42no—
143C2×C4×C197C2×D414yes—
144C2×C4×C197C2412yes—
145C2×C4×C197C2412yes—
146C2×C4×C197C2414yes—
147C2×C4×C197C2411yescharacteristic
148C2×C4×C197C2411yescharacteristic
149C22×C197—22no—
150C22×C197—24no—
151C22×C197—28no—
152C22×C197C22×D412yes—
153C22×C197C2511yescharacteristic
154C2×D197—24no—
155C2×D197—24no—
156C2×D197—28no—
157C2×D197—44no—
158C2×D197—416no—
159C2×D197C22×D414yes—
160C4×C197—24no—
161C4×C197—22no—
162C4×C197—22no—
163C4×C197C22×D412yes—
164C4×C197C22×D412yes—
165Dic197—24no—
166Dic197—24no—
167Dic197C22×D414yes—
168C2×C197—22no—
169C2×C197—44no—
170C2×C197C23×D411yescharacteristic
171C2×C197#58512yes—
172D197—24no—
173D197—44no—
174C197#344911yescharacteristic, complemented, Sylow
175#3449—1971nomaximal, Sylow
176C22×#41—1972no—
177#537—1974no—
178#536—1972no—
179#536—1972no—
180#535—1972no—
181#535—1972no—
182#534—1974no—
183#532—1972no—
184#532—1971no—
185#532D19711yescharacteristic, complemented
186#525—1978no—
187C2×#117—1971no—
188C2×#41—39416no—
189C2×#41—1972no—
190C2×#41—1972no—
191C2×#41—1974no—
192C2×#41—1974no—
193C2×#41—1972no—
194C2×#41C2×D19712yescomplemented
195C22×Q8—1972no—
196C22×D4—1974no—
197C22×D4—1972no—
198C23×C4—1974no—
199C23×C4—1971no—
200C4⋊Q8—1972no—
201C4⋊Q8—1971no—
202C4⋊Q8C2×D19711yescharacteristic, complemented
203#127—1972no—
204#127—1971no—
205#127C2×D19711yescharacteristic, complemented
206#124—1972no—
207#124—1972no—
208#124—1972no—
209#124C2×D19712yescomplemented
210#123—1978no—
211#123—1978no—
212C22⋊Q8—1978no—
213C22⋊Q8—1978no—
214#121—1978no—
215#121—1978no—
216#121—1974no—
217#121—1978no—
218#121C2×D19714yescomplemented
219#120—1978no—
220#120—1978no—
221C4×D4—1978no—
222C4×D4—1974no—
223C4×D4C2×D19714yescomplemented
224#117—1974no—
225#117—1972no—
226#117—1972no—
227C2×C4⋊C4—1972no—
228C2×C22⋊C4—1972no—
229C2×C42—1971no—
230C2×C42C2×D19711yescharacteristic, complemented
231C24—1974no—
232C24—1971no—
233#41—28no—
234#41—39416no—
235#41—39416no—
236#41—3948no—
237#41—3948no—
238#41—3948no—
239C2×Q8—3948no—
240C2×Q8—1974no—
241C2×Q8—1972no—
242C2×Q8C22×D19712yes—
243C2×D4—39416no—
244C2×D4—3948no—
245C2×D4—1978no—
246C2×D4—1974no—
247C2×D4—1974no—
248C2×D4—1974no—
249C2×D4—1974no—
250C2×D4—1974no—
251C2×D4—1972no—
252C2×D4C22×D19714yes—
253C2×D4C22×D19712yes—
254C2×D4C22×D19714yes—
255C22×C4—39416no—
256C22×C4—1972no—
257C22×C4—1974no—
258C22×C4—1974no—
259C22×C4—1974no—
260C22×C4—1974no—
261C22×C4—1971no—
262C22×C4—1978no—
263C22×C4—1972no—
264C22×C4—1972no—
265C22×C4C22×D19714yes—
266C22×C4C22×D19712yes—
267C22×C4C22×D19711yescharacteristic
268C4⋊C4—1974no—
269C4⋊C4—1974no—
270C4⋊C4—1978no—
271C4⋊C4—1974no—
272C4⋊C4C22×D19714yes—
273C22⋊C4—1978no—
274C22⋊C4—1974no—
275C22⋊C4—1974no—
276C22⋊C4—1978no—
277C22⋊C4—1978no—
278C22⋊C4C22×D19718yes—
279C42—1972no—
280C42—1972no—
281C42C22×D19711yescharacteristic
282C42C22×D19711yescharacteristic
283C42C22×D19712yes—
284C23—7882no—
285C23—7888no—
286C23—3948no—
287C23—1974no—
288C23—1974no—
289C23—1972no—
290C23C23×D19714yes—
291C23C23×D19711yescharacteristic
292Q8—24no—
293Q8—3948no—
294Q8—3944no—
295D4—28no—
296D4—24no—
297D4—44no—
298D4—7884no—
299D4—7888no—
300D4—3948no—
301D4—3948no—
302D4—3948no—
303D4—3948no—
304D4—3944no—
305C2×C4—28no—
306C2×C4—42no—
307C2×C4—7882no—
308C2×C4—3948no—
309C2×C4—3948no—
310C2×C4—3948no—
311C2×C4—3948no—
312C2×C4—1974no—
313C2×C4—1974no—
314C2×C4—1974no—
315C2×C4—1972no—
316C2×C4—1978no—
317C2×C4—1978no—
318C2×C4D4×D19714yes—
319C2×C4C23×D19712yes—
320C2×C4C23×D19712yes—
321C2×C4C23×D19714yes—
322C2×C4C23×D19711yescharacteristic
323C2×C4C23×D19711yescharacteristic
324C22—22no—
325C22—24no—
326C22—28no—
327C22—7884no—
328C22—78816no—
329C22—3944no—
330C22—3948no—
331C22—3944no—
332C22—1974no—
333C22C2×D4×D19712yes—
334C22C24×D19711yescharacteristic
335C4—24no—
336C4—22no—
337C4—22no—
338C4—3944no—
339C4—3944no—
340C4—1974no—
341C4C2×D4×D19712yes—
342C4C2×D4×D19712yes—
343C2—22no—
344C2—44no—
345C2—7884no—
346C2—3944no—
347C2C22×D4×D19711yescharacteristic
348C2#2716191512yes—
subgroupG/H
centerC2×C4C23×D197
commutator, socle, cosocleC22×C197C25
FrattiniC22C24×D197
Fitting#532×C197C2
hypercenter#532D197
hypocenterC197#3449
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#41×D197C2semidirect
2. #27161656C2semidirect
3. #27161654C2semidirect
4. #27161649C2semidirect
5. #27161647C2semidirect
6. #27161642C2semidirect
7. #27161637C2semidirect
8. #27161629C2semidirect
9. #27161625C2semidirect
10. #27161588C2semidirect
11. C2×#20263283C2semidirect
12. #532×C197C2semidirect
15. C2×#15691358C22semidirect
17. C2×#15691357C22semidirect
19. C2×Q8×D197C22semidirect
21. C2×D4×D197C22semidirect
23. C22×C4×D197C22semidirect
24. #20263326C22semidirect
25. #20263324C22semidirect
26. #20263323C22semidirect
27. #20263322C22semidirect
28. #20263315C22semidirect
29. #20263314C22semidirect
30. #20263313C22semidirect
31. #20263311C22semidirect
32. #20263308C22semidirect
33. #20263304C22semidirect
34. #20263303C22semidirect
35. #20263300C22semidirect
36. #20263299C22semidirect
37. #20263298C22semidirect
38. #20263298C22semidirect
39. #20263296C22semidirect
40. #20263294C22semidirect
41. C4×D788C22semidirect
42. C4×C22⋊D197C22semidirect
43. #20263283C22semidirect
44. #20263283C22semidirect
45. #20263283C22semidirect
46. C2×#15691337C22semidirect
48. C2×C4×Dic197C22semidirect
49. C2×#41×C197C22semidirect
50. C4⋊Q8×C197C22semidirect
51. #127×C197C22semidirect
52. #124×C197C22semidirect
53. #121×C197C22semidirect
54. C4×D4×C197C22semidirect
55. C2×C42×C197C22semidirect
79. C2×C4×D197D4semidirect
174. C197#3449semidirect
185. #532D197semidirect
194. C2×#41C2×D197semidirect
202. C4⋊Q8C2×D197semidirect
205. #127C2×D197semidirect
209. #124C2×D197semidirect
218. #121C2×D197semidirect
223. C4×D4C2×D197semidirect
230. C2×C42C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1972
#271616562
#271616542
#271616492
#271616472
#271616422
#271616372
#271616292
#271616252
#271615882
C2×#202632832
#532×C1972
#3449197