Finite Group Databasebeta

← #38751940

Subgroups — #38751940

Order 25216.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D197C211yesmaximal, characteristic, complemented
2C2×#20263353C212yesmaximal, complemented
3#27161592C211yesmaximal, characteristic, complemented
4#27161229C211yesmaximal, characteristic, complemented
5#27161228C212yesmaximal, complemented
6#27161215C211yesmaximal, characteristic, complemented
7#27161214C212yesmaximal, complemented
8#27161213C211yesmaximal, characteristic, complemented
9#27161082C212yesmaximal, complemented
10#27161041C211yesmaximal, characteristic, complemented
11#450×C197C211yesmaximal, characteristic, complemented
12C24×D197—21no—
13C2×#15691359C2211yescharacteristic, complemented
14C22×D788C2211yescharacteristic
15C22×C22⋊D197—21no—
16C2×D4×D197—28no—
17C2×D4×D197—21no—
18C2×D4×D197—21no—
19C2×D4×D197C2212yes—
20C2×D4×D197C2212yes—
21C22×C4×D197C2211yescharacteristic
22#20263353—28no—
23C2×#15691345C2212yescomplemented
24C2×D1576C2212yescomplemented
25C2×#15691343C2212yescomplemented
26C2×#15691341C2212yescomplemented
27#20263311—21no—
28#20263302C2211yescharacteristic, complemented
29#20263301C2211yescharacteristic, complemented
30#20263300C2211yescharacteristic, complemented
31#20263296—21no—
32#20263288—21no—
33#20263286—21no—
34#20263284C2211yescharacteristic, complemented
35#20263248C2212yescomplemented
36#20263247C2212yescomplemented
37#20263246C2212yescomplemented
38C22⋊#13096802C2211yescharacteristic, complemented
39#20263238C2212yescomplemented
40C22×D4×C197C2211yescharacteristic
41C2×SD16×C197C2212yescomplemented
42C22⋊Q8×C197C2211yescharacteristic, complemented
43#102×C197C2212yescomplemented
44C22⋊C8×C197C2211yescharacteristic, complemented
45C23×D197—28no—
46C23×D197—22no—
47C23×D197—22no—
48C23×D197—21no—
49C23×D197—44no—
50C23×D197—44no—
51C23×D197D411yescharacteristic
52#15691359—24no—
53C2×D788—24no—
54C2×D788—21no—
55C2×D788—21no—
56C2×D788—21no—
57C2×D788C2312yes—
58C2×D788C2311yescharacteristic, complemented
59C2×C22⋊D197—28no—
60C2×C22⋊D197—22no—
61C2×C22⋊D197—22no—
62C2×C22⋊D197—21no—
63D4×D197—28no—
64D4×D197—28no—
65D4×D197—24no—
66D4×D197—24no—
67D4×D197—24no—
68D4×D197D418yescomplemented
69C22×Dic197D411yescharacteristic
70C2×C4×D197—24no—
71C2×C4×D197—21no—
72C2×C4×D197D411yescharacteristic
73C2×C4×D197D411yescharacteristic
74C2×C4×D197C2311yescharacteristic
75C2×C4×D197C2311yescharacteristic
76#15691345—24no—
77D1576—24no—
78#15691343—24no—
79#15691341—24no—
80C2×#13096802C2312yescomplemented
81C4⋊Dic197C2311yescharacteristic, complemented
82#15691337—21no—
83#15691335—21no—
84#15691335—21no—
85#15691335—21no—
86#15691335—21no—
87C4×Dic197—21no—
88C24×C197—21no—
89C2×Q8×C197C2311yescharacteristic, complemented
90C2×D4×C197—24no—
91C2×D4×C197—21no—
92C2×D4×C197C2312yes—
93C22×C4×C197C2311yescharacteristic
94SD16×C197—24no—
95C2×C8×C197C2312yescomplemented
96C4⋊C4×C197—21no—
97C4⋊C4×C197C2311yescharacteristic, complemented
98C22⋊C4×C197—21no—
99C22×D197—22no—
100C22×D197—22no—
101C22×D197—22no—
102C22×D197—28no—
103C22×D197—24no—
104C22×D197—22no—
105C22×D197—28no—
106C22×D197—21no—
107C22×D197—432no—
108C22×D197—44no—
109C22×D197—44no—
110C22×D197—44no—
111C22×D197—42no—
112C22×D197—42no—
113C22×D197—44no—
114C22×D197C2×D411yescharacteristic
115C22×D197C2×D411yescharacteristic
116D788—24no—
117D788—22no—
118D788—22no—
119D788—42no—
120D788—42no—
121D788C2×D414yes—
122C22⋊D197—28no—
123C22⋊D197—28no—
124C22⋊D197—42no—
125C2×Dic197—22no—
126C2×Dic197—21no—
127C2×Dic197C2×D411yescharacteristic
128C2×Dic197C2×D411yescharacteristic
129C4×D197—22no—
130C4×D197—22no—
131C4×D197—22no—
132C4×D197—42no—
133C4×D197C2×D414yes—
134#13096802—22no—
135C23×C197—24no—
136C23×C197—22no—
137C23×C197—42no—
138C23×C197—42no—
139C23×C197C2×D411yescharacteristic
140Q8×C197—22no—
141D4×C197—24no—
142D4×C197—22no—
143D4×C197C2×D414yes—
144C2×C4×C197—22no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2411yescharacteristic
149C8×C197—22no—
150C22×C197—21no—
151C22×C197—21no—
152C22×C197—24no—
153C22×C197—48no—
154C22×C197—42no—
155C22×C197—42no—
156C22×C197AGL(1,ℤ/8)12yescomplemented
157C22×C197C22×D411yescharacteristic
158C2×D197—28no—
159C2×D197—21no—
160C2×D197—21no—
161C2×D197—21no—
162C2×D197—21no—
163C2×D197—24no—
164C2×D197—416no—
165C2×D197—416no—
166C2×D197—48no—
167C2×D197—42no—
168C2×D197—42no—
169C2×D197—41no—
170C2×D197—41no—
171C2×D197—41no—
172C2×D197#12012yes—
173C2×D197AGL(1,ℤ/8)12yescomplemented
174C4×C197—21no—
175C4×C197—41no—
176C4×C197—41no—
177C4×C197C22×D412yes—
178Dic197—21no—
179Dic197—41no—
180Dic197#12012yes—
181C2×C197—22no—
182C2×C197—44no—
183C2×C197C2×#12011yescharacteristic
184C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
186D197—22no—
187D197—22no—
188D197—44no—
189D197—81no—
190C197#300311yescharacteristic, complemented, Sylow
191#3003—1971nomaximal, Sylow
192C23×D4—1971no—
193C2×AGL(1,ℤ/8)—1972no—
194#525—1971no—
195#450—1972no—
196#450—1971no—
197#450D19711yescharacteristic, complemented
198#447—1971no—
199#447—1972no—
200#447—1971no—
201#418—1972no—
202#407—1971no—
203C25—3941no—
204C2×#41—1971no—
205C22×D4—3948no—
206C22×D4—3941no—
207C22×D4—3941no—
208C22×D4—3941no—
209C22×D4—1972no—
210C22×D4—1972no—
211C22×D4—1971no—
212C22×D4C2×D19711yescharacteristic
213C23×C4—1971no—
214AGL(1,ℤ/8)—3948no—
215C2×SD16—1972no—
216C2×SD16C2×D19712yescomplemented
217C2×D8—1972no—
218C2×D8—1972no—
219C2×M16—1972no—
220#123—3941no—
221C22⋊Q8—1971no—
222C22⋊Q8C2×D19711yescharacteristic, complemented
223#121—1971no—
224#121—1971no—
225#120—3941no—
226C4×D4—3941no—
227C4×D4—3941no—
228#117—1971no—
229#102—1972no—
230#102—1972no—
231#102—1972no—
232#102C2×D19712yescomplemented
233C22⋊C8—1971no—
234C22⋊C8—1972no—
235C22⋊C8C2×D19711yescharacteristic, complemented
236C24—21no—
237C24—7884no—
238C24—7884no—
239C24—3948no—
240C24—3942no—
241C24—3942no—
242C24—3941no—
243C24—1971no—
244#41—3944no—
245C2×Q8C22×D19711yescharacteristic, complemented
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—3941no—
249C2×D4—3941no—
250C2×D4—3944no—
251C2×D4—3948no—
252C2×D4—3948no—
253C2×D4—3944no—
254C2×D4—3941no—
255C2×D4—3948no—
256C2×D4—3944no—
257C2×D4—3944no—
258C2×D4—3942no—
259C2×D4—3942no—
260C2×D4—3941no—
261C2×D4—1978no—
262C2×D4—1971no—
263C2×D4—1972no—
264C2×D4C22×D19712yes—
265C22×C4—3941no—
266C22×C4—3944no—
267C22×C4—1971no—
268C22×C4—1971no—
269C22×C4—1971no—
270C22×C4—1971no—
271C22×C4—1971no—
272C22×C4C22×D19711yescharacteristic
273SD16—24no—
274SD16—3944no—
275D8—3944no—
276D8—3944no—
277M16—3944no—
278C2×C8—1972no—
279C2×C8C22×D19712yescomplemented
280C4⋊C4—21no—
281C4⋊C4—3941no—
282C4⋊C4—1971no—
283C4⋊C4C22×D19711yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—3941no—
286C22⋊C4—3941no—
287C22⋊C4—3941no—
288C22⋊C4—3941no—
289C42—3941no—
290C23—24no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—7882no—
295C23—78832no—
296C23—7884no—
297C23—7884no—
298C23—7884no—
299C23—7884no—
300C23—7882no—
301C23—3944no—
302C23—3948no—
303C23—3948no—
304C23—3942no—
305C23—3942no—
306C23—3941no—
307C23—3942no—
308C23—3942no—
309C23—1971no—
310C23—1971no—
311C23D4×D19711yescharacteristic
312Q8—22no—
313D4—24no—
314D4—22no—
315D4—7882no—
316D4—7882no—
317D4—7882no—
318D4—3944no—
319D4—3948no—
320D4—3948no—
321D4—3942no—
322D4—3942no—
323D4—1974no—
324D4D4×D19714yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—7882no—
329C2×C4—3942no—
330C2×C4—3942no—
331C2×C4—3941no—
332C2×C4—3942no—
333C2×C4—3942no—
334C2×C4—1971no—
335C2×C4—1971no—
336C2×C4—1974no—
337C2×C4D4×D19711yescharacteristic
338C2×C4C23×D19711yescharacteristic
339C8—22no—
340C8—3942no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—7888no—
348C22—78816no—
349C22—78816no—
350C22—7882no—
351C22—7882no—
352C22—7881no—
353C22—7881no—
354C22—7881no—
355C22—3948no—
356C22—3944no—
357C22—3941no—
358C22—3941no—
359C22—3941no—
360C22—3941no—
361C22—1972no—
362C22—1972no—
363C22#2026335312yescomplemented
364C22C2×D4×D19711yescharacteristic
365C4—21no—
366C4—41no—
367C4—41no—
368C4—7881no—
369C4—3941no—
370C4—1972no—
371C4C2×D4×D19712yes—
372C2—22no—
373C2—44no—
374C2—7884no—
375C2—15761no—
376C2—3942no—
377C2—3942no—
378C2#120×D19711yescharacteristic
379C2AGL(1,ℤ/8)×D19711yescharacteristic
380C2C2×#2026335311yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#450×C197C2
socleC22×C197C22×D4
hypercenter#450D197
hypocenterC197#3003
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×D4×D197C2semidirect
2. C2×#20263353C2semidirect
3. #27161592C2semidirect
4. #27161229C2semidirect
5. #27161228C2semidirect
6. #27161215C2semidirect
7. #27161214C2semidirect
8. #27161213C2semidirect
9. #27161082C2semidirect
10. #27161041C2semidirect
11. #450×C197C2semidirect
13. C2×#15691359C22semidirect
23. C2×#15691345C22semidirect
24. C2×D1576C22semidirect
25. C2×#15691343C22semidirect
26. C2×#15691341C22semidirect
28. #20263302C22semidirect
29. #20263301C22semidirect
30. #20263300C22semidirect
34. #20263284C22semidirect
35. #20263248C22semidirect
36. #20263247C22semidirect
37. #20263246C22semidirect
38. C22⋊#13096802C22semidirect
39. #20263238C22semidirect
41. C2×SD16×C197C22semidirect
42. C22⋊Q8×C197C22semidirect
43. #102×C197C22semidirect
44. C22⋊C8×C197C22semidirect
58. C2×D788C23semidirect
68. D4×D197D4semidirect
80. C2×#13096802C23semidirect
81. C4⋊Dic197C23semidirect
89. C2×Q8×C197C23semidirect
95. C2×C8×C197C23semidirect
97. C4⋊C4×C197C23semidirect
156. C22×C197AGL(1,ℤ/8)semidirect
173. C2×D197AGL(1,ℤ/8)semidirect
190. C197#3003semidirect
197. #450D197semidirect
216. C2×SD16C2×D197semidirect
222. C22⋊Q8C2×D197semidirect
232. #102C2×D197semidirect
235. C22⋊C8C2×D197semidirect
245. C2×Q8C22×D197semidirect
279. C2×C8C22×D197semidirect
283. C4⋊C4C22×D197semidirect
363. C22#20263353semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1972
C2×#202633532
#271615922
#271612292
#271612282
#271612152
#271612142
#271612132
#271610822
#271610412
#450×C1972
#3003197