Finite Group Databasebeta

← #38751936

Subgroups — #38751936

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×#20263350C212yesmaximal, complemented
3#27161589C211yesmaximal, characteristic, complemented
4#27161232C211yesmaximal, characteristic, complemented
5#27161231C212yesmaximal, complemented
6#27161230C211yesmaximal, characteristic, complemented
7#27161212C211yesmaximal, characteristic, complemented
8#27161211C212yesmaximal, complemented
9#27161082C212yesmaximal, complemented
10#27161041C211yesmaximal, characteristic, complemented
11#447×C197C211yesmaximal, characteristic, complemented
12C24×D197—21no—
13C2×#15691357C2211yescharacteristic, 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#20263350—28no—
23C2×#15691347C2212yescomplemented
24C2×#15691346C2212yescomplemented
25C2×#15691343C2212yescomplemented
26C2×#15691341C2212yescomplemented
27#20263308—21no—
28#20263306C2211yescharacteristic, complemented
29#20263305C2211yescharacteristic, complemented
30#20263297C2211yescharacteristic, complemented
31#20263294—21no—
32D4×Dic197—21no—
33#20263286—21no—
34#20263284C2211yescharacteristic, complemented
35#20263248C2212yescomplemented
36#20263247C2212yescomplemented
37#20263246C2212yescomplemented
38C22⋊#13096802C2211yescharacteristic, complemented
39#20263238C2212yescomplemented
40C22×D4×C197C2211yescharacteristic
41C2×D8×C197C2212yescomplemented
42#121×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#15691357—24no—
53C2×Dic394C2311yescharacteristic, complemented
54C2×D788—24no—
55C2×D788—21no—
56C2×D788C2312yes—
57C2×C22⋊D197—28no—
58C2×C22⋊D197—21no—
59C2×C22⋊D197—21no—
60C2×C22⋊D197—22no—
61C2×C22⋊D197—22no—
62D4×D197—28no—
63D4×D197—28no—
64D4×D197—24no—
65D4×D197—24no—
66D4×D197—24no—
67D4×D197D418yescomplemented
68C22×Dic197—21no—
69C22×Dic197D411yescharacteristic
70C2×C4×D197—24no—
71C2×C4×D197D411yescharacteristic
72C2×C4×D197D411yescharacteristic
73C2×C4×D197C2311yescharacteristic
74C2×C4×D197C2311yescharacteristic
75#15691347—24no—
76#15691346—24no—
77#15691343—24no—
78#15691341—24no—
79C2×#13096802C2312yescomplemented
80C4⋊Dic197—21no—
81C4⋊Dic197C2311yescharacteristic, complemented
82#15691337—21no—
83C22⋊Dic197—21no—
84C22⋊Dic197—21no—
85#15691335—21no—
86#15691335—21no—
87C4×Dic197—21no—
88C24×C197—21no—
89C2×D4×C197—24no—
90C2×D4×C197—21no—
91C2×D4×C197—21no—
92C2×D4×C197C2312yes—
93C2×D4×C197C2311yescharacteristic, complemented
94C22×C4×C197C2311yescharacteristic
95D8×C197—24no—
96C2×C8×C197C2312yescomplemented
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—432no—
107C22×D197—44no—
108C22×D197—44no—
109C22×D197—44no—
110C22×D197—42no—
111C22×D197—42no—
112C22×D197—44no—
113C22×D197C2×D411yescharacteristic
114C22×D197C2×D411yescharacteristic
115Dic394—22no—
116D788—24no—
117D788—22no—
118D788C2×D414yes—
119C22⋊D197—28no—
120C22⋊D197—28no—
121C22⋊D197—42no—
122C22⋊D197—42no—
123C2×Dic197—22no—
124C2×Dic197—21no—
125C2×Dic197—21no—
126C2×Dic197—42no—
127C2×Dic197C2×D411yescharacteristic
128C2×Dic197C2×D411yescharacteristic
129C4×D197—22no—
130C4×D197—22no—
131C4×D197—22no—
132C4×D197C2×D414yes—
133#13096802—22no—
134C23×C197—24no—
135C23×C197—22no—
136C23×C197—21no—
137C23×C197—42no—
138C23×C197—42no—
139C23×C197C2×D411yescharacteristic
140D4×C197—24no—
141D4×C197—22no—
142D4×C197—22no—
143D4×C197—42no—
144D4×C197C2×D414yes—
145C2×C4×C197—22no—
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×C197—41no—
157C22×C197—41no—
158C22×C197—41no—
159C22×C197AGL(1,ℤ/8)12yescomplemented
160C22×C197C22×D411yescharacteristic
161C2×D197—28no—
162C2×D197—21no—
163C2×D197—21no—
164C2×D197—21no—
165C2×D197—21no—
166C2×D197—24no—
167C2×D197—416no—
168C2×D197—416no—
169C2×D197—48no—
170C2×D197—42no—
171C2×D197—42no—
172C2×D197#12012yes—
173C2×D197AGL(1,ℤ/8)12yescomplemented
174C4×C197—21no—
175C4×C197—41no—
176C4×C197C22×D412yes—
177Dic197—21no—
178Dic197—41no—
179Dic197—41no—
180Dic197#12012yes—
181C2×C197—22no—
182C2×C197—44no—
183C2×C197—81no—
184C2×C197C2×#12011yescharacteristic
185C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
186C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
187D197—22no—
188D197—22no—
189D197—44no—
190C197#300311yescharacteristic, complemented, Sylow
191#3003—1971nomaximal, Sylow
192C23×D4—1971no—
193C2×AGL(1,ℤ/8)—1972no—
194#525—1971no—
195#450—1971no—
196#450—1972no—
197#450—1971no—
198#447—1972no—
199#447—1971no—
200#447D19711yescharacteristic, complemented
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×SD16—1972no—
217C2×D8—1972no—
218C2×D8C2×D19712yescomplemented
219C2×M16—1972no—
220#123—3941no—
221C22⋊Q8—1971no—
222C22⋊Q8—1971no—
223#121—1971no—
224#121C2×D19711yescharacteristic, complemented
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×Q8—1971no—
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—3944no—
250C2×D4—3948no—
251C2×D4—3948no—
252C2×D4—3944no—
253C2×D4—3941no—
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—1972no—
263C2×D4C22×D19712yes—
264C2×D4C22×D19711yescharacteristic, complemented
265C22×C4—3944no—
266C22×C4—3941no—
267C22×C4—1971no—
268C22×C4—1971no—
269C22×C4—1971no—
270C22×C4—1971no—
271C22×C4—1971no—
272C22×C4C22×D19711yescharacteristic
273SD16—3944no—
274SD16—3944no—
275D8—24no—
276D8—3944no—
277M16—3944no—
278C2×C8—1972no—
279C2×C8C22×D19712yescomplemented
280C4⋊C4—3941no—
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—21no—
293C23—42no—
294C23—42no—
295C23—7882no—
296C23—78832no—
297C23—7884no—
298C23—7884no—
299C23—7884no—
300C23—7884no—
301C23—7882no—
302C23—3944no—
303C23—3948no—
304C23—3948no—
305C23—3942no—
306C23—3942no—
307C23—3942no—
308C23—3942no—
309C23—1971no—
310C23—1971no—
311C23D4×D19711yescharacteristic
312Q8—3942no—
313D4—24no—
314D4—22no—
315D4—22no—
316D4—42no—
317D4—7882no—
318D4—7882no—
319D4—3944no—
320D4—3948no—
321D4—3948no—
322D4—3942no—
323D4—1974no—
324D4D4×D19714yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—7882no—
328C2×C4—3942no—
329C2×C4—3942no—
330C2×C4—3941no—
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—41no—
348C22—41no—
349C22—41no—
350C22—7888no—
351C22—78816no—
352C22—78816no—
353C22—7882no—
354C22—7882no—
355C22—3948no—
356C22—3944no—
357C22—3941no—
358C22—3941no—
359C22—3941no—
360C22—3941no—
361C22—1972no—
362C22—1972no—
363C22#2026335012yescomplemented
364C22C2×D4×D19711yescharacteristic
365C4—21no—
366C4—41no—
367C4—7881no—
368C4—7881no—
369C4—3941no—
370C4—1972no—
371C4C2×D4×D19712yes—
372C2—22no—
373C2—44no—
374C2—81no—
375C2—7884no—
376C2—3942no—
377C2—3942no—
378C2#120×D19711yescharacteristic
379C2AGL(1,ℤ/8)×D19711yescharacteristic
380C2C2×#2026335011yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#447×C197C2
socleC22×C197C22×D4
hypercenter#447D197
hypocenterC197#3003
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×D4×D197C2semidirect
2. C2×#20263350C2semidirect
3. #27161589C2semidirect
4. #27161232C2semidirect
5. #27161231C2semidirect
6. #27161230C2semidirect
7. #27161212C2semidirect
8. #27161211C2semidirect
9. #27161082C2semidirect
10. #27161041C2semidirect
11. #447×C197C2semidirect
13. C2×#15691357C22semidirect
23. C2×#15691347C22semidirect
24. C2×#15691346C22semidirect
25. C2×#15691343C22semidirect
26. C2×#15691341C22semidirect
28. #20263306C22semidirect
29. #20263305C22semidirect
30. #20263297C22semidirect
34. #20263284C22semidirect
35. #20263248C22semidirect
36. #20263247C22semidirect
37. #20263246C22semidirect
38. C22⋊#13096802C22semidirect
39. #20263238C22semidirect
41. C2×D8×C197C22semidirect
42. #121×C197C22semidirect
43. #102×C197C22semidirect
44. C22⋊C8×C197C22semidirect
53. C2×Dic394C23semidirect
67. D4×D197D4semidirect
79. C2×#13096802C23semidirect
81. C4⋊Dic197C23semidirect
93. C2×D4×C197C23semidirect
96. C2×C8×C197C23semidirect
97. C4⋊C4×C197C23semidirect
159. C22×C197AGL(1,ℤ/8)semidirect
173. C2×D197AGL(1,ℤ/8)semidirect
190. C197#3003semidirect
200. #447D197semidirect
218. C2×D8C2×D197semidirect
224. #121C2×D197semidirect
232. #102C2×D197semidirect
235. C22⋊C8C2×D197semidirect
264. C2×D4C22×D197semidirect
279. C2×C8C22×D197semidirect
283. C4⋊C4C22×D197semidirect
363. C22#20263350semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1972
C2×#202633502
#271615892
#271612322
#271612312
#271612302
#271612122
#271612112
#271610822
#271610412
#447×C1972
#3003197