Finite Group Databasebeta

← #450×D197 #38751891

Subgroups — #450×D197 #38751891

Order 25216.

996 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 155 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
2C2×SD16×D197C212yesmaximal, complemented
3C22⋊Q8×D197C211yesmaximal, characteristic, complemented
4#27161232C211yesmaximal, characteristic, complemented
5#27161231C212yesmaximal, complemented
6#27161230C211yesmaximal, characteristic, complemented
7#27161229C211yesmaximal, characteristic, complemented
8#27161228C212yesmaximal, complemented
9#102×D197C212yesmaximal, complemented
10C22⋊C8×D197C211yesmaximal, characteristic, complemented
11#450×C197C211yesmaximal, characteristic, complemented
12C24×D197—21no—
13C2×Q8×D197C2211yescharacteristic, 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
22C2×#15691347C2212yescomplemented
23C2×#15691346C2212yescomplemented
24C2×#15691345C2212yescomplemented
25SD16×D197—28no—
26C2×C8×D197C2212yescomplemented
27#20263306C2211yescharacteristic, complemented
28#20263305C2211yescharacteristic, complemented
29#20263304—21no—
30#20263303—21no—
31#20263302C2211yescharacteristic, complemented
32C4⋊C4×D197—21no—
33C4⋊C4×D197C2211yescharacteristic, complemented
34C22⋊C4×D197—21no—
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
52C2×Dic394—21no—
53C2×Dic394C2311yescharacteristic, complemented
54Q8×D197—24no—
55C2×D788—24no—
56C2×D788—21no—
57C2×D788C2312yes—
58C2×C22⋊D197—28no—
59C2×C22⋊D197—22no—
60C2×C22⋊D197—22no—
61D4×D197—28no—
62D4×D197—28no—
63D4×D197—24no—
64D4×D197—24no—
65D4×D197—24no—
66D4×D197D418yes—
67C22×Dic197D411yescharacteristic
68C2×C4×D197—24no—
69C2×C4×D197—21no—
70C2×C4×D197—21no—
71C2×C4×D197D411yescharacteristic
72C2×C4×D197D411yescharacteristic
73C2×C4×D197C2311yescharacteristic
74C2×C4×D197C2311yescharacteristic
75#15691347—24no—
76#15691346—24no—
77#15691345—24no—
78C2×#13096802C2312yescomplemented
79C8×D197—24no—
80C4⋊Dic197—21no—
81C4⋊Dic197C2311yescharacteristic, complemented
82#15691337—21no—
83#15691337—21no—
84#15691337—21no—
85C22⋊Dic197—21no—
86#15691335—21no—
87#15691335—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—28no—
102C22×D197—24no—
103C22×D197—22no—
104C22×D197—28no—
105C22×D197—44no—
106C22×D197—432no—
107C22×D197—42no—
108C22×D197—42no—
109C22×D197—44no—
110C22×D197—44no—
111C22×D197—44no—
112C22×D197SD1614yescomplemented
113C22×D197C2×D411yescharacteristic
114C22×D197C2×D411yescharacteristic
115Dic394—22no—
116Dic394—42no—
117D788—24no—
118D788—22no—
119D788C2×D414yes—
120C22⋊D197—28no—
121C22⋊D197—28no—
122C2×Dic197—22no—
123C2×Dic197—21no—
124C2×Dic197—21no—
125C2×Dic197C2×D411yescharacteristic
126C2×Dic197C2×D411yescharacteristic
127C4×D197—22no—
128C4×D197—22no—
129C4×D197—22no—
130C4×D197—42no—
131C4×D197—42no—
132C4×D197C2×D414yes—
133#13096802—22no—
134C23×C197—24no—
135C23×C197—22no—
136C23×C197—42no—
137C23×C197—42no—
138C23×C197C2×D411yescharacteristic
139Q8×C197—22no—
140D4×C197—24no—
141D4×C197—22no—
142D4×C197C2×D414yes—
143C2×C4×C197—22no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2411yescharacteristic
148C8×C197—22no—
149C22×C197—21no—
150C22×C197—21no—
151C22×C197—24no—
152C22×C197—48no—
153C22×C197—42no—
154C22×C197—42no—
155C22×C197C2×SD1612yescomplemented
156C22×C197C22×D411yescharacteristic
157C2×D197—21no—
158C2×D197—21no—
159C2×D197—28no—
160C2×D197—24no—
161C2×D197—416no—
162C2×D197—48no—
163C2×D197—42no—
164C2×D197—42no—
165C2×D197—416no—
166C2×D197#12012yes—
167C2×D197C2×SD1612yes—
168C2×D197C2×SD1612yescomplemented
169C2×D197AGL(1,ℤ/8)12yes—
170C4×C197—21no—
171C4×C197—41no—
172C4×C197—41no—
173C4×C197C22×D412yes—
174Dic197—21no—
175Dic197—41no—
176Dic197—41no—
177Dic197#12012yes—
178C2×C197—22no—
179C2×C197—44no—
180C2×C197C2×#12011yescharacteristic
181C2×C197C22×SD1611yescharacteristic
182C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
183D197—22no—
184D197—44no—
185D197#45014yesdirect product factor
186C197C2×#45011yescharacteristic, complemented, Sylow
187C2×#450—1971nomaximal, Sylow
188C23×D4—1971no—
189C22×SD16—1972no—
190C2×C22⋊Q8—1971no—
191#450—1971no—
192#450—1972no—
193#450—1971no—
194#450—1972no—
195#450—1971no—
196#450D19711yescharacteristic, direct product factor
197C2×#102—1972no—
198C2×C22⋊C8—1971no—
199C25—3941no—
200C22×Q8—1971no—
201C22×D4—3948no—
202C22×D4—3941no—
203C22×D4—3941no—
204C22×D4—3941no—
205C22×D4—1972no—
206C22×D4—1972no—
207C22×D4—1971no—
208C22×D4C2×D19711yescharacteristic
209C23×C4—1971no—
210C2×SD16—3948no—
211C2×SD16—1972no—
212C2×SD16—1972no—
213C2×SD16—1972no—
214C2×SD16C2×D19712yescomplemented
215C22×C8—1972no—
216C22⋊Q8—3941no—
217C22⋊Q8—3941no—
218C22⋊Q8—1971no—
219C22⋊Q8—1971no—
220C22⋊Q8—1971no—
221C22⋊Q8C2×D19711yescharacteristic, complemented
222C2×C4⋊C4—3941no—
223C2×C4⋊C4—1971no—
224C2×C22⋊C4—3941no—
225#102—1972no—
226#102—1972no—
227#102—1972no—
228#102C2×D19712yescomplemented
229C22⋊C8—1971no—
230C22⋊C8—1972no—
231C22⋊C8C2×D19711yescharacteristic, complemented
232C24—21no—
233C24—7884no—
234C24—7884no—
235C24—3948no—
236C24—3942no—
237C24—3942no—
238C24—3941no—
239C24—1971no—
240C2×Q8—3941no—
241C2×Q8—3944no—
242C2×Q8—1971no—
243C2×Q8C22×D19711yescharacteristic, complemented
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—3944no—
247C2×D4—3948no—
248C2×D4—3948no—
249C2×D4—3944no—
250C2×D4—3948no—
251C2×D4—3944no—
252C2×D4—3944no—
253C2×D4—3942no—
254C2×D4—3942no—
255C2×D4—3941no—
256C2×D4—1978no—
257C2×D4—1972no—
258C2×D4C22×D19712yes—
259C22×C4—3941no—
260C22×C4—3941no—
261C22×C4—3944no—
262C22×C4—1971no—
263C22×C4—1971no—
264C22×C4—1971no—
265C22×C4—1971no—
266C22×C4—1971no—
267C22×C4C22×D19711yescharacteristic
268SD16—24no—
269SD16—3944no—
270SD16—3944no—
271SD16—3944no—
272C2×C8—3944no—
273C2×C8—1972no—
274C2×C8C22×D19712yescomplemented
275C4⋊C4—21no—
276C4⋊C4—3941no—
277C4⋊C4—3941no—
278C4⋊C4—3941no—
279C4⋊C4—3941no—
280C4⋊C4—1971no—
281C4⋊C4C22×D19711yescharacteristic, complemented
282C22⋊C4—21no—
283C22⋊C4—3941no—
284C22⋊C4—3941no—
285C22⋊C4—3941no—
286C23—24no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—7882no—
291C23—78832no—
292C23—7884no—
293C23—7884no—
294C23—7884no—
295C23—7884no—
296C23—7882no—
297C23—3944no—
298C23—3948no—
299C23—3948no—
300C23—3942no—
301C23—3942no—
302C23—3942no—
303C23—1974no—
304C23—1971no—
305C23—1971no—
306C23D4×D19711yescharacteristic
307Q8—22no—
308Q8—7882no—
309Q8—3942no—
310D4—24no—
311D4—22no—
312D4—3944no—
313D4—3948no—
314D4—3948no—
315D4—3942no—
316D4—1974no—
317D4D4×D19714yes—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—7882no—
322C2×C4—7882no—
323C2×C4—3942no—
324C2×C4—3942no—
325C2×C4—3941no—
326C2×C4—3941no—
327C2×C4—3942no—
328C2×C4—3942no—
329C2×C4—1971no—
330C2×C4—1971no—
331C2×C4—1974no—
332C2×C4D4×D19711yescharacteristic
333C2×C4C23×D19711yescharacteristic
334C8—22no—
335C8—3942no—
336C22—21no—
337C22—21no—
338C22—24no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22—7888no—
343C22—78816no—
344C22—78816no—
345C22—7882no—
346C22—7882no—
347C22—3948no—
348C22—3944no—
349C22—3941no—
350C22—3941no—
351C22—1972no—
352C22—1972no—
353C22—1972no—
354C22—1972no—
355C22SD16×D19712yescomplemented
356C22C2×D4×D19711yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—7881no—
361C4—7881no—
362C4—3941no—
363C4—1972no—
364C4C2×D4×D19712yes—
365C2—22no—
366C2—44no—
367C2—7884no—
368C2—3942no—
369C2—1974no—
370C2#120×D19711yescharacteristic
371C2C2×SD16×D19711yescharacteristic
372C2AGL(1,ℤ/8)×D19711yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#450×C197C2
socleC22×C197C22×D4
prime radicalC22×D4×D197C2
hypercenter#450D197
hypocenterC197C2×#450
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
2. C2×SD16×D197C2semidirect
3. C22⋊Q8×D197C2semidirect
4. #27161232C2semidirect
5. #27161231C2semidirect
6. #27161230C2semidirect
7. #27161229C2semidirect
8. #27161228C2semidirect
9. #102×D197C2semidirect
10. C22⋊C8×D197C2semidirect
11. #450×C197C2semidirect
13. C2×Q8×D197C22semidirect
22. C2×#15691347C22semidirect
23. C2×#15691346C22semidirect
24. C2×#15691345C22semidirect
26. C2×C8×D197C22semidirect
27. #20263306C22semidirect
28. #20263305C22semidirect
31. #20263302C22semidirect
33. C4⋊C4×D197C22semidirect
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
53. C2×Dic394C23semidirect
78. C2×#13096802C23semidirect
81. C4⋊Dic197C23semidirect
89. C2×Q8×C197C23semidirect
95. C2×C8×C197C23semidirect
97. C4⋊C4×C197C23semidirect
112. C22×D197SD16semidirect
155. C22×C197C2×SD16semidirect
168. C2×D197C2×SD16semidirect
185. D197#450direct (N × H)
186. C197C2×#450semidirect
196. #450D197direct (N × H)
214. C2×SD16C2×D197semidirect
221. C22⋊Q8C2×D197semidirect
228. #102C2×D197semidirect
231. C22⋊C8C2×D197semidirect
243. C2×Q8C22×D197semidirect
274. C2×C8C22×D197semidirect
281. C4⋊C4C22×D197semidirect
355. C22SD16×D197semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1972
C2×SD16×D1972
C22⋊Q8×D1972
#271612322
#271612312
#271612302
#271612292
#271612282
#102×D1972
C22⋊C8×D1972
#450×C1972
C2×#450197