Finite Group Databasebeta

← #38753160

Subgroups — #38753160

Order 25216.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C2×#20263354C211yesmaximal, characteristic, complemented
2C2×#20263352C211yesmaximal, characteristic, complemented
3C2×#20263351C211yesmaximal, characteristic, complemented
4#27161625C211yesmaximal, characteristic, complemented
5C2×#20263322C211yesmaximal, characteristic, complemented
6#27161467C212yesmaximal, complemented
7#27161465C212yesmaximal, complemented
8#27161454C212yesmaximal, complemented
9#27161450C212yesmaximal, complemented
10M16×Dic197C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2211yescharacteristic
13C22×D788C2211yescharacteristic
14C22×C22⋊D197—22no—
15#20263354—24no—
16#20263352—24no—
17#20263351—24no—
18C2×#15691347C2212yescomplemented
19C2×#15691346C2212yescomplemented
20C2×#15691345C2212yescomplemented
21C2×D1576C2212yescomplemented
22C2×#15691343C2212yescomplemented
23C2×#15691343C2212yescomplemented
24C2×#15691342C2211yescharacteristic
25#20263324C2211yescharacteristic, complemented
26#20263322—22no—
27#20263322C2212yes—
28#20263322C2212yes—
29#20263322C2211yescharacteristic, complemented
30#20263315C2211yescharacteristic, complemented
31#20263313C2211yescharacteristic, complemented
32#20263298—22no—
33C4×C22⋊D197—22no—
34C2×C4×Dic197C2211yescharacteristic
35#20263236C2212yescomplemented
36C8×Dic197C2212yescomplemented
37C2×#41×C197C2211yescharacteristic
38C22×D4×C197C2211yescharacteristic
39AGL(1,ℤ/8)×C197—24no—
40C2×SD16×C197C2212yescomplemented
41C2×D8×C197C2212yescomplemented
42C2×M16×C197C2211yescharacteristic
43C23×D197—21no—
44#15691358—22no—
45#15691358—22no—
46C2×Dic394C2311yescharacteristic
47C2×D788—22no—
48C2×D788—22no—
49C2×D788—21no—
50C2×D788C2311yescharacteristic
51C2×D788C2312yes—
52C2×C22⋊D197—22no—
53C2×C22⋊D197—24no—
54C2×C22⋊D197—24no—
55C2×C22⋊D197—21no—
56C2×C22⋊D197—48no—
57C22×Dic197D412yes—
58C2×C4×D197—21no—
59#15691347—24no—
60#15691346—24no—
61#15691345—24no—
62D1576—24no—
63#15691343—24no—
64#15691343—24no—
65#15691342D414yescomplemented
66C2×#13096802C2312yes—
67#15691337—22no—
68C22⋊Dic197—22no—
69#15691335—22no—
70C4×Dic197D412yes—
71C4×Dic197C2311yescharacteristic
72C4×Dic197C2311yescharacteristic
73C24×C197—21no—
74#41×C197—22no—
75#41×C197—22no—
76C2×Q8×C197C2311yescharacteristic
77C2×D4×C197—21no—
78C2×D4×C197—22no—
79C2×D4×C197—22no—
80C2×D4×C197—21no—
81C2×D4×C197C2311yescharacteristic
82C2×D4×C197C2312yes—
83C22×C4×C197—21no—
84C22×C4×C197C2311yescharacteristic
85SD16×C197—24no—
86D8×C197—24no—
87M16×C197D414yescomplemented
88C2×C8×C197C2312yes—
89C22×D197—21no—
90C22×D197—22no—
91C22×D197—44no—
92C22×D197—42no—
93Dic394—21no—
94Dic394—21no—
95D788—21no—
96D788—22no—
97D788—21no—
98D788—22no—
99D788—42no—
100C22⋊D197—48no—
101C22⋊D197—42no—
102C22⋊D197—44no—
103C22⋊D197—48no—
104C2×Dic197—24no—
105C2×Dic197—21no—
106C2×Dic197C2×D412yes—
107C2×Dic197C2×D412yes—
108C4×D197—42no—
109#13096802C2×D414yescomplemented
110C23×C197—21no—
111C23×C197—22no—
112C23×C197—44no—
113C23×C197—42no—
114C23×C197C2×D411yescharacteristic
115Q8×C197—21no—
116Q8×C197—21no—
117D4×C197—21no—
118D4×C197—22no—
119D4×C197—22no—
120D4×C197—21no—
121D4×C197—42no—
122D4×C197—42no—
123C2×C4×C197—21no—
124C2×C4×C197—42no—
125C2×C4×C197C2×D411yescharacteristic
126C2×C4×C197C2×D412yes—
127C2×C4×C197C2×D412yes—
128C2×C4×C197C2411yescharacteristic
129C8×C197C2×D414yescomplemented
130C22×C197—22no—
131C22×C197—41no—
132C22×C197—42no—
133C22×C197—44no—
134C22×C197—42no—
135C22×C197—84no—
136C22×C197#12712yes—
137C22×C197C22×D411yescharacteristic
138C2×D197—41no—
139C2×D197—42no—
140C2×D197—44no—
141C2×D197—42no—
142C2×D197—84no—
143C4×C197—41no—
144C4×C197#12712yes—
145C4×C197C22×D411yescharacteristic
146C4×C197C22×D411yescharacteristic
147Dic197—22no—
148Dic197—41no—
149Dic197AGL(1,ℤ/8)14yescomplemented
150C2×C197—22no—
151C2×C197—81no—
152C2×C197—82no—
153C2×C197C2×#12711yescharacteristic
154C2×C197C2×AGL(1,ℤ/8)12yes—
155D197—81no—
156D197—82no—
157C197#315111yescharacteristic, complemented, Sylow
158#3151—1971nomaximal, Sylow
159C2×AGL(1,ℤ/8)—1971no—
160C2×AGL(1,ℤ/8)—1971no—
161C2×AGL(1,ℤ/8)—1971no—
162C2×AGL(1,ℤ/8)D19711yescharacteristic, complemented
163#532—1971no—
164C2×#127—1971no—
165#496—1972no—
166#496—1972no—
167#493—1972no—
168#492—1972no—
169C4×M16—1971no—
170C2×#41—1971no—
171C2×#41C2×D19711yescharacteristic
172C22×D4—3942no—
173C22×D4—1971no—
174C22×D4C2×D19711yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—3944no—
177AGL(1,ℤ/8)—3944no—
178AGL(1,ℤ/8)—3944no—
179C2×SD16—1972no—
180C2×SD16—1972no—
181C2×SD16—1972no—
182C2×SD16C2×D19712yescomplemented
183C2×D8—1972no—
184C2×D8—1972no—
185C2×D8—1972no—
186C2×D8C2×D19712yescomplemented
187C2×M16—1971no—
188C2×M16C2×D19711yescharacteristic
189C4⋊Q8—1971no—
190#127—3942no—
191#127—1972no—
192#127—1972no—
193#127—1971no—
194#124—1971no—
195#124—1971no—
196#121—3942no—
197C4×D4—3942no—
198C2×C42—1971no—
199#97—1972no—
200C4×C8—1972no—
201C24—21no—
202C24—3941no—
203#41—22no—
204#41—22no—
205#41—3942no—
206#41—3942no—
207C2×Q8—1971no—
208C2×Q8C22×D19711yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—7888no—
214C2×D4—3941no—
215C2×D4—3942no—
216C2×D4—3944no—
217C2×D4—3944no—
218C2×D4—3941no—
219C2×D4—3942no—
220C2×D4—3942no—
221C2×D4—1971no—
222C2×D4—1972no—
223C2×D4C22×D19711yescharacteristic
224C2×D4C22×D19712yes—
225C22×C4—21no—
226C22×C4—3941no—
227C22×C4—1972no—
228C22×C4C22×D19711yescharacteristic
229SD16—24no—
230SD16—3944no—
231SD16—3944no—
232SD16—3944no—
233D8—24no—
234D8—3944no—
235D8—3944no—
236D8—3944no—
237M16—1974no—
238M16C22⋊D19714yescomplemented
239C2×C8—1972no—
240C2×C8C22×D19712yes—
241C4⋊C4—3942no—
242C22⋊C4—3942no—
243C22⋊C4—3942no—
244C42—1972no—
245C42—1971no—
246C42—1971no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—7884no—
252C23—7882no—
253C23—3941no—
254C23—3942no—
255C23D4×D19711yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—3941no—
259Q8—3941no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—7882no—
267D4—7888no—
268D4—7888no—
269D4—7884no—
270D4—7882no—
271D4—3941no—
272D4—3942no—
273D4—3941no—
274D4—3942no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—7882no—
278C2×C4—3941no—
279C2×C4—3944no—
280C2×C4—1972no—
281C2×C4—1972no—
282C2×C4D4×D19711yescharacteristic
283C2×C4D4×D19712yes—
284C2×C4C2×C22⋊D19712yes—
285C2×C4C23×D19711yescharacteristic
286C8—1974no—
287C8C2×C22⋊D19714yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—7884no—
295C22—7882no—
296C22—7881no—
297C22—7882no—
298C22—15764no—
299C22#2026332212yes—
300C22C2×D4×D19711yescharacteristic
301C4—41no—
302C4—7881no—
303C4—3942no—
304C4—1974no—
305C4#2026332212yes—
306C4C2×D4×D19711yescharacteristic
307C4C22×C22⋊D19711yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—15762no—
312C2—15761no—
313C2C2×#2026332211yescharacteristic
314C2AGL(1,ℤ/8)×D19712yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×AGL(1,ℤ/8)×C197C2
socleC22×C197C22×D4
hypercenterC2×AGL(1,ℤ/8)D197
hypocenterC197#3151
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20263354C2semidirect
2. C2×#20263352C2semidirect
3. C2×#20263351C2semidirect
4. #27161625C2semidirect
5. C2×#20263322C2semidirect
6. #27161467C2semidirect
7. #27161465C2semidirect
8. #27161454C2semidirect
9. #27161450C2semidirect
10. M16×Dic197C2semidirect
11. C2×AGL(1,ℤ/8)×C197C2semidirect
18. C2×#15691347C22semidirect
19. C2×#15691346C22semidirect
20. C2×#15691345C22semidirect
21. C2×D1576C22semidirect
22. C2×#15691343C22semidirect
23. C2×#15691343C22semidirect
25. #20263324C22semidirect
29. #20263322C22semidirect
30. #20263315C22semidirect
31. #20263313C22semidirect
35. #20263236C22semidirect
36. C8×Dic197C22semidirect
40. C2×SD16×C197C22semidirect
41. C2×D8×C197C22semidirect
65. #15691342D4semidirect
87. M16×C197D4semidirect
109. #13096802C2×D4semidirect
129. C8×C197C2×D4semidirect
149. Dic197AGL(1,ℤ/8)semidirect
157. C197#3151semidirect
162. C2×AGL(1,ℤ/8)D197semidirect
182. C2×SD16C2×D197semidirect
186. C2×D8C2×D197semidirect
238. M16C22⋊D197semidirect
287. C8C2×C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633542
C2×#202633522
C2×#202633512
#271616252
C2×#202633222
#271614672
#271614652
#271614542
#271614502
M16×Dic1972
C2×AGL(1,ℤ/8)×C1972
#3151197