Finite Group Databasebeta

← #483×D197 #38752586

Subgroups — #483×D197 #38752586

Order 25216.

460 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 135 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⋊Q8×D197C211yesmaximal, characteristic
2C2×C4⋊C4×D197C211yesmaximal, characteristic
3#27161411C212yesmaximal, complemented
4#27161410C211yesmaximal, characteristic, complemented
5#27161409C211yesmaximal, characteristic, complemented
6#27161408C212yesmaximal, complemented
7#27161407C211yesmaximal, characteristic, complemented
8#106×D197C212yesmaximal, complemented
9#103×D197C212yesmaximal, complemented
10C22⋊C8×D197C211yesmaximal, characteristic
11#483×C197C211yesmaximal, characteristic, complemented
12C2×Q8×D197C2211yescharacteristic
13C22×C4×D197—21no—
14C22×C4×D197C2211yescharacteristic
15C2×C8×D197C2212yes—
16#20263306C2211yescharacteristic
17#20263305C2211yescharacteristic
18#20263304—21no—
19#20263303—21no—
20#20263302C2211yescharacteristic
21C2×C4⋊Dic197C2211yescharacteristic
22C2×#15691337—21no—
23C4⋊C4×D197—21no—
24C4⋊C4×D197—21no—
25C4⋊C4×D197—21no—
26C4⋊C4×D197C2211yescharacteristic
27C4⋊C4×D197C2212yes—
28C4⋊C4×D197C2212yes—
29C22⋊C4×D197—21no—
30#20263258C2212yescomplemented
31#20263257C2212yescomplemented
32#20263257C2212yescomplemented
33#20263251C2212yescomplemented
34#20263250C2212yescomplemented
35#20263249C2212yescomplemented
36C22⋊#13096802C2211yescharacteristic
37#20263238C2212yes—
38C22⋊Q8×C197C2211yescharacteristic
39C2×C4⋊C4×C197C2211yescharacteristic
40#106×C197C2212yescomplemented
41#103×C197C2212yescomplemented
42C22⋊C8×C197C2211yescharacteristic
43C23×D197D411yescharacteristic
44C2×Dic394—21no—
45C2×Dic394C2311yescharacteristic
46Q8×D197—24no—
47C22×Dic197—21no—
48C22×Dic197D411yescharacteristic
49C2×C4×D197—24no—
50C2×C4×D197—21no—
51C2×C4×D197—21no—
52C2×C4×D197—22no—
53C2×C4×D197—22no—
54C2×C4×D197—44no—
55C2×C4×D197D411yescharacteristic
56C2×C4×D197D411yescharacteristic
57C2×C4×D197C2311yescharacteristic
58C2×C4×D197C2311yescharacteristic
59C2×#13096802C2312yes—
60C8×D197—24no—
61C4⋊Dic197—21no—
62C4⋊Dic197—21no—
63C4⋊Dic197C2312yes—
64C4⋊Dic197C2311yescharacteristic
65#15691337—22no—
66#15691337—22no—
67#15691337—21no—
68#15691337—21no—
69#15691337—21no—
70C22⋊Dic197—21no—
71#15691335—21no—
72#15691335—21no—
73C2×Q8×C197C2311yescharacteristic
74C22×C4×C197—21no—
75C22×C4×C197C2311yescharacteristic
76C2×C8×C197C2312yes—
77C4⋊C4×C197—21no—
78C4⋊C4×C197—21no—
79C4⋊C4×C197C2312yes—
80C4⋊C4×C197C2311yescharacteristic
81C22⋊C4×C197—21no—
82C22×D197—22no—
83C22×D197—22no—
84C22×D197SD1614yes—
85C22×D197C2×D411yescharacteristic
86C22×D197C2×D411yescharacteristic
87Dic394—22no—
88Dic394—42no—
89C2×Dic197—22no—
90C2×Dic197—21no—
91C2×Dic197—21no—
92C2×Dic197—22no—
93C2×Dic197—42no—
94C2×Dic197C2×D411yescharacteristic
95C2×Dic197C2×D411yescharacteristic
96C4×D197—22no—
97C4×D197—22no—
98C4×D197—22no—
99C4×D197—42no—
100C4×D197—44no—
101C4×D197—44no—
102C4×D197—42no—
103C4×D197#4114yes—
104#13096802—22no—
105C23×C197C2×D411yescharacteristic
106Q8×C197—22no—
107C2×C4×C197—22no—
108C2×C4×C197—21no—
109C2×C4×C197—21no—
110C2×C4×C197—22no—
111C2×C4×C197—42no—
112C2×C4×C197C2×D411yescharacteristic
113C2×C4×C197C2411yescharacteristic
114C8×C197—22no—
115C22×C197—21no—
116C22×C197—21no—
117C22×C197C2×SD1612yes—
118C22×C197C22×D411yescharacteristic
119C2×D197—21no—
120C2×D197—21no—
121C2×D197—28no—
122C2×D197#12312yes—
123C2×D197C2×SD1612yes—
124C2×D197C2×SD1612yes—
125C2×D197#13712yes—
126C4×C197—21no—
127C4×C197—41no—
128C4×C197—42no—
129C4×C197—41no—
130C4×C197C2×#4112yes—
131Dic197—21no—
132Dic197—41no—
133Dic197—42no—
134Dic197—41no—
135Dic197#12312yes—
136C2×C197—22no—
137C2×C197C2×#12311yescharacteristic
138C2×C197C22×SD1611yescharacteristic
139C2×C197C2×#13711yescharacteristic
140D197—22no—
141D197#48314yesdirect product factor
142C197C2×#48311yescharacteristic, complemented, Sylow
143C2×#483—1971nomaximal, Sylow
144C2×C22⋊Q8—1971no—
145C22×C4⋊C4—1971no—
146#483—1971no—
147#483—1971no—
148#483—1972no—
149#483—1972no—
150#483—1971no—
151#483D19711yescharacteristic, direct product factor
152C2×#106—1972no—
153C2×#103—1972no—
154C2×C22⋊C8—1971no—
155C22×Q8—1971no—
156C23×C4—3941no—
157C23×C4—1971no—
158C22×C8—1972no—
159C22⋊Q8—3941no—
160C22⋊Q8—3941no—
161C22⋊Q8—1971no—
162C22⋊Q8—1971no—
163C22⋊Q8—1971no—
164C22⋊Q8C2×D19711yescharacteristic
165C2×C4⋊C4—3941no—
166C2×C4⋊C4—3941no—
167C2×C4⋊C4—3941no—
168C2×C4⋊C4—3941no—
169C2×C4⋊C4—1971no—
170C2×C4⋊C4—1972no—
171C2×C4⋊C4—1972no—
172C2×C4⋊C4—1971no—
173C2×C4⋊C4C2×D19711yescharacteristic
174C2×C22⋊C4—3941no—
175#106—1972no—
176#106—1972no—
177#106—1972no—
178#106C2×D19712yescomplemented
179#103—1972no—
180#103—1972no—
181#103—1972no—
182#103C2×D19712yescomplemented
183C22⋊C8—1971no—
184C22⋊C8—1972no—
185C22⋊C8C2×D19711yescharacteristic
186C24—1971no—
187C2×Q8—3941no—
188C2×Q8—3944no—
189C2×Q8—1971no—
190C2×Q8C22×D19711yescharacteristic
191C22×C4—21no—
192C22×C4—7884no—
193C22×C4—3941no—
194C22×C4—3941no—
195C22×C4—3942no—
196C22×C4—3942no—
197C22×C4—3944no—
198C22×C4—3941no—
199C22×C4—1971no—
200C22×C4—1971no—
201C22×C4—1971no—
202C22×C4—1971no—
203C22×C4—1971no—
204C22×C4C22×D19711yescharacteristic
205C2×C8—3944no—
206C2×C8—1972no—
207C2×C8C22×D19712yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—3942no—
211C4⋊C4—3942no—
212C4⋊C4—3941no—
213C4⋊C4—3941no—
214C4⋊C4—3941no—
215C4⋊C4—3941no—
216C4⋊C4—3941no—
217C4⋊C4—1972no—
218C4⋊C4—1971no—
219C4⋊C4C22×D19712yes—
220C4⋊C4C22×D19711yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—3941no—
223C22⋊C4—3941no—
224C22⋊C4—3941no—
225C23—3942no—
226C23—3942no—
227C23—1974no—
228C23—1971no—
229C23—1971no—
230C23D4×D19711yescharacteristic
231Q8—22no—
232Q8—7882no—
233Q8—3942no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—7884no—
240C2×C4—7884no—
241C2×C4—7882no—
242C2×C4—7882no—
243C2×C4—7882no—
244C2×C4—3942no—
245C2×C4—3942no—
246C2×C4—3941no—
247C2×C4—3941no—
248C2×C4—3942no—
249C2×C4—3942no—
250C2×C4—3942no—
251C2×C4—1971no—
252C2×C4—1971no—
253C2×C4—1974no—
254C2×C4D4×D19711yescharacteristic
255C2×C4C23×D19711yescharacteristic
256C8—22no—
257C8—3942no—
258C22—21no—
259C22—21no—
260C22—3948no—
261C22—3941no—
262C22—3941no—
263C22—1972no—
264C22—1972no—
265C22—1972no—
266C22—1972no—
267C22SD16×D19712yes—
268C22C2×D4×D19711yescharacteristic
269C4—21no—
270C4—41no—
271C4—41no—
272C4—42no—
273C4—7881no—
274C4—7881no—
275C4—7882no—
276C4—3941no—
277C4—1972no—
278C4#41×D19712yes—
279C2—22no—
280C2—3942no—
281C2—1974no—
282C2#123×D19711yescharacteristic
283C2C2×SD16×D19711yescharacteristic
284C2#137×D19711yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#483×C197C2
socleC22×C197C22×D4
prime radicalC23×D197D4
hypercenter#483D197
hypocenterC197C2×#483
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #27161411C2semidirect
4. #27161410C2semidirect
5. #27161409C2semidirect
6. #27161408C2semidirect
7. #27161407C2semidirect
8. #106×D197C2semidirect
9. #103×D197C2semidirect
11. #483×C197C2semidirect
30. #20263258C22semidirect
31. #20263257C22semidirect
32. #20263257C22semidirect
33. #20263251C22semidirect
34. #20263250C22semidirect
35. #20263249C22semidirect
40. #106×C197C22semidirect
41. #103×C197C22semidirect
141. D197#483direct (N × H)
142. C197C2×#483semidirect
151. #483D197direct (N × H)
178. #106C2×D197semidirect
182. #103C2×D197semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1972
C2×C4⋊C4×D1972
#271614112
#271614102
#271614092
#271614082
#271614072
#106×D1972
#103×D1972
C22⋊C8×D1972
#483×C1972
C2×#483197