Finite Group Databasebeta

← #489×D197 #38753038

Subgroups — #489×D197 #38753038

Order 25216.

436 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C4⋊Q8×D197C211yesmaximal, characteristic, complemented
2#124×D197C211yesmaximal, characteristic
3#27161441C211yesmaximal, characteristic, complemented
4#27161440C212yesmaximal, complemented
5#27161439C211yesmaximal, characteristic, complemented
6#27161438C211yesmaximal, characteristic, complemented
7#27161437C212yesmaximal, complemented
8#103×D197C212yesmaximal, complemented
9#102×D197C212yesmaximal
10#97×D197C211yesmaximal, characteristic, complemented
11#489×C197C211yesmaximal, characteristic, complemented
12C2×Q8×D197—21no—
13C2×Q8×D197C2211yescharacteristic
14C2×D4×D197C2211yescharacteristic
15C2×C8×D197C2212yes—
16#20263326C2211yescharacteristic, complemented
17#20263325—21no—
18#20263325C2212yescomplemented
19#20263324—21no—
20#20263315C2211yescharacteristic
21#20263314C2211yescharacteristic
22#20263313C2211yescharacteristic
23#20263312—21no—
24#20263312—21no—
25C4⋊C4×D197—21no—
26C4⋊C4×D197C2212yes—
27C22⋊C4×D197—21no—
28C22⋊C4×D197—21no—
29C42×D197C2211yescharacteristic
30#20263251C2212yescomplemented
31#20263250C2212yescomplemented
32#20263249C2212yescomplemented
33#20263248C2212yes—
34#20263247C2212yes—
35#20263246C2212yes—
36#20263237C2211yescharacteristic, complemented
37#20263236C2212yescomplemented
38C4⋊Q8×C197C2211yescharacteristic, complemented
39#124×C197C2211yescharacteristic
40#103×C197C2212yescomplemented
41#102×C197C2212yes—
42#97×C197C2211yescharacteristic, complemented
43C23×D197—21no—
44C2×Dic394—21no—
45C2×Dic394—22no—
46C2×Dic394—21no—
47C2×Dic394C2311yescharacteristic
48Q8×D197—24no—
49Q8×D197—44no—
50C2×D788C2311yescharacteristic
51C2×C22⋊D197—21no—
52D4×D197—24no—
53C2×C4×D197—21no—
54C2×C4×D197—22no—
55C2×C4×D197D411yescharacteristic
56C2×C4×D197D411yescharacteristic
57C2×C4×D197C2311yescharacteristic
58C2×#13096802C2312yes—
59C8×D197—24no—
60C4⋊Dic197—21no—
61C4⋊Dic197C2312yes—
62#15691337—22no—
63#15691337—22no—
64C22⋊Dic197—21no—
65C22⋊Dic197—21no—
66#15691335—21no—
67#15691335—21no—
68#15691335—21no—
69#15691335—21no—
70C4×Dic197D411yescharacteristic
71C4×Dic197D411yescharacteristic
72C4×Dic197C2311yescharacteristic
73C2×Q8×C197—21no—
74C2×Q8×C197C2311yescharacteristic
75C2×D4×C197C2311yescharacteristic
76C2×C8×C197C2312yes—
77C4⋊C4×C197—21no—
78C4⋊C4×C197C2312yes—
79C22⋊C4×C197—21no—
80C22⋊C4×C197—21no—
81C42×C197C2311yescharacteristic
82C22×D197—21no—
83C22×D197—42no—
84C22×D197—42no—
85C22×D197—42no—
86C22×D197C2×D411yescharacteristic
87Dic394—22no—
88Dic394—44no—
89Dic394—42no—
90Dic394—42no—
91D788—22no—
92C22⋊D197—42no—
93C2×Dic197—21no—
94C2×Dic197—22no—
95C2×Dic197C2×D411yescharacteristic
96C2×Dic197C2×D411yescharacteristic
97C2×Dic197C2×D411yescharacteristic
98C4×D197—22no—
99C4×D197—22no—
100C4×D197—42no—
101C4×D197—44no—
102C4×D197#4114yes—
103#13096802—22no—
104C23×C197—21no—
105Q8×C197—22no—
106Q8×C197—42no—
107D4×C197—22no—
108C2×C4×C197—21no—
109C2×C4×C197—22no—
110C2×C4×C197C2×D411yescharacteristic
111C2×C4×C197C2×D411yescharacteristic
112C2×C4×C197C2411yescharacteristic
113C8×C197—22no—
114C22×C197—41no—
115C22×C197—41no—
116C22×C197—41no—
117C22×C197C22×D411yescharacteristic
118C2×D197—41no—
119C2×D197—41no—
120C2×D197—41no—
121C2×D197—84no—
122C2×D197#12412yes—
123C2×D197AGL(1,ℤ/8)12yes—
124C2×D197#13712yes—
125C4×C197—21no—
126C4×C197—21no—
127C4×C197—41no—
128C4×C197—42no—
129C4×C197C2×#4112yes—
130Dic197—21no—
131Dic197—21no—
132Dic197—41no—
133Dic197—42no—
134Dic197#12412yes—
135C2×C197—81no—
136C2×C197C2×#12411yescharacteristic
137C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C197C2×#13711yescharacteristic
139D197—81no—
140D197#48914yesdirect product factor
141C197C2×#48911yescharacteristic, complemented, Sylow
142C2×#489—1971nomaximal, Sylow
143C2×C4⋊Q8—1971no—
144C2×#124—1971no—
145#489—1971no—
146#489—1971no—
147#489—1972no—
148#489—1972no—
149#489—1971no—
150#489D19711yescharacteristic, direct product factor
151C2×#103—1972no—
152C2×#102—1972no—
153C2×#97—1971no—
154C22×Q8—3941no—
155C22×Q8—1971no—
156C22×D4—1971no—
157C22×C8—1972no—
158C4⋊Q8—3941no—
159C4⋊Q8—3941no—
160C4⋊Q8—1972no—
161C4⋊Q8—1971no—
162C4⋊Q8C2×D19711yescharacteristic, complemented
163#124—3941no—
164#124—3941no—
165#124—1971no—
166#124—1971no—
167#124—1971no—
168#124C2×D19711yescharacteristic
169C2×C4⋊C4—3941no—
170C2×C4⋊C4—1972no—
171C2×C22⋊C4—3941no—
172C2×C22⋊C4—3941no—
173C2×C42—1971no—
174#103—1972no—
175#103—1972no—
176#103—1972no—
177#103C2×D19712yescomplemented
178#102—1972no—
179#102—1972no—
180#102—1972no—
181#102C2×D19712yes—
182#97—1972no—
183#97—1971no—
184#97C2×D19711yescharacteristic, complemented
185C24—3941no—
186C2×Q8—21no—
187C2×Q8—7884no—
188C2×Q8—3941no—
189C2×Q8—3942no—
190C2×Q8—3944no—
191C2×Q8—3941no—
192C2×Q8—1971no—
193C2×Q8C22×D19711yescharacteristic
194C2×D4—3944no—
195C2×D4—3941no—
196C2×D4—1971no—
197C2×D4C22×D19711yescharacteristic
198C22×C4—3941no—
199C22×C4—3942no—
200C22×C4—1971no—
201C22×C4—1971no—
202C22×C4—1971no—
203C2×C8—3944no—
204C2×C8—1972no—
205C2×C8C22×D19712yes—
206C4⋊C4—21no—
207C4⋊C4—3942no—
208C4⋊C4—3942no—
209C4⋊C4—3941no—
210C4⋊C4—1972no—
211C4⋊C4C22×D19712yes—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—3941no—
215C22⋊C4—3941no—
216C22⋊C4—3941no—
217C22⋊C4—3941no—
218C22⋊C4—3941no—
219C22⋊C4—3941no—
220C42—1971no—
221C42—1971no—
222C42—1971no—
223C42C22×D19711yescharacteristic
224C23—21no—
225C23—7882no—
226C23—7882no—
227C23—7882no—
228C23—3941no—
229C23—1971no—
230Q8—22no—
231Q8—42no—
232Q8—7882no—
233Q8—7884no—
234Q8—7882no—
235Q8—3942no—
236D4—22no—
237D4—7882no—
238D4—3942no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—7884no—
242C2×C4—7882no—
243C2×C4—3942no—
244C2×C4—3942no—
245C2×C4—3941no—
246C2×C4—3942no—
247C2×C4—1971no—
248C2×C4—1971no—
249C2×C4—1971no—
250C2×C4—1974no—
251C2×C4D4×D19711yescharacteristic
252C2×C4D4×D19711yescharacteristic
253C2×C4C23×D19711yescharacteristic
254C8—22no—
255C8—3942no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—7881no—
260C22—7881no—
261C22—7881no—
262C22—15764no—
263C22—1972no—
264C22—1972no—
265C22—1972no—
266C22C2×D4×D19711yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—41no—
271C4—7882no—
272C4—7881no—
273C4—3941no—
274C4—3941no—
275C4—1972no—
276C4#41×D19712yes—
277C2—81no—
278C2—15761no—
279C2—1974no—
280C2#124×D19711yescharacteristic
281C2AGL(1,ℤ/8)×D19711yescharacteristic
282C2#137×D19711yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#489×C197C2
socleC22×C197C22×D4
prime radicalC2×D4×D197C22
hypercenter#489D197
hypocenterC197C2×#489
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D197C2semidirect
3. #27161441C2semidirect
4. #27161440C2semidirect
5. #27161439C2semidirect
6. #27161438C2semidirect
7. #27161437C2semidirect
8. #103×D197C2semidirect
10. #97×D197C2semidirect
11. #489×C197C2semidirect
16. #20263326C22semidirect
18. #20263325C22semidirect
30. #20263251C22semidirect
31. #20263250C22semidirect
32. #20263249C22semidirect
36. #20263237C22semidirect
37. #20263236C22semidirect
38. C4⋊Q8×C197C22semidirect
40. #103×C197C22semidirect
42. #97×C197C22semidirect
140. D197#489direct (N × H)
141. C197C2×#489semidirect
150. #489D197direct (N × H)
162. C4⋊Q8C2×D197semidirect
177. #103C2×D197semidirect
184. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1972
#124×D1972
#271614412
#271614402
#271614392
#271614382
#271614372
#103×D1972
#102×D1972
#97×D1972
#489×C1972
C2×#489197