Finite Group Databasebeta

← #469×D197 #38752304

Subgroups — #469×D197 #38752304

Order 25216.

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

normal subgroups only

This group has 596 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
1C2×D8×D197C211yesmaximal, characteristic, complemented
2C2×M16×D197C211yesmaximal, characteristic, complemented
3#121×D197C212yesmaximal, complemented
4#27161333C211yesmaximal, characteristic, complemented
5#27161332C211yesmaximal, characteristic, complemented
6#27161331C212yesmaximal, complemented
7#27161330C212yesmaximal, complemented
8#27161329C211yesmaximal, characteristic, complemented
9#106×D197C211yesmaximal, characteristic, complemented
10#102×D197C212yesmaximal, complemented
11#469×C197C211yesmaximal, characteristic, complemented
12C2×D4×D197—22no—
13C2×D4×D197C2212yes—
14C22×C4×D197C2211yescharacteristic
15C2×D1576C2211yescharacteristic, complemented
16C2×#15691343C2212yescomplemented
17D8×D197—24no—
18C2×#15691342C2211yescharacteristic, complemented
19C2×#15691341C2211yescharacteristic, complemented
20C2×#15691341C2211yescharacteristic, complemented
21M16×D197—24no—
22C2×C8×D197C2211yescharacteristic
23C2×C8×D197C2211yescharacteristic
24#20263301C2212yescomplemented
25#20263300C2212yescomplemented
26#20263299—22no—
27#20263298—22no—
28#20263297C2212yescomplemented
29C4⋊C4×D197C2212yes—
30C22⋊C4×D197—22no—
31#20263258C2211yescharacteristic, complemented
32#20263257C2212yescomplemented
33#20263248C2212yescomplemented
34#20263247C2212yescomplemented
35#20263246C2212yescomplemented
36C2×D8×C197C2211yescharacteristic, complemented
37C2×M16×C197C2211yescharacteristic, complemented
38#121×C197C2212yescomplemented
39#106×C197C2211yescharacteristic, complemented
40#102×C197C2212yescomplemented
41C23×D197—22no—
42C23×D197D411yescharacteristic
43C2×D788—22no—
44C2×D788C2312yes—
45C2×C22⋊D197—22no—
46C2×C22⋊D197—22no—
47C2×C22⋊D197—22no—
48D4×D197—24no—
49D4×D197—24no—
50D4×D197—48no—
51C22×Dic197D411yescharacteristic
52C2×C4×D197—22no—
53C2×C4×D197—22no—
54C2×C4×D197—22no—
55C2×C4×D197D411yescharacteristic
56C2×C4×D197D411yescharacteristic
57C2×C4×D197C2311yescharacteristic
58C2×C4×D197C2311yescharacteristic
59D1576—22no—
60#15691343—24no—
61#15691342—22no—
62#15691341—22no—
63#15691341—22no—
64C2×#13096802C2311yescharacteristic
65C2×#13096802C2311yescharacteristic
66C8×D197—22no—
67C8×D197D414yescomplemented
68C4⋊Dic197C2312yes—
69#15691337—22no—
70C22⋊Dic197—22no—
71#15691335—22no—
72#15691335—22no—
73C2×D4×C197—22no—
74C2×D4×C197C2312yes—
75C22×C4×C197C2311yescharacteristic
76D8×C197—22no—
77M16×C197—22no—
78C2×C8×C197C2311yescharacteristic
79C2×C8×C197C2311yescharacteristic
80C4⋊C4×C197C2312yes—
81C22⋊C4×C197—22no—
82C22×D197—22no—
83C22×D197—24no—
84C22×D197—22no—
85C22×D197—44no—
86C22×D197—44no—
87C22×D197—44no—
88C22×D197C2×D411yescharacteristic
89C22×D197C2×D411yescharacteristic
90D788—22no—
91D788—22no—
92D788—44no—
93C22⋊D197—44no—
94C22⋊D197—44no—
95C22⋊D197—44no—
96C2×Dic197—22no—
97C2×Dic197—22no—
98C2×Dic197C2×D411yescharacteristic
99C2×Dic197C2×D411yescharacteristic
100C4×D197—22no—
101C4×D197—21no—
102C4×D197—21no—
103C4×D197—22no—
104C4×D197—44no—
105C4×D197C2×D412yes—
106C4×D197#4112yes—
107#13096802—21no—
108#13096802C2×D412yescomplemented
109C23×C197—22no—
110C23×C197C2×D411yescharacteristic
111D4×C197—22no—
112D4×C197—22no—
113D4×C197—44no—
114C2×C4×C197—21no—
115C2×C4×C197—21no—
116C2×C4×C197—22no—
117C2×C4×C197C2×D411yescharacteristic
118C2×C4×C197C2411yescharacteristic
119C8×C197—21no—
120C8×C197C2×D412yescomplemented
121C22×C197—21no—
122C22×C197—22no—
123C22×C197—42no—
124C22×C197—42no—
125C22×C197—42no—
126C22×C197C22×D411yescharacteristic
127C2×D197—21no—
128C2×D197—22no—
129C2×D197—44no—
130C2×D197—42no—
131C2×D197—42no—
132C2×D197—42no—
133C2×D197—88no—
134C2×D197#12112yes—
135C2×D197AGL(1,ℤ/8)14yes—
136C4×C197—21no—
137C4×C197—42no—
138C4×C197C22×D411yescharacteristic
139C4×C197C2×#4111yescharacteristic
140Dic197—21no—
141Dic197—42no—
142Dic197#12112yes—
143C2×C197—41no—
144C2×C197—82no—
145C2×C197C2×#12111yescharacteristic
146C2×C197C2×AGL(1,ℤ/8)12yes—
147D197—41no—
148D197—82no—
149D197#46914yesdirect product factor
150C197C2×#46911yescharacteristic, complemented, Sylow
151C2×#469—1971nomaximal, Sylow
152C22×D8—1971no—
153C22×M16—1971no—
154C2×#121—1972no—
155#469—1971no—
156#469—1971no—
157#469—1972no—
158#469—1971no—
159#469—1972no—
160#469D19711yescharacteristic, direct product factor
161C2×#106—1971no—
162C2×#102—1972no—
163C22×D4—3942no—
164C22×D4—1972no—
165C23×C4—1971no—
166C2×D8—3944no—
167C2×D8—1972no—
168C2×D8—1971no—
169C2×D8C2×D19711yescharacteristic, complemented
170C2×M16—3944no—
171C2×M16—1971no—
172C2×M16—1971no—
173C2×M16—1971no—
174C2×M16C2×D19711yescharacteristic, complemented
175C22×C8—1971no—
176C22×C8—1971no—
177#121—3942no—
178#121—3942no—
179#121—1972no—
180#121—1972no—
181#121—1972no—
182#121C2×D19712yescomplemented
183C2×C4⋊C4—1972no—
184C2×C22⋊C4—3942no—
185#106—1971no—
186#106—1972no—
187#106C2×D19711yescharacteristic, complemented
188#102—1972no—
189#102—1972no—
190#102—1972no—
191#102C2×D19712yescomplemented
192C24—3942no—
193C24—1971no—
194C2×D4—22no—
195C2×D4—7888no—
196C2×D4—3942no—
197C2×D4—3942no—
198C2×D4—3942no—
199C2×D4—3944no—
200C2×D4—3944no—
201C2×D4—3942no—
202C2×D4—1972no—
203C2×D4C22×D19712yes—
204C22×C4—3942no—
205C22×C4—3942no—
206C22×C4—3942no—
207C22×C4—1971no—
208C22×C4—1971no—
209C22×C4—1971no—
210C22×C4—1971no—
211C22×C4—1971no—
212C22×C4C22×D19711yescharacteristic
213D8—22no—
214D8—3944no—
215D8—3942no—
216M16—22no—
217M16—3942no—
218M16—3942no—
219M16—3942no—
220C2×C8—3942no—
221C2×C8—1971no—
222C2×C8—1971no—
223C2×C8—1974no—
224C2×C8C22×D19711yescharacteristic
225C2×C8C22×D19711yescharacteristic
226C4⋊C4—3942no—
227C4⋊C4—1972no—
228C4⋊C4C22×D19712yes—
229C22⋊C4—22no—
230C22⋊C4—3942no—
231C22⋊C4—3942no—
232C22⋊C4—3942no—
233C23—22no—
234C23—7884no—
235C23—7884no—
236C23—7884no—
237C23—3942no—
238C23—3944no—
239C23—3942no—
240C23—1971no—
241C23—1971no—
242C23D4×D19711yescharacteristic
243D4—22no—
244D4—22no—
245D4—44no—
246D4—7884no—
247D4—7884no—
248D4—7884no—
249D4—7884no—
250D4—3942no—
251D4—3942no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—7884no—
256C2×C4—3942no—
257C2×C4—3942no—
258C2×C4—3942no—
259C2×C4—3942no—
260C2×C4—3941no—
261C2×C4—3941no—
262C2×C4—1971no—
263C2×C4—1971no—
264C2×C4—1972no—
265C2×C4—1972no—
266C2×C4D4×D19711yescharacteristic
267C2×C4C23×D19711yescharacteristic
268C8—21no—
269C8—3941no—
270C8—1972no—
271C8D4×D19712yescomplemented
272C22—22no—
273C22—21no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—7884no—
278C22—7882no—
279C22—7882no—
280C22—7882no—
281C22—15768no—
282C22—3942no—
283C22—3941no—
284C22—1974no—
285C22—1972no—
286C22C2×D4×D19711yescharacteristic
287C4—21no—
288C4—42no—
289C4—7882no—
290C4—3941no—
291C4—1972no—
292C4C2×D4×D19711yescharacteristic
293C4#41×D19711yescharacteristic
294C2—41no—
295C2—82no—
296C2—7881no—
297C2—15762no—
298C2—1974no—
299C2#121×D19711yescharacteristic
300C2AGL(1,ℤ/8)×D19712yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#469×C197C2
socleC22×C197C22×D4
hypercenter#469D197
hypocenterC197C2×#469
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D197C2semidirect
2. C2×M16×D197C2semidirect
3. #121×D197C2semidirect
4. #27161333C2semidirect
5. #27161332C2semidirect
6. #27161331C2semidirect
7. #27161330C2semidirect
8. #27161329C2semidirect
9. #106×D197C2semidirect
10. #102×D197C2semidirect
11. #469×C197C2semidirect
15. C2×D1576C22semidirect
16. C2×#15691343C22semidirect
18. C2×#15691342C22semidirect
19. C2×#15691341C22semidirect
20. C2×#15691341C22semidirect
24. #20263301C22semidirect
25. #20263300C22semidirect
28. #20263297C22semidirect
31. #20263258C22semidirect
32. #20263257C22semidirect
33. #20263248C22semidirect
34. #20263247C22semidirect
35. #20263246C22semidirect
36. C2×D8×C197C22semidirect
37. C2×M16×C197C22semidirect
38. #121×C197C22semidirect
39. #106×C197C22semidirect
40. #102×C197C22semidirect
67. C8×D197D4semidirect
108. #13096802C2×D4semidirect
120. C8×C197C2×D4semidirect
149. D197#469direct (N × H)
150. C197C2×#469semidirect
160. #469D197direct (N × H)
169. C2×D8C2×D197semidirect
174. C2×M16C2×D197semidirect
182. #121C2×D197semidirect
187. #106C2×D197semidirect
191. #102C2×D197semidirect
271. C8D4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1972
C2×M16×D1972
#121×D1972
#271613332
#271613322
#271613312
#271613302
#271613292
#106×D1972
#102×D1972
#469×C1972
C2×#469197