Finite Group Databasebeta

← #38754314

Subgroups — #38754314

Order 25216.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#20263354C212yesmaximal, complemented
2#27161734C211yesmaximal, characteristic, complemented
3D4×D788C211yesmaximal, characteristic, complemented
4#27161452C211yesmaximal, characteristic, complemented
5#27161325C212yesmaximal, complemented
6#27161285C211yesmaximal, characteristic, complemented
7#27161275C211yesmaximal, characteristic, complemented
8#27161232C212yesmaximal, complemented
9#27161226C212yesmaximal, complemented
10C4×D1576C211yesmaximal, characteristic, complemented
11#436×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2212yescomplemented
13C22×D788C2212yes—
14C2×D4×D197—21no—
15#20263354—28no—
16C2×#15691347C2212yes—
17C2×#15691347C2212yescomplemented
18C2×#15691347C2212yescomplemented
19C2×D1576C2211yescharacteristic
20C2×D1576C2212yescomplemented
21#20263326C2211yescharacteristic, complemented
22#20263323C2211yescharacteristic
23#20263308—22no—
24#20263305C2212yescomplemented
25#20263301—21no—
26#20263300C2212yes—
27#20263297—21no—
28#20263296—22no—
29C4×D788C2211yescharacteristic
30C4×D788C2211yescharacteristic, complemented
31C4⋊C4×D197—21no—
32#20263260C2211yescharacteristic, complemented
33#20263251C2212yescomplemented
34#20263248C2211yescharacteristic
35#20263248C2211yescharacteristic, complemented
36#20263248C2212yescomplemented
37C2×M16×C197C2212yescomplemented
38C4×D4×C197C2211yescharacteristic
39C4⋊C8×C197C2211yescharacteristic, complemented
40C22⋊C8×C197C2212yescomplemented
41C4×C8×C197C2211yescharacteristic, complemented
42C23×D197—22no—
43#15691358—24no—
44#15691358—42no—
45C2×Dic394C2312yes—
46C2×D788—24no—
47C2×D788—22no—
48C2×D788—21no—
49C2×D788—21no—
50C2×D788—42no—
51C2×D788C2312yes—
52C2×D788C2311yescharacteristic
53C2×D788C2311yescharacteristic
54C2×C22⋊D197—22no—
55C2×C22⋊D197—22no—
56D4×D197—44no—
57C2×C4×D197—21no—
58C2×C4×D197—22no—
59C2×C4×D197—21no—
60#15691347—24no—
61#15691347—24no—
62#15691347—42no—
63D1576—24no—
64D1576D414yescomplemented
65C4⋊Dic197—21no—
66C4⋊Dic197—21no—
67C4⋊Dic197C2311yescharacteristic
68C4⋊Dic197C2311yescharacteristic
69#15691337—22no—
70C22⋊Dic197—22no—
71#15691335—21no—
72#15691335—21no—
73#15691335—22no—
74C2×D4×C197D411yescharacteristic
75C22×C4×C197C2312yes—
76M16×C197—24no—
77C2×C8×C197C2311yescharacteristic
78C2×C8×C197C2312yes—
79C2×C8×C197C2311yescharacteristic
80C4⋊C4×C197D411yescharacteristic
81C22⋊C4×C197D412yes—
82C42×C197C2311yescharacteristic
83C22×D197—21no—
84C22×D197—21no—
85C22×D197—22no—
86C22×D197—44no—
87C22×D197—44no—
88C22×D197—82no—
89Dic394—22no—
90Dic394—22no—
91D788—22no—
92D788—21no—
93D788—21no—
94D788—22no—
95D788—44no—
96D788—42no—
97D788—81no—
98D788C2×D412yes—
99D788C2×D412yescomplemented
100C22⋊D197—44no—
101C22⋊D197—44no—
102C2×Dic197—21no—
103C2×Dic197—21no—
104C2×Dic197—22no—
105C4×D197—44no—
106C4×D197—42no—
107C4×D197—42no—
108C23×C197C2×D412yes—
109D4×C197—22no—
110C2×C4×C197—22no—
111C2×C4×C197—22no—
112C2×C4×C197C2×D411yescharacteristic
113C2×C4×C197C2×D412yes—
114C2×C4×C197C2×D411yescharacteristic
115C2×C4×C197C2411yescharacteristic
116C8×C197—22no—
117C8×C197—21no—
118C8×C197C2×D412yescomplemented
119C22×C197—22no—
120C22×C197—22no—
121C22×C197—22no—
122C22×C197C22×D411yescharacteristic
123C2×D197—22no—
124C2×D197—22no—
125C2×D197—41no—
126C2×D197—41no—
127C2×D197—41no—
128C2×D197—41no—
129C2×D197—42no—
130C2×D197—42no—
131C2×D197—42no—
132C2×D197—88no—
133C4×C197—22no—
134C4×C197—21no—
135C4×C197AGL(1,ℤ/8)12yescomplemented
136C4×C197C22×D411yescharacteristic
137C4×C197H(5,𝔽2)11yescharacteristic
138Dic197—42no—
139Dic197—41no—
140Dic197—41no—
141C2×C197—42no—
142C2×C197D4211yescharacteristic
143C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C197#57711yescharacteristic
145D197—42no—
146D197—42no—
147D197—82no—
148C197#327211yescharacteristic, complemented, Sylow
149#3272—1971nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1972no—
151#547—1971no—
152D42—1971no—
153#492—1971no—
154#468—1972no—
155#461—1971no—
156#459—1971no—
157#450—1972no—
158#449—1972no—
159C4×D8—1971no—
160#436D19711yescharacteristic, complemented
161C2×#41—1972no—
162C22×D4—3941no—
163C22×D4—1972no—
164AGL(1,ℤ/8)—3948no—
165C2×SD16—1972no—
166C2×SD16—1972no—
167C2×SD16—1972no—
168C2×D8—1972no—
169C2×D8—1971no—
170C2×M16C2×D19712yescomplemented
171C4⋊Q8—1971no—
172#127—1971no—
173#123—3942no—
174C22⋊Q8—1972no—
175#121—3941no—
176#121—3941no—
177#121—1972no—
178#120—3942no—
179C4×D4—1971no—
180C4×D4—1971no—
181C4×D4C2×D19711yescharacteristic
182C2×C4⋊C4—3941no—
183#107—1971no—
184C4⋊C8C2×D19711yescharacteristic, complemented
185#103—1972no—
186#102—1971no—
187#102—1971no—
188#102—1972no—
189C22⋊C8C2×D19712yescomplemented
190C4×C8C2×D19711yescharacteristic, complemented
191C24—3942no—
192#41—7882no—
193#41—3944no—
194C2×Q8—1972no—
195C2×D4—7882no—
196C2×D4—7884no—
197C2×D4—3944no—
198C2×D4—3942no—
199C2×D4—3942no—
200C2×D4—3942no—
201C2×D4—3941no—
202C2×D4—3941no—
203C2×D4—1972no—
204C2×D4—1971no—
205C2×D4—1971no—
206C2×D4D78811yescharacteristic
207C22×C4—3941no—
208C22×C4—3942no—
209C22×C4—3941no—
210C22×C4C22×D19712yes—
211SD16—7882no—
212SD16—3944no—
213SD16—3944no—
214D8—3944no—
215D8—1974no—
216M16—24no—
217C2×C8C22×D19711yescharacteristic
218C2×C8C22×D19712yes—
219C2×C8C22×D19711yescharacteristic
220C4⋊C4—3942no—
221C4⋊C4—3941no—
222C4⋊C4—3941no—
223C4⋊C4—1971no—
224C4⋊C4—1971no—
225C4⋊C4D78811yescharacteristic
226C22⋊C4—3942no—
227C22⋊C4—3941no—
228C22⋊C4—3941no—
229C22⋊C4—3942no—
230C22⋊C4D78812yes—
231C42C22×D19711yescharacteristic
232C23—7884no—
233C23—7884no—
234C23—15762no—
235C23—3942no—
236C23—3941no—
237C23—3941no—
238C23C2×D78812yes—
239Q8—3942no—
240Q8—3942no—
241D4—22no—
242D4—7884no—
243D4—7884no—
244D4—7884no—
245D4—7882no—
246D4—15761no—
247D4—3942no—
248D4—3942no—
249D4—3941no—
250D4—3941no—
251D4—1972no—
252D4—1972no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—7884no—
256C2×C4—7882no—
257C2×C4—7882no—
258C2×C4—3941no—
259C2×C4—3941no—
260C2×C4—3942no—
261C2×C4C2×D78811yescharacteristic
262C2×C4C2×D78812yes—
263C2×C4C2×D78811yescharacteristic
264C2×C4C23×D19711yescharacteristic
265C8—22no—
266C8—21no—
267C8D4×D19712yescomplemented
268C22—22no—
269C22—22no—
270C22—22no—
271C22—7881no—
272C22—7881no—
273C22—7881no—
274C22—7881no—
275C22—7882no—
276C22—7882no—
277C22—7882no—
278C22—15768no—
279C22—3942no—
280C22—3942no—
281C22C22×D78811yescharacteristic
282C4—21no—
283C4—22no—
284C4—7881no—
285C4—7881no—
286C4—7882no—
287C4#2026335412yescomplemented
288C4C2×D4×D19711yescharacteristic
289C4#2026337211yescharacteristic
290C2—42no—
291C2—7882no—
292C2—7882no—
293C2—15762no—
294C2D4×D78811yescharacteristic
295C2C2×#2026335411yescharacteristic
296C2#2716189111yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#436×C197C2
socleC22×C197C22×D4
hypercenter#436D197
hypocenterC197#3272
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#20263354C2semidirect
2. #27161734C2semidirect
3. D4×D788C2semidirect
4. #27161452C2semidirect
5. #27161325C2semidirect
6. #27161285C2semidirect
7. #27161275C2semidirect
8. #27161232C2semidirect
9. #27161226C2semidirect
10. C4×D1576C2semidirect
11. #436×C197C2semidirect
12. C2×#15691358C22semidirect
17. C2×#15691347C22semidirect
18. C2×#15691347C22semidirect
20. C2×D1576C22semidirect
21. #20263326C22semidirect
24. #20263305C22semidirect
30. C4×D788C22semidirect
32. #20263260C22semidirect
33. #20263251C22semidirect
35. #20263248C22semidirect
36. #20263248C22semidirect
37. C2×M16×C197C22semidirect
39. C4⋊C8×C197C22semidirect
40. C22⋊C8×C197C22semidirect
41. C4×C8×C197C22semidirect
64. D1576D4semidirect
99. D788C2×D4semidirect
118. C8×C197C2×D4semidirect
135. C4×C197AGL(1,ℤ/8)semidirect
148. C197#3272semidirect
160. #436D197semidirect
170. C2×M16C2×D197semidirect
184. C4⋊C8C2×D197semidirect
189. C22⋊C8C2×D197semidirect
190. C4×C8C2×D197semidirect
267. C8D4×D197semidirect
287. C4#20263354semidirect

Maximal subgroups

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

subgroupindex
C2×#202633542
#271617342
D4×D7882
#271614522
#271613252
#271612852
#271612752
#271612322
#271612262
C4×D15762
#436×C1972
#3272197