Finite Group Databasebeta

← #38754307

Subgroups — #38754307

Order 25216.

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

normal subgroups only

This group has 546 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
1C2×#20263350C212yesmaximal, complemented
2#27161733C211yesmaximal, characteristic, complemented
3#27161714C211yesmaximal, characteristic, complemented
4#27161451C211yesmaximal, characteristic, complemented
5#27161324C212yesmaximal, complemented
6#27161285C211yesmaximal, characteristic, complemented
7#27161231C212yesmaximal, complemented
8#27161221C212yesmaximal, complemented
9C4×#15691343C211yesmaximal, characteristic, complemented
10#27161156C211yesmaximal, characteristic, complemented
11#459×C197C211yesmaximal, characteristic, complemented
12C2×#15691357C2212yescomplemented
13C2×D4×D197—21no—
14C2×D4×D197C2212yes—
15#20263350—28no—
16C2×#15691347C2212yes—
17C2×#15691346C2212yescomplemented
18C2×#15691346C2212yescomplemented
19C2×#15691343C2211yescharacteristic
20C2×#15691341C2212yescomplemented
21#20263326C2211yescharacteristic, complemented
22#20263310—22no—
23#20263306C2212yescomplemented
24#20263300—21no—
25#20263297—21no—
26#20263297C2212yes—
27#20263294—22no—
28C4×D788C2211yescharacteristic
29C4×D788C2211yescharacteristic
30C2×C4⋊Dic197—21no—
31#20263259C2211yescharacteristic, complemented
32#20263251C2212yescomplemented
33#20263247C2211yescharacteristic, complemented
34#20263246C2211yescharacteristic
35#20263238C2212yescomplemented
36C4×#13096802C2211yescharacteristic, complemented
37C2×D8×C197C2212yescomplemented
38#127×C197C2211yescharacteristic
39C4×D4×C197C2211yescharacteristic, complemented
40C4⋊C8×C197C2211yescharacteristic, complemented
41#102×C197C2212yescomplemented
42C23×D197—22no—
43#15691357—24no—
44#15691357—42no—
45C2×Dic394C2312yes—
46C2×D788D411yescharacteristic
47C2×D788C2311yescharacteristic
48C2×C22⋊D197—22no—
49C2×C22⋊D197—22no—
50C2×C22⋊D197—22no—
51D4×D197—24no—
52D4×D197—42no—
53D4×D197—44no—
54C22×Dic197—22no—
55C2×C4×D197—21no—
56C2×C4×D197C2312yes—
57#15691347—24no—
58#15691346—24no—
59#15691346—42no—
60#15691343D414yescomplemented
61#15691341—24no—
62C2×#13096802C2311yescharacteristic
63C2×#13096802C2311yescharacteristic
64C4⋊Dic197—22no—
65C4⋊Dic197—21no—
66C4⋊Dic197—21no—
67C4⋊Dic197D411yescharacteristic
68C4⋊Dic197C2311yescharacteristic
69C22⋊Dic197—22no—
70#15691335—22no—
71#15691335—21no—
72#15691335D412yes—
73C2×D4×C197—21no—
74C2×D4×C197—21no—
75C2×D4×C197C2312yes—
76C2×D4×C197C2311yescharacteristic
77C22×C4×C197—21no—
78D8×C197—24no—
79C2×C8×C197C2312yes—
80C4⋊C4×C197C2311yescharacteristic
81C22⋊C4×C197—21no—
82C42×C197C2311yescharacteristic
83C22×D197—21no—
84C22×D197—44no—
85C22×D197—44no—
86C22×D197—82no—
87C22×D197C2×D412yes—
88Dic394—22no—
89Dic394—22no—
90D788—22no—
91D788—21no—
92D788C2×D412yes—
93C22⋊D197—44no—
94C22⋊D197—44no—
95C22⋊D197—44no—
96C2×Dic197—21no—
97C2×Dic197—22no—
98C2×Dic197—44no—
99C2×Dic197C2×D412yes—
100C4×D197—22no—
101C4×D197—22no—
102C4×D197—42no—
103#13096802—21no—
104#13096802C2×D412yescomplemented
105C23×C197—21no—
106C23×C197—22no—
107D4×C197—22no—
108D4×C197—21no—
109D4×C197—22no—
110D4×C197—42no—
111D4×C197—81no—
112D4×C197C2×D412yescomplemented
113C2×C4×C197—21no—
114C2×C4×C197—42no—
115C2×C4×C197C2×D411yescharacteristic
116C2×C4×C197C2×D411yescharacteristic
117C2×C4×C197C2411yescharacteristic
118C8×C197—22no—
119C22×C197—22no—
120C22×C197—41no—
121C22×C197—41no—
122C22×C197—42no—
123C22×C197—42no—
124C22×C197—42no—
125C22×C197C22×D411yescharacteristic
126C2×D197—22no—
127C2×D197—22no—
128C2×D197—22no—
129C2×D197—22no—
130C2×D197—41no—
131C2×D197—41no—
132C2×D197—88no—
133C4×C197—21no—
134C4×C197—41no—
135C4×C197AGL(1,ℤ/8)12yescomplemented
136C4×C197C22×D411yescharacteristic
137C4×C197H(5,𝔽2)11yescharacteristic
138Dic197—22no—
139Dic197—42no—
140Dic197—41no—
141C2×C197—42no—
142C2×C197—82no—
143C2×C197D4211yescharacteristic
144C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C197#57711yescharacteristic
146D197—42no—
147D197—42no—
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#459D19711yescharacteristic, complemented
157#450—1972no—
158#449—1972no—
159C4×D8—1971no—
160#436—1971no—
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—1971no—
169C2×D8C2×D19712yescomplemented
170C2×M16—1972no—
171C4⋊Q8—1971no—
172#127C2×D19711yescharacteristic
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, complemented
182C2×C4⋊C4—3941no—
183#107—1971no—
184C4⋊C8C2×D19711yescharacteristic, complemented
185#103—1972no—
186#102—1971no—
187#102—1971no—
188#102C2×D19712yescomplemented
189C22⋊C8—1972no—
190C4×C8—1971no—
191C24—3942no—
192#41—7882no—
193#41—3944no—
194C2×Q8—1972no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—7884no—
198C2×D4—7882no—
199C2×D4—3944no—
200C2×D4—3942no—
201C2×D4—3942no—
202C2×D4—3942no—
203C2×D4—1971no—
204C2×D4—1971no—
205C2×D4C22×D19712yes—
206C2×D4C22×D19711yescharacteristic
207C22×C4—21no—
208C22×C4—3942no—
209C22×C4—3941no—
210C22×C4—1972no—
211SD16—7882no—
212SD16—3944no—
213SD16—3944no—
214D8—24no—
215D8—1974no—
216M16—3944no—
217C2×C8—1971no—
218C2×C8—1971no—
219C2×C8C22×D19712yes—
220C4⋊C4—3942no—
221C4⋊C4—3941no—
222C4⋊C4—3941no—
223C4⋊C4—1971no—
224C4⋊C4—1971no—
225C4⋊C4C22×D19711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—3942no—
228C22⋊C4—3942no—
229C22⋊C4—3941no—
230C22⋊C4—1972no—
231C42C22×D19711yescharacteristic
232C23—21no—
233C23—22no—
234C23—7884no—
235C23—7884no—
236C23—15762no—
237C23—3941no—
238C23—1972no—
239Q8—3942no—
240Q8—3942no—
241D4—22no—
242D4—21no—
243D4—22no—
244D4—42no—
245D4—81no—
246D4—7884no—
247D4—7884no—
248D4—7884no—
249D4—3942no—
250D4—3941no—
251D4—1972no—
252D4D4×D19712yescomplemented
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—7884no—
256C2×C4—7882no—
257C2×C4—3941no—
258C2×C4—3942no—
259C2×C4—3942no—
260C2×C4—3942no—
261C2×C4—1972no—
262C2×C4D4×D19711yescharacteristic
263C2×C4D4×D19711yescharacteristic
264C2×C4C23×D19711yescharacteristic
265C8—22no—
266C8—3941no—
267C8—1972no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—7881no—
275C22—7881no—
276C22—15768no—
277C22—3942no—
278C22—3942no—
279C22—3942no—
280C22—3942no—
281C22C2×D4×D19711yescharacteristic
282C4—21no—
283C4—41no—
284C4—7881no—
285C4—7882no—
286C4—3942no—
287C4#2026335012yescomplemented
288C4C2×D4×D19711yescharacteristic
289C4#2026337211yescharacteristic
290C2—42no—
291C2—82no—
292C2—7882no—
293C2—7882no—
294C2#2716171311yescharacteristic
295C2C2×#2026335011yescharacteristic
296C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#459×C197C2
socleC22×C197C22×D4
hypercenter#459D197
hypocenterC197#3272
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#20263350C2semidirect
2. #27161733C2semidirect
3. #27161714C2semidirect
4. #27161451C2semidirect
5. #27161324C2semidirect
6. #27161285C2semidirect
7. #27161231C2semidirect
8. #27161221C2semidirect
9. C4×#15691343C2semidirect
10. #27161156C2semidirect
11. #459×C197C2semidirect
12. C2×#15691357C22semidirect
17. C2×#15691346C22semidirect
18. C2×#15691346C22semidirect
20. C2×#15691341C22semidirect
21. #20263326C22semidirect
23. #20263306C22semidirect
31. #20263259C22semidirect
32. #20263251C22semidirect
33. #20263247C22semidirect
35. #20263238C22semidirect
36. C4×#13096802C22semidirect
37. C2×D8×C197C22semidirect
39. C4×D4×C197C22semidirect
40. C4⋊C8×C197C22semidirect
41. #102×C197C22semidirect
60. #15691343D4semidirect
104. #13096802C2×D4semidirect
112. D4×C197C2×D4semidirect
135. C4×C197AGL(1,ℤ/8)semidirect
148. C197#3272semidirect
156. #459D197semidirect
169. C2×D8C2×D197semidirect
181. C4×D4C2×D197semidirect
184. C4⋊C8C2×D197semidirect
188. #102C2×D197semidirect
252. D4D4×D197semidirect
287. C4#20263350semidirect

Maximal subgroups

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

subgroupindex
C2×#202633502
#271617332
#271617142
#271614512
#271613242
#271612852
#271612312
#271612212
C4×#156913432
#271611562
#459×C1972
#3272197