Finite Group Databasebeta

← #38754394

Subgroups — #38754394

Order 25216.

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

normal subgroups only

This group has 526 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×#20263349C211yesmaximal, characteristic, complemented
2C22×D1576C211yesmaximal, characteristic, complemented
3#27161720C212yesmaximal, complemented
4#27161464C211yesmaximal, characteristic, complemented
5#27161316C211yesmaximal, characteristic, complemented
6#27161297C212yesmaximal, complemented
7#27161229C212yesmaximal, complemented
8#27161227C212yesmaximal, complemented
9#27161161C211yesmaximal, characteristic, complemented
10C8×C22⋊D197C211yesmaximal, characteristic, complemented
11#467×C197C211yesmaximal, characteristic, complemented
12C2×#15691359C2212yescomplemented
13C22×D788C2212yes—
14#20263349—24no—
15C2×#15691345C2212yescomplemented
16C2×#15691345C2212yescomplemented
17C2×D1576—24no—
18C2×D1576C2211yescharacteristic
19C2×D1576C2212yes—
20C2×D1576C2211yescharacteristic
21C2×C8×D197C2211yescharacteristic, complemented
22#20263315C2212yescomplemented
23#20263311—22no—
24#20263301—22no—
25#20263301C2212yescomplemented
26#20263296—22no—
27#20263288C2212yescomplemented
28C4×C22⋊D197C2211yescharacteristic
29C22⋊C4×D197—22no—
30#20263255C2211yescharacteristic, complemented
31#20263246C2212yescomplemented
32#20263246C2212yescomplemented
33C22⋊#13096802C2211yescharacteristic
34#20263238C2211yescharacteristic, complemented
35C8×Dic197C2211yescharacteristic, complemented
36C2×Q16×C197C2211yescharacteristic, complemented
37C22×C8×C197C2211yescharacteristic
38C22⋊Q8×C197C2212yes—
39#107×C197C2211yescharacteristic, complemented
40#103×C197C2212yescomplemented
41C23×D197—22no—
42#15691359—24no—
43#15691359—42no—
44C2×D788—24no—
45C2×D788—22no—
46C2×D788—22no—
47C2×D788—22no—
48C2×D788—42no—
49C2×D788C2312yes—
50C2×D788C2312yes—
51C2×C22⋊D197D411yescharacteristic
52C2×C4×D197—22no—
53C2×C4×D197—22no—
54C2×C4×D197C2311yescharacteristic
55#15691345—24no—
56#15691345—42no—
57D1576—24no—
58D1576—22no—
59D1576D414yescomplemented
60C2×#13096802C2311yescharacteristic
61C2×#13096802C2311yescharacteristic
62C8×D197—22no—
63#15691337D411yescharacteristic
64C22⋊Dic197D411yescharacteristic
65#15691335—22no—
66#15691335—22no—
67#15691335—22no—
68#15691335—22no—
69#15691335D411yescharacteristic
70C4×Dic197C2311yescharacteristic
71C2×Q8×C197C2312yes—
72C22×C4×C197C2311yescharacteristic
73Q16×C197—22no—
74C2×C8×C197—22no—
75C2×C8×C197C2311yescharacteristic
76C2×C8×C197C2311yescharacteristic
77C4⋊C4×C197—22no—
78C4⋊C4×C197C2312yes—
79C22⋊C4×C197—22no—
80C22×D197—22no—
81C22×D197—22no—
82C22×D197—44no—
83C22×D197—44no—
84C22×D197—82no—
85C22×D197C2×D411yescharacteristic
86D788—22no—
87D788—22no—
88D788—22no—
89D788—44no—
90D788—44no—
91D788—82no—
92D788C2×D414yes—
93C22⋊D197—22no—
94C2×Dic197C2×D411yescharacteristic
95C2×Dic197C2×D411yescharacteristic
96C2×Dic197C2×D411yescharacteristic
97C4×D197—21no—
98C4×D197—21no—
99C4×D197—44no—
100C4×D197—82no—
101#13096802—21no—
102#13096802—21no—
103C23×C197C2×D411yescharacteristic
104Q8×C197—22no—
105Q8×C197—22no—
106C2×C4×C197—21no—
107C2×C4×C197—21no—
108C2×C4×C197—22no—
109C2×C4×C197—22no—
110C2×C4×C197C2×D411yescharacteristic
111C2×C4×C197C2411yescharacteristic
112C8×C197—21no—
113C8×C197C2×D412yes—
114C22×C197—21no—
115C22×C197—21no—
116C22×C197#13512yescomplemented
117C22×C197C22×D411yescharacteristic
118C2×D197—21no—
119C2×D197—24no—
120C2×D197—21no—
121C2×D197—21no—
122C2×D197—42no—
123C2×D197—42no—
124C2×D197—42no—
125C2×D197—42no—
126C2×D197—42no—
127C2×D197—88no—
128C4×C197—21no—
129C4×C197—42no—
130C4×C197—42no—
131C4×C197C22×D411yescharacteristic
132C4×C197H(5,𝔽2)11yescharacteristic
133Dic197—21no—
134Dic197—21no—
135Dic197—21no—
136C2×C197—22no—
137C2×C197D4211yescharacteristic
138C2×C197C2×#13511yescharacteristic
139C2×C197#57611yescharacteristic
140D197—41no—
141D197—44no—
142D197—82no—
143C197#328011yescharacteristic, complemented, Sylow
144#3280—1971nomaximal, Sylow
145C2×#135—1971no—
146C22×D8—1971no—
147#546—1972no—
148#495—1971no—
149#467D19711yescharacteristic, complemented
150#466—1971no—
151#463—1972no—
152#450—1972no—
153#449—1972no—
154C4×D8—1971no—
155C8×D4—1971no—
156C2×#41—1972no—
157C22×D4—1972no—
158#135—3944no—
159C2×Q16C2×D19711yescharacteristic, complemented
160C2×SD16—1972no—
161C2×SD16—1972no—
162C2×D8—3944no—
163C2×D8—1972no—
164C2×D8—1971no—
165C2×D8—1971no—
166C22×C8—1971no—
167C22×C8C2×D19711yescharacteristic
168#124—1972no—
169#123—3942no—
170C22⋊Q8C2×D19712yes—
171#121—3942no—
172#121—1972no—
173#120—3942no—
174C4×D4—1972no—
175C4×D4—1971no—
176C2×C22⋊C4—3942no—
177#107C2×D19711yescharacteristic, complemented
178C4⋊C8—1971no—
179#103C2×D19712yescomplemented
180#102—1972no—
181#102—1972no—
182C22⋊C8—1971no—
183C22⋊C8—1971no—
184C4×C8—1971no—
185C24—3942no—
186#41—7882no—
187#41—3944no—
188C2×Q8C22×D19712yes—
189C2×D4—7882no—
190C2×D4—3942no—
191C2×D4—3944no—
192C2×D4—3942no—
193C2×D4—3942no—
194C2×D4—1971no—
195C2×D4—1972no—
196C2×D4—1972no—
197C22×C4—3942no—
198C22×C4—3942no—
199C22×C4—1971no—
200C22×C4C22×D19711yescharacteristic
201Q16—22no—
202SD16—7882no—
203SD16—3944no—
204D8—3944no—
205D8—3942no—
206D8—1974no—
207C2×C8—22no—
208C2×C8—3942no—
209C2×C8—1971no—
210C2×C8—1971no—
211C2×C8C22×D19711yescharacteristic
212C2×C8C22×D19711yescharacteristic
213C4⋊C4—22no—
214C4⋊C4—1971no—
215C4⋊C4C22×D19712yes—
216C22⋊C4—22no—
217C22⋊C4—3942no—
218C22⋊C4—3942no—
219C22⋊C4—3942no—
220C22⋊C4—3942no—
221C22⋊C4—1971no—
222C22⋊C4—1971no—
223C42—1971no—
224C23—7884no—
225C23—7884no—
226C23—15762no—
227C23—3942no—
228C23—3942no—
229C23—1971no—
230C23D4×D19711yescharacteristic
231Q8—22no—
232Q8—22no—
233D4—7884no—
234D4—7884no—
235D4—15762no—
236D4—3942no—
237D4—3942no—
238D4—3942no—
239D4—3942no—
240D4—1974no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—7884no—
246C2×C4—15762no—
247C2×C4—3941no—
248C2×C4—3941no—
249C2×C4—1971no—
250C2×C4—1971no—
251C2×C4—1971no—
252C2×C4D4×D19711yescharacteristic
253C2×C4C23×D19711yescharacteristic
254C8—21no—
255C8—3941no—
256C8—3941no—
257C8D4×D19712yes—
258C22—21no—
259C22—21no—
260C22—7882no—
261C22—7882no—
262C22—7882no—
263C22—7882no—
264C22—7882no—
265C22—15768no—
266C22—3941no—
267C22—3941no—
268C22—3944no—
269C22—3941no—
270C22#2026334912yescomplemented
271C22C2×D4×D19711yescharacteristic
272C4—21no—
273C4—42no—
274C4—42no—
275C4—3941no—
276C4—3941no—
277C4—3941no—
278C4C2×D4×D19711yescharacteristic
279C4#2026337211yescharacteristic
280C2—22no—
281C2—7884no—
282C2—7881no—
283C2—15762no—
284C2#2716171311yescharacteristic
285C2C2×#2026334911yescharacteristic
286C2#2716188111yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#467×C197C2
socleC22×C197C22×D4
hypercenter#467D197
hypocenterC197#3280
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#20263349C2semidirect
2. C22×D1576C2semidirect
3. #27161720C2semidirect
4. #27161464C2semidirect
5. #27161316C2semidirect
6. #27161297C2semidirect
7. #27161229C2semidirect
8. #27161227C2semidirect
9. #27161161C2semidirect
10. C8×C22⋊D197C2semidirect
11. #467×C197C2semidirect
12. C2×#15691359C22semidirect
15. C2×#15691345C22semidirect
16. C2×#15691345C22semidirect
21. C2×C8×D197C22semidirect
22. #20263315C22semidirect
25. #20263301C22semidirect
27. #20263288C22semidirect
30. #20263255C22semidirect
31. #20263246C22semidirect
32. #20263246C22semidirect
34. #20263238C22semidirect
35. C8×Dic197C22semidirect
36. C2×Q16×C197C22semidirect
39. #107×C197C22semidirect
40. #103×C197C22semidirect
59. D1576D4semidirect
116. C22×C197#135semidirect
143. C197#3280semidirect
149. #467D197semidirect
159. C2×Q16C2×D197semidirect
177. #107C2×D197semidirect
179. #103C2×D197semidirect
270. C22#20263349semidirect

Maximal subgroups

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

subgroupindex
C2×#202633492
C22×D15762
#271617202
#271614642
#271613162
#271612972
#271612292
#271612272
#271611612
C8×C22⋊D1972
#467×C1972
#3280197