Finite Group Databasebeta

← #38752850

Subgroups — #38752850

Order 25216.

450 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#27161613C211yesmaximal, characteristic, complemented
2#27161409C211yesmaximal, characteristic, complemented
3#27161392C211yesmaximal, characteristic, complemented
4#27161379C212yesmaximal, complemented
5#27161294C212yesmaximal, complemented
6#27161230C211yesmaximal, characteristic, complemented
7#27161213C211yesmaximal, characteristic
8#27161189C212yesmaximal, complemented
9#27161179C212yesmaximal, complemented
10#27161136C211yesmaximal, characteristic, complemented
11C2×C4×D4×C197C211yesmaximal, characteristic, complemented
12C2×#15691346C2212yes—
13C2×#15691343C2212yes—
14#20263320—21no—
15#20263318C2211yescharacteristic, complemented
16#20263314C2211yescharacteristic, complemented
17#20263309—21no—
18#20263305C2211yescharacteristic
19#20263300C2211yescharacteristic
20C4×Dic394C2211yescharacteristic, complemented
21C4×D788C2211yescharacteristic, complemented
22C4×C22⋊D197—21no—
23#20263283—21no—
24#20263259C2212yescomplemented
25#20263257C2212yescomplemented
26C4⋊#13096802C2212yescomplemented
27#20263249C2212yescomplemented
28#20263247C2212yes—
29#20263247C2212yes—
30#20263246C2212yescomplemented
31C22⋊#13096802C2211yescharacteristic
32C22⋊#13096802C2211yescharacteristic
33#20263237C2212yescomplemented
34C22×D4×C197C2211yescharacteristic
35C23×C4×C197—21no—
36C4×D4×C197—21no—
37C4×D4×C197—21no—
38C4×D4×C197C2212yes—
39C4×D4×C197C2212yes—
40C2×C4⋊C4×C197C2211yescharacteristic
41C2×C22⋊C4×C197—21no—
42C2×C42×C197C2211yescharacteristic
43C2×Dic394C2311yescharacteristic
44C2×D788C2311yescharacteristic
45C2×C22⋊D197—21no—
46C2×C4×D197—21no—
47#15691346—24no—
48#15691343—24no—
49C2×#13096802C2312yes—
50C2×#13096802C2312yes—
51C4⋊Dic197C2311yescharacteristic
52C4⋊Dic197C2311yescharacteristic
53#15691337—21no—
54#15691337—21no—
55#15691337—21no—
56#15691337—21no—
57C22⋊Dic197—21no—
58#15691335—21no—
59#15691335—21no—
60#15691335—21no—
61#15691335—21no—
62C4×Dic197—21no—
63C24×C197—21no—
64C2×D4×C197—24no—
65C2×D4×C197—21no—
66C2×D4×C197C2312yes—
67C22×C4×C197—21no—
68C22×C4×C197—22no—
69C22×C4×C197—22no—
70C22×C4×C197—44no—
71C22×C4×C197D411yescharacteristic
72C22×C4×C197D411yescharacteristic
73C22×C4×C197C2311yescharacteristic
74C4⋊C4×C197—21no—
75C4⋊C4×C197C2312yes—
76C22⋊C4×C197—22no—
77C22⋊C4×C197—22no—
78C42×C197D411yescharacteristic
79C42×C197D411yescharacteristic
80C42×C197C2311yescharacteristic
81C42×C197C2311yescharacteristic
82C22×D197—21no—
83Dic394—22no—
84D788—22no—
85C22⋊D197—42no—
86C2×Dic197—21no—
87C2×Dic197—21no—
88C2×Dic197—21no—
89C4×D197—42no—
90#13096802—22no—
91#13096802—22no—
92C23×C197—24no—
93C23×C197—22no—
94C23×C197—42no—
95C23×C197—42no—
96C23×C197C2×D411yescharacteristic
97D4×C197—24no—
98D4×C197—22no—
99D4×C197#4114yes—
100C2×C4×C197—22no—
101C2×C4×C197—22no—
102C2×C4×C197—22no—
103C2×C4×C197—41no—
104C2×C4×C197—44no—
105C2×C4×C197—44no—
106C2×C4×C197—42no—
107C2×C4×C197C2×D411yescharacteristic
108C2×C4×C197C2×D411yescharacteristic
109C2×C4×C197C2×D411yescharacteristic
110C2×C4×C197C2×D411yescharacteristic
111C2×C4×C197C2×D411yescharacteristic
112C2×C4×C197C2411yescharacteristic
113C22×C197—21no—
114C22×C197—21no—
115C22×C197—24no—
116C22×C197—48no—
117C22×C197—42no—
118C22×C197—42no—
119C22×C197AGL(1,ℤ/8)12yes—
120C22×C197C22×D411yescharacteristic
121C2×D197—41no—
122C2×D197—41no—
123C2×D197—41no—
124C4×C197—21no—
125C4×C197—21no—
126C4×C197—21no—
127C4×C197—21no—
128C4×C197—21no—
129C4×C197—42no—
130C4×C197C2×#4112yes—
131Dic197—41no—
132Dic197—41no—
133Dic197—41no—
134C2×C197—22no—
135C2×C197—44no—
136C2×C197#52511yescharacteristic
137C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C197#57511yescharacteristic
139D197—81no—
140C197#311411yescharacteristic, complemented, Sylow
141#3114—1971nomaximal, Sylow
142#529—1971no—
143C2×C4×D4D19711yescharacteristic, complemented
144#483—1971no—
145#480—1971no—
146#478—1972no—
147#463—1972no—
148#450—1971no—
149#447—1971no—
150#442—1972no—
151#440—1972no—
152#432—1971no—
153C22×D4C2×D19711yescharacteristic
154C23×C4—21no—
155C2×SD16—1972no—
156C2×D8—1972no—
157#126—3941no—
158#125—1971no—
159#124—1971no—
160#123—3941no—
161C22⋊Q8—1971no—
162#121—1971no—
163C4×Q8—1971no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—3941no—
167C4×D4—1971no—
168C4×D4C2×D19712yes—
169C4×D4C2×D19712yes—
170#117—3941no—
171C2×C4⋊C4C2×D19711yescharacteristic
172C2×C22⋊C4—21no—
173C2×C42C2×D19711yescharacteristic
174#107—1972no—
175#106—1972no—
176C4⋊C8—1972no—
177#103—1972no—
178#102—1972no—
179#102—1972no—
180#102—1972no—
181C22⋊C8—1971no—
182C22⋊C8—1971no—
183#97—1972no—
184C24—21no—
185C2×Q8—1971no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—3941no—
189C2×D4—1971no—
190C2×D4C22×D19712yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—3941no—
196C22×C4C22⋊D19711yescharacteristic
197C22×C4C22⋊D19711yescharacteristic
198C22×C4C22×D19711yescharacteristic
199SD16—3944no—
200D8—3944no—
201C2×C8—1972no—
202C2×C8—1972no—
203C4⋊C4—21no—
204C4⋊C4—3941no—
205C4⋊C4—3941no—
206C4⋊C4—3941no—
207C4⋊C4—3941no—
208C4⋊C4—1971no—
209C4⋊C4—1971no—
210C4⋊C4C22×D19712yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—3941no—
214C22⋊C4—3941no—
215C22⋊C4—3941no—
216C22⋊C4—3941no—
217C22⋊C4—3941no—
218C42—3941no—
219C42C22⋊D19711yescharacteristic
220C42C22⋊D19711yescharacteristic
221C42C22×D19711yescharacteristic
222C42C22×D19711yescharacteristic
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—3941no—
228C23C2×C22⋊D19711yescharacteristic
229Q8—3942no—
230D4—24no—
231D4—22no—
232D4—7882no—
233D4—3942no—
234D4#1569135814yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—7882no—
243C2×C4—3941no—
244C2×C4—3941no—
245C2×C4—3941no—
246C2×C4C2×C22⋊D19711yescharacteristic
247C2×C4C2×C22⋊D19711yescharacteristic
248C2×C4C2×C22⋊D19711yescharacteristic
249C2×C4C2×C22⋊D19711yescharacteristic
250C2×C4C2×C22⋊D19711yescharacteristic
251C2×C4C23×D19711yescharacteristic
252C8—3942no—
253C8—3942no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—7881no—
261C22—7881no—
262C22—7881no—
263C22#2026335212yes—
264C22C22×C22⋊D19711yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—7881no—
272C4—7881no—
273C4—7881no—
274C4C2×#1569135812yes—
275C2—22no—
276C2—44no—
277C2—15761no—
278C2#2716159011yescharacteristic
279C2C2×#2026335211yescharacteristic
280C2#2716187711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×C4×D4×C197C2
socleC22×C197C22×D4
prime radical#27161213C2
hypercenterC2×C4×D4D197
hypocenterC197#3114
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #27161613C2semidirect
2. #27161409C2semidirect
3. #27161392C2semidirect
4. #27161379C2semidirect
5. #27161294C2semidirect
6. #27161230C2semidirect
8. #27161189C2semidirect
9. #27161179C2semidirect
10. #27161136C2semidirect
11. C2×C4×D4×C197C2semidirect
15. #20263318C22semidirect
16. #20263314C22semidirect
20. C4×Dic394C22semidirect
21. C4×D788C22semidirect
24. #20263259C22semidirect
25. #20263257C22semidirect
26. C4⋊#13096802C22semidirect
27. #20263249C22semidirect
30. #20263246C22semidirect
33. #20263237C22semidirect
140. C197#3114semidirect
143. C2×C4×D4D197semidirect

Maximal subgroups

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

subgroupindex
#271616132
#271614092
#271613922
#271613792
#271612942
#271612302
#271612132
#271611892
#271611792
#271611362
C2×C4×D4×C1972
#3114197