Finite Group Databasebeta

← #38752662

Subgroups — #38752662

Order 25216.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#20263306C211yesmaximal, characteristic
2#117×D197C211yesmaximal, characteristic
3#27161423C212yesmaximal, complemented
4#27161422C212yesmaximal, complemented
5#27161416C212yesmaximal, complemented
6#27161411C212yesmaximal, complemented
7#27161119C211yesmaximal, characteristic, complemented
8#27161089C211yesmaximal, characteristic, complemented
9#27161087C211yesmaximal, characteristic, complemented
10#27161042C211yesmaximal, characteristic
11#428×C197C211yesmaximal, characteristic, complemented
12C22×Dic394C2211yescharacteristic
13C22×C4×D197C2211yescharacteristic
14C2×#15691342C2211yescharacteristic
15#20263306—21no—
16#20263306—21no—
17#20263306C2212yes—
18#20263306C2212yes—
19#20263285C2211yescharacteristic
20#20263284C2211yescharacteristic
21#20263284C2211yescharacteristic
22#20263283—21no—
23#20263282—21no—
24C2×C4⋊Dic197—21no—
25C4⋊C4×D197C2211yescharacteristic
26C4⋊C4×D197C2211yescharacteristic
27C2×#15691335—21no—
28C22⋊C4×D197—21no—
29C42×D197—21no—
30#20263259C2212yescomplemented
31#20263257C2212yescomplemented
32#20263251C2212yescomplemented
33#20263251C2212yescomplemented
34#20263249C2212yescomplemented
35#20263249C2212yescomplemented
36#20263238C2212yes—
37#20263238C2212yes—
38C2×M16×C197C2211yescharacteristic
39#117×C197C2211yescharacteristic
40C2×C4⋊C4×C197C2211yescharacteristic
41#107×C197C2212yescomplemented
42#106×C197C2212yescomplemented
43C23×D197D411yescharacteristic
44C2×Dic394—21no—
45C2×Dic394—22no—
46C2×Dic394—22no—
47C2×Dic394C2312yes—
48C22×Dic197—21no—
49C22×Dic197D411yescharacteristic
50C2×C4×D197—21no—
51C2×C4×D197—21no—
52C2×C4×D197—22no—
53C2×C4×D197—22no—
54C2×C4×D197D411yescharacteristic
55C2×C4×D197D411yescharacteristic
56C2×C4×D197C2311yescharacteristic
57C2×C4×D197C2311yescharacteristic
58#15691342—22no—
59C2×#13096802C2312yes—
60C4⋊Dic197—22no—
61C4⋊Dic197—22no—
62C4⋊Dic197C2311yescharacteristic
63C4⋊Dic197C2311yescharacteristic
64#15691337—21no—
65#15691337—21no—
66C22⋊Dic197—21no—
67#15691335—21no—
68#15691335—21no—
69#15691335—22no—
70#15691335—22no—
71C4×Dic197—21no—
72C4×Dic197—21no—
73C4×Dic197—21no—
74C22×C4×C197—21no—
75C22×C4×C197C2311yescharacteristic
76M16×C197—22no—
77C2×C8×C197C2312yes—
78C4⋊C4×C197—21no—
79C4⋊C4×C197C2311yescharacteristic
80C4⋊C4×C197C2312yes—
81C4⋊C4×C197C2311yescharacteristic
82C22⋊C4×C197—21no—
83C42×C197—21no—
84C22×D197—22no—
85C22×D197—22no—
86C22×D197—41no—
87C22×D197C2×D411yescharacteristic
88C22×D197C2×D411yescharacteristic
89Dic394—22no—
90Dic394—22no—
91Dic394—42no—
92C2×Dic197—22no—
93C2×Dic197—21no—
94C2×Dic197—21no—
95C2×Dic197—22no—
96C2×Dic197—42no—
97C2×Dic197C2×D411yescharacteristic
98C2×Dic197C2×D411yescharacteristic
99C4×D197—22no—
100C4×D197—22no—
101C4×D197—21no—
102C4×D197—21no—
103C4×D197—21no—
104C4×D197—21no—
105C4×D197—42no—
106C4×D197—42no—
107#13096802—22no—
108C23×C197C2×D411yescharacteristic
109C2×C4×C197—21no—
110C2×C4×C197—21no—
111C2×C4×C197—22no—
112C2×C4×C197—42no—
113C2×C4×C197C2×D411yescharacteristic
114C2×C4×C197#4112yes—
115C2×C4×C197#4112yes—
116C2×C4×C197C2411yescharacteristic
117C8×C197—22no—
118C22×C197—21no—
119C22×C197—21no—
120C22×C197#12312yes—
121C22×C197C22×D411yescharacteristic
122C2×D197—21no—
123C2×D197—21no—
124C2×D197—21no—
125C2×D197—21no—
126C2×D197—21no—
127C2×D197—44no—
128C2×D197#13712yes—
129C4×C197—41no—
130C4×C197—42no—
131C4×C197—41no—
132C4×C197#12312yes—
133C4×C197C2×#4111yescharacteristic
134C4×C197C2×#4111yescharacteristic
135Dic197—21no—
136Dic197—21no—
137Dic197—42no—
138Dic197—41no—
139Dic197—41no—
140C2×C197—22no—
141C2×C197C2×#12311yescharacteristic
142C2×C197C2×#13711yescharacteristic
143C2×C197#57511yescharacteristic
144D197—22no—
145D197—41no—
146C197#309511yescharacteristic, complemented, Sylow
147#3095—1971nomaximal, Sylow
148C2×C22⋊Q8—1971no—
149C2×#117—1971no—
150#485—1972no—
151#485—1972no—
152#484—1972no—
153#483—1972no—
154#428—1971no—
155#428D19711yescharacteristic, complemented
156#419—1971no—
157#419—1971no—
158#407—1971no—
159C22×Q8—1971no—
160C23×C4—1971no—
161C2×M16—1971no—
162C2×M16C2×D19711yescharacteristic
163C22⋊Q8—3941no—
164C22⋊Q8—3941no—
165C22⋊Q8—1972no—
166C22⋊Q8—1972no—
167#117—3941no—
168#117—3941no—
169#117—1971no—
170#117—1971no—
171#117—1971no—
172#117C2×D19711yescharacteristic
173C2×C4⋊C4—3941no—
174C2×C4⋊C4—1971no—
175C2×C4⋊C4—1971no—
176C2×C4⋊C4C2×D19711yescharacteristic
177C2×C22⋊C4—3941no—
178C2×C22⋊C4—3941no—
179C2×C42—3941no—
180#107—1972no—
181#107C2×D19712yescomplemented
182#106—1972no—
183#106C2×D19712yescomplemented
184#103—1972no—
185#103—1972no—
186#103—1972no—
187#103—1972no—
188C22⋊C8—1972no—
189C22⋊C8—1972no—
190C24—1971no—
191C2×Q8—3942no—
192C2×Q8—3942no—
193C2×Q8—3941no—
194C2×Q8—1972no—
195C22×C4—21no—
196C22×C4—3941no—
197C22×C4—3941no—
198C22×C4—3941no—
199C22×C4—3942no—
200C22×C4—3942no—
201C22×C4—1971no—
202C22×C4—1971no—
203C22×C4—1971no—
204C22×C4—1971no—
205C22×C4—1971no—
206C22×C4C22×D19711yescharacteristic
207M16—22no—
208M16—3942no—
209C2×C8—1972no—
210C2×C8C22×D19712yes—
211C4⋊C4—21no—
212C4⋊C4—3941no—
213C4⋊C4—3941no—
214C4⋊C4—3942no—
215C4⋊C4—3942no—
216C4⋊C4—1971no—
217C4⋊C4—1971no—
218C4⋊C4C22×D19711yescharacteristic
219C4⋊C4C22×D19711yescharacteristic
220C4⋊C4C22×D19712yes—
221C22⋊C4—21no—
222C22⋊C4—3941no—
223C22⋊C4—3941no—
224C22⋊C4—3942no—
225C22⋊C4—3942no—
226C22⋊C4—3941no—
227C42—21no—
228C42—3941no—
229C42—3941no—
230C42—3941no—
231C23—7881no—
232C23—3942no—
233C23—3942no—
234C23—1971no—
235C23—1971no—
236C23D4×D19711yescharacteristic
237Q8—7882no—
238Q8—3942no—
239Q8—3942no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—7882no—
245C2×C4—7882no—
246C2×C4—7882no—
247C2×C4—3941no—
248C2×C4—3941no—
249C2×C4—3942no—
250C2×C4—3942no—
251C2×C4—3942no—
252C2×C4—3942no—
253C2×C4—3941no—
254C2×C4—3941no—
255C2×C4—3941no—
256C2×C4—3941no—
257C2×C4—1971no—
258C2×C4—1971no—
259C2×C4D4×D19711yescharacteristic
260C2×C4#1569135812yes—
261C2×C4#1569135912yes—
262C2×C4C23×D19711yescharacteristic
263C8—22no—
264C8—3942no—
265C22—21no—
266C22—21no—
267C22—7884no—
268C22—3941no—
269C22—3941no—
270C22—3941no—
271C22—3941no—
272C22—3941no—
273C22—1972no—
274C22#2026331112yes—
275C22C2×D4×D19711yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—7881no—
280C4—7881no—
281C4—7882no—
282C4—3941no—
283C4—3941no—
284C4#2026331112yes—
285C4C2×#1569135811yescharacteristic
286C4C2×#1569135911yescharacteristic
287C2—22no—
288C2—7881no—
289C2—3942no—
290C2C2×#2026331111yescharacteristic
291C2#137×D19711yescharacteristic
292C2#2716187511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#428×C197C2
socleC22×C197C22×D4
prime radicalC23×D197D4
hypercenter#428D197
hypocenterC197#3095
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #27161423C2semidirect
4. #27161422C2semidirect
5. #27161416C2semidirect
6. #27161411C2semidirect
7. #27161119C2semidirect
8. #27161089C2semidirect
9. #27161087C2semidirect
11. #428×C197C2semidirect
30. #20263259C22semidirect
31. #20263257C22semidirect
32. #20263251C22semidirect
33. #20263251C22semidirect
34. #20263249C22semidirect
35. #20263249C22semidirect
41. #107×C197C22semidirect
42. #106×C197C22semidirect
146. C197#3095semidirect
155. #428D197semidirect
181. #107C2×D197semidirect
183. #106C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633062
#117×D1972
#271614232
#271614222
#271614162
#271614112
#271611192
#271610892
#271610872
#271610422
#428×C1972
#3095197