Finite Group Databasebeta

← #38752634

Subgroups — #38752634

Order 25216.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#27161589C211yesmaximal, characteristic
2C2×C4⋊C4×D197C211yesmaximal, characteristic, complemented
3#27161415C211yesmaximal, characteristic, complemented
4#27161414C211yesmaximal, characteristic, complemented
5#27161411C212yesmaximal, complemented
6#27161395C211yesmaximal, characteristic, complemented
7#27161391C212yesmaximal, complemented
8#27161118C212yesmaximal, complemented
9#27161077C212yesmaximal, complemented
10#27161041C211yesmaximal, characteristic, complemented
11#481×C197C211yesmaximal, characteristic, complemented
12C2×#15691357C2211yescharacteristic
13C22×C4×D197—21no—
14C22×C4×D197C2211yescharacteristic
15C2×#15691341C2212yes—
16#20263308—21no—
17#20263306C2211yescharacteristic
18#20263305C2211yescharacteristic
19#20263297C2211yescharacteristic
20#20263294—21no—
21D4×Dic197—21no—
22#20263286—21no—
23#20263284C2211yescharacteristic
24C2×C4⋊Dic197C2211yescharacteristic
25C2×#15691337—21no—
26C4⋊C4×D197—21no—
27C4⋊C4×D197—21no—
28C4⋊C4×D197C2212yes—
29C4⋊C4×D197C2212yes—
30#20263260C2212yescomplemented
31#20263259C2212yescomplemented
32#20263257C2212yescomplemented
33#20263251C2212yescomplemented
34#20263249C2212yescomplemented
35#20263247C2212yescomplemented
36C22⋊#13096802C2211yescharacteristic
37#20263238C2212yes—
38#121×C197C2211yescharacteristic
39C2×C4⋊C4×C197C2211yescharacteristic
40#106×C197C2212yescomplemented
41#102×C197C2212yescomplemented
42C22⋊C8×C197C2211yescharacteristic
43C23×D197D411yescharacteristic
44#15691357—24no—
45C2×Dic394C2311yescharacteristic
46C2×C22⋊D197—21no—
47C2×C22⋊D197—21no—
48C22×Dic197—21no—
49C22×Dic197—21no—
50C22×Dic197D411yescharacteristic
51C2×C4×D197—24no—
52C2×C4×D197—22no—
53C2×C4×D197—22no—
54C2×C4×D197—44no—
55C2×C4×D197D411yescharacteristic
56C2×C4×D197D411yescharacteristic
57C2×C4×D197C2311yescharacteristic
58C2×C4×D197C2311yescharacteristic
59#15691341—24no—
60C2×#13096802C2312yes—
61C4⋊Dic197—21no—
62C4⋊Dic197—21no—
63C4⋊Dic197C2312yes—
64C4⋊Dic197C2311yescharacteristic
65#15691337—22no—
66#15691337—22no—
67#15691337—21no—
68C22⋊Dic197—21no—
69C22⋊Dic197—21no—
70#15691335—21no—
71#15691335—21no—
72C4×Dic197—21no—
73C2×D4×C197—21no—
74C2×D4×C197C2311yescharacteristic
75C22×C4×C197—21no—
76C22×C4×C197C2311yescharacteristic
77C2×C8×C197C2312yes—
78C4⋊C4×C197—21no—
79C4⋊C4×C197C2312yes—
80C4⋊C4×C197C2311yescharacteristic
81C22⋊C4×C197—21no—
82C22×D197—22no—
83C22×D197—22no—
84C22×D197—22no—
85C22×D197C2×D411yescharacteristic
86C22×D197C2×D411yescharacteristic
87Dic394—22no—
88C22⋊D197—42no—
89C22⋊D197—42no—
90C2×Dic197—22no—
91C2×Dic197—21no—
92C2×Dic197—21no—
93C2×Dic197—22no—
94C2×Dic197—42no—
95C2×Dic197—42no—
96C2×Dic197C2×D411yescharacteristic
97C2×Dic197C2×D411yescharacteristic
98C4×D197—22no—
99C4×D197—22no—
100C4×D197—22no—
101C4×D197—44no—
102C4×D197—44no—
103C4×D197#4114yes—
104#13096802—22no—
105C23×C197—21no—
106C23×C197C2×D411yescharacteristic
107D4×C197—22no—
108D4×C197—42no—
109C2×C4×C197—22no—
110C2×C4×C197—21no—
111C2×C4×C197—22no—
112C2×C4×C197—42no—
113C2×C4×C197C2×D411yescharacteristic
114C2×C4×C197C2411yescharacteristic
115C8×C197—22no—
116C22×C197—21no—
117C22×C197—21no—
118C22×C197—41no—
119C22×C197—41no—
120C22×C197—41no—
121C22×C197#13712yes—
122C22×C197C22×D411yescharacteristic
123C2×D197—28no—
124C2×D197—21no—
125C2×D197—21no—
126C2×D197—21no—
127C2×D197—21no—
128C2×D197#12312yes—
129C2×D197AGL(1,ℤ/8)12yes—
130C4×C197—21no—
131C4×C197—42no—
132C4×C197—41no—
133C4×C197C2×#4112yes—
134Dic197—21no—
135Dic197—41no—
136Dic197—42no—
137Dic197—41no—
138Dic197#12312yes—
139C2×C197—22no—
140C2×C197—81no—
141C2×C197C2×#12311yescharacteristic
142C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C197C2×#13711yescharacteristic
144D197—22no—
145D197—22no—
146C197#309211yescharacteristic, complemented, Sylow
147#3092—1971nomaximal, Sylow
148#525—1971no—
149C22×C4⋊C4—1971no—
150#484—1971no—
151#484—1971no—
152#483—1972no—
153#481—1971no—
154#481D19711yescharacteristic, complemented
155#480—1972no—
156#428—1972no—
157#417—1972no—
158#407—1971no—
159C2×#41—1971no—
160C23×C4—3941no—
161C23×C4—1971no—
162C2×M16—1972no—
163#123—3941no—
164C22⋊Q8—1971no—
165C22⋊Q8—1971no—
166#121—1971no—
167#121C2×D19711yescharacteristic
168#120—3941no—
169C4×D4—3941no—
170C4×D4—3941no—
171#117—1971no—
172C2×C4⋊C4—3941no—
173C2×C4⋊C4—3941no—
174C2×C4⋊C4—3941no—
175C2×C4⋊C4—1971no—
176C2×C4⋊C4—1972no—
177C2×C4⋊C4—1972no—
178C2×C4⋊C4C2×D19711yescharacteristic
179#107—1972no—
180#107—1972no—
181#106—1972no—
182#106C2×D19712yescomplemented
183#103—1972no—
184#103—1972no—
185#102—1972no—
186#102C2×D19712yescomplemented
187C22⋊C8—1971no—
188C22⋊C8—1972no—
189C22⋊C8C2×D19711yescharacteristic
190C24—1971no—
191#41—3944no—
192C2×Q8—1971no—
193C2×D4—21no—
194C2×D4—3941no—
195C2×D4—3941no—
196C2×D4C22×D19711yescharacteristic
197C22×C4—21no—
198C22×C4—7884no—
199C22×C4—3942no—
200C22×C4—3942no—
201C22×C4—3944no—
202C22×C4—3941no—
203C22×C4—3941no—
204C22×C4—1971no—
205C22×C4—1971no—
206C22×C4—1971no—
207C22×C4—1971no—
208C22×C4—1971no—
209C22×C4C22×D19711yescharacteristic
210M16—3944no—
211C2×C8—1972no—
212C2×C8C22×D19712yes—
213C4⋊C4—21no—
214C4⋊C4—3942no—
215C4⋊C4—3942no—
216C4⋊C4—3941no—
217C4⋊C4—3941no—
218C4⋊C4—3941no—
219C4⋊C4—1972no—
220C4⋊C4—1971no—
221C4⋊C4C22×D19712yes—
222C4⋊C4C22×D19711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—3941no—
225C22⋊C4—3941no—
226C22⋊C4—3941no—
227C22⋊C4—3941no—
228C42—3941no—
229C23—21no—
230C23—3942no—
231C23—3942no—
232C23—3942no—
233C23—1971no—
234C23—1971no—
235C23D4×D19711yescharacteristic
236Q8—3942no—
237D4—22no—
238D4—42no—
239D4—7882no—
240D4—7882no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—7884no—
246C2×C4—7884no—
247C2×C4—7882no—
248C2×C4—7882no—
249C2×C4—3942no—
250C2×C4—3942no—
251C2×C4—3941no—
252C2×C4—3941no—
253C2×C4—3942no—
254C2×C4—3942no—
255C2×C4—3942no—
256C2×C4—1971no—
257C2×C4—1971no—
258C2×C4—1974no—
259C2×C4D4×D19711yescharacteristic
260C2×C4C23×D19711yescharacteristic
261C8—22no—
262C8—3942no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—3948no—
269C22—3941no—
270C22—3941no—
271C22—3941no—
272C22—3941no—
273C22—1972no—
274C22—1972no—
275C22#2026335512yes—
276C22C2×D4×D19711yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—7882no—
281C4—7881no—
282C4—7881no—
283C4—3941no—
284C4—1972no—
285C4#41×D19712yes—
286C2—22no—
287C2—81no—
288C2—3942no—
289C2—3942no—
290C2#123×D19711yescharacteristic
291C2AGL(1,ℤ/8)×D19711yescharacteristic
292C2C2×#2026335511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#481×C197C2
socleC22×C197C22×D4
prime radical#27161589C2
hypercenter#481D197
hypocenterC197#3092
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D197C2semidirect
3. #27161415C2semidirect
4. #27161414C2semidirect
5. #27161411C2semidirect
6. #27161395C2semidirect
7. #27161391C2semidirect
8. #27161118C2semidirect
9. #27161077C2semidirect
10. #27161041C2semidirect
11. #481×C197C2semidirect
30. #20263260C22semidirect
31. #20263259C22semidirect
32. #20263257C22semidirect
33. #20263251C22semidirect
34. #20263249C22semidirect
35. #20263247C22semidirect
40. #106×C197C22semidirect
41. #102×C197C22semidirect
146. C197#3092semidirect
154. #481D197semidirect
182. #106C2×D197semidirect
186. #102C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271615892
C2×C4⋊C4×D1972
#271614152
#271614142
#271614112
#271613952
#271613912
#271611182
#271610772
#271610412
#481×C1972
#3092197