Finite Group Databasebeta

← #38751037

Subgroups — #38751037

Order 25216.

582 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 153 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×#20263310C211yesmaximal, characteristic
2C22×C4×Dic197C211yesmaximal, characteristic, complemented
3#27160969C212yesmaximal, complemented
4#27160968C212yesmaximal, complemented
5#27160966C212yesmaximal, complemented
6#27160965C212yesmaximal, complemented
7#27160958C211yesmaximal, characteristic
8#27160956C211yesmaximal, characteristic
9#27160951C211yesmaximal, characteristic, complemented
10#27160940C211yesmaximal, characteristic, complemented
11#385×C197C211yesmaximal, characteristic, complemented
12C22×C22⋊D197C2211yescharacteristic
13C23×Dic197C2211yescharacteristic
14C23×Dic197C2211yescharacteristic
15#20263310—24no—
16C2×C4⋊Dic197C2211yescharacteristic
17C2×C4⋊Dic197C2211yescharacteristic
18C2×#15691337C2212yes—
19C2×C22⋊Dic197C2211yescharacteristic
20C2×#15691335C2212yes—
21C2×#15691335C2212yes—
22C2×#15691335C2211yescharacteristic
23C2×#15691335C2211yescharacteristic
24C2×C4×Dic197—24no—
25C2×C4×Dic197C2212yes—
26C2×C4×Dic197C2212yes—
27#20263233C2212yes—
28#20263233C2212yes—
29#20263233C2212yes—
30#20263233C2212yes—
31#20263233C2212yes—
32C23×C4×C197C2211yescharacteristic
33C2×C4⋊C4×C197C2212yes—
34C2×C22⋊C4×C197C2211yescharacteristic
35C2×C22⋊C4×C197C2211yescharacteristic
36#95×C197C2212yes—
37C23×D197C2311yescharacteristic
38C2×C22⋊D197—22no—
39C2×C22⋊D197—22no—
40C2×C22⋊D197—22no—
41C22×Dic197—24no—
42C22×Dic197—22no—
43C22×Dic197—22no—
44C22×Dic197—22no—
45C22×Dic197D414yes—
46C22×Dic197C2312yes—
47C22×Dic197C2311yescharacteristic
48C22×Dic197C2311yescharacteristic
49C22×Dic197C2311yescharacteristic
50C22×Dic197C2312yes—
51C4⋊Dic197—22no—
52C4⋊Dic197—22no—
53#15691337—24no—
54C22⋊Dic197—22no—
55#15691335—22no—
56#15691335—22no—
57#15691335—24no—
58#15691335—24no—
59C4×Dic197—24no—
60C4×Dic197—24no—
61C24×C197C2311yescharacteristic
62C22×C4×C197—22no—
63C22×C4×C197—24no—
64C22×C4×C197C2311yescharacteristic
65C22×C4×C197C2312yes—
66C22×C4×C197C2311yescharacteristic
67C22×C4×C197C2312yes—
68C4⋊C4×C197—24no—
69C22⋊C4×C197—22no—
70C22⋊C4×C197—22no—
71C22×D197—21no—
72C22×D197—22no—
73C22×D197—22no—
74C22×D197—21no—
75C22×D197—21no—
76C22⋊D197—44no—
77C2×Dic197—28no—
78C2×Dic197—28no—
79C2×Dic197—22no—
80C2×Dic197—22no—
81C2×Dic197—21no—
82C2×Dic197—21no—
83C2×Dic197—21no—
84C2×Dic197—21no—
85C2×Dic197—21no—
86C2×Dic197—22no—
87C2×Dic197—22no—
88C2×Dic197—22no—
89C2×Dic197—21no—
90C2×Dic197—21no—
91C2×Dic197C2×D412yes—
92C2×Dic197C2×D412yes—
93C2×Dic197#4114yes—
94C23×C197—22no—
95C23×C197—22no—
96C23×C197C2×D412yes—
97C23×C197#4112yes—
98C23×C197#4112yes—
99C23×C197C2411yescharacteristic
100C2×C4×C197—28no—
101C2×C4×C197—22no—
102C2×C4×C197—22no—
103C2×C4×C197—21no—
104C2×C4×C197—21no—
105C2×C4×C197—22no—
106C2×C4×C197—22no—
107C2×C4×C197—22no—
108C2×C4×C197—22no—
109C2×C4×C197—22no—
110C2×C4×C197#4114yes—
111C22×C197—22no—
112C22×C197—22no—
113C22×C197—24no—
114C22×C197—24no—
115C22×C197#12314yes—
116C22×C197C22×D411yescharacteristic
117C22×C197C2×#4112yes—
118C22×C197C2×#4112yes—
119C22×C197C2×#4111yescharacteristic
120C22×C197C2×#4111yescharacteristic
121C2×D197—41no—
122C2×D197—41no—
123C2×D197—41no—
124C2×D197—42no—
125C2×D197—42no—
126C4×C197—24no—
127C4×C197—41no—
128C4×C197—41no—
129C4×C197—42no—
130Dic197—22no—
131Dic197—22no—
132Dic197—24no—
133Dic197—41no—
134Dic197—42no—
135C2×C197—24no—
136C2×C197C2×#12311yescharacteristic
137C2×C197#52511yescharacteristic
138C2×C197#52511yescharacteristic
139C2×C197#52912yes—
140C2×C197#52912yes—
141D197—81no—
142C197#285311yescharacteristic, complemented, Sylow
143#2853—1971nomaximal, Sylow
144C2×#123—1971no—
145C22×C42—1971no—
146#388—1972no—
147#388—1972no—
148#387—1972no—
149#387—1972no—
150#386—1971no—
151#386—1971no—
152#385—1971no—
153#385D19711yescharacteristic, complemented
154#381—1971no—
155C22×D4—1971no—
156C23×C4—1971no—
157C23×C4—1971no—
158C23×C4C2×D19711yescharacteristic
159#123—3944no—
160C2×C4⋊C4—1971no—
161C2×C4⋊C4—1971no—
162C2×C4⋊C4—1972no—
163C2×C4⋊C4C2×D19712yes—
164C2×C22⋊C4—1971no—
165C2×C22⋊C4—1971no—
166C2×C22⋊C4—1971no—
167C2×C22⋊C4—1972no—
168C2×C22⋊C4—1972no—
169C2×C22⋊C4C2×D19711yescharacteristic
170C2×C22⋊C4C2×D19711yescharacteristic
171C2×C42—3944no—
172C2×C42—1972no—
173C2×C42—1972no—
174#95—1972no—
175#95—1972no—
176#95—1972no—
177#95—1972no—
178#95—1972no—
179#95C2×D19712yes—
180C24—1971no—
181C24C22×D19711yescharacteristic
182C2×D4—3942no—
183C2×D4—3942no—
184C2×D4—3942no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—3942no—
188C22×C4—3942no—
189C22×C4—3942no—
190C22×C4—3944no—
191C22×C4—1972no—
192C22×C4—1971no—
193C22×C4—1974no—
194C22×C4—1972no—
195C22×C4—1971no—
196C22×C4—1971no—
197C22×C4C22×D19712yes—
198C22×C4C22×D19711yescharacteristic
199C22×C4C22×D19711yescharacteristic
200C22×C4C22×D19712yes—
201C4⋊C4—24no—
202C4⋊C4—3944no—
203C4⋊C4—3942no—
204C4⋊C4—3942no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—3944no—
208C22⋊C4—3942no—
209C22⋊C4—3942no—
210C22⋊C4—3942no—
211C22⋊C4—3944no—
212C42—3944no—
213C42—3944no—
214C23—22no—
215C23—22no—
216C23—3941no—
217C23—3942no—
218C23—3942no—
219C23—3941no—
220C23—3941no—
221C23D4×D19712yes—
222C23#1569135812yes—
223C23#1569135912yes—
224C23C23×D19711yescharacteristic
225D4—7884no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—3941no—
237C2×C4—3942no—
238C2×C4—3942no—
239C2×C4—3941no—
240C2×C4—3942no—
241C2×C4—3941no—
242C2×C4—3948no—
243C2×C4—3948no—
244C2×C4—3942no—
245C2×C4—3942no—
246C2×C4—3941no—
247C2×C4—3941no—
248C2×C4—3941no—
249C2×C4—3941no—
250C2×C4—1972no—
251C2×C4—1972no—
252C2×C4—1974no—
253C2×C4#1569135714yes—
254C22—24no—
255C22—24no—
256C22—22no—
257C22—22no—
258C22—7882no—
259C22—7882no—
260C22—7881no—
261C22—7881no—
262C22—7881no—
263C22#2026331114yes—
264C22C2×D4×D19711yescharacteristic
265C22#41×D19712yes—
266C22C2×#1569135712yes—
267C22C2×#1569135811yescharacteristic
268C22C2×#1569135911yescharacteristic
269C4—24no—
270C4—42no—
271C4—41no—
272C4—41no—
273C4—7882no—
274C4—7881no—
275C4—3942no—
276C4—3942no—
277C4—3944no—
278C2—24no—
279C2—15761no—
280C2C2×#2026331111yescharacteristic
281C2#2716158511yescharacteristic
282C2#2716158711yescharacteristic
283C2#2716160912yes—
284C2#2716161512yes—
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#385×C197C2
prime radicalC22×C22⋊D197C22
hypercenter#385D197
hypocenterC197#2853
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic197C2semidirect
3. #27160969C2semidirect
4. #27160968C2semidirect
5. #27160966C2semidirect
6. #27160965C2semidirect
9. #27160951C2semidirect
10. #27160940C2semidirect
11. #385×C197C2semidirect
142. C197#2853semidirect
153. #385D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633102
C22×C4×Dic1972
#271609692
#271609682
#271609662
#271609652
#271609582
#271609562
#271609512
#271609402
#385×C1972
#2853197