Finite Group Databasebeta

← #38745953

Subgroups — #38745953

Order 25216.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×#20263288C212yesmaximal, complemented
2C2×#20263288C211yesmaximal, characteristic, complemented
3C2×#20263284C211yesmaximal, characteristic, complemented
4#27161005C211yesmaximal, characteristic, complemented
5#27161001C212yesmaximal, complemented
6#27160997C212yesmaximal, complemented
7#27160996C211yesmaximal, characteristic, complemented
8#27160989C211yesmaximal, characteristic
9#27160930C211yesmaximal, characteristic, complemented
10#27160926C212yesmaximal, complemented
11#398×C197C211yesmaximal, characteristic, complemented
12C22×D788C2211yescharacteristic
13C22×D788C2212yes—
14C22×C4×D197C2212yes—
15C22×C4×D197C2211yescharacteristic
16#20263288—28no—
17#20263288—24no—
18#20263284—24no—
19C2×C4⋊Dic197C2211yescharacteristic
20C2×#15691337C2212yes—
21C2×#15691335C2212yes—
22C2×#15691335C2212yes—
23C2×#15691335C2211yescharacteristic
24C2×#15691335C2212yes—
25C2×#15691335C2211yescharacteristic
26C2×#15691335C2212yes—
27C2×#15691335C2212yes—
28C2×C4×Dic197C2211yescharacteristic
29C2×C4×Dic197C2212yes—
30C2×C4×Dic197C2211yescharacteristic
31#20263233C2212yes—
32#20263233C2211yescharacteristic
33C2×C4⋊C4×C197C2211yescharacteristic
34C2×C4⋊C4×C197C2212yes—
35C2×C4⋊C4×C197C2211yescharacteristic
36#95×C197C2212yes—
37#95×C197C2211yescharacteristic
38C23×D197C2311yescharacteristic
39C23×D197C2312yes—
40C2×D788—24no—
41C2×D788—24no—
42C2×D788—24no—
43C2×D788—22no—
44C2×D788—24no—
45C22×Dic197C2312yes—
46C22×Dic197C2311yescharacteristic
47C22×Dic197C2311yescharacteristic
48C22×Dic197C2311yescharacteristic
49C2×C4×D197—22no—
50C2×C4×D197—22no—
51C2×C4×D197—24no—
52C2×C4×D197—24no—
53C2×C4×D197—24no—
54C2×C4×D197D414yescomplemented
55C4⋊Dic197—22no—
56#15691337—24no—
57#15691335—24no—
58#15691335—24no—
59#15691335—22no—
60#15691335—24no—
61#15691335—22no—
62#15691335—24no—
63#15691335—24no—
64C4×Dic197—22no—
65C4×Dic197—24no—
66C4×Dic197D414yescomplemented
67C22×C4×C197C2311yescharacteristic
68C22×C4×C197C2312yes—
69C22×C4×C197C2311yescharacteristic
70C22×C4×C197C2311yescharacteristic
71C22×C4×C197C2312yes—
72C4⋊C4×C197—22no—
73C4⋊C4×C197—24no—
74C4⋊C4×C197—22no—
75C22×D197—21no—
76C22×D197—22no—
77C22×D197—22no—
78C22×D197—22no—
79C22×D197—21no—
80C22×D197—22no—
81C22×D197—22no—
82C22×D197—22no—
83C22×D197—22no—
84C22×D197—22no—
85C22×D197C2×D412yes—
86C22×D197#4114yes—
87D788—28no—
88D788—216no—
89C2×Dic197—22no—
90C2×Dic197—22no—
91C2×Dic197—21no—
92C2×Dic197—22no—
93C2×Dic197—22no—
94C2×Dic197—22no—
95C2×Dic197C2×D412yes—
96C2×Dic197C2×D412yes—
97C2×Dic197C2×D412yes—
98C2×Dic197#4112yes—
99C4×D197—28no—
100C4×D197—48no—
101C23×C197C2411yescharacteristic
102C2×C4×C197—22no—
103C2×C4×C197—21no—
104C2×C4×C197—21no—
105C2×C4×C197—22no—
106C2×C4×C197—22no—
107C2×C4×C197—21no—
108C2×C4×C197—22no—
109C2×C4×C197—22no—
110C2×C4×C197—22no—
111C2×C4×C197—22no—
112C2×C4×C197C2×D412yes—
113C2×C4×C197C2×D412yes—
114C2×C4×C197#4114yes—
115C22×C197C22×D411yescharacteristic
116C22×C197C22×D411yescharacteristic
117C22×C197C2×#4111yescharacteristic
118C22×C197C2×#4112yes—
119C22×C197C2×#4112yes—
120C2×D197—22no—
121C2×D197—24no—
122C2×D197—24no—
123C2×D197—24no—
124C2×D197—24no—
125C2×D197—42no—
126C2×D197—41no—
127C2×D197—42no—
128C2×D197—42no—
129C2×D197—42no—
130C2×D197—41no—
131C2×D197—42no—
132C4×C197—22no—
133C4×C197—22no—
134C4×C197—24no—
135C4×C197—42no—
136C4×C197—41no—
137Dic197—22no—
138Dic197—42no—
139Dic197—41no—
140Dic197#12114yes—
141C2×C197C2×#12111yescharacteristic
142C2×C197#52511yescharacteristic
143C2×C197#52512yes—
144C2×C197#52912yes—
145C2×C197#53211yescharacteristic
146D197—42no—
147D197—44no—
148C197#239211yescharacteristic, complemented, Sylow
149#2392—1971nomaximal, Sylow
150C2×C4×D4—1972no—
151C2×C4×D4—1971no—
152C2×#117—1971no—
153#398D19711yescharacteristic, complemented
154#397—1971no—
155#396—1972no—
156#394—1971no—
157#394—1972no—
158#392—1971no—
159C4×C4⋊C4—1971no—
160C4×C22⋊C4—1972no—
161C22×D4—1971no—
162C22×D4—1972no—
163C23×C4—1972no—
164C23×C4—1971no—
165C4×D4—3948no—
166C4×D4—3944no—
167#117—3944no—
168C2×C4⋊C4—1971no—
169C2×C4⋊C4—1972no—
170C2×C4⋊C4C2×D19711yescharacteristic
171C2×C4⋊C4C2×D19712yes—
172C2×C4⋊C4C2×D19711yescharacteristic
173C2×C22⋊C4—1971no—
174C2×C22⋊C4—1971no—
175C2×C22⋊C4—1972no—
176C2×C22⋊C4—1972no—
177C2×C22⋊C4—1972no—
178C2×C22⋊C4—1972no—
179C2×C22⋊C4—1972no—
180C2×C42—1971no—
181C2×C42—1972no—
182C2×C42—1971no—
183#95—1972no—
184#95—1971no—
185#95C2×D19712yes—
186#95C2×D19711yescharacteristic
187C24—1971no—
188C24—1972no—
189C2×D4—3944no—
190C2×D4—3944no—
191C2×D4—3944no—
192C2×D4—3942no—
193C2×D4—3944no—
194C22×C4—3944no—
195C22×C4—3944no—
196C22×C4—3944no—
197C22×C4—3942no—
198C22×C4—3942no—
199C22×C4—1972no—
200C22×C4—1971no—
201C22×C4—1971no—
202C22×C4—1974no—
203C22×C4—1971no—
204C22×C4C22×D19712yes—
205C22×C4C22×D19711yescharacteristic
206C22×C4C22×D19711yescharacteristic
207C22×C4C22×D19711yescharacteristic
208C22×C4C22×D19712yes—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—3942no—
213C4⋊C4—3944no—
214C22⋊C4—3944no—
215C22⋊C4—3944no—
216C22⋊C4—3942no—
217C22⋊C4—3944no—
218C22⋊C4—3942no—
219C22⋊C4—3944no—
220C22⋊C4—3944no—
221C42—3942no—
222C42—3944no—
223C42—1974no—
224C23—3941no—
225C23—3942no—
226C23—3942no—
227C23—3941no—
228C23—3942no—
229C23—3942no—
230C23—3942no—
231C23—3942no—
232C23—3942no—
233C23—3942no—
234C23—1972no—
235C23—1974no—
236C23C23×D19711yescharacteristic
237D4—3948no—
238D4—39416no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—7888no—
250C2×C4—3948no—
251C2×C4—3942no—
252C2×C4—3941no—
253C2×C4—3942no—
254C2×C4—3942no—
255C2×C4—3942no—
256C2×C4—3942no—
257C2×C4—1972no—
258C2×C4—1972no—
259C2×C4—1972no—
260C2×C4—1972no—
261C2×C4D4×D19712yes—
262C2×C4D4×D19712yes—
263C2×C4#1569135914yes—
264C22—7882no—
265C22—7882no—
266C22—7881no—
267C22—7882no—
268C22—7882no—
269C22—7881no—
270C22—7882no—
271C22—3944no—
272C22—3942no—
273C22—3944no—
274C22—3944no—
275C22—3944no—
276C22C2×D4×D19711yescharacteristic
277C22C2×D4×D19711yescharacteristic
278C22#41×D19711yescharacteristic
279C22#41×D19712yes—
280C22C2×#1569135912yes—
281C4—22no—
282C4—22no—
283C4—24no—
284C4—42no—
285C4—41no—
286C4—7881no—
287C4—7882no—
288C4—3942no—
289C4—1974no—
290C2—7882no—
291C2—7884no—
292C2#121×D19711yescharacteristic
293C2#2716158711yescharacteristic
294C2#2716159212yes—
295C2#2716161412yes—
296C2#2716163011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#398×C197C2
prime radical#27160989C2
hypercenter#398D197
hypocenterC197#2392
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#20263288C2semidirect
2. C2×#20263288C2semidirect
3. C2×#20263284C2semidirect
4. #27161005C2semidirect
5. #27161001C2semidirect
6. #27160997C2semidirect
7. #27160996C2semidirect
9. #27160930C2semidirect
10. #27160926C2semidirect
11. #398×C197C2semidirect
54. C2×C4×D197D4semidirect
66. C4×Dic197D4semidirect
148. C197#2392semidirect
153. #398D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632882
C2×#202632842
#271610052
#271610012
#271609972
#271609962
#271609892
#271609302
#271609262
#398×C1972
#2392197