Finite Group Databasebeta

← #38751040

Subgroups — #38751040

Order 25216.

688 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 165 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, complemented
2C2×#20263307C211yesmaximal, characteristic, complemented
3C2×#20263298C211yesmaximal, characteristic
4C22×C4×Dic197C211yesmaximal, characteristic, complemented
5#27160979C212yesmaximal, complemented
6#27160976C212yesmaximal, complemented
7#27160972C212yesmaximal, complemented
8#27160968C212yesmaximal, complemented
9#27160958C211yesmaximal, characteristic, complemented
10#27160937C211yesmaximal, characteristic, complemented
11#386×C197C211yesmaximal, characteristic, complemented
12C22×C22⋊D197C2211yescharacteristic
13C22×C22⋊D197C2211yescharacteristic
14C23×Dic197C2211yescharacteristic
15C23×Dic197C2211yescharacteristic
16#20263310—24no—
17#20263307—24no—
18#20263298—24no—
19C2×C4⋊Dic197C2212yes—
20C2×#15691337C2212yes—
21C2×#15691337C2212yes—
22C2×#15691337C2211yescharacteristic
23C2×C22⋊Dic197C2211yescharacteristic
24C2×C22⋊Dic197C2211yescharacteristic
25C2×C22⋊Dic197C2212yes—
26C2×#15691335C2212yes—
27C2×#15691335C2211yescharacteristic
28C2×#15691335C2212yes—
29C2×C4×Dic197—24no—
30C2×C4×Dic197C2212yes—
31C2×C4×Dic197C2212yes—
32#20263233C2212yes—
33#20263233C2212yes—
34C22×D4×C197C2211yescharacteristic
35C23×C4×C197C2211yescharacteristic
36C2×C22⋊C4×C197C2211yescharacteristic
37C2×C22⋊C4×C197C2212yes—
38#95×C197C2212yes—
39C23×D197C2311yescharacteristic
40C2×C22⋊D197—22no—
41C2×C22⋊D197—24no—
42C2×C22⋊D197—22no—
43C2×C22⋊D197—22no—
44C2×C22⋊D197—22no—
45C22×Dic197—24no—
46C22×Dic197—22no—
47C22×Dic197—22no—
48C22×Dic197D414yes—
49C22×Dic197D414yescomplemented
50C22×Dic197C2312yes—
51C22×Dic197C2311yescharacteristic
52C22×Dic197C2311yescharacteristic
53C22×Dic197C2312yes—
54C22×Dic197C2311yescharacteristic
55C4⋊Dic197—24no—
56#15691337—22no—
57#15691337—24no—
58#15691337—24no—
59C22⋊Dic197—22no—
60C22⋊Dic197—24no—
61C22⋊Dic197—22no—
62#15691335—22no—
63#15691335—24no—
64#15691335—24no—
65C4×Dic197—24no—
66C4×Dic197—24no—
67C24×C197C2311yescharacteristic
68C24×C197C2311yescharacteristic
69C2×D4×C197—22no—
70C2×D4×C197—22no—
71C2×D4×C197—22no—
72C22×C4×C197—22no—
73C22×C4×C197—24no—
74C22×C4×C197C2311yescharacteristic
75C22×C4×C197C2312yes—
76C22×C4×C197C2312yes—
77C22⋊C4×C197—22no—
78C22⋊C4×C197—24no—
79C22×D197—21no—
80C22×D197—21no—
81C22×D197—22no—
82C22×D197—22no—
83C22×D197—21no—
84C22⋊D197—44no—
85C22⋊D197—44no—
86C2×Dic197—28no—
87C2×Dic197—22no—
88C2×Dic197—22no—
89C2×Dic197—22no—
90C2×Dic197—28no—
91C2×Dic197—22no—
92C2×Dic197—21no—
93C2×Dic197—21no—
94C2×Dic197—22no—
95C2×Dic197—22no—
96C2×Dic197—21no—
97C2×Dic197C2×D412yes—
98C2×Dic197C2×D414yes—
99C2×Dic197C2×D412yes—
100C2×Dic197#4114yes—
101C23×C197—22no—
102C23×C197—22no—
103C23×C197—21no—
104C23×C197—21no—
105C23×C197—22no—
106C23×C197—21no—
107C23×C197—22no—
108C23×C197C2×D412yes—
109C23×C197C2×D412yes—
110C23×C197#4112yes—
111C23×C197C2411yescharacteristic
112D4×C197—44no—
113C2×C4×C197—28no—
114C2×C4×C197—22no—
115C2×C4×C197—22no—
116C2×C4×C197—22no—
117C2×C4×C197—21no—
118C2×C4×C197—21no—
119C2×C4×C197—22no—
120C2×C4×C197—21no—
121C2×C4×C197—22no—
122C2×C4×C197#4114yes—
123C22×C197—22no—
124C22×C197—22no—
125C22×C197—24no—
126C22×C197—24no—
127C22×C197—41no—
128C22×C197—41no—
129C22×C197—41no—
130C22×C197—42no—
131C22×C197—42no—
132C22×C197#12114yes—
133C22×C197C22×D411yescharacteristic
134C22×C197C22×D411yescharacteristic
135C22×C197C2×#4112yes—
136C22×C197C2×#4112yes—
137C22×C197C2×#4111yescharacteristic
138C2×D197—41no—
139C2×D197—41no—
140C2×D197—41no—
141C2×D197—42no—
142C2×D197—42no—
143C4×C197—24no—
144C4×C197—41no—
145C4×C197—42no—
146Dic197—24no—
147Dic197—22no—
148Dic197—41no—
149Dic197—42no—
150Dic197#12314yescomplemented
151C2×C197—24no—
152C2×C197—81no—
153C2×C197C2×#12111yescharacteristic
154C2×C197C2×#12311yescharacteristic
155C2×C197#52511yescharacteristic
156C2×C197#52912yes—
157C2×C197#53212yes—
158D197—81no—
159C197#285411yescharacteristic, complemented, Sylow
160#2854—1971nomaximal, Sylow
161C2×#123—1971no—
162C2×#123—1971no—
163C2×#121—1971no—
164C22×C42—1971no—
165#390—1972no—
166#389—1972no—
167#388—1972no—
168#388—1972no—
169#386—1971no—
170#386D19711yescharacteristic, complemented
171#380—1971no—
172C22×D4—1971no—
173C22×D4—1971no—
174C22×D4C2×D19711yescharacteristic
175C23×C4—1971no—
176C23×C4—1971no—
177C23×C4C2×D19711yescharacteristic
178#123—3944no—
179#123—3944no—
180#121—3944no—
181C2×C4⋊C4—1971no—
182C2×C4⋊C4—1972no—
183C2×C4⋊C4—1972no—
184C2×C4⋊C4—1972no—
185C2×C22⋊C4—1971no—
186C2×C22⋊C4—1971no—
187C2×C22⋊C4—1972no—
188C2×C22⋊C4—1971no—
189C2×C22⋊C4—1972no—
190C2×C22⋊C4—1972no—
191C2×C22⋊C4C2×D19711yescharacteristic
192C2×C22⋊C4C2×D19712yes—
193C2×C42—3944no—
194C2×C42—1972no—
195C2×C42—1972no—
196#95—1972no—
197#95—1972no—
198#95C2×D19712yes—
199C24—1971no—
200C24C22×D19711yescharacteristic
201C24C22×D19711yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—3942no—
206C2×D4—3942no—
207C2×D4—3942no—
208C2×D4—3942no—
209C2×D4—3944no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—3942no—
213C22×C4—3942no—
214C22×C4—3944no—
215C22×C4—1972no—
216C22×C4—1971no—
217C22×C4—1974no—
218C22×C4—1974no—
219C22×C4—1971no—
220C22×C4—1972no—
221C22×C4—1971no—
222C22×C4C22×D19712yes—
223C22×C4C22×D19711yescharacteristic
224C22×C4C22×D19712yes—
225C4⋊C4—3944no—
226C4⋊C4—3942no—
227C4⋊C4—3944no—
228C4⋊C4—3944no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—3942no—
232C22⋊C4—3944no—
233C22⋊C4—3942no—
234C22⋊C4—3942no—
235C22⋊C4—3944no—
236C22⋊C4—3944no—
237C42—3944no—
238C42—3944no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23—21no—
246C23—3941no—
247C23—3941no—
248C23—3942no—
249C23—3942no—
250C23—3941no—
251C23D4×D19712yes—
252C23D4×D19712yes—
253C23#1569135812yes—
254C23C23×D19711yescharacteristic
255D4—44no—
256D4—7884no—
257D4—7884no—
258C2×C4—28no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—3942no—
268C2×C4—3941no—
269C2×C4—3941no—
270C2×C4—3942no—
271C2×C4—3942no—
272C2×C4—3941no—
273C2×C4—3948no—
274C2×C4—3942no—
275C2×C4—3942no—
276C2×C4—3942no—
277C2×C4—3948no—
278C2×C4—1972no—
279C2×C4—1974no—
280C2×C4—1972no—
281C2×C4—1974no—
282C2×C4#1569135714yes—
283C22—24no—
284C22—24no—
285C22—22no—
286C22—22no—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—7882no—
293C22—7882no—
294C22—7881no—
295C22—7881no—
296C22—7881no—
297C22#2026329914yes—
298C22C2×D4×D19711yescharacteristic
299C22C2×D4×D19711yescharacteristic
300C22#41×D19712yes—
301C22C2×#1569135712yes—
302C22C2×#1569135811yescharacteristic
303C4—24no—
304C4—42no—
305C4—41no—
306C4—7882no—
307C4—7881no—
308C4—3944no—
309C4—3942no—
310C4—1974no—
311C2—24no—
312C2—81no—
313C2—15761no—
314C2C2×#2026329911yescharacteristic
315C2#123×D19711yescharacteristic
316C2#2716158511yescharacteristic
317C2#2716160912yes—
318C2#2716162412yes—
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#386×C197C2
prime radicalC2×#20263298C2
hypercenter#386D197
hypocenterC197#2854
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#20263310C2semidirect
2. C2×#20263307C2semidirect
4. C22×C4×Dic197C2semidirect
5. #27160979C2semidirect
6. #27160976C2semidirect
7. #27160972C2semidirect
8. #27160968C2semidirect
9. #27160958C2semidirect
10. #27160937C2semidirect
11. #386×C197C2semidirect
49. C22×Dic197D4semidirect
150. Dic197#123semidirect
159. C197#2854semidirect
170. #386D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633102
C2×#202633072
C2×#202632982
C22×C4×Dic1972
#271609792
#271609762
#271609722
#271609682
#271609582
#271609372
#386×C1972
#2854197