Finite Group Databasebeta

← #38753188

Subgroups — #38753188

Order 25216.

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

normal subgroups only

This group has 564 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×#20263359C211yesmaximal, characteristic, complemented
2C2×#20263354C211yesmaximal, characteristic, complemented
3C2×#20263348C211yesmaximal, characteristic, complemented
4C22×#15691347C211yesmaximal, characteristic, complemented
5#27161655C211yesmaximal, characteristic, complemented
6#27161648C211yesmaximal, characteristic, complemented
7#27161473C212yesmaximal, complemented
8#27161468C212yesmaximal, complemented
9#27161463C212yesmaximal, complemented
10#27161452C212yesmaximal, complemented
11#405×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2211yescharacteristic
13C2×#15691358C2211yescharacteristic
14C22×Dic394C2211yescharacteristic
15C22×D788C2211yescharacteristic
16#20263359—24no—
17#20263354—24no—
18#20263348—24no—
19C2×Dic788C2211yescharacteristic, complemented
20C2×Dic788C2212yescomplemented
21C2×#15691347—24no—
22C2×#15691347C2212yes—
23C2×#15691347C2212yes—
24C2×#15691347C2212yes—
25C2×#15691347C2211yescharacteristic, complemented
26C2×#15691347C2211yescharacteristic, complemented
27C2×#15691347C2212yescomplemented
28C2×D1576C2212yes—
29C2×D1576C2211yescharacteristic, complemented
30#20263326C2212yescomplemented
31#20263323C2212yes—
32#20263314C2212yes—
33#20263314C2212yescomplemented
34#20263310—22no—
35#20263306—22no—
36#20263301—22no—
37#20263296—22no—
38C2×M16×C197C2211yescharacteristic
39C22×C8×C197C2211yescharacteristic
40#117×C197C2211yescharacteristic
41#97×C197C2212yescomplemented
42C4×C8×C197C2212yescomplemented
43C23×D197—21no—
44#15691358—22no—
45#15691358—22no—
46#15691358—22no—
47#15691358—22no—
48C2×Dic394—21no—
49C2×Dic394—22no—
50C2×Dic394—22no—
51C2×Dic394C2311yescharacteristic
52C2×Dic394C2312yes—
53C2×Dic394C2311yescharacteristic
54C2×D788—22no—
55C2×D788—22no—
56C2×D788—21no—
57C2×D788—24no—
58C2×D788C2312yes—
59C2×D788C2311yescharacteristic
60C2×D788C2311yescharacteristic
61C2×C22⋊D197—21no—
62C2×C22⋊D197—21no—
63C22×Dic197—21no—
64C2×C4×D197—21no—
65C2×C4×D197—21no—
66Dic788—22no—
67Dic788—24no—
68#15691347—24no—
69#15691347—24no—
70#15691347—24no—
71#15691347—22no—
72#15691347—22no—
73#15691347—24no—
74D1576—24no—
75D1576—22no—
76C4⋊Dic197—24no—
77#15691335—24no—
78#15691335—24no—
79C22×C4×C197C2311yescharacteristic
80M16×C197D414yescomplemented
81C2×C8×C197D414yescomplemented
82C2×C8×C197C2311yescharacteristic
83C2×C8×C197C2312yes—
84C2×C8×C197C2311yescharacteristic
85C4⋊C4×C197D412yes—
86C22⋊C4×C197D412yes—
87C42×C197C2312yes—
88C22×D197—21no—
89C22×D197—22no—
90C22×D197—21no—
91C22×D197—42no—
92C22×D197—82no—
93Dic394—22no—
94Dic394—21no—
95Dic394—22no—
96Dic394—21no—
97Dic394—21no—
98Dic394—21no—
99Dic394—42no—
100D788—21no—
101D788—22no—
102D788—21no—
103D788—21no—
104D788—22no—
105D788—21no—
106D788—42no—
107D788—84no—
108C22⋊D197—42no—
109C22⋊D197—42no—
110C2×Dic197—21no—
111C2×Dic197—22no—
112C2×Dic197—21no—
113C2×Dic197—42no—
114C4×D197—42no—
115C4×D197—42no—
116C23×C197C2×D411yescharacteristic
117C2×C4×C197C2×D414yes—
118C2×C4×C197C2×D411yescharacteristic
119C2×C4×C197C2×D412yes—
120C2×C4×C197C2×D412yes—
121C2×C4×C197C2411yescharacteristic
122C8×C197C2×D412yes—
123C8×C197C2×D412yes—
124C8×C197C2×D414yes—
125C22×C197—22no—
126C22×C197#12712yes—
127C22×C197C22×D411yescharacteristic
128C2×D197—41no—
129C2×D197—41no—
130C2×D197—42no—
131C2×D197—42no—
132C2×D197—44no—
133C2×D197—42no—
134C2×D197—84no—
135C4×C197—24no—
136C4×C197#12712yes—
137C4×C197C22×D411yescharacteristic
138C4×C197C22×D411yescharacteristic
139Dic197—41no—
140Dic197—41no—
141Dic197—42no—
142C2×C197—22no—
143C2×C197C2×#12711yescharacteristic
144C2×C197#57712yes—
145D197—81no—
146D197—81no—
147D197—82no—
148C197#315411yescharacteristic, complemented, Sylow
149#3154—1971nomaximal, Sylow
150C2×#137—1971no—
151C2×AGL(1,ℤ/8)—1971no—
152C2×#135—1971no—
153C22×SD16—1971no—
154#536—1971no—
155#535—1971no—
156#497—1972no—
157#496—1972no—
158#495—1972no—
159#492—1972no—
160#405D19711yescharacteristic, complemented
161C2×#41—1971no—
162C2×#41—1971no—
163C22×Q8—1971no—
164C22×D4—1971no—
165#137—3944no—
166AGL(1,ℤ/8)—3944no—
167#135—3944no—
168C2×Q16—1971no—
169C2×Q16—1972no—
170C2×SD16—3944no—
171C2×SD16—1972no—
172C2×SD16—1972no—
173C2×SD16—1971no—
174C2×SD16—1971no—
175C2×SD16—1972no—
176C2×SD16—1972no—
177C2×D8—1971no—
178C2×D8—1972no—
179C2×M16C2×D19711yescharacteristic
180C22×C8C2×D19711yescharacteristic
181C4⋊Q8—1972no—
182#127—1972no—
183#124—1972no—
184#124—1972no—
185#123—3942no—
186C22⋊Q8—3942no—
187#121—3942no—
188#120—3942no—
189#117C2×D19711yescharacteristic
190#97C2×D19712yescomplemented
191C4×C8C2×D19712yescomplemented
192C24—3941no—
193#41—3942no—
194#41—3942no—
195#41—3942no—
196#41—3942no—
197C2×Q8—3942no—
198C2×Q8—3942no—
199C2×Q8—3941no—
200C2×Q8—1971no—
201C2×Q8—1972no—
202C2×Q8—1971no—
203C2×D4—3941no—
204C2×D4—3941no—
205C2×D4—3941no—
206C2×D4—3942no—
207C2×D4—3942no—
208C2×D4—3944no—
209C2×D4—1971no—
210C2×D4—1972no—
211C2×D4—1971no—
212C22×C4—3941no—
213C22×C4—3941no—
214C22×C4—3941no—
215C22×C4C22×D19711yescharacteristic
216Q16—3942no—
217Q16—3944no—
218SD16—3944no—
219SD16—3944no—
220SD16—3942no—
221SD16—3942no—
222SD16—3944no—
223SD16—3944no—
224D8—3942no—
225D8—3944no—
226M16D78814yescomplemented
227C2×C8D78814yescomplemented
228C2×C8C22×D19711yescharacteristic
229C2×C8C22×D19712yes—
230C2×C8C22×D19711yescharacteristic
231C4⋊C4—3944no—
232C4⋊C4D78812yes—
233C22⋊C4—3944no—
234C22⋊C4—3944no—
235C22⋊C4D78812yes—
236C42C22×D19712yes—
237C23—7882no—
238C23—15762no—
239C23—3941no—
240C23—3942no—
241C23—3941no—
242C23C2×D78811yescharacteristic
243Q8—7882no—
244Q8—3942no—
245Q8—3941no—
246Q8—3942no—
247Q8—3941no—
248Q8—3941no—
249Q8—3941no—
250D4—7882no—
251D4—7882no—
252D4—7882no—
253D4—15764no—
254D4—3941no—
255D4—3942no—
256D4—3941no—
257D4—3941no—
258D4—3942no—
259D4—3941no—
260C2×C4—7882no—
261C2×C4—7882no—
262C2×C4—7882no—
263C2×C4—3941no—
264C2×C4—3942no—
265C2×C4—3941no—
266C2×C4C2×D78814yes—
267C2×C4C2×D78811yescharacteristic
268C2×C4C2×D78812yes—
269C2×C4C2×D78812yes—
270C2×C4C23×D19711yescharacteristic
271C8C2×D78812yes—
272C8C2×D78812yes—
273C8C2×D78814yes—
274C22—22no—
275C22—7881no—
276C22—7881no—
277C22—7882no—
278C22—7882no—
279C22—7884no—
280C22—7882no—
281C22—15764no—
282C22#2026332312yes—
283C22C22×D78811yescharacteristic
284C4—24no—
285C4—7881no—
286C4—7881no—
287C4—7882no—
288C4#2026332312yes—
289C4C22×D78811yescharacteristic
290C4C22×D78811yescharacteristic
291C2—22no—
292C2—15762no—
293C2—15761no—
294C2—15761no—
295C2C2×#2026332311yescharacteristic
296C2#2716189112yes—
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#405×C197C2
socleC22×C197C22×D4
hypercenter#405D197
hypocenterC197#3154
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#20263359C2semidirect
2. C2×#20263354C2semidirect
3. C2×#20263348C2semidirect
4. C22×#15691347C2semidirect
5. #27161655C2semidirect
6. #27161648C2semidirect
7. #27161473C2semidirect
8. #27161468C2semidirect
9. #27161463C2semidirect
10. #27161452C2semidirect
11. #405×C197C2semidirect
19. C2×Dic788C22semidirect
20. C2×Dic788C22semidirect
25. C2×#15691347C22semidirect
26. C2×#15691347C22semidirect
27. C2×#15691347C22semidirect
29. C2×D1576C22semidirect
30. #20263326C22semidirect
33. #20263314C22semidirect
41. #97×C197C22semidirect
42. C4×C8×C197C22semidirect
80. M16×C197D4semidirect
81. C2×C8×C197D4semidirect
148. C197#3154semidirect
160. #405D197semidirect
190. #97C2×D197semidirect
191. C4×C8C2×D197semidirect
226. M16D788semidirect
227. C2×C8D788semidirect

Maximal subgroups

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

subgroupindex
C2×#202633592
C2×#202633542
C2×#202633482
C22×#156913472
#271616552
#271616482
#271614732
#271614682
#271614632
#271614522
#405×C1972
#3154197