Finite Group Databasebeta

← #38752065

Subgroups — #38752065

Order 25216.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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#27161915C211yesmaximal, characteristic, complemented
2#27161879C212yesmaximal, complemented
3#27161625C211yesmaximal, characteristic, complemented
4#27161264C212yesmaximal, complemented
5#27161259C212yesmaximal, complemented
6#27161254C212yesmaximal, complemented
7#27161249C212yesmaximal, complemented
8C2×#20263252C212yesmaximal, complemented
9#413×C197C211yesmaximal, characteristic, complemented
10#20263374C2212yes—
11#20263373—22no—
12#20263372—22no—
13#20263371—22no—
14#20263371C2212yes—
15C2×#15691358—22no—
16C2×#15691358C2212yes—
17C2×#15691358C2211yescharacteristic, complemented
18#41×D197—24no—
19#20263359—22no—
20#20263359C2214yescomplemented
21#20263354—22no—
22#20263354C2214yescomplemented
23#20263348—24no—
24#20263324C2211yescharacteristic, complemented
25#20263322C2211yescharacteristic, complemented
26#20263315C2211yescharacteristic, complemented
27#20263313C2211yescharacteristic, complemented
28#20263298—22no—
29C4×C22⋊D197—22no—
30C2×C4×Dic197C2211yescharacteristic, complemented
31#20263252C2214yescomplemented
32#20263252C2214yescomplemented
33C2×#41×C197C2211yescharacteristic
34C2×M16×C197C2212yescomplemented
35#101×C197C2212yescomplemented
36#100×C197C2212yescomplemented
37#15691359—24no—
38#15691358—24no—
39#15691358—24no—
40#15691358—22no—
41#15691358—24no—
42#15691358—22no—
43#15691358—22no—
44#15691358—22no—
45#15691358D414yescomplemented
46#15691358C2314yes—
47#15691357—28no—
48#15691357—24no—
49C2×Dic394—22no—
50C2×Dic394D412yescomplemented
51C2×Dic394C2311yescharacteristic
52Q8×D197—24no—
53C2×D788—22no—
54C2×D788D412yescomplemented
55C2×D788C2311yescharacteristic
56C2×C22⋊D197—24no—
57C2×C22⋊D197—22no—
58C2×C22⋊D197—22no—
59C2×C22⋊D197—21no—
60D4×D197—28no—
61D4×D197—24no—
62C22×Dic197—21no—
63C2×C4×D197—24no—
64C2×C4×D197—22no—
65C2×C4×D197—21no—
66Dic788—24no—
67#15691347—24no—
68#15691347—24no—
69D1576—24no—
70#15691337—22no—
71C22⋊Dic197—22no—
72#15691335—22no—
73C4×Dic197—21no—
74C4×Dic197C2311yescharacteristic
75C4×Dic197C2311yescharacteristic
76#41×C197—22no—
77#41×C197—22no—
78C2×Q8×C197C2311yescharacteristic
79C2×D4×C197D412yes—
80C2×D4×C197C2311yescharacteristic
81C22×C4×C197D412yes—
82C22×C4×C197C2311yescharacteristic
83M16×C197—22no—
84M16×C197C2314yes—
85C2×C8×C197—22no—
86C22×D197—24no—
87C22×D197—22no—
88C22×D197—21no—
89Dic394—24no—
90Dic394—22no—
91Dic394—22no—
92Dic394C2×D414yes—
93D788—24no—
94D788—22no—
95D788—22no—
96D788C2×D414yes—
97C22⋊D197—28no—
98C22⋊D197—24no—
99C22⋊D197—24no—
100C22⋊D197—44no—
101C22⋊D197—42no—
102C2×Dic197—24no—
103C2×Dic197—22no—
104C2×Dic197—21no—
105C2×Dic197—21no—
106C2×Dic197—21no—
107C2×Dic197—42no—
108C4×D197—28no—
109C4×D197—24no—
110C4×D197—24no—
111C4×D197—41no—
112C4×D197—41no—
113C23×C197C2×D412yes—
114C23×C197C2×D411yescharacteristic
115Q8×C197—22no—
116D4×C197—24no—
117D4×C197—22no—
118C2×C4×C197—22no—
119C2×C4×C197—22no—
120C2×C4×C197C2×D412yes—
121C2×C4×C197C2×D412yes—
122C2×C4×C197C2×D412yes—
123C2×C4×C197C2×D411yescharacteristic
124C2×C4×C197C2411yescharacteristic
125C8×C197—24no—
126C22×C197—22no—
127C22×C197—41no—
128C22×C197—42no—
129C22×C197#12012yes—
130C22×C197C22×D411yescharacteristic
131C2×D197—24no—
132C2×D197—42no—
133C2×D197—41no—
134C2×D197—82no—
135C2×D197—84no—
136C4×C197—22no—
137C4×C197#12011yescharacteristic
138C4×C197#12011yescharacteristic
139C4×C197C22×D412yes—
140Dic197—22no—
141Dic197—24no—
142Dic197—41no—
143Dic197—41no—
144C2×C197—21no—
145C2×C197—22no—
146C2×C197—42no—
147C2×C197C2×#12011yescharacteristic
148D197—44no—
149D197—81no—
150C197#301811yescharacteristic, complemented, Sylow
151#3018—1971nomaximal, Sylow
152#585—1971no—
153#575—1972no—
154#532—1971no—
155#456—1972no—
156#455—1972no—
157#454—1972no—
158#453—1972no—
159C2×#104—1972no—
160#413D19711yescharacteristic, complemented
1612−1+4—3942no—
1622−1+4—1972no—
163H(5,𝔽2)—3942no—
164H(5,𝔽2)—3942no—
165H(5,𝔽2)—1972no—
166C2×#41—3944no—
167C2×#41—3942no—
168C2×#41—1971no—
169C2×#41—1972no—
170C2×#41C2×D19711yescharacteristic
171#137—3942no—
172#137—1974no—
173AGL(1,ℤ/8)—3942no—
174AGL(1,ℤ/8)—1974no—
175#135—3944no—
176C2×M16C2×D19712yescomplemented
177C4⋊Q8—1971no—
178#127—1971no—
179#124—1971no—
180#124—1971no—
181#121—3942no—
182C4×D4—3942no—
183C2×C42—1971no—
184#104—1974no—
185#104—1974no—
186#101C2×D19712yescomplemented
187#100C2×D19712yescomplemented
188#41—22no—
189#41—22no—
190#41—3944no—
191#41—3944no—
192#41—3942no—
193#41—3942no—
194#41—3948no—
195#41—3944no—
196#41—3944no—
197#41—3942no—
198#41—3942no—
199#41—3944no—
200#41—1974no—
201#41—1974no—
202C2×Q8—3942no—
203C2×Q8—3944no—
204C2×Q8—1972no—
205C2×Q8—1971no—
206C2×Q8C22×D19711yescharacteristic
207C2×D4—3942no—
208C2×D4—3948no—
209C2×D4—3944no—
210C2×D4—3942no—
211C2×D4—3941no—
212C2×D4—3944no—
213C2×D4—3942no—
214C2×D4—1972no—
215C2×D4—1971no—
216C2×D4D78812yes—
217C2×D4C22×D19711yescharacteristic
218C22×C4—3944no—
219C22×C4—3941no—
220C22×C4—3941no—
221C22×C4—3942no—
222C22×C4D78812yes—
223C22×C4C22×D19711yescharacteristic
224Q16—3944no—
225SD16—3944no—
226SD16—3944no—
227D8—3944no—
228M16—22no—
229M16C22×D19714yes—
230C2×C8—22no—
231C4⋊C4—3942no—
232C22⋊C4—3942no—
233C22⋊C4—3942no—
234C42—3941no—
235C42—1971no—
236C42—1971no—
237C23—3944no—
238C23—3941no—
239C23—3942no—
240C23C2×D78812yes—
241C23C2×D78811yescharacteristic
242Q8—22no—
243Q8—3944no—
244Q8—3942no—
245Q8—3942no—
246Q8—1974no—
247D4—24no—
248D4—22no—
249D4—7882no—
250D4—7884no—
251D4—3944no—
252D4—3948no—
253D4—3944no—
254D4—3944no—
255D4—3942no—
256D4—3942no—
257D4—1974no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—7882no—
261C2×C4—7881no—
262C2×C4—7881no—
263C2×C4—3948no—
264C2×C4—3941no—
265C2×C4—3941no—
266C2×C4—3944no—
267C2×C4—3942no—
268C2×C4—3941no—
269C2×C4—3944no—
270C2×C4—3944no—
271C2×C4D4×D19712yes—
272C2×C4D4×D19712yes—
273C2×C4C2×D78812yes—
274C2×C4C2×D78811yescharacteristic
275C2×C4C23×D19711yescharacteristic
276C8—24no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—7882no—
281C22—7881no—
282C22—15762no—
283C22—15764no—
284C22—3944no—
285C22#2026329612yes—
286C22C22×D78811yescharacteristic
287C4—22no—
288C4—7881no—
289C4—7881no—
290C4—3942no—
291C4—3944no—
292C4#2026329611yescharacteristic
293C4#2026329611yescharacteristic
294C4C2×D4×D19712yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—7884no—
299C2—15761no—
300C2C2×#2026329611yescharacteristic
subgroupG/H
centerC4#20263296
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#413×C197C2
socleC2×C197C2×#120
hypercenter#413D197
hypocenterC197#3018
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #27161915C2semidirect
2. #27161879C2semidirect
3. #27161625C2semidirect
4. #27161264C2semidirect
5. #27161259C2semidirect
6. #27161254C2semidirect
7. #27161249C2semidirect
8. C2×#20263252C2semidirect
9. #413×C197C2semidirect
17. C2×#15691358C22semidirect
20. #20263359C22semidirect
22. #20263354C22semidirect
24. #20263324C22semidirect
25. #20263322C22semidirect
26. #20263315C22semidirect
27. #20263313C22semidirect
30. C2×C4×Dic197C22semidirect
31. #20263252C22semidirect
32. #20263252C22semidirect
34. C2×M16×C197C22semidirect
35. #101×C197C22semidirect
36. #100×C197C22semidirect
45. #15691358D4semidirect
50. C2×Dic394D4semidirect
54. C2×D788D4semidirect
150. C197#3018semidirect
160. #413D197semidirect
176. C2×M16C2×D197semidirect
186. #101C2×D197semidirect
187. #100C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271619152
#271618792
#271616252
#271612642
#271612592
#271612542
#271612492
C2×#202632522
#413×C1972
#3018197