Finite Group Databasebeta

← #38751957

Subgroups — #38751957

Order 25216.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#15691358C211yesmaximal, characteristic, complemented
2C2×#20263359C211yesmaximal, characteristic, complemented
3C2×#20263354C211yesmaximal, characteristic, complemented
4#27161590C211yesmaximal, characteristic, complemented
5#27161243C214yesmaximal, complemented
6#27161226C214yesmaximal, complemented
7#27161089C211yesmaximal, characteristic, complemented
8#27161085C211yesmaximal, characteristic, complemented
9#407×C197C211yesmaximal, characteristic, complemented
10C2×#15691358—24no—
11C2×#15691358—24no—
12C2×#15691358—22no—
13C2×#15691358C2211yescharacteristic
14C2×#15691358C2214yes—
15C22×Dic394C2211yescharacteristic
16C22×D788C2211yescharacteristic
17C22×C22⋊D197—21no—
18C22×C4×D197—21no—
19#20263359—24no—
20#20263354—24no—
21C2×Dic788C2212yescomplemented
22C2×#15691347C2212yescomplemented
23C2×#15691347C2212yescomplemented
24C2×D1576C2212yescomplemented
25#20263309—21no—
26#20263305C2212yescomplemented
27#20263300C2212yescomplemented
28#20263295—21no—
29C4×C22⋊D197—22no—
30#20263285C2211yescharacteristic
31#20263251C2214yescomplemented
32#20263248C2214yescomplemented
33C23×C4×C197C2211yescharacteristic
34C2×M16×C197C2211yescharacteristic
35C2×M16×C197C2211yescharacteristic
36C22⋊C8×C197C2214yescomplemented
37C23×D197—21no—
38#15691358—216no—
39#15691358—28no—
40#15691358—28no—
41#15691358—22no—
42#15691358—22no—
43C2×Dic394—22no—
44C2×Dic394—21no—
45C2×Dic394D414yescomplemented
46C2×Dic394C2311yescharacteristic
47C2×Dic394C2312yes—
48C2×D788—22no—
49C2×D788—21no—
50C2×D788D414yescomplemented
51C2×D788C2311yescharacteristic
52C2×D788C2312yes—
53C2×C22⋊D197—28no—
54C2×C22⋊D197—24no—
55C2×C22⋊D197—22no—
56C2×C22⋊D197—21no—
57C22×Dic197—21no—
58C2×C4×D197—28no—
59C2×C4×D197—24no—
60C2×C4×D197—21no—
61Dic788—24no—
62#15691347—24no—
63#15691347—24no—
64D1576—24no—
65C4⋊Dic197C2312yes—
66#15691337—22no—
67C22⋊Dic197—21no—
68C22⋊Dic197—22no—
69#15691335—22no—
70C4×Dic197—21no—
71C24×C197D411yescharacteristic
72C22×C4×C197—22no—
73C22×C4×C197—22no—
74C22×C4×C197D412yes—
75C22×C4×C197D411yescharacteristic
76C22×C4×C197C2312yes—
77C22×C4×C197C2311yescharacteristic
78M16×C197—22no—
79M16×C197—22no—
80C2×C8×C197C2312yes—
81C2×C8×C197C2312yes—
82C22×D197—24no—
83C22×D197—22no—
84C22×D197—21no—
85C22×D197—44no—
86Dic394—24no—
87Dic394—22no—
88Dic394—21no—
89Dic394—21no—
90Dic394C2×D414yes—
91D788—24no—
92D788—22no—
93D788—21no—
94D788—21no—
95D788C2×D414yes—
96C22⋊D197—216no—
97C22⋊D197—44no—
98C22⋊D197—42no—
99C2×Dic197—24no—
100C2×Dic197—22no—
101C2×Dic197—22no—
102C2×Dic197—21no—
103C4×D197—216no—
104C4×D197—42no—
105C23×C197—22no—
106C23×C197—42no—
107C23×C197C2×D412yes—
108C23×C197C2×D411yescharacteristic
109C2×C4×C197—22no—
110C2×C4×C197—24no—
111C2×C4×C197—22no—
112C2×C4×C197—22no—
113C2×C4×C197C2×D412yes—
114C2×C4×C197C2×D412yes—
115C2×C4×C197C2×D412yes—
116C2×C4×C197C2×D411yescharacteristic
117C2×C4×C197C2411yescharacteristic
118C8×C197—22no—
119C8×C197—22no—
120C22×C197—22no—
121C22×C197—22no—
122C22×C197—22no—
123C22×C197—22no—
124C22×C197—44no—
125C22×C197#12012yes—
126C22×C197C22×D411yescharacteristic
127C2×D197—24no—
128C2×D197—48no—
129C2×D197—44no—
130C2×D197—41no—
131C2×D197—42no—
132C4×C197—22no—
133C4×C197#12012yes—
134C4×C197C22×D411yescharacteristic
135C4×C197C22×D411yescharacteristic
136Dic197—24no—
137Dic197—41no—
138Dic197—42no—
139C2×C197—22no—
140C2×C197—42no—
141C2×C197C2×#12011yescharacteristic
142C2×C197#57512yes—
143D197—44no—
144D197—81no—
145C197#300511yescharacteristic, complemented, Sylow
146#3005—1971nomaximal, Sylow
147C22×#41—1971no—
148C2×#137—1971no—
149C2×AGL(1,ℤ/8)—1971no—
150#525—1971no—
151#452—1974no—
152#449—1974no—
153#419—1971no—
154#418—1971no—
155#407D19711yescharacteristic, complemented
156C2×#41—3944no—
157C2×#41—3944no—
158C2×#41—3942no—
159C2×#41—1971no—
160C2×#41—1974no—
161C22×Q8—1971no—
162C22×D4—3941no—
163C22×D4—1971no—
164C23×C4—3941no—
165C23×C4C2×D19711yescharacteristic
166#137—3944no—
167AGL(1,ℤ/8)—3944no—
168C2×Q16—1972no—
169C2×SD16—1972no—
170C2×SD16—1972no—
171C2×D8—1972no—
172C2×M16C2×D19711yescharacteristic
173C2×M16C2×D19711yescharacteristic
174#123—3941no—
175C22⋊Q8—1972no—
176#121—1972no—
177#120—3941no—
178C4×D4—3942no—
179#117—1971no—
180#103—1974no—
181#102—1974no—
182C22⋊C8C2×D19714yescomplemented
183C24—3941no—
184C24D78811yescharacteristic
185#41—39416no—
186#41—3948no—
187#41—3948no—
188#41—3942no—
189#41—3942no—
190C2×Q8—3942no—
191C2×Q8—3941no—
192C2×Q8—1974no—
193C2×Q8—1971no—
194C2×Q8—1972no—
195C2×D4—3942no—
196C2×D4—3941no—
197C2×D4—3948no—
198C2×D4—3944no—
199C2×D4—3942no—
200C2×D4—3941no—
201C2×D4—1974no—
202C2×D4—1971no—
203C2×D4—1972no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—3948no—
207C22×C4—3941no—
208C22×C4—3941no—
209C22×C4—3944no—
210C22×C4D78812yes—
211C22×C4D78811yescharacteristic
212C22×C4C22×D19712yes—
213C22×C4C22×D19711yescharacteristic
214Q16—3944no—
215SD16—3944no—
216SD16—3944no—
217D8—3944no—
218M16—22no—
219M16—22no—
220C2×C8C22×D19712yes—
221C2×C8C22×D19712yes—
222C4⋊C4—3942no—
223C4⋊C4—1972no—
224C22⋊C4—3941no—
225C22⋊C4—3942no—
226C22⋊C4—3942no—
227C42—3941no—
228C23—22no—
229C23—42no—
230C23—7884no—
231C23—3941no—
232C23—3944no—
233C23—3942no—
234C23C2×D78812yes—
235C23C2×D78811yescharacteristic
236Q8—3941no—
237Q8—3941no—
238Q8—3944no—
239Q8—3942no—
240Q8—1974no—
241D4—7884no—
242D4—7882no—
243D4—39416no—
244D4—3944no—
245D4—3941no—
246D4—3941no—
247D4—3942no—
248D4—1974no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—7882no—
254C2×C4—39416no—
255C2×C4—3942no—
256C2×C4—3941no—
257C2×C4—3944no—
258C2×C4—3942no—
259C2×C4D4×D19712yes—
260C2×C4D4×D19712yes—
261C2×C4C2×D78812yes—
262C2×C4C2×D78811yescharacteristic
263C2×C4C23×D19711yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—7888no—
272C22—7884no—
273C22—7882no—
274C22—7881no—
275C22—3944no—
276C22#2026329612yes—
277C22C22×D78811yescharacteristic
278C4—22no—
279C4—7882no—
280C4—7881no—
281C4—3944no—
282C4#2026329612yes—
283C4C2×D4×D19711yescharacteristic
284C4C2×D4×D19711yescharacteristic
285C2—22no—
286C2—42no—
287C2—7884no—
288C2—15761no—
289C2C2×#2026329611yescharacteristic
290C2#2716187912yes—
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#407×C197C2
socleC22×C197C22×D4
hypercenter#407D197
hypocenterC197#3005
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#15691358C2semidirect
2. C2×#20263359C2semidirect
3. C2×#20263354C2semidirect
4. #27161590C2semidirect
5. #27161243C2semidirect
6. #27161226C2semidirect
7. #27161089C2semidirect
8. #27161085C2semidirect
9. #407×C197C2semidirect
21. C2×Dic788C22semidirect
22. C2×#15691347C22semidirect
23. C2×#15691347C22semidirect
24. C2×D1576C22semidirect
26. #20263305C22semidirect
27. #20263300C22semidirect
31. #20263251C22semidirect
32. #20263248C22semidirect
36. C22⋊C8×C197C22semidirect
45. C2×Dic394D4semidirect
50. C2×D788D4semidirect
145. C197#3005semidirect
155. #407D197semidirect
182. C22⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
C22×#156913582
C2×#202633592
C2×#202633542
#271615902
#271612432
#271612262
#271610892
#271610852
#407×C1972
#3005197