Finite Group Databasebeta

← #38751952

Subgroups — #38751952

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×#20263356C211yesmaximal, characteristic, complemented
3C2×#20263351C211yesmaximal, characteristic, complemented
4#27161239C214yesmaximal, complemented
5#27161222C214yesmaximal, complemented
6#27161087C211yesmaximal, characteristic, complemented
7#27161083C211yesmaximal, characteristic, complemented
8#27161043C211yesmaximal, characteristic, complemented
9#525×C197C211yesmaximal, characteristic, complemented
10C2×#15691358—24no—
11C2×#15691358—24no—
12C2×#15691358—22no—
13C2×#15691358C2214yes—
14C22×Dic394C2211yescharacteristic
15C22×D788C2211yescharacteristic
16C22×C22⋊D197—21no—
17C22×C4×D197—21no—
18#20263356—24no—
19#20263351—24no—
20C2×#15691348C2212yescomplemented
21C2×#15691346C2212yescomplemented
22C2×#15691345C2212yescomplemented
23C2×#15691343C2212yescomplemented
24C2×#15691342C2211yescharacteristic
25C2×#15691342C2211yescharacteristic
26#20263249C2214yescomplemented
27#20263246C2214yescomplemented
28C22⋊#13096802C2214yescomplemented
29C2×#41×C197C2211yescharacteristic
30C23×C4×C197C2211yescharacteristic
31#123×C197—21no—
32C22⋊Q8×C197C2212yescomplemented
33#121×C197C2212yescomplemented
34#120×C197—21no—
35C4×D4×C197—22no—
36#117×C197C2211yescharacteristic
37C23×D197—21no—
38#15691358—216no—
39#15691358—28no—
40#15691358—28no—
41C2×Dic394—22no—
42C2×Dic394—21no—
43C2×Dic394D414yescomplemented
44C2×Dic394C2312yes—
45C2×D788—22no—
46C2×D788—21no—
47C2×D788D414yescomplemented
48C2×D788C2312yes—
49C2×C22⋊D197—28no—
50C2×C22⋊D197—24no—
51C22×Dic197—21no—
52C2×C4×D197—28no—
53C2×C4×D197—24no—
54#15691348—24no—
55#15691346—24no—
56#15691345—24no—
57#15691343—24no—
58#15691342—22no—
59#15691342—22no—
60C2×#13096802C2312yes—
61C2×#13096802C2312yes—
62C24×C197D411yescharacteristic
63#41×C197—22no—
64#41×C197—22no—
65C2×Q8×C197C2311yescharacteristic
66C2×D4×C197—22no—
67C2×D4×C197—21no—
68C2×D4×C197C2311yescharacteristic
69C22×C4×C197—22no—
70C22×C4×C197—22no—
71C22×C4×C197—21no—
72C22×C4×C197D412yes—
73C22×C4×C197D411yescharacteristic
74C22×C4×C197C2312yes—
75C22×C4×C197C2311yescharacteristic
76C4⋊C4×C197—22no—
77C4⋊C4×C197C2312yes—
78C22⋊C4×C197—21no—
79C22⋊C4×C197—22no—
80C22⋊C4×C197—22no—
81C42×C197—21no—
82C22×D197—24no—
83C22×D197—22no—
84C22×D197—44no—
85Dic394—24no—
86Dic394—22no—
87Dic394C2×D414yes—
88D788—24no—
89D788—22no—
90D788C2×D414yes—
91C22⋊D197—216no—
92C2×Dic197—24no—
93C2×Dic197—22no—
94C4×D197—216no—
95#13096802—22no—
96#13096802—22no—
97C23×C197—22no—
98C23×C197—21no—
99C23×C197—42no—
100C23×C197C2×D412yes—
101C23×C197C2×D411yescharacteristic
102Q8×C197—21no—
103Q8×C197—21no—
104D4×C197—21no—
105D4×C197—21no—
106D4×C197—44no—
107D4×C197—42no—
108C2×C4×C197—22no—
109C2×C4×C197—24no—
110C2×C4×C197—22no—
111C2×C4×C197—22no—
112C2×C4×C197—22no—
113C2×C4×C197—21no—
114C2×C4×C197—42no—
115C2×C4×C197C2×D412yes—
116C2×C4×C197C2×D412yes—
117C2×C4×C197C2×D412yes—
118C2×C4×C197C2×D411yescharacteristic
119C2×C4×C197C2411yescharacteristic
120C22×C197—22no—
121C22×C197—22no—
122C22×C197—22no—
123C22×C197—22no—
124C22×C197—44no—
125C22×C197—41no—
126C22×C197—42no—
127C22×C197#12012yes—
128C22×C197C22×D411yescharacteristic
129C2×D197—24no—
130C2×D197—48no—
131C2×D197—44no—
132C4×C197—22no—
133C4×C197—41no—
134C4×C197—42no—
135C4×C197#12012yes—
136C4×C197C22×D411yescharacteristic
137C4×C197C22×D411yescharacteristic
138Dic197—24no—
139C2×C197—22no—
140C2×C197—42no—
141C2×C197—81no—
142C2×C197C2×#12011yescharacteristic
143C2×C197#57512yes—
144D197—44no—
145C197#300511yescharacteristic, complemented, Sylow
146#3005—1971nomaximal, Sylow
147C22×#41—1971no—
148C2×#137—1971no—
149C2×AGL(1,ℤ/8)—1971no—
150#525D19711yescharacteristic, complemented
151#452—1974no—
152#449—1974no—
153#419—1971no—
154#418—1971no—
155#407—1971no—
156C2×#41—3944no—
157C2×#41—3944no—
158C2×#41—3942no—
159C2×#41—1974no—
160C2×#41C2×D19711yescharacteristic
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×M16—1971no—
173C2×M16—1971no—
174#123—21no—
175C22⋊Q8C2×D19712yescomplemented
176#121C2×D19712yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D19711yescharacteristic
180#103—1974no—
181#102—1974no—
182C22⋊C8—1974no—
183C24—3941no—
184C24C22⋊D19711yescharacteristic
185#41—22no—
186#41—22no—
187#41—39416no—
188#41—3948no—
189#41—3948no—
190C2×Q8—3942no—
191C2×Q8—3941no—
192C2×Q8—1974no—
193C2×Q8—1972no—
194C2×Q8C22×D19711yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—3948no—
198C2×D4—3944no—
199C2×D4—3942no—
200C2×D4—3941no—
201C2×D4—1974no—
202C2×D4—1972no—
203C2×D4C22×D19711yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—3948no—
208C22×C4—3941no—
209C22×C4—3944no—
210C22×C4C22⋊D19712yes—
211C22×C4C22⋊D19711yescharacteristic
212C22×C4C22×D19712yes—
213C22×C4C22×D19711yescharacteristic
214Q16—3944no—
215SD16—3944no—
216SD16—3944no—
217D8—3944no—
218M16—3942no—
219M16—3942no—
220C2×C8—1972no—
221C2×C8—1972no—
222C4⋊C4—22no—
223C4⋊C4C22×D19712yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—7884no—
232C23—3944no—
233C23—3942no—
234C23C2×C22⋊D19712yes—
235C23C2×C22⋊D19711yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—3944no—
239Q8—3942no—
240Q8—1974no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—39416no—
246D4—3944no—
247D4—3942no—
248D4—1974no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—39416no—
257C2×C4—3944no—
258C2×C4—3942no—
259C2×C4D4×D19712yes—
260C2×C4D4×D19712yes—
261C2×C4C2×C22⋊D19712yes—
262C2×C4C2×C22⋊D19711yescharacteristic
263C2×C4C23×D19711yescharacteristic
264C8—3942no—
265C8—3942no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—42no—
272C22—41no—
273C22—7888no—
274C22—7884no—
275C22—3944no—
276C22#2026329412yes—
277C22C22×C22⋊D19711yescharacteristic
278C4—22no—
279C4—42no—
280C4—41no—
281C4—3944no—
282C4#2026329412yes—
283C4C2×D4×D19711yescharacteristic
284C4C2×D4×D19711yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—7884no—
289C2C2×#2026329411yescharacteristic
290C2#2716187712yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#525×C197C2
socleC22×C197C22×D4
hypercenter#525D197
hypocenterC197#3005
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#15691358C2semidirect
2. C2×#20263356C2semidirect
3. C2×#20263351C2semidirect
4. #27161239C2semidirect
5. #27161222C2semidirect
6. #27161087C2semidirect
7. #27161083C2semidirect
8. #27161043C2semidirect
9. #525×C197C2semidirect
20. C2×#15691348C22semidirect
21. C2×#15691346C22semidirect
22. C2×#15691345C22semidirect
23. C2×#15691343C22semidirect
26. #20263249C22semidirect
27. #20263246C22semidirect
28. C22⋊#13096802C22semidirect
32. C22⋊Q8×C197C22semidirect
33. #121×C197C22semidirect
43. C2×Dic394D4semidirect
47. C2×D788D4semidirect
145. C197#3005semidirect
150. #525D197semidirect
175. C22⋊Q8C2×D197semidirect
176. #121C2×D197semidirect

Maximal subgroups

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

subgroupindex
C22×#156913582
C2×#202633562
C2×#202633512
#271612392
#271612222
#271610872
#271610832
#271610432
#525×C1972
#3005197