Finite Group Databasebeta

← #38751954

Subgroups — #38751954

Order 25216.

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

normal subgroups only

This group has 758 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×#20263357C211yesmaximal, characteristic, complemented
2C2×#20263352C211yesmaximal, characteristic, complemented
3#27161590C211yesmaximal, characteristic, complemented
4#27161241C214yesmaximal, complemented
5#27161224C214yesmaximal, complemented
6#27161088C211yesmaximal, characteristic, complemented
7#27161084C211yesmaximal, characteristic, complemented
8#27161043C211yesmaximal, characteristic, complemented
9C22×#41×C197C211yesmaximal, characteristic, complemented
10C2×#15691358C2211yescharacteristic
11#20263357—24no—
12#20263352—24no—
13C2×#15691348C2212yescomplemented
14C2×#15691346C2212yescomplemented
15C2×#15691345C2212yescomplemented
16C2×#15691343C2212yescomplemented
17C2×#15691342C2211yescharacteristic
18C2×#15691342C2211yescharacteristic
19#20263309—21no—
20#20263305C2212yescomplemented
21#20263300C2212yescomplemented
22#20263295—21no—
23C4×C22⋊D197—22no—
24#20263285C2211yescharacteristic
25#20263250C2214yescomplemented
26#20263247C2214yescomplemented
27C22⋊#13096802C2214yescomplemented
28C2×#41×C197—24no—
29C2×#41×C197—24no—
30C2×#41×C197—22no—
31C2×#41×C197C2214yes—
32C22×Q8×C197C2211yescharacteristic
33C22×D4×C197—21no—
34C22×D4×C197C2211yescharacteristic
35C23×C4×C197—21no—
36C23×C4×C197C2211yescharacteristic
37#15691358—22no—
38#15691358—22no—
39C2×Dic394C2311yescharacteristic
40C2×D788C2311yescharacteristic
41C2×C22⋊D197—22no—
42C2×C22⋊D197—21no—
43C2×C4×D197—21no—
44#15691348—24no—
45#15691346—24no—
46#15691345—24no—
47#15691343—24no—
48#15691342—22no—
49#15691342—22no—
50C2×#13096802C2312yes—
51C2×#13096802C2312yes—
52C4⋊Dic197C2312yes—
53#15691337—22no—
54C22⋊Dic197—21no—
55C22⋊Dic197—22no—
56#15691335—22no—
57C4×Dic197—21no—
58C24×C197—21no—
59C24×C197D411yescharacteristic
60#41×C197—216no—
61#41×C197—28no—
62#41×C197—28no—
63C2×Q8×C197—22no—
64C2×Q8×C197—21no—
65C2×Q8×C197D414yescomplemented
66C2×Q8×C197C2312yes—
67C2×D4×C197—28no—
68C2×D4×C197—24no—
69C2×D4×C197—21no—
70C2×D4×C197—22no—
71C2×D4×C197D414yescomplemented
72C2×D4×C197C2312yes—
73C22×C4×C197—28no—
74C22×C4×C197—21no—
75C22×C4×C197—24no—
76C22×C4×C197—22no—
77C22×C4×C197—22no—
78C22×C4×C197D411yescharacteristic
79C22×C4×C197D412yes—
80C22×C4×C197C2312yes—
81C22×C4×C197C2311yescharacteristic
82C22×D197—21no—
83Dic394—21no—
84Dic394—21no—
85D788—21no—
86D788—21no—
87C22⋊D197—44no—
88C22⋊D197—42no—
89C2×Dic197—22no—
90C2×Dic197—21no—
91C4×D197—42no—
92#13096802—22no—
93#13096802—22no—
94C23×C197—22no—
95C23×C197—24no—
96C23×C197—22no—
97C23×C197—42no—
98C23×C197—44no—
99C23×C197C2×D412yes—
100C23×C197C2×D411yescharacteristic
101Q8×C197—24no—
102Q8×C197—22no—
103Q8×C197C2×D414yes—
104D4×C197—216no—
105D4×C197—24no—
106D4×C197—22no—
107D4×C197C2×D414yes—
108C2×C4×C197—216no—
109C2×C4×C197—22no—
110C2×C4×C197—24no—
111C2×C4×C197—22no—
112C2×C4×C197—24no—
113C2×C4×C197—22no—
114C2×C4×C197—22no—
115C2×C4×C197C2×D412yes—
116C2×C4×C197C2×D411yescharacteristic
117C2×C4×C197C2×D412yes—
118C2×C4×C197C2×D412yes—
119C2×C4×C197C2411yescharacteristic
120C22×C197—22no—
121C22×C197—22no—
122C22×C197—24no—
123C22×C197—22no—
124C22×C197—22no—
125C22×C197—44no—
126C22×C197—48no—
127C22×C197—44no—
128C22×C197#12012yes—
129C22×C197C22×D411yescharacteristic
130C2×D197—41no—
131C2×D197—42no—
132C4×C197—22no—
133C4×C197—24no—
134C4×C197#12012yes—
135C4×C197C22×D411yescharacteristic
136C4×C197C22×D411yescharacteristic
137Dic197—41no—
138Dic197—42no—
139C2×C197—22no—
140C2×C197—42no—
141C2×C197—44no—
142C2×C197C2×#12011yescharacteristic
143C2×C197#57512yes—
144D197—81no—
145C197#300511yescharacteristic, complemented, Sylow
146#3005—1971nomaximal, Sylow
147C22×#41D19711yescharacteristic, complemented
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#407—1971no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—1971no—
160C2×#41C2×D19714yes—
161C22×Q8C2×D19711yescharacteristic
162C22×D4—21no—
163C22×D4C2×D19711yescharacteristic
164C23×C4—21no—
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—3941no—
175C22⋊Q8—1972no—
176#121—1972no—
177#120—3941no—
178C4×D4—3942no—
179#117—1971no—
180#103—1974no—
181#102—1974no—
182C22⋊C8—1974no—
183C24—21no—
184C24C22⋊D19711yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188#41—3942no—
189#41—3942no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—1971no—
193C2×Q8C22⋊D19714yescomplemented
194C2×Q8C22×D19712yes—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—3942no—
200C2×D4—3941no—
201C2×D4—1971no—
202C2×D4C22⋊D19714yescomplemented
203C2×D4C22×D19712yes—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—3941no—
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—3942no—
223C4⋊C4—1972no—
224C22⋊C4—3941no—
225C22⋊C4—3942no—
226C22⋊C4—3942no—
227C42—3941no—
228C23—22no—
229C23—24no—
230C23—22no—
231C23—42no—
232C23—44no—
233C23—3941no—
234C23C2×C22⋊D19712yes—
235C23C2×C22⋊D19711yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—3941no—
239Q8—3941no—
240Q8C2×C22⋊D19714yes—
241D4—216no—
242D4—24no—
243D4—22no—
244D4—7884no—
245D4—7882no—
246D4—3941no—
247D4—3941no—
248D4C2×C22⋊D19714yes—
249C2×C4—216no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—7882no—
257C2×C4—3942no—
258C2×C4—3941no—
259C2×C4C2×C22⋊D19712yes—
260C2×C4C2×C22⋊D19711yescharacteristic
261C2×C4C2×C22⋊D19712yes—
262C2×C4C2×C22⋊D19712yes—
263C2×C4C23×D19711yescharacteristic
264C8—3942no—
265C8—3942no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—7882no—
275C22—7881no—
276C22#2026329512yes—
277C22C22×C22⋊D19711yescharacteristic
278C4—22no—
279C4—24no—
280C4—7882no—
281C4—7881no—
282C4#2026329512yes—
283C4C22×C22⋊D19711yescharacteristic
284C4C22×C22⋊D19711yescharacteristic
285C2—22no—
286C2—42no—
287C2—44no—
288C2—15761no—
289C2C2×#2026329511yescharacteristic
290C2#2716187712yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC22×#41×C197C2
socleC22×C197C22×D4
hypercenterC22×#41D197
hypocenterC197#3005
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#20263357C2semidirect
2. C2×#20263352C2semidirect
3. #27161590C2semidirect
4. #27161241C2semidirect
5. #27161224C2semidirect
6. #27161088C2semidirect
7. #27161084C2semidirect
8. #27161043C2semidirect
9. C22×#41×C197C2semidirect
13. C2×#15691348C22semidirect
14. C2×#15691346C22semidirect
15. C2×#15691345C22semidirect
16. C2×#15691343C22semidirect
20. #20263305C22semidirect
21. #20263300C22semidirect
25. #20263250C22semidirect
26. #20263247C22semidirect
27. C22⋊#13096802C22semidirect
65. C2×Q8×C197D4semidirect
71. C2×D4×C197D4semidirect
145. C197#3005semidirect
147. C22×#41D197semidirect
193. C2×Q8C22⋊D197semidirect
202. C2×D4C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633572
C2×#202633522
#271615902
#271612412
#271612242
#271610882
#271610842
#271610432
C22×#41×C1972
#3005197