Finite Group Databasebeta

← #38751953

Subgroups — #38751953

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×#15691359C211yesmaximal, characteristic, complemented
2C2×#20263351C211yesmaximal, characteristic, complemented
3#27161586C211yesmaximal, characteristic, complemented
4#27161240C214yesmaximal, complemented
5#27161223C214yesmaximal, complemented
6#27161088C211yesmaximal, characteristic, complemented
7#27161085C211yesmaximal, characteristic, complemented
8#27161042C211yesmaximal, characteristic, complemented
9C2×#137×C197C211yesmaximal, characteristic, complemented
10C2×#15691359—24no—
11C2×#15691359—22no—
12C2×#15691359—24no—
13C2×#15691359C2214yes—
14C22×D788—21no—
15C22×D788C2211yescharacteristic
16C22×C4×D197—21no—
17C22×C4×D197C2211yescharacteristic
18#20263351—24no—
19C2×#15691345C2212yescomplemented
20C2×#15691343C2212yescomplemented
21C2×#15691342C2211yescharacteristic
22#20263307—21no—
23#20263302C2212yescomplemented
24#20263297C2212yescomplemented
25#20263294—21no—
26C4×C22⋊D197—22no—
27#20263285C2211yescharacteristic
28#20263250C2214yescomplemented
29#20263248C2214yescomplemented
30#20263238C2214yescomplemented
31C2×#41×C197C2211yescharacteristic
32C22×Q8×C197C2211yescharacteristic
33#137×C197—24no—
34C2×Q16×C197C2212yescomplemented
35C2×SD16×C197C2212yescomplemented
36C2×M16×C197C2211yescharacteristic
37C23×D197—21no—
38C23×D197D411yescharacteristic
39#15691359—216no—
40#15691359—28no—
41#15691359—28no—
42C2×D788—28no—
43C2×D788—24no—
44C2×D788—21no—
45C2×D788—22no—
46C2×D788D414yescomplemented
47C2×D788C2312yes—
48C2×C22⋊D197—22no—
49C22×Dic197D411yescharacteristic
50C2×C4×D197—28no—
51C2×C4×D197—24no—
52C2×C4×D197—22no—
53C2×C4×D197—22no—
54C2×C4×D197D412yes—
55C2×C4×D197C2312yes—
56#15691345—24no—
57#15691343—24no—
58#15691342—22no—
59C2×#13096802C2312yes—
60C4⋊Dic197C2312yes—
61#15691337—22no—
62C22⋊Dic197—22no—
63C22⋊Dic197—21no—
64#15691335—22no—
65C4×Dic197—21no—
66#41×C197—22no—
67#41×C197—22no—
68C2×Q8×C197—22no—
69C2×Q8×C197—21no—
70C2×Q8×C197D414yescomplemented
71C2×Q8×C197C2311yescharacteristic
72C2×Q8×C197C2312yes—
73C2×D4×C197—21no—
74C2×D4×C197C2311yescharacteristic
75C22×C4×C197—21no—
76C22×C4×C197—21no—
77C22×C4×C197C2311yescharacteristic
78Q16×C197—24no—
79SD16×C197—24no—
80M16×C197—22no—
81C2×C8×C197C2312yes—
82C22×D197—22no—
83C22×D197—24no—
84C22×D197—22no—
85C22×D197—42no—
86C22×D197—44no—
87C22×D197C2×D412yes—
88D788—216no—
89D788—24no—
90D788—22no—
91D788C2×D414yes—
92C22⋊D197—44no—
93C2×Dic197—22no—
94C2×Dic197—22no—
95C2×Dic197C2×D412yes—
96C4×D197—216no—
97C4×D197—24no—
98C4×D197—22no—
99C4×D197—22no—
100#13096802—22no—
101C23×C197—21no—
102C23×C197C2×D411yescharacteristic
103Q8×C197—24no—
104Q8×C197—22no—
105Q8×C197—21no—
106Q8×C197—21no—
107Q8×C197C2×D414yes—
108D4×C197—21no—
109D4×C197—21no—
110D4×C197—42no—
111C2×C4×C197—24no—
112C2×C4×C197—22no—
113C2×C4×C197—21no—
114C2×C4×C197—42no—
115C2×C4×C197C2×D411yescharacteristic
116C2×C4×C197C2×D412yes—
117C2×C4×C197C2×D412yes—
118C2×C4×C197C2411yescharacteristic
119C8×C197—22no—
120C22×C197—22no—
121C22×C197—41no—
122C22×C197—42no—
123C22×C197#12012yes—
124C22×C197C22×D411yescharacteristic
125C2×D197—22no—
126C2×D197—24no—
127C2×D197—22no—
128C2×D197—22no—
129C2×D197—44no—
130C2×D197—48no—
131C2×D197—44no—
132C4×C197—24no—
133C4×C197—41no—
134C4×C197#12012yes—
135C4×C197C22×D411yescharacteristic
136C4×C197C22×D411yescharacteristic
137Dic197—22no—
138Dic197—42no—
139C2×C197—22no—
140C2×C197—81no—
141C2×C197C2×#12011yescharacteristic
142C2×C197#57512yes—
143D197—42no—
144D197—44no—
145C197#300511yescharacteristic, complemented, Sylow
146#3005—1971nomaximal, Sylow
147C22×#41—1971no—
148C2×#137D19711yescharacteristic, complemented
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—3944no—
157C2×#41—3944no—
158C2×#41—3942no—
159C2×#41—1974no—
160C2×#41C2×D19711yescharacteristic
161C22×Q8C2×D19711yescharacteristic
162C22×D4—3941no—
163C22×D4—1971no—
164C23×C4—3941no—
165C23×C4—1971no—
166#137—24no—
167AGL(1,ℤ/8)—3944no—
168C2×Q16C2×D19712yescomplemented
169C2×SD16—1972no—
170C2×SD16C2×D19712yescomplemented
171C2×D8—1972no—
172C2×M16—1971no—
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⋊C8—1974no—
183C24—3941no—
184C24—1971no—
185#41—22no—
186#41—22no—
187#41—39416no—
188#41—3948no—
189#41—3948no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8C22⋊D19714yescomplemented
193C2×Q8C22×D19711yescharacteristic
194C2×Q8C22×D19712yes—
195C2×D4—21no—
196C2×D4—3948no—
197C2×D4—3944no—
198C2×D4—3942no—
199C2×D4—3941no—
200C2×D4—3942no—
201C2×D4—1974no—
202C2×D4—1972no—
203C2×D4C22×D19711yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—3948no—
207C22×C4—3944no—
208C22×C4—3942no—
209C22×C4—3942no—
210C22×C4—1971no—
211C22×C4—1972no—
212C22×C4—1972no—
213C22×C4C22×D19711yescharacteristic
214Q16—24no—
215SD16—24no—
216SD16—3944no—
217D8—3944no—
218M16—22no—
219M16—3942no—
220C2×C8—1972no—
221C2×C8C22×D19712yes—
222C4⋊C4—3942no—
223C4⋊C4—1972no—
224C22⋊C4—3942no—
225C22⋊C4—3941no—
226C22⋊C4—3942no—
227C42—3941no—
228C23—21no—
229C23—7884no—
230C23—7882no—
231C23—3942no—
232C23—3944no—
233C23—3942no—
234C23—1972no—
235C23D4×D19711yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—21no—
239Q8—21no—
240Q8C2×C22⋊D19714yes—
241D4—21no—
242D4—21no—
243D4—42no—
244D4—7884no—
245D4—39416no—
246D4—3944no—
247D4—3942no—
248D4—1974no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—39416no—
254C2×C4—3942no—
255C2×C4—3942no—
256C2×C4—3944no—
257C2×C4—3942no—
258C2×C4—3942no—
259C2×C4—1972no—
260C2×C4D4×D19711yescharacteristic
261C2×C4D4×D19712yes—
262C2×C4C2×C22⋊D19712yes—
263C2×C4C23×D19711yescharacteristic
264C8—22no—
265C8—3942no—
266C22—22no—
267C22—42no—
268C22—41no—
269C22—7888no—
270C22—7884no—
271C22—7884no—
272C22—3942no—
273C22—3942no—
274C22—3942no—
275C22—3944no—
276C22#2026329412yes—
277C22C2×D4×D19711yescharacteristic
278C4—24no—
279C4—41no—
280C4—7882no—
281C4—3942no—
282C4#2026329412yes—
283C4C2×D4×D19711yescharacteristic
284C4C22×C22⋊D19711yescharacteristic
285C2—22no—
286C2—81no—
287C2—7884no—
288C2—7882no—
289C2C2×#2026329411yescharacteristic
290C2#2716187812yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#137×C197C2
socleC22×C197C22×D4
hypercenterC2×#137D197
hypocenterC197#3005
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#15691359C2semidirect
2. C2×#20263351C2semidirect
3. #27161586C2semidirect
4. #27161240C2semidirect
5. #27161223C2semidirect
6. #27161088C2semidirect
7. #27161085C2semidirect
8. #27161042C2semidirect
9. C2×#137×C197C2semidirect
19. C2×#15691345C22semidirect
20. C2×#15691343C22semidirect
23. #20263302C22semidirect
24. #20263297C22semidirect
28. #20263250C22semidirect
29. #20263248C22semidirect
30. #20263238C22semidirect
34. C2×Q16×C197C22semidirect
35. C2×SD16×C197C22semidirect
46. C2×D788D4semidirect
70. C2×Q8×C197D4semidirect
145. C197#3005semidirect
148. C2×#137D197semidirect
168. C2×Q16C2×D197semidirect
170. C2×SD16C2×D197semidirect
192. C2×Q8C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C22×#156913592
C2×#202633512
#271615862
#271612402
#271612232
#271610882
#271610852
#271610422
C2×#137×C1972
#3005197