Finite Group Databasebeta

← #38751964

Subgroups — #38751964

Order 25216.

806 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 169 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×#20263351C212yesmaximal, complemented
2C2×#20263300C211yesmaximal, characteristic, complemented
3#27161224C214yesmaximal, complemented
4#27161218C212yesmaximal, complemented
5#27161213C212yesmaximal, complemented
6#27161080C212yesmaximal, complemented
7#27161043C211yesmaximal, characteristic, complemented
8C22×#41×C197C211yesmaximal, characteristic, complemented
9C22×D788C2211yescharacteristic, complemented
10C22×C22⋊D197—21no—
11#20263351—28no—
12C2×#15691345C2214yescomplemented
13C2×#15691343C2214yescomplemented
14C2×#15691342C2212yescomplemented
15#20263300—21no—
16#20263300—21no—
17#20263300C2212yes—
18#20263300C2212yescomplemented
19C2×C4⋊Dic197C2211yescharacteristic, complemented
20C2×#15691335—21no—
21#20263250C2214yescomplemented
22#20263247C2214yescomplemented
23C22⋊#13096802C2212yescomplemented
24C22⋊#13096802C2212yescomplemented
25C2×#41×C197—28no—
26C2×#41×C197—21no—
27C2×#41×C197—21no—
28C2×#41×C197C2212yes—
29C2×#41×C197C2212yescomplemented
30C22×Q8×C197C2211yescharacteristic, complemented
31C22×D4×C197—21no—
32C22×D4×C197C2211yescharacteristic, complemented
33C23×C4×C197—21no—
34C23×C4×C197C2211yescharacteristic
35C23×D197—21no—
36C2×D788—24no—
37C2×D788—21no—
38C2×D788C2312yes—
39C2×C22⋊D197—22no—
40C2×C22⋊D197—22no—
41C2×C22⋊D197—44no—
42C22×Dic197—21no—
43#15691345—28no—
44#15691343—28no—
45#15691342—24no—
46C2×#13096802C2314yes—
47C4⋊Dic197—21no—
48C4⋊Dic197C2312yes—
49#15691335—22no—
50#15691335—22no—
51C24×C197—21no—
52C24×C197D411yescharacteristic
53#41×C197—28no—
54#41×C197—28no—
55#41×C197—24no—
56#41×C197—24no—
57#41×C197—24no—
58#41×C197D418yescomplemented
59C2×Q8×C197—24no—
60C2×Q8×C197—21no—
61C2×Q8×C197C2312yes—
62C2×D4×C197—28no—
63C2×D4×C197—22no—
64C2×D4×C197—22no—
65C2×D4×C197—24no—
66C2×D4×C197—21no—
67C2×D4×C197C2312yes—
68C22×C4×C197—28no—
69C22×C4×C197—21no—
70C22×C4×C197—22no—
71C22×C4×C197—22no—
72C22×C4×C197—24no—
73C22×C4×C197D411yescharacteristic
74C22×C4×C197D411yescharacteristic
75C22×C4×C197D411yescharacteristic
76C22×C4×C197C2311yescharacteristic
77C22×C4×C197C2311yescharacteristic
78C22×C4×C197C2311yescharacteristic
79C22×D197—22no—
80C22×D197—42no—
81C22×D197—42no—
82C22×D197—42no—
83D788—24no—
84D788—42no—
85C22⋊D197—44no—
86C22⋊D197—44no—
87C2×Dic197—22no—
88C2×Dic197—42no—
89#13096802—24no—
90C23×C197—24no—
91C23×C197—22no—
92C23×C197—22no—
93C23×C197—22no—
94C23×C197—41no—
95C23×C197—42no—
96C23×C197—42no—
97C23×C197C2×D411yescharacteristic
98C23×C197C2×D411yescharacteristic
99C23×C197C2×D411yescharacteristic
100Q8×C197—24no—
101Q8×C197—22no—
102Q8×C197C2×D414yescomplemented
103D4×C197—28no—
104D4×C197—24no—
105D4×C197—28no—
106D4×C197—22no—
107D4×C197C2×D414yescomplemented
108C2×C4×C197—24no—
109C2×C4×C197—28no—
110C2×C4×C197—22no—
111C2×C4×C197—22no—
112C2×C4×C197—22no—
113C2×C4×C197—28no—
114C2×C4×C197—22no—
115C2×C4×C197—22no—
116C2×C4×C197—22no—
117C2×C4×C197C2×D411yescharacteristic
118C2×C4×C197C2×D411yescharacteristic
119C2×C4×C197C2×D411yescharacteristic
120C2×C4×C197C2×D414yes—
121C2×C4×C197C2411yescharacteristic
122C22×C197—21no—
123C22×C197—21no—
124C22×C197—21no—
125C22×C197—21no—
126C22×C197—21no—
127C22×C197—24no—
128C22×C197—21no—
129C22×C197—21no—
130C22×C197—44no—
131C22×C197—48no—
132C22×C197—42no—
133C22×C197—42no—
134C22×C197#12012yes—
135C22×C197AGL(1,ℤ/8)12yescomplemented
136C22×C197C22×D411yescharacteristic
137C2×D197—42no—
138C2×D197—42no—
139C2×D197—42no—
140C2×D197—84no—
141C4×C197—21no—
142C4×C197—24no—
143C4×C197—21no—
144C4×C197#12012yes—
145C4×C197C22×D412yes—
146Dic197—42no—
147C2×C197—22no—
148C2×C197—22no—
149C2×C197—41no—
150C2×C197—44no—
151C2×C197C2×#12011yescharacteristic
152C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C197#57511yescharacteristic
154D197—82no—
155C197#300611yescharacteristic, complemented, Sylow
156#3006—1971nomaximal, Sylow
157C22×#41D19711yescharacteristic, complemented
158C2×AGL(1,ℤ/8)—1972no—
159C2×#121—1971no—
160#449—1974no—
161#448—1972no—
162#447—1972no—
163#417—1972no—
164#407—1971no—
165C2×#41—28no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41C2×D19712yes—
169C2×#41C2×D19712yescomplemented
170C22×Q8C2×D19711yescharacteristic, complemented
171C22×D4—21no—
172C22×D4—3941no—
173C22×D4—1971no—
174C22×D4C2×D19711yescharacteristic, complemented
175C23×C4—21no—
176C23×C4C2×D19711yescharacteristic
177AGL(1,ℤ/8)—3948no—
178C2×SD16—1974no—
179C2×D8—1974no—
180C2×M16—1972no—
181#121—3941no—
182#121—3941no—
183#121—1972no—
184#121—1972no—
185C2×C4⋊C4—1971no—
186C2×C22⋊C4—3941no—
187#103—1974no—
188#102—1974no—
189C22⋊C8—1972no—
190C22⋊C8—1972no—
191C24—21no—
192C24—3941no—
193C24C22⋊D19711yescharacteristic
194#41—28no—
195#41—28no—
196#41—24no—
197#41—24no—
198#41—24no—
199#41C22⋊D19718yescomplemented
200C2×Q8—24no—
201C2×Q8—21no—
202C2×Q8C22×D19712yes—
203C2×D4—28no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—7884no—
209C2×D4—3942no—
210C2×D4—3942no—
211C2×D4—3941no—
212C2×D4—3944no—
213C2×D4—1972no—
214C2×D4C22×D19712yes—
215C22×C4—28no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—3941no—
221C22×C4C22⋊D19711yescharacteristic
222C22×C4C22⋊D19711yescharacteristic
223C22×C4C22⋊D19711yescharacteristic
224C22×C4C22×D19711yescharacteristic
225C22×C4C22×D19711yescharacteristic
226C22×C4C22×D19711yescharacteristic
227SD16—3948no—
228D8—3948no—
229M16—3944no—
230C2×C8—1974no—
231C4⋊C4—3941no—
232C4⋊C4—1972no—
233C22⋊C4—3942no—
234C22⋊C4—3942no—
235C23—24no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—41no—
240C23—42no—
241C23—42no—
242C23—7882no—
243C23—7882no—
244C23—7882no—
245C23—3942no—
246C23C2×C22⋊D19711yescharacteristic
247C23C2×C22⋊D19711yescharacteristic
248C23C2×C22⋊D19711yescharacteristic
249Q8—24no—
250Q8—22no—
251Q8C2×C22⋊D19714yescomplemented
252D4—28no—
253D4—24no—
254D4—28no—
255D4—22no—
256D4—7884no—
257D4—7884no—
258D4—7882no—
259D4—3944no—
260D4C2×C22⋊D19714yescomplemented
261C2×C4—24no—
262C2×C4—28no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—7882no—
271C2×C4—3942no—
272C2×C4C2×C22⋊D19711yescharacteristic
273C2×C4C2×C22⋊D19711yescharacteristic
274C2×C4C2×C22⋊D19711yescharacteristic
275C2×C4C2×C22⋊D19714yes—
276C2×C4C23×D19711yescharacteristic
277C8—3944no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—24no—
286C22—44no—
287C22—48no—
288C22—42no—
289C22—42no—
290C22—7882no—
291C22—7882no—
292C22—7882no—
293C22—15764no—
294C22#2026329512yes—
295C22#2026335112yescomplemented
296C22C22×C22⋊D19711yescharacteristic
297C4—21no—
298C4—24no—
299C4—21no—
300C4—7882no—
301C4#2026329512yes—
302C4C22×C22⋊D19712yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—44no—
307C2—15762no—
308C2C2×#2026329511yescharacteristic
309C2C2×#2026335111yescharacteristic
310C2#2716187711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC22×#41×C197C2
socleC22×C197C22×D4
hypercenterC22×#41D197
hypocenterC197#3006
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×#20263351C2semidirect
2. C2×#20263300C2semidirect
3. #27161224C2semidirect
4. #27161218C2semidirect
5. #27161213C2semidirect
6. #27161080C2semidirect
7. #27161043C2semidirect
8. C22×#41×C197C2semidirect
9. C22×D788C22semidirect
12. C2×#15691345C22semidirect
13. C2×#15691343C22semidirect
14. C2×#15691342C22semidirect
18. #20263300C22semidirect
19. C2×C4⋊Dic197C22semidirect
21. #20263250C22semidirect
22. #20263247C22semidirect
23. C22⋊#13096802C22semidirect
24. C22⋊#13096802C22semidirect
29. C2×#41×C197C22semidirect
30. C22×Q8×C197C22semidirect
32. C22×D4×C197C22semidirect
58. #41×C197D4semidirect
102. Q8×C197C2×D4semidirect
107. D4×C197C2×D4semidirect
135. C22×C197AGL(1,ℤ/8)semidirect
155. C197#3006semidirect
157. C22×#41D197semidirect
169. C2×#41C2×D197semidirect
170. C22×Q8C2×D197semidirect
174. C22×D4C2×D197semidirect
199. #41C22⋊D197semidirect
251. Q8C2×C22⋊D197semidirect
260. D4C2×C22⋊D197semidirect
295. C22#20263351semidirect

Maximal subgroups

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

subgroupindex
C2×#202633512
C2×#202633002
#271612242
#271612182
#271612132
#271610802
#271610432
C22×#41×C1972
#3006197