Finite Group Databasebeta

← #38751978

Subgroups — #38751978

Order 25216.

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

normal subgroups only

This group has 784 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
1C2×#20263372C211yesmaximal, characteristic, complemented
2C2×#20263352C211yesmaximal, characteristic, complemented
3C22×#15691343C211yesmaximal, characteristic, complemented
4#27161222C214yesmaximal, complemented
5#27161212C214yesmaximal, complemented
6#27161083C211yesmaximal, characteristic, complemented
7#27161073C211yesmaximal, characteristic, complemented
8#27161046C211yesmaximal, characteristic, complemented
9#535×C197C211yesmaximal, characteristic, complemented
10#20263372—24no—
11#20263372—24no—
12C2×#15691359—21no—
13C2×#15691358C2211yescharacteristic
14C2×#15691358C2212yescomplemented
15C22×D788C2211yescharacteristic
16C22×D788C2212yescomplemented
17C2×D4×D197—22no—
18C2×D4×D197—21no—
19#20263352—24no—
20C2×#15691346C2212yescomplemented
21C2×#15691343—24no—
22C2×#15691343C2212yescomplemented
23C2×#15691343C2214yescomplemented
24C2×#15691342C2211yescharacteristic, complemented
25C22×#13096802C2211yescharacteristic, complemented
26#20263249C2212yescomplemented
27#20263246C2212yescomplemented
28#20263246C2214yescomplemented
29C22⋊#13096802C2214yescomplemented
30C2×#41×C197C2211yescharacteristic
31C22×D4×C197C2211yescharacteristic, complemented
32#127×C197—21no—
33#124×C197—21no—
34#121×C197C2214yescomplemented
35#120×C197—22no—
36#117×C197C2211yescharacteristic, complemented
37C23×D197—21no—
38C23×D197—22no—
39#15691359—28no—
40#15691358—24no—
41#15691358—24no—
42#15691358—22no—
43#15691358D414yescomplemented
44C2×Dic394C2311yescharacteristic
45C2×D788—24no—
46C2×D788—24no—
47C2×D788—21no—
48C2×D788—22no—
49C2×D788—22no—
50C2×D788D414yescomplemented
51C2×D788C2312yes—
52C2×D788C2311yescharacteristic
53C2×C22⋊D197—21no—
54C2×C22⋊D197—22no—
55D4×D197—216no—
56D4×D197—28no—
57C2×C4×D197—21no—
58C2×C4×D197—22no—
59#15691346—24no—
60#15691343—28no—
61#15691343—24no—
62#15691342—22no—
63C2×#13096802—22no—
64C2×#13096802C2312yes—
65C2×#13096802C2312yes—
66C24×C197—21no—
67#41×C197—22no—
68#41×C197—22no—
69C2×Q8×C197D411yescharacteristic
70C2×D4×C197—24no—
71C2×D4×C197—22no—
72C2×D4×C197—22no—
73C2×D4×C197—21no—
74C2×D4×C197D412yes—
75C2×D4×C197D411yescharacteristic
76C2×D4×C197C2312yes—
77C22×C4×C197C2312yes—
78C22×C4×C197C2311yescharacteristic
79C4⋊C4×C197C2312yes—
80C22⋊C4×C197—21no—
81C22⋊C4×C197—24no—
82C42×C197—21no—
83C22×D197—21no—
84C22×D197—22no—
85C22×D197—28no—
86C22×D197—24no—
87C22×D197—82no—
88C22×D197—84no—
89Dic394—21no—
90Dic394C2×D412yes—
91D788—28no—
92D788—24no—
93D788—22no—
94D788—21no—
95D788C2×D412yes—
96D788C2×D414yes—
97C22⋊D197—28no—
98C22⋊D197—24no—
99C2×Dic197—21no—
100C4×D197—28no—
101C4×D197—24no—
102#13096802—22no—
103#13096802—22no—
104C23×C197—22no—
105C23×C197—42no—
106C23×C197—82no—
107C23×C197C2×D412yes—
108C23×C197C2×D411yescharacteristic
109Q8×C197—22no—
110D4×C197—24no—
111D4×C197—22no—
112D4×C197—22no—
113D4×C197—22no—
114D4×C197—42no—
115D4×C197—84no—
116C2×C4×C197—22no—
117C2×C4×C197—22no—
118C2×C4×C197—22no—
119C2×C4×C197C2×D412yes—
120C2×C4×C197C2×D412yes—
121C2×C4×C197C2×D412yes—
122C2×C4×C197C2×D411yescharacteristic
123C2×C4×C197C2411yescharacteristic
124C22×C197—22no—
125C22×C197—22no—
126C22×C197—24no—
127C22×C197—42no—
128C22×C197—44no—
129C22×C197—84no—
130C22×C197#12012yes—
131C22×C197C22×D411yescharacteristic
132C2×D197—24no—
133C2×D197—22no—
134C2×D197—44no—
135C2×D197—42no—
136C2×D197—84no—
137C2×D197—88no—
138C4×C197—22no—
139C4×C197—42no—
140C4×C197#12012yes—
141C4×C197C22×D411yescharacteristic
142C4×C197C22×D411yescharacteristic
143Dic197—22no—
144C2×C197—22no—
145C2×C197—42no—
146C2×C197—82no—
147C2×C197C2×#12011yescharacteristic
148C2×C197#57612yes—
149D197—44no—
150D197—42no—
151C197#300811yescharacteristic, complemented, Sylow
152#3008—1971nomaximal, Sylow
153C2×H(5,𝔽2)—1971no—
154C2×AGL(1,ℤ/8)—1971no—
155C22×D8—1971no—
156#535D19711yescharacteristic, complemented
157#449—1974no—
158#447—1974no—
159#418—1971no—
160#416—1971no—
161#408—1971no—
162H(5,𝔽2)—3944no—
163H(5,𝔽2)—3944no—
164C2×#41—3941no—
165C2×#41—1972no—
166C2×#41—1971no—
167C2×#41C2×D19711yescharacteristic
168C22×D4—3942no—
169C22×D4—3941no—
170C22×D4—1972no—
171C22×D4—1971no—
172C22×D4C2×D19711yescharacteristic, complemented
173AGL(1,ℤ/8)—3944no—
174C2×SD16—1972no—
175C2×D8—3944no—
176C2×D8—1974no—
177C2×D8—1972no—
178C2×M16—1971no—
179C22×C8—1971no—
180#127—21no—
181#124—21no—
182#121C2×D19714yescomplemented
183#120—22no—
184#117C2×D19711yescharacteristic, complemented
185#103—1972no—
186#102—1972no—
187#102—1974no—
188C22⋊C8—1974no—
189C24—21no—
190C24—3942no—
191C24—3941no—
192#41—22no—
193#41—22no—
194#41—3944no—
195#41—3944no—
196#41—3948no—
197#41—3942no—
198#41—1974no—
199C2×Q8—1971no—
200C2×Q8C22⋊D19711yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—3944no—
206C2×D4—3944no—
207C2×D4—39416no—
208C2×D4—3948no—
209C2×D4—3942no—
210C2×D4—3941no—
211C2×D4—3942no—
212C2×D4—3942no—
213C2×D4—3941no—
214C2×D4—1974no—
215C2×D4—1971no—
216C2×D4—1972no—
217C2×D4C22⋊D19712yes—
218C2×D4C22⋊D19711yescharacteristic
219C2×D4C22×D19712yes—
220C22×C4—3942no—
221C22×C4—3941no—
222C22×C4C22×D19712yes—
223C22×C4C22×D19711yescharacteristic
224SD16—3944no—
225D8—3948no—
226D8—3944no—
227M16—3942no—
228C2×C8—3942no—
229C2×C8—1972no—
230C2×C8—1972no—
231C4⋊C4C22×D19712yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—15762no—
239C23—15764no—
240C23—3948no—
241C23—3942no—
242C23—3941no—
243C23—3944no—
244C23C2×C22⋊D19712yes—
245C23C2×C22⋊D19711yescharacteristic
246Q8—22no—
247Q8—3941no—
248Q8—1972no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—3948no—
256D4—3948no—
257D4—3944no—
258D4—3944no—
259D4—3942no—
260D4—3941no—
261D4—1972no—
262D4—1974no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—3948no—
267C2×C4—3941no—
268C2×C4—3944no—
269C2×C4D4×D19712yes—
270C2×C4D4×D19712yes—
271C2×C4C2×C22⋊D19712yes—
272C2×C4C2×C22⋊D19711yescharacteristic
273C2×C4C23×D19711yescharacteristic
274C8—3942no—
275C8—3942no—
276C22—22no—
277C22—24no—
278C22—22no—
279C22—44no—
280C22—42no—
281C22—84no—
282C22—7884no—
283C22—7882no—
284C22—15764no—
285C22—15768no—
286C22—3944no—
287C22—3942no—
288C22#2026329412yes—
289C22C22×C22⋊D19711yescharacteristic
290C4—22no—
291C4—42no—
292C4—3942no—
293C4#2026329412yes—
294C4C2×D4×D19711yescharacteristic
295C4C2×D4×D19711yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—7884no—
300C2—7882no—
301C2C2×#2026329411yescharacteristic
302C2#2716188212yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#535×C197C2
socleC22×C197C22×D4
hypercenter#535D197
hypocenterC197#3008
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#20263372C2semidirect
2. C2×#20263352C2semidirect
3. C22×#15691343C2semidirect
4. #27161222C2semidirect
5. #27161212C2semidirect
6. #27161083C2semidirect
7. #27161073C2semidirect
8. #27161046C2semidirect
9. #535×C197C2semidirect
14. C2×#15691358C22semidirect
16. C22×D788C22semidirect
20. C2×#15691346C22semidirect
22. C2×#15691343C22semidirect
23. C2×#15691343C22semidirect
24. C2×#15691342C22semidirect
25. C22×#13096802C22semidirect
26. #20263249C22semidirect
27. #20263246C22semidirect
28. #20263246C22semidirect
29. C22⋊#13096802C22semidirect
31. C22×D4×C197C22semidirect
34. #121×C197C22semidirect
36. #117×C197C22semidirect
43. #15691358D4semidirect
50. C2×D788D4semidirect
151. C197#3008semidirect
156. #535D197semidirect
172. C22×D4C2×D197semidirect
182. #121C2×D197semidirect
184. #117C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633722
C2×#202633522
C22×#156913432
#271612222
#271612122
#271610832
#271610732
#271610462
#535×C1972
#3008197