Finite Group Databasebeta

← #38751979

Subgroups — #38751979

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
2C22×#15691343C211yesmaximal, characteristic, complemented
3#27161646C211yesmaximal, characteristic, complemented
4#27161223C214yesmaximal, complemented
5#27161211C214yesmaximal, complemented
6#27161085C211yesmaximal, characteristic, complemented
7#27161075C211yesmaximal, characteristic, complemented
8#27161047C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C197C211yesmaximal, characteristic, complemented
10#20263372—24no—
11#20263372—24no—
12C2×#15691359C2212yescomplemented
13C2×#15691358—21no—
14C2×#15691358C2211yescharacteristic
15C22×D788—21no—
16C22×D788C2211yescharacteristic
17C2×D4×D197—22no—
18C2×D4×D197C2212yescomplemented
19C2×#15691343—24no—
20C2×#15691343C2214yescomplemented
21C22×#13096802C2211yescharacteristic, complemented
22#20263322—21no—
23#20263313—21no—
24#20263297C2214yescomplemented
25#20263295—22no—
26#20263285C2211yescharacteristic, complemented
27#20263250C2212yescomplemented
28#20263248C2214yescomplemented
29#20263247C2212yescomplemented
30#20263238C2214yescomplemented
31C2×#41×C197C2211yescharacteristic
32C22×D4×C197C2211yescharacteristic, complemented
33AGL(1,ℤ/8)×C197—24no—
34C2×SD16×C197C2212yescomplemented
35C2×D8×C197C2212yescomplemented
36C2×M16×C197C2211yescharacteristic, complemented
37C23×D197—22no—
38C23×D197—21no—
39#15691359—24no—
40#15691359—24no—
41#15691358—28no—
42#15691358—22no—
43#15691358—22no—
44C2×Dic394D411yescharacteristic
45C2×D788—28no—
46C2×D788—22no—
47C2×D788—21no—
48C2×D788—22no—
49C2×D788D411yescharacteristic
50C2×D788D414yescomplemented
51C2×D788C2312yes—
52C2×C22⋊D197—24no—
53C2×C22⋊D197—22no—
54C2×C22⋊D197D412yes—
55D4×D197—216no—
56D4×D197—24no—
57D4×D197—24no—
58C2×C4×D197—22no—
59C2×C4×D197C2312yes—
60#15691343—28no—
61C2×#13096802—22no—
62C2×#13096802C2312yes—
63C4⋊Dic197C2312yes—
64C22⋊Dic197—24no—
65C22⋊Dic197—21no—
66C4×Dic197—21no—
67C24×C197—21no—
68#41×C197—22no—
69#41×C197D414yescomplemented
70C2×Q8×C197C2311yescharacteristic
71C2×D4×C197—21no—
72C2×D4×C197—22no—
73C2×D4×C197—22no—
74C2×D4×C197—21no—
75C2×D4×C197C2312yes—
76C2×D4×C197C2311yescharacteristic
77C22×C4×C197—21no—
78C22×C4×C197C2311yescharacteristic
79SD16×C197—24no—
80D8×C197—24no—
81M16×C197—22no—
82C2×C8×C197C2312yes—
83C22×D197—22no—
84C22×D197—28no—
85C22×D197—24no—
86C22×D197—84no—
87C22×D197—82no—
88C22×D197C2×D412yes—
89Dic394—22no—
90D788—28no—
91D788—22no—
92D788—24no—
93D788—22no—
94D788C2×D414yes—
95C22⋊D197—28no—
96C22⋊D197—24no—
97C22⋊D197—84no—
98C2×Dic197—22no—
99C2×Dic197C2×D412yes—
100C4×D197—28no—
101C4×D197—22no—
102C4×D197—22no—
103#13096802—22no—
104C23×C197—21no—
105C23×C197—22no—
106C23×C197—42no—
107C23×C197—82no—
108C23×C197C2×D411yescharacteristic
109Q8×C197—21no—
110Q8×C197C2×D412yes—
111D4×C197—24no—
112D4×C197—21no—
113D4×C197—22no—
114D4×C197—22no—
115D4×C197—42no—
116D4×C197C2×D412yes—
117C2×C4×C197—21no—
118C2×C4×C197—24no—
119C2×C4×C197C2×D411yescharacteristic
120C2×C4×C197C2×D412yes—
121C2×C4×C197C2×D412yes—
122C2×C4×C197C2411yescharacteristic
123C8×C197—22no—
124C22×C197—22no—
125C22×C197—22no—
126C22×C197—42no—
127C22×C197—42no—
128C22×C197—44no—
129C22×C197—84no—
130C22×C197#12012yes—
131C22×C197C22×D411yescharacteristic
132C2×D197—22no—
133C2×D197—24no—
134C2×D197—24no—
135C2×D197—44no—
136C2×D197—84no—
137C2×D197—88no—
138C4×C197—22no—
139C4×C197#12012yes—
140C4×C197C22×D411yescharacteristic
141C4×C197C22×D411yescharacteristic
142Dic197—22no—
143Dic197—42no—
144C2×C197—22no—
145C2×C197—42no—
146C2×C197—82no—
147C2×C197C2×#12011yescharacteristic
148C2×C197#57612yes—
149D197—42no—
150D197—44no—
151C197#300811yescharacteristic, complemented, Sylow
152#3008—1971nomaximal, Sylow
153C2×H(5,𝔽2)—1971no—
154C2×AGL(1,ℤ/8)D19711yescharacteristic, complemented
155C22×D8—1971no—
156#535—1971no—
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—3941no—
169C22×D4—3942no—
170C22×D4—1972no—
171C22×D4—1971no—
172C22×D4C2×D19711yescharacteristic, complemented
173AGL(1,ℤ/8)—24no—
174C2×SD16C2×D19712yescomplemented
175C2×D8—3944no—
176C2×D8—1974no—
177C2×D8C2×D19712yescomplemented
178C2×M16C2×D19711yescharacteristic, complemented
179C22×C8—1971no—
180#127—3941no—
181#124—3941no—
182#121—1974no—
183#120—3942no—
184#117—1971no—
185#103—1972no—
186#102—1972no—
187#102—1974no—
188C22⋊C8—1974no—
189C24—21no—
190C24—3942no—
191C24—3941no—
192#41—22no—
193#41—3948no—
194#41—3944no—
195#41—3944no—
196#41—3942no—
197#41—3942no—
198#41C22⋊D19714yescomplemented
199C2×Q8—1971no—
200C2×Q8C22×D19711yescharacteristic
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—3948no—
206C2×D4—3942no—
207C2×D4—39416no—
208C2×D4—3944no—
209C2×D4—3944no—
210C2×D4—3944no—
211C2×D4—3942no—
212C2×D4—3941no—
213C2×D4—3942no—
214C2×D4—1972no—
215C2×D4—1971no—
216C2×D4—1974no—
217C2×D4—1972no—
218C2×D4C22×D19712yes—
219C2×D4C22×D19711yescharacteristic
220C22×C4—21no—
221C22×C4—3942no—
222C22×C4—1972no—
223C22×C4C22×D19711yescharacteristic
224SD16—24no—
225D8—24no—
226D8—3948no—
227M16—22no—
228C2×C8—3942no—
229C2×C8—1972no—
230C2×C8C22×D19712yes—
231C4⋊C4—1972no—
232C22⋊C4—3941no—
233C22⋊C4—3944no—
234C42—3941no—
235C23—21no—
236C23—22no—
237C23—42no—
238C23—82no—
239C23—15762no—
240C23—15764no—
241C23—3948no—
242C23—3942no—
243C23—3944no—
244C23—1972no—
245C23D4×D19711yescharacteristic
246Q8—21no—
247Q8—3942no—
248Q8C2×C22⋊D19712yes—
249D4—24no—
250D4—21no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—15764no—
255D4—3948no—
256D4—3948no—
257D4—3944no—
258D4—3942no—
259D4—3944no—
260D4—3942no—
261D4—1974no—
262D4C2×C22⋊D19712yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—3948no—
266C2×C4—3942no—
267C2×C4—3942no—
268C2×C4—3942no—
269C2×C4—1972no—
270C2×C4D4×D19711yescharacteristic
271C2×C4D4×D19712yes—
272C2×C4C2×C22⋊D19712yes—
273C2×C4C23×D19711yescharacteristic
274C8—22no—
275C8—3942no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—44no—
280C22—42no—
281C22—84no—
282C22—7884no—
283C22—15764no—
284C22—15768no—
285C22—3944no—
286C22—3942no—
287C22—3944no—
288C22#2026329412yes—
289C22C2×D4×D19711yescharacteristic
290C4—22no—
291C4—7882no—
292C4—3942no—
293C4#2026329412yes—
294C4C2×D4×D19711yescharacteristic
295C4C22×C22⋊D19711yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—7884no—
300C2—7882no—
301C2C2×#2026329411yescharacteristic
302C2#2716188312yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×AGL(1,ℤ/8)×C197C2
socleC22×C197C22×D4
hypercenterC2×AGL(1,ℤ/8)D197
hypocenterC197#3008
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20263372C2semidirect
2. C22×#15691343C2semidirect
3. #27161646C2semidirect
4. #27161223C2semidirect
5. #27161211C2semidirect
6. #27161085C2semidirect
7. #27161075C2semidirect
8. #27161047C2semidirect
9. C2×AGL(1,ℤ/8)×C197C2semidirect
12. C2×#15691359C22semidirect
18. C2×D4×D197C22semidirect
20. C2×#15691343C22semidirect
21. C22×#13096802C22semidirect
24. #20263297C22semidirect
26. #20263285C22semidirect
27. #20263250C22semidirect
28. #20263248C22semidirect
29. #20263247C22semidirect
30. #20263238C22semidirect
32. C22×D4×C197C22semidirect
34. C2×SD16×C197C22semidirect
35. C2×D8×C197C22semidirect
36. C2×M16×C197C22semidirect
50. C2×D788D4semidirect
69. #41×C197D4semidirect
151. C197#3008semidirect
154. C2×AGL(1,ℤ/8)D197semidirect
172. C22×D4C2×D197semidirect
174. C2×SD16C2×D197semidirect
177. C2×D8C2×D197semidirect
178. C2×M16C2×D197semidirect
198. #41C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633722
C22×#156913432
#271616462
#271612232
#271612112
#271610852
#271610752
#271610472
C2×AGL(1,ℤ/8)×C1972
#3008197