Finite Group Databasebeta

← #38754391

Subgroups — #38754391

Order 25216.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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×D8×D197C211yesmaximal, characteristic, complemented
2#27161717C212yesmaximal, complemented
3#27161464C211yesmaximal, characteristic, complemented
4#27161317C211yesmaximal, characteristic, complemented
5#27161315C211yesmaximal, characteristic, complemented
6#27161296C212yesmaximal, complemented
7#27161228C212yesmaximal, complemented
8#27161224C212yesmaximal, complemented
9D8×Dic197C211yesmaximal, characteristic, complemented
10C8×C22⋊D197C211yesmaximal, characteristic, complemented
11C2×#135×C197C211yesmaximal, characteristic, complemented
12C2×D4×D197C2212yes—
13C2×#15691345C2212yescomplemented
14C2×D1576C2211yescharacteristic
15C2×#15691343C2212yes—
16D8×D197—24no—
17C2×C8×D197C2211yescharacteristic
18#20263315C2212yescomplemented
19#20263309—22no—
20#20263302C2212yes—
21#20263300C2212yescomplemented
22#20263298—22no—
23#20263294—22no—
24D4×Dic197C2212yescomplemented
25C4×C22⋊D197C2211yescharacteristic
26C2×#15691335—22no—
27#20263260C2211yescharacteristic, complemented
28#20263255C2211yescharacteristic, complemented
29#20263250C2212yescomplemented
30#20263248C2212yescomplemented
31#20263247C2212yescomplemented
32C22⋊#13096802C2211yescharacteristic, complemented
33#20263238C2211yescharacteristic
34C8×Dic197C2211yescharacteristic, complemented
35C2×#41×C197C2212yescomplemented
36#135×C197—24no—
37C2×Q16×C197C2211yescharacteristic, complemented
38C2×SD16×C197C2212yescomplemented
39C2×D8×C197C2211yescharacteristic
40C22×C8×C197C2211yescharacteristic, complemented
41C23×D197—22no—
42C2×D788C2312yes—
43C2×C22⋊D197—22no—
44C2×C22⋊D197—22no—
45C2×C22⋊D197D411yescharacteristic
46D4×D197—24no—
47D4×D197—42no—
48C22×Dic197—22no—
49C2×C4×D197C2311yescharacteristic
50#15691345—42no—
51D1576—22no—
52#15691343—24no—
53C2×#13096802C2311yescharacteristic
54C2×#13096802C2311yescharacteristic
55C8×D197—22no—
56C4⋊Dic197C2312yes—
57#15691337—22no—
58#15691337D411yescharacteristic
59C22⋊Dic197—22no—
60C22⋊Dic197D411yescharacteristic
61#15691335—22no—
62#15691335—22no—
63#15691335—22no—
64#15691335—22no—
65#15691335D411yescharacteristic
66C4×Dic197C2311yescharacteristic
67#41×C197—24no—
68#41×C197—42no—
69C2×Q8×C197C2312yes—
70C2×D4×C197—22no—
71C2×D4×C197C2312yes—
72C22×C4×C197—22no—
73C22×C4×C197C2311yescharacteristic
74Q16×C197—22no—
75SD16×C197—24no—
76D8×C197D414yescomplemented
77C2×C8×C197—22no—
78C2×C8×C197C2311yescharacteristic
79C2×C8×C197C2311yescharacteristic
80C22×D197—22no—
81C22×D197—44no—
82C22×D197—44no—
83C22×D197—82no—
84C22×D197C2×D411yescharacteristic
85D788—22no—
86D788—22no—
87C22⋊D197—22no—
88C22⋊D197—44no—
89C22⋊D197—82no—
90C2×Dic197—22no—
91C2×Dic197—82no—
92C2×Dic197C2×D411yescharacteristic
93C2×Dic197C2×D411yescharacteristic
94C2×Dic197C2×D411yescharacteristic
95C4×D197—21no—
96C4×D197—21no—
97#13096802—21no—
98#13096802—21no—
99C23×C197—22no—
100C23×C197C2×D411yescharacteristic
101Q8×C197—22no—
102Q8×C197—22no—
103D4×C197—22no—
104D4×C197—44no—
105D4×C197C2×D414yes—
106C2×C4×C197—21no—
107C2×C4×C197—21no—
108C2×C4×C197—22no—
109C2×C4×C197—44no—
110C2×C4×C197C2×D411yescharacteristic
111C2×C4×C197C2411yescharacteristic
112C8×C197—21no—
113C8×C197C2×D412yes—
114C22×C197—21no—
115C22×C197—24no—
116C22×C197—21no—
117C22×C197—21no—
118C22×C197—42no—
119C22×C197—42no—
120C22×C197C22×D411yescharacteristic
121C2×D197—21no—
122C2×D197—21no—
123C2×D197—42no—
124C2×D197—42no—
125C2×D197—42no—
126C2×D197—88no—
127C2×D197#13512yescomplemented
128C4×C197—21no—
129C4×C197—42no—
130C4×C197C22×D411yescharacteristic
131C4×C197H(5,𝔽2)11yescharacteristic
132Dic197—21no—
133Dic197—21no—
134Dic197—21no—
135Dic197—42no—
136C2×C197—44no—
137C2×C197—41no—
138C2×C197D4211yescharacteristic
139C2×C197C2×#13511yescharacteristic
140C2×C197#57611yescharacteristic
141D197—22no—
142D197—82no—
143C197#328011yescharacteristic, complemented, Sylow
144#3280—1971nomaximal, Sylow
145C2×#135D19711yescharacteristic, complemented
146C22×D8—1971no—
147#546—1972no—
148#495—1971no—
149#467—1971no—
150#466—1971no—
151#463—1972no—
152#450—1972no—
153#449—1972no—
154C4×D8—1971no—
155C8×D4—1971no—
156C2×#41C2×D19712yescomplemented
157C22×D4—1972no—
158#135—24no—
159C2×Q16C2×D19711yescharacteristic, complemented
160C2×SD16—1972no—
161C2×SD16C2×D19712yescomplemented
162C2×D8—3944no—
163C2×D8—1972no—
164C2×D8—1971no—
165C2×D8C2×D19711yescharacteristic
166C22×C8—1971no—
167C22×C8C2×D19711yescharacteristic, complemented
168#124—1972no—
169#123—3942no—
170C22⋊Q8—1972no—
171#121—3942no—
172#121—1972no—
173#120—3942no—
174C4×D4—1972no—
175C4×D4—1971no—
176C2×C22⋊C4—3942no—
177#107—1971no—
178C4⋊C8—1971no—
179#103—1972no—
180#102—1972no—
181#102—1972no—
182C22⋊C8—1971no—
183C22⋊C8—1971no—
184C4×C8—1971no—
185C24—3942no—
186#41—24no—
187#41—42no—
188C2×Q8C22×D19712yes—
189C2×D4—22no—
190C2×D4—7882no—
191C2×D4—3944no—
192C2×D4—3942no—
193C2×D4—3942no—
194C2×D4—1971no—
195C2×D4—1972no—
196C2×D4C22×D19712yes—
197C22×C4—22no—
198C22×C4—3942no—
199C22×C4—1971no—
200C22×C4C22×D19711yescharacteristic
201Q16—22no—
202SD16—24no—
203SD16—7882no—
204D8—3944no—
205D8—3942no—
206D8C22⋊D19714yescomplemented
207C2×C8—22no—
208C2×C8—3942no—
209C2×C8—1971no—
210C2×C8—1971no—
211C2×C8C22×D19711yescharacteristic
212C2×C8C22×D19711yescharacteristic
213C4⋊C4—3942no—
214C4⋊C4—1972no—
215C4⋊C4—1971no—
216C22⋊C4—3942no—
217C22⋊C4—3942no—
218C22⋊C4—3942no—
219C22⋊C4—3942no—
220C22⋊C4—3942no—
221C22⋊C4—1971no—
222C22⋊C4—1971no—
223C42—1971no—
224C23—22no—
225C23—7884no—
226C23—7884no—
227C23—15762no—
228C23—3942no—
229C23—1971no—
230C23D4×D19711yescharacteristic
231Q8—22no—
232Q8—22no—
233D4—22no—
234D4—44no—
235D4—7884no—
236D4—15762no—
237D4—3942no—
238D4—3942no—
239D4—3942no—
240D4C2×C22⋊D19714yes—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—44no—
245C2×C4—15762no—
246C2×C4—3942no—
247C2×C4—3941no—
248C2×C4—3941no—
249C2×C4—1971no—
250C2×C4—1971no—
251C2×C4—1971no—
252C2×C4D4×D19711yescharacteristic
253C2×C4C23×D19711yescharacteristic
254C8—21no—
255C8—3941no—
256C8—3941no—
257C8C2×C22⋊D19712yes—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—24no—
262C22—42no—
263C22—42no—
264C22—7882no—
265C22—7882no—
266C22—7882no—
267C22—15768no—
268C22—3941no—
269C22—3941no—
270C22—1972no—
271C22C2×D4×D19711yescharacteristic
272C4—21no—
273C4—42no—
274C4—7882no—
275C4—3941no—
276C4—3941no—
277C4—3941no—
278C4C22×C22⋊D19711yescharacteristic
279C4#2026337111yescharacteristic
280C2—41no—
281C2—44no—
282C2—15762no—
283C2—3942no—
284C2D4×C22⋊D19711yescharacteristic
285C2#135×D19711yescharacteristic
286C2#2716188111yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#135×C197C2
socleC22×C197C22×D4
hypercenterC2×#135D197
hypocenterC197#3280
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×D8×D197C2semidirect
2. #27161717C2semidirect
3. #27161464C2semidirect
4. #27161317C2semidirect
5. #27161315C2semidirect
6. #27161296C2semidirect
7. #27161228C2semidirect
8. #27161224C2semidirect
9. D8×Dic197C2semidirect
10. C8×C22⋊D197C2semidirect
11. C2×#135×C197C2semidirect
13. C2×#15691345C22semidirect
18. #20263315C22semidirect
21. #20263300C22semidirect
24. D4×Dic197C22semidirect
27. #20263260C22semidirect
28. #20263255C22semidirect
29. #20263250C22semidirect
30. #20263248C22semidirect
31. #20263247C22semidirect
32. C22⋊#13096802C22semidirect
34. C8×Dic197C22semidirect
35. C2×#41×C197C22semidirect
37. C2×Q16×C197C22semidirect
38. C2×SD16×C197C22semidirect
40. C22×C8×C197C22semidirect
76. D8×C197D4semidirect
127. C2×D197#135semidirect
143. C197#3280semidirect
145. C2×#135D197semidirect
156. C2×#41C2×D197semidirect
159. C2×Q16C2×D197semidirect
161. C2×SD16C2×D197semidirect
167. C22×C8C2×D197semidirect
206. D8C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1972
#271617172
#271614642
#271613172
#271613152
#271612962
#271612282
#271612242
D8×Dic1972
C8×C22⋊D1972
C2×#135×C1972
#3280197