Finite Group Databasebeta

← #38754411

Subgroups — #38754411

Order 25216.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×SD16×D197C212yesmaximal, complemented
2#27161744C211yesmaximal, characteristic, complemented
3D4×D788C211yesmaximal, characteristic
4#27161453C211yesmaximal, characteristic, complemented
5#27161311C212yesmaximal, complemented
6#27161280C211yesmaximal, characteristic, complemented
7#27161228C212yesmaximal, complemented
8#27161219C212yesmaximal, complemented
9C4×#15691346C211yesmaximal, characteristic, complemented
10#27161144C211yesmaximal, characteristic, complemented
11#461×C197C211yesmaximal, characteristic, complemented
12C2×Q8×D197C2212yescomplemented
13C22×D788—21no—
14C2×D4×D197C2212yes—
15C2×#15691347C2212yescomplemented
16C2×#15691346C2211yescharacteristic
17C2×#15691345C2212yescomplemented
18C2×#15691345C2212yescomplemented
19SD16×D197—28no—
20C2×C8×D197C2212yescomplemented
21#20263325—21no—
22#20263323C2211yescharacteristic
23#20263304—22no—
24#20263302C2212yescomplemented
25#20263301C2212yes—
26#20263300—21no—
27#20263296—22no—
28C4×Dic394C2211yescharacteristic, complemented
29C4×D788C2211yescharacteristic
30#20263288—21no—
31#20263257C2211yescharacteristic, complemented
32#20263249C2211yescharacteristic, complemented
33#20263248C2212yescomplemented
34#20263247C2211yescharacteristic
35#20263238C2212yescomplemented
36C4×#13096802C2211yescharacteristic, complemented
37C2×SD16×C197C2212yes—
38C4⋊Q8×C197C2211yescharacteristic, complemented
39C4×D4×C197C2211yescharacteristic
40C4⋊C8×C197C2211yescharacteristic, complemented
41#103×C197C2212yescomplemented
42C23×D197—22no—
43C2×Dic394—22no—
44C2×Dic394C2311yescharacteristic
45Q8×D197—24no—
46Q8×D197—42no—
47C2×D788—22no—
48C2×D788—21no—
49C2×D788—21no—
50C2×D788—44no—
51C2×D788D411yescharacteristic
52C2×D788C2312yes—
53C2×C22⋊D197—22no—
54D4×D197—24no—
55D4×D197—42no—
56C2×C4×D197—22no—
57C2×C4×D197C2312yes—
58#15691347—24no—
59#15691346D414yescomplemented
60#15691345—24no—
61#15691345—42no—
62C2×#13096802C2311yescharacteristic
63C2×#13096802C2311yescharacteristic
64C8×D197—24no—
65C4⋊Dic197D411yescharacteristic
66C4⋊Dic197C2311yescharacteristic
67#15691337—22no—
68#15691337—21no—
69#15691335—22no—
70#15691335—22no—
71#15691335D412yes—
72C4×Dic197—21no—
73C2×Q8×C197C2312yes—
74C2×D4×C197C2311yescharacteristic
75C22×C4×C197—21no—
76SD16×C197—24no—
77C2×C8×C197C2312yes—
78C4⋊C4×C197—21no—
79C4⋊C4×C197—21no—
80C4⋊C4×C197C2311yescharacteristic
81C22⋊C4×C197—21no—
82C42×C197C2311yescharacteristic
83C22×D197—22no—
84C22×D197—44no—
85C22×D197—44no—
86C22×D197—82no—
87C22×D197C2×D412yes—
88Dic394—21no—
89Dic394—44no—
90Dic394C2×D412yescomplemented
91D788—22no—
92D788—22no—
93D788—44no—
94D788—42no—
95D788—81no—
96D788SD1614yescomplemented
97C22⋊D197—44no—
98C2×Dic197—21no—
99C2×Dic197—21no—
100C2×Dic197C2×D412yes—
101C4×D197—22no—
102C4×D197—22no—
103C4×D197—44no—
104#13096802—21no—
105#13096802C2×D412yescomplemented
106C23×C197—21no—
107Q8×C197—22no—
108Q8×C197—22no—
109D4×C197—21no—
110D4×C197C2×D412yes—
111C2×C4×C197—21no—
112C2×C4×C197—22no—
113C2×C4×C197—42no—
114C2×C4×C197C2×D411yescharacteristic
115C2×C4×C197C2×D411yescharacteristic
116C2×C4×C197C2411yescharacteristic
117C8×C197—22no—
118C22×C197—22no—
119C22×C197—41no—
120C22×C197—41no—
121C22×C197C22×D411yescharacteristic
122C2×D197—22no—
123C2×D197—22no—
124C2×D197—42no—
125C2×D197—42no—
126C2×D197—42no—
127C2×D197—88no—
128C2×D197C2×SD1614yescomplemented
129C4×C197—21no—
130C4×C197—42no—
131C4×C197—41no—
132C4×C197C2×SD1612yescomplemented
133C4×C197C22×D411yescharacteristic
134C4×C197H(5,𝔽2)11yescharacteristic
135Dic197—22no—
136Dic197—22no—
137Dic197—41no—
138C2×C197—42no—
139C2×C197D4211yescharacteristic
140C2×C197C22×SD1611yescharacteristic
141C2×C197#57711yescharacteristic
142D197—24no—
143D197—82no—
144C197D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1971nomaximal, Sylow
146C22×SD16—1972no—
147D4×Q8—1971no—
148D42—1971no—
149#492—1971no—
150#465—1972no—
151#461D19711yescharacteristic, complemented
152#460—1971no—
153#450—1972no—
154#448—1972no—
155C4×SD16—1971no—
156C8×D4—1971no—
157C22×Q8—1972no—
158C22×D4—3941no—
159C22×D4—1972no—
160C2×SD16—3948no—
161C2×SD16—1972no—
162C2×SD16—1972no—
163C2×SD16—1971no—
164C2×SD16—1972no—
165C2×SD16C2×D19712yes—
166C22×C8—1972no—
167C4⋊Q8—3941no—
168C4⋊Q8C2×D19711yescharacteristic, complemented
169#127—1971no—
170C22⋊Q8—3942no—
171C22⋊Q8—1972no—
172#121—3941no—
173#121—1972no—
174#120—3942no—
175C4×Q8—1971no—
176C4×D4—3941no—
177C4×D4—1971no—
178C4×D4C2×D19711yescharacteristic
179#106—1971no—
180C4⋊C8C2×D19711yescharacteristic, complemented
181#103—1971no—
182#103C2×D19712yescomplemented
183#102—1971no—
184#102—1972no—
185C22⋊C8—1972no—
186C4×C8—1971no—
187C24—3942no—
188C2×Q8—7882no—
189C2×Q8—3942no—
190C2×Q8—3944no—
191C2×Q8—1971no—
192C2×Q8C22×D19712yes—
193C2×D4—7884no—
194C2×D4—7882no—
195C2×D4—3942no—
196C2×D4—3944no—
197C2×D4—3942no—
198C2×D4—3941no—
199C2×D4—3941no—
200C2×D4—1971no—
201C2×D4—1972no—
202C2×D4C22×D19711yescharacteristic
203C22×C4—21no—
204C22×C4—3942no—
205C22×C4—1972no—
206SD16—24no—
207SD16—7882no—
208SD16—3944no—
209SD16—3944no—
210SD16—1974no—
211C2×C8—3944no—
212C2×C8—1971no—
213C2×C8—1971no—
214C2×C8C22×D19712yes—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—3942no—
218C4⋊C4—3941no—
219C4⋊C4—1971no—
220C4⋊C4—1971no—
221C4⋊C4C22×D19711yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—3942no—
224C22⋊C4—3942no—
225C22⋊C4—1972no—
226C42—3941no—
227C42C22×D19711yescharacteristic
228C23—21no—
229C23—7884no—
230C23—7884no—
231C23—15762no—
232C23—3942no—
233C23—1972no—
234Q8—22no—
235Q8—22no—
236Q8—7884no—
237Q8—3941no—
238Q8—1972no—
239D4—21no—
240D4—7884no—
241D4—7884no—
242D4—7882no—
243D4—15761no—
244D4—3942no—
245D4—3942no—
246D4—1974no—
247D4D4×D19712yes—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—42no—
251C2×C4—7884no—
252C2×C4—3941no—
253C2×C4—3941no—
254C2×C4—3942no—
255C2×C4—3942no—
256C2×C4—1972no—
257C2×C4D4×D19711yescharacteristic
258C2×C4D4×D19711yescharacteristic
259C2×C4C23×D19711yescharacteristic
260C8—22no—
261C8—3941no—
262C8—1972no—
263C22—22no—
264C22—41no—
265C22—41no—
266C22—7882no—
267C22—7882no—
268C22—7882no—
269C22—15768no—
270C22—3942no—
271C22—3942no—
272C22—1974no—
273C22C2×D4×D19711yescharacteristic
274C4—21no—
275C4—42no—
276C4—41no—
277C4—7881no—
278C4—3942no—
279C4—3942no—
280C4SD16×D19712yescomplemented
281C4C2×D4×D19711yescharacteristic
282C4#2026337211yescharacteristic
283C2—42no—
284C2—15762no—
285C2—3944no—
286C2#2716171311yescharacteristic
287C2C2×SD16×D19711yescharacteristic
288C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#461×C197C2
socleC22×C197C22×D4
prime radicalD4×D788C2
hypercenter#461D197
hypocenterC197D4×SD16
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D197C2semidirect
2. #27161744C2semidirect
4. #27161453C2semidirect
5. #27161311C2semidirect
6. #27161280C2semidirect
7. #27161228C2semidirect
8. #27161219C2semidirect
9. C4×#15691346C2semidirect
10. #27161144C2semidirect
11. #461×C197C2semidirect
12. C2×Q8×D197C22semidirect
15. C2×#15691347C22semidirect
17. C2×#15691345C22semidirect
18. C2×#15691345C22semidirect
20. C2×C8×D197C22semidirect
24. #20263302C22semidirect
28. C4×Dic394C22semidirect
31. #20263257C22semidirect
32. #20263249C22semidirect
33. #20263248C22semidirect
35. #20263238C22semidirect
36. C4×#13096802C22semidirect
38. C4⋊Q8×C197C22semidirect
40. C4⋊C8×C197C22semidirect
41. #103×C197C22semidirect
59. #15691346D4semidirect
90. Dic394C2×D4semidirect
96. D788SD16semidirect
105. #13096802C2×D4semidirect
128. C2×D197C2×SD16semidirect
132. C4×C197C2×SD16semidirect
144. C197D4×SD16semidirect
151. #461D197semidirect
168. C4⋊Q8C2×D197semidirect
180. C4⋊C8C2×D197semidirect
182. #103C2×D197semidirect
280. C4SD16×D197semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1972
#271617442
D4×D7882
#271614532
#271613112
#271612802
#271612282
#271612192
C4×#156913462
#271611442
#461×C1972
D4×SD16197