Finite Group Databasebeta

← #38754403

Subgroups — #38754403

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
2Q8×D788C211yesmaximal, characteristic, complemented
3#27161714C211yesmaximal, characteristic
4#27161451C211yesmaximal, characteristic, complemented
5#27161307C212yesmaximal, complemented
6#27161285C211yesmaximal, characteristic, complemented
7#27161231C212yesmaximal, complemented
8#27161216C212yesmaximal, complemented
9C4×#15691345C211yesmaximal, characteristic, complemented
10#27161144C211yesmaximal, characteristic, complemented
11#460×C197C211yesmaximal, characteristic, complemented
12C2×Q8×D197C2212yescomplemented
13C2×D4×D197—21no—
14C2×D4×D197C2212yes—
15C2×#15691347C2212yes—
16C2×#15691346C2212yescomplemented
17C2×#15691346C2212yescomplemented
18C2×#15691345C2211yescharacteristic
19SD16×D197—28no—
20C2×C8×D197C2212yescomplemented
21#20263326—21no—
22#20263326C2211yescharacteristic, complemented
23#20263306—22no—
24#20263306C2212yescomplemented
25#20263297—21no—
26#20263297C2212yes—
27#20263294—22no—
28C4×D788—21no—
29C4×D788C2211yescharacteristic
30C4×D788C2211yescharacteristic
31#20263257C2211yescharacteristic, complemented
32#20263251C2212yescomplemented
33#20263250C2211yescharacteristic, complemented
34#20263246C2211yescharacteristic
35#20263238C2212yescomplemented
36C4×#13096802C2211yescharacteristic, complemented
37C2×SD16×C197C2212yescomplemented
38#127×C197C2211yescharacteristic
39C4×Q8×C197C2211yescharacteristic, complemented
40C4⋊C8×C197C2211yescharacteristic, complemented
41#102×C197C2212yescomplemented
42C23×D197—22no—
43C2×Dic394—22no—
44C2×Dic394C2312yes—
45Q8×D197—24no—
46Q8×D197—42no—
47C2×D788D411yescharacteristic
48C2×D788C2311yescharacteristic
49C2×C22⋊D197—22no—
50C2×C22⋊D197—22no—
51D4×D197—24no—
52D4×D197—42no—
53D4×D197—44no—
54C2×C4×D197—22no—
55C2×C4×D197—21no—
56C2×C4×D197C2312yes—
57#15691347—24no—
58#15691346—24no—
59#15691346—42no—
60#15691345D414yescomplemented
61C2×#13096802C2311yescharacteristic
62C2×#13096802C2311yescharacteristic
63C8×D197—24no—
64C4⋊Dic197—22no—
65C4⋊Dic197—21no—
66C4⋊Dic197—21no—
67C4⋊Dic197D411yescharacteristic
68C4⋊Dic197C2311yescharacteristic
69C22⋊Dic197—22no—
70#15691335—22no—
71#15691335—21no—
72#15691335D412yes—
73C2×Q8×C197C2311yescharacteristic
74C2×D4×C197—21no—
75C2×D4×C197—21no—
76C2×D4×C197C2312yes—
77SD16×C197—24no—
78C2×C8×C197C2312yes—
79C4⋊C4×C197—21no—
80C4⋊C4×C197C2311yescharacteristic
81C42×C197—21no—
82C42×C197C2311yescharacteristic
83C22×D197—21no—
84C22×D197—44no—
85C22×D197—44no—
86C22×D197—82no—
87C22×D197C2×D412yes—
88Dic394—22no—
89Dic394—22no—
90Dic394—44no—
91D788—21no—
92D788SD1614yescomplemented
93D788C2×D412yes—
94C22⋊D197—44no—
95C22⋊D197—44no—
96C2×Dic197—21no—
97C2×Dic197—22no—
98C2×Dic197C2×D412yes—
99C4×D197—22no—
100C4×D197—22no—
101C4×D197—44no—
102C4×D197—42no—
103#13096802—21no—
104#13096802C2×D412yescomplemented
105C23×C197—22no—
106Q8×C197—21no—
107Q8×C197C2×D412yescomplemented
108D4×C197—22no—
109D4×C197—22no—
110D4×C197—42no—
111D4×C197—81no—
112C2×C4×C197—21no—
113C2×C4×C197—21no—
114C2×C4×C197C2×D411yescharacteristic
115C2×C4×C197C2×D411yescharacteristic
116C2×C4×C197C2411yescharacteristic
117C8×C197—22no—
118C22×C197—42no—
119C22×C197—42no—
120C22×C197—42no—
121C22×C197C22×D411yescharacteristic
122C2×D197—22no—
123C2×D197—22no—
124C2×D197—22no—
125C2×D197—41no—
126C2×D197—41no—
127C2×D197—88no—
128C2×D197C2×SD1614yescomplemented
129C4×C197—22no—
130C4×C197—21no—
131C4×C197—41no—
132C4×C197C2×SD1612yescomplemented
133C4×C197C22×D411yescharacteristic
134C4×C197H(5,𝔽2)11yescharacteristic
135Dic197—22no—
136Dic197—42no—
137Dic197—41no—
138C2×C197—82no—
139C2×C197D4211yescharacteristic
140C2×C197C22×SD1611yescharacteristic
141C2×C197#57711yescharacteristic
142D197—24no—
143D197—42no—
144C197D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1971nomaximal, Sylow
146C22×SD16—1972no—
147D4×Q8—1971no—
148D42—1971no—
149#492—1971no—
150#465—1972no—
151#461—1971no—
152#460D19711yescharacteristic, complemented
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×D19712yescomplemented
166C22×C8—1972no—
167C4⋊Q8—3941no—
168C4⋊Q8—1971no—
169#127C2×D19711yescharacteristic
170C22⋊Q8—3942no—
171C22⋊Q8—1972no—
172#121—3941no—
173#121—1972no—
174#120—3942no—
175C4×Q8C2×D19711yescharacteristic, complemented
176C4×D4—3941no—
177C4×D4—1971no—
178C4×D4—1971no—
179#106—1971no—
180C4⋊C8C2×D19711yescharacteristic, complemented
181#103—1971no—
182#103—1972no—
183#102—1971no—
184#102C2×D19712yescomplemented
185C22⋊C8—1972no—
186C4×C8—1971no—
187C24—3942no—
188C2×Q8—7882no—
189C2×Q8—3942no—
190C2×Q8—3944no—
191C2×Q8—1972no—
192C2×Q8C22×D19711yescharacteristic
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—7884no—
196C2×D4—7882no—
197C2×D4—3944no—
198C2×D4—3942no—
199C2×D4—3942no—
200C2×D4—1971no—
201C2×D4—1971no—
202C2×D4C22×D19712yes—
203C22×C4—3942no—
204C22×C4—3941no—
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—3942no—
217C4⋊C4—3941no—
218C4⋊C4—3941no—
219C4⋊C4—1971no—
220C4⋊C4—1971no—
221C4⋊C4C22×D19711yescharacteristic
222C22⋊C4—3942no—
223C22⋊C4—3942no—
224C22⋊C4—3941no—
225C22⋊C4—1972no—
226C42—21no—
227C42C22×D19711yescharacteristic
228C23—22no—
229C23—7884no—
230C23—7884no—
231C23—15762no—
232C23—3941no—
233C23—1972no—
234Q8—21no—
235Q8—7884no—
236Q8—3942no—
237Q8—3942no—
238Q8D4×D19712yescomplemented
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—7884no—
244D4—7884no—
245D4—3941no—
246D4—1974no—
247D4—1972no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—7884no—
251C2×C4—7882no—
252C2×C4—3941no—
253C2×C4—3942no—
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—42no—
264C22—42no—
265C22—42no—
266C22—7881no—
267C22—7881no—
268C22—15768no—
269C22—3942no—
270C22—3942no—
271C22—3942no—
272C22—1974no—
273C22C2×D4×D19711yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—7882no—
278C4—7881no—
279C4—3942no—
280C4SD16×D19712yescomplemented
281C4C2×D4×D19711yescharacteristic
282C4#2026337211yescharacteristic
283C2—82no—
284C2—7882no—
285C2—3944no—
286C2#2716171311yescharacteristic
287C2C2×SD16×D19711yescharacteristic
288C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#460×C197C2
socleC22×C197C22×D4
prime radical#27161714C2
hypercenter#460D197
hypocenterC197D4×SD16
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D197C2semidirect
2. Q8×D788C2semidirect
4. #27161451C2semidirect
5. #27161307C2semidirect
6. #27161285C2semidirect
7. #27161231C2semidirect
8. #27161216C2semidirect
9. C4×#15691345C2semidirect
10. #27161144C2semidirect
11. #460×C197C2semidirect
12. C2×Q8×D197C22semidirect
16. C2×#15691346C22semidirect
17. C2×#15691346C22semidirect
20. C2×C8×D197C22semidirect
22. #20263326C22semidirect
24. #20263306C22semidirect
31. #20263257C22semidirect
32. #20263251C22semidirect
33. #20263250C22semidirect
35. #20263238C22semidirect
36. C4×#13096802C22semidirect
37. C2×SD16×C197C22semidirect
39. C4×Q8×C197C22semidirect
40. C4⋊C8×C197C22semidirect
41. #102×C197C22semidirect
60. #15691345D4semidirect
92. D788SD16semidirect
104. #13096802C2×D4semidirect
107. Q8×C197C2×D4semidirect
128. C2×D197C2×SD16semidirect
132. C4×C197C2×SD16semidirect
144. C197D4×SD16semidirect
152. #460D197semidirect
165. C2×SD16C2×D197semidirect
175. C4×Q8C2×D197semidirect
180. C4⋊C8C2×D197semidirect
184. #102C2×D197semidirect
238. Q8D4×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
Q8×D7882
#271617142
#271614512
#271613072
#271612852
#271612312
#271612162
C4×#156913452
#271611442
#460×C1972
D4×SD16197