Finite Group Databasebeta

← #38751989

Subgroups — #38751989

Order 25216.

752 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 165 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×#20263356C211yesmaximal, characteristic, complemented
2C22×#15691346C211yesmaximal, characteristic, complemented
3#27161650C211yesmaximal, characteristic, complemented
4#27161241C214yesmaximal, complemented
5#27161230C214yesmaximal, complemented
6#27161084C211yesmaximal, characteristic, complemented
7#27161075C211yesmaximal, characteristic, complemented
8#27161046C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C197C211yesmaximal, characteristic
10C22×Dic394C2211yescharacteristic, complemented
11#20263356—24no—
12C2×#15691348C2212yescomplemented
13C2×#15691346—24no—
14C2×#15691346C2212yescomplemented
15C2×#15691346C2214yescomplemented
16C2×#15691342C2211yescharacteristic, complemented
17C22×#13096802C2211yescharacteristic, complemented
18#20263324—21no—
19#20263313—21no—
20#20263307—22no—
21#20263305C2214yescomplemented
22#20263285C2211yescharacteristic, complemented
23#20263250C2212yescomplemented
24#20263247C2212yescomplemented
25#20263247C2214yescomplemented
26C22⋊#13096802C2214yescomplemented
27H(5,𝔽2)×C197—24no—
28H(5,𝔽2)×C197—24no—
29C2×#41×C197—21no—
30C2×#41×C197C2212yes—
31C2×#41×C197C2211yescharacteristic
32C2×#41×C197C2211yescharacteristic
33C22×D4×C197—22no—
34C22×D4×C197—21no—
35C22×D4×C197C2212yes—
36C22×D4×C197C2211yescharacteristic
37C2×Dic394—21no—
38C2×Dic394—22no—
39C2×Dic394—22no—
40C2×Dic394C2312yes—
41C22×Dic197—21no—
42#15691348—24no—
43#15691346—28no—
44#15691346—24no—
45#15691342—22no—
46C2×#13096802—22no—
47C2×#13096802C2312yes—
48C2×#13096802C2312yes—
49C4⋊Dic197C2312yes—
50#15691337—24no—
51C22⋊Dic197—21no—
52C22⋊Dic197—24no—
53C4×Dic197—21no—
54C24×C197—22no—
55C24×C197—21no—
56#41×C197—28no—
57#41×C197—24no—
58#41×C197—24no—
59#41×C197—22no—
60#41×C197—22no—
61#41×C197—22no—
62#41×C197D414yes—
63C2×Q8×C197D411yescharacteristic
64C2×Q8×C197C2311yescharacteristic
65C2×D4×C197—216no—
66C2×D4×C197—28no—
67C2×D4×C197—22no—
68C2×D4×C197—21no—
69C2×D4×C197—24no—
70C2×D4×C197—24no—
71C2×D4×C197—22no—
72C2×D4×C197—21no—
73C2×D4×C197—22no—
74C2×D4×C197D412yes—
75C2×D4×C197D411yescharacteristic
76C2×D4×C197D414yes—
77C2×D4×C197C2311yescharacteristic
78C2×D4×C197C2312yes—
79C22×C4×C197—22no—
80C22×C4×C197—21no—
81C22×C4×C197C2312yes—
82C22×C4×C197C2311yescharacteristic
83Dic394—22no—
84Dic394—22no—
85Dic394—42no—
86C2×Dic197—22no—
87C2×Dic197—22no—
88C2×Dic197—42no—
89#13096802—22no—
90#13096802—22no—
91C23×C197—28no—
92C23×C197—22no—
93C23×C197—21no—
94C23×C197—24no—
95C23×C197—82no—
96C23×C197—84no—
97C23×C197C2×D412yes—
98C23×C197C2×D411yescharacteristic
99Q8×C197—22no—
100Q8×C197—21no—
101Q8×C197C2×D412yes—
102D4×C197—28no—
103D4×C197—28no—
104D4×C197—24no—
105D4×C197—24no—
106D4×C197—22no—
107D4×C197—24no—
108D4×C197—22no—
109D4×C197—21no—
110D4×C197C2×D412yes—
111D4×C197C2×D414yes—
112C2×C4×C197—28no—
113C2×C4×C197—21no—
114C2×C4×C197—24no—
115C2×C4×C197—22no—
116C2×C4×C197—22no—
117C2×C4×C197C2×D412yes—
118C2×C4×C197C2×D411yescharacteristic
119C2×C4×C197C2×D412yes—
120C2×C4×C197C2×D412yes—
121C2×C4×C197C2411yescharacteristic
122C22×C197—22no—
123C22×C197—22no—
124C22×C197—24no—
125C22×C197—22no—
126C22×C197—24no—
127C22×C197—44no—
128C22×C197—42no—
129C22×C197—84no—
130C22×C197—88no—
131C22×C197#12012yes—
132C22×C197C22×D411yescharacteristic
133C4×C197—22no—
134C4×C197—22no—
135C4×C197#12012yes—
136C4×C197C22×D411yescharacteristic
137C4×C197C22×D411yescharacteristic
138Dic197—42no—
139Dic197—42no—
140C2×C197—22no—
141C2×C197—42no—
142C2×C197—44no—
143C2×C197—42no—
144C2×C197C2×#12011yescharacteristic
145C2×C197#57712yes—
146C197#300911yescharacteristic, complemented, Sylow
147#3009—1971nomaximal, Sylow
148C2×H(5,𝔽2)D19711yescharacteristic
149C2×#137—1971no—
150C22×SD16—1971no—
151#536—1971no—
152#452—1974no—
153#450—1974no—
154#418—1971no—
155#416—1971no—
156#408—1971no—
157H(5,𝔽2)—24no—
158H(5,𝔽2)—24no—
159C2×#41—21no—
160C2×#41C2×D19712yes—
161C2×#41C2×D19711yescharacteristic
162C2×#41C2×D19711yescharacteristic
163C22×Q8—1971no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4C2×D19712yes—
167C22×D4C2×D19711yescharacteristic
168#137—3944no—
169C2×Q16—1972no—
170C2×SD16—3944no—
171C2×SD16—1974no—
172C2×SD16—1972no—
173C2×M16—1971no—
174C22×C8—1971no—
175C4⋊Q8—3941no—
176#124—3941no—
177#123—3942no—
178C22⋊Q8—1974no—
179#117—1971no—
180#103—1972no—
181#102—1972no—
182#102—1974no—
183C22⋊C8—1974no—
184C24—22no—
185C24—21no—
186#41—28no—
187#41—24no—
188#41—24no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41C22⋊D19714yes—
193C2×Q8—3942no—
194C2×Q8—3942no—
195C2×Q8—3941no—
196C2×Q8—1972no—
197C2×Q8C22⋊D19711yescharacteristic
198C2×Q8C22×D19711yescharacteristic
199C2×D4—216no—
200C2×D4—28no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4C22⋊D19712yes—
209C2×D4C22⋊D19711yescharacteristic
210C2×D4C22⋊D19714yes—
211C2×D4C22×D19711yescharacteristic
212C2×D4C22×D19712yes—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—3941no—
216C22×C4C22×D19712yes—
217C22×C4C22×D19711yescharacteristic
218Q16—3944no—
219SD16—3948no—
220SD16—3944no—
221M16—3942no—
222C2×C8—3942no—
223C2×C8—1972no—
224C2×C8—1972no—
225C4⋊C4—3944no—
226C4⋊C4—1972no—
227C22⋊C4—3941no—
228C22⋊C4—3944no—
229C42—3941no—
230C23—28no—
231C23—22no—
232C23—21no—
233C23—24no—
234C23—82no—
235C23—84no—
236C23C2×C22⋊D19712yes—
237C23C2×C22⋊D19711yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—7882no—
241Q8—3942no—
242Q8—3942no—
243Q8C2×C22⋊D19712yes—
244D4—28no—
245D4—28no—
246D4—24no—
247D4—24no—
248D4—22no—
249D4—24no—
250D4—22no—
251D4—21no—
252D4C2×C22⋊D19712yes—
253D4C2×C22⋊D19714yes—
254C2×C4—28no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—7882no—
260C2×C4—3942no—
261C2×C4—3942no—
262C2×C4C2×C22⋊D19712yes—
263C2×C4C2×C22⋊D19711yescharacteristic
264C2×C4C2×C22⋊D19712yes—
265C2×C4C2×C22⋊D19712yes—
266C2×C4C23×D19711yescharacteristic
267C8—3942no—
268C8—3942no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—24no—
273C22—22no—
274C22—44no—
275C22—42no—
276C22—84no—
277C22—88no—
278C22#2026329512yes—
279C22C22×C22⋊D19711yescharacteristic
280C4—22no—
281C4—22no—
282C4—7882no—
283C4—7882no—
284C4#2026329512yes—
285C4C22×C22⋊D19711yescharacteristic
286C4C22×C22⋊D19711yescharacteristic
287C2—22no—
288C2—42no—
289C2—44no—
290C2—42no—
291C2C2×#2026329511yescharacteristic
292C2#2716189012yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting, prime radicalC2×H(5,𝔽2)×C197C2
socleC22×C197C22×D4
hypercenterC2×H(5,𝔽2)D197
hypocenterC197#3009
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#20263356C2semidirect
2. C22×#15691346C2semidirect
3. #27161650C2semidirect
4. #27161241C2semidirect
5. #27161230C2semidirect
6. #27161084C2semidirect
7. #27161075C2semidirect
8. #27161046C2semidirect
10. C22×Dic394C22semidirect
12. C2×#15691348C22semidirect
14. C2×#15691346C22semidirect
15. C2×#15691346C22semidirect
16. C2×#15691342C22semidirect
17. C22×#13096802C22semidirect
21. #20263305C22semidirect
22. #20263285C22semidirect
23. #20263250C22semidirect
24. #20263247C22semidirect
25. #20263247C22semidirect
26. C22⋊#13096802C22semidirect
146. C197#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#202633562
C22×#156913462
#271616502
#271612412
#271612302
#271610842
#271610752
#271610462
C2×H(5,𝔽2)×C1972
#3009197