Finite Group Databasebeta

← #38751988

Subgroups — #38751988

Order 25216.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 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
2C22×#15691345C211yesmaximal, characteristic, complemented
3#27161653C211yesmaximal, characteristic, complemented
4#27161240C214yesmaximal, complemented
5#27161228C214yesmaximal, complemented
6#27161085C211yesmaximal, characteristic, complemented
7#27161075C211yesmaximal, characteristic, complemented
8#27161047C211yesmaximal, characteristic, complemented
9C2×#137×C197C211yesmaximal, characteristic, complemented
10#20263372—24no—
11#20263372—24no—
12C2×#15691359C2212yes—
13C2×#15691358—21no—
14C2×#15691358C2211yescharacteristic
15C22×D788—21no—
16C22×D788C2211yescharacteristic
17C2×D4×D197—22no—
18C2×D4×D197C2212yes—
19C2×#15691345—24no—
20C2×#15691345C2214yescomplemented
21C22×#13096802C2211yescharacteristic, complemented
22#20263324—21no—
23#20263315—21no—
24#20263309—22no—
25#20263302C2214yescomplemented
26#20263285C2211yescharacteristic, complemented
27#20263250C2212yescomplemented
28#20263248C2214yescomplemented
29#20263247C2212yescomplemented
30#20263238C2214yescomplemented
31C2×#41×C197C2211yescharacteristic
32C22×Q8×C197C2211yescharacteristic, complemented
33#137×C197—24no—
34C2×Q16×C197C2212yescomplemented
35C2×SD16×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×D788D414yes—
51C2×D788C2312yes—
52C2×C22⋊D197—22no—
53C2×C22⋊D197D412yes—
54D4×D197—216no—
55D4×D197—24no—
56D4×D197—24no—
57C2×C4×D197—22no—
58C2×C4×D197C2312yes—
59#15691345—28no—
60C2×#13096802—22no—
61C2×#13096802C2312yes—
62C4⋊Dic197C2312yes—
63#15691337—24no—
64C22⋊Dic197—21no—
65#15691335—24no—
66C4×Dic197—21no—
67#41×C197—22no—
68#41×C197D414yes—
69C2×Q8×C197—21no—
70C2×Q8×C197—22no—
71C2×Q8×C197—22no—
72C2×Q8×C197C2312yes—
73C2×Q8×C197C2311yescharacteristic
74C2×D4×C197—21no—
75C2×D4×C197C2311yescharacteristic
76C22×C4×C197—21no—
77C22×C4×C197—21no—
78C22×C4×C197C2311yescharacteristic
79Q16×C197—24no—
80SD16×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—
97C2×Dic197—22no—
98C2×Dic197C2×D412yes—
99C4×D197—28no—
100C4×D197—22no—
101C4×D197—22no—
102#13096802—22no—
103C23×C197—21no—
104C23×C197C2×D411yescharacteristic
105Q8×C197—21no—
106Q8×C197—22no—
107Q8×C197—22no—
108Q8×C197—42no—
109Q8×C197C2×D412yes—
110D4×C197—24no—
111D4×C197—21no—
112D4×C197C2×D412yes—
113C2×C4×C197—21no—
114C2×C4×C197—24no—
115C2×C4×C197—22no—
116C2×C4×C197—42no—
117C2×C4×C197C2×D411yescharacteristic
118C2×C4×C197C2×D412yes—
119C2×C4×C197C2×D412yes—
120C2×C4×C197C2411yescharacteristic
121C8×C197—22no—
122C22×C197—22no—
123C22×C197—22no—
124C22×C197—42no—
125C22×C197#12012yes—
126C22×C197C22×D411yescharacteristic
127C2×D197—22no—
128C2×D197—24no—
129C2×D197—24no—
130C2×D197—44no—
131C2×D197—84no—
132C2×D197—88no—
133C4×C197—22no—
134C4×C197—42no—
135C4×C197#12012yes—
136C4×C197C22×D411yescharacteristic
137C4×C197C22×D411yescharacteristic
138Dic197—22no—
139Dic197—42no—
140C2×C197—22no—
141C2×C197—42no—
142C2×C197C2×#12011yescharacteristic
143C2×C197#57712yes—
144D197—42no—
145D197—44no—
146C197#300911yescharacteristic, complemented, Sylow
147#3009—1971nomaximal, Sylow
148C2×H(5,𝔽2)—1971no—
149C2×#137D19711yescharacteristic, complemented
150C22×SD16—1971no—
151#536—1971no—
152#452—1974no—
153#450—1974no—
154#418—1971no—
155#416—1971no—
156#408—1971no—
157H(5,𝔽2)—3944no—
158H(5,𝔽2)—3944no—
159C2×#41—3941no—
160C2×#41—1972no—
161C2×#41—1971no—
162C2×#41C2×D19711yescharacteristic
163C22×Q8C2×D19711yescharacteristic, complemented
164C22×D4—3941no—
165C22×D4—3942no—
166C22×D4—1972no—
167C22×D4—1971no—
168#137—24no—
169C2×Q16C2×D19712yescomplemented
170C2×SD16—3944no—
171C2×SD16—1974no—
172C2×SD16C2×D19712yescomplemented
173C2×M16C2×D19711yescharacteristic, complemented
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—3942no—
185C24—3941no—
186#41—22no—
187#41—3948no—
188#41—3944no—
189#41—3944no—
190#41—3942no—
191#41—3942no—
192#41C22⋊D19714yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1971no—
197C2×Q8C22×D19712yes—
198C2×Q8C22×D19711yescharacteristic
199C2×D4—21no—
200C2×D4—3948no—
201C2×D4—3942no—
202C2×D4—39416no—
203C2×D4—3944no—
204C2×D4—3944no—
205C2×D4—3942no—
206C2×D4—3941no—
207C2×D4—3942no—
208C2×D4—1972no—
209C2×D4—1971no—
210C2×D4—1974no—
211C2×D4—1972no—
212C2×D4C22×D19711yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—3942no—
216C22×C4—1972no—
217C22×C4C22×D19711yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—3948no—
221M16—22no—
222C2×C8—3942no—
223C2×C8—1972no—
224C2×C8C22×D19712yes—
225C4⋊C4—3944no—
226C4⋊C4—1972no—
227C22⋊C4—3944no—
228C22⋊C4—3941no—
229C42—3941no—
230C23—21no—
231C23—15762no—
232C23—15764no—
233C23—3948no—
234C23—3942no—
235C23—3944no—
236C23—1972no—
237C23D4×D19711yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—3942no—
243Q8C2×C22⋊D19712yes—
244D4—24no—
245D4—21no—
246D4—3948no—
247D4—3948no—
248D4—3944no—
249D4—3942no—
250D4—3944no—
251D4—3942no—
252D4—1974no—
253D4C2×C22⋊D19712yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—3948no—
259C2×C4—3942no—
260C2×C4—3942no—
261C2×C4—3942no—
262C2×C4—1972no—
263C2×C4D4×D19711yescharacteristic
264C2×C4D4×D19712yes—
265C2×C4C2×C22⋊D19712yes—
266C2×C4C23×D19711yescharacteristic
267C8—22no—
268C8—3942no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—7884no—
273C22—15764no—
274C22—15768no—
275C22—3944no—
276C22—3942no—
277C22—3944no—
278C22#2026329412yes—
279C22C2×D4×D19711yescharacteristic
280C4—22no—
281C4—42no—
282C4—7882no—
283C4—3942no—
284C4#2026329412yes—
285C4C2×D4×D19711yescharacteristic
286C4C22×C22⋊D19711yescharacteristic
287C2—22no—
288C2—42no—
289C2—7884no—
290C2—7882no—
291C2C2×#2026329411yescharacteristic
292C2#2716188812yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#137×C197C2
socleC22×C197C22×D4
prime radicalC2×#20263372C2
hypercenterC2×#137D197
hypocenterC197#3009
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#15691345C2semidirect
3. #27161653C2semidirect
4. #27161240C2semidirect
5. #27161228C2semidirect
6. #27161085C2semidirect
7. #27161075C2semidirect
8. #27161047C2semidirect
9. C2×#137×C197C2semidirect
20. C2×#15691345C22semidirect
21. C22×#13096802C22semidirect
25. #20263302C22semidirect
26. #20263285C22semidirect
27. #20263250C22semidirect
28. #20263248C22semidirect
29. #20263247C22semidirect
30. #20263238C22semidirect
32. C22×Q8×C197C22semidirect
34. C2×Q16×C197C22semidirect
35. C2×SD16×C197C22semidirect
36. C2×M16×C197C22semidirect
146. C197#3009semidirect
149. C2×#137D197semidirect
163. C22×Q8C2×D197semidirect
169. C2×Q16C2×D197semidirect
172. C2×SD16C2×D197semidirect
173. C2×M16C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633722
C22×#156913452
#271616532
#271612402
#271612282
#271610852
#271610752
#271610472
C2×#137×C1972
#3009197