Finite Group Databasebeta

← #38751990

Subgroups — #38751990

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×#20263371C211yesmaximal, characteristic
2C2×#20263359C211yesmaximal, characteristic, complemented
3C22×#15691346C211yesmaximal, characteristic, complemented
4#27161655C211yesmaximal, characteristic, complemented
5#27161242C214yesmaximal, complemented
6#27161231C214yesmaximal, complemented
7#27161073C211yesmaximal, characteristic, complemented
8#27161047C211yesmaximal, characteristic, complemented
9#418×C197C211yesmaximal, characteristic, complemented
10#20263371—24no—
11#20263371—24no—
12C2×#15691358C2211yescharacteristic
13C2×#15691358C2211yescharacteristic
14C2×#15691357—21no—
15C2×#15691357C2212yes—
16C22×Dic394C2211yescharacteristic, complemented
17C22×C22⋊D197—22no—
18C2×D4×D197—21no—
19C2×D4×D197C2212yes—
20#20263359—24no—
21C2×Dic788C2212yescomplemented
22C2×#15691347C2212yescomplemented
23C2×#15691346—24no—
24C2×#15691346C2214yescomplemented
25C22×#13096802C2211yescharacteristic, complemented
26#20263326—21no—
27#20263314—21no—
28#20263310—22no—
29#20263306C2214yescomplemented
30#20263249C2212yescomplemented
31#20263246C2212yescomplemented
32#20263238C2214yescomplemented
33C22×D4×C197C2211yescharacteristic
34C2×M16×C197C2211yescharacteristic, complemented
35#117×C197C2211yescharacteristic, complemented
36#102×C197C2214yescomplemented
37C23×D197—22no—
38#15691358—22no—
39#15691358—22no—
40#15691358—22no—
41#15691358D414yes—
42#15691357—28no—
43#15691357—24no—
44#15691357—24no—
45C2×Dic394—21no—
46C2×Dic394—22no—
47C2×Dic394—22no—
48C2×Dic394D411yescharacteristic
49C2×Dic394C2312yes—
50C2×Dic394C2311yescharacteristic
51C2×D788D411yescharacteristic
52C2×D788C2311yescharacteristic
53C2×C22⋊D197—216no—
54C2×C22⋊D197—22no—
55C2×C22⋊D197—22no—
56C2×C22⋊D197—21no—
57C2×C22⋊D197D412yes—
58D4×D197—28no—
59D4×D197—24no—
60D4×D197—24no—
61C22×Dic197—22no—
62C22×Dic197—21no—
63C2×C4×D197—21no—
64C2×C4×D197C2312yes—
65Dic788—24no—
66#15691347—24no—
67#15691346—28no—
68C2×#13096802—22no—
69C2×#13096802C2312yes—
70C4⋊Dic197—24no—
71#15691335—24no—
72C24×C197—21no—
73C2×D4×C197—21no—
74C2×D4×C197—22no—
75C2×D4×C197D414yes—
76C2×D4×C197C2312yes—
77C22×C4×C197C2311yescharacteristic
78M16×C197—22no—
79C2×C8×C197C2312yes—
80C4⋊C4×C197C2312yes—
81C22⋊C4×C197—21no—
82C42×C197—21no—
83C22×D197—21no—
84C22×D197—28no—
85C22×D197—84no—
86C22×D197C2×D412yes—
87Dic394—22no—
88Dic394—21no—
89Dic394—22no—
90Dic394—22no—
91Dic394—42no—
92Dic394C2×D412yes—
93D788—22no—
94D788—21no—
95D788C2×D412yes—
96C22⋊D197—28no—
97C22⋊D197—28no—
98C22⋊D197—24no—
99C22⋊D197—24no—
100C2×Dic197—28no—
101C2×Dic197—21no—
102C2×Dic197—22no—
103C2×Dic197—42no—
104C2×Dic197C2×D412yes—
105C4×D197—24no—
106C4×D197—22no—
107C4×D197—22no—
108#13096802—22no—
109C23×C197—22no—
110C23×C197—24no—
111C23×C197—82no—
112C23×C197C2×D411yescharacteristic
113D4×C197—24no—
114D4×C197—22no—
115D4×C197C2×D414yes—
116C2×C4×C197—22no—
117C2×C4×C197C2×D411yescharacteristic
118C2×C4×C197C2×D412yes—
119C2×C4×C197C2×D412yes—
120C2×C4×C197C2411yescharacteristic
121C8×C197—22no—
122C22×C197—22no—
123C22×C197—24no—
124C22×C197—44no—
125C22×C197—84no—
126C22×C197#12012yes—
127C22×C197C22×D411yescharacteristic
128C2×D197—22no—
129C2×D197—22no—
130C2×D197—24no—
131C2×D197—42no—
132C2×D197—88no—
133C4×C197—42no—
134C4×C197#12012yes—
135C4×C197C22×D411yescharacteristic
136C4×C197C22×D411yescharacteristic
137Dic197—22no—
138Dic197—22no—
139Dic197—42no—
140C2×C197—22no—
141C2×C197—44no—
142C2×C197C2×#12011yescharacteristic
143C2×C197#57712yes—
144D197—42no—
145D197—42no—
146C197#300911yescharacteristic, complemented, Sylow
147#3009—1971nomaximal, Sylow
148C2×H(5,𝔽2)—1971no—
149C2×#137—1971no—
150C22×SD16—1971no—
151#536—1971no—
152#452—1974no—
153#450—1974no—
154#418D19711yescharacteristic, complemented
155#416—1971no—
156#408—1971no—
157H(5,𝔽2)—3944no—
158H(5,𝔽2)—3944no—
159C2×#41—3941no—
160C2×#41—1972no—
161C2×#41—1971no—
162C2×#41—1971no—
163C22×Q8—1971no—
164C22×D4—3941no—
165C22×D4—3942no—
166C22×D4—1972no—
167C22×D4C2×D19711yescharacteristic
168#137—3944no—
169C2×Q16—1972no—
170C2×SD16—3944no—
171C2×SD16—1974no—
172C2×SD16—1972no—
173C2×M16C2×D19711yescharacteristic, complemented
174C22×C8—1971no—
175C4⋊Q8—3941no—
176#124—3941no—
177#123—3942no—
178C22⋊Q8—1974no—
179#117C2×D19711yescharacteristic, complemented
180#103—1972no—
181#102—1972no—
182#102C2×D19714yescomplemented
183C22⋊C8—1974no—
184C24—21no—
185C24—3942no—
186#41—3948no—
187#41—3944no—
188#41—3944no—
189#41—3942no—
190#41—3942no—
191#41—3942no—
192#41—1974no—
193C2×Q8—3942no—
194C2×Q8—3942no—
195C2×Q8—3941no—
196C2×Q8—1972no—
197C2×Q8—1971no—
198C2×Q8—1971no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—3948no—
202C2×D4—3944no—
203C2×D4—39416no—
204C2×D4—3942no—
205C2×D4—3941no—
206C2×D4—3944no—
207C2×D4—3942no—
208C2×D4—1972no—
209C2×D4—1971no—
210C2×D4—1971no—
211C2×D4D78814yes—
212C2×D4C22×D19712yes—
213C22×C4—3941no—
214C22×C4—3942no—
215C22×C4—3941no—
216C22×C4—1972no—
217C22×C4C22×D19711yescharacteristic
218Q16—3944no—
219SD16—3944no—
220SD16—3948no—
221M16—22no—
222C2×C8—3942no—
223C2×C8—1972no—
224C2×C8C22×D19712yes—
225C4⋊C4—3944no—
226C4⋊C4C22×D19712yes—
227C22⋊C4—21no—
228C22⋊C4—3944no—
229C42—21no—
230C23—22no—
231C23—24no—
232C23—82no—
233C23—15764no—
234C23—3948no—
235C23—3941no—
236C23—1972no—
237C23D4×D19711yescharacteristic
238Q8—7882no—
239Q8—3942no—
240Q8—3942no—
241Q8—3942no—
242Q8—3941no—
243Q8—1972no—
244D4—24no—
245D4—22no—
246D4—3948no—
247D4—3948no—
248D4—3944no—
249D4—3944no—
250D4—3942no—
251D4—3941no—
252D4—1972no—
253D4C2×D78814yes—
254C2×C4—22no—
255C2×C4—7882no—
256C2×C4—3942no—
257C2×C4—3948no—
258C2×C4—3941no—
259C2×C4—3944no—
260C2×C4—3942no—
261C2×C4—3942no—
262C2×C4—1972no—
263C2×C4D4×D19711yescharacteristic
264C2×C4D4×D19712yes—
265C2×C4C2×D78812yes—
266C2×C4C23×D19711yescharacteristic
267C8—22no—
268C8—3942no—
269C22—22no—
270C22—24no—
271C22—44no—
272C22—84no—
273C22—7882no—
274C22—15768no—
275C22—3944no—
276C22—3942no—
277C22—3942no—
278C22#2026329612yes—
279C22C2×D4×D19711yescharacteristic
280C4—42no—
281C4—7882no—
282C4—3942no—
283C4—3942no—
284C4#2026329612yes—
285C4C2×D4×D19711yescharacteristic
286C4C22×D78811yescharacteristic
287C2—22no—
288C2—44no—
289C2—7882no—
290C2—7882no—
291C2C2×#2026329611yescharacteristic
292C2#2716188912yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#418×C197C2
socleC22×C197C22×D4
prime radicalC2×#20263371C2
hypercenter#418D197
hypocenterC197#3009
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C2×#20263359C2semidirect
3. C22×#15691346C2semidirect
4. #27161655C2semidirect
5. #27161242C2semidirect
6. #27161231C2semidirect
7. #27161073C2semidirect
8. #27161047C2semidirect
9. #418×C197C2semidirect
16. C22×Dic394C22semidirect
21. C2×Dic788C22semidirect
22. C2×#15691347C22semidirect
24. C2×#15691346C22semidirect
25. C22×#13096802C22semidirect
29. #20263306C22semidirect
30. #20263249C22semidirect
31. #20263246C22semidirect
32. #20263238C22semidirect
34. C2×M16×C197C22semidirect
35. #117×C197C22semidirect
36. #102×C197C22semidirect
146. C197#3009semidirect
154. #418D197semidirect
173. C2×M16C2×D197semidirect
179. #117C2×D197semidirect
182. #102C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633712
C2×#202633592
C22×#156913462
#271616552
#271612422
#271612312
#271610732
#271610472
#418×C1972
#3009197