Finite Group Databasebeta

← #38752108

Subgroups — #38752108

Order 25216.

800 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 145 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, complemented
2#27161638C212yesmaximal, complemented
3#27161268C214yesmaximal, complemented
4#27161265C214yesmaximal, complemented
5C2×#20263241C211yesmaximal, characteristic, complemented
6C2×#20263240C212yesmaximal, complemented
7#534×C197C211yesmaximal, characteristic, complemented
8#20263371—24no—
9#20263371—22no—
10#20263371C2214yes—
11C2×#15691358—21no—
12C2×#15691357—22no—
13C22×C22⋊D197—21no—
14C22×C22⋊D197C2212yes—
15C2×D4×D197—22no—
16#20263309C2214yescomplemented
17#20263307—22no—
18#20263298—24no—
19#20263295C2214yescomplemented
20#20263294—22no—
21C2×C22⋊Dic197C2212yescomplemented
22#20263241C2214yescomplemented
23#20263240C2218yescomplemented
24C22×D4×C197C2211yescharacteristic
25C22×D4×C197C2211yescharacteristic, complemented
26#123×C197—21no—
27#123×C197C2212yescomplemented
28#121×C197—22no—
29#120×C197—21no—
30#120×C197C2212yescomplemented
31C2×C22⋊C4×C197C2211yescharacteristic, complemented
32C23×D197—21no—
33C23×D197D412yescomplemented
34#15691358—24no—
35#15691358—24no—
36#15691357—28no—
37#15691357—28no—
38C2×Dic394—21no—
39C2×D788—21no—
40C2×C22⋊D197—22no—
41C2×C22⋊D197—24no—
42C2×C22⋊D197—24no—
43C2×C22⋊D197—22no—
44C2×C22⋊D197—22no—
45C2×C22⋊D197—24no—
46C2×C22⋊D197—22no—
47C2×C22⋊D197—22no—
48C2×C22⋊D197D414yescomplemented
49C2×C22⋊D197C2314yes—
50D4×D197—28no—
51D4×D197—28no—
52C22×Dic197—21no—
53C22×Dic197—22no—
54C22×Dic197D412yescomplemented
55C2×C4×D197—22no—
56#15691337—24no—
57C22⋊Dic197—24no—
58C22⋊Dic197—22no—
59C22⋊Dic197C2314yescomplemented
60#15691335—24no—
61C24×C197D411yescharacteristic, complemented
62C24×C197C2311yescharacteristic
63C24×C197C2311yescharacteristic, complemented
64C2×D4×C197—22no—
65C2×D4×C197—22no—
66C2×D4×C197—21no—
67C2×D4×C197—21no—
68C2×D4×C197—22no—
69C2×D4×C197—22no—
70C2×D4×C197—22no—
71C2×D4×C197D412yescomplemented
72C2×D4×C197C2312yes—
73C22×C4×C197—21no—
74C22×C4×C197D411yescharacteristic, complemented
75C22×C4×C197C2311yescharacteristic, complemented
76C4⋊C4×C197—22no—
77C22⋊C4×C197—21no—
78C22⋊C4×C197—22no—
79C22⋊C4×C197—22no—
80C22⋊C4×C197C2312yescomplemented
81C22×D197—22no—
82C22×D197—24no—
83C22×D197—22no—
84C22×D197—82no—
85C22×D197—82no—
86C22×D197C2×D414yes—
87Dic394—24no—
88D788—24no—
89C22⋊D197—28no—
90C22⋊D197—24no—
91C22⋊D197—24no—
92C22⋊D197—24no—
93C22⋊D197—24no—
94C22⋊D197—84no—
95C2×Dic197—24no—
96C2×Dic197—22no—
97C2×Dic197—22no—
98C2×Dic197—24no—
99C2×Dic197—82no—
100C2×Dic197C2×D414yes—
101C4×D197—28no—
102C23×C197—21no—
103C23×C197—21no—
104C23×C197—21no—
105C23×C197—22no—
106C23×C197—81no—
107C23×C197—81no—
108C23×C197—81no—
109C23×C197C2×D412yes—
110C23×C197C2×D412yes—
111C23×C197C2×D412yes—
112C23×C197C2×D411yescharacteristic
113C23×C197C2×D411yescharacteristic
114C23×C197C2411yescharacteristic
115D4×C197—24no—
116D4×C197—24no—
117D4×C197—41no—
118D4×C197—41no—
119D4×C197—42no—
120D4×C197—82no—
121C2×C4×C197—21no—
122C2×C4×C197—21no—
123C2×C4×C197—22no—
124C2×C4×C197—21no—
125C2×C4×C197—22no—
126C2×C4×C197—81no—
127C2×C4×C197C2×D412yes—
128C22×C197—22no—
129C22×C197—41no—
130C22×C197—41no—
131C22×C197—41no—
132C22×C197—42no—
133C22×C197—42no—
134C22×C197—42no—
135C22×C197—42no—
136C22×C197—42no—
137C22×C197—44no—
138C22×C197—41no—
139C22×C197—81no—
140C22×C197#12011yescharacteristic
141C22×C197#12011yescharacteristic
142C22×C197#12012yes—
143C22×C197C22×D412yes—
144C22×C197C22×D411yescharacteristic
145C2×D197—24no—
146C2×D197—44no—
147C2×D197—44no—
148C2×D197—82no—
149C2×D197—84no—
150C4×C197—22no—
151C4×C197—42no—
152C4×C197—41no—
153Dic197—24no—
154Dic197—44no—
155C2×C197—22no—
156C2×C197—21no—
157C2×C197—21no—
158C2×C197—22no—
159C2×C197—21no—
160C2×C197—42no—
161C2×C197—81no—
162C2×C197C2×#12011yescharacteristic
163D197—44no—
164C197#302611yescharacteristic, complemented, Sylow
165#3026—1971nomaximal, Sylow
166C2×H(5,𝔽2)—1971no—
167#534—1972no—
168#534D19711yescharacteristic, complemented
169#458—1974no—
170B(4,𝔽2)—1974no—
171C2×#99—1971no—
172C2×#99—1972no—
173H(5,𝔽2)—3944no—
174H(5,𝔽2)—3942no—
175H(5,𝔽2)—1974no—
176C2×#41—3942no—
177C2×#41—3941no—
178C22×D4—3942no—
179C22×D4—3941no—
180C22×D4—1972no—
181C22×D4C2×D19711yescharacteristic
182C22×D4C2×D19711yescharacteristic, complemented
183#123—21no—
184#123—3942no—
185#123—1974no—
186#123C2×D19712yescomplemented
187#121—22no—
188#121—3944no—
189#120—21no—
190#120—3942no—
191#120—1974no—
192#120C2×D19712yescomplemented
193C2×C22⋊C4—1972no—
194C2×C22⋊C4C2×D19711yescharacteristic, complemented
195#99—1974no—
196#99—1978no—
197C24—3941no—
198C24—1972no—
199C24C22⋊D19711yescharacteristic, complemented
200C24C22×D19711yescharacteristic
201C24C22×D19711yescharacteristic, complemented
202#41—3944no—
203#41—3948no—
204#41—3948no—
205#41—3944no—
206C2×Q8—3941no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—22no—
214C2×D4—3941no—
215C2×D4—3948no—
216C2×D4—3948no—
217C2×D4—3944no—
218C2×D4—3942no—
219C2×D4—3944no—
220C2×D4—3942no—
221C2×D4—3942no—
222C2×D4—3942no—
223C2×D4—3944no—
224C2×D4—3942no—
225C2×D4—1974no—
226C2×D4—1974no—
227C2×D4C22⋊D19712yescomplemented
228C2×D4C22×D19712yes—
229C22×C4—21no—
230C22×C4—3942no—
231C22×C4—3942no—
232C22×C4—3941no—
233C22×C4—1972no—
234C22×C4C22⋊D19711yescharacteristic, complemented
235C22×C4C22×D19711yescharacteristic, complemented
236C4⋊C4—22no—
237C4⋊C4—3944no—
238C22⋊C4—21no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—3944no—
242C22⋊C4—3942no—
243C22⋊C4—3944no—
244C22⋊C4—1974no—
245C22⋊C4C22×D19712yescomplemented
246C23—21no—
247C23—21no—
248C23—22no—
249C23—21no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—15762no—
254C23—15762no—
255C23—3944no—
256C23—3942no—
257C23—3942no—
258C23—1974no—
259C23D4×D19712yes—
260C23D4×D19712yes—
261C23C2×C22⋊D19712yes—
262C23C2×C22⋊D19711yescharacteristic
263C23C2×C22⋊D19711yescharacteristic
264C23C23×D19711yescharacteristic
265Q8—3944no—
266D4—24no—
267D4—24no—
268D4—41no—
269D4—41no—
270D4—42no—
271D4—82no—
272D4—15764no—
273D4—3944no—
274D4—3948no—
275D4—3944no—
276D4—3944no—
277D4—3944no—
278D4—3944no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—15762no—
286C2×C4—3948no—
287C2×C4—3944no—
288C2×C4—3944no—
289C2×C4—3942no—
290C2×C4—3942no—
291C2×C4—1974no—
292C2×C4C2×C22⋊D19712yes—
293C22—22no—
294C22—41no—
295C22—42no—
296C22—41no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—44no—
303C22—41no—
304C22—81no—
305C22—7884no—
306C22—7884no—
307C22—15762no—
308C22—15764no—
309C22—3944no—
310C22#2026329411yescharacteristic
311C22#2026329411yescharacteristic
312C22#2026329412yes—
313C22C2×D4×D19712yes—
314C22C22×C22⋊D19711yescharacteristic
315C4—22no—
316C4—42no—
317C4—41no—
318C4—7884no—
319C4—3944no—
320C2—21no—
321C2—22no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—81no—
327C2—7884no—
328C2C2×#2026329411yescharacteristic
subgroupG/H
centerC2C2×#20263294
commutator, cosocleC23×C197C24
FrattiniC23C23×D197
Fitting#534×C197C2
socleC2×C197C2×#120
hypercenter#534D197
hypocenterC197#3026
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#20263371C2semidirect
2. #27161638C2semidirect
3. #27161268C2semidirect
4. #27161265C2semidirect
5. C2×#20263241C2semidirect
6. C2×#20263240C2semidirect
7. #534×C197C2semidirect
16. #20263309C22semidirect
19. #20263295C22semidirect
21. C2×C22⋊Dic197C22semidirect
22. #20263241C22semidirect
23. #20263240C22semidirect
25. C22×D4×C197C22semidirect
27. #123×C197C22semidirect
30. #120×C197C22semidirect
31. C2×C22⋊C4×C197C22semidirect
33. C23×D197D4semidirect
48. C2×C22⋊D197D4semidirect
54. C22×Dic197D4semidirect
59. C22⋊Dic197C23semidirect
61. C24×C197D4semidirect
63. C24×C197C23semidirect
71. C2×D4×C197D4semidirect
74. C22×C4×C197D4semidirect
75. C22×C4×C197C23semidirect
80. C22⋊C4×C197C23semidirect
164. C197#3026semidirect
168. #534D197semidirect
182. C22×D4C2×D197semidirect
186. #123C2×D197semidirect
192. #120C2×D197semidirect
194. C2×C22⋊C4C2×D197semidirect
199. C24C22⋊D197semidirect
201. C24C22×D197semidirect
227. C2×D4C22⋊D197semidirect
234. C22×C4C22⋊D197semidirect
235. C22×C4C22×D197semidirect
245. C22⋊C4C22×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633712
#271616382
#271612682
#271612652
C2×#202632412
C2×#202632402
#534×C1972
#3026197