Finite Group Databasebeta

← #38751982

Subgroups — #38751982

Order 25216.

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

normal subgroups only

This group has 784 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, complemented
2C2×#20263354C211yesmaximal, characteristic, complemented
3C22×D1576C211yesmaximal, characteristic, complemented
4#27161643C211yesmaximal, characteristic, complemented
5#27161226C214yesmaximal, complemented
6#27161215C214yesmaximal, complemented
7#27161085C211yesmaximal, characteristic, complemented
8#27161076C211yesmaximal, characteristic, complemented
9#408×C197C211yesmaximal, characteristic, complemented
10#20263372—24no—
11#20263372—24no—
12C2×#15691359—21no—
13C2×#15691358C2211yescharacteristic
14C2×#15691358C2212yescomplemented
15C22×D788C2211yescharacteristic
16C22×D788C2211yescharacteristic, complemented
17C22×D788C2212yescomplemented
18C2×D4×D197—22no—
19C2×D4×D197—21no—
20#20263354—24no—
21C2×#15691347C2212yescomplemented
22C2×D1576—24no—
23C2×D1576C2212yescomplemented
24C2×D1576C2214yescomplemented
25#20263322—21no—
26#20263315—21no—
27#20263300C2214yescomplemented
28#20263294—22no—
29#20263285C2211yescharacteristic, complemented
30#20263251C2212yescomplemented
31#20263248C2212yescomplemented
32#20263248C2214yescomplemented
33C2×#41×C197C2211yescharacteristic
34C2×M16×C197C2211yescharacteristic, complemented
35C22×C8×C197C2211yescharacteristic, complemented
36C22⋊C8×C197C2214yescomplemented
37C23×D197—21no—
38C23×D197—22no—
39C23×D197—21no—
40#15691359—28no—
41#15691358—24no—
42#15691358—24no—
43#15691358—22no—
44#15691358D414yescomplemented
45C2×Dic394C2311yescharacteristic
46C2×D788—24no—
47C2×D788—24no—
48C2×D788—21no—
49C2×D788—22no—
50C2×D788—22no—
51C2×D788—22no—
52C2×D788—22no—
53C2×D788—21no—
54C2×D788D414yescomplemented
55C2×D788C2312yes—
56C2×D788C2312yes—
57C2×D788C2311yescharacteristic
58C2×C22⋊D197—21no—
59C2×C22⋊D197—22no—
60C2×C22⋊D197—24no—
61D4×D197—216no—
62D4×D197—28no—
63C2×C4×D197—21no—
64C2×C4×D197—22no—
65#15691347—24no—
66D1576—28no—
67D1576—24no—
68C4⋊Dic197C2312yes—
69C22⋊Dic197—21no—
70#15691335—24no—
71C4×Dic197—21no—
72#41×C197—22no—
73#41×C197—22no—
74C2×Q8×C197D411yescharacteristic
75C2×D4×C197D412yes—
76C2×D4×C197D411yescharacteristic
77C22×C4×C197C2312yes—
78C22×C4×C197C2311yescharacteristic
79M16×C197—22no—
80C2×C8×C197—22no—
81C2×C8×C197C2312yes—
82C2×C8×C197C2312yes—
83C22×D197—21no—
84C22×D197—22no—
85C22×D197—28no—
86C22×D197—24no—
87C22×D197—22no—
88C22×D197—42no—
89C22×D197—82no—
90C22×D197—84no—
91C22×D197—82no—
92Dic394—21no—
93Dic394C2×D412yes—
94D788—28no—
95D788—24no—
96D788—22no—
97D788—21no—
98D788—22no—
99D788—22no—
100D788—42no—
101D788C2×D412yes—
102D788C2×D414yes—
103C22⋊D197—28no—
104C22⋊D197—24no—
105C22⋊D197—84no—
106C2×Dic197—21no—
107C2×Dic197—22no—
108C4×D197—28no—
109C4×D197—24no—
110C23×C197C2×D412yes—
111C23×C197C2×D411yescharacteristic
112Q8×C197—22no—
113D4×C197—24no—
114D4×C197—22no—
115C2×C4×C197—22no—
116C2×C4×C197—22no—
117C2×C4×C197C2×D412yes—
118C2×C4×C197C2×D412yes—
119C2×C4×C197C2×D412yes—
120C2×C4×C197C2×D411yescharacteristic
121C2×C4×C197C2411yescharacteristic
122C8×C197—22no—
123C8×C197—22no—
124C22×C197—22no—
125C22×C197—22no—
126C22×C197—24no—
127C22×C197#12012yes—
128C22×C197C22×D411yescharacteristic
129C2×D197—24no—
130C2×D197—22no—
131C2×D197—44no—
132C2×D197—42no—
133C2×D197—42no—
134C2×D197—44no—
135C2×D197—84no—
136C2×D197—88no—
137C2×D197—84no—
138C4×C197—22no—
139C4×C197#12012yes—
140C4×C197C22×D411yescharacteristic
141C4×C197C22×D411yescharacteristic
142Dic197—22no—
143Dic197—42no—
144C2×C197—22no—
145C2×C197—42no—
146C2×C197C2×#12011yescharacteristic
147C2×C197#57612yes—
148D197—44no—
149D197—42no—
150D197—82no—
151C197#300811yescharacteristic, complemented, Sylow
152#3008—1971nomaximal, Sylow
153C2×H(5,𝔽2)—1971no—
154C2×AGL(1,ℤ/8)—1971no—
155C22×D8—1971no—
156#535—1971no—
157#449—1974no—
158#447—1974no—
159#418—1971no—
160#416—1971no—
161#408D19711yescharacteristic, complemented
162H(5,𝔽2)—3944no—
163H(5,𝔽2)—3944no—
164C2×#41—3941no—
165C2×#41—1972no—
166C2×#41—1971no—
167C2×#41C2×D19711yescharacteristic
168C22×D4—3942no—
169C22×D4—3941no—
170C22×D4—1971no—
171C22×D4—1972no—
172C22×D4—1971no—
173AGL(1,ℤ/8)—3944no—
174C2×SD16—1972no—
175C2×D8—3944no—
176C2×D8—1974no—
177C2×D8—1972no—
178C2×M16C2×D19711yescharacteristic, complemented
179C22×C8C2×D19711yescharacteristic, complemented
180#127—3941no—
181#124—3941no—
182#121—1974no—
183#120—3942no—
184#117—1971no—
185#103—1972no—
186#102—1972no—
187#102—1974no—
188C22⋊C8C2×D19714yescomplemented
189C24—3941no—
190C24—3942no—
191C24—3941no—
192#41—22no—
193#41—22no—
194#41—3944no—
195#41—3944no—
196#41—3948no—
197#41—3942no—
198#41—1974no—
199C2×Q8—1971no—
200C2×Q8D78811yescharacteristic
201C2×D4—3944no—
202C2×D4—3944no—
203C2×D4—39416no—
204C2×D4—3948no—
205C2×D4—3944no—
206C2×D4—3942no—
207C2×D4—3941no—
208C2×D4—3942no—
209C2×D4—3942no—
210C2×D4—3942no—
211C2×D4—3941no—
212C2×D4—3942no—
213C2×D4—3941no—
214C2×D4—1974no—
215C2×D4—1972no—
216C2×D4—1971no—
217C2×D4—1972no—
218C2×D4D78812yes—
219C2×D4D78811yescharacteristic
220C22×C4—3942no—
221C22×C4—3941no—
222C22×C4C22×D19712yes—
223C22×C4C22×D19711yescharacteristic
224SD16—3944no—
225D8—3948no—
226D8—3944no—
227M16—22no—
228C2×C8—22no—
229C2×C8C22×D19712yes—
230C2×C8C22×D19712yes—
231C4⋊C4—1972no—
232C22⋊C4—3941no—
233C22⋊C4—3944no—
234C42—3941no—
235C23—7882no—
236C23—15762no—
237C23—15762no—
238C23—15764no—
239C23—3948no—
240C23—3942no—
241C23—3942no—
242C23—3941no—
243C23—3944no—
244C23C2×D78812yes—
245C23C2×D78811yescharacteristic
246Q8—22no—
247Q8—3941no—
248Q8—1972no—
249D4—24no—
250D4—22no—
251D4—7882no—
252D4—15764no—
253D4—3948no—
254D4—3948no—
255D4—3944no—
256D4—3944no—
257D4—3942no—
258D4—3942no—
259D4—3942no—
260D4—3941no—
261D4—1972no—
262D4—1974no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—3948no—
266C2×C4—3942no—
267C2×C4—3941no—
268C2×C4—3944no—
269C2×C4D4×D19712yes—
270C2×C4D4×D19712yes—
271C2×C4C2×D78812yes—
272C2×C4C2×D78811yescharacteristic
273C2×C4C23×D19711yescharacteristic
274C8—22no—
275C8—22no—
276C22—22no—
277C22—24no—
278C22—22no—
279C22—7884no—
280C22—7882no—
281C22—7884no—
282C22—7882no—
283C22—15764no—
284C22—15764no—
285C22—15768no—
286C22—3944no—
287C22—3942no—
288C22#2026329612yes—
289C22C22×D78811yescharacteristic
290C4—22no—
291C4—7882no—
292C4—3942no—
293C4#2026329612yes—
294C4C2×D4×D19711yescharacteristic
295C4C2×D4×D19711yescharacteristic
296C2—22no—
297C2—42no—
298C2—7884no—
299C2—7882no—
300C2—15762no—
301C2C2×#2026329611yescharacteristic
302C2#2716188412yes—
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#408×C197C2
socleC22×C197C22×D4
hypercenter#408D197
hypocenterC197#3008
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#20263372C2semidirect
2. C2×#20263354C2semidirect
3. C22×D1576C2semidirect
4. #27161643C2semidirect
5. #27161226C2semidirect
6. #27161215C2semidirect
7. #27161085C2semidirect
8. #27161076C2semidirect
9. #408×C197C2semidirect
14. C2×#15691358C22semidirect
16. C22×D788C22semidirect
17. C22×D788C22semidirect
21. C2×#15691347C22semidirect
23. C2×D1576C22semidirect
24. C2×D1576C22semidirect
27. #20263300C22semidirect
29. #20263285C22semidirect
30. #20263251C22semidirect
31. #20263248C22semidirect
32. #20263248C22semidirect
34. C2×M16×C197C22semidirect
35. C22×C8×C197C22semidirect
36. C22⋊C8×C197C22semidirect
44. #15691358D4semidirect
54. C2×D788D4semidirect
151. C197#3008semidirect
161. #408D197semidirect
178. C2×M16C2×D197semidirect
179. C22×C8C2×D197semidirect
188. C22⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633722
C2×#202633542
C22×D15762
#271616432
#271612262
#271612152
#271610852
#271610762
#408×C1972
#3008197