Finite Group Databasebeta

← #38753191

Subgroups — #38753191

Order 25216.

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

normal subgroups only

This group has 564 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×#20263354C211yesmaximal, characteristic, complemented
2C2×#20263346C211yesmaximal, characteristic, complemented
3C22×#15691345C211yesmaximal, characteristic, complemented
4#27161653C211yesmaximal, characteristic, complemented
5#27161643C211yesmaximal, characteristic, complemented
6#27161472C212yesmaximal, complemented
7#27161465C212yesmaximal, complemented
8#27161464C212yesmaximal, complemented
9#27161450C212yesmaximal, complemented
10#27161037C211yesmaximal, characteristic, complemented
11C2×#137×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2211yescharacteristic
13C22×D788C2211yescharacteristic
14#20263354—24no—
15#20263346—24no—
16C2×#15691348C2211yescharacteristic, complemented
17C2×#15691347C2212yes—
18C2×#15691346C2211yescharacteristic, complemented
19C2×#15691345—24no—
20C2×#15691345C2212yes—
21C2×#15691345C2212yes—
22C2×#15691345C2211yescharacteristic, complemented
23C2×D1576C2212yes—
24C2×#15691343C2211yescharacteristic, complemented
25C22×#13096802C2211yescharacteristic
26#20263324C2212yescomplemented
27#20263322C2212yes—
28#20263315C2212yes—
29#20263315C2212yescomplemented
30#20263309—22no—
31#20263302—22no—
32#20263300—22no—
33#20263294—22no—
34#20263285C2211yescharacteristic
35#20263236C2212yescomplemented
36C8×Dic197C2212yescomplemented
37C2×#41×C197C2211yescharacteristic
38C22×Q8×C197C2211yescharacteristic
39#137×C197—24no—
40C2×Q16×C197C2212yescomplemented
41C2×SD16×C197C2212yescomplemented
42C2×M16×C197C2211yescharacteristic
43C23×D197—21no—
44#15691358—22no—
45#15691358—22no—
46C2×Dic394C2311yescharacteristic
47C2×D788—22no—
48C2×D788—22no—
49C2×D788—21no—
50C2×D788C2312yes—
51C2×D788C2311yescharacteristic
52C2×C22⋊D197—24no—
53C2×C22⋊D197—21no—
54C2×C4×D197—21no—
55#15691348—22no—
56#15691347—24no—
57#15691346—22no—
58#15691345—24no—
59#15691345—24no—
60#15691345—22no—
61D1576—24no—
62#15691343—22no—
63C2×#13096802D414yescomplemented
64C2×#13096802C2311yescharacteristic
65C2×#13096802C2311yescharacteristic
66C4⋊Dic197D412yes—
67#15691337—24no—
68C22⋊Dic197D412yes—
69#15691335—24no—
70#15691335—24no—
71C4×Dic197C2312yes—
72#41×C197—22no—
73#41×C197—22no—
74C2×Q8×C197—21no—
75C2×Q8×C197—22no—
76C2×Q8×C197—22no—
77C2×Q8×C197C2312yes—
78C2×Q8×C197C2311yescharacteristic
79C2×D4×C197—21no—
80C2×D4×C197C2311yescharacteristic
81C22×C4×C197—21no—
82C22×C4×C197—21no—
83C22×C4×C197C2311yescharacteristic
84Q16×C197—24no—
85SD16×C197—24no—
86M16×C197D414yescomplemented
87C2×C8×C197C2312yes—
88C22×D197—21no—
89C22×D197—22no—
90C22×D197—42no—
91C22×D197—82no—
92Dic394—21no—
93Dic394—21no—
94D788—21no—
95D788—21no—
96D788—22no—
97D788—22no—
98D788—42no—
99C22⋊D197—42no—
100C22⋊D197—84no—
101C2×Dic197—21no—
102C2×Dic197C2×D414yes—
103C4×D197—42no—
104#13096802C2×D412yes—
105#13096802C2×D412yes—
106C23×C197—21no—
107C23×C197C2×D411yescharacteristic
108Q8×C197—22no—
109Q8×C197—21no—
110Q8×C197—22no—
111Q8×C197—21no—
112Q8×C197—42no—
113D4×C197—21no—
114D4×C197—21no—
115D4×C197—42no—
116C2×C4×C197—21no—
117C2×C4×C197—22no—
118C2×C4×C197—42no—
119C2×C4×C197—42no—
120C2×C4×C197C2×D411yescharacteristic
121C2×C4×C197C2×D412yes—
122C2×C4×C197C2×D412yes—
123C2×C4×C197C2411yescharacteristic
124C8×C197C2×D414yes—
125C22×C197—22no—
126C22×C197—41no—
127C22×C197—42no—
128C22×C197#12712yes—
129C22×C197C22×D411yescharacteristic
130C2×D197—41no—
131C2×D197—42no—
132C2×D197—44no—
133C2×D197—42no—
134C2×D197—84no—
135C4×C197—41no—
136C4×C197—42no—
137C4×C197#12712yes—
138C4×C197C22×D411yescharacteristic
139C4×C197C22×D411yescharacteristic
140Dic197—24no—
141Dic197—41no—
142C2×C197—22no—
143C2×C197—81no—
144C2×C197C2×#12711yescharacteristic
145C2×C197#57712yes—
146D197—81no—
147D197—82no—
148C197#315411yescharacteristic, complemented, Sylow
149#3154—1971nomaximal, Sylow
150C2×#137D19711yescharacteristic, complemented
151C2×AGL(1,ℤ/8)—1971no—
152C2×#135—1971no—
153C22×SD16—1971no—
154#536—1971no—
155#535—1971no—
156#497—1972no—
157#496—1972no—
158#495—1972no—
159#492—1972no—
160#405—1971no—
161C2×#41—1971no—
162C2×#41C2×D19711yescharacteristic
163C22×Q8C2×D19711yescharacteristic
164C22×D4—1971no—
165#137—24no—
166AGL(1,ℤ/8)—3944no—
167#135—3944no—
168C2×Q16—1971no—
169C2×Q16C2×D19712yescomplemented
170C2×SD16—3944no—
171C2×SD16—1972no—
172C2×SD16—1972no—
173C2×SD16—1971no—
174C2×SD16—1971no—
175C2×SD16—1972no—
176C2×SD16C2×D19712yescomplemented
177C2×D8—1971no—
178C2×D8—1972no—
179C2×M16C2×D19711yescharacteristic
180C22×C8—1971no—
181C4⋊Q8—1972no—
182#127—1972no—
183#124—1972no—
184#124—1972no—
185#123—3942no—
186C22⋊Q8—3942no—
187#121—3942no—
188#120—3942no—
189#117—1971no—
190#97—1972no—
191C4×C8—1972no—
192C24—3941no—
193#41—22no—
194#41—22no—
195#41—3942no—
196#41—3942no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—1971no—
201C2×Q8C22×D19711yescharacteristic
202C2×Q8C22×D19712yes—
203C2×D4—21no—
204C2×D4—3941no—
205C2×D4—3944no—
206C2×D4—3941no—
207C2×D4—3942no—
208C2×D4—3942no—
209C2×D4—1971no—
210C2×D4—1972no—
211C2×D4C22×D19711yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—3941no—
215C22×C4C22×D19711yescharacteristic
216Q16—24no—
217Q16—3942no—
218SD16—24no—
219SD16—3942no—
220SD16—3944no—
221SD16—3944no—
222SD16—3942no—
223SD16—3944no—
224D8—3942no—
225D8—3944no—
226M16C22⋊D19714yescomplemented
227C2×C8—1974no—
228C2×C8—1971no—
229C2×C8—1971no—
230C2×C8C22×D19712yes—
231C4⋊C4—3944no—
232C4⋊C4—1972no—
233C22⋊C4—3944no—
234C22⋊C4—3944no—
235C22⋊C4—1972no—
236C42—1972no—
237C23—21no—
238C23—7882no—
239C23—15762no—
240C23—3941no—
241C23—3942no—
242C23D4×D19711yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—3941no—
249Q8—3941no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—7882no—
254D4—7882no—
255D4—15764no—
256D4—3941no—
257D4—3942no—
258D4—3941no—
259D4—3942no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—7882no—
265C2×C4—3941no—
266C2×C4—1974no—
267C2×C4D4×D19711yescharacteristic
268C2×C4D4×D19712yes—
269C2×C4C2×C22⋊D19712yes—
270C2×C4C23×D19711yescharacteristic
271C8—1972no—
272C8—1972no—
273C8C2×C22⋊D19714yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—7881no—
278C22—7882no—
279C22—7884no—
280C22—7882no—
281C22—15764no—
282C22#2026332212yes—
283C22C2×D4×D19711yescharacteristic
284C4—41no—
285C4—42no—
286C4—7881no—
287C4—3944no—
288C4#2026332212yes—
289C4C2×D4×D19711yescharacteristic
290C4C22×C22⋊D19711yescharacteristic
291C2—22no—
292C2—81no—
293C2—15762no—
294C2—15761no—
295C2C2×#2026332211yescharacteristic
296C2#2716188812yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#137×C197C2
socleC22×C197C22×D4
hypercenterC2×#137D197
hypocenterC197#3154
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
1. C2×#20263354C2semidirect
2. C2×#20263346C2semidirect
3. C22×#15691345C2semidirect
4. #27161653C2semidirect
5. #27161643C2semidirect
6. #27161472C2semidirect
7. #27161465C2semidirect
8. #27161464C2semidirect
9. #27161450C2semidirect
10. #27161037C2semidirect
11. C2×#137×C197C2semidirect
16. C2×#15691348C22semidirect
18. C2×#15691346C22semidirect
22. C2×#15691345C22semidirect
24. C2×#15691343C22semidirect
26. #20263324C22semidirect
29. #20263315C22semidirect
35. #20263236C22semidirect
36. C8×Dic197C22semidirect
40. C2×Q16×C197C22semidirect
41. C2×SD16×C197C22semidirect
63. C2×#13096802D4semidirect
86. M16×C197D4semidirect
148. C197#3154semidirect
150. C2×#137D197semidirect
169. C2×Q16C2×D197semidirect
176. C2×SD16C2×D197semidirect
226. M16C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633542
C2×#202633462
C22×#156913452
#271616532
#271616432
#271614722
#271614652
#271614642
#271614502
#271610372
C2×#137×C1972
#3154197