Finite Group Databasebeta

← #38751937

Subgroups — #38751937

Order 25216.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×D4×D197C211yesmaximal, characteristic, complemented
2C2×#20263351C211yesmaximal, characteristic, complemented
3#27161586C211yesmaximal, characteristic, complemented
4#27161228C214yesmaximal, complemented
5#27161211C214yesmaximal, complemented
6#27161085C211yesmaximal, characteristic, complemented
7#27161084C211yesmaximal, characteristic, complemented
8#27161042C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C197C211yesmaximal, characteristic, complemented
10C24×D197—21no—
11C22×D788C2211yescharacteristic
12C22×C22⋊D197—21no—
13C2×D4×D197—24no—
14C2×D4×D197—22no—
15C2×D4×D197—24no—
16C2×D4×D197C2214yes—
17C22×C4×D197C2211yescharacteristic
18#20263351—24no—
19C2×#15691345C2212yescomplemented
20C2×#15691343C2212yescomplemented
21C2×#15691342C2211yescharacteristic, complemented
22#20263307—21no—
23#20263302C2212yescomplemented
24#20263297C2212yescomplemented
25#20263294—21no—
26C4×C22⋊D197—22no—
27#20263285C2211yescharacteristic, complemented
28#20263248C2214yescomplemented
29#20263247C2214yescomplemented
30#20263238C2214yescomplemented
31C2×#41×C197C2211yescharacteristic, complemented
32C22×D4×C197C2211yescharacteristic
33AGL(1,ℤ/8)×C197—24no—
34C2×SD16×C197C2212yescomplemented
35C2×D8×C197C2212yescomplemented
36C2×M16×C197C2211yescharacteristic, complemented
37C23×D197—28no—
38C23×D197—24no—
39C23×D197—21no—
40C23×D197—48no—
41C23×D197D411yescharacteristic
42C2×D788—21no—
43C2×D788—22no—
44C2×D788D414yescomplemented
45C2×D788C2312yes—
46C2×C22⋊D197—28no—
47C2×C22⋊D197—22no—
48C2×C22⋊D197—24no—
49D4×D197—216no—
50D4×D197—28no—
51D4×D197—28no—
52C22×Dic197D411yescharacteristic
53C2×C4×D197—22no—
54C2×C4×D197—22no—
55C2×C4×D197D412yes—
56C2×C4×D197C2312yes—
57#15691345—24no—
58#15691343—24no—
59#15691342—22no—
60C2×#13096802C2312yescomplemented
61C4⋊Dic197C2312yescomplemented
62#15691337—22no—
63C22⋊Dic197—22no—
64C22⋊Dic197—21no—
65#15691335—22no—
66C4×Dic197—21no—
67C24×C197—21no—
68#41×C197—22no—
69#41×C197—22no—
70C2×Q8×C197C2311yescharacteristic, complemented
71C2×D4×C197—21no—
72C2×D4×C197—21no—
73C2×D4×C197—22no—
74C2×D4×C197D414yescomplemented
75C2×D4×C197C2312yes—
76C2×D4×C197C2311yescharacteristic, complemented
77C22×C4×C197—21no—
78C22×C4×C197C2311yescharacteristic
79SD16×C197—24no—
80D8×C197—24no—
81M16×C197—22no—
82C2×C8×C197C2312yescomplemented
83C22×D197—22no—
84C22×D197—24no—
85C22×D197—216no—
86C22×D197—24no—
87C22×D197—22no—
88C22×D197—432no—
89C22×D197—48no—
90C22×D197—44no—
91C22×D197—48no—
92C22×D197C2×D412yes—
93D788—24no—
94D788—22no—
95D788C2×D414yes—
96C22⋊D197—216no—
97C22⋊D197—44no—
98C2×Dic197—22no—
99C2×Dic197—22no—
100C2×Dic197C2×D412yes—
101C4×D197—24no—
102C4×D197—22no—
103C4×D197—22no—
104#13096802—22no—
105C23×C197—24no—
106C23×C197—22no—
107C23×C197—21no—
108C23×C197—44no—
109C23×C197C2×D411yescharacteristic
110Q8×C197—21no—
111Q8×C197—21no—
112D4×C197—24no—
113D4×C197—22no—
114D4×C197—21no—
115D4×C197—21no—
116D4×C197—42no—
117D4×C197C2×D414yes—
118C2×C4×C197—21no—
119C2×C4×C197—42no—
120C2×C4×C197C2×D411yescharacteristic
121C2×C4×C197C2×D412yes—
122C2×C4×C197C2×D412yes—
123C2×C4×C197C2411yescharacteristic
124C8×C197—22no—
125C22×C197—22no—
126C22×C197—24no—
127C22×C197—48no—
128C22×C197—44no—
129C22×C197—41no—
130C22×C197—42no—
131C22×C197#12012yes—
132C22×C197C22×D411yescharacteristic
133C2×D197—28no—
134C2×D197—22no—
135C2×D197—22no—
136C2×D197—24no—
137C2×D197—432no—
138C2×D197—48no—
139C2×D197—44no—
140C2×D197AGL(1,ℤ/8)14yescomplemented
141C4×C197—41no—
142C4×C197#12012yes—
143C4×C197C22×D411yescharacteristic
144C4×C197C22×D411yescharacteristic
145Dic197—22no—
146Dic197—42no—
147C2×C197—22no—
148C2×C197—44no—
149C2×C197—81no—
150C2×C197C2×#12011yescharacteristic
151C2×C197C2×AGL(1,ℤ/8)12yes—
152D197—24no—
153D197—44no—
154C197#300311yescharacteristic, complemented, Sylow
155#3003—1971nomaximal, Sylow
156C23×D4—1971no—
157C2×AGL(1,ℤ/8)—1971no—
158C2×AGL(1,ℤ/8)D19711yescharacteristic, complemented
159#525—1971no—
160#450—1974no—
161#447—1974no—
162#418—1971no—
163#418—1971no—
164#407—1971no—
165C25—3941no—
166C2×#41C2×D19711yescharacteristic, complemented
167C22×D4—3944no—
168C22×D4—3944no—
169C22×D4—3942no—
170C22×D4—3941no—
171C22×D4—1974no—
172C22×D4—1971no—
173C22×D4C2×D19711yescharacteristic
174C23×C4—1971no—
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—3944no—
177C2×SD16—1972no—
178C2×SD16C2×D19712yescomplemented
179C2×D8—1972no—
180C2×D8C2×D19712yescomplemented
181C2×M16—1971no—
182C2×M16C2×D19711yescharacteristic, complemented
183#123—3941no—
184C22⋊Q8—1972no—
185#121—1972no—
186#120—3941no—
187C4×D4—3942no—
188#117—1971no—
189#102—1974no—
190#102—1974no—
191C22⋊C8—1974no—
192C24—21no—
193C24—7888no—
194C24—3948no—
195C24—3944no—
196C24—3941no—
197C24—1971no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D19711yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—39416no—
205C2×D4—3948no—
206C2×D4—3942no—
207C2×D4—3948no—
208C2×D4—3948no—
209C2×D4—3944no—
210C2×D4—3941no—
211C2×D4—3942no—
212C2×D4—1974no—
213C2×D4—1972no—
214C2×D4C22⋊D19714yescomplemented
215C2×D4C22×D19712yes—
216C2×D4C22×D19711yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—3942no—
219C22×C4—3942no—
220C22×C4—1971no—
221C22×C4—1972no—
222C22×C4—1972no—
223C22×C4C22×D19711yescharacteristic
224SD16—24no—
225SD16—3944no—
226D8—24no—
227D8—3944no—
228M16—22no—
229M16—3942no—
230C2×C8—1972no—
231C2×C8C22×D19712yescomplemented
232C4⋊C4—3942no—
233C4⋊C4—1972no—
234C22⋊C4—3942no—
235C22⋊C4—3941no—
236C22⋊C4—3942no—
237C42—3941no—
238C23—24no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—7884no—
243C23—78832no—
244C23—7888no—
245C23—7888no—
246C23—39416no—
247C23—3944no—
248C23—3944no—
249C23—3942no—
250C23—3942no—
251C23—1972no—
252C23D4×D19711yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—24no—
256D4—22no—
257D4—21no—
258D4—21no—
259D4—42no—
260D4—7884no—
261D4—39416no—
262D4—3944no—
263D4—3942no—
264D4—1974no—
265D4C2×C22⋊D19714yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—3942no—
269C2×C4—3942no—
270C2×C4—3944no—
271C2×C4—3942no—
272C2×C4—3942no—
273C2×C4—1972no—
274C2×C4D4×D19711yescharacteristic
275C2×C4D4×D19712yes—
276C2×C4C2×C22⋊D19712yes—
277C2×C4C23×D19711yescharacteristic
278C8—22no—
279C8—3942no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—42no—
285C22—41no—
286C22—7888no—
287C22—78832no—
288C22—7884no—
289C22—3948no—
290C22—3944no—
291C22—3942no—
292C22—3942no—
293C22—1974no—
294C22#2026329412yes—
295C22C2×D4×D19711yescharacteristic
296C4—41no—
297C4—7882no—
298C4—3942no—
299C4#2026329412yes—
300C4C2×D4×D19711yescharacteristic
301C4C22×C22⋊D19711yescharacteristic
302C2—22no—
303C2—44no—
304C2—81no—
305C2—7884no—
306C2—3944no—
307C2C2×#2026329411yescharacteristic
308C2AGL(1,ℤ/8)×D19712yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×AGL(1,ℤ/8)×C197C2
socleC22×C197C22×D4
hypercenterC2×AGL(1,ℤ/8)D197
hypocenterC197#3003
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×D4×D197C2semidirect
2. C2×#20263351C2semidirect
3. #27161586C2semidirect
4. #27161228C2semidirect
5. #27161211C2semidirect
6. #27161085C2semidirect
7. #27161084C2semidirect
8. #27161042C2semidirect
9. C2×AGL(1,ℤ/8)×C197C2semidirect
19. C2×#15691345C22semidirect
20. C2×#15691343C22semidirect
21. C2×#15691342C22semidirect
23. #20263302C22semidirect
24. #20263297C22semidirect
27. #20263285C22semidirect
28. #20263248C22semidirect
29. #20263247C22semidirect
30. #20263238C22semidirect
31. C2×#41×C197C22semidirect
34. C2×SD16×C197C22semidirect
35. C2×D8×C197C22semidirect
36. C2×M16×C197C22semidirect
44. C2×D788D4semidirect
60. C2×#13096802C23semidirect
61. C4⋊Dic197C23semidirect
70. C2×Q8×C197C23semidirect
74. C2×D4×C197D4semidirect
76. C2×D4×C197C23semidirect
82. C2×C8×C197C23semidirect
140. C2×D197AGL(1,ℤ/8)semidirect
154. C197#3003semidirect
158. C2×AGL(1,ℤ/8)D197semidirect
166. C2×#41C2×D197semidirect
178. C2×SD16C2×D197semidirect
180. C2×D8C2×D197semidirect
182. C2×M16C2×D197semidirect
200. C2×Q8C22×D197semidirect
214. C2×D4C22⋊D197semidirect
216. C2×D4C22×D197semidirect
231. C2×C8C22×D197semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1972
C2×#202633512
#271615862
#271612282
#271612112
#271610852
#271610842
#271610422
C2×AGL(1,ℤ/8)×C1972
#3003197