Finite Group Databasebeta

← #38753195

Subgroups — #38753195

Order 25216.

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

normal subgroups only

This group has 596 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×#20263352C211yesmaximal, characteristic, complemented
2C2×#20263351C211yesmaximal, characteristic, complemented
3C2×#20263346C211yesmaximal, characteristic, complemented
4C22×#15691343C211yesmaximal, characteristic, complemented
5#27161648C211yesmaximal, characteristic, complemented
6#27161469C212yesmaximal, complemented
7#27161466C212yesmaximal, complemented
8#27161461C212yesmaximal, complemented
9#27161455C212yesmaximal, complemented
10#27161035C211yesmaximal, characteristic, complemented
11#535×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2211yescharacteristic
13C22×D788C2211yescharacteristic
14#20263352—24no—
15#20263351—24no—
16#20263346—24no—
17C2×#15691348C2211yescharacteristic, complemented
18C2×#15691346C2211yescharacteristic, complemented
19C2×#15691346C2212yescomplemented
20C2×#15691345C2211yescharacteristic, complemented
21C2×#15691345C2212yescomplemented
22C2×#15691343—24no—
23C2×#15691343C2212yes—
24C2×#15691343C2212yes—
25C2×#15691343C2211yescharacteristic, complemented
26C2×#15691343C2212yescomplemented
27C2×#15691343C2212yescomplemented
28C2×#15691342C2211yescharacteristic
29C22×#13096802C2211yescharacteristic
30#20263323C2212yescomplemented
31#20263314C2212yescomplemented
32#20263301—22no—
33#20263296—22no—
34#20263237C2212yescomplemented
35C4×#13096802C2212yescomplemented
36C2×#41×C197C2211yescharacteristic
37C22×D4×C197C2211yescharacteristic
38#127×C197C2212yescomplemented
39#124×C197C2212yescomplemented
40#121×C197—22no—
41#120×C197—22no—
42#117×C197C2211yescharacteristic
43C23×D197—21no—
44#15691358—22no—
45#15691358—22no—
46C2×Dic394C2311yescharacteristic
47C2×D788—22no—
48C2×D788—22no—
49C2×D788—21no—
50C2×D788—24no—
51C2×D788C2312yes—
52C2×D788C2311yescharacteristic
53C2×C22⋊D197—21no—
54C2×C4×D197—21no—
55#15691348—22no—
56#15691346—22no—
57#15691346—24no—
58#15691345—22no—
59#15691345—24no—
60#15691343—24no—
61#15691343—24no—
62#15691343—22no—
63#15691343—24no—
64#15691343—24no—
65#15691342D414yescomplemented
66C2×#13096802D414yescomplemented
67C2×#13096802C2311yescharacteristic
68C2×#13096802C2312yes—
69C2×#13096802C2311yescharacteristic
70#15691335—24no—
71C24×C197—21no—
72#41×C197—22no—
73#41×C197—22no—
74C2×Q8×C197C2311yescharacteristic
75C2×D4×C197—21no—
76C2×D4×C197—22no—
77C2×D4×C197—22no—
78C2×D4×C197—21no—
79C2×D4×C197—24no—
80C2×D4×C197C2312yes—
81C2×D4×C197C2311yescharacteristic
82C22×C4×C197—21no—
83C22×C4×C197C2311yescharacteristic
84C4⋊C4×C197D412yes—
85C22⋊C4×C197—24no—
86C22⋊C4×C197D412yes—
87C42×C197C2312yes—
88C22×D197—22no—
89C22×D197—21no—
90C22×D197—42no—
91C22×D197—82no—
92Dic394—21no—
93Dic394—21no—
94D788—22no—
95D788—21no—
96D788—22no—
97D788—21no—
98D788—42no—
99D788—84no—
100C22⋊D197—42no—
101C2×Dic197—21no—
102C4×D197—42no—
103#13096802C2×D412yes—
104#13096802C2×D412yes—
105#13096802C2×D414yes—
106C23×C197—22no—
107C23×C197—21no—
108C23×C197—42no—
109C23×C197—82no—
110C23×C197C2×D411yescharacteristic
111Q8×C197—21no—
112Q8×C197—21no—
113D4×C197—22no—
114D4×C197—22no—
115D4×C197—21no—
116D4×C197—21no—
117D4×C197—42no—
118D4×C197—42no—
119D4×C197—84no—
120C2×C4×C197—21no—
121C2×C4×C197—42no—
122C2×C4×C197C2×D412yes—
123C2×C4×C197C2×D412yes—
124C2×C4×C197C2×D414yes—
125C2×C4×C197C2×D411yescharacteristic
126C2×C4×C197C2411yescharacteristic
127C22×C197—22no—
128C22×C197—41no—
129C22×C197—42no—
130C22×C197—42no—
131C22×C197—44no—
132C22×C197—84no—
133C22×C197#12712yes—
134C22×C197C22×D411yescharacteristic
135C2×D197—41no—
136C2×D197—42no—
137C2×D197—42no—
138C2×D197—44no—
139C2×D197—84no—
140C4×C197—24no—
141C4×C197—41no—
142C4×C197#12712yes—
143C4×C197C22×D411yescharacteristic
144C4×C197C22×D411yescharacteristic
145Dic197—41no—
146C2×C197—22no—
147C2×C197—81no—
148C2×C197—82no—
149C2×C197C2×#12711yescharacteristic
150C2×C197#57612yes—
151D197—81no—
152D197—82no—
153C197#315511yescharacteristic, complemented, Sylow
154#3155—1971nomaximal, Sylow
155C2×AGL(1,ℤ/8)—1971no—
156C2×AGL(1,ℤ/8)—1971no—
157C2×#135—1971no—
158C22×D8—1971no—
159#535—1971no—
160#535D19711yescharacteristic, complemented
161#496—1972no—
162#496—1972no—
163#495—1972no—
164#493—1972no—
165#405—1971no—
166C2×#41—1971no—
167C2×#41C2×D19711yescharacteristic
168C22×D4—1971no—
169C22×D4C2×D19711yescharacteristic
170AGL(1,ℤ/8)—3944no—
171AGL(1,ℤ/8)—3944no—
172#135—3944no—
173C2×Q16—1971no—
174C2×SD16—1971no—
175C2×SD16—1972no—
176C2×SD16—1971no—
177C2×SD16—1972no—
178C2×D8—3944no—
179C2×D8—1972no—
180C2×D8—1972no—
181C2×D8—1971no—
182C2×D8—1972no—
183C2×D8—1972no—
184C2×M16—1971no—
185C22×C8—1971no—
186#127—1972no—
187#127C2×D19712yescomplemented
188#124—1972no—
189#124C2×D19712yescomplemented
190#121—22no—
191#121—3942no—
192#120—22no—
193#120—3942no—
194#117C2×D19711yescharacteristic
195#97—1972no—
196C4×C8—1972no—
197C24—21no—
198C24—3941no—
199#41—22no—
200#41—22no—
201#41—3942no—
202#41—3942no—
203C2×Q8—1971no—
204C2×Q8C22×D19711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—24no—
210C2×D4—3941no—
211C2×D4—3942no—
212C2×D4—3942no—
213C2×D4—3941no—
214C2×D4—3944no—
215C2×D4—1972no—
216C2×D4—1971no—
217C2×D4C22×D19712yes—
218C2×D4C22×D19711yescharacteristic
219C22×C4—21no—
220C22×C4—3941no—
221C22×C4C22×D19711yescharacteristic
222Q16—3942no—
223SD16—3942no—
224SD16—3944no—
225SD16—3942no—
226SD16—3944no—
227D8—3944no—
228D8—3944no—
229D8—3942no—
230D8—3944no—
231D8—3944no—
232M16—1974no—
233C2×C8—1974no—
234C2×C8—1971no—
235C2×C8—1972no—
236C2×C8—1971no—
237C4⋊C4C22⋊D19712yes—
238C22⋊C4—24no—
239C22⋊C4—3944no—
240C22⋊C4C22⋊D19712yes—
241C42C22×D19712yes—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—7882no—
247C23—15762no—
248C23—3942no—
249C23—3941no—
250C23C2×C22⋊D19711yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—3941no—
254Q8—3941no—
255D4—22no—
256D4—21no—
257D4—22no—
258D4—21no—
259D4—42no—
260D4—42no—
261D4—84no—
262D4—7882no—
263D4—7882no—
264D4—15764no—
265D4—3942no—
266D4—3941no—
267D4—3942no—
268D4—3941no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—7882no—
272C2×C4—3941no—
273C2×C4D4×D19712yes—
274C2×C4D4×D19712yes—
275C2×C4C2×C22⋊D19714yes—
276C2×C4C2×C22⋊D19711yescharacteristic
277C2×C4C23×D19711yescharacteristic
278C8—1972no—
279C8—1972no—
280C8—1974no—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—7881no—
288C22—7882no—
289C22—7882no—
290C22—7884no—
291C22—15764no—
292C22#2026332212yes—
293C22C22×C22⋊D19711yescharacteristic
294C4—24no—
295C4—41no—
296C4—7881no—
297C4#2026332212yes—
298C4C2×D4×D19711yescharacteristic
299C4C2×D4×D19711yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—15762no—
304C2—15761no—
305C2C2×#2026332211yescharacteristic
306C2#2716188212yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#535×C197C2
socleC22×C197C22×D4
hypercenter#535D197
hypocenterC197#3155
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#20263352C2semidirect
2. C2×#20263351C2semidirect
3. C2×#20263346C2semidirect
4. C22×#15691343C2semidirect
5. #27161648C2semidirect
6. #27161469C2semidirect
7. #27161466C2semidirect
8. #27161461C2semidirect
9. #27161455C2semidirect
10. #27161035C2semidirect
11. #535×C197C2semidirect
17. C2×#15691348C22semidirect
18. C2×#15691346C22semidirect
19. C2×#15691346C22semidirect
20. C2×#15691345C22semidirect
21. C2×#15691345C22semidirect
25. C2×#15691343C22semidirect
26. C2×#15691343C22semidirect
27. C2×#15691343C22semidirect
30. #20263323C22semidirect
31. #20263314C22semidirect
34. #20263237C22semidirect
35. C4×#13096802C22semidirect
38. #127×C197C22semidirect
39. #124×C197C22semidirect
65. #15691342D4semidirect
66. C2×#13096802D4semidirect
153. C197#3155semidirect
160. #535D197semidirect
187. #127C2×D197semidirect
189. #124C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633522
C2×#202633512
C2×#202633462
C22×#156913432
#271616482
#271614692
#271614662
#271614612
#271614552
#271610352
#535×C1972
#3155197