Finite Group Databasebeta

← #38753184

Subgroups — #38753184

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×#20263356C211yesmaximal, characteristic, complemented
2C2×#20263352C211yesmaximal, characteristic, complemented
3C2×#20263346C211yesmaximal, characteristic, complemented
4C22×#15691346C211yesmaximal, characteristic, complemented
5#27161655C211yesmaximal, characteristic, complemented
6#27161474C212yesmaximal, complemented
7#27161466C212yesmaximal, complemented
8#27161461C212yesmaximal, complemented
9#27161451C212yesmaximal, complemented
10#27161035C211yesmaximal, characteristic, complemented
11#535×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2211yescharacteristic
13C22×Dic394C2211yescharacteristic
14#20263356—24no—
15#20263352—24no—
16#20263346—24no—
17C2×#15691348C2211yescharacteristic, complemented
18C2×#15691348C2212yescomplemented
19C2×#15691346—24no—
20C2×#15691346C2212yes—
21C2×#15691346C2212yes—
22C2×#15691346C2212yes—
23C2×#15691346C2211yescharacteristic, complemented
24C2×#15691346C2212yescomplemented
25C2×#15691345C2211yescharacteristic, complemented
26C2×#15691343C2212yes—
27C2×#15691343C2211yescharacteristic, complemented
28C2×#15691342C2211yescharacteristic
29C22×#13096802C2211yescharacteristic
30#20263326C2212yescomplemented
31#20263314C2212yescomplemented
32#20263310—22no—
33#20263306—22no—
34#20263237C2212yescomplemented
35C4×#13096802C2212yescomplemented
36C2×#41×C197C2211yescharacteristic
37C22×D4×C197C2211yescharacteristic
38#127×C197C2212yes—
39#124×C197C2212yes—
40#121×C197—22no—
41#120×C197—22no—
42#117×C197C2211yescharacteristic
43#15691358—22no—
44#15691358—22no—
45C2×Dic394—21no—
46C2×Dic394—22no—
47C2×Dic394—22no—
48C2×Dic394C2312yes—
49C2×Dic394C2311yescharacteristic
50C2×D788C2311yescharacteristic
51C2×C22⋊D197—21no—
52C22×Dic197—21no—
53C2×C4×D197—21no—
54#15691348—22no—
55#15691348—24no—
56#15691346—24no—
57#15691346—24no—
58#15691346—24no—
59#15691346—22no—
60#15691346—24no—
61#15691345—22no—
62#15691343—24no—
63#15691343—22no—
64#15691342D414yescomplemented
65C2×#13096802D414yescomplemented
66C2×#13096802C2311yescharacteristic
67C2×#13096802C2312yes—
68C2×#13096802C2311yescharacteristic
69C4⋊Dic197—24no—
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—21no—
89Dic394—22no—
90Dic394—21no—
91Dic394—22no—
92Dic394—21no—
93Dic394—42no—
94D788—21no—
95D788—21no—
96C22⋊D197—42no—
97C2×Dic197—22no—
98C2×Dic197—21no—
99C2×Dic197—42no—
100C4×D197—42no—
101#13096802C2×D412yes—
102#13096802C2×D412yes—
103#13096802C2×D414yes—
104C23×C197—22no—
105C23×C197—21no—
106C23×C197—42no—
107C23×C197—82no—
108C23×C197C2×D411yescharacteristic
109Q8×C197—21no—
110Q8×C197—21no—
111D4×C197—22no—
112D4×C197—22no—
113D4×C197—21no—
114D4×C197—21no—
115D4×C197—42no—
116D4×C197—42no—
117D4×C197—84no—
118C2×C4×C197—21no—
119C2×C4×C197—42no—
120C2×C4×C197C2×D412yes—
121C2×C4×C197C2×D412yes—
122C2×C4×C197C2×D414yes—
123C2×C4×C197C2×D411yescharacteristic
124C2×C4×C197C2411yescharacteristic
125C22×C197—22no—
126C22×C197—41no—
127C22×C197—42no—
128C22×C197—42no—
129C22×C197—44no—
130C22×C197—84no—
131C22×C197#12712yes—
132C22×C197C22×D411yescharacteristic
133C2×D197—41no—
134C2×D197—42no—
135C4×C197—24no—
136C4×C197—41no—
137C4×C197#12712yes—
138C4×C197C22×D411yescharacteristic
139C4×C197C22×D411yescharacteristic
140Dic197—41no—
141Dic197—42no—
142C2×C197—22no—
143C2×C197—81no—
144C2×C197—82no—
145C2×C197C2×#12711yescharacteristic
146C2×C197#57712yes—
147D197—81no—
148C197#315411yescharacteristic, complemented, Sylow
149#3154—1971nomaximal, Sylow
150C2×#137—1971no—
151C2×AGL(1,ℤ/8)—1971no—
152C2×#135—1971no—
153C22×SD16—1971no—
154#536—1971no—
155#535D19711yescharacteristic, complemented
156#497—1972no—
157#496—1972no—
158#495—1972no—
159#492—1972no—
160#405—1971no—
161C2×#41—1971no—
162C2×#41C2×D19711yescharacteristic
163C22×Q8—1971no—
164C22×D4C2×D19711yescharacteristic
165#137—3944no—
166AGL(1,ℤ/8)—3944no—
167#135—3944no—
168C2×Q16—1971no—
169C2×Q16—1972no—
170C2×SD16—3944no—
171C2×SD16—1971no—
172C2×SD16—1972no—
173C2×SD16—1972no—
174C2×SD16—1971no—
175C2×SD16—1972no—
176C2×SD16—1972no—
177C2×D8—1971no—
178C2×D8—1972no—
179C2×M16—1971no—
180C22×C8—1971no—
181C4⋊Q8—1972no—
182#127C2×D19712yes—
183#124—1972no—
184#124C2×D19712yes—
185#123—3942no—
186C22⋊Q8—3942no—
187#121—22no—
188#120—22no—
189#117C2×D19711yescharacteristic
190#97—1972no—
191C4×C8—1972no—
192C24—21no—
193#41—22no—
194#41—22no—
195#41—3942no—
196#41—3942no—
197C2×Q8—3942no—
198C2×Q8—3942no—
199C2×Q8—3941no—
200C2×Q8—1972no—
201C2×Q8—1971no—
202C2×Q8C22×D19711yescharacteristic
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—24no—
208C2×D4—3941no—
209C2×D4—1971no—
210C2×D4C22×D19712yes—
211C2×D4C22×D19711yescharacteristic
212C22×C4—21no—
213C22×C4—3941no—
214C22×C4—3941no—
215C22×C4C22×D19711yescharacteristic
216Q16—3942no—
217Q16—3944no—
218SD16—3944no—
219SD16—3944no—
220SD16—3942no—
221SD16—3944no—
222SD16—3944no—
223SD16—3942no—
224D8—3942no—
225D8—3944no—
226M16—1974no—
227C2×C8—1974no—
228C2×C8—1971no—
229C2×C8—1972no—
230C2×C8—1971no—
231C4⋊C4—3944no—
232C4⋊C4C22⋊D19712yes—
233C22⋊C4—24no—
234C22⋊C4—3944no—
235C22⋊C4C22⋊D19712yes—
236C42C22×D19712yes—
237C23—22no—
238C23—21no—
239C23—42no—
240C23—82no—
241C23—3941no—
242C23C2×C22⋊D19711yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—7882no—
246Q8—3942no—
247Q8—3941no—
248Q8—3942no—
249Q8—3941no—
250D4—22no—
251D4—21no—
252D4—22no—
253D4—21no—
254D4—42no—
255D4—42no—
256D4—84no—
257D4—7882no—
258D4—3941no—
259D4—3941no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—7882no—
263C2×C4—7882no—
264C2×C4—3942no—
265C2×C4—3941no—
266C2×C4D4×D19712yes—
267C2×C4D4×D19712yes—
268C2×C4C2×C22⋊D19714yes—
269C2×C4C2×C22⋊D19711yescharacteristic
270C2×C4C23×D19711yescharacteristic
271C8—1972no—
272C8—1972no—
273C8—1974no—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—44no—
279C22—84no—
280C22—7881no—
281C22—7882no—
282C22#2026332212yes—
283C22C22×C22⋊D19711yescharacteristic
284C4—24no—
285C4—41no—
286C4—7881no—
287C4—7882no—
288C4#2026332212yes—
289C4C2×D4×D19711yescharacteristic
290C4C2×D4×D19711yescharacteristic
291C2—22no—
292C2—82no—
293C2—81no—
294C2—15761no—
295C2C2×#2026332211yescharacteristic
296C2#2716189012yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#535×C197C2
socleC22×C197C22×D4
hypercenter#535D197
hypocenterC197#3154
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×#20263356C2semidirect
2. C2×#20263352C2semidirect
3. C2×#20263346C2semidirect
4. C22×#15691346C2semidirect
5. #27161655C2semidirect
6. #27161474C2semidirect
7. #27161466C2semidirect
8. #27161461C2semidirect
9. #27161451C2semidirect
10. #27161035C2semidirect
11. #535×C197C2semidirect
17. C2×#15691348C22semidirect
18. C2×#15691348C22semidirect
23. C2×#15691346C22semidirect
24. C2×#15691346C22semidirect
25. C2×#15691345C22semidirect
27. C2×#15691343C22semidirect
30. #20263326C22semidirect
31. #20263314C22semidirect
34. #20263237C22semidirect
35. C4×#13096802C22semidirect
64. #15691342D4semidirect
65. C2×#13096802D4semidirect
148. C197#3154semidirect
155. #535D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633562
C2×#202633522
C2×#202633462
C22×#156913462
#271616552
#271614742
#271614662
#271614612
#271614512
#271610352
#535×C1972
#3154197