Finite Group Databasebeta

← #38752202

Subgroups — #38752202

Order 25216.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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×#20263357C211yesmaximal, characteristic, complemented
2C2×#20263352C211yesmaximal, characteristic
3C4×#15691358C211yesmaximal, characteristic, complemented
4#27161298C214yesmaximal, complemented
5#27161291C214yesmaximal, complemented
6#27161103C211yesmaximal, characteristic, complemented
7#27161088C211yesmaximal, characteristic, complemented
8#27161084C211yesmaximal, characteristic, complemented
9C2×#124×C197C211yesmaximal, characteristic, complemented
10C2×#15691358C2211yescharacteristic
11#20263357—24no—
12#20263352—24no—
13C2×#15691348C2212yescomplemented
14C2×#15691346C2212yes—
15C2×#15691345C2212yescomplemented
16C2×#15691343C2212yes—
17C2×#15691342C2211yescharacteristic
18C2×#15691342C2211yescharacteristic
19C4×Dic394C2212yescomplemented
20C4×D788C2212yescomplemented
21C4×C22⋊D197—22no—
22#20263285C2211yescharacteristic
23#20263283—21no—
24C42×D197—21no—
25C4⋊#13096802C2214yescomplemented
26#20263250C2214yescomplemented
27#20263247C2214yes—
28C22×Q8×C197C2211yescharacteristic
29C22×D4×C197C2211yescharacteristic
30#124×C197—21no—
31#124×C197—21no—
32#124×C197C2214yes—
33C2×C22⋊C4×C197—22no—
34C2×C42×C197C2211yescharacteristic
35#15691358—22no—
36#15691358D414yes—
37C2×Dic394C2311yescharacteristic
38C2×D788C2311yescharacteristic
39C2×C22⋊D197—21no—
40C2×C4×D197—21no—
41C2×C4×D197—22no—
42#15691348—24no—
43#15691346—24no—
44#15691345—24no—
45#15691343—24no—
46#15691342—22no—
47#15691342—22no—
48C2×#13096802C2312yes—
49C2×#13096802C2312yes—
50C4⋊Dic197C2312yes—
51#15691337—22no—
52C22⋊Dic197—21no—
53#15691335—22no—
54C4×Dic197—21no—
55C4×Dic197—22no—
56C24×C197—21no—
57C2×Q8×C197—21no—
58C2×Q8×C197—22no—
59C2×Q8×C197—22no—
60C2×Q8×C197C2312yes—
61C2×D4×C197—22no—
62C2×D4×C197—21no—
63C2×D4×C197—22no—
64C2×D4×C197C2312yes—
65C22×C4×C197—21no—
66C22×C4×C197D412yes—
67C22×C4×C197C2311yescharacteristic
68C22⋊C4×C197—24no—
69C22⋊C4×C197—24no—
70C42×C197D411yescharacteristic
71C42×C197D411yescharacteristic
72C42×C197C2312yes—
73C22×D197—21no—
74Dic394—21no—
75Dic394C2×D412yes—
76D788—21no—
77D788C2×D412yes—
78C22⋊D197—24no—
79C2×Dic197—21no—
80C2×Dic197—22no—
81C4×D197—24no—
82C4×D197—44no—
83#13096802—22no—
84#13096802—22no—
85C23×C197—22no—
86C23×C197—44no—
87C23×C197—42no—
88C23×C197C2×D411yescharacteristic
89Q8×C197—22no—
90Q8×C197—22no—
91Q8×C197—42no—
92D4×C197—22no—
93D4×C197—22no—
94D4×C197—42no—
95C2×C4×C197—22no—
96C2×C4×C197—42no—
97C2×C4×C197—42no—
98C2×C4×C197C2×D412yes—
99C2×C4×C197C2×D412yes—
100C2×C4×C197C2×D412yes—
101C2×C4×C197C2×D411yescharacteristic
102C2×C4×C197#4112yes—
103C2×C4×C197C2411yescharacteristic
104C22×C197—22no—
105C22×C197—42no—
106C22×C197—44no—
107C22×C197—84no—
108C22×C197#12112yes—
109C22×C197C22×D411yescharacteristic
110C2×D197—22no—
111C2×D197—42no—
112C4×C197—22no—
113C4×C197—22no—
114C4×C197—42no—
115C4×C197#12112yes—
116C4×C197C22×D411yescharacteristic
117C4×C197C2×#4111yescharacteristic
118Dic197—22no—
119Dic197—42no—
120C2×C197—22no—
121C2×C197—82no—
122C2×C197C2×#12111yescharacteristic
123C2×C197#57512yes—
124D197—42no—
125C197#304011yescharacteristic, complemented, Sylow
126#3040—1971nomaximal, Sylow
127C2×#137—1971no—
128C2×AGL(1,ℤ/8)—1971no—
129C2×#124D19711yescharacteristic, complemented
130C4×#41—1971no—
131#464—1974no—
132#463—1974no—
133#423—1971no—
134#419—1971no—
135#418—1971no—
136C2×#41—1971no—
137C22×Q8C2×D19711yescharacteristic
138C22×D4C2×D19711yescharacteristic
139#137—3944no—
140AGL(1,ℤ/8)—3944no—
141C2×Q16—1972no—
142C2×SD16—1972no—
143C2×SD16—1972no—
144C2×D8—1972no—
145C2×M16—1971no—
146C2×M16—1971no—
147#124—21no—
148#124—21no—
149#124C2×D19714yes—
150C4×Q8—1972no—
151C4×D4—3942no—
152C4×D4—1972no—
153#117—3941no—
154#117—1971no—
155C2×C22⋊C4—22no—
156C2×C42—3941no—
157C2×C42C2×D19711yescharacteristic
158C4⋊C8—1974no—
159#103—1974no—
160#102—1974no—
161C24—21no—
162#41—3942no—
163#41—1974no—
164C2×Q8—21no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—1971no—
168C2×Q8C22×D19712yes—
169C2×D4—22no—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—3941no—
173C2×D4—1971no—
174C2×D4C22×D19712yes—
175C22×C4—21no—
176C22×C4—3942no—
177C22×C4—3941no—
178C22×C4C22⋊D19712yes—
179C22×C4C22×D19711yescharacteristic
180Q16—3944no—
181SD16—3944no—
182SD16—3944no—
183D8—3944no—
184M16—3942no—
185M16—3942no—
186C2×C8—1972no—
187C2×C8—1972no—
188C4⋊C4—3942no—
189C4⋊C4—1972no—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—3941no—
193C22⋊C4—3942no—
194C42—3941no—
195C42—3942no—
196C42C22⋊D19711yescharacteristic
197C42C22⋊D19711yescharacteristic
198C42C22×D19712yes—
199C23—22no—
200C23—44no—
201C23—42no—
202C23—3941no—
203C23C2×C22⋊D19711yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—42no—
207Q8—3941no—
208Q8—1972no—
209D4—22no—
210D4—22no—
211D4—42no—
212D4—3944no—
213D4—3941no—
214D4—1972no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—7884no—
219C2×C4—3942no—
220C2×C4—3941no—
221C2×C4—3944no—
222C2×C4D4×D19712yes—
223C2×C4C2×C22⋊D19712yes—
224C2×C4C2×C22⋊D19712yes—
225C2×C4C2×C22⋊D19711yescharacteristic
226C2×C4#1569135712yes—
227C2×C4C23×D19711yescharacteristic
228C8—3942no—
229C8—3942no—
230C22—22no—
231C22—44no—
232C22—42no—
233C22—84no—
234C22—7882no—
235C22—3942no—
236C22#2026329712yes—
237C22C22×C22⋊D19711yescharacteristic
238C4—22no—
239C4—22no—
240C4—42no—
241C4—7882no—
242C4—3942no—
243C4#2026329712yes—
244C4C2×D4×D19711yescharacteristic
245C4C2×#1569135711yescharacteristic
246C2—22no—
247C2—82no—
248C2—7882no—
249C2C2×#2026329711yescharacteristic
250C2#2716187712yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#124×C197C2
socleC22×C197C22×D4
prime radicalC2×#20263352C2
hypercenterC2×#124D197
hypocenterC197#3040
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#20263357C2semidirect
3. C4×#15691358C2semidirect
4. #27161298C2semidirect
5. #27161291C2semidirect
6. #27161103C2semidirect
7. #27161088C2semidirect
8. #27161084C2semidirect
9. C2×#124×C197C2semidirect
13. C2×#15691348C22semidirect
15. C2×#15691345C22semidirect
19. C4×Dic394C22semidirect
20. C4×D788C22semidirect
25. C4⋊#13096802C22semidirect
26. #20263250C22semidirect
125. C197#3040semidirect
129. C2×#124D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633572
C2×#202633522
C4×#156913582
#271612982
#271612912
#271611032
#271610882
#271610842
C2×#124×C1972
#3040197