Finite Group Databasebeta

← #38751939

Subgroups — #38751939

Order 25216.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 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×#20263352C212yesmaximal, complemented
2#27161590C211yesmaximal, characteristic, complemented
3#27161230C214yesmaximal, complemented
4#27161213C214yesmaximal, complemented
5#27161084C212yesmaximal, complemented
6#27161043C211yesmaximal, characteristic, complemented
7C23×D4×C197C211yesmaximal, characteristic, complemented
8C2×#15691358C2211yescharacteristic, complemented
9#20263352—28no—
10C2×#15691346C2214yescomplemented
11C2×#15691343C2214yescomplemented
12C2×#15691342C2212yescomplemented
13#20263309—21no—
14#20263305C2212yescomplemented
15#20263300C2212yescomplemented
16#20263295—21no—
17C4×C22⋊D197—22no—
18#20263285C2211yescharacteristic, complemented
19#20263247C2218yescomplemented
20C22⋊#13096802C2214yescomplemented
21C25×C197—21no—
22C22×D4×C197—28no—
23C22×D4×C197—22no—
24C22×D4×C197—21no—
25C22×D4×C197C2214yes—
26C22×D4×C197C2212yes—
27C23×C4×C197C2211yescharacteristic
28#15691358—24no—
29C2×Dic394C2311yescharacteristic, complemented
30C2×D788C2311yescharacteristic, complemented
31C2×C22⋊D197—22no—
32C2×C22⋊D197—21no—
33C2×C4×D197—21no—
34#15691346—28no—
35#15691343—28no—
36#15691342—24no—
37C2×#13096802C2314yescomplemented
38C4⋊Dic197C2312yescomplemented
39#15691337—22no—
40C22⋊Dic197—21no—
41C22⋊Dic197—22no—
42#15691335—22no—
43C4×Dic197—21no—
44C24×C197—28no—
45C24×C197—24no—
46C24×C197—22no—
47C24×C197—48no—
48C24×C197D411yescharacteristic
49C2×D4×C197—28no—
50C2×D4×C197—216no—
51C2×D4×C197—216no—
52C2×D4×C197—24no—
53C2×D4×C197—22no—
54C2×D4×C197—24no—
55C2×D4×C197D418yescomplemented
56C2×D4×C197C2314yes—
57C22×C4×C197—24no—
58C22×C4×C197D412yes—
59C22×C4×C197D411yescharacteristic
60C22×C4×C197C2312yes—
61C22×C4×C197C2311yescharacteristic
62C22×D197—21no—
63Dic394—22no—
64D788—22no—
65C22⋊D197—42no—
66C22⋊D197—44no—
67C2×Dic197—22no—
68C2×Dic197—21no—
69C4×D197—42no—
70#13096802—24no—
71C23×C197—22no—
72C23×C197—24no—
73C23×C197—216no—
74C23×C197—28no—
75C23×C197—24no—
76C23×C197—432no—
77C23×C197—48no—
78C23×C197—48no—
79C23×C197—48no—
80C23×C197C2×D411yescharacteristic
81C23×C197C2×D412yes—
82D4×C197—216no—
83D4×C197—28no—
84D4×C197—24no—
85D4×C197C2×D418yes—
86C2×C4×C197—22no—
87C2×C4×C197—24no—
88C2×C4×C197—24no—
89C2×C4×C197C2×D412yes—
90C2×C4×C197C2×D411yescharacteristic
91C2×C4×C197C2×D414yes—
92C2×C4×C197C2411yescharacteristic
93C22×C197—28no—
94C22×C197—22no—
95C22×C197—22no—
96C22×C197—22no—
97C22×C197—28no—
98C22×C197—416no—
99C22×C197—432no—
100C22×C197—48no—
101C22×C197#12012yes—
102C22×C197AGL(1,ℤ/8)14yescomplemented
103C22×C197C22×D411yescharacteristic
104C2×D197—41no—
105C2×D197—42no—
106C4×C197—22no—
107C4×C197#12012yes—
108C4×C197C22×D412yes—
109Dic197—41no—
110Dic197—42no—
111C2×C197—22no—
112C2×C197—24no—
113C2×C197—48no—
114C2×C197C2×#12011yescharacteristic
115C2×C197C2×AGL(1,ℤ/8)12yes—
116D197—81no—
117C197#300311yescharacteristic, complemented, Sylow
118#3003—1971nomaximal, Sylow
119C23×D4D19711yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—1972no—
121#525—1971no—
122#450—1974no—
123#447—1974no—
124#418—1972no—
125#407—1971no—
126C25—21no—
127C2×#41—1971no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D19714yes—
132C22×D4C2×D19712yes—
133C23×C4C2×D19711yescharacteristic
134AGL(1,ℤ/8)—3948no—
135C2×SD16—1974no—
136C2×D8—1974no—
137C2×M16—1972no—
138#123—3941no—
139C22⋊Q8—1972no—
140#121—1972no—
141#120—3941no—
142C4×D4—3942no—
143#117—1971no—
144#102—1978no—
145C22⋊C8—1974no—
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D19711yescharacteristic
151#41—3944no—
152C2×Q8—1971no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—3942no—
160C2×D4—3941no—
161C2×D4—1971no—
162C2×D4C22⋊D19718yescomplemented
163C2×D4C22×D19714yes—
164C22×C4—24no—
165C22×C4—3941no—
166C22×C4C22⋊D19712yes—
167C22×C4C22⋊D19711yescharacteristic
168C22×C4C22×D19712yes—
169C22×C4C22×D19711yescharacteristic
170SD16—3948no—
171D8—3948no—
172M16—3944no—
173C2×C8—1974no—
174C4⋊C4—3942no—
175C4⋊C4—1972no—
176C22⋊C4—3941no—
177C22⋊C4—3942no—
178C22⋊C4—3942no—
179C42—3941no—
180C23—22no—
181C23—24no—
182C23—216no—
183C23—28no—
184C23—24no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—3941no—
190C23C2×C22⋊D19711yescharacteristic
191C23C2×C22⋊D19712yes—
192Q8—3942no—
193D4—216no—
194D4—28no—
195D4—24no—
196D4—7882no—
197D4—7884no—
198D4—3942no—
199D4C2×C22⋊D19718yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—7882no—
204C2×C4—3942no—
205C2×C4—3941no—
206C2×C4C2×C22⋊D19712yes—
207C2×C4C2×C22⋊D19711yescharacteristic
208C2×C4C2×C22⋊D19714yes—
209C2×C4C23×D19711yescharacteristic
210C8—3944no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—416no—
217C22—432no—
218C22—48no—
219C22—7882no—
220C22—7881no—
221C22#2026329512yes—
222C22#2026335214yescomplemented
223C22C22×C22⋊D19711yescharacteristic
224C4—22no—
225C4—7882no—
226C4—7881no—
227C4#2026329512yes—
228C4C22×C22⋊D19712yes—
229C2—22no—
230C2—24no—
231C2—48no—
232C2—15761no—
233C2C2×#2026329511yescharacteristic
234C2C2×#2026335212yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC23×D4×C197C2
socleC22×C197C22×D4
hypercenterC23×D4D197
hypocenterC197#3003
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#20263352C2semidirect
2. #27161590C2semidirect
3. #27161230C2semidirect
4. #27161213C2semidirect
5. #27161084C2semidirect
6. #27161043C2semidirect
7. C23×D4×C197C2semidirect
8. C2×#15691358C22semidirect
10. C2×#15691346C22semidirect
11. C2×#15691343C22semidirect
12. C2×#15691342C22semidirect
14. #20263305C22semidirect
15. #20263300C22semidirect
18. #20263285C22semidirect
19. #20263247C22semidirect
20. C22⋊#13096802C22semidirect
29. C2×Dic394C23semidirect
30. C2×D788C23semidirect
37. C2×#13096802C23semidirect
38. C4⋊Dic197C23semidirect
55. C2×D4×C197D4semidirect
102. C22×C197AGL(1,ℤ/8)semidirect
117. C197#3003semidirect
119. C23×D4D197semidirect
162. C2×D4C22⋊D197semidirect
222. C22#20263352semidirect

Maximal subgroups

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

subgroupindex
C2×#202633522
#271615902
#271612302
#271612132
#271610842
#271610432
C23×D4×C1972
#3003197