Finite Group Databasebeta

← #38752635

Subgroups — #38752635

Order 25216.

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

normal subgroups only

This group has 458 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
1#27161590C211yesmaximal, characteristic
2#27161414C212yesmaximal, complemented
3#27161409C212yesmaximal, complemented
4#27161397C212yesmaximal, complemented
5#27161392C212yesmaximal, complemented
6#27161119C212yesmaximal, complemented
7#27161078C212yesmaximal, complemented
8#27161043C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C197C211yesmaximal, characteristic, complemented
10C2×#15691358C2211yescharacteristic
11C2×#15691342C2212yes—
12#20263309—21no—
13#20263305C2211yescharacteristic
14#20263305C2211yescharacteristic
15#20263300C2211yescharacteristic
16#20263300C2211yescharacteristic
17#20263295—21no—
18C4×C22⋊D197—21no—
19C4×C22⋊D197—21no—
20#20263285C2211yescharacteristic
21#20263259C2214yescomplemented
22#20263257C2214yescomplemented
23#20263249C2214yescomplemented
24#20263246C2214yescomplemented
25C22⋊#13096802C2212yes—
26C22⋊#13096802C2212yes—
27C23×C4×C197—21no—
28C23×C4×C197C2211yescharacteristic
29C2×C4⋊C4×C197—21no—
30C2×C4⋊C4×C197—21no—
31C2×C4⋊C4×C197—21no—
32C2×C4⋊C4×C197C2212yes—
33C2×C4⋊C4×C197C2212yes—
34C2×C4⋊C4×C197C2212yes—
35#15691358—24no—
36C2×Dic394C2311yescharacteristic
37C2×D788C2311yescharacteristic
38C2×C22⋊D197—21no—
39C2×C22⋊D197—21no—
40C2×C22⋊D197—21no—
41C2×C4×D197—21no—
42#15691342—24no—
43C2×#13096802C2314yes—
44C4⋊Dic197C2311yescharacteristic
45C4⋊Dic197C2311yescharacteristic
46#15691337—21no—
47#15691337—21no—
48C22⋊Dic197—21no—
49C22⋊Dic197—21no—
50C22⋊Dic197—21no—
51#15691335—21no—
52#15691335—21no—
53C4×Dic197—21no—
54C24×C197D411yescharacteristic
55C22×C4×C197—24no—
56C22×C4×C197—22no—
57C22×C4×C197—22no—
58C22×C4×C197—22no—
59C22×C4×C197—44no—
60C22×C4×C197D411yescharacteristic
61C22×C4×C197D411yescharacteristic
62C22×C4×C197D411yescharacteristic
63C22×C4×C197C2311yescharacteristic
64C22×C4×C197C2311yescharacteristic
65C22×C4×C197C2311yescharacteristic
66C4⋊C4×C197—22no—
67C4⋊C4×C197—22no—
68C4⋊C4×C197—22no—
69C4⋊C4×C197C2314yes—
70C22×D197—21no—
71Dic394—22no—
72D788—22no—
73C22⋊D197—42no—
74C22⋊D197—42no—
75C22⋊D197—42no—
76C2×Dic197—21no—
77C2×Dic197—21no—
78C2×Dic197—21no—
79C4×D197—42no—
80#13096802—24no—
81C23×C197—22no—
82C23×C197—22no—
83C23×C197—22no—
84C23×C197C2×D411yescharacteristic
85C23×C197C2×D411yescharacteristic
86C23×C197C2×D411yescharacteristic
87C2×C4×C197—22no—
88C2×C4×C197—22no—
89C2×C4×C197—22no—
90C2×C4×C197—22no—
91C2×C4×C197—22no—
92C2×C4×C197—24no—
93C2×C4×C197—44no—
94C2×C4×C197—44no—
95C2×C4×C197—44no—
96C2×C4×C197C2×D411yescharacteristic
97C2×C4×C197C2×D411yescharacteristic
98C2×C4×C197C2×D411yescharacteristic
99C2×C4×C197#4114yes—
100C2×C4×C197C2411yescharacteristic
101C22×C197—28no—
102C22×C197—21no—
103C22×C197—21no—
104C22×C197—21no—
105C22×C197—21no—
106C22×C197—21no—
107C22×C197—21no—
108C22×C197#12312yes—
109C22×C197AGL(1,ℤ/8)12yes—
110C22×C197#13712yes—
111C22×C197C22×D411yescharacteristic
112C2×D197—41no—
113C2×D197—41no—
114C2×D197—41no—
115C4×C197—21no—
116C4×C197—21no—
117C4×C197—44no—
118C4×C197#12312yes—
119C4×C197C2×#4112yes—
120Dic197—41no—
121Dic197—41no—
122Dic197—41no—
123C2×C197—22no—
124C2×C197—22no—
125C2×C197—22no—
126C2×C197C2×#12311yescharacteristic
127C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C197C2×#13711yescharacteristic
129D197—81no—
130C197#309211yescharacteristic, complemented, Sylow
131#3092—1971nomaximal, Sylow
132#525—1971no—
133C22×C4⋊C4D19711yescharacteristic, complemented
134#484—1972no—
135#483—1972no—
136#481—1972no—
137#480—1972no—
138#428—1972no—
139#417—1972no—
140#407—1971no—
141C2×#41—1971no—
142C23×C4—21no—
143C23×C4C2×D19711yescharacteristic
144C2×M16—1972no—
145#123—3941no—
146C22⋊Q8—1971no—
147C22⋊Q8—1971no—
148#121—1971no—
149#121—1971no—
150#120—3941no—
151C4×D4—3941no—
152C4×D4—3941no—
153#117—1971no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×D19712yes—
158C2×C4⋊C4C2×D19712yes—
159C2×C4⋊C4C2×D19712yes—
160#107—1974no—
161#106—1974no—
162#103—1974no—
163#102—1974no—
164C22⋊C8—1972no—
165C22⋊C8—1972no—
166C24C22⋊D19711yescharacteristic
167#41—3944no—
168C2×Q8—1971no—
169C2×D4—3941no—
170C2×D4—3941no—
171C2×D4—3941no—
172C2×D4—1971no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—44no—
178C22×C4—3941no—
179C22×C4C22⋊D19711yescharacteristic
180C22×C4C22⋊D19711yescharacteristic
181C22×C4C22⋊D19711yescharacteristic
182C22×C4C22×D19711yescharacteristic
183C22×C4C22×D19711yescharacteristic
184C22×C4C22×D19711yescharacteristic
185M16—3944no—
186C2×C8—1974no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—3941no—
191C4⋊C4—3941no—
192C4⋊C4—1971no—
193C4⋊C4—1971no—
194C4⋊C4C22×D19714yes—
195C22⋊C4—3941no—
196C22⋊C4—3941no—
197C22⋊C4—3941no—
198C22⋊C4—3941no—
199C22⋊C4—3941no—
200C42—3941no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—3941no—
205C23C2×C22⋊D19711yescharacteristic
206C23C2×C22⋊D19711yescharacteristic
207C23C2×C22⋊D19711yescharacteristic
208Q8—3942no—
209D4—7882no—
210D4—7882no—
211D4—7882no—
212D4—3942no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—7882no—
223C2×C4—3941no—
224C2×C4—3941no—
225C2×C4—3941no—
226C2×C4C2×C22⋊D19711yescharacteristic
227C2×C4C2×C22⋊D19711yescharacteristic
228C2×C4C2×C22⋊D19711yescharacteristic
229C2×C4#1569135814yes—
230C2×C4C23×D19711yescharacteristic
231C8—3944no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—7881no—
240C22—7881no—
241C22—7881no—
242C22#2026330912yes—
243C22#2026335212yes—
244C22#2026335712yes—
245C22C22×C22⋊D19711yescharacteristic
246C4—21no—
247C4—21no—
248C4—44no—
249C4—7881no—
250C4—7881no—
251C4—7881no—
252C4#2026330912yes—
253C4C2×#1569135812yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—15761no—
258C2C2×#2026330911yescharacteristic
259C2C2×#2026335211yescharacteristic
260C2C2×#2026335711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC22×C4⋊C4×C197C2
socleC22×C197C22×D4
prime radical#27161590C2
hypercenterC22×C4⋊C4D197
hypocenterC197#3092
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #27161414C2semidirect
3. #27161409C2semidirect
4. #27161397C2semidirect
5. #27161392C2semidirect
6. #27161119C2semidirect
7. #27161078C2semidirect
8. #27161043C2semidirect
9. C22×C4⋊C4×C197C2semidirect
21. #20263259C22semidirect
22. #20263257C22semidirect
23. #20263249C22semidirect
24. #20263246C22semidirect
130. C197#3092semidirect
133. C22×C4⋊C4D197semidirect

Maximal subgroups

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

subgroupindex
#271615902
#271614142
#271614092
#271613972
#271613922
#271611192
#271610782
#271610432
C22×C4⋊C4×C1972
#3092197