Finite Group Databasebeta

← #38745978

Subgroups — #38745978

Order 25216.

560 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 143 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×#20263288C211yesmaximal, characteristic
2C2×#20263284C211yesmaximal, characteristic, complemented
3#27161014C211yesmaximal, characteristic, complemented
4#27161009C211yesmaximal, characteristic, complemented
5#27161008C212yesmaximal, complemented
6#27161006C211yesmaximal, characteristic, complemented
7#27160997C212yesmaximal
8#27160930C212yesmaximal, complemented
9#27160926C211yesmaximal, characteristic, complemented
10#27160926C212yesmaximal, complemented
11#400×C197C211yesmaximal, characteristic, complemented
12C22×D788C2211yescharacteristic
13C22×C4×D197C2211yescharacteristic
14C22×C4×D197C2211yescharacteristic
15#20263288—24no—
16#20263284—24no—
17C2×C4⋊Dic197C2211yescharacteristic
18C2×#15691337C2212yes—
19C2×#15691337C2212yes—
20C2×#15691335C2212yes—
21C2×#15691335C2212yes—
22C2×#15691335C2212yes—
23C2×#15691335C2212yes—
24C2×#15691335C2211yescharacteristic
25C2×#15691335C2211yescharacteristic
26C2×C4×Dic197C2212yes—
27C2×C4×Dic197C2212yes—
28C2×C4×Dic197C2211yescharacteristic
29#20263233C2212yes—
30#20263233C2211yescharacteristic
31#20263233C2212yes—
32C2×C4⋊C4×C197C2211yescharacteristic
33C2×C4⋊C4×C197C2211yescharacteristic
34#95×C197C2211yescharacteristic
35#95×C197C2212yes—
36#95×C197C2212yes—
37C23×D197C2311yescharacteristic
38C23×D197C2311yescharacteristic
39C2×D788—22no—
40C2×D788—22no—
41C2×D788—22no—
42C22×Dic197C2311yescharacteristic
43C22×Dic197C2312yes—
44C22×Dic197C2312yes—
45C22×Dic197C2311yescharacteristic
46C2×C4×D197—24no—
47C2×C4×D197—22no—
48C2×C4×D197—24no—
49C2×C4×D197D414yes—
50C4⋊Dic197—22no—
51#15691337—24no—
52#15691337—24no—
53#15691335—24no—
54#15691335—22no—
55#15691335—24no—
56#15691335—24no—
57#15691335—24no—
58#15691335—22no—
59C4×Dic197—24no—
60C4×Dic197—22no—
61C4×Dic197—24no—
62C22×C4×C197C2312yes—
63C22×C4×C197C2311yescharacteristic
64C22×C4×C197C2312yes—
65C22×C4×C197C2311yescharacteristic
66C22×C4×C197C2311yescharacteristic
67C4⋊C4×C197—22no—
68C4⋊C4×C197—22no—
69C22×D197—21no—
70C22×D197—22no—
71C22×D197—22no—
72C22×D197—22no—
73C22×D197—22no—
74C22×D197—21no—
75C22×D197—21no—
76C22×D197—21no—
77C22×D197C2×D412yes—
78C22×D197#4112yes—
79D788—28no—
80C2×Dic197—22no—
81C2×Dic197—22no—
82C2×Dic197—21no—
83C2×Dic197—22no—
84C2×Dic197—22no—
85C2×Dic197—22no—
86C2×Dic197—22no—
87C2×Dic197C2×D412yes—
88C2×Dic197#4114yes—
89C2×Dic197#4114yes—
90C4×D197—28no—
91C4×D197—44no—
92C23×C197C2411yescharacteristic
93C2×C4×C197—21no—
94C2×C4×C197—22no—
95C2×C4×C197—21no—
96C2×C4×C197—22no—
97C2×C4×C197—21no—
98C2×C4×C197—22no—
99C2×C4×C197—22no—
100C2×C4×C197—22no—
101C2×C4×C197—22no—
102C2×C4×C197—22no—
103C2×C4×C197—22no—
104C2×C4×C197C2×D412yes—
105C2×C4×C197#4112yes—
106C22×C197C22×D411yescharacteristic
107C22×C197C2×#4112yes—
108C22×C197C2×#4111yescharacteristic
109C22×C197C2×#4112yes—
110C22×C197C2×#4111yescharacteristic
111C2×D197—22no—
112C2×D197—22no—
113C2×D197—22no—
114C2×D197—24no—
115C2×D197—22no—
116C2×D197—41no—
117C2×D197—42no—
118C2×D197—42no—
119C2×D197—42no—
120C2×D197—41no—
121C4×C197—22no—
122C4×C197—22no—
123C4×C197—41no—
124C4×C197—42no—
125C4×C197—42no—
126Dic197—24no—
127Dic197—42no—
128Dic197—41no—
129Dic197#12314yes—
130C2×C197C2×#12311yescharacteristic
131C2×C197#52511yescharacteristic
132C2×C197#52912yes—
133C2×C197#52912yes—
134C2×C197#53211yescharacteristic
135D197—42no—
136D197—42no—
137C197#239511yescharacteristic, complemented, Sylow
138#2395—1971nomaximal, Sylow
139C2×C4×D4—1971no—
140C2×#117—1971no—
141#400D19711yescharacteristic, complemented
142#399—1971no—
143#398—1971no—
144#397—1971no—
145#397—1972no—
146#394—1972no—
147C4×C4⋊C4—1972no—
148C4×C22⋊C4—1972no—
149C4×C22⋊C4—1971no—
150C22×D4—1971no—
151C23×C4—1971no—
152C23×C4—1971no—
153C4×D4—3944no—
154#117—3944no—
155C2×C4⋊C4—1971no—
156C2×C4⋊C4—1972no—
157C2×C4⋊C4—1972no—
158C2×C4⋊C4C2×D19711yescharacteristic
159C2×C4⋊C4C2×D19711yescharacteristic
160C2×C22⋊C4—1972no—
161C2×C22⋊C4—1971no—
162C2×C22⋊C4—1971no—
163C2×C22⋊C4—1972no—
164C2×C22⋊C4—1972no—
165C2×C22⋊C4—1972no—
166C2×C42—1972no—
167C2×C42—1971no—
168C2×C42—1972no—
169#95—1972no—
170#95—1971no—
171#95—1972no—
172#95C2×D19712yes—
173#95C2×D19711yescharacteristic
174#95C2×D19712yes—
175C24—1971no—
176C24—1971no—
177C2×D4—3942no—
178C2×D4—3942no—
179C2×D4—3942no—
180C22×C4—3942no—
181C22×C4—3944no—
182C22×C4—3944no—
183C22×C4—1971no—
184C22×C4—1972no—
185C22×C4—1972no—
186C22×C4—1974no—
187C22×C4—1971no—
188C22×C4C22×D19711yescharacteristic
189C22×C4C22×D19712yes—
190C22×C4C22×D19712yes—
191C22×C4C22×D19711yescharacteristic
192C22×C4C22×D19711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—3944no—
196C4⋊C4—3942no—
197C4⋊C4—3944no—
198C22⋊C4—3944no—
199C22⋊C4—3944no—
200C22⋊C4—3942no—
201C22⋊C4—3944no—
202C22⋊C4—3944no—
203C22⋊C4—3942no—
204C42—3944no—
205C42—3944no—
206C42—3942no—
207C23—3941no—
208C23—3942no—
209C23—3942no—
210C23—3942no—
211C23—3942no—
212C23—3941no—
213C23—3941no—
214C23—3941no—
215C23—1972no—
216C23—1972no—
217C23C23×D19711yescharacteristic
218D4—3948no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—7884no—
231C2×C4—3948no—
232C2×C4—3941no—
233C2×C4—3942no—
234C2×C4—3942no—
235C2×C4—3942no—
236C2×C4—3942no—
237C2×C4—3942no—
238C2×C4—3942no—
239C2×C4—1974no—
240C2×C4—1972no—
241C2×C4—1974no—
242C2×C4D4×D19712yes—
243C2×C4#1569135912yes—
244C22—7882no—
245C22—7882no—
246C22—7882no—
247C22—7881no—
248C22—7881no—
249C22—3944no—
250C22—3942no—
251C22—3942no—
252C22—3942no—
253C22—3942no—
254C22C2×D4×D19711yescharacteristic
255C22#41×D19711yescharacteristic
256C22#41×D19712yes—
257C22#41×D19712yes—
258C22C2×#1569135911yescharacteristic
259C4—22no—
260C4—22no—
261C4—42no—
262C4—42no—
263C4—41no—
264C4—7881no—
265C4—7882no—
266C4—3944no—
267C4—1974no—
268C2—7882no—
269C2—7882no—
270C2#123×D19711yescharacteristic
271C2#2716159211yescharacteristic
272C2#2716161212yes—
273C2#2716161412yes—
274C2#2716162811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#400×C197C2
prime radicalC22×D788C22
hypercenter#400D197
hypocenterC197#2395
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#20263284C2semidirect
3. #27161014C2semidirect
4. #27161009C2semidirect
5. #27161008C2semidirect
6. #27161006C2semidirect
8. #27160930C2semidirect
9. #27160926C2semidirect
10. #27160926C2semidirect
11. #400×C197C2semidirect
137. C197#2395semidirect
141. #400D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632882
C2×#202632842
#271610142
#271610092
#271610082
#271610062
#271609972
#271609302
#271609262
#400×C1972
#2395197