Finite Group Databasebeta

← #38745974

Subgroups — #38745974

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×#20263286C211yesmaximal, characteristic
2C2×#20263284C211yesmaximal, characteristic, complemented
3#27161019C211yesmaximal, characteristic, complemented
4#27161015C211yesmaximal, characteristic, complemented
5#27161010C211yesmaximal, characteristic, complemented
6#27161008C212yesmaximal, complemented
7#27160994C212yesmaximal
8#27160930C212yesmaximal, complemented
9C22⋊C4×Dic197C212yesmaximal, complemented
10#27160926C211yesmaximal, characteristic, complemented
11#397×C197C211yesmaximal, characteristic, complemented
12C22×C22⋊D197C2211yescharacteristic
13C23×Dic197C2211yescharacteristic
14C22×C4×D197C2211yescharacteristic
15#20263286—24no—
16#20263284—24no—
17C2×C4⋊Dic197C2211yescharacteristic
18C2×#15691337C2212yes—
19C2×#15691337C2212yes—
20C2×#15691337C2211yescharacteristic
21C2×C22⋊Dic197C2212yes—
22C2×#15691335C2212yes—
23C2×#15691335C2212yes—
24C2×#15691335C2211yescharacteristic
25C2×C4×Dic197C2212yes—
26C2×C4×Dic197C2212yes—
27C2×C4×Dic197C2211yescharacteristic
28#20263233C2211yescharacteristic
29#20263233C2212yes—
30#20263233C2212yes—
31#20263233C2212yes—
32C2×C4⋊C4×C197C2211yescharacteristic
33C2×C22⋊C4×C197C2212yes—
34C2×C22⋊C4×C197C2211yescharacteristic
35#95×C197C2211yescharacteristic
36#95×C197C2212yes—
37C23×D197C2311yescharacteristic
38C2×C22⋊D197—22no—
39C2×C22⋊D197—22no—
40C2×C22⋊D197—22no—
41C22×Dic197—22no—
42C22×Dic197—24no—
43C22×Dic197C2311yescharacteristic
44C22×Dic197C2312yes—
45C22×Dic197C2312yes—
46C22×Dic197C2311yescharacteristic
47C22×Dic197C2311yescharacteristic
48C2×C4×D197—24no—
49C2×C4×D197D414yes—
50C4⋊Dic197—22no—
51#15691337—24no—
52#15691337—24no—
53#15691337—22no—
54C22⋊Dic197—24no—
55#15691335—24no—
56#15691335—22no—
57#15691335—24no—
58C4×Dic197—24no—
59C4×Dic197—22no—
60C4×Dic197—24no—
61C24×C197C2311yescharacteristic
62C22×C4×C197C2312yes—
63C22×C4×C197C2311yescharacteristic
64C22×C4×C197C2312yes—
65C22×C4×C197C2311yescharacteristic
66C4⋊C4×C197—22no—
67C22⋊C4×C197—24no—
68C22⋊C4×C197—22no—
69C22×D197—22no—
70C22×D197—22no—
71C22×D197—21no—
72C22×D197—21no—
73C22×D197C2×D412yes—
74C22⋊D197—28no—
75C2×Dic197—21no—
76C2×Dic197—22no—
77C2×Dic197—22no—
78C2×Dic197—21no—
79C2×Dic197—22no—
80C2×Dic197—21no—
81C2×Dic197—22no—
82C2×Dic197—22no—
83C2×Dic197—22no—
84C2×Dic197—44no—
85C2×Dic197C2×D412yes—
86C2×Dic197#4114yes—
87C2×Dic197#4112yes—
88C2×Dic197#4114yes—
89C4×D197—28no—
90C23×C197—21no—
91C23×C197—22no—
92C23×C197—22no—
93C23×C197—21no—
94C23×C197#4112yes—
95C23×C197C2411yescharacteristic
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×C197—22no—
105C2×C4×C197C2×D412yes—
106C22×C197—22no—
107C22×C197—24no—
108C22×C197—41no—
109C22×C197—42no—
110C22×C197—41no—
111C22×C197C22×D411yescharacteristic
112C22×C197C2×#4112yes—
113C22×C197C2×#4111yescharacteristic
114C22×C197C2×#4112yes—
115C22×C197C2×#4111yescharacteristic
116C2×D197—22no—
117C2×D197—22no—
118C2×D197—22no—
119C2×D197—42no—
120C2×D197—42no—
121C4×C197—22no—
122C4×C197—42no—
123C4×C197—42no—
124C4×C197—41no—
125Dic197—22no—
126Dic197—24no—
127Dic197—41no—
128Dic197—42no—
129Dic197#12314yes—
130C2×C197—42no—
131C2×C197C2×#12311yescharacteristic
132C2×C197#52511yescharacteristic
133C2×C197#52912yes—
134C2×C197#52912yes—
135C2×C197#53211yescharacteristic
136D197—42no—
137C197#239511yescharacteristic, complemented, Sylow
138#2395—1971nomaximal, Sylow
139C2×C4×D4—1971no—
140C2×#117—1971no—
141#400—1971no—
142#399—1971no—
143#398—1971no—
144#397—1972no—
145#397D19711yescharacteristic, complemented
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—1971no—
157C2×C4⋊C4—1972no—
158C2×C4⋊C4—1972no—
159C2×C4⋊C4C2×D19711yescharacteristic
160C2×C22⋊C4—1972no—
161C2×C22⋊C4—1972no—
162C2×C22⋊C4—1971no—
163C2×C22⋊C4—1972no—
164C2×C22⋊C4C2×D19711yescharacteristic
165C2×C22⋊C4C2×D19712yes—
166C2×C42—1972no—
167C2×C42—1971no—
168C2×C42—1972no—
169#95—1972no—
170#95—1972no—
171#95—1971no—
172#95—1972no—
173#95C2×D19711yescharacteristic
174#95C2×D19712yes—
175C24—1971no—
176C24C22×D19711yescharacteristic
177C2×D4—3942no—
178C2×D4—3942no—
179C2×D4—3942no—
180C22×C4—3944no—
181C22×C4—3942no—
182C22×C4—3944no—
183C22×C4—1971no—
184C22×C4—1972no—
185C22×C4—1972no—
186C22×C4—1971no—
187C22×C4—1974no—
188C22×C4—1971no—
189C22×C4C22×D19712yes—
190C22×C4C22×D19712yes—
191C22×C4C22×D19711yescharacteristic
192C22×C4C22×D19711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—3944no—
195C4⋊C4—3944no—
196C4⋊C4—3942no—
197C4⋊C4—3942no—
198C22⋊C4—24no—
199C22⋊C4—22no—
200C22⋊C4—3944no—
201C22⋊C4—3942no—
202C22⋊C4—3944no—
203C22⋊C4—3944no—
204C42—3944no—
205C42—3944no—
206C42—3942no—
207C23—21no—
208C23—22no—
209C23—22no—
210C23—21no—
211C23—3942no—
212C23—3942no—
213C23—3941no—
214C23—3941no—
215C23—1972no—
216C23#1569135712yes—
217C23C23×D19711yescharacteristic
218D4—3948no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—7884no—
229C2×C4—3948no—
230C2×C4—3942no—
231C2×C4—3941no—
232C2×C4—3942no—
233C2×C4—3942no—
234C2×C4—3942no—
235C2×C4—3941no—
236C2×C4—3942no—
237C2×C4—3942no—
238C2×C4—3941no—
239C2×C4—1974no—
240C2×C4—1972no—
241C2×C4—1972no—
242C2×C4—1974no—
243C2×C4D4×D19712yes—
244C22—24no—
245C22—22no—
246C22—42no—
247C22—41no—
248C22—41no—
249C22—7882no—
250C22—7882no—
251C22—3942no—
252C22—3942no—
253C22—3942no—
254C22C2×D4×D19711yescharacteristic
255C22#41×D19712yes—
256C22#41×D19712yes—
257C22#41×D19711yescharacteristic
258C22C2×#1569135711yescharacteristic
259C4—22no—
260C4—42no—
261C4—41no—
262C4—42no—
263C4—7882no—
264C4—7881no—
265C4—3944no—
266C4—3942no—
267C4—1974no—
268C2—42no—
269C2—7882no—
270C2#123×D19711yescharacteristic
271C2#2716158911yescharacteristic
272C2#2716160612yes—
273C2#2716161212yes—
274C2#2716162811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#397×C197C2
prime radicalC22×C22⋊D197C22
hypercenter#397D197
hypocenterC197#2395
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#20263284C2semidirect
3. #27161019C2semidirect
4. #27161015C2semidirect
5. #27161010C2semidirect
6. #27161008C2semidirect
8. #27160930C2semidirect
9. C22⋊C4×Dic197C2semidirect
10. #27160926C2semidirect
11. #397×C197C2semidirect
137. C197#2395semidirect
145. #397D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632862
C2×#202632842
#271610192
#271610152
#271610102
#271610082
#271609942
#271609302
C22⋊C4×Dic1972
#271609262
#397×C1972
#2395197