Finite Group Databasebeta

← #38745969

Subgroups — #38745969

Order 25216.

560 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 157 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×C4×C22⋊D197C211yesmaximal, characteristic
2#27161017C211yesmaximal, characteristic, complemented
3#27161014C211yesmaximal, characteristic, complemented
4#27161010C211yesmaximal, characteristic, complemented
5#27161007C212yesmaximal, complemented
6#27161005C211yesmaximal, characteristic, complemented
7#27160994C212yesmaximal
8C4×#15691337C212yesmaximal, complemented
9C4×#15691335C212yesmaximal, complemented
10C4×C22⋊Dic197C211yesmaximal, characteristic, complemented
11C2×#117×C197C211yesmaximal, characteristic, complemented
12C22×C22⋊D197C2211yescharacteristic
13C22×C4×D197C2211yescharacteristic
14C4×C22⋊D197—24no—
15C2×#15691337C2212yes—
16C2×#15691337C2212yes—
17C2×#15691337C2211yescharacteristic
18C2×C22⋊Dic197C2212yes—
19C2×C22⋊Dic197C2211yescharacteristic
20C2×#15691335C2212yes—
21C2×#15691335C2212yes—
22C2×#15691335C2211yescharacteristic
23C2×C4×Dic197C2212yes—
24C2×C4×Dic197C2211yescharacteristic
25#20263233C2211yescharacteristic
26#20263233C2211yescharacteristic
27#20263233C2212yes—
28#20263233C2212yes—
29#20263233C2212yes—
30#20263233C2212yes—
31C23×C4×C197C2211yescharacteristic
32#117×C197—24no—
33C2×C4⋊C4×C197C2211yescharacteristic
34C2×C4⋊C4×C197C2211yescharacteristic
35C2×C22⋊C4×C197C2212yes—
36C2×C42×C197C2212yes—
37C23×D197C2311yescharacteristic
38C2×C22⋊D197—22no—
39C2×C22⋊D197—22no—
40C2×C22⋊D197—22no—
41C22×Dic197C2312yes—
42C22×Dic197C2311yescharacteristic
43C22×Dic197C2311yescharacteristic
44C22×Dic197C2312yes—
45C22×Dic197C2311yescharacteristic
46C2×C4×D197—22no—
47C2×C4×D197—24no—
48#15691337—24no—
49#15691337—24no—
50#15691337—22no—
51C22⋊Dic197—24no—
52C22⋊Dic197—22no—
53#15691335—24no—
54#15691335—24no—
55#15691335—22no—
56C4×Dic197—22no—
57C4×Dic197—24no—
58C24×C197C2311yescharacteristic
59C22×C4×C197—24no—
60C22×C4×C197D414yes—
61C22×C4×C197C2312yes—
62C22×C4×C197C2312yes—
63C22×C4×C197C2311yescharacteristic
64C22×C4×C197C2311yescharacteristic
65C4⋊C4×C197—22no—
66C4⋊C4×C197—22no—
67C22⋊C4×C197—24no—
68C42×C197—24no—
69C22×D197—21no—
70C22×D197—22no—
71C22×D197—22no—
72C22×D197—21no—
73C22×D197#4112yes—
74C22⋊D197—28no—
75C2×Dic197—21no—
76C2×Dic197—22no—
77C2×Dic197—22no—
78C2×Dic197—21no—
79C2×Dic197—21no—
80C2×Dic197—21no—
81C2×Dic197—22no—
82C2×Dic197—22no—
83C2×Dic197—22no—
84C2×Dic197—22no—
85C2×Dic197—22no—
86C2×Dic197#4114yes—
87C2×Dic197#4112yes—
88C4×D197—44no—
89C23×C197—22no—
90C23×C197—22no—
91C23×C197—21no—
92C23×C197—21no—
93C23×C197C2×D412yes—
94C23×C197C2411yescharacteristic
95C2×C4×C197—22no—
96C2×C4×C197—28no—
97C2×C4×C197—22no—
98C2×C4×C197—22no—
99C2×C4×C197—22no—
100C2×C4×C197—22no—
101C2×C4×C197—22no—
102C2×C4×C197—22no—
103C2×C4×C197C2×D412yes—
104C2×C4×C197C2×D412yes—
105C2×C4×C197#4114yes—
106C22×C197—22no—
107C22×C197—22no—
108C22×C197—22no—
109C22×C197—42no—
110C22×C197—42no—
111C22×C197C22×D411yescharacteristic
112C22×C197C2×#4111yescharacteristic
113C22×C197C2×#4111yescharacteristic
114C22×C197C2×#4112yes—
115C22×C197C2×#4112yes—
116C2×D197—22no—
117C2×D197—24no—
118C2×D197—41no—
119C2×D197—42no—
120C2×D197—41no—
121C4×C197—22no—
122C4×C197—42no—
123C4×C197—42no—
124C4×C197#12314yes—
125Dic197—22no—
126Dic197—24no—
127Dic197—41no—
128Dic197—42no—
129Dic197—41no—
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×#117D19711yescharacteristic, complemented
141#400—1971no—
142#399—1971no—
143#398—1971no—
144#397—1971no—
145#397—1972no—
146#394—1972no—
147C4×C4⋊C4—1972no—
148C4×C22⋊C4—1971no—
149C4×C22⋊C4—1972no—
150C22×D4—1971no—
151C23×C4—1971no—
152C23×C4C2×D19711yescharacteristic
153C4×D4—3944no—
154#117—24no—
155C2×C4⋊C4—1971no—
156C2×C4⋊C4—1972no—
157C2×C4⋊C4—1972no—
158C2×C4⋊C4C2×D19711yescharacteristic
159C2×C4⋊C4C2×D19711yescharacteristic
160C2×C22⋊C4—1971no—
161C2×C22⋊C4—1972no—
162C2×C22⋊C4—1971no—
163C2×C22⋊C4—1972no—
164C2×C22⋊C4—1972no—
165C2×C22⋊C4C2×D19712yes—
166C2×C42—1971no—
167C2×C42—1972no—
168C2×C42C2×D19712yes—
169#95—1972no—
170#95—1971no—
171#95—1972no—
172#95—1971no—
173#95—1972no—
174#95—1972no—
175C24—1971no—
176C24C22×D19711yescharacteristic
177C2×D4—3942no—
178C2×D4—3942no—
179C2×D4—3942no—
180C22×C4—24no—
181C22×C4—3942no—
182C22×C4—3944no—
183C22×C4—1972no—
184C22×C4—1971no—
185C22×C4—1971no—
186C22×C4—1972no—
187C22×C4—1971no—
188C22×C4C22⋊D19714yes—
189C22×C4C22×D19712yes—
190C22×C4C22×D19712yes—
191C22×C4C22×D19711yescharacteristic
192C22×C4C22×D19711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—3944no—
196C4⋊C4—3944no—
197C4⋊C4—3942no—
198C22⋊C4—24no—
199C22⋊C4—3944no—
200C22⋊C4—3944no—
201C22⋊C4—3942no—
202C22⋊C4—3944no—
203C22⋊C4—3942no—
204C42—24no—
205C42—3942no—
206C42—3944no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—3941no—
212C23—3942no—
213C23—3941no—
214C23—3942no—
215C23—1972no—
216C23C2×C22⋊D19712yes—
217C23C23×D19711yescharacteristic
218D4—3948no—
219C2×C4—22no—
220C2×C4—28no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—7884no—
228C2×C4—3941no—
229C2×C4—3941no—
230C2×C4—3942no—
231C2×C4—3942no—
232C2×C4—3942no—
233C2×C4—3942no—
234C2×C4—3942no—
235C2×C4—3941no—
236C2×C4—3942no—
237C2×C4—3941no—
238C2×C4—3942no—
239C2×C4—1974no—
240C2×C4—1972no—
241C2×C4C2×C22⋊D19712yes—
242C2×C4C2×C22⋊D19712yes—
243C2×C4#1569135814yes—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—7882no—
250C22—7881no—
251C22—7881no—
252C22—3944no—
253C22—3942no—
254C22C22×C22⋊D19711yescharacteristic
255C22#41×D19712yes—
256C22#41×D19711yescharacteristic
257C22#41×D19711yescharacteristic
258C22C2×#1569135812yes—
259C4—22no—
260C4—42no—
261C4—42no—
262C4—7882no—
263C4—7881no—
264C4—7881no—
265C4—3944no—
266C4—3942no—
267C4#2026330914yes—
268C2—42no—
269C2—7882no—
270C2C2×#2026330911yescharacteristic
271C2#2716158611yescharacteristic
272C2#2716160712yes—
273C2#2716160712yes—
274C2#2716162511yescharacteristic
subgroupG/H
centerC22×C4C22×D197
commutator, socle, cosocleC23×C197C24
FrattiniC23C23×D197
FittingC2×#117×C197C2
prime radicalC22×C22⋊D197C22
hypercenterC2×#117D197
hypocenterC197#2395
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #27161017C2semidirect
3. #27161014C2semidirect
4. #27161010C2semidirect
5. #27161007C2semidirect
6. #27161005C2semidirect
8. C4×#15691337C2semidirect
9. C4×#15691335C2semidirect
10. C4×C22⋊Dic197C2semidirect
11. C2×#117×C197C2semidirect
137. C197#2395semidirect
140. C2×#117D197semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1972
#271610172
#271610142
#271610102
#271610072
#271610052
#271609942
C4×#156913372
C4×#156913352
C4×C22⋊Dic1972
C2×#117×C1972
#2395197