Finite Group Databasebeta

← #38751049

Subgroups — #38751049

Order 25216.

564 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 169 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×#20263303C211yesmaximal, characteristic
2C22×C4×Dic197C211yesmaximal, characteristic
3#27160983C212yesmaximal, complemented
4#27160976C212yesmaximal, complemented
5#27160966C212yesmaximal, complemented
6#27160962C212yesmaximal, complemented
7#27160954C211yesmaximal, characteristic
8#27160951C211yesmaximal, characteristic
9#27160942C211yesmaximal, characteristic
10#27160937C211yesmaximal, characteristic
11#381×C197C211yesmaximal, characteristic
12C22×Dic394C2211yescharacteristic
13C23×Dic197C2211yescharacteristic
14C23×Dic197C2211yescharacteristic
15#20263303—24no—
16C2×C4⋊Dic197C2212yes—
17C2×C4⋊Dic197C2211yescharacteristic
18C2×#15691337C2212yes—
19C2×#15691337C2212yes—
20C2×#15691337C2211yescharacteristic
21C2×#15691337C2211yescharacteristic
22C2×C22⋊Dic197C2211yescharacteristic
23C2×C22⋊Dic197C2211yescharacteristic
24C2×C4×Dic197—24no—
25C2×C4×Dic197C2212yes—
26C2×C4×Dic197C2212yes—
27#20263233C2212yes—
28#20263233C2212yes—
29#20263233C2212yes—
30#20263233C2212yes—
31#20263233C2212yes—
32C23×C4×C197C2211yescharacteristic
33C2×C22⋊C4×C197C2211yescharacteristic
34C2×C22⋊C4×C197C2211yescharacteristic
35#95×C197C2212yes—
36#95×C197C2212yes—
37C2×Dic394—22no—
38C2×Dic394—24no—
39C22×Dic197—24no—
40C22×Dic197—24no—
41C22×Dic197—22no—
42C22×Dic197D414yes—
43C22×Dic197C2312yes—
44C22×Dic197C2311yescharacteristic
45C22×Dic197C2311yescharacteristic
46C22×Dic197C2311yescharacteristic
47C22×Dic197C2312yes—
48C22×Dic197C2311yescharacteristic
49C4⋊Dic197—22no—
50C4⋊Dic197—24no—
51#15691337—22no—
52#15691337—22no—
53#15691337—24no—
54#15691337—24no—
55C22⋊Dic197—22no—
56C22⋊Dic197—22no—
57C4×Dic197—24no—
58C4×Dic197—24no—
59C24×C197C2311yescharacteristic
60C22×C4×C197—22no—
61C22×C4×C197—22no—
62C22×C4×C197Q814yescomplemented
63C22×C4×C197C2312yes—
64C22×C4×C197C2311yescharacteristic
65C22×C4×C197C2311yescharacteristic
66C22×C4×C197C2312yes—
67C22⋊C4×C197—22no—
68C22⋊C4×C197—22no—
69Dic394—44no—
70C2×Dic197—28no—
71C2×Dic197—22no—
72C2×Dic197—22no—
73C2×Dic197—22no—
74C2×Dic197—28no—
75C2×Dic197—21no—
76C2×Dic197—21no—
77C2×Dic197—22no—
78C2×Dic197—22no—
79C2×Dic197—22no—
80C2×Dic197—22no—
81C2×Dic197—22no—
82C2×Dic197C2×D412yes—
83C2×Dic197C2×D412yes—
84C2×Dic197#4114yes—
85C2×Dic197#4114yes—
86C23×C197—22no—
87C23×C197—22no—
88C23×C197C2×D412yes—
89C23×C197C2×Q812yes—
90C23×C197#4112yes—
91C23×C197C2411yescharacteristic
92C2×C4×C197—28no—
93C2×C4×C197—22no—
94C2×C4×C197—22no—
95C2×C4×C197—21no—
96C2×C4×C197—21no—
97C2×C4×C197—22no—
98C2×C4×C197—22no—
99C2×C4×C197—22no—
100C2×C4×C197—22no—
101C2×C4×C197—22no—
102C2×C4×C197C2×Q814yes—
103C22×C197—22no—
104C22×C197—22no—
105C22×C197—24no—
106C22×C197—24no—
107C22×C197C22⋊Q814yes—
108C22×C197C22×D411yescharacteristic
109C22×C197C22×Q811yescharacteristic
110C22×C197C2×#4112yes—
111C22×C197C2×#4112yes—
112C22×C197C2×#4111yescharacteristic
113C4×C197—24no—
114C4×C197—41no—
115C4×C197—41no—
116C4×C197—42no—
117Dic197—22no—
118Dic197—24no—
119Dic197—41no—
120Dic197—41no—
121Dic197—42no—
122Dic197#12314yes—
123C2×C197—24no—
124C2×C197C2×C22⋊Q811yescharacteristic
125C2×C197C2×#12311yescharacteristic
126C2×C197#52511yescharacteristic
127C2×C197#52912yes—
128C2×C197#53312yes—
129C197#285511yescharacteristic, complemented, Sylow
130#2855—1971nomaximal, Sylow
131C2×C22⋊Q8—1971no—
132C22×C42—1971no—
133#391—1972no—
134#389—1972no—
135#387—1972no—
136#387—1972no—
137#385—1971no—
138#385—1971no—
139#381—1971no—
140#381D19711yescharacteristic
141#380—1971no—
142C22×Q8—1971no—
143C23×C4—1971no—
144C23×C4—1971no—
145C23×C4C2×D19711yescharacteristic
146C22⋊Q8—3944no—
147C2×C4⋊C4—1971no—
148C2×C4⋊C4—1971no—
149C2×C4⋊C4—1971no—
150C2×C4⋊C4—1972no—
151C2×C4⋊C4—1972no—
152C2×C4⋊C4—1972no—
153C2×C22⋊C4—1971no—
154C2×C22⋊C4—1971no—
155C2×C22⋊C4C2×D19711yescharacteristic
156C2×C22⋊C4C2×D19711yescharacteristic
157C2×C42—3944no—
158C2×C42—1972no—
159C2×C42—1972no—
160#95—1972no—
161#95—1972no—
162#95—1972no—
163#95—1972no—
164#95—1972no—
165#95C2×D19712yes—
166#95C2×D19712yes—
167C24C22×D19711yescharacteristic
168C2×Q8—3942no—
169C2×Q8—3944no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—3942no—
173C22×C4—3944no—
174C22×C4—3944no—
175C22×C4—1972no—
176C22×C4—1971no—
177C22×C4—1971no—
178C22×C4—1974no—
179C22×C4—1972no—
180C22×C4—1971no—
181C22×C4—1971no—
182C22×C4Dic39414yescomplemented
183C22×C4C22×D19712yes—
184C22×C4C22×D19711yescharacteristic
185C22×C4C22×D19711yescharacteristic
186C22×C4C22×D19712yes—
187C4⋊C4—3944no—
188C4⋊C4—3944no—
189C4⋊C4—3942no—
190C4⋊C4—3942no—
191C4⋊C4—3944no—
192C4⋊C4—3942no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—3942no—
196C22⋊C4—3942no—
197C42—3944no—
198C42—3944no—
199C23—22no—
200C23—22no—
201C23D4×D19712yes—
202C23C2×Dic39412yes—
203C23#1569135712yes—
204C23C23×D19711yescharacteristic
205Q8—7884no—
206C2×C4—28no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—3941no—
217C2×C4—3941no—
218C2×C4—3942no—
219C2×C4—3942no—
220C2×C4—3942no—
221C2×C4—3942no—
222C2×C4—3942no—
223C2×C4—3948no—
224C2×C4—3942no—
225C2×C4—3942no—
226C2×C4—3942no—
227C2×C4—3948no—
228C2×C4—1972no—
229C2×C4—1974no—
230C2×C4—1972no—
231C2×C4—1974no—
232C2×C4C2×Dic39414yes—
233C22—24no—
234C22—24no—
235C22—22no—
236C22—22no—
237C22#2026330314yes—
238C22C2×D4×D19711yescharacteristic
239C22C22×Dic39411yescharacteristic
240C22#41×D19712yes—
241C22#41×D19712yes—
242C22C2×#1569135711yescharacteristic
243C4—24no—
244C4—42no—
245C4—41no—
246C4—41no—
247C4—7882no—
248C4—7881no—
249C4—7881no—
250C4—3942no—
251C4—3944no—
252C4—1974no—
253C2—24no—
254C2C2×#2026330311yescharacteristic
255C2#123×D19711yescharacteristic
256C2#2716158711yescharacteristic
257C2#2716160612yes—
258C2#2716163212yes—
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#381×C197C2
prime radicalC24×C197C23
hypercenter#381D197
hypocenterC197#2855
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #27160983C2semidirect
4. #27160976C2semidirect
5. #27160966C2semidirect
6. #27160962C2semidirect
62. C22×C4×C197Q8semidirect
129. C197#2855semidirect
182. C22×C4Dic394semidirect

Maximal subgroups

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

subgroupindex
C2×#202633032
C22×C4×Dic1972
#271609832
#271609762
#271609662
#271609622
#271609542
#271609512
#271609422
#271609372
#381×C1972
#2855197