Finite Group Databasebeta

← #20278992

Subgroups — #20278992

Order 6784.

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

normal subgroups only

This group has 700 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
1C2×C22⋊C4×D53C212yesmaximal
2#16810551C212yesmaximal, complemented
3#16810548C211yesmaximal, characteristic, complemented
4#16810497C212yesmaximal, complemented
5#16810494C212yesmaximal, complemented
6#16810485C212yesmaximal, complemented
7#16810485C212yesmaximal, complemented
8#16810478C211yesmaximal, characteristic
9C53×#397C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C22×C4×D53C2212yes—
12C22×C4×D53C2212yes—
13C2×C4⋊Dic53C2211yescharacteristic, complemented
14C2×#11719919C2212yescomplemented
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2211yescharacteristic
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C22⋊C4×D53—28no—
22#13133124C2212yescomplemented
23#13133124C2211yescharacteristic, complemented
24#13133124C2212yescomplemented
25#13133124C2212yescomplemented
26#13133124C2212yescomplemented
27C2×C4⋊C4×C53C2211yescharacteristic, complemented
28C2×C22⋊C4×C53C2212yes—
29C2×C22⋊C4×C53C2211yescharacteristic
30#95×C53C2212yescomplemented
31#95×C53C2211yescharacteristic, complemented
32C24×C53C2311yescharacteristic
33C23×D53—24no—
34C23×D53—24no—
35C23×D53—41no—
36C23×D53—41no—
37C23×D53D414yes—
38C23×D53C2312yes—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C2×C4×D53—24no—
47C2×C4×D53—24no—
48C2×C4×D53—24no—
49C2×C4×D53—24no—
50C2×C4×D53—42no—
51C2×C4×D53—42no—
52C4⋊C4×C53—41no—
53C22⋊C4×C53—24no—
54C22⋊C4×C53—41no—
55C22⋊Dic53—24no—
56C22⋊Dic53—41no—
57#11719921—24no—
58#11719921—24no—
59#11719921—24no—
60#11719921—24no—
61C4⋊Dic53—41no—
62#11719919—42no—
63C23×C53—22no—
64C23×C53—22no—
65C23×C53—21no—
66C23×C53—21no—
67C23×C53C2×D412yes—
68C23×C53C2411yescharacteristic
69C22×D53—22no—
70C22×D53—22no—
71C22×D53—22no—
72C22×D53—22no—
73C22×D53—216no—
74C22×D53—44no—
75C22×D53—48no—
76C22×D53—44no—
77C22×D53—44no—
78C22×D53C2×D414yes—
79C22×D53#4114yes—
80C22×D53#4114yes—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C4×D53—48no—
101C4×D53—48no—
102C22×C53—24no—
103C22×C53—22no—
104C22×C53—42no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53C22×D411yescharacteristic
108C22×C53C2×#4112yes—
109C22×C53C2×#4112yes—
110C22×C53H(5,𝔽2)11yescharacteristic
111C22×C53H(5,𝔽2)11yescharacteristic
112C2×D53—24no—
113C2×D53—24no—
114C2×D53—24no—
115C2×D53—24no—
116C2×D53—24no—
117C2×D53—24no—
118C2×D53—416no—
119C2×D53#12318yes—
120C4×C53—42no—
121C4×C53—42no—
122C4×C53—41no—
123C4×C53—41no—
124Dic53—42no—
125Dic53—42no—
126Dic53—42no—
127#2483—531nomaximal, Sylow
128C2×C53—42no—
129C2×C53C2×#12312yes—
130C2×C53#53411yescharacteristic
131C2×C53#55112yes—
132C2×C53#55111yescharacteristic
133C2×C53#55111yescharacteristic
134D53—28no—
135C22×C22⋊C4—532no—
136#397—531no—
137#397—532no—
138#397D5311yescharacteristic, complemented
139#385—532no—
140#385—532no—
141#381—532no—
142#381—532no—
143#379—531no—
144C53#248311yescharacteristic, complemented, Sylow
145C25—531no—
146C23×C4—532no—
147C23×C4—532no—
148C2×C4⋊C4—532no—
149C2×C4⋊C4—531no—
150C2×C4⋊C4C2×D5311yescharacteristic, complemented
151C2×C22⋊C4—532no—
152C2×C22⋊C4—532no—
153C2×C22⋊C4—532no—
154C2×C22⋊C4—532no—
155C2×C22⋊C4—532no—
156C2×C22⋊C4—531no—
157C2×C22⋊C4—1068no—
158C2×C22⋊C4C2×D5312yes—
159C2×C22⋊C4C2×D5311yescharacteristic
160#95—531no—
161#95—532no—
162#95—532no—
163#95—532no—
164#95—532no—
165#95C2×D5311yescharacteristic, complemented
166#95C2×D5312yescomplemented
167C24—532no—
168C24—534no—
169C24—1064no—
170C24—1064no—
171C24—2121no—
172C24—2121no—
173C24C22×D5311yescharacteristic
174C22×C4—532no—
175C22×C4—532no—
176C22×C4—532no—
177C22×C4—1064no—
178C22×C4—1064no—
179C22×C4—1064no—
180C22×C4—1064no—
181C22×C4—2122no—
182C22×C4—2122no—
183C22×C4C22×D5312yes—
184C22×C4C22×D5311yescharacteristic
185C22×C4C22×D5311yescharacteristic
186C22×C4C22×D5312yes—
187C4⋊C4—41no—
188C4⋊C4—2122no—
189C4⋊C4—2121no—
190C22⋊C4—24no—
191C22⋊C4—41no—
192C22⋊C4—1064no—
193C22⋊C4—1064no—
194C22⋊C4—1064no—
195C22⋊C4—1064no—
196C22⋊C4—1064no—
197C22⋊C4—2121no—
198C23—22no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—534no—
203C23—534no—
204C23—534no—
205C23—10616no—
206C23—1062no—
207C23—1062no—
208C23—1062no—
209C23—1062no—
210C23—2124no—
211C23—2128no—
212C23—2124no—
213C23—2124no—
214C23D4×D5312yes—
215C23C23×D5311yescharacteristic
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—2128no—
236C2×C4—2128no—
237C22—24no—
238C22—22no—
239C22—42no—
240C22—41no—
241C22—41no—
242C22—538no—
243C22—1064no—
244C22—1064no—
245C22—1064no—
246C22—1064no—
247C22—1064no—
248C22—1064no—
249C22—21216no—
250C22C2×D4×D5311yescharacteristic
251C22#41×D5312yes—
252C22#41×D5312yes—
253C22#1313326211yescharacteristic
254C22#1313326311yescharacteristic
255C4—41no—
256C4—41no—
257C4—42no—
258C4—42no—
259C4—2122no—
260C4—2122no—
261C4—2122no—
262C2—42no—
263C2—1068no—
264C2#123×D5312yes—
265C2#1681118111yescharacteristic
266C2#1681129812yes—
267C2#1681130111yescharacteristic
268C2#1681130311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2483
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #16810551C2semidirect
3. #16810548C2semidirect
4. #16810497C2semidirect
5. #16810494C2semidirect
6. #16810485C2semidirect
7. #16810485C2semidirect
9. C53×#397C2semidirect
13. C2×C4⋊Dic53C22semidirect
14. C2×#11719919C22semidirect
22. #13133124C22semidirect
23. #13133124C22semidirect
24. #13133124C22semidirect
25. #13133124C22semidirect
26. #13133124C22semidirect
27. C2×C4⋊C4×C53C22semidirect
30. #95×C53C22semidirect
31. #95×C53C22semidirect
138. #397D53semidirect
144. C53#2483semidirect
150. C2×C4⋊C4C2×D53semidirect
165. #95C2×D53semidirect
166. #95C2×D53semidirect