Finite Group Databasebeta

← #20277852

Subgroups — #20277852

Order 6784.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13133185C211yesmaximal, characteristic
2#16810523C212yesmaximal, complemented
3#16810514C212yesmaximal, complemented
4#16810513C212yesmaximal, complemented
5#16810511C212yesmaximal, complemented
6#16810503C211yesmaximal, characteristic, complemented
7#16810500C212yesmaximal, complemented
8#16810478C212yesmaximal, complemented
9C53×#390C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C23×Dic53C2211yescharacteristic, complemented
13#13133185C418yescomplemented
14C2×#11719919C2212yescomplemented
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2211yescharacteristic
17C2×C22⋊Dic53C2211yescharacteristic, complemented
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×C4×Dic53C2212yescomplemented
24#13133124C2212yescomplemented
25#13133124C2212yescomplemented
26#13133124C2212yescomplemented
27C22×D4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic, complemented
29C2×C22⋊C4×C53C2212yes—
30C2×C22⋊C4×C53C2212yescomplemented
31C2×C42×C53C2211yescharacteristic, complemented
32C24×C53C2311yescharacteristic
33C24×C53C2311yescharacteristic
34C23×D53—42no—
35C23×D53—42no—
36C23×D53—41no—
37C23×D53—41no—
38C23×D53C2×C414yescomplemented
39C23×D53C2312yes—
40C2×D4×C53—41no—
41C2×D4×C53—41no—
42C2×D4×C53C2×C414yes—
43C22×C4×C53C2312yes—
44C22×C4×C53C2312yes—
45C22×C4×C53C2311yescharacteristic
46C2×C22⋊D53—42no—
47C2×C22⋊D53—42no—
48C2×C22⋊D53C2×C418yes—
49C22×Dic53—41no—
50C22×Dic53—42no—
51C22×Dic53C2312yes—
52C22×Dic53C2312yes—
53C22×Dic53C2312yes—
54C4⋊C4×C53—41no—
55C22⋊C4×C53—42no—
56C22⋊C4×C53—42no—
57C42×C53—41no—
58C22⋊Dic53—42no—
59C22⋊Dic53—41no—
60C22⋊Dic53C2×C414yescomplemented
61#11719921—42no—
62#11719921—42no—
63#11719921—42no—
64#11719921—42no—
65#11719921C2×C418yescomplemented
66#11719919—42no—
67C4×Dic53—42no—
68C23×C53—22no—
69C23×C53—21no—
70C23×C53—22no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53—21no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53C22×C412yes—
77C23×C53C22×C412yes—
78C23×C53C2411yescharacteristic
79C22×D53—22no—
80C22×D53—22no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—48no—
86C22×D53—44no—
87C22×D53—44no—
88C22×D53—48no—
89C22×D53—44no—
90C22×D53C22×C414yes—
91D4×C53—44no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53C22×C412yes—
101C22⋊D53—48no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—44no—
111C2×Dic53C22×C414yes—
112C22×C53—24no—
113C22×C53—22no—
114C22×C53—22no—
115C22×C53—24no—
116C22×C53—41no—
117C22×C53—42no—
118C22×C53—41no—
119C22×C53—42no—
120C22×C53—41no—
121C22×C53—41no—
122C22×C53C23×C411yescharacteristic
123C22×C53H(5,𝔽2)12yes—
124C22×C53H(5,𝔽2)11yescharacteristic
125C22×C53H(5,𝔽2)11yescharacteristic
126C22×C53H(5,𝔽2)12yes—
127C2×D53—24no—
128C2×D53—24no—
129C2×D53—24no—
130C2×D53—416no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—42no—
135C4×C53—22no—
136C4×C53—42no—
137C4×C53—42no—
138Dic53—24no—
139Dic53—42no—
140Dic53—42no—
141#2375—531nomaximal, Sylow
142C2×C53—42no—
143C2×C53—42no—
144C2×C53#51812yes—
145C2×C53#51811yescharacteristic
146C2×C53#56011yescharacteristic
147C2×C53#56012yes—
148C2×C53H(3,𝔽4)11yescharacteristic
149D53—44no—
150C2×#120—531no—
151#390—532no—
152#390D5311yescharacteristic, complemented
153#388—532no—
154#388—532no—
155#388—532no—
156#386—531no—
157#386—532no—
158#379—532no—
159C53#237511yescharacteristic, complemented, Sylow
160C25—531no—
161C22×D4—532no—
162C22×D4C2×D5311yescharacteristic
163C23×C4—531no—
164#120—538no—
165C2×C4⋊C4—532no—
166C2×C4⋊C4C2×D5311yescharacteristic, complemented
167C2×C22⋊C4—531no—
168C2×C22⋊C4—532no—
169C2×C22⋊C4—532no—
170C2×C22⋊C4—531no—
171C2×C22⋊C4—532no—
172C2×C22⋊C4—532no—
173C2×C22⋊C4—532no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4C2×D5312yes—
176C2×C22⋊C4C2×D5312yescomplemented
177C2×C42—532no—
178C2×C42C2×D5311yescharacteristic, complemented
179#95—532no—
180#95—532no—
181#95—532no—
182C24—532no—
183C24—534no—
184C24—2122no—
185C24—2121no—
186C24—2121no—
187C24—2122no—
188C24C22×D5311yescharacteristic
189C24C22×D5311yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—538no—
193C2×D4—2122no—
194C2×D4—2122no—
195C2×D4C4×D5314yes—
196C22×C4—532no—
197C22×C4—532no—
198C22×C4—532no—
199C22×C4—2121no—
200C22×C4—2122no—
201C22×C4C22×D5312yes—
202C22×C4C22×D5312yes—
203C22×C4C22×D5311yescharacteristic
204C4⋊C4—41no—
205C4⋊C4—2122no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—534no—
209C22⋊C4—538no—
210C22⋊C4—2122no—
211C22⋊C4—2122no—
212C22⋊C4—2122no—
213C22⋊C4—2122no—
214C22⋊C4—2121no—
215C22⋊C4—2122no—
216C42—41no—
217C42—2122no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—534no—
227C23—1062no—
228C23—1062no—
229C23—1062no—
230C23—1062no—
231C23—1062no—
232C23—1062no—
233C23—2128no—
234C23—2124no—
235C23—2124no—
236C23—2128no—
237C23—2124no—
238C23C2×C4×D5312yes—
239C23C2×C4×D5312yes—
240C23C23×D5311yescharacteristic
241D4—44no—
242D4—2128no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—534no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—2124no—
261C2×C4C2×C4×D5312yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—1064no—
273C22—1064no—
274C22—1064no—
275C22—21216no—
276C22—2122no—
277C22—2122no—
278C22—2122no—
279C22—2122no—
280C22C22×C4×D5311yescharacteristic
281C22#1313326212yes—
282C22#1313326211yescharacteristic
283C22#1313326211yescharacteristic
284C22#1313326312yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—1064no—
289C4—2122no—
290C4—2122no—
291C2—42no—
292C2—42no—
293C2—2124no—
294C2#1681109512yes—
295C2#1681109611yescharacteristic
296C2#1681136011yescharacteristic
297C2#1681136112yes—
298C2#1681136511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#390C2
socleC23×C53C24
hypercenter#390D53
hypocenterC53#2375
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #16810523C2semidirect
3. #16810514C2semidirect
4. #16810513C2semidirect
5. #16810511C2semidirect
6. #16810503C2semidirect
7. #16810500C2semidirect
8. #16810478C2semidirect
9. C53×#390C2semidirect
12. C23×Dic53C22semidirect
13. #13133185C4semidirect
14. C2×#11719919C22semidirect
17. C2×C22⋊Dic53C22semidirect
23. C2×C4×Dic53C22semidirect
24. #13133124C22semidirect
25. #13133124C22semidirect
26. #13133124C22semidirect
28. C2×C4⋊C4×C53C22semidirect
30. C2×C22⋊C4×C53C22semidirect
31. C2×C42×C53C22semidirect
38. C23×D53C2×C4semidirect
60. C22⋊Dic53C2×C4semidirect
65. #11719921C2×C4semidirect
152. #390D53semidirect
159. C53#2375semidirect
166. C2×C4⋊C4C2×D53semidirect
176. C2×C22⋊C4C2×D53semidirect
178. C2×C42C2×D53semidirect