Finite Group Databasebeta

← #20284803

Subgroups — #20284803

Order 6784.

488 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 119 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
1#16811168C211yesmaximal, characteristic, complemented
2C2×#13133179C211yesmaximal, characteristic, complemented
3#16810941C211yesmaximal, characteristic, complemented
4#16810912C212yesmaximal, complemented
5#16810817C212yesmaximal, complemented
6#16810759C211yesmaximal, characteristic, complemented
7#16810756C211yesmaximal, characteristic, complemented
8#16810734C212yesmaximal, complemented
9#16810732C212yesmaximal, complemented
10#16810677C211yesmaximal, characteristic, complemented
11C53×#481C211yesmaximal, characteristic, complemented
12C2×#11719947—21no—
13C22×D212C2211yescharacteristic
14C22×C4×D53—21no—
15C2×D424C2212yes—
16C2×#11719922C2212yes—
17#13133216C2211yescharacteristic, complemented
18#13133213C2211yescharacteristic, complemented
19#13133203—21no—
20#13133191C2211yescharacteristic
21#13133189—21no—
22D4×Dic53C2211yescharacteristic, complemented
23#13133179—21no—
24#13133179—21no—
25#13133179C2212yes—
26#13133179C2212yes—
27#13133179C2211yescharacteristic, complemented
28#13133177—21no—
29C2×#11719921—21no—
30C2×C4×Dic53C2211yescharacteristic
31#13133148C2212yescomplemented
32#13133146C2212yescomplemented
33#13133139C2212yes—
34#13133137C2212yes—
35#13133137C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37#13133127C2212yescomplemented
38#121×C53C2211yescharacteristic
39C2×C4⋊C4×C53C2211yescharacteristic
40#106×C53C2212yescomplemented
41#102×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C23×D53—21no—
44C2×D4×C53—21no—
45C2×D4×C53C2311yescharacteristic
46C22×C4×C53—21no—
47C22×C4×C53C2311yescharacteristic
48C2×C22⋊D53—21no—
49C2×C22⋊D53—21no—
50C2×C22⋊D53—21no—
51C22×Dic53—21no—
52C22×Dic53D411yescharacteristic
53C22×Dic53D411yescharacteristic
54#11719947—44no—
55C2×D212—21no—
56C2×D212—24no—
57C2×D212C2311yescharacteristic
58C2×D212C2312yes—
59C2×C4×D53—21no—
60C2×C4×D53—22no—
61C2×C4×D53—22no—
62C2×C4×D53—44no—
63C2×Dic106—21no—
64C2×C8×C53C2312yes—
65C4⋊C4×C53—21no—
66C4⋊C4×C53C2312yes—
67C4⋊C4×C53C2311yescharacteristic
68C22⋊C4×C53—21no—
69C22⋊Dic53—21no—
70#11719922—24no—
71#11719921—21no—
72#11719921—22no—
73#11719921—22no—
74#11719921—21no—
75C4⋊Dic53C2311yescharacteristic
76#11719919—21no—
77C4×Dic53D411yescharacteristic
78C4×Dic53D411yescharacteristic
79C4×Dic53C2311yescharacteristic
80C4×Dic53C2311yescharacteristic
81C2×#88322C2312yes—
82D424—24no—
83C23×C53—21no—
84C23×C53C2×D411yescharacteristic
85C22×D53—24no—
86C22×D53—22no—
87C22×D53—21no—
88C22×D53—42no—
89C22×D53—42no—
90D4×C53—22no—
91D4×C53—42no—
92C2×C4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—42no—
96C2×C4×C53C2×D411yescharacteristic
97C2×C4×C53C2411yescharacteristic
98C22⋊D53—42no—
99C22⋊D53—42no—
100C22⋊D53—42no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—21no—
104C2×Dic53—42no—
105C2×Dic53C2×D411yescharacteristic
106C2×Dic53C2×D411yescharacteristic
107C2×Dic53C2×D411yescharacteristic
108C2×Dic53C2×D411yescharacteristic
109D212—24no—
110D212—22no—
111D212—22no—
112D212#4114yescomplemented
113C4×D53—44no—
114C4×D53—42no—
115C4×D53—44no—
116Dic106—42no—
117C8×C53—22no—
118#88322—22no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53AGL(1,ℤ/8)12yes—
125C22×C53C22×D411yescharacteristic
126C2×D53—24no—
127C2×D53—48no—
128C2×D53—42no—
129C2×D53—42no—
130C2×D53—41no—
131C2×D53—41no—
132C2×D53—41no—
133C4×C53—21no—
134C4×C53—41no—
135C4×C53—42no—
136C4×C53C2×#4112yes—
137Dic53—21no—
138Dic53—21no—
139Dic53—21no—
140Dic53—41no—
141Dic53AGL(1,ℤ/8)12yes—
142#3109—531nomaximal, Sylow
143C2×C53—22no—
144C2×C53—81no—
145C2×C53#52511yescharacteristic
146C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
148D53—44no—
149D53—81no—
150#532—531no—
151C2×C4×D4—531no—
152#481—531no—
153#481D5311yescharacteristic, complemented
154#476—532no—
155#459—532no—
156#447—531no—
157#447—531no—
158#442—532no—
159#442—532no—
160#432—531no—
161C53#310911yescharacteristic, complemented, Sylow
162C2×#41—1061no—
163C22×D4—531no—
164C23×C4—1061no—
165C2×D8—532no—
166C2×D8—532no—
167C4⋊Q8—531no—
168#127—531no—
169#124—1061no—
170#121—531no—
171#121—1061no—
172#121C2×D5311yescharacteristic
173C4×D4—531no—
174C4×D4—532no—
175C4×D4—532no—
176C4×D4—531no—
177C4×D4—1061no—
178C4×D4—1061no—
179C4×D4—1061no—
180C2×C4⋊C4C2×D5311yescharacteristic
181C2×C22⋊C4—1061no—
182C2×C42—531no—
183#106—532no—
184#106C2×D5312yescomplemented
185C4⋊C8—532no—
186#102—532no—
187#102—532no—
188#102—532no—
189#102C2×D5312yescomplemented
190C22⋊C8—531no—
191C22⋊C8C2×D5311yescharacteristic
192#97—532no—
193C24—1061no—
194#41—2124no—
195C2×Q8—1061no—
196C2×D4—21no—
197C2×D4—532no—
198C2×D4—531no—
199C2×D4—1064no—
200C2×D4—1061no—
201C2×D4—1061no—
202C2×D4—1061no—
203C2×D4—1061no—
204C2×D4C22×D5311yescharacteristic
205C22×C4—21no—
206C22×C4—531no—
207C22×C4—531no—
208C22×C4—1062no—
209C22×C4—1062no—
210C22×C4—1061no—
211C22×C4—1061no—
212C22×C4—2124no—
213C22×C4C22×D5311yescharacteristic
214D8—1064no—
215D8—1064no—
216C2×C8—532no—
217C2×C8C22×D5312yes—
218C4⋊C4—21no—
219C4⋊C4—531no—
220C4⋊C4—1061no—
221C4⋊C4C22×D5311yescharacteristic
222C4⋊C4C22×D5312yes—
223C22⋊C4—21no—
224C22⋊C4—1062no—
225C22⋊C4—1062no—
226C22⋊C4—1061no—
227C22⋊C4—1061no—
228C22⋊C4—1061no—
229C42—531no—
230C42—531no—
231C42—531no—
232C42—531no—
233C23—21no—
234C23—1064no—
235C23—1062no—
236C23—1061no—
237C23—2122no—
238C23—2122no—
239C23D4×D5311yescharacteristic
240Q8—2122no—
241D4—22no—
242D4—42no—
243D4—534no—
244D4—1064no—
245D4—1062no—
246D4—1062no—
247D4—2122no—
248D4—2122no—
249D4—2122no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—531no—
255C2×C4—531no—
256C2×C4—531no—
257C2×C4—531no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—1061no—
261C2×C4—2124no—
262C2×C4—2124no—
263C2×C4—2122no—
264C2×C4—2122no—
265C2×C4D4×D5311yescharacteristic
266C2×C4C23×D5311yescharacteristic
267C8—22no—
268C8—1062no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—1064no—
275C22—2128no—
276C22—2121no—
277C22—2121no—
278C22—2121no—
279C22—2122no—
280C22—2122no—
281C22#1313324412yes—
282C22C2×D4×D5311yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—532no—
287C4—1061no—
288C4—1061no—
289C4—1061no—
290C4—2121no—
291C4#41×D5312yes—
292C2—22no—
293C2—81no—
294C2—4241no—
295C2—2124no—
296C2#1681113111yescharacteristic
297C2AGL(1,ℤ/8)×D5311yescharacteristic
298C2C2×#1313324411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#481C2
socleC22×C53C22×D4
hypercenter#481D53
hypocenterC53#3109
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #16811168C2semidirect
2. C2×#13133179C2semidirect
3. #16810941C2semidirect
4. #16810912C2semidirect
5. #16810817C2semidirect
6. #16810759C2semidirect
7. #16810756C2semidirect
8. #16810734C2semidirect
9. #16810732C2semidirect
10. #16810677C2semidirect
11. C53×#481C2semidirect
17. #13133216C22semidirect
18. #13133213C22semidirect
22. D4×Dic53C22semidirect
27. #13133179C22semidirect
31. #13133148C22semidirect
32. #13133146C22semidirect
35. #13133137C22semidirect
37. #13133127C22semidirect
40. #106×C53C22semidirect
41. #102×C53C22semidirect
112. D212#41semidirect
153. #481D53semidirect
161. C53#3109semidirect
184. #106C2×D53semidirect
189. #102C2×D53semidirect