Finite Group Databasebeta

← #20285734

Subgroups — #20285734

Order 6784.

490 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 113 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#127×D53C211yesmaximal, characteristic, complemented
2#16811158C211yesmaximal, characteristic, complemented
3#16811012C211yesmaximal, characteristic, complemented
4#16811010C211yesmaximal, characteristic, complemented
5#16810989C211yesmaximal, characteristic, complemented
6#16810822C212yesmaximal, complemented
7#16810817C212yesmaximal, complemented
8#16810775C212yesmaximal, complemented
9#16810758C212yesmaximal, complemented
10#16810678C211yesmaximal, characteristic, complemented
11C53×#490C211yesmaximal, characteristic, complemented
12C2×D4×D53—21no—
13C2×D4×D53—21no—
14C2×D4×D53C2212yes—
15C2×#11719913C2212yescomplemented
16C2×#11719923C2212yescomplemented
17C2×D424C2212yescomplemented
18C2×#11719922C2212yescomplemented
19#13133214C2211yescharacteristic
20#13133213—21no—
21#13133213—21no—
22#13133213C2212yes—
23#13133212—21no—
24#13133205C2211yescharacteristic, complemented
25#13133202—21no—
26#13133197C2211yescharacteristic
27#13133192C2211yescharacteristic
28#13133182C2211yescharacteristic, complemented
29#13133179—21no—
30#13133179C2211yescharacteristic, complemented
31#13133175—21no—
32C42×D53C2211yescharacteristic
33#13133146C2212yescomplemented
34#13133137C2212yescomplemented
35#13133137C2212yescomplemented
36#13133129C2212yes—
37#13133128C2211yescharacteristic, complemented
38#127×C53C2211yescharacteristic
39#125×C53C2211yescharacteristic, complemented
40#102×C53C2212yescomplemented
41#102×C53C2212yescomplemented
42#97×C53C2211yescharacteristic, complemented
43C23×D53—22no—
44C2×D4×C53—21no—
45C2×D4×C53—21no—
46C2×D4×C53C2312yes—
47C2×C22⋊D53—22no—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—22no—
50D4×D53—42no—
51D4×D53—42no—
52D4×D53—44no—
53D4×D53—44no—
54C2×D212—21no—
55C2×D212—21no—
56C2×D212—21no—
57C2×D212C2312yes—
58C2×D212C2311yescharacteristic
59C2×C4×D53—21no—
60C2×C4×D53D411yescharacteristic
61C2×C4×D53D411yescharacteristic
62C2×C4×D53C2311yescharacteristic
63C2×Dic106C2311yescharacteristic
64C2×C8×C53C2312yes—
65C4⋊C4×C53—21no—
66C4⋊C4×C53—21no—
67C4⋊C4×C53C2311yescharacteristic
68C4⋊C4×C53C2311yescharacteristic
69C42×C53C2311yescharacteristic
70#11719923—42no—
71#11719922—42no—
72#11719921—21no—
73#11719921—21no—
74#11719921—21no—
75#11719921—21no—
76#11719921—21no—
77C4⋊Dic53—21no—
78#11719919—21no—
79C4×Dic53—21no—
80C4×Dic53D411yescharacteristic
81C4×Dic53D411yescharacteristic
82C4×Dic53C2311yescharacteristic
83C2×#88322C2312yes—
84D424—42no—
85#11719913—42no—
86C23×C53—22no—
87C22×D53—21no—
88C22×D53—22no—
89C22×D53—44no—
90C22×D53—44no—
91C22×D53—44no—
92C22×D53C2×D411yescharacteristic
93D4×C53—22no—
94D4×C53—22no—
95D4×C53—42no—
96D4×C53—42no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53C2×D411yescharacteristic
100C2×C4×C53C2×D411yescharacteristic
101C2×C4×C53C2411yescharacteristic
102C22⋊D53—44no—
103C22⋊D53—44no—
104C22⋊D53—44no—
105C2×Dic53—21no—
106C2×Dic53C2×D411yescharacteristic
107C2×Dic53C2×D411yescharacteristic
108C2×Dic53C2×D411yescharacteristic
109D212—22no—
110D212—22no—
111D212—22no—
112D212—42no—
113D212—42no—
114D212—42no—
115C4×D53—22no—
116C4×D53—22no—
117C4×D53—22no—
118C4×D53—42no—
119Dic106—22no—
120C8×C53—22no—
121#88322—22no—
122C22×C53—42no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53C22×D411yescharacteristic
126C2×D53—21no—
127C2×D53—21no—
128C2×D53—42no—
129C2×D53—41no—
130C2×D53—41no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—41no—
134C2×D53—88no—
135C2×D53AGL(1,ℤ/8)12yes—
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—41no—
139C4×C53—41no—
140C4×C53H(5,𝔽2)12yes—
141Dic53—21no—
142Dic53—21no—
143Dic53—41no—
144Dic53AGL(1,ℤ/8)12yes—
145#3217—531nomaximal, Sylow
146C2×C53—82no—
147C2×C53#53511yescharacteristic
148C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
150D53—22no—
151D53—81no—
152D53—82no—
153C2×#127—531no—
154#529—531no—
155#496—531no—
156#496—531no—
157#490—531no—
158#490D5311yescharacteristic, complemented
159#460—532no—
160#459—532no—
161#450—532no—
162#447—532no—
163#432—531no—
164C53#321711yescharacteristic, complemented, Sylow
165C22×D4—532no—
166C22×D4—1061no—
167C22×D4—1061no—
168C2×SD16—532no—
169C2×SD16—532no—
170C2×D8—532no—
171C2×D8—532no—
172#127—532no—
173#127—531no—
174#127—1061no—
175#127—1061no—
176#127C2×D5311yescharacteristic
177#126—1061no—
178#125C2×D5311yescharacteristic, complemented
179#124—531no—
180#123—1061no—
181C22⋊Q8—531no—
182#121—531no—
183C4×Q8—531no—
184C4×D4—531no—
185C4×D4—1061no—
186#117—1061no—
187C2×C42—531no—
188C4⋊C8—532no—
189#102—532no—
190#102—532no—
191#102C2×D5312yescomplemented
192#102C2×D5312yescomplemented
193C22⋊C8—532no—
194#97—531no—
195#97C2×D5311yescharacteristic, complemented
196C24—1062no—
197C2×Q8—531no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—532no—
201C2×D4—531no—
202C2×D4—1061no—
203C2×D4—1062no—
204C2×D4—1062no—
205C2×D4—1062no—
206C2×D4—1061no—
207C2×D4—1061no—
208C2×D4—2122no—
209C2×D4—2122no—
210C2×D4—2124no—
211C2×D4—2124no—
212C2×D4C22×D5312yes—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—1061no—
217SD16—2122no—
218SD16—2122no—
219D8—2122no—
220D8—2122no—
221C2×C8—532no—
222C2×C8C22×D5312yes—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—1061no—
226C4⋊C4—1061no—
227C4⋊C4C22×D5311yescharacteristic
228C4⋊C4C22×D5311yescharacteristic
229C22⋊C4—1061no—
230C22⋊C4—1061no—
231C22⋊C4—1061no—
232C22⋊C4—1061no—
233C22⋊C4—1061no—
234C42—531no—
235C42—531no—
236C42—531no—
237C42—1061no—
238C42C22×D5311yescharacteristic
239C23—22no—
240C23—531no—
241C23—1061no—
242C23—1062no—
243C23—2124no—
244C23—2124no—
245C23—2124no—
246Q8—1062no—
247D4—22no—
248D4—22no—
249D4—42no—
250D4—42no—
251D4—1062no—
252D4—1062no—
253D4—1062no—
254D4—2122no—
255D4—2122no—
256D4—2122no—
257D4—2124no—
258D4—2124no—
259D4—2124no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—531no—
263C2×C4—531no—
264C2×C4—531no—
265C2×C4—1062no—
266C2×C4—1062no—
267C2×C4—1062no—
268C2×C4—1061no—
269C2×C4—2122no—
270C2×C4D4×D5311yescharacteristic
271C2×C4D4×D5311yescharacteristic
272C2×C4C23×D5311yescharacteristic
273C8—22no—
274C8—1062no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—532no—
279C22—1061no—
280C22—1061no—
281C22—4248no—
282C22—2121no—
283C22—2122no—
284C22—2122no—
285C22—2121no—
286C22—2122no—
287C22—2121no—
288C22C2×D4×D5311yescharacteristic
289C4—21no—
290C4—21no—
291C4—41no—
292C4—41no—
293C4—532no—
294C4—1061no—
295C4—1061no—
296C4—2121no—
297C4#1313326312yes—
298C2—82no—
299C2—1062no—
300C2—4242no—
301C2—4241no—
302C2#1681119111yescharacteristic
303C2AGL(1,ℤ/8)×D5311yescharacteristic
304C2AGL(1,ℤ/8)×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#490C2
socleC22×C53C22×D4
hypercenter#490D53
hypocenterC53#3217
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #127×D53C2semidirect
2. #16811158C2semidirect
3. #16811012C2semidirect
4. #16811010C2semidirect
5. #16810989C2semidirect
6. #16810822C2semidirect
7. #16810817C2semidirect
8. #16810775C2semidirect
9. #16810758C2semidirect
10. #16810678C2semidirect
11. C53×#490C2semidirect
15. C2×#11719913C22semidirect
16. C2×#11719923C22semidirect
17. C2×D424C22semidirect
18. C2×#11719922C22semidirect
24. #13133205C22semidirect
28. #13133182C22semidirect
30. #13133179C22semidirect
33. #13133146C22semidirect
34. #13133137C22semidirect
35. #13133137C22semidirect
37. #13133128C22semidirect
39. #125×C53C22semidirect
40. #102×C53C22semidirect
41. #102×C53C22semidirect
42. #97×C53C22semidirect
158. #490D53semidirect
164. C53#3217semidirect
178. #125C2×D53semidirect
191. #102C2×D53semidirect
192. #102C2×D53semidirect
195. #97C2×D53semidirect