Finite Group Databasebeta

← #20285655

Subgroups — #20285655

Order 6784.

488 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 123 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#16811177C211yesmaximal, characteristic, complemented
2#127×D53C211yesmaximal, characteristic
3#16810994C212yesmaximal, complemented
4#16810968C212yesmaximal, complemented
5#16810824C211yesmaximal, characteristic, complemented
6#16810823C211yesmaximal, characteristic, complemented
7#16810820C211yesmaximal, characteristic, complemented
8#16810775C212yesmaximal, complemented
9#16810772C212yesmaximal, complemented
10#16810675C211yesmaximal, characteristic, complemented
11C53×#460C211yesmaximal, characteristic, complemented
12C2×D4×D53—21no—
13C2×D4×D53—21no—
14C2×D4×D53C2212yes—
15C2×#11719913C2212yescomplemented
16C2×#11719923C2212yescomplemented
17C2×#11719924C2212yescomplemented
18#13133216C2211yescharacteristic, complemented
19#13133215C2211yescharacteristic, complemented
20#13133214C2211yescharacteristic
21#13133213—21no—
22#13133213—21no—
23#13133213C2212yes—
24#13133207—21no—
25#13133197C2211yescharacteristic
26#13133195—21no—
27#13133193C2211yescharacteristic
28C4×Dic106—21no—
29C4×Dic106C2211yescharacteristic, complemented
30#13133174—21no—
31C42×D53C2211yescharacteristic
32C4⋊#88322C2211yescharacteristic, complemented
33#13133139C2212yescomplemented
34#13133138C2212yescomplemented
35#13133137C2212yescomplemented
36#13133129C2212yes—
37C8×Dic53C2212yescomplemented
38C2×SD16×C53C2212yescomplemented
39#127×C53C2211yescharacteristic
40C4×Q8×C53C2211yescharacteristic, complemented
41C4⋊C8×C53C2211yescharacteristic, complemented
42#102×C53C2212yescomplemented
43C23×D53—22no—
44C2×Q8×C53C2311yescharacteristic
45C2×D4×C53—21no—
46C2×D4×C53—21no—
47C2×D4×C53C2312yes—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—22no—
50C2×C22⋊D53—22no—
51D4×D53—42no—
52D4×D53—42no—
53D4×D53—44no—
54D4×D53—44no—
55C2×D212—21no—
56C2×D212—21no—
57C2×D212C2312yes—
58C2×C4×D53D411yescharacteristic
59C2×C4×D53D411yescharacteristic
60C2×C4×D53C2311yescharacteristic
61C2×Dic106—21no—
62C2×Dic106C2311yescharacteristic
63SD16×C53—42no—
64C2×C8×C53C2312yes—
65C4⋊C4×C53—21no—
66C4⋊C4×C53C2311yescharacteristic
67C42×C53—21no—
68C42×C53C2311yescharacteristic
69#11719924—42no—
70#11719923—42no—
71#11719921—21no—
72#11719921—21no—
73C4⋊Dic53—21no—
74C4⋊Dic53C2311yescharacteristic
75#11719919—21no—
76#11719919—21no—
77#11719919—21no—
78#11719919—21no—
79#11719919—21no—
80C4×Dic53—21no—
81C4×Dic53D411yescharacteristic
82C4×Dic53D411yescharacteristic
83C4×Dic53C2311yescharacteristic
84C2×#88322C2312yes—
85#11719913—42no—
86C23×C53—22no—
87C22×D53—22no—
88C22×D53—44no—
89C22×D53—44no—
90C22×D53—44no—
91C22×D53C2×D411yescharacteristic
92Q8×C53—21no—
93Q8×C53—21no—
94D4×C53—22no—
95D4×C53—22no—
96D4×C53—42no—
97D4×C53—42no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53C2×D411yescharacteristic
101C2×C4×C53C2×D411yescharacteristic
102C2×C4×C53C2411yescharacteristic
103C22⋊D53—44no—
104C22⋊D53—44no—
105C22⋊D53—44no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53C2×D411yescharacteristic
109C2×Dic53C2×D411yescharacteristic
110C2×Dic53C2×D411yescharacteristic
111D212—22no—
112D212—22no—
113D212—42no—
114D212—42no—
115C4×D53—21no—
116C4×D53—21no—
117C4×D53—22no—
118C4×D53SD1614yescomplemented
119Dic106—21no—
120Dic106—21no—
121Dic106—42no—
122C8×C53—22no—
123#88322—22no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53C22×D411yescharacteristic
128C2×D53—21no—
129C2×D53—21no—
130C2×D53—42no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—88no—
134C2×D53C2×SD1612yes—
135C4×C53—21no—
136C4×C53—41no—
137C4×C53—41no—
138C4×C53C2×SD1612yes—
139C4×C53H(5,𝔽2)11yescharacteristic
140C4×C53H(5,𝔽2)11yescharacteristic
141Dic53—21no—
142Dic53—41no—
143Dic53—41no—
144Dic53C2×SD1612yes—
145Dic53AGL(1,ℤ/8)12yes—
146#3208—531nomaximal, Sylow
147C2×C53—82no—
148C2×C53#53511yescharacteristic
149C2×C53C22×SD1611yescharacteristic
150C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
151D53—22no—
152D53—82no—
153#533—531no—
154C2×#127—531no—
155#492—532no—
156#486—532no—
157#460—531no—
158#460—531no—
159#460—531no—
160#460D5311yescharacteristic, complemented
161#450—532no—
162#450—532no—
163#431—531no—
164C53#320811yescharacteristic, complemented, Sylow
165C22×D4—532no—
166C22×D4—1061no—
167C22×D4—1061no—
168C2×SD16—532no—
169C2×SD16—532no—
170C2×SD16—532no—
171C2×SD16C2×D5312yescomplemented
172C4⋊Q8—531no—
173C4⋊Q8—531no—
174#127—532no—
175#127—531no—
176#127—1061no—
177#127—1061no—
178#127C2×D5311yescharacteristic
179#125—1061no—
180C22⋊Q8—531no—
181C22⋊Q8—531no—
182C22⋊Q8—1061no—
183C4×Q8—531no—
184C4×Q8—1061no—
185C4×Q8C2×D5311yescharacteristic, complemented
186#117—1061no—
187C2×C42—531no—
188C4⋊C8—531no—
189C4⋊C8C2×D5311yescharacteristic, complemented
190#102—532no—
191#102—532no—
192#102—532no—
193#102C2×D5312yescomplemented
194C22⋊C8—532no—
195C4×C8—532no—
196C24—1062no—
197C2×Q8—531no—
198C2×Q8—1061no—
199C2×Q8C22×D5311yescharacteristic
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—532no—
203C2×D4—1062no—
204C2×D4—1062no—
205C2×D4—1062no—
206C2×D4—1061no—
207C2×D4—1061no—
208C2×D4—2124no—
209C2×D4—2122no—
210C2×D4—2122no—
211C2×D4—2124no—
212C2×D4C22×D5312yes—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—531no—
216SD16—42no—
217SD16—2122no—
218SD16—2122no—
219SD16—2122no—
220C2×C8—532no—
221C2×C8C22×D5312yes—
222C4⋊C4—21no—
223C4⋊C4—531no—
224C4⋊C4—1061no—
225C4⋊C4—1061no—
226C4⋊C4—1061no—
227C4⋊C4—1061no—
228C4⋊C4—1061no—
229C4⋊C4—1061no—
230C4⋊C4C22×D5311yescharacteristic
231C22⋊C4—1061no—
232C22⋊C4—1061no—
233C42—21no—
234C42—531no—
235C42—531no—
236C42—531no—
237C42—1061no—
238C42C22×D5311yescharacteristic
239C23—22no—
240C23—531no—
241C23—1062no—
242C23—2124no—
243C23—2124no—
244C23—2124no—
245Q8—21no—
246Q8—21no—
247Q8—1061no—
248Q8—1061no—
249Q8—2122no—
250D4—22no—
251D4—22no—
252D4—42no—
253D4—42no—
254D4—1062no—
255D4—1062no—
256D4—2122no—
257D4—2124no—
258D4—2124no—
259D4—2124no—
260D4—2122no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—531no—
264C2×C4—531no—
265C2×C4—534no—
266C2×C4—531no—
267C2×C4—1062no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—1061no—
272C2×C4D4×D5311yescharacteristic
273C2×C4D4×D5311yescharacteristic
274C2×C4C23×D5311yescharacteristic
275C8—22no—
276C8—1062no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—532no—
281C22—1061no—
282C22—1061no—
283C22—4248no—
284C22—2122no—
285C22—2122no—
286C22—2122no—
287C22C2×D4×D5311yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—532no—
292C4—532no—
293C4—1061no—
294C4—2121no—
295C4—2121no—
296C4SD16×D5312yes—
297C4#1313326211yescharacteristic
298C4#1313326311yescharacteristic
299C2—82no—
300C2—1062no—
301C2—4242no—
302C2#1681118911yescharacteristic
303C2C2×SD16×D5311yescharacteristic
304C2AGL(1,ℤ/8)×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#460C2
socleC22×C53C22×D4
hypercenter#460D53
hypocenterC53#3208
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #16811177C2semidirect
3. #16810994C2semidirect
4. #16810968C2semidirect
5. #16810824C2semidirect
6. #16810823C2semidirect
7. #16810820C2semidirect
8. #16810775C2semidirect
9. #16810772C2semidirect
10. #16810675C2semidirect
11. C53×#460C2semidirect
15. C2×#11719913C22semidirect
16. C2×#11719923C22semidirect
17. C2×#11719924C22semidirect
18. #13133216C22semidirect
19. #13133215C22semidirect
29. C4×Dic106C22semidirect
32. C4⋊#88322C22semidirect
33. #13133139C22semidirect
34. #13133138C22semidirect
35. #13133137C22semidirect
37. C8×Dic53C22semidirect
38. C2×SD16×C53C22semidirect
40. C4×Q8×C53C22semidirect
41. C4⋊C8×C53C22semidirect
42. #102×C53C22semidirect
118. C4×D53SD16semidirect
160. #460D53semidirect
164. C53#3208semidirect
171. C2×SD16C2×D53semidirect
185. C4×Q8C2×D53semidirect
189. C4⋊C8C2×D53semidirect
193. #102C2×D53semidirect