Finite Group Databasebeta

← #20285654

Subgroups — #20285654

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#16811175C211yesmaximal, characteristic, complemented
2C2×#13133213C211yesmaximal, characteristic
3#16810994C212yesmaximal, complemented
4#16810968C212yesmaximal, complemented
5#16810822C212yesmaximal, complemented
6#16810821C212yesmaximal, complemented
7#16810776C211yesmaximal, characteristic, complemented
8#16810774C211yesmaximal, characteristic, complemented
9#16810773C211yesmaximal, characteristic, complemented
10#16810673C211yesmaximal, characteristic, complemented
11C53×#450C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53—21no—
14C22×C22⋊D53—21no—
15C2×#11719913C2212yescomplemented
16C2×#11719923C2212yescomplemented
17C2×#11719924C2212yescomplemented
18#13133216C2211yescharacteristic, complemented
19#13133215C2211yescharacteristic, complemented
20#13133213—21no—
21#13133213—21no—
22#13133213C2212yes—
23#13133213C2212yes—
24#13133207—21no—
25#13133196C2211yescharacteristic
26#13133194—21no—
27Q8×Dic53C2211yescharacteristic, complemented
28#13133182—21no—
29#13133182C2211yescharacteristic, complemented
30#13133173—21no—
31C2×C4×Dic53C2211yescharacteristic
32#13133146C2212yescomplemented
33#13133139C2212yescomplemented
34#13133138C2212yescomplemented
35#13133137C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37C8×Dic53C2212yescomplemented
38C22×D4×C53C2211yescharacteristic
39C2×SD16×C53C2212yescomplemented
40C22⋊Q8×C53C2211yescharacteristic
41#102×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C24×C53—21no—
44C23×D53—21no—
45C2×Q8×C53C2311yescharacteristic
46C2×D4×C53—21no—
47C2×D4×C53—42no—
48C2×D4×C53C2312yes—
49C22×C4×C53C2311yescharacteristic
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—22no—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—44no—
55C2×C22⋊D53—44no—
56C22×Dic53D411yescharacteristic
57C22×Dic53D411yescharacteristic
58C2×D212—21no—
59C2×D212—42no—
60C2×D212C2312yes—
61C2×Dic106—21no—
62C2×Dic106C2311yescharacteristic
63SD16×C53—42no—
64C2×C8×C53C2312yes—
65C4⋊C4×C53—21no—
66C4⋊C4×C53C2311yescharacteristic
67C22⋊C4×C53—21no—
68C22⋊Dic53—21no—
69#11719924—42no—
70#11719923—42no—
71C4⋊Dic53—21no—
72C4⋊Dic53C2311yescharacteristic
73#11719919—21no—
74#11719919—21no—
75#11719919—21no—
76#11719919—21no—
77#11719919—21no—
78C4×Dic53—21no—
79C4×Dic53—21no—
80C4×Dic53D411yescharacteristic
81C4×Dic53D411yescharacteristic
82C4×Dic53C2311yescharacteristic
83C4×Dic53C2311yescharacteristic
84C2×#88322C2312yes—
85#11719913—42no—
86C23×C53—22no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53C2×D411yescharacteristic
91C22×D53—22no—
92C22×D53—42no—
93C22×D53—42no—
94C22×D53—42no—
95Q8×C53—22no—
96D4×C53—22no—
97D4×C53—22no—
98D4×C53—42no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53C2×D411yescharacteristic
103C2×C4×C53C2411yescharacteristic
104C22⋊D53—44no—
105C22⋊D53—44no—
106C22⋊D53—44no—
107C22⋊D53—44no—
108C2×Dic53—22no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53SD1614yescomplemented
112C2×Dic53C2×D411yescharacteristic
113C2×Dic53C2×D411yescharacteristic
114C2×Dic53C2×D411yescharacteristic
115C2×Dic53C2×D411yescharacteristic
116D212—22no—
117D212—22no—
118D212—42no—
119Dic106—22no—
120Dic106—42no—
121C8×C53—22no—
122#88322—22no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—84no—
129C22×C53C2×SD1612yes—
130C22×C53C22×D411yescharacteristic
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—84no—
135C4×C53—21no—
136C4×C53—41no—
137C4×C53—41no—
138C4×C53H(5,𝔽2)12yes—
139Dic53—21no—
140Dic53—41no—
141Dic53—41no—
142Dic53C2×SD1612yes—
143Dic53C2×SD1612yes—
144Dic53AGL(1,ℤ/8)12yes—
145#3208—531nomaximal, Sylow
146C2×C53—22no—
147C2×C53—82no—
148C2×C53#53511yescharacteristic
149C2×C53C22×SD1611yescharacteristic
150C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
151D53—82no—
152#533—531no—
153C2×#127—531no—
154#492—532no—
155#486—532no—
156#460—532no—
157#460—532no—
158#450—531no—
159#450—531no—
160#450—531no—
161#450D5311yescharacteristic, complemented
162#431—531no—
163C53#320811yescharacteristic, complemented, Sylow
164C22×D4—531no—
165C22×D4—1061no—
166C22×D4—1061no—
167C22×D4C2×D5311yescharacteristic
168C2×SD16—532no—
169C2×SD16—532no—
170C2×SD16—532no—
171C2×SD16C2×D5312yescomplemented
172C4⋊Q8—531no—
173C4⋊Q8—531no—
174#127—532no—
175#127—532no—
176#127—1061no—
177#127—1061no—
178#125—1061no—
179C22⋊Q8—531no—
180C22⋊Q8—1061no—
181C22⋊Q8C2×D5311yescharacteristic
182C4×Q8—531no—
183C4×Q8—531no—
184C4×Q8—1061no—
185#117—1061no—
186C2×C42—531no—
187C4⋊C8—532no—
188#102—532no—
189#102—532no—
190#102—532no—
191#102C2×D5312yescomplemented
192C22⋊C8—531no—
193C22⋊C8C2×D5311yescharacteristic
194C4×C8—532no—
195C24—21no—
196C24—1061no—
197C2×Q8—531no—
198C2×Q8—1061no—
199C2×Q8C22×D5311yescharacteristic
200C2×D4—21no—
201C2×D4—42no—
202C2×D4—532no—
203C2×D4—1062no—
204C2×D4—1062no—
205C2×D4—1062no—
206C2×D4—1062no—
207C2×D4—1061no—
208C2×D4—2122no—
209C2×D4—2124no—
210C2×D4—2124no—
211C2×D4C22×D5312yes—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4C22×D5311yescharacteristic
215SD16—42no—
216SD16—2122no—
217SD16—2122no—
218SD16—2122no—
219C2×C8—532no—
220C2×C8C22×D5312yes—
221C4⋊C4—21no—
222C4⋊C4—531no—
223C4⋊C4—1061no—
224C4⋊C4—1061no—
225C4⋊C4—1061no—
226C4⋊C4—1061no—
227C4⋊C4—1061no—
228C4⋊C4—1061no—
229C4⋊C4C22×D5311yescharacteristic
230C22⋊C4—21no—
231C22⋊C4—1061no—
232C42—531no—
233C42—531no—
234C42—531no—
235C42—531no—
236C42—1061no—
237C42—1061no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—1062no—
243C23—2122no—
244C23—2122no—
245C23—2122no—
246C23D4×D5311yescharacteristic
247Q8—22no—
248Q8—1062no—
249Q8—2122no—
250D4—22no—
251D4—22no—
252D4—42no—
253D4—1062no—
254D4—1062no—
255D4—2122no—
256D4—2124no—
257D4—2124no—
258D4—2124no—
259D4—2124no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—534no—
264C2×C4—531no—
265C2×C4—531no—
266C2×C4—531no—
267C2×C4—531no—
268C2×C4—1062no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4D4×D5311yescharacteristic
272C2×C4C23×D5311yescharacteristic
273C8—22no—
274C8—1062no—
275C22—21no—
276C22—21no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—84no—
281C22—4244no—
282C22—2122no—
283C22—2122no—
284C22—2122no—
285C22SD16×D5312yes—
286C22C2×D4×D5311yescharacteristic
287C4—21no—
288C4—41no—
289C4—41no—
290C4—532no—
291C4—532no—
292C4—532no—
293C4—1061no—
294C4—2121no—
295C4—2121no—
296C4#1313326212yes—
297C2—22no—
298C2—82no—
299C2—4242no—
300C2#1681118811yescharacteristic
301C2C2×SD16×D5311yescharacteristic
302C2AGL(1,ℤ/8)×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#450C2
socleC22×C53C22×D4
hypercenter#450D53
hypocenterC53#3208
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #16811175C2semidirect
3. #16810994C2semidirect
4. #16810968C2semidirect
5. #16810822C2semidirect
6. #16810821C2semidirect
7. #16810776C2semidirect
8. #16810774C2semidirect
9. #16810773C2semidirect
10. #16810673C2semidirect
11. C53×#450C2semidirect
15. C2×#11719913C22semidirect
16. C2×#11719923C22semidirect
17. C2×#11719924C22semidirect
18. #13133216C22semidirect
19. #13133215C22semidirect
27. Q8×Dic53C22semidirect
29. #13133182C22semidirect
32. #13133146C22semidirect
33. #13133139C22semidirect
34. #13133138C22semidirect
35. #13133137C22semidirect
37. C8×Dic53C22semidirect
39. C2×SD16×C53C22semidirect
41. #102×C53C22semidirect
111. C2×Dic53SD16semidirect
161. #450D53semidirect
163. C53#3208semidirect
171. C2×SD16C2×D53semidirect
191. #102C2×D53semidirect