Finite Group Databasebeta

← #20284765

Subgroups — #20284765

Order 6784.

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

normal subgroups only

This group has 454 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#16811159C211yesmaximal, characteristic, complemented
2C2×#13133179C211yesmaximal, characteristic, complemented
3#16810941C211yesmaximal, characteristic, complemented
4#16810926C212yesmaximal, complemented
5#16810841C212yesmaximal, complemented
6#16810773C211yesmaximal, characteristic, complemented
7#16810759C211yesmaximal, characteristic
8#16810708C212yesmaximal, complemented
9#16810705C212yesmaximal, complemented
10#16810673C211yesmaximal, characteristic, complemented
11C53×#484C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C4×D53—21no—
14C2×#11719924C2212yes—
15C2×D424C2212yes—
16#13133212—21no—
17#13133208C2211yescharacteristic, complemented
18#13133206C2211yescharacteristic, complemented
19#13133201—21no—
20#13133191C2211yescharacteristic
21Q8×Dic53C2211yescharacteristic, complemented
22#13133179—21no—
23#13133179—21no—
24#13133179C2212yes—
25#13133179C2212yes—
26#13133179C2211yescharacteristic, complemented
27#13133177—21no—
28#13133175—21no—
29C2×#11719921—21no—
30C2×C4×Dic53C2211yescharacteristic
31#13133148C2212yescomplemented
32#13133146C2212yescomplemented
33#13133139C2212yes—
34#13133137C2212yes—
35#13133137C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37C8×Dic53C2212yescomplemented
38C22⋊Q8×C53C2211yescharacteristic
39C2×C4⋊C4×C53C2211yescharacteristic
40#107×C53C2212yescomplemented
41#103×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C23×D53—21no—
44C2×Q8×C53C2311yescharacteristic
45C22×C4×C53—21no—
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—21no—
48C22×Dic53D411yescharacteristic
49C22×Dic53D411yescharacteristic
50C2×D212—21no—
51C2×D212—24no—
52C2×D212C2311yescharacteristic
53C2×D212C2312yes—
54C2×C4×D53—21no—
55C2×C4×D53—22no—
56C2×C4×D53—22no—
57C2×C4×D53—44no—
58C2×C8×C53C2312yes—
59C4⋊C4×C53—21no—
60C4⋊C4×C53—21no—
61C4⋊C4×C53C2312yes—
62C4⋊C4×C53C2311yescharacteristic
63C22⋊C4×C53—21no—
64#11719924—24no—
65#11719921—21no—
66#11719921—22no—
67#11719921—22no—
68#11719921—21no—
69#11719921—21no—
70#11719921—21no—
71C4⋊Dic53—21no—
72C4⋊Dic53—21no—
73C4⋊Dic53C2311yescharacteristic
74#11719919—21no—
75C4×Dic53—21no—
76C4×Dic53D411yescharacteristic
77C4×Dic53D411yescharacteristic
78C4×Dic53C2311yescharacteristic
79C4×Dic53C2311yescharacteristic
80C2×#88322C2312yes—
81D424—24no—
82C23×C53C2×D411yescharacteristic
83C22×D53—24no—
84C22×D53—22no—
85C22×D53—21no—
86C22×D53—42no—
87C22×D53—42no—
88Q8×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—42no—
94C2×C4×C53C2×D411yescharacteristic
95C2×C4×C53C2411yescharacteristic
96C22⋊D53—42no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—21no—
100C2×Dic53C2×D411yescharacteristic
101C2×Dic53C2×D411yescharacteristic
102C2×Dic53C2×D411yescharacteristic
103C2×Dic53C2×D411yescharacteristic
104D212—24no—
105D212—22no—
106D212—22no—
107D212#4114yes—
108C4×D53—44no—
109C4×D53—42no—
110C4×D53—44no—
111C8×C53—22no—
112#88322—22no—
113C22×C53—21no—
114C22×C53—21no—
115C22×C53#13512yes—
116C22×C53C22×D411yescharacteristic
117C2×D53—24no—
118C2×D53—48no—
119C2×D53—42no—
120C2×D53—42no—
121C2×D53—41no—
122C2×D53—41no—
123C2×D53—41no—
124C4×C53—21no—
125C4×C53—41no—
126C4×C53—42no—
127C4×C53—41no—
128C4×C53C2×#4112yes—
129Dic53—21no—
130Dic53—21no—
131Dic53—21no—
132Dic53—41no—
133Dic53AGL(1,ℤ/8)12yes—
134#3105—531nomaximal, Sylow
135C2×C53—22no—
136C2×C53#52511yescharacteristic
137C2×C53C2×#13511yescharacteristic
138C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
139D53—44no—
140D53—81no—
141#529—531no—
142C2×C4×D4—531no—
143#484D5311yescharacteristic, complemented
144#481—531no—
145#478—532no—
146#463—532no—
147#450—531no—
148#447—531no—
149C4×SD16—532no—
150C4×D8—532no—
151#431—531no—
152C53#310511yescharacteristic, complemented, Sylow
153C22×D4—531no—
154C23×C4—1061no—
155C2×SD16—532no—
156C2×D8—532no—
157#126—1061no—
158#125—531no—
159#124—531no—
160#123—1061no—
161C22⋊Q8C2×D5311yescharacteristic
162#121—531no—
163C4×Q8—531no—
164C4×D4—532no—
165C4×D4—532no—
166C4×D4—531no—
167C4×D4—1061no—
168C4×D4—1061no—
169C4×D4—1061no—
170#117—1061no—
171C2×C4⋊C4C2×D5311yescharacteristic
172C2×C22⋊C4—1061no—
173C2×C42—531no—
174#107C2×D5312yescomplemented
175#106—532no—
176C4⋊C8—532no—
177#103C2×D5312yescomplemented
178#102—532no—
179#102—532no—
180#102—532no—
181C22⋊C8—531no—
182C22⋊C8C2×D5311yescharacteristic
183C4×C8—532no—
184C24—1061no—
185C2×Q8C22×D5311yescharacteristic
186C2×D4—532no—
187C2×D4—531no—
188C2×D4—1064no—
189C2×D4—1061no—
190C2×D4—1061no—
191C22×C4—21no—
192C22×C4—531no—
193C22×C4—531no—
194C22×C4—1061no—
195C22×C4—1062no—
196C22×C4—1062no—
197C22×C4—2124no—
198C22×C4C22×D5311yescharacteristic
199SD16—1064no—
200D8—1064no—
201C2×C8—532no—
202C2×C8C22×D5312yes—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—531no—
206C4⋊C4—1061no—
207C4⋊C4—1061no—
208C4⋊C4—1061no—
209C4⋊C4C22×D5311yescharacteristic
210C4⋊C4C22×D5312yes—
211C22⋊C4—21no—
212C22⋊C4—1062no—
213C22⋊C4—1062no—
214C22⋊C4—1061no—
215C22⋊C4—1061no—
216C22⋊C4—1061no—
217C22⋊C4—1061no—
218C42—531no—
219C42—531no—
220C42—531no—
221C42—531no—
222C42—1061no—
223C23—1064no—
224C23—1062no—
225C23—1061no—
226C23—2122no—
227C23—2122no—
228C23D4×D5311yescharacteristic
229Q8—22no—
230D4—534no—
231D4—1064no—
232D4—1062no—
233D4—1062no—
234D4—2122no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—531no—
241C2×C4—531no—
242C2×C4—531no—
243C2×C4—531no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—1061no—
247C2×C4—2122no—
248C2×C4—2124no—
249C2×C4—2124no—
250C2×C4D4×D5311yescharacteristic
251C2×C4C23×D5311yescharacteristic
252C8—22no—
253C8—1062no—
254C22—21no—
255C22—21no—
256C22—1064no—
257C22—2128no—
258C22—2121no—
259C22—2121no—
260C22—2121no—
261C22—2122no—
262C22—2122no—
263C22#1313324012yes—
264C22C2×D4×D5311yescharacteristic
265C4—21no—
266C4—41no—
267C4—42no—
268C4—41no—
269C4—532no—
270C4—1061no—
271C4—1061no—
272C4—1061no—
273C4—2121no—
274C4#41×D5312yes—
275C2—22no—
276C2—4241no—
277C2—2124no—
278C2#1681113111yescharacteristic
279C2C2×#1313324011yescharacteristic
280C2AGL(1,ℤ/8)×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#484C2
socleC22×C53C22×D4
hypercenter#484D53
hypocenterC53#3105
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #16811159C2semidirect
2. C2×#13133179C2semidirect
3. #16810941C2semidirect
4. #16810926C2semidirect
5. #16810841C2semidirect
6. #16810773C2semidirect
8. #16810708C2semidirect
9. #16810705C2semidirect
10. #16810673C2semidirect
11. C53×#484C2semidirect
17. #13133208C22semidirect
18. #13133206C22semidirect
21. Q8×Dic53C22semidirect
26. #13133179C22semidirect
31. #13133148C22semidirect
32. #13133146C22semidirect
35. #13133137C22semidirect
37. C8×Dic53C22semidirect
40. #107×C53C22semidirect
41. #103×C53C22semidirect
143. #484D53semidirect
152. C53#3105semidirect
174. #107C2×D53semidirect
177. #103C2×D53semidirect