Finite Group Databasebeta

← #20284811

Subgroups — #20284811

Order 6784.

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

normal subgroups only

This group has 444 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#16811175C211yesmaximal, characteristic, complemented
2C2×#13133179C211yesmaximal, characteristic
3#16810958C211yesmaximal, characteristic, complemented
4#16810902C212yesmaximal, complemented
5#16810827C212yesmaximal, complemented
6#16810776C211yesmaximal, characteristic
7#16810773C211yesmaximal, characteristic
8#16810726C212yesmaximal, complemented
9#16810722C212yesmaximal, complemented
10#16810677C211yesmaximal, characteristic, complemented
11C53×#484C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C4×D53—21no—
14C2×#11719913C2212yes—
15C2×#11719924C2212yes—
16#13133216C2211yescharacteristic, complemented
17#13133215C2211yescharacteristic, complemented
18#13133207—21no—
19#13133196C2211yescharacteristic
20#13133194—21no—
21Q8×Dic53C2211yescharacteristic, complemented
22#13133182—21no—
23#13133182C2211yescharacteristic, complemented
24#13133179—21no—
25#13133179—21no—
26#13133179C2212yes—
27#13133179C2212yes—
28#13133173—21no—
29C2×#11719921—21no—
30C2×C4×Dic53C2211yescharacteristic
31#13133150C2212yescomplemented
32#13133146C2212yescomplemented
33#13133140C2212yescomplemented
34#13133139C2212yes—
35#13133137C2212yes—
36C22⋊#88322C2211yescharacteristic
37#13133127C2212yescomplemented
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
47C22×Dic53D411yescharacteristic
48C22×Dic53D411yescharacteristic
49C2×D212—21no—
50C2×D212—24no—
51C2×D212C2312yes—
52C2×C4×D53—22no—
53C2×C4×D53—22no—
54C2×C4×D53—44no—
55C2×Dic106—21no—
56C2×Dic106C2311yescharacteristic
57C2×C8×C53C2312yes—
58C4⋊C4×C53—21no—
59C4⋊C4×C53—21no—
60C4⋊C4×C53C2312yes—
61C4⋊C4×C53C2311yescharacteristic
62C22⋊C4×C53—21no—
63C22⋊Dic53—21no—
64#11719924—24no—
65#11719921—22no—
66#11719921—22no—
67C4⋊Dic53—21no—
68C4⋊Dic53C2311yescharacteristic
69#11719919—21no—
70#11719919—21no—
71#11719919—21no—
72#11719919—21no—
73#11719919—21no—
74C4×Dic53—21no—
75C4×Dic53—21no—
76C4×Dic53D411yescharacteristic
77C4×Dic53D411yescharacteristic
78C4×Dic53C2311yescharacteristic
79C4×Dic53C2311yescharacteristic
80C2×#88322C2312yes—
81#11719913—24no—
82C23×C53C2×D411yescharacteristic
83C22×D53—24no—
84C22×D53—22no—
85C22×D53—42no—
86C22×D53—42no—
87Q8×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—21no—
92C2×C4×C53—42no—
93C2×C4×C53C2×D411yescharacteristic
94C2×C4×C53C2411yescharacteristic
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53C2×D411yescharacteristic
100C2×Dic53C2×D411yescharacteristic
101C2×Dic53C2×D411yescharacteristic
102C2×Dic53C2×D411yescharacteristic
103D212—24no—
104D212—22no—
105D212#4114yes—
106C4×D53—44no—
107C4×D53—44no—
108Dic106—22no—
109Dic106—42no—
110C8×C53—22no—
111#88322—22no—
112C22×C53—21no—
113C22×C53—21no—
114C22×C53#13712yes—
115C22×C53C22×D411yescharacteristic
116C2×D53—24no—
117C2×D53—48no—
118C2×D53—42no—
119C2×D53—42no—
120C4×C53—21no—
121C4×C53—41no—
122C4×C53—41no—
123C4×C53—42no—
124C4×C53C2×#4112yes—
125Dic53—21no—
126Dic53—21no—
127Dic53—21no—
128Dic53—41no—
129Dic53—41no—
130Dic53AGL(1,ℤ/8)12yes—
131#3110—531nomaximal, Sylow
132C2×C53—22no—
133C2×C53#52511yescharacteristic
134C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
135C2×C53C2×#13711yescharacteristic
136D53—44no—
137#533—531no—
138C2×C4×D4—531no—
139#484—531no—
140#484D5311yescharacteristic, complemented
141#474—532no—
142#461—532no—
143#450—531no—
144#450—531no—
145#440—532no—
146#440—532no—
147#432—531no—
148C53#311011yescharacteristic, complemented, Sylow
149C22×D4—531no—
150C23×C4—1061no—
151C2×SD16—532no—
152C2×SD16—532no—
153C4⋊Q8—531no—
154C4⋊Q8—531no—
155#125—1061no—
156C22⋊Q8—531no—
157C22⋊Q8—1061no—
158C22⋊Q8C2×D5311yescharacteristic
159C4×Q8—531no—
160C4×Q8—531no—
161C4×Q8—1061no—
162C4×D4—532no—
163C4×D4—532no—
164C4×D4—1061no—
165C4×D4—1061no—
166#117—1061no—
167C2×C4⋊C4C2×D5311yescharacteristic
168C2×C22⋊C4—1061no—
169C2×C42—531no—
170#107—532no—
171#107C2×D5312yescomplemented
172C4⋊C8—532no—
173#103—532no—
174#103C2×D5312yescomplemented
175#102—532no—
176#102—532no—
177C22⋊C8—531no—
178C22⋊C8C2×D5311yescharacteristic
179#97—532no—
180C24—1061no—
181C2×Q8—531no—
182C2×Q8—1061no—
183C2×Q8C22×D5311yescharacteristic
184C2×D4—532no—
185C2×D4—1064no—
186C2×D4—1061no—
187C22×C4—21no—
188C22×C4—531no—
189C22×C4—531no—
190C22×C4—1062no—
191C22×C4—1062no—
192C22×C4—2124no—
193C22×C4C22×D5311yescharacteristic
194SD16—1064no—
195SD16—1064no—
196C2×C8—532no—
197C2×C8C22×D5312yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—531no—
201C4⋊C4—1061no—
202C4⋊C4—1061no—
203C4⋊C4—1061no—
204C4⋊C4—1061no—
205C4⋊C4—1061no—
206C4⋊C4—1061no—
207C4⋊C4C22×D5311yescharacteristic
208C4⋊C4C22×D5312yes—
209C22⋊C4—21no—
210C22⋊C4—1062no—
211C22⋊C4—1062no—
212C22⋊C4—1061no—
213C42—531no—
214C42—531no—
215C42—531no—
216C42—531no—
217C42—1061no—
218C42—1061no—
219C23—1064no—
220C23—1062no—
221C23—2122no—
222C23—2122no—
223C23D4×D5311yescharacteristic
224Q8—22no—
225Q8—1062no—
226Q8—2122no—
227D4—534no—
228D4—1064no—
229D4—1062no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—531no—
236C2×C4—531no—
237C2×C4—531no—
238C2×C4—531no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—1061no—
242C2×C4—1061no—
243C2×C4—2124no—
244C2×C4—2124no—
245C2×C4D4×D5311yescharacteristic
246C2×C4C23×D5311yescharacteristic
247C8—22no—
248C8—1062no—
249C22—21no—
250C22—21no—
251C22—1064no—
252C22—2128no—
253C22—2122no—
254C22—2122no—
255C22#1313324912yes—
256C22C2×D4×D5311yescharacteristic
257C4—21no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—532no—
262C4—1061no—
263C4—1061no—
264C4—1061no—
265C4—2121no—
266C4—2121no—
267C4#41×D5312yes—
268C2—22no—
269C2—2124no—
270C2#1681113111yescharacteristic
271C2AGL(1,ℤ/8)×D5311yescharacteristic
272C2C2×#1313324911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#484C2
socleC22×C53C22×D4
hypercenter#484D53
hypocenterC53#3110
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. #16811175C2semidirect
3. #16810958C2semidirect
4. #16810902C2semidirect
5. #16810827C2semidirect
8. #16810726C2semidirect
9. #16810722C2semidirect
10. #16810677C2semidirect
11. C53×#484C2semidirect
16. #13133216C22semidirect
17. #13133215C22semidirect
21. Q8×Dic53C22semidirect
23. #13133182C22semidirect
31. #13133150C22semidirect
32. #13133146C22semidirect
33. #13133140C22semidirect
37. #13133127C22semidirect
40. #107×C53C22semidirect
41. #103×C53C22semidirect
140. #484D53semidirect
148. C53#3110semidirect
171. #107C2×D53semidirect
174. #103C2×D53semidirect