Finite Group Databasebeta

← #20283999

Subgroups — #20283999

Order 6784.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#13133264C211yesmaximal, characteristic, complemented
2C2×#13133243C211yesmaximal, characteristic, complemented
3C22×#11719923C211yesmaximal, characteristic, complemented
4#16810766C214yesmaximal, complemented
5#16810761C214yesmaximal, complemented
6#16810631C211yesmaximal, characteristic, complemented
7#16810617C211yesmaximal, characteristic, complemented
8#16810590C211yesmaximal, characteristic, complemented
9C53×#535C211yesmaximal, characteristic, complemented
10#13133264—24no—
11#13133264—24no—
12C2×#11719945C2211yescharacteristic
13C2×#11719945C2212yescomplemented
14C2×#11719947—21no—
15C2×#11719947—22no—
16C22×Dic106C2211yescharacteristic
17C22×Dic106C2212yescomplemented
18C2×Q8×D53—21no—
19#13133243—24no—
20C2×#11719923—24no—
21C2×#11719923C2212yes—
22C2×#11719923C2214yes—
23C2×#11719922C2212yes—
24C2×#11719917C2211yescharacteristic
25C22×#88322C2211yescharacteristic
26#13133140C2212yescomplemented
27#13133140C2214yescomplemented
28#13133137C2212yescomplemented
29C22⋊#88322C2214yescomplemented
30C2×#41×C53C2211yescharacteristic
31C22×D4×C53C2211yescharacteristic
32#127×C53—21no—
33#124×C53—21no—
34#121×C53C2214yes—
35#120×C53—22no—
36#117×C53C2211yescharacteristic
37C24×C53—21no—
38#41×C53—22no—
39#41×C53—22no—
40C2×Q8×C53D411yescharacteristic
41C2×D4×C53—24no—
42C2×D4×C53—22no—
43C2×D4×C53—22no—
44C2×D4×C53—21no—
45C2×D4×C53D412yes—
46C2×D4×C53D411yescharacteristic
47C2×D4×C53C2312yes—
48C22×C4×C53C2312yes—
49C22×C4×C53C2311yescharacteristic
50C2×C22⋊D53—21no—
51C2×C22⋊D53—22no—
52C22×Dic53—21no—
53C22×Dic53—22no—
54Q8×D53—28no—
55#11719947—216no—
56#11719947—28no—
57#11719945—22no—
58#11719945—24no—
59#11719945—24no—
60#11719945D414yescomplemented
61C2×D212C2311yescharacteristic
62C2×C4×D53—21no—
63C2×C4×D53—22no—
64C2×Dic106—21no—
65C2×Dic106—22no—
66C2×Dic106—22no—
67C2×Dic106—24no—
68C2×Dic106—24no—
69C2×Dic106D414yescomplemented
70C2×Dic106C2312yes—
71C2×Dic106C2311yescharacteristic
72C4⋊C4×C53C2312yes—
73C22⋊C4×C53—21no—
74C22⋊C4×C53—24no—
75C42×C53—21no—
76#11719923—28no—
77#11719923—24no—
78#11719922—24no—
79#11719917—22no—
80C2×#88322—22no—
81C2×#88322C2312yes—
82C2×#88322C2312yes—
83C23×C53—22no—
84C23×C53—42no—
85C23×C53—82no—
86C23×C53C2×D412yes—
87C23×C53C2×D411yescharacteristic
88C22×D53—21no—
89Q8×C53—22no—
90D4×C53—24no—
91D4×C53—22no—
92D4×C53—22no—
93D4×C53—22no—
94D4×C53—42no—
95D4×C53—84no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53C2×D412yes—
100C2×C4×C53C2×D412yes—
101C2×C4×C53C2×D412yes—
102C2×C4×C53C2×D411yescharacteristic
103C2×C4×C53C2411yescharacteristic
104C22⋊D53—28no—
105C22⋊D53—24no—
106C2×Dic53—28no—
107C2×Dic53—24no—
108C2×Dic53—22no—
109C2×Dic53—21no—
110D212—21no—
111D212C2×D412yes—
112C4×D53—28no—
113C4×D53—24no—
114Dic106—28no—
115Dic106—24no—
116Dic106—22no—
117Dic106—21no—
118Dic106C2×D412yes—
119Dic106C2×D414yes—
120#88322—22no—
121#88322—22no—
122C22×C53—22no—
123C22×C53—22no—
124C22×C53—24no—
125C22×C53—42no—
126C22×C53—44no—
127C22×C53—84no—
128C22×C53#12012yes—
129C22×C53C22×D411yescharacteristic
130C2×D53—22no—
131C2×D53—42no—
132C4×C53—22no—
133C4×C53—42no—
134C4×C53#12012yes—
135C4×C53C22×D411yescharacteristic
136C4×C53C22×D411yescharacteristic
137Dic53—22no—
138Dic53—24no—
139#3010—531nomaximal, Sylow
140C2×C53—22no—
141C2×C53—42no—
142C2×C53—82no—
143C2×C53C2×#12011yescharacteristic
144C2×C53#57712yes—
145D53—42no—
146C2×2−1+4—531no—
147C2×AGL(1,ℤ/8)—531no—
148C22×SD16—531no—
149#535D5311yescharacteristic, complemented
150#449—534no—
151#448—534no—
152#419—531no—
153#416—531no—
154#408—531no—
155C53#301011yescharacteristic, complemented, Sylow
1562−1+4—1064no—
1572−1+4—1064no—
158C2×#41—531no—
159C2×#41—532no—
160C2×#41—1062no—
161C2×#41—1061no—
162C2×#41C2×D5311yescharacteristic
163C22×Q8—531no—
164C22×Q8—532no—
165C22×Q8—1061no—
166C22×D4C2×D5311yescharacteristic
167AGL(1,ℤ/8)—1064no—
168C2×SD16—534no—
169C2×SD16—532no—
170C2×SD16—1064no—
171C2×D8—532no—
172C2×M16—531no—
173C22×C8—531no—
174#127—21no—
175#124—21no—
176#121C2×D5314yes—
177#120—22no—
178#117C2×D5311yescharacteristic
179#103—532no—
180#103—534no—
181#102—532no—
182C22⋊C8—534no—
183C24—21no—
184#41—22no—
185#41—22no—
186#41—534no—
187#41—1064no—
188#41—1064no—
189#41—10616no—
190#41—1068no—
191#41—1062no—
192C2×Q8—534no—
193C2×Q8—532no—
194C2×Q8—531no—
195C2×Q8—1064no—
196C2×Q8—1064no—
197C2×Q8—1068no—
198C2×Q8—1062no—
199C2×Q8—1062no—
200C2×Q8—1061no—
201C2×Q8C22⋊D5311yescharacteristic
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—531no—
207C2×D4—1062no—
208C2×D4—1061no—
209C2×D4C22⋊D5312yes—
210C2×D4C22⋊D5311yescharacteristic
211C2×D4C22×D5312yes—
212C22×C4—1062no—
213C22×C4—1061no—
214C22×C4—1062no—
215C22×C4—1061no—
216C22×C4C22×D5312yes—
217C22×C4C22×D5311yescharacteristic
218SD16—1068no—
219SD16—1064no—
220D8—1064no—
221M16—1062no—
222C2×C8—532no—
223C2×C8—532no—
224C2×C8—1062no—
225C4⋊C4C22×D5312yes—
226C22⋊C4—21no—
227C22⋊C4—24no—
228C42—21no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—1061no—
233C23C2×C22⋊D5312yes—
234C23C2×C22⋊D5311yescharacteristic
235Q8—22no—
236Q8—534no—
237Q8—532no—
238Q8—1068no—
239Q8—1064no—
240Q8—1062no—
241Q8—1061no—
242D4—24no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—84no—
248D4—532no—
249D4—1068no—
250D4—1064no—
251D4—1061no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—1068no—
256C2×C4—1068no—
257C2×C4—1064no—
258C2×C4—1062no—
259C2×C4—1061no—
260C2×C4—1064no—
261C2×C4D4×D5312yes—
262C2×C4D4×D5312yes—
263C2×C4C2×C22⋊D5312yes—
264C2×C4C2×C22⋊D5311yescharacteristic
265C2×C4C23×D5311yescharacteristic
266C8—1062no—
267C8—1062no—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—42no—
272C22—44no—
273C22—84no—
274C22—1062no—
275C22—2122no—
276C22#1313318512yes—
277C22C22×C22⋊D5311yescharacteristic
278C4—22no—
279C4—42no—
280C4—1064no—
281C4—1062no—
282C4#1313318512yes—
283C4C2×D4×D5311yescharacteristic
284C4C2×D4×D5311yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—2122no—
289C2C2×#1313318511yescharacteristic
290C2#1681143412yes—
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#535C2
socleC22×C53C22×D4
hypercenter#535D53
hypocenterC53#3010
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#13133264C2semidirect
2. C2×#13133243C2semidirect
3. C22×#11719923C2semidirect
4. #16810766C2semidirect
5. #16810761C2semidirect
6. #16810631C2semidirect
7. #16810617C2semidirect
8. #16810590C2semidirect
9. C53×#535C2semidirect
13. C2×#11719945C22semidirect
17. C22×Dic106C22semidirect
26. #13133140C22semidirect
27. #13133140C22semidirect
28. #13133137C22semidirect
29. C22⋊#88322C22semidirect
60. #11719945D4semidirect
69. C2×Dic106D4semidirect
149. #535D53semidirect
155. C53#3010semidirect