Finite Group Databasebeta

← #20281091

Subgroups — #20281091

Order 6784.

688 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13133185C211yesmaximal, characteristic
2#16810551C212yesmaximal, complemented
3#16810538C212yesmaximal, complemented
4#16810534C212yesmaximal, complemented
5#16810514C212yesmaximal, complemented
6#16810503C211yesmaximal, characteristic, complemented
7#16810500C212yesmaximal, complemented
8#16810478C212yesmaximal, complemented
9C2×#124×C53C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C23×Dic53C2211yescharacteristic, complemented
13#13133185—24no—
14C2×#11719919C2212yescomplemented
15C2×C22⋊Dic53C2211yescharacteristic
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2211yescharacteristic, complemented
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23#13133124C2212yescomplemented
24#13133124C2212yescomplemented
25#13133124C2212yescomplemented
26#13133124C2212yescomplemented
27C22×Q8×C53C2211yescharacteristic, complemented
28C22×D4×C53C2211yescharacteristic
29#124×C53—24no—
30C2×C22⋊C4×C53C2212yes—
31C2×C22⋊C4×C53C2212yescomplemented
32C2×C42×C53C2211yescharacteristic, complemented
33C24×C53C2311yescharacteristic
34C24×C53C2311yescharacteristic
35C23×D53—22no—
36C23×D53—24no—
37C23×D53—42no—
38C23×D53—41no—
39C23×D53—41no—
40C23×D53C2312yes—
41C2×Q8×C53—22no—
42C2×Q8×C53—41no—
43C2×Q8×C53—41no—
44C2×D4×C53—41no—
45C2×D4×C53—41no—
46C2×D4×C53D414yes—
47C22×C4×C53C2312yes—
48C22×C4×C53C2312yes—
49C22×C4×C53C2311yescharacteristic
50C2×C22⋊D53—24no—
51C2×C22⋊D53—24no—
52C2×C22⋊D53—42no—
53C22×Dic53—41no—
54C22×Dic53—42no—
55C22×Dic53C2312yes—
56C22×Dic53C2312yes—
57C22×Dic53C2312yes—
58C22⋊C4×C53—24no—
59C22⋊C4×C53—24no—
60C42×C53—22no—
61C22⋊Dic53—22no—
62C22⋊Dic53—42no—
63C22⋊Dic53—41no—
64#11719921—24no—
65#11719921—24no—
66#11719921—42no—
67#11719921—42no—
68#11719921—42no—
69#11719919—42no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53—21no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—22no—
78C23×C53C2×D412yes—
79C23×C53C2×D412yes—
80C23×C53C2411yescharacteristic
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—44no—
88C22×D53—48no—
89C22×D53—48no—
90C22×D53—44no—
91C22×D53—44no—
92C22×D53#4114yes—
93Q8×C53—44no—
94D4×C53—44no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53C2×D412yes—
104C22⋊D53—48no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—44no—
115C22×C53—22no—
116C22×C53—22no—
117C22×C53—24no—
118C22×C53—24no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53—41no—
125C22×C53C22×D411yescharacteristic
126C22×C53C2×#4112yes—
127C22×C53H(5,𝔽2)11yescharacteristic
128C22×C53H(5,𝔽2)11yescharacteristic
129C22×C53H(5,𝔽2)12yes—
130C2×D53—24no—
131C2×D53—24no—
132C2×D53—24no—
133C2×D53—416no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C4×C53—22no—
139C4×C53—42no—
140C4×C53—42no—
141Dic53—42no—
142Dic53—42no—
143Dic53—42no—
144#2684—531nomaximal, Sylow
145C2×C53—42no—
146C2×C53—42no—
147C2×C53#53411yescharacteristic
148C2×C53#53812yes—
149C2×C53#54612yes—
150C2×C53#55111yescharacteristic
151C2×C53H(3,𝔽4)11yescharacteristic
152D53—44no—
153C2×#124D5311yescharacteristic, complemented
154C2×#120—531no—
155#397—532no—
156#394—532no—
157#393—532no—
158#388—532no—
159#386—531no—
160#386—532no—
161#379—532no—
162C53#268411yescharacteristic, complemented, Sylow
163C25—531no—
164C22×Q8C2×D5311yescharacteristic, complemented
165C22×D4—532no—
166C22×D4C2×D5311yescharacteristic
167C23×C4—531no—
168#124—24no—
169#120—1064no—
170C2×C4⋊C4—532no—
171C2×C22⋊C4—531no—
172C2×C22⋊C4—532no—
173C2×C22⋊C4—532no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4—531no—
176C2×C22⋊C4—532no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4—532no—
179C2×C22⋊C4C2×D5312yes—
180C2×C22⋊C4C2×D5312yescomplemented
181C2×C42C2×D5311yescharacteristic, complemented
182#95—532no—
183#95—532no—
184#95—532no—
185#95—532no—
186C24—532no—
187C24—1064no—
188C24—1062no—
189C24—2121no—
190C24—2121no—
191C24—2122no—
192C24C22×D5311yescharacteristic
193C24C22×D5311yescharacteristic
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—1064no—
200C2×D4—1064no—
201C2×D4—2122no—
202C2×D4C22⋊D5314yes—
203C22×C4—532no—
204C22×C4—532no—
205C22×C4—532no—
206C22×C4—2121no—
207C22×C4—2122no—
208C22×C4C22×D5312yes—
209C22×C4C22×D5312yes—
210C22×C4C22×D5311yescharacteristic
211C4⋊C4—2122no—
212C22⋊C4—24no—
213C22⋊C4—24no—
214C22⋊C4—1064no—
215C22⋊C4—1062no—
216C22⋊C4—1064no—
217C22⋊C4—2122no—
218C22⋊C4—2122no—
219C22⋊C4—2122no—
220C22⋊C4—2121no—
221C22⋊C4—2122no—
222C42—22no—
223C23—22no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—22no—
231C23—534no—
232C23—1062no—
233C23—1062no—
234C23—1062no—
235C23—1062no—
236C23—1062no—
237C23—1062no—
238C23—2128no—
239C23—2124no—
240C23—2124no—
241C23—2128no—
242C23—2124no—
243C23C2×C22⋊D5312yes—
244C23C2×C22⋊D5312yes—
245C23C23×D5311yescharacteristic
246Q8—44no—
247D4—44no—
248D4—2128no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—2124no—
267C2×C4C2×C22⋊D5312yes—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—24no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—1064no—
279C22—1064no—
280C22—1064no—
281C22—21216no—
282C22—2122no—
283C22—2122no—
284C22—2122no—
285C22—2122no—
286C22C22×C22⋊D5311yescharacteristic
287C22#41×D5312yes—
288C22#1313326211yescharacteristic
289C22#1313326211yescharacteristic
290C22#1313326312yes—
291C4—22no—
292C4—42no—
293C4—42no—
294C4—2122no—
295C4—2122no—
296C4—2122no—
297C2—42no—
298C2—42no—
299C2—2124no—
300C2#1681118211yescharacteristic
301C2#1681120712yes—
302C2#1681126112yes—
303C2#1681130311yescharacteristic
304C2#1681136511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#124×C53C2
hypercenterC2×#124D53
hypocenterC53#2684
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #16810551C2semidirect
3. #16810538C2semidirect
4. #16810534C2semidirect
5. #16810514C2semidirect
6. #16810503C2semidirect
7. #16810500C2semidirect
8. #16810478C2semidirect
9. C2×#124×C53C2semidirect
12. C23×Dic53C22semidirect
14. C2×#11719919C22semidirect
17. C2×C22⋊Dic53C22semidirect
23. #13133124C22semidirect
24. #13133124C22semidirect
25. #13133124C22semidirect
26. #13133124C22semidirect
27. C22×Q8×C53C22semidirect
31. C2×C22⋊C4×C53C22semidirect
32. C2×C42×C53C22semidirect
153. C2×#124D53semidirect
162. C53#2684semidirect
164. C22×Q8C2×D53semidirect
180. C2×C22⋊C4C2×D53semidirect
181. C2×C42C2×D53semidirect