Finite Group Databasebeta

← #20278988

Subgroups — #20278988

Order 6784.

672 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 145 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×#13133173C211yesmaximal, characteristic, complemented
2C22×#11719921C211yesmaximal, characteristic
3#16810558C212yesmaximal, complemented
4#16810552C212yesmaximal, complemented
5#16810513C212yesmaximal, complemented
6#16810512C212yesmaximal, complemented
7#16810496C211yesmaximal, characteristic, complemented
8#16810484C211yesmaximal, characteristic, complemented
9#16810478C211yesmaximal, characteristic
10#16810478C211yesmaximal, characteristic
11C53×#385C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C23×Dic53C2211yescharacteristic
14#13133173—24no—
15C2×#11719919C2211yescharacteristic, complemented
16C2×#11719919C2211yescharacteristic, complemented
17C2×#11719919C2212yescomplemented
18C2×C22⋊Dic53C2211yescharacteristic
19C2×C22⋊Dic53C2211yescharacteristic
20C2×#11719921—24no—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×C4×Dic53C2211yescharacteristic, complemented
28C2×C4×Dic53C2211yescharacteristic, complemented
29#13133124C2212yescomplemented
30#13133124C2212yescomplemented
31#13133124C2212yescomplemented
32C23×C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2212yescomplemented
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2211yescharacteristic
36#95×C53C2212yescomplemented
37C24×C53C2311yescharacteristic
38C23×D53—22no—
39C23×D53—24no—
40C23×D53—22no—
41C23×D53—22no—
42C23×D53—42no—
43C23×D53C2312yes—
44C22×C4×C53—22no—
45C22×C4×C53—42no—
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2312yes—
48C22×C4×C53C2312yes—
49C22×C4×C53C2311yescharacteristic
50C22×Dic53—22no—
51C22×Dic53—22no—
52C22×Dic53D414yes—
53C22×Dic53C2312yes—
54C22×Dic53C2311yescharacteristic
55C22×Dic53C2311yescharacteristic
56C22×Dic53C2311yescharacteristic
57C22×Dic53C2311yescharacteristic
58C4⋊C4×C53—42no—
59C22⋊C4×C53—22no—
60C22⋊C4×C53—22no—
61C22⋊Dic53—22no—
62C22⋊Dic53—22no—
63#11719921—24no—
64#11719921—24no—
65#11719921—24no—
66#11719921—24no—
67#11719921—42no—
68#11719921—42no—
69#11719919—22no—
70#11719919—22no—
71#11719919—42no—
72C4×Dic53—22no—
73C4×Dic53—22no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53C2×D412yes—
77C23×C53#4112yes—
78C23×C53#4112yes—
79C23×C53C2411yescharacteristic
80C22×D53—28no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—48no—
88C22×D53—48no—
89C22×D53—44no—
90C22×D53—44no—
91C22×D53#4114yes—
92C2×C4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—44no—
103C2×Dic53—28no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53C2×D412yes—
116C2×Dic53C2×D412yes—
117C22×C53—22no—
118C22×C53—22no—
119C22×C53—24no—
120C22×C53—24no—
121C22×C53#12314yes—
122C22×C53C22×D411yescharacteristic
123C22×C53C2×#4112yes—
124C22×C53C2×#4111yescharacteristic
125C22×C53C2×#4111yescharacteristic
126C22×C53H(5,𝔽2)12yes—
127C2×D53—24no—
128C2×D53—24no—
129C2×D53—24no—
130C2×D53—416no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—42no—
135C4×C53—41no—
136C4×C53—41no—
137C4×C53—42no—
138C4×C53—42no—
139Dic53—22no—
140Dic53—22no—
141Dic53—41no—
142Dic53—41no—
143Dic53—42no—
144#2482—531nomaximal, Sylow
145C2×C53—24no—
146C2×C53C2×#12311yescharacteristic
147C2×C53#52511yescharacteristic
148C2×C53#53511yescharacteristic
149C2×C53#55112yes—
150C2×C53#55112yes—
151D53—44no—
152C2×#117—531no—
153C22×C22⋊C4—531no—
154#399—532no—
155#397—532no—
156#388—532no—
157#388—532no—
158#385—531no—
159#385D5311yescharacteristic, complemented
160#381—531no—
161#379—531no—
162#379—531no—
163C53#248211yescharacteristic, complemented, Sylow
164C25—531no—
165C23×C4—531no—
166C23×C4C2×D5311yescharacteristic
167#117—1064no—
168C2×C4⋊C4—531no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4C2×D5312yescomplemented
172C2×C22⋊C4—531no—
173C2×C22⋊C4—531no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4—532no—
176C2×C22⋊C4—532no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4—532no—
179C2×C22⋊C4—532no—
180C2×C22⋊C4—1064no—
181C2×C22⋊C4C2×D5311yescharacteristic
182C2×C22⋊C4C2×D5311yescharacteristic
183C2×C42—531no—
184C2×C42—531no—
185#95—532no—
186#95—532no—
187#95—532no—
188#95C2×D5312yescomplemented
189C24—532no—
190C24—1064no—
191C24—1062no—
192C24—1062no—
193C24—1062no—
194C24—2122no—
195C24C22×D5311yescharacteristic
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—534no—
199C22×C4—531no—
200C22×C4—531no—
201C22×C4—532no—
202C22×C4—531no—
203C22×C4—531no—
204C22×C4—1062no—
205C22×C4—1062no—
206C22×C4C22×D5312yes—
207C22×C4C22×D5312yes—
208C22×C4C22×D5311yescharacteristic
209C22×C4C22×D5311yescharacteristic
210C4⋊C4—42no—
211C4⋊C4—1062no—
212C4⋊C4—1062no—
213C4⋊C4—2122no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—1064no—
217C22⋊C4—1064no—
218C22⋊C4—1064no—
219C22⋊C4—1064no—
220C22⋊C4—1062no—
221C22⋊C4—1062no—
222C22⋊C4—2122no—
223C22⋊C4—2122no—
224C42—1062no—
225C42—1062no—
226C23—22no—
227C23—22no—
228C23—534no—
229C23—1068no—
230C23—1062no—
231C23—1062no—
232C23—1062no—
233C23—1062no—
234C23—1062no—
235C23—1062no—
236C23—2128no—
237C23—2128no—
238C23—2124no—
239C23—2124no—
240C23#1171994512yes—
241C23D4×D5312yes—
242C23#1171994912yes—
243C23C23×D5311yescharacteristic
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—44no—
255C2×C4—532no—
256C2×C4—532no—
257C2×C4—1068no—
258C2×C4—1061no—
259C2×C4—1061no—
260C2×C4—1061no—
261C2×C4—1061no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—1062no—
267C2×C4—1062no—
268C2×C4—1062no—
269C22—24no—
270C22—24no—
271C22—22no—
272C22—22no—
273C22—1064no—
274C22—1064no—
275C22—1064no—
276C22—21216no—
277C22—2122no—
278C22—2122no—
279C22—2122no—
280C22—2122no—
281C22#1313320214yes—
282C22C2×D4×D5311yescharacteristic
283C22#41×D5312yes—
284C22C2×#1171994511yescharacteristic
285C22C2×#1171994911yescharacteristic
286C22#1313326212yes—
287C4—42no—
288C4—41no—
289C4—42no—
290C4—41no—
291C4—1062no—
292C4—1062no—
293C4—2122no—
294C4—2121no—
295C4—2121no—
296C2—24no—
297C2—2124no—
298C2C2×#1313320211yescharacteristic
299C2#1681113111yescharacteristic
300C2#1681118811yescharacteristic
301C2#1681130012yes—
302C2#1681130212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#385C2
hypercenter#385D53
hypocenterC53#2482
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#13133173C2semidirect
3. #16810558C2semidirect
4. #16810552C2semidirect
5. #16810513C2semidirect
6. #16810512C2semidirect
7. #16810496C2semidirect
8. #16810484C2semidirect
11. C53×#385C2semidirect
15. C2×#11719919C22semidirect
16. C2×#11719919C22semidirect
17. C2×#11719919C22semidirect
27. C2×C4×Dic53C22semidirect
28. C2×C4×Dic53C22semidirect
29. #13133124C22semidirect
30. #13133124C22semidirect
31. #13133124C22semidirect
33. C2×C4⋊C4×C53C22semidirect
36. #95×C53C22semidirect
159. #385D53semidirect
163. C53#2482semidirect
171. C2×C4⋊C4C2×D53semidirect
188. #95C2×D53semidirect