Finite Group Databasebeta

← #20278990

Subgroups — #20278990

Order 6784.

672 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 139 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×#13133175C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D53C211yesmaximal, characteristic
3#16810558C211yesmaximal, characteristic, complemented
4#16810550C211yesmaximal, characteristic, complemented
5#16810548C211yesmaximal, characteristic, complemented
6#16810517C212yesmaximal, complemented
7#16810513C212yesmaximal, complemented
8#16810494C212yesmaximal, complemented
9#16810485C211yesmaximal, characteristic, complemented
10#16810478C212yesmaximal
11C53×#399C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15#13133175—24no—
16C2×C4⋊Dic53C2211yescharacteristic, complemented
17C2×#11719919C2212yescomplemented
18C2×C22⋊Dic53C2211yescharacteristic
19C2×C22⋊Dic53C2212yes—
20C2×#11719921C2211yescharacteristic
21C2×#11719921C2212yes—
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C22⋊C4×D53—24no—
27C2×C4×Dic53C2212yescomplemented
28#13133124C2212yescomplemented
29#13133124C2212yescomplemented
30#13133124C2212yescomplemented
31#13133124C2211yescharacteristic, complemented
32C2×C4⋊C4×C53C2211yescharacteristic, complemented
33C2×C4⋊C4×C53C2212yescomplemented
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic, complemented
37C24×C53C2311yescharacteristic
38C23×D53—22no—
39C23×D53—22no—
40C23×D53—22no—
41C23×D53—24no—
42C23×D53—42no—
43C23×D53C2311yescharacteristic
44C23×D53C2311yescharacteristic
45C22×C4×C53C2312yes—
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2312yes—
48C22×C4×C53C2311yescharacteristic
49C22×Dic53C2312yes—
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2311yescharacteristic
53C2×C4×D53—24no—
54C2×C4×D53—22no—
55C2×C4×D53—42no—
56C2×C4×D53D414yes—
57C4⋊C4×C53—22no—
58C4⋊C4×C53—42no—
59C22⋊C4×C53—22no—
60C22⋊C4×C53—42no—
61C22⋊Dic53—24no—
62C22⋊Dic53—22no—
63#11719921—24no—
64#11719921—24no—
65#11719921—24no—
66#11719921—22no—
67#11719921—22no—
68#11719921—42no—
69C4⋊Dic53—22no—
70#11719919—42no—
71C4×Dic53—24no—
72C23×C53—21no—
73C23×C53—21no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53#4112yes—
77C23×C53C2411yescharacteristic
78C22×D53—21no—
79C22×D53—21no—
80C22×D53—22no—
81C22×D53—28no—
82C22×D53—21no—
83C22×D53—22no—
84C22×D53—21no—
85C22×D53—22no—
86C22×D53—48no—
87C22×D53—48no—
88C22×D53—44no—
89C22×D53—44no—
90C22×D53C2×D412yes—
91C22×D53#4112yes—
92C22×D53#4114yes—
93C2×C4×C53—22no—
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×C53C2×D412yes—
103C2×Dic53—22no—
104C2×Dic53—21no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53C2×D412yes—
113C4×D53—28no—
114C4×D53—44no—
115C22×C53—22no—
116C22×C53—24no—
117C22×C53—41no—
118C22×C53—41no—
119C22×C53—42no—
120C22×C53C22×D411yescharacteristic
121C22×C53C2×#4112yes—
122C22×C53C2×#4111yescharacteristic
123C22×C53C2×#4111yescharacteristic
124C22×C53H(5,𝔽2)12yes—
125C2×D53—22no—
126C2×D53—22no—
127C2×D53—24no—
128C2×D53—24no—
129C2×D53—22no—
130C2×D53—24no—
131C2×D53—416no—
132C2×D53—41no—
133C2×D53—41no—
134C2×D53—42no—
135C2×D53#12314yes—
136C4×C53—22no—
137C4×C53—41no—
138C4×C53—42no—
139C4×C53—42no—
140Dic53—22no—
141Dic53—42no—
142Dic53—41no—
143Dic53—42no—
144#2482—531nomaximal, Sylow
145C2×C53—42no—
146C2×C53C2×#12311yescharacteristic
147C2×C53#52511yescharacteristic
148C2×C53#53511yescharacteristic
149C2×C53#55112yes—
150C2×C53#55112yes—
151D53—24no—
152D53—42no—
153C2×#117—531no—
154C22×C22⋊C4—531no—
155#399—531no—
156#399D5311yescharacteristic, complemented
157#397—531no—
158#397—531no—
159#388—532no—
160#388—532no—
161#385—532no—
162#381—531no—
163#379—532no—
164C53#248211yescharacteristic, complemented, Sylow
165C25—531no—
166C23×C4—531no—
167C23×C4—531no—
168#117—1064no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4C2×D5311yescharacteristic, complemented
172C2×C4⋊C4C2×D5312yescomplemented
173C2×C22⋊C4—531no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4—531no—
176C2×C22⋊C4—531no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4—532no—
179C2×C22⋊C4—532no—
180C2×C22⋊C4—532no—
181C2×C22⋊C4—1064no—
182C2×C22⋊C4C2×D5311yescharacteristic
183C2×C22⋊C4C2×D5312yes—
184C2×C42—532no—
185#95—532no—
186#95—531no—
187#95—532no—
188#95—532no—
189#95C2×D5311yescharacteristic, complemented
190C24—531no—
191C24—531no—
192C24—1062no—
193C24—1062no—
194C24—1064no—
195C24—1062no—
196C24—2122no—
197C24C22×D5311yescharacteristic
198C22×C4—531no—
199C22×C4—534no—
200C22×C4—532no—
201C22×C4—532no—
202C22×C4—531no—
203C22×C4—1064no—
204C22×C4—1062no—
205C22×C4—2122no—
206C22×C4C22×D5312yes—
207C22×C4C22×D5311yescharacteristic
208C22×C4C22×D5312yes—
209C22×C4C22×D5311yescharacteristic
210C4⋊C4—22no—
211C4⋊C4—42no—
212C4⋊C4—1062no—
213C4⋊C4—2122no—
214C22⋊C4—22no—
215C22⋊C4—42no—
216C22⋊C4—1062no—
217C22⋊C4—1064no—
218C22⋊C4—1064no—
219C22⋊C4—1064no—
220C22⋊C4—1062no—
221C22⋊C4—1064no—
222C22⋊C4—1062no—
223C22⋊C4—2122no—
224C42—1064no—
225C23—21no—
226C23—22no—
227C23—21no—
228C23—22no—
229C23—532no—
230C23—534no—
231C23—532no—
232C23—1068no—
233C23—1061no—
234C23—1061no—
235C23—1062no—
236C23—1061no—
237C23—1062no—
238C23—1061no—
239C23—1062no—
240C23—2128no—
241C23—2124no—
242C23—2128no—
243C23—2124no—
244C23#1171994912yes—
245C23C23×D5311yescharacteristic
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—532no—
256C2×C4—1068no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1061no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—2124no—
267C2×C4D4×D5312yes—
268C22—24no—
269C22—22no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—534no—
274C22—1064no—
275C22—1064no—
276C22—1062no—
277C22—1062no—
278C22—1064no—
279C22—1062no—
280C22—21216no—
281C22—2122no—
282C22—2121no—
283C22—2121no—
284C22C2×D4×D5311yescharacteristic
285C22#41×D5312yes—
286C22#41×D5311yescharacteristic
287C22C2×#1171994911yescharacteristic
288C22#1313326212yes—
289C4—22no—
290C4—42no—
291C4—41no—
292C4—42no—
293C4—1062no—
294C4—2122no—
295C4—2122no—
296C4—2121no—
297C2—42no—
298C2—1064no—
299C2—2122no—
300C2#123×D5311yescharacteristic
301C2#1681113611yescharacteristic
302C2#1681118611yescharacteristic
303C2#1681130012yes—
304C2#1681130112yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2482
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#13133175C2semidirect
3. #16810558C2semidirect
4. #16810550C2semidirect
5. #16810548C2semidirect
6. #16810517C2semidirect
7. #16810513C2semidirect
8. #16810494C2semidirect
9. #16810485C2semidirect
11. C53×#399C2semidirect
16. C2×C4⋊Dic53C22semidirect
17. C2×#11719919C22semidirect
27. C2×C4×Dic53C22semidirect
28. #13133124C22semidirect
29. #13133124C22semidirect
30. #13133124C22semidirect
31. #13133124C22semidirect
32. C2×C4⋊C4×C53C22semidirect
33. C2×C4⋊C4×C53C22semidirect
36. #95×C53C22semidirect
156. #399D53semidirect
164. C53#2482semidirect
171. C2×C4⋊C4C2×D53semidirect
172. C2×C4⋊C4C2×D53semidirect
189. #95C2×D53semidirect