Finite Group Databasebeta

← #20282709

Subgroups — #20282709

Order 6784.

712 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13133185C212yesmaximal, complemented
2#16810552C212yesmaximal, complemented
3#16810548C211yesmaximal, characteristic, complemented
4#16810539C212yesmaximal, complemented
5#16810538C212yesmaximal, complemented
6#16810537C212yesmaximal, complemented
7#16810531C212yesmaximal, complemented
8#16810478C211yesmaximal, characteristic, complemented
9C53×#392C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C22×C22⋊D53C2212yes—
13#13133185—28no—
14C2×C4⋊Dic53C2211yescharacteristic, complemented
15C2×#11719919C2212yescomplemented
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2211yescharacteristic
18C2×C22⋊Dic53C2212yescomplemented
19C2×C22⋊Dic53C2212yescomplemented
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24#13133124C2212yescomplemented
25#13133124C2212yescomplemented
26#13133124C2211yescharacteristic, complemented
27C22×D4×C53C2212yes—
28C22×D4×C53C2211yescharacteristic, complemented
29C2×C22⋊C4×C53C2211yescharacteristic
30C2×C22⋊C4×C53C2212yescomplemented
31#95×C53C2211yescharacteristic, complemented
32C24×C53C2312yes—
33C24×C53C2311yescharacteristic
34C23×D53—41no—
35C23×D53—41no—
36C23×D53—42no—
37C23×D53—42no—
38C23×D53D414yescomplemented
39C23×D53C2312yes—
40C2×D4×C53—24no—
41C2×D4×C53—42no—
42C2×D4×C53—42no—
43C2×D4×C53—42no—
44C2×D4×C53—41no—
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2312yes—
47C22×C4×C53C2311yescharacteristic
48C2×C22⋊D53—24no—
49C2×C22⋊D53—24no—
50C2×C22⋊D53—42no—
51C2×C22⋊D53—42no—
52C2×C22⋊D53—42no—
53C2×C22⋊D53—42no—
54C22×Dic53C2312yes—
55C22×Dic53C2312yes—
56C22×Dic53C2312yes—
57C22⋊C4×C53—41no—
58C22⋊C4×C53—42no—
59C22⋊Dic53—24no—
60C22⋊Dic53—41no—
61C22⋊Dic53—42no—
62C22⋊Dic53—42no—
63#11719921—24no—
64#11719921—24no—
65#11719921—42no—
66#11719921—42no—
67C4⋊Dic53—41no—
68#11719919—42no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53C2×D412yes—
81C23×C53C2411yescharacteristic
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—44no—
89C22×D53—48no—
90C22×D53—44no—
91C22×D53—48no—
92C22×D53—44no—
93C22×D53C2×D414yes—
94D4×C53—48no—
95D4×C53—44no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C22⋊D53—48no—
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—
114C22×C53—24no—
115C22×C53—22no—
116C22×C53—41no—
117C22×C53—42no—
118C22×C53—41no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—42no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53C22×D411yescharacteristic
127C22×C53H(5,𝔽2)12yes—
128C22×C53H(5,𝔽2)11yescharacteristic
129C22×C53H(5,𝔽2)12yes—
130C22×C53H(5,𝔽2)11yescharacteristic
131C2×D53—24no—
132C2×D53—24no—
133C2×D53—24no—
134C2×D53—416no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—42no—
139C4×C53—42no—
140C4×C53—41no—
141C4×C53—41no—
142Dic53—42no—
143Dic53—42no—
144Dic53—42no—
145#2812—531nomaximal, Sylow
146C2×C53—42no—
147C2×C53—82no—
148C2×C53#53411yescharacteristic
149C2×C53#53412yes—
150C2×C53#56011yescharacteristic
151C2×C53#56012yes—
152C2×C53H(3,𝔽4)11yescharacteristic
153D53—44no—
154C2×#120—532no—
155#397—531no—
156#397—532no—
157#394—532no—
158#394—532no—
159#394—532no—
160#392—532no—
161#392D5311yescharacteristic, complemented
162#379—531no—
163C53#281211yescharacteristic, complemented, Sylow
164C25—531no—
165C22×D4—532no—
166C22×D4—532no—
167C22×D4C2×D5312yes—
168C22×D4C2×D5311yescharacteristic, complemented
169#120—1068no—
170C2×C4⋊C4—532no—
171C2×C4⋊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⋊C4—532no—
180C2×C22⋊C4C2×D5311yescharacteristic
181C2×C22⋊C4C2×D5312yescomplemented
182#95—532no—
183#95—531no—
184#95—532no—
185#95C2×D5311yescharacteristic, complemented
186C24—532no—
187C24—534no—
188C24—2121no—
189C24—2122no—
190C24—2122no—
191C24—2121no—
192C24C22×D5312yes—
193C24C22×D5311yescharacteristic
194C2×D4—24no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—1064no—
200C2×D4—1064no—
201C2×D4—2122no—
202C2×D4—2122no—
203C2×D4—2122no—
204C2×D4—2122no—
205C22×C4—532no—
206C22×C4—532no—
207C22×C4—532no—
208C22×C4C22×D5311yescharacteristic
209C22×C4C22×D5312yes—
210C22×C4C22×D5311yescharacteristic
211C4⋊C4—2122no—
212C4⋊C4—2121no—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—1064no—
216C22⋊C4—1064no—
217C22⋊C4—1064no—
218C22⋊C4—2122no—
219C22⋊C4—2122no—
220C22⋊C4—2122no—
221C22⋊C4—2121no—
222C22⋊C4—2122no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—534no—
235C23—1062no—
236C23—1062no—
237C23—1062no—
238C23—1062no—
239C23—1062no—
240C23—1062no—
241C23—2124no—
242C23—2128no—
243C23—2124no—
244C23—2128no—
245C23—2124no—
246C23D4×D5312yes—
247C23C23×D5311yescharacteristic
248D4—48no—
249D4—44no—
250D4—2128no—
251D4—2128no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—1062no—
267C2×C4—1062no—
268C22—22no—
269C22—24no—
270C22—41no—
271C22—42no—
272C22—41no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—1064no—
281C22—1064no—
282C22—1064no—
283C22—21216no—
284C22—2122no—
285C22—2122no—
286C22—2122no—
287C22—2122no—
288C22C2×D4×D5311yescharacteristic
289C22#1313326212yes—
290C22#1313326211yescharacteristic
291C22#1313326212yes—
292C22#1313326311yescharacteristic
293C4—41no—
294C4—41no—
295C4—42no—
296C4—2122no—
297C4—2122no—
298C4—2122no—
299C2—42no—
300C2—82no—
301C2—2124no—
302C2#1681118111yescharacteristic
303C2#1681118312yes—
304C2#1681136011yescharacteristic
305C2#1681136112yes—
306C2#1681136611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2812
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#13133185C2semidirect
2. #16810552C2semidirect
3. #16810548C2semidirect
4. #16810539C2semidirect
5. #16810538C2semidirect
6. #16810537C2semidirect
7. #16810531C2semidirect
8. #16810478C2semidirect
9. C53×#392C2semidirect
14. C2×C4⋊Dic53C22semidirect
15. C2×#11719919C22semidirect
18. C2×C22⋊Dic53C22semidirect
19. C2×C22⋊Dic53C22semidirect
24. #13133124C22semidirect
25. #13133124C22semidirect
26. #13133124C22semidirect
28. C22×D4×C53C22semidirect
30. C2×C22⋊C4×C53C22semidirect
31. #95×C53C22semidirect
38. C23×D53D4semidirect
161. #392D53semidirect
163. C53#2812semidirect
168. C22×D4C2×D53semidirect
181. C2×C22⋊C4C2×D53semidirect
185. #95C2×D53semidirect