Finite Group Databasebeta

← #20281277

Subgroups — #20281277

Order 6784.

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

normal subgroups only

This group has 680 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×#13133198C211yesmaximal, characteristic, complemented
2C2×#13133185C211yesmaximal, characteristic
3#16810548C212yesmaximal, complemented
4#16810545C212yesmaximal, complemented
5#16810539C212yesmaximal, complemented
6#16810514C212yesmaximal, complemented
7#16810500C212yesmaximal, complemented
8#16810478C212yesmaximal, complemented
9C53×#390C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C23×Dic53C2211yescharacteristic, complemented
13#13133198—24no—
14#13133185—24no—
15C2×C4⋊Dic53C2212yescomplemented
16C2×#11719919C2212yescomplemented
17C2×C22⋊Dic53C2211yescharacteristic
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2211yescharacteristic, complemented
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25#13133124C2212yescomplemented
26#13133124C2212yescomplemented
27#13133124C2212yescomplemented
28C22×D4×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2211yescharacteristic, complemented
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×D4×C53—41no—
42C2×D4×C53—41no—
43C2×D4×C53D414yes—
44C22×C4×C53C2312yes—
45C22×C4×C53C2312yes—
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—24no—
48C2×C22⋊D53—24no—
49C2×C22⋊D53—42no—
50C22×Dic53—22no—
51C22×Dic53—41no—
52C22×Dic53—41no—
53C22×Dic53C2312yes—
54C22×Dic53C2312yes—
55C22×Dic53C2312yes—
56C4⋊C4×C53—41no—
57C22⋊C4×C53—42no—
58C22⋊C4×C53—42no—
59C42×C53—41no—
60C22⋊Dic53—22no—
61C22⋊Dic53—24no—
62C22⋊Dic53—22no—
63#11719921—24no—
64#11719921—24no—
65#11719921—42no—
66#11719921—42no—
67#11719921—42no—
68C4⋊Dic53—42no—
69#11719919—24no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53—21no—
73C23×C53—21no—
74C23×C53—22no—
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—
93D4×C53—44no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
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—
103C22⋊D53—48no—
104C2×Dic53—22no—
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—44no—
114C22×C53—22no—
115C22×C53—22no—
116C22×C53—24no—
117C22×C53—24no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53—41no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53C22×D411yescharacteristic
125C22×C53C2×#4112yes—
126C22×C53H(5,𝔽2)11yescharacteristic
127C22×C53H(5,𝔽2)11yescharacteristic
128C22×C53H(5,𝔽2)12yes—
129C2×D53—24no—
130C2×D53—24no—
131C2×D53—24no—
132C2×D53—416no—
133C2×D53—42no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—42no—
137C4×C53—22no—
138C4×C53—42no—
139C4×C53—42no—
140Dic53—42no—
141Dic53—42no—
142Dic53—42no—
143#2697—531nomaximal, Sylow
144C2×C53—42no—
145C2×C53—42no—
146C2×C53#53411yescharacteristic
147C2×C53#53812yes—
148C2×C53#54612yes—
149C2×C53#55111yescharacteristic
150C2×C53#56011yescharacteristic
151D53—44no—
152C2×#123—531no—
153C2×#120—531no—
154#397—532no—
155#396—532no—
156#394—532no—
157#390D5311yescharacteristic, complemented
158#388—532no—
159#386—532no—
160#379—532no—
161C53#269711yescharacteristic, complemented, Sylow
162C25—531no—
163C22×D4—532no—
164C22×D4C2×D5311yescharacteristic
165C23×C4—531no—
166#123—1064no—
167#120—1064no—
168C2×C4⋊C4—532no—
169C2×C4⋊C4—532no—
170C2×C4⋊C4C2×D5311yescharacteristic, complemented
171C2×C22⋊C4—531no—
172C2×C22⋊C4—532no—
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⋊C4C2×D5312yes—
180C2×C22⋊C4C2×D5312yescomplemented
181C2×C42C2×D5311yescharacteristic, complemented
182#95—532no—
183#95—532no—
184#95—532no—
185C24—532no—
186C24—1064no—
187C24—1062no—
188C24—2122no—
189C24—2121no—
190C24—2121no—
191C24C22×D5311yescharacteristic
192C24C22×D5311yescharacteristic
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—1064no—
196C2×D4—1064no—
197C2×D4—2122no—
198C2×D4D21214yes—
199C22×C4—532no—
200C22×C4—532no—
201C22×C4—532no—
202C22×C4—1062no—
203C22×C4—2121no—
204C22×C4—2121no—
205C22×C4C22×D5312yes—
206C22×C4C22×D5312yes—
207C22×C4C22×D5311yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—1064no—
210C4⋊C4—2122no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—1064no—
214C22⋊C4—1062no—
215C22⋊C4—1062no—
216C22⋊C4—1064no—
217C22⋊C4—1064no—
218C22⋊C4—2122no—
219C22⋊C4—2122no—
220C22⋊C4—2122no—
221C42—41no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—534no—
231C23—1062no—
232C23—1062no—
233C23—1062no—
234C23—1062no—
235C23—1062no—
236C23—1062no—
237C23—2128no—
238C23—2124no—
239C23—2124no—
240C23—2128no—
241C23—2124no—
242C23C2×D21212yes—
243C23C2×D21212yes—
244C23C23×D5311yescharacteristic
245D4—44no—
246D4—2128no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—2124no—
265C2×C4C2×D21212yes—
266C22—24no—
267C22—24no—
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—1064no—
277C22—1064no—
278C22—1064no—
279C22—21216no—
280C22—2122no—
281C22—2122no—
282C22—2122no—
283C22—2122no—
284C22C22×D21211yescharacteristic
285C22#41×D5312yes—
286C22#1313326212yes—
287C22#1313326211yescharacteristic
288C22#1313326211yescharacteristic
289C4—22no—
290C4—42no—
291C4—42no—
292C4—2122no—
293C4—2122no—
294C4—2122no—
295C2—42no—
296C2—42no—
297C2—2124no—
298C2#1681118511yescharacteristic
299C2#1681120512yes—
300C2#1681126812yes—
301C2#1681130311yescharacteristic
302C2#1681136011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#390C2
hypercenter#390D53
hypocenterC53#2697
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#13133198C2semidirect
3. #16810548C2semidirect
4. #16810545C2semidirect
5. #16810539C2semidirect
6. #16810514C2semidirect
7. #16810500C2semidirect
8. #16810478C2semidirect
9. C53×#390C2semidirect
12. C23×Dic53C22semidirect
15. C2×C4⋊Dic53C22semidirect
16. C2×#11719919C22semidirect
19. C2×C22⋊Dic53C22semidirect
25. #13133124C22semidirect
26. #13133124C22semidirect
27. #13133124C22semidirect
29. C2×C4⋊C4×C53C22semidirect
31. C2×C22⋊C4×C53C22semidirect
32. C2×C42×C53C22semidirect
157. #390D53semidirect
161. C53#2697semidirect
170. C2×C4⋊C4C2×D53semidirect
180. C2×C22⋊C4C2×D53semidirect
181. C2×C42C2×D53semidirect