Finite Group Databasebeta

← #20281761

Subgroups — #20281761

Order 6784.

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

normal subgroups only

This group has 560 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
1#16810551C212yesmaximal, complemented
2#16810548C211yesmaximal, characteristic, complemented
3#16810535C212yesmaximal, complemented
4#16810535C211yesmaximal, characteristic, complemented
5#16810517C212yesmaximal, complemented
6#16810516C212yesmaximal, complemented
7#16810511C212yesmaximal, complemented
8#16810479C212yesmaximal
9C53×#379C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic, complemented
11C2×C4⋊Dic53C2211yescharacteristic, complemented
12C2×#11719919C2212yescomplemented
13C2×C22⋊Dic53C2212yes—
14C2×C22⋊Dic53C2212yes—
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×C22⋊Dic53C2212yes—
19C2×C4×Dic53C2212yescomplemented
20#13133124C2212yescomplemented
21#13133124C2212yescomplemented
22#13133124C2212yescomplemented
23#13133124C2211yescharacteristic, complemented
24#13133124C2212yescomplemented
25#13133124C2211yescharacteristic, complemented
26C25×C53C2211yescharacteristic
27C2×C22⋊C4×C53C2212yes—
28C2×C22⋊C4×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2212yes—
30C2×C22⋊C4×C53C2211yescharacteristic
31C24×C53—22no—
32C24×C53—24no—
33C24×C53—42no—
34C24×C53—41no—
35C24×C53—41no—
36C24×C53C2312yes—
37C24×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C2×Dic106—42no—
46C2×Dic106—41no—
47C22⋊C4×C53—24no—
48C22⋊C4×C53—42no—
49C22⋊C4×C53—41no—
50C22⋊C4×C53—41no—
51C22⋊Dic53—24no—
52C22⋊Dic53—24no—
53C22⋊Dic53—42no—
54C22⋊Dic53—42no—
55C22⋊Dic53—42no—
56C22⋊Dic53—42no—
57C4⋊Dic53—41no—
58#11719919—42no—
59C4×Dic53—42no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53—22no—
63C23×C53—22no—
64C23×C53—22no—
65C23×C53—22no—
66C23×C53—22no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53—22no—
70C23×C53—48no—
71C23×C53—44no—
72C23×C53—48no—
73C23×C53—44no—
74C23×C53—44no—
75C23×C53#4112yes—
76C23×C53#4114yes—
77C23×C53C2411yescharacteristic
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—21no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97Dic106—44no—
98C22×C53—24no—
99C22×C53—22no—
100C22×C53—24no—
101C22×C53—24no—
102C22×C53—24no—
103C22×C53—416no—
104C22×C53—42no—
105C22×C53—42no—
106C22×C53—42no—
107C22×C53—42no—
108C22×C53—41no—
109C22×C53—42no—
110C22×C53—41no—
111C22×C53C2×#4112yes—
112C22×C53C2×#4111yescharacteristic
113C22×C53H(5,𝔽2)11yescharacteristic
114C22×C53H(5,𝔽2)12yes—
115C22×C53H(5,𝔽2)11yescharacteristic
116C4×C53—42no—
117C4×C53—41no—
118C4×C53—41no—
119Dic53—42no—
120Dic53—42no—
121Dic53—42no—
122Dic53—42no—
123#2736—531nomaximal, Sylow
124C2×C53—42no—
125C2×C53—44no—
126C2×C53#53812yes—
127C2×C53#53811yescharacteristic
128C2×C53#55112yes—
129C2×C53#55111yescharacteristic
130C2×C53H(3,𝔽4)11yescharacteristic
131#397—531no—
132#397—532no—
133#393—531no—
134#393—532no—
135#388—532no—
136#388—532no—
137#388—532no—
138#379—532no—
139#379D5311yescharacteristic
140C53#273611yescharacteristic, complemented, Sylow
141C25C2×D5311yescharacteristic
142C22×Q8—531no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4—531no—
145C2×C22⋊C4—532no—
146C2×C22⋊C4—532no—
147C2×C22⋊C4—532no—
148C2×C22⋊C4—532no—
149C2×C22⋊C4—532no—
150C2×C22⋊C4—532no—
151C2×C22⋊C4C2×D5312yes—
152C2×C22⋊C4C2×D5311yescharacteristic
153C2×C22⋊C4C2×D5312yes—
154C2×C22⋊C4C2×D5311yescharacteristic
155C2×C42—532no—
156#95—532no—
157#95—531no—
158#95—532no—
159#95—532no—
160#95—532no—
161#95—531no—
162C24—22no—
163C24—24no—
164C24—42no—
165C24—41no—
166C24—41no—
167C24C22×D5311yescharacteristic
168C24C22×D5312yes—
169C2×Q8—2122no—
170C2×Q8—2121no—
171C22×C4—532no—
172C22×C4—532no—
173C22×C4—532no—
174C22×C4—532no—
175C22×C4C22×D5311yescharacteristic
176C22×C4C22×D5311yescharacteristic
177C22×C4C22×D5312yes—
178C4⋊C4—2122no—
179C4⋊C4—2121no—
180C22⋊C4—24no—
181C22⋊C4—41no—
182C22⋊C4—42no—
183C22⋊C4—41no—
184C22⋊C4—1064no—
185C22⋊C4—1064no—
186C22⋊C4—2122no—
187C22⋊C4—2122no—
188C22⋊C4—2122no—
189C22⋊C4—2122no—
190C42—2122no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—21no—
196C23—21no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—48no—
202C23—44no—
203C23—48no—
204C23—44no—
205C23—44no—
206C23#1171994512yes—
207C23#1171994714yes—
208C23C23×D5311yescharacteristic
209Q8—2124no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1062no—
229C22—24no—
230C22—24no—
231C22—24no—
232C22—24no—
233C22—22no—
234C22—416no—
235C22—42no—
236C22—41no—
237C22—41no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22C2×#1171994712yes—
243C22C2×#1171994511yescharacteristic
244C22#1313326212yes—
245C22#1313326211yescharacteristic
246C22#1313326211yescharacteristic
247C4—41no—
248C4—41no—
249C4—42no—
250C4—2122no—
251C4—2122no—
252C4—2122no—
253C4—2122no—
254C2—42no—
255C2—44no—
256C2#1681120812yes—
257C2#1681121111yescharacteristic
258C2#1681129912yes—
259C2#1681130411yescharacteristic
260C2#1681136611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#379C2
hypercenter#379D53
hypocenterC53#2736
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. #16810551C2semidirect
2. #16810548C2semidirect
3. #16810535C2semidirect
4. #16810535C2semidirect
5. #16810517C2semidirect
6. #16810516C2semidirect
7. #16810511C2semidirect
10. C22×Dic106C22semidirect
11. C2×C4⋊Dic53C22semidirect
12. C2×#11719919C22semidirect
19. C2×C4×Dic53C22semidirect
20. #13133124C22semidirect
21. #13133124C22semidirect
22. #13133124C22semidirect
23. #13133124C22semidirect
24. #13133124C22semidirect
25. #13133124C22semidirect
140. C53#2736semidirect