Finite Group Databasebeta

← #20282834

Subgroups — #20282834

Order 6784.

508 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 117 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#16810562C211yesmaximal, characteristic, complemented
2#16810551C212yesmaximal, complemented
3#16810540C211yesmaximal, characteristic, complemented
4#16810539C212yesmaximal, complemented
5#16810538C212yesmaximal, complemented
6#16810532C212yesmaximal, complemented
7#16810531C212yesmaximal, complemented
8#16810514C212yesmaximal, complemented
9C53×#390C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C22×C22⋊D53C2212yes—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919C2212yes—
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2211yescharacteristic
25#13133124C2211yescharacteristic
26C22×D4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2212yes—
29C2×C22⋊C4×C53C2212yes—
30C2×C42×C53C2211yescharacteristic
31C24×C53C2312yes—
32C23×D53C2312yes—
33C2×D4×C53—22no—
34C2×D4×C53—41no—
35C2×D4×C53—41no—
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C2×C22⋊D53—42no—
41C2×C22⋊D53—42no—
42C2×C22⋊D53—42no—
43C2×C22⋊D53—42no—
44C2×C22⋊D53—42no—
45C2×C22⋊D53—42no—
46C22×Dic53C2312yes—
47C22×Dic53C2312yes—
48C22×Dic53C2312yes—
49C2×D212—42no—
50C2×D212—41no—
51C4⋊C4×C53—41no—
52C22⋊C4×C53—42no—
53C22⋊C4×C53—42no—
54C42×C53—41no—
55C22⋊Dic53—42no—
56C22⋊Dic53—42no—
57#11719921—24no—
58#11719921—42no—
59#11719921—42no—
60#11719921—42no—
61C4⋊Dic53—41no—
62#11719919—42no—
63C23×C53—22no—
64C23×C53—22no—
65C23×C53—22no—
66C23×C53—22no—
67C23×C53—22no—
68C23×C53—22no—
69C23×C53—22no—
70C23×C53C2411yescharacteristic
71C22×D53—22no—
72C22×D53—22no—
73C22×D53—22no—
74C22×D53—22no—
75C22×D53—22no—
76C22×D53—22no—
77C22×D53—22no—
78D4×C53—44no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53#4112yes—
89C22⋊D53—48no—
90C22⋊D53—48no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100D212—44no—
101C22×C53—42no—
102C22×C53—42no—
103C22×C53—42no—
104C22×C53—42no—
105C22×C53—42no—
106C22×C53—42no—
107C22×C53—42no—
108C22×C53C2×#4111yescharacteristic
109C22×C53H(5,𝔽2)11yescharacteristic
110C22×C53H(5,𝔽2)11yescharacteristic
111C22×C53H(5,𝔽2)12yes—
112C22×C53H(5,𝔽2)12yes—
113C2×D53—42no—
114C2×D53—42no—
115C2×D53—42no—
116C2×D53—42no—
117C2×D53—42no—
118C2×D53—42no—
119C2×D53—42no—
120C4×C53—22no—
121C4×C53—41no—
122C4×C53—41no—
123C4×C53—42no—
124Dic53—42no—
125Dic53—42no—
126Dic53—42no—
127#2828—531nomaximal, Sylow
128C2×C53—82no—
129C2×C53#53811yescharacteristic
130C2×C53#54012yes—
131C2×C53#56011yescharacteristic
132C2×C53#56012yes—
133C2×C53H(3,𝔽4)11yescharacteristic
134D53—82no—
135#400—531no—
136#397—532no—
137#394—531no—
138#394—532no—
139#394—532no—
140#392—532no—
141#392—532no—
142#390D5311yescharacteristic, complemented
143#388—532no—
144C53#282811yescharacteristic, complemented, Sylow
145C22×D4—532no—
146C22×D4—532no—
147C22×D4—531no—
148C22×D4C2×D5311yescharacteristic
149C2×C4⋊C4—532no—
150C2×C4⋊C4—531no—
151C2×C4⋊C4C2×D5311yescharacteristic
152C2×C22⋊C4—532no—
153C2×C22⋊C4—532no—
154C2×C22⋊C4—532no—
155C2×C22⋊C4—532no—
156C2×C22⋊C4—532no—
157C2×C22⋊C4—532no—
158C2×C22⋊C4C2×D5312yes—
159C2×C22⋊C4C2×D5312yes—
160C2×C42C2×D5311yescharacteristic
161#95—532no—
162#95—532no—
163#95—532no—
164#95—531no—
165#95—531no—
166C24—532no—
167C24C22×D5312yes—
168C2×D4—22no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—2122no—
172C2×D4—2121no—
173C2×D4—2122no—
174C2×D4—2122no—
175C2×D4—2122no—
176C2×D4—2122no—
177C2×D4—2122no—
178C2×D4—2122no—
179C22×C4—532no—
180C22×C4—532no—
181C22×C4—532no—
182C22×C4C22×D5311yescharacteristic
183C22×C4C22×D5312yes—
184C22×C4C22×D5311yescharacteristic
185C22×C4C22×D5311yescharacteristic
186C4⋊C4—41no—
187C4⋊C4—2122no—
188C4⋊C4—2121no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—1064no—
192C22⋊C4—2122no—
193C22⋊C4—2122no—
194C22⋊C4—2122no—
195C22⋊C4—2122no—
196C22⋊C4—2122no—
197C42—41no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—1062no—
206C23—1062no—
207C23—1062no—
208C23—1062no—
209C23—1062no—
210C23—1062no—
211C23—1062no—
212C23C23×D5311yescharacteristic
213D4—44no—
214D4—2124no—
215D4—2128no—
216D4—2128no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4#1171994512yes—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—2122no—
244C22—2122no—
245C22—2122no—
246C22—2122no—
247C22—2122no—
248C22—2122no—
249C22—2122no—
250C22C2×#1171994511yescharacteristic
251C22#1313326212yes—
252C22#1313326211yescharacteristic
253C22#1313326211yescharacteristic
254C22#1313326312yes—
255C4—22no—
256C4—41no—
257C4—41no—
258C4—42no—
259C4—2122no—
260C4—2122no—
261C4—2122no—
262C2—82no—
263C2—4242no—
264C2#1681121111yescharacteristic
265C2#1681122412yes—
266C2#1681136011yescharacteristic
267C2#1681136112yes—
268C2#1681136511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#390C2
hypercenter#390D53
hypocenterC53#2828
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #16810562C2semidirect
2. #16810551C2semidirect
3. #16810540C2semidirect
4. #16810539C2semidirect
5. #16810538C2semidirect
6. #16810532C2semidirect
7. #16810531C2semidirect
8. #16810514C2semidirect
9. C53×#390C2semidirect
142. #390D53semidirect
144. C53#2828semidirect