Finite Group Databasebeta

← #20282795

Subgroups — #20282795

Order 6784.

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

normal subgroups only

This group has 472 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#16810561C212yesmaximal, complemented
2#16810552C212yesmaximal, complemented
3#16810540C212yesmaximal, complemented
4#16810538C212yesmaximal, complemented
5#16810536C212yesmaximal, complemented
6#16810535C211yesmaximal, characteristic, complemented
7#16810532C211yesmaximal, characteristic
8#16810502C212yesmaximal, complemented
9C53×#385C211yesmaximal, characteristic, complemented
10C22×Dic106C2211yescharacteristic
11C22×D212C2211yescharacteristic
12C22×C22⋊D53C2212yes—
13C2×#11719919C2212yes—
14C2×C22⋊Dic53C2212yes—
15C2×#11719921C2212yes—
16C2×#11719921C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2211yescharacteristic
23#13133124C2212yes—
24#13133124C2211yescharacteristic
25#13133124C2212yes—
26C23×C4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2212yes—
28C2×C22⋊C4×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2211yescharacteristic
30#95×C53C2212yes—
31C24×C53C2311yescharacteristic
32C23×D53C2312yes—
33C22×C4×C53—41no—
34C22×C4×C53—42no—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2312yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C2×C22⋊D53—24no—
40C2×C22⋊D53—42no—
41C2×C22⋊D53—42no—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C2×D212—41no—
46C2×D212—42no—
47C2×Dic106—41no—
48C2×Dic106—42no—
49C4⋊C4×C53—42no—
50C22⋊C4×C53—22no—
51C22⋊C4×C53—41no—
52C22⋊Dic53—42no—
53#11719921—42no—
54#11719921—42no—
55#11719921—42no—
56#11719921—42no—
57#11719921—42no—
58#11719919—42no—
59C23×C53—22no—
60C23×C53—22no—
61C23×C53—21no—
62C23×C53—21no—
63C23×C53#4112yes—
64C23×C53C2411yescharacteristic
65C22×D53—22no—
66C22×D53—22no—
67C22×D53—22no—
68C22×D53—22no—
69C22×D53—22no—
70C22×D53—22no—
71C22×D53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—44no—
83C22⋊D53—48no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93D212—44no—
94Dic106—44no—
95C22×C53—24no—
96C22×C53—22no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—42no—
100C22×C53C2×#4111yescharacteristic
101C22×C53H(5,𝔽2)12yes—
102C22×C53H(5,𝔽2)11yescharacteristic
103C22×C53H(5,𝔽2)12yes—
104C22×C532−1+411yescharacteristic
105C2×D53—42no—
106C2×D53—42no—
107C2×D53—42no—
108C2×D53—42no—
109C2×D53—42no—
110C2×D53—42no—
111C2×D53—42no—
112C4×C53—42no—
113C4×C53—41no—
114C4×C53—41no—
115C4×C53—42no—
116Dic53—42no—
117Dic53—42no—
118Dic53—42no—
119#2824—531nomaximal, Sylow
120C2×C53—42no—
121C2×C53#53812yes—
122C2×C53#53911yescharacteristic
123C2×C53#56011yescharacteristic
124C2×C53H(3,𝔽4)11yescharacteristic
125C2×C53#56212yes—
126D53—82no—
127#400—532no—
128#397—532no—
129#394—532no—
130#394—532no—
131#393—531no—
132#393—532no—
133#392—531no—
134#386—532no—
135#385D5311yescharacteristic, complemented
136C53#282411yescharacteristic, complemented, Sylow
137C22×Q8—531no—
138C22×D4—532no—
139C22×D4—531no—
140C23×C4C2×D5311yescharacteristic
141C2×C4⋊C4—532no—
142C2×C4⋊C4C2×D5312yes—
143C2×C22⋊C4—532no—
144C2×C22⋊C4—532no—
145C2×C22⋊C4—532no—
146C2×C22⋊C4—532no—
147C2×C22⋊C4—532no—
148C2×C22⋊C4—532no—
149C2×C22⋊C4C2×D5311yescharacteristic
150C2×C22⋊C4C2×D5311yescharacteristic
151#95—532no—
152#95—532no—
153#95—531no—
154#95—532no—
155#95—532no—
156#95—531no—
157#95C2×D5312yes—
158C24—532no—
159C24C22×D5311yescharacteristic
160C2×Q8—2121no—
161C2×Q8—2122no—
162C2×D4—1064no—
163C2×D4—2121no—
164C2×D4—2122no—
165C2×D4—2122no—
166C2×D4—2122no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—532no—
170C22×C4—532no—
171C22×C4—532no—
172C22×C4C22×D5311yescharacteristic
173C22×C4C22×D5312yes—
174C22×C4C22×D5311yescharacteristic
175C22×C4C22×D5312yes—
176C4⋊C4—42no—
177C4⋊C4—2122no—
178C22⋊C4—22no—
179C22⋊C4—41no—
180C22⋊C4—2122no—
181C22⋊C4—2122no—
182C22⋊C4—2122no—
183C22⋊C4—2122no—
184C22⋊C4—2122no—
185C22⋊C4—2122no—
186C23—21no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—1062no—
191C23—1062no—
192C23—1062no—
193C23—1062no—
194C23—1062no—
195C23—1062no—
196C23—1062no—
197C23#1171994512yes—
198C23C23×D5311yescharacteristic
199Q8—2124no—
200D4—2124no—
201D4—2128no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—44no—
213C2×C4—1062no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1062no—
222C22—24no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—2122no—
228C22—2122no—
229C22—2122no—
230C22—2122no—
231C22—2122no—
232C22—2122no—
233C22—2122no—
234C22C2×#1171994511yescharacteristic
235C22#1313326211yescharacteristic
236C22#1313326212yes—
237C22#1313326312yes—
238C22#1313326511yescharacteristic
239C4—41no—
240C4—42no—
241C4—41no—
242C4—42no—
243C4—2122no—
244C4—2122no—
245C4—2122no—
246C2—42no—
247C2—4242no—
248C2#1681120612yes—
249C2#1681122011yescharacteristic
250C2#1681136311yescharacteristic
251C2#1681136611yescharacteristic
252C2#1681136912yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#385C2
hypercenter#385D53
hypocenterC53#2824
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. #16810561C2semidirect
2. #16810552C2semidirect
3. #16810540C2semidirect
4. #16810538C2semidirect
5. #16810536C2semidirect
6. #16810535C2semidirect
8. #16810502C2semidirect
9. C53×#385C2semidirect
135. #385D53semidirect
136. C53#2824semidirect