Finite Group Databasebeta

← #20281979

Subgroups — #20281979

Order 6784.

484 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 121 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#16810559C211yesmaximal, characteristic, complemented
2#16810552C212yesmaximal, complemented
3#16810541C212yesmaximal, complemented
4#16810532C211yesmaximal, characteristic
5#16810523C212yesmaximal, complemented
6#16810515C212yesmaximal, complemented
7#16810505C212yesmaximal, complemented
8#16810502C212yesmaximal, complemented
9C53×#381C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×C22⋊Dic53C2212yes—
16C2×#11719921C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×C4×Dic53C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2211yescharacteristic
25#13133124C2211yescharacteristic
26C23×C4×C53C2211yescharacteristic
27C2×C22⋊C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2211yescharacteristic
29#95×C53C2212yes—
30#95×C53C2212yes—
31C24×C53C2311yescharacteristic
32C23×D53C2312yes—
33C22×C4×C53—22no—
34C22×C4×C53—41no—
35C22×C4×C53—41no—
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×C4×C53C2312yes—
40C2×C22⋊D53—24no—
41C2×C22⋊D53—24no—
42C2×C22⋊D53—42no—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C2×D212—42no—
47C2×D212—41no—
48C22⋊C4×C53—41no—
49C22⋊C4×C53—41no—
50C22⋊Dic53—42no—
51#11719921—24no—
52#11719921—42no—
53#11719921—42no—
54#11719921—42no—
55#11719921—42no—
56C4⋊Dic53—41no—
57#11719919—24no—
58#11719919—42no—
59C4×Dic53—42no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53—21no—
63C23×C53—21no—
64C23×C53#4112yes—
65C23×C53C2411yescharacteristic
66C22×D53—22no—
67C22×D53—22no—
68C22×D53—22no—
69C22×D53—22no—
70C22×D53—22no—
71C22×D53—22no—
72C22×D53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—44no—
84C22⋊D53—48no—
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#4114yes—
94D212—44no—
95C22×C53—22no—
96C22×C53—24no—
97C22×C53—42no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53C2×#4112yes—
101C22×C53C2×#4111yescharacteristic
102C22×C53H(5,𝔽2)11yescharacteristic
103C22×C53H(5,𝔽2)11yescharacteristic
104C22×C53H(5,𝔽2)12yes—
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—42no—
114C4×C53—41no—
115C4×C53—41no—
116Dic53—24no—
117Dic53—42no—
118Dic53—42no—
119#2752—531nomaximal, Sylow
120C2×C53—42no—
121C2×C53#53811yescharacteristic
122C2×C53#54012yes—
123C2×C53#55112yes—
124C2×C53#55911yescharacteristic
125C2×C53#56011yescharacteristic
126D53—82no—
127#399—531no—
128#397—532no—
129#394—532no—
130#392—531no—
131#390—532no—
132#388—532no—
133#387—532no—
134#386—532no—
135#381D5311yescharacteristic, complemented
136C53#275211yescharacteristic, complemented, Sylow
137C22×D4—532no—
138C22×D4—531no—
139C23×C4C2×D5311yescharacteristic
140C2×C4⋊C4—532no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—531no—
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
151C2×C42—532no—
152#95—532no—
153#95—532no—
154#95—531no—
155#95—531no—
156#95C2×D5312yes—
157#95C2×D5312yes—
158C24—532no—
159C24C22×D5311yescharacteristic
160C2×D4—1064no—
161C2×D4—1064no—
162C2×D4—2121no—
163C2×D4—2122no—
164C2×D4—2122no—
165C22×C4—22no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—532no—
169C22×C4—532no—
170C22×C4—532no—
171C22×C4C22×D5312yes—
172C22×C4C22×D5311yescharacteristic
173C22×C4C22×D5311yescharacteristic
174C22×C4C22×D5312yes—
175C4⋊C4—1064no—
176C4⋊C4—2122no—
177C4⋊C4—2121no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—1064no—
181C22⋊C4—2122no—
182C22⋊C4—2122no—
183C22⋊C4—2122no—
184C22⋊C4—2122no—
185C22⋊C4—2122no—
186C42—2122no—
187C23—22no—
188C23—21no—
189C23—21no—
190C23—22no—
191C23—1062no—
192C23—1062no—
193C23—1062no—
194C23—1062no—
195C23—1062no—
196C23—1062no—
197C23—1062no—
198C23#1171994512yes—
199C23C23×D5311yescharacteristic
200D4—2124no—
201D4—2128no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—44no—
213C2×C4—534no—
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—
234C22#41×D5312yes—
235C22C2×#1171994511yescharacteristic
236C22#1313326211yescharacteristic
237C22#1313326211yescharacteristic
238C22#1313326312yes—
239C4—41no—
240C4—41no—
241C4—42no—
242C4—42no—
243C4—1064no—
244C4—2122no—
245C4—2122no—
246C2—42no—
247C2—4242no—
248C2#1681121111yescharacteristic
249C2#1681122512yes—
250C2#1681129712yes—
251C2#1681135711yescharacteristic
252C2#1681136311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#381C2
hypercenter#381D53
hypocenterC53#2752
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. #16810559C2semidirect
2. #16810552C2semidirect
3. #16810541C2semidirect
5. #16810523C2semidirect
6. #16810515C2semidirect
7. #16810505C2semidirect
8. #16810502C2semidirect
9. C53×#381C2semidirect
135. #381D53semidirect
136. C53#2752semidirect