Finite Group Databasebeta

← #20280451

Subgroups — #20280451

Order 6784.

824 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13133185C212yesmaximal, complemented
2C22×#11719921C211yesmaximal, characteristic, complemented
3#16810549C212yesmaximal, complemented
4#16810539C214yesmaximal, complemented
5#16810531C212yesmaximal, complemented
6#16810503C212yesmaximal, complemented
7#16810483C211yesmaximal, characteristic, complemented
8C2×#121×C53C211yesmaximal, characteristic, complemented
9C24×D53C2211yescharacteristic
10C22×C22⋊D53C2214yes—
11C23×Dic53C2211yescharacteristic
12#13133185—28no—
13C2×C4⋊Dic53C2212yescomplemented
14C2×C22⋊Dic53C2212yes—
15C2×C22⋊Dic53C2214yescomplemented
16C2×#11719921—24no—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2214yes—
20#13133124C2214yescomplemented
21#13133124C2212yescomplemented
22C22×D4×C53C2212yes—
23C22×D4×C53C2211yescharacteristic, complemented
24C23×C4×C53C2211yescharacteristic
25#121×C53—24no—
26C2×C4⋊C4×C53C2211yescharacteristic, complemented
27C2×C22⋊C4×C53C2212yescomplemented
28C24×C53C2312yes—
29C24×C53C2311yescharacteristic
30C23×D53—22no—
31C23×D53—22no—
32C23×D53—42no—
33C23×D53—42no—
34C23×D53D414yescomplemented
35C23×D53C2312yes—
36C2×D4×C53—24no—
37C2×D4×C53—24no—
38C2×D4×C53—24no—
39C2×D4×C53—22no—
40C2×D4×C53—42no—
41C22×C4×C53—42no—
42C22×C4×C53D414yescomplemented
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C2×C22⋊D53—28no—
47C2×C22⋊D53—44no—
48C2×C22⋊D53—44no—
49C22×Dic53—22no—
50C22×Dic53—42no—
51C22×Dic53C2312yes—
52C22×Dic53C2314yes—
53C4⋊C4×C53—22no—
54C22⋊C4×C53—24no—
55C22⋊Dic53—24no—
56C22⋊Dic53—44no—
57#11719921—28no—
58#11719921—24no—
59#11719921—24no—
60C4⋊Dic53—42no—
61C23×C53—22no—
62C23×C53—22no—
63C23×C53—22no—
64C23×C53—22no—
65C23×C53—22no—
66C23×C53—24no—
67C23×C53—24no—
68C23×C53C2×D412yes—
69C23×C53C2×D412yes—
70C23×C53#4112yes—
71C23×C53C2411yescharacteristic
72C22×D53—28no—
73C22×D53—22no—
74C22×D53—22no—
75C22×D53—24no—
76C22×D53—24no—
77C22×D53—48no—
78C22×D53—44no—
79C22×D53—44no—
80C22×D53—48no—
81C22×D53C2×D414yes—
82D4×C53—44no—
83D4×C53—48no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—44no—
90C2×C4×C53C2×D412yes—
91C2×C4×C53C2×D412yes—
92C22⋊D53—416no—
93C2×Dic53—22no—
94C2×Dic53—24no—
95C2×Dic53—24no—
96C2×Dic53—24no—
97C2×Dic53—24no—
98C2×Dic53—44no—
99C22×C53—22no—
100C22×C53—24no—
101C22×C53—22no—
102C22×C53—24no—
103C22×C53—42no—
104C22×C53—44no—
105C22×C53—42no—
106C22×C53—42no—
107C22×C53—44no—
108C22×C53#12114yes—
109C22×C53C22×D411yescharacteristic
110C22×C53C22×D411yescharacteristic
111C22×C53C2×#4111yescharacteristic
112C22×C53H(5,𝔽2)12yes—
113C22×C53H(5,𝔽2)12yes—
114C2×D53—24no—
115C2×D53—28no—
116C2×D53—416no—
117C2×D53—42no—
118C2×D53—42no—
119C2×D53—44no—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—42no—
123Dic53—44no—
124Dic53—42no—
125#2617—531nomaximal, Sylow
126C2×C53—24no—
127C2×C53—82no—
128C2×C53C2×#12111yescharacteristic
129C2×C53#53412yes—
130C2×C53#53411yescharacteristic
131C2×C53#53511yescharacteristic
132C2×C53#53812yes—
133D53—44no—
134C2×#121D5311yescharacteristic, complemented
135C2×#120—532no—
136C22×C22⋊C4—531no—
137#397—532no—
138#394—534no—
139#392—532no—
140#386—532no—
141#380—531no—
142C53#261711yescharacteristic, complemented, Sylow
143C25—531no—
144C22×D4—534no—
145C22×D4C2×D5312yes—
146C22×D4C2×D5311yescharacteristic, complemented
147C23×C4—531no—
148C23×C4C2×D5311yescharacteristic
149#121—24no—
150#120—1068no—
151C2×C4⋊C4—532no—
152C2×C4⋊C4C2×D5311yescharacteristic, complemented
153C2×C22⋊C4—532no—
154C2×C22⋊C4—532no—
155C2×C22⋊C4—532no—
156C2×C22⋊C4—534no—
157C2×C22⋊C4—534no—
158C2×C22⋊C4—1064no—
159C2×C22⋊C4C2×D5312yescomplemented
160#95—532no—
161#95—534no—
162C24—532no—
163C24—534no—
164C24—1062no—
165C24—1062no—
166C24—2122no—
167C24—2122no—
168C24C22×D5312yes—
169C24C22×D5311yescharacteristic
170C2×D4—24no—
171C2×D4—24no—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—42no—
175C2×D4—1068no—
176C2×D4—2124no—
177C2×D4—2124no—
178C22×C4—42no—
179C22×C4—532no—
180C22×C4—534no—
181C22×C4—1062no—
182C22×C4—2122no—
183C22×C4C22⋊D5314yescomplemented
184C22×C4C22×D5312yes—
185C22×C4C22×D5311yescharacteristic
186C22×C4C22×D5311yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—2122no—
189C22⋊C4—24no—
190C22⋊C4—1068no—
191C22⋊C4—1064no—
192C22⋊C4—1064no—
193C22⋊C4—1064no—
194C22⋊C4—2124no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—24no—
201C23—24no—
202C23—534no—
203C23—1068no—
204C23—1062no—
205C23—1062no—
206C23—1064no—
207C23—1064no—
208C23—2128no—
209C23—2124no—
210C23—2124no—
211C23—2128no—
212C23D4×D5312yes—
213C23#1171994712yes—
214C23C2×C22⋊D5312yes—
215C23C23×D5311yescharacteristic
216D4—44no—
217D4—48no—
218D4—21216no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—44no—
225C2×C4—1064no—
226C2×C4—1062no—
227C2×C4—1064no—
228C2×C4—1064no—
229C2×C4—1064no—
230C2×C4—2124no—
231C2×C4C2×C22⋊D5312yes—
232C2×C4C2×C22⋊D5312yes—
233C22—24no—
234C22—22no—
235C22—22no—
236C22—24no—
237C22—42no—
238C22—44no—
239C22—42no—
240C22—42no—
241C22—44no—
242C22—1064no—
243C22—1068no—
244C22—21216no—
245C22—2122no—
246C22—2122no—
247C22—2124no—
248C22#1313318814yes—
249C22C2×D4×D5311yescharacteristic
250C22C22×C22⋊D5311yescharacteristic
251C22C2×#1171994711yescharacteristic
252C22#1313326212yes—
253C22#1313326312yes—
254C4—22no—
255C4—22no—
256C4—42no—
257C4—2124no—
258C4—2122no—
259C2—24no—
260C2—82no—
261C2—2124no—
262C2C2×#1313318811yescharacteristic
263C2#1681118112yes—
264C2#1681118211yescharacteristic
265C2#1681118711yescharacteristic
266C2#1681121012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#121×C53C2
hypercenterC2×#121D53
hypocenterC53#2617
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#13133185C2semidirect
2. C22×#11719921C2semidirect
3. #16810549C2semidirect
4. #16810539C2semidirect
5. #16810531C2semidirect
6. #16810503C2semidirect
7. #16810483C2semidirect
8. C2×#121×C53C2semidirect
13. C2×C4⋊Dic53C22semidirect
15. C2×C22⋊Dic53C22semidirect
20. #13133124C22semidirect
21. #13133124C22semidirect
23. C22×D4×C53C22semidirect
26. C2×C4⋊C4×C53C22semidirect
27. C2×C22⋊C4×C53C22semidirect
34. C23×D53D4semidirect
42. C22×C4×C53D4semidirect
134. C2×#121D53semidirect
142. C53#2617semidirect
146. C22×D4C2×D53semidirect
152. C2×C4⋊C4C2×D53semidirect
159. C2×C22⋊C4C2×D53semidirect
183. C22×C4C22⋊D53semidirect