Finite Group Databasebeta

← #20279647

Subgroups — #20279647

Order 6784.

900 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 169 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×#13133198C212yesmaximal, complemented
2C2×#13133185C212yesmaximal, complemented
3C22×#11719921C211yesmaximal, characteristic, complemented
4#16810558C212yesmaximal, complemented
5#16810531C212yesmaximal, complemented
6#16810523C214yesmaximal, complemented
7C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
8C2×#121×C53C211yesmaximal, characteristic, complemented
9C24×D53C2211yescharacteristic
10C22×C22⋊D53C2214yes—
11C23×Dic53C2211yescharacteristic
12#13133198—28no—
13#13133185—28no—
14C2×#11719919C2214yescomplemented
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2214yescomplemented
17C2×#11719921—24no—
18C2×#11719921C2212yes—
19C2×#11719921C2214yes—
20C2×#11719921C2212yes—
21C2×C4×Dic53C2212yescomplemented
22#13133124C2212yescomplemented
23C22×D4×C53C2212yes—
24C22×D4×C53C2211yescharacteristic, complemented
25C23×C4×C53C2211yescharacteristic
26#121×C53—24no—
27C2×C4⋊C4×C53C2211yescharacteristic, complemented
28C2×C22⋊C4×C53C2212yescomplemented
29C24×C53C2312yes—
30C24×C53C2311yescharacteristic
31C23×D53—22no—
32C23×D53—24no—
33C23×D53—24no—
34C23×D53—42no—
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—28no—
48C2×C22⋊D53—44no—
49C22×Dic53—22no—
50C22×Dic53—24no—
51C22×Dic53C2312yes—
52C22×Dic53C2314yes—
53C4⋊C4×C53—22no—
54C22⋊C4×C53—24no—
55C22⋊Dic53—28no—
56C22⋊Dic53D418yescomplemented
57#11719921—24no—
58#11719921—24no—
59#11719921—28no—
60#11719919—28no—
61C4×Dic53—42no—
62C23×C53—22no—
63C23×C53—22no—
64C23×C53—22no—
65C23×C53—22no—
66C23×C53—22no—
67C23×C53—24no—
68C23×C53—24no—
69C23×C53C2×D414yes—
70C23×C53C2×D412yes—
71C23×C53C2411yescharacteristic
72C22×D53—28no—
73C22×D53—24no—
74C22×D53—22no—
75C22×D53—22no—
76C22×D53—24no—
77C22×D53—416no—
78C22×D53—44no—
79C22×D53—44no—
80C22×D53#4114yes—
81D4×C53—44no—
82D4×C53—48no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—44no—
89C2×C4×C53C2×D412yes—
90C2×C4×C53C2×D412yes—
91C22⋊D53—416no—
92C2×Dic53—24no—
93C2×Dic53—24no—
94C2×Dic53—24no—
95C2×Dic53—22no—
96C2×Dic53—44no—
97C2×Dic53C2×D418yes—
98C22×C53—22no—
99C22×C53—22no—
100C22×C53—28no—
101C22×C53—42no—
102C22×C53—42no—
103C22×C53—42no—
104C22×C53—44no—
105C22×C53—44no—
106C22×C53#12714yes—
107C22×C53C22×D412yes—
108C22×C53C22×D411yescharacteristic
109C22×C53C2×#4112yes—
110C22×C53H(5,𝔽2)12yes—
111C2×D53—24no—
112C2×D53—28no—
113C2×D53—416no—
114C2×D53—42no—
115C2×D53—42no—
116C2×D53—44no—
117C4×C53—22no—
118C4×C53—22no—
119C4×C53—42no—
120Dic53—28no—
121Dic53—42no—
122#2539—531nomaximal, Sylow
123C2×C53—24no—
124C2×C53—82no—
125C2×C53#52511yescharacteristic
126C2×C53C2×#12711yescharacteristic
127C2×C53#53411yescharacteristic
128C2×C53#54614yes—
129D53—44no—
130C2×#123—532no—
131C2×#121D5311yescharacteristic, complemented
132C2×#120—532no—
133C22×C22⋊C4—531no—
134#399—532no—
135#392—532no—
136#390—534no—
137C4×C22⋊C4—531no—
138C53#253911yescharacteristic, complemented, Sylow
139C25—531no—
140C22×D4—534no—
141C22×D4C2×D5312yes—
142C22×D4C2×D5311yescharacteristic, complemented
143C23×C4—531no—
144C23×C4C2×D5311yescharacteristic
145#123—1068no—
146#121—24no—
147#120—1068no—
148C2×C4⋊C4—534no—
149C2×C4⋊C4C2×D5311yescharacteristic, complemented
150C2×C22⋊C4—532no—
151C2×C22⋊C4—534no—
152C2×C22⋊C4—532no—
153C2×C22⋊C4—532no—
154C2×C22⋊C4—534no—
155C2×C22⋊C4—1064no—
156C2×C22⋊C4C2×D5312yescomplemented
157C2×C42—532no—
158#95—532no—
159C24—532no—
160C24—1064no—
161C24—1062no—
162C24—1064no—
163C24—2122no—
164C24C22×D5312yes—
165C24C22×D5311yescharacteristic
166C2×D4—24no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—22no—
170C2×D4—42no—
171C2×D4—1068no—
172C2×D4—1068no—
173C2×D4—2124no—
174C22×C4—42no—
175C22×C4—534no—
176C22×C4—532no—
177C22×C4—1064no—
178C22×C4—1062no—
179C22×C4C22⋊D5314yescomplemented
180C22×C4C22×D5312yes—
181C22×C4C22×D5311yescharacteristic
182C22×C4C22×D5311yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—1068no—
185C22⋊C4—24no—
186C22⋊C4—538no—
187C22⋊C4—1068no—
188C22⋊C4—1068no—
189C22⋊C4—1064no—
190C22⋊C4—1064no—
191C42—2122no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—24no—
198C23—24no—
199C23—534no—
200C23—1068no—
201C23—1064no—
202C23—1062no—
203C23—1062no—
204C23—1064no—
205C23—21216no—
206C23—2124no—
207C23—2124no—
208C23D4×D5314yes—
209C23C2×C22⋊D5312yes—
210C23C23×D5311yescharacteristic
211D4—44no—
212D4—48no—
213D4—21216no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—44no—
220C2×C4—538no—
221C2×C4—1064no—
222C2×C4—1064no—
223C2×C4—1064no—
224C2×C4—1062no—
225C2×C4—2124no—
226C2×C4C2×C22⋊D5312yes—
227C2×C4C2×C22⋊D5312yes—
228C22—22no—
229C22—22no—
230C22—28no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—44no—
235C22—44no—
236C22—1064no—
237C22—1068no—
238C22—21216no—
239C22—2122no—
240C22—2122no—
241C22—2124no—
242C22#1313321314yes—
243C22C2×D4×D5312yes—
244C22C22×C22⋊D5311yescharacteristic
245C22#41×D5312yes—
246C22#1313326212yes—
247C4—22no—
248C4—22no—
249C4—42no—
250C4—1068no—
251C4—2122no—
252C2—24no—
253C2—82no—
254C2—2124no—
255C2#1681113011yescharacteristic
256C2C2×#1313321311yescharacteristic
257C2#1681118211yescharacteristic
258C2#1681126214yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#121×C53C2
hypercenterC2×#121D53
hypocenterC53#2539
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#13133198C2semidirect
2. C2×#13133185C2semidirect
3. C22×#11719921C2semidirect
4. #16810558C2semidirect
5. #16810531C2semidirect
6. #16810523C2semidirect
7. C4×C22⋊Dic53C2semidirect
8. C2×#121×C53C2semidirect
14. C2×#11719919C22semidirect
16. C2×C22⋊Dic53C22semidirect
21. C2×C4×Dic53C22semidirect
22. #13133124C22semidirect
24. C22×D4×C53C22semidirect
27. C2×C4⋊C4×C53C22semidirect
28. C2×C22⋊C4×C53C22semidirect
42. C22×C4×C53D4semidirect
56. C22⋊Dic53D4semidirect
131. C2×#121D53semidirect
138. C53#2539semidirect
142. C22×D4C2×D53semidirect
149. C2×C4⋊C4C2×D53semidirect
156. C2×C22⋊C4C2×D53semidirect
179. C22×C4C22⋊D53semidirect