Finite Group Databasebeta

← #20278043

Subgroups — #20278043

Order 6784.

1116 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×C22⋊D53C211yesmaximal, characteristic
2C2×#13133201C212yesmaximal, complemented
3C22×C22⋊Dic53C211yesmaximal, characteristic, complemented
4#16810539C214yesmaximal, complemented
5#16810500C212yesmaximal, complemented
6#16810498C212yesmaximal, complemented
7#16810478C212yesmaximal, complemented
8C53×#379C211yesmaximal, characteristic, complemented
9C24×D53C2211yescharacteristic
10C22×C22⋊D53—24no—
11C22×C22⋊D53—28no—
12C22×C22⋊D53C2212yes—
13C22×C22⋊D53C2212yes—
14C23×Dic53C2211yescharacteristic
15C23×Dic53C2211yescharacteristic, complemented
16C23×Dic53C2211yescharacteristic, complemented
17#13133201—28no—
18C2×C4⋊Dic53C2214yescomplemented
19C2×C22⋊Dic53—24no—
20C2×C22⋊Dic53C2212yes—
21C2×C22⋊Dic53C2212yescomplemented
22C2×#11719921C2214yescomplemented
23C2×#11719921C2214yescomplemented
24#13133124C2214yescomplemented
25C25×C53C2211yescharacteristic
26C2×C22⋊C4×C53C2212yes—
27C2×C22⋊C4×C53C2212yescomplemented
28C2×C22⋊C4×C53C2212yescomplemented
29C24×C53—24no—
30C24×C53—24no—
31C24×C53—42no—
32C24×C53D414yes—
33C24×C53C2311yescharacteristic
34C24×C53C2311yescharacteristic
35C24×C53C2311yescharacteristic
36C23×D53—22no—
37C23×D53—24no—
38C23×D53—24no—
39C23×D53—42no—
40C23×D53C2312yes—
41C22×C4×C53C2314yes—
42C2×C22⋊D53—232no—
43C2×C22⋊D53—24no—
44C2×C22⋊D53—24no—
45C2×C22⋊D53—24no—
46C2×C22⋊D53—24no—
47C2×C22⋊D53—24no—
48C2×C22⋊D53D418yes—
49C22×Dic53—22no—
50C22×Dic53—24no—
51C22×Dic53—22no—
52C22×Dic53—22no—
53C22×Dic53—24no—
54C22×Dic53—42no—
55C22×Dic53C2312yes—
56C22×Dic53C2312yes—
57C22×Dic53C2312yes—
58C22⋊C4×C53—24no—
59C22⋊C4×C53—24no—
60C22⋊C4×C53—42no—
61C22⋊Dic53—24no—
62C22⋊Dic53—24no—
63#11719921—28no—
64#11719921—44no—
65C4⋊Dic53—28no—
66C23×C53—22no—
67C23×C53—22no—
68C23×C53—22no—
69C23×C53—22no—
70C23×C53—28no—
71C23×C53—28no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—44no—
76C23×C53—416no—
77C23×C53C2×D412yes—
78C23×C53C2×D412yes—
79C23×C53C2×D412yes—
80C23×C53C2×D414yes—
81C23×C53#4114yes—
82C23×C53C2411yescharacteristic
83C22×D53—28no—
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—24no—
87C22×D53—24no—
88C22×D53—416no—
89C22×D53—44no—
90C22×D53—44no—
91C22×D53C2×D414yes—
92C2×C4×C53—24no—
93C2×C4×C53—24no—
94C2×C4×C53—24no—
95C22⋊D53—216no—
96C22⋊D53—216no—
97C2×Dic53—28no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—24no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—24no—
104C2×Dic53—44no—
105C2×Dic53—44no—
106C2×Dic53C2×D414yes—
107C22×C53—22no—
108C22×C53—22no—
109C22×C53—22no—
110C22×C53—22no—
111C22×C53—28no—
112C22×C53—28no—
113C22×C53—22no—
114C22×C53—22no—
115C22×C53—22no—
116C22×C53—416no—
117C22×C53—44no—
118C22×C53#12014yes—
119C22×C53#12314yescomplemented
120C22×C53C22×D412yes—
121C22×C53C22×D411yescharacteristic
122C22×C53C2×#4112yes—
123C22×C53H(5,𝔽2)12yes—
124C2×D53—24no—
125C2×D53—28no—
126C2×D53—416no—
127C2×D53—42no—
128C2×D53—42no—
129C2×D53—44no—
130C4×C53—44no—
131Dic53—24no—
132Dic53—42no—
133Dic53—42no—
134#2405—531nomaximal, Sylow
135C2×C53—24no—
136C2×C53—24no—
137C2×C53—42no—
138C2×C53C2×#12011yescharacteristic
139C2×C53C2×#12311yescharacteristic
140C2×C53#53411yescharacteristic
141C2×C53#54614yes—
142D53—44no—
143C23×D4—531no—
144C2×#123—532no—
145C22×C22⋊C4—531no—
146#394—534no—
147#386—532no—
148#385—532no—
149#379—532no—
150#379D5311yescharacteristic, complemented
151C53#240511yescharacteristic, complemented, Sylow
152C25—531no—
153C25C2×D5311yescharacteristic
154C22×D4—532no—
155C22×D4—532no—
156C22×D4—1068no—
157C22×D4—1064no—
158C23×C4—531no—
159C23×C4—531no—
160C23×C4—531no—
161#123—1068no—
162C2×C4⋊C4—534no—
163C2×C22⋊C4—532no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4—534no—
166C2×C22⋊C4—534no—
167C2×C22⋊C4—1064no—
168C2×C22⋊C4C2×D5312yes—
169C2×C22⋊C4C2×D5312yescomplemented
170C2×C22⋊C4C2×D5312yescomplemented
171#95—534no—
172C24—24no—
173C24—24no—
174C24—42no—
175C24—532no—
176C24—1064no—
177C24—1062no—
178C24—1064no—
179C24—2122no—
180C24D21214yes—
181C24C22×D5311yescharacteristic
182C24C22×D5311yescharacteristic
183C24C22×D5311yescharacteristic
184C2×D4—538no—
185C2×D4—1064no—
186C2×D4—1064no—
187C2×D4—10632no—
188C2×D4—1064no—
189C2×D4—1064no—
190C2×D4—1064no—
191C22×C4—532no—
192C22×C4—532no—
193C22×C4—532no—
194C22×C4—1064no—
195C22×C4—1062no—
196C22×C4—1062no—
197C22×C4—1062no—
198C22×C4—1064no—
199C22×C4—2122no—
200C22×C4C22×D5314yes—
201C4⋊C4—1068no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—42no—
205C22⋊C4—1068no—
206C22⋊C4—1064no—
207C22⋊C4—1064no—
208C22⋊C4—2124no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—44no—
219C23—416no—
220C23—534no—
221C23—1068no—
222C23—1062no—
223C23—1062no—
224C23—1064no—
225C23—1064no—
226C23—21216no—
227C23—2124no—
228C23—2124no—
229C23C2×D21212yes—
230C23C2×D21212yes—
231C23C2×D21212yes—
232C23D4×D5314yes—
233C23#1171994714yes—
234C23C23×D5311yescharacteristic
235D4—10616no—
236D4—10616no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—534no—
241C2×C4—1068no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1064no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—1064no—
248C2×C4—2124no—
249C2×C4—2124no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—28no—
255C22—28no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—416no—
260C22—44no—
261C22—1064no—
262C22—1068no—
263C22—21216no—
264C22—2122no—
265C22—2122no—
266C22—2124no—
267C22#1313318714yes—
268C22#1313320114yescomplemented
269C22C2×D4×D5312yes—
270C22C22×D21211yescharacteristic
271C22C2×#1171994712yes—
272C22#1313326212yes—
273C4—44no—
274C4—1064no—
275C4—2122no—
276C4—2122no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—2124no—
281C2C2×#1313318711yescharacteristic
282C2C2×#1313320111yescharacteristic
283C2#1681118511yescharacteristic
284C2#1681126014yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#379C2
hypercenter#379D53
hypocenterC53#2405
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#13133201C2semidirect
3. C22×C22⋊Dic53C2semidirect
4. #16810539C2semidirect
5. #16810500C2semidirect
6. #16810498C2semidirect
7. #16810478C2semidirect
8. C53×#379C2semidirect
15. C23×Dic53C22semidirect
16. C23×Dic53C22semidirect
18. C2×C4⋊Dic53C22semidirect
21. C2×C22⋊Dic53C22semidirect
22. C2×#11719921C22semidirect
23. C2×#11719921C22semidirect
24. #13133124C22semidirect
27. C2×C22⋊C4×C53C22semidirect
28. C2×C22⋊C4×C53C22semidirect
119. C22×C53#123semidirect
150. #379D53semidirect
151. C53#2405semidirect
169. C2×C22⋊C4C2×D53semidirect
170. C2×C22⋊C4C2×D53semidirect
268. C22#13133201semidirect