Finite Group Databasebeta

← #20278038

Subgroups — #20278038

Order 6784.

1116 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 197 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
2C22×C22⋊Dic53C211yesmaximal, characteristic, complemented
3#16810537C214yesmaximal, complemented
4#16810503C212yesmaximal, complemented
5#16810496C212yesmaximal, complemented
6#16810479C212yesmaximal, complemented
7#16810479C211yesmaximal, characteristic, complemented
8C23×D4×C53C211yesmaximal, characteristic
9C23×Dic53C2211yescharacteristic, complemented
10C23×Dic53C2211yescharacteristic, complemented
11#13133198—28no—
12C2×#11719919C2214yescomplemented
13C2×C22⋊Dic53—24no—
14C2×C22⋊Dic53C2212yes—
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2212yescomplemented
17C2×C22⋊Dic53C2212yescomplemented
18C2×C22⋊Dic53C2214yescomplemented
19C2×C22⋊Dic53C2214yescomplemented
20C2×C22⋊Dic53C2212yescomplemented
21#13133124C2214yescomplemented
22C25×C53C2211yescharacteristic
23C25×C53C2211yescharacteristic
24C22×D4×C53—24no—
25C22×D4×C53—28no—
26C22×D4×C53C2212yes—
27C22×D4×C53C2212yes—
28C23×C4×C53C2211yescharacteristic
29C24×C53—24no—
30C24×C53—24no—
31C24×C53—22no—
32C24×C53—24no—
33C24×C53—24no—
34C24×C53—42no—
35C24×C53—42no—
36C24×C53D414yes—
37C24×C53C2311yescharacteristic
38C24×C53C2312yes—
39C24×C53C2311yescharacteristic
40C24×C53C2311yescharacteristic
41C2×D4×C53—232no—
42C2×D4×C53—24no—
43C2×D4×C53—24no—
44C2×D4×C53—24no—
45C2×D4×C53—24no—
46C2×D4×C53—24no—
47C2×D4×C53D418yes—
48C22×C4×C53—22no—
49C22×C4×C53—24no—
50C22×C4×C53C2312yes—
51C22×Dic53—22no—
52C22×Dic53—22no—
53C22×Dic53—24no—
54C22×Dic53—42no—
55C22×Dic53C2312yes—
56C22×Dic53C2314yes—
57C22×Dic53C2312yes—
58C22⋊Dic53—24no—
59C22⋊Dic53—24no—
60C22⋊Dic53—28no—
61C22⋊Dic53—24no—
62C22⋊Dic53—24no—
63C22⋊Dic53—44no—
64C22⋊Dic53—42no—
65#11719919—28no—
66C23×C53—22no—
67C23×C53—22no—
68C23×C53—22no—
69C23×C53—22no—
70C23×C53—28no—
71C23×C53—28no—
72C23×C53—28no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—24no—
78C23×C53—22no—
79C23×C53—24no—
80C23×C53—44no—
81C23×C53—416no—
82C23×C53—416no—
83C23×C53—44no—
84C23×C53—44no—
85C23×C53C2×D412yes—
86C23×C53C2×D414yes—
87C23×C53C2×D412yes—
88C23×C53C2×D414yes—
89C23×C53C2×D412yes—
90C23×C53#4114yes—
91C23×C53C2411yescharacteristic
92D4×C53—216no—
93D4×C53—216no—
94C2×C4×C53—28no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53C2×D414yes—
98C2×Dic53—24no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—24no—
102C2×Dic53—24no—
103C2×Dic53—24no—
104C2×Dic53—24no—
105C2×Dic53—44no—
106C2×Dic53—44no—
107C22×C53—22no—
108C22×C53—22no—
109C22×C53—22no—
110C22×C53—22no—
111C22×C53—28no—
112C22×C53—28no—
113C22×C53—22no—
114C22×C53—24no—
115C22×C53—22no—
116C22×C53—22no—
117C22×C53—28no—
118C22×C53—416no—
119C22×C53—416no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—44no—
123C22×C53—44no—
124C22×C53#12014yes—
125C22×C53#12314yescomplemented
126C22×C53C22×D411yescharacteristic
127C22×C53C22×D412yes—
128C22×C53C2×#4112yes—
129C22×C53H(5,𝔽2)12yes—
130C4×C53—24no—
131Dic53—42no—
132Dic53—42no—
133Dic53—44no—
134#2405—531nomaximal, Sylow
135C2×C53—24no—
136C2×C53—24no—
137C2×C53—42no—
138C2×C53—44no—
139C2×C53C2×#12011yescharacteristic
140C2×C53C2×#12311yescharacteristic
141C2×C53#53411yescharacteristic
142C2×C53#54614yes—
143C23×D4D5311yescharacteristic
144C2×#123—532no—
145C22×C22⋊C4—531no—
146#394—534no—
147#386—532no—
148#385—532no—
149#379—532no—
150#379—531no—
151C53#240511yescharacteristic, complemented, Sylow
152C25C2×D5311yescharacteristic
153C25C2×D5311yescharacteristic
154C22×D4—24no—
155C22×D4—28no—
156C22×D4C2×D5312yes—
157C22×D4C2×D5312yes—
158C23×C4—531no—
159C23×C4—531no—
160C23×C4C2×D5311yescharacteristic
161#123—1068no—
162C2×C4⋊C4—534no—
163C2×C22⋊C4—532no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4—532no—
166C2×C22⋊C4—534no—
167C2×C22⋊C4—532no—
168C2×C22⋊C4—534no—
169C2×C22⋊C4—532no—
170C2×C22⋊C4—1064no—
171#95—534no—
172C24—24no—
173C24—24no—
174C24—24no—
175C24—22no—
176C24—24no—
177C24—42no—
178C24—42no—
179C24C22⋊D5314yes—
180C24C22×D5311yescharacteristic
181C24C22×D5312yes—
182C24C22×D5311yescharacteristic
183C24C22×D5311yescharacteristic
184C2×D4—232no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4C22⋊D5318yes—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—532no—
194C22×C4—534no—
195C22×C4—532no—
196C22×C4—1062no—
197C22×C4—1062no—
198C22×C4—1064no—
199C22×C4—2122no—
200C22×C4C22×D5312yes—
201C4⋊C4—1068no—
202C22⋊C4—1064no—
203C22⋊C4—1064no—
204C22⋊C4—1064no—
205C22⋊C4—1064no—
206C22⋊C4—1068no—
207C22⋊C4—2122no—
208C22⋊C4—2124no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—28no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—24no—
220C23—22no—
221C23—24no—
222C23—22no—
223C23—44no—
224C23—416no—
225C23—416no—
226C23—44no—
227C23—44no—
228C23#1171994714yes—
229C23C2×C22⋊D5312yes—
230C23C2×C22⋊D5314yes—
231C23C2×C22⋊D5314yes—
232C23C2×C22⋊D5312yes—
233C23C2×C22⋊D5312yes—
234C23C23×D5311yescharacteristic
235D4—216no—
236D4—216no—
237C2×C4—28no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—1064no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1064no—
244C2×C4—1064no—
245C2×C4—1064no—
246C2×C4—1064no—
247C2×C4—2124no—
248C2×C4—2124no—
249C2×C4C2×C22⋊D5314yes—
250C22—28no—
251C22—28no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—28no—
257C22—22no—
258C22—24no—
259C22—22no—
260C22—22no—
261C22—416no—
262C22—416no—
263C22—44no—
264C22—44no—
265C22—42no—
266C22—42no—
267C22#1313318614yes—
268C22#1313319814yescomplemented
269C22C22×C22⋊D5312yes—
270C22C22×C22⋊D5311yescharacteristic
271C22C2×#1171994712yes—
272C22#1313326212yes—
273C4—24no—
274C4—2124no—
275C4—2122no—
276C4—2122no—
277C2—24no—
278C2—24no—
279C2—44no—
280C2—42no—
281C2C2×#1313318611yescharacteristic
282C2C2×#1313319811yescharacteristic
283C2#1681118211yescharacteristic
284C2#1681126714yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC23×D4×C53C2
hypercenterC23×D4D53
hypocenterC53#2405
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#13133198C2semidirect
2. C22×C22⋊Dic53C2semidirect
3. #16810537C2semidirect
4. #16810503C2semidirect
5. #16810496C2semidirect
6. #16810479C2semidirect
7. #16810479C2semidirect
9. C23×Dic53C22semidirect
10. C23×Dic53C22semidirect
12. C2×#11719919C22semidirect
16. C2×C22⋊Dic53C22semidirect
17. C2×C22⋊Dic53C22semidirect
18. C2×C22⋊Dic53C22semidirect
19. C2×C22⋊Dic53C22semidirect
20. C2×C22⋊Dic53C22semidirect
21. #13133124C22semidirect
125. C22×C53#123semidirect
151. C53#2405semidirect
268. C22#13133198semidirect