Finite Group Databasebeta

← #20283004

Subgroups — #20283004

Order 6784.

1102 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1102 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×C4×D53C211yesmaximal, characteristic, complemented
2C2×#13133198C211yesmaximal, characteristic, complemented
3C2×#13133185C211yesmaximal, characteristic
4#16810499C214yesmaximal, complemented
5#16810494C214yesmaximal, complemented
6#16810486C211yesmaximal, characteristic, complemented
7#16810484C211yesmaximal, characteristic, complemented
8#16810478C211yesmaximal, characteristic, complemented
9C2×#123×C53C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C23×Dic53C2211yescharacteristic
13C22×C4×D53—24no—
14C22×C4×D53—24no—
15C22×C4×D53—24no—
16C22×C4×D53C2214yes—
17#13133198—24no—
18#13133185—24no—
19C2×#11719919C2212yescomplemented
20C2×C22⋊Dic53C2211yescharacteristic
21C2×C22⋊Dic53C2211yescharacteristic
22C2×C22⋊Dic53C2212yescomplemented
23C2×#11719921C2214yes—
24C2×#11719921C2212yes—
25#13133124C2214yescomplemented
26#13133124C2214yescomplemented
27C22×D4×C53C2211yescharacteristic
28C23×C4×C53C2211yescharacteristic
29#123×C53—24no—
30C2×C4⋊C4×C53C2212yescomplemented
31C2×C22⋊C4×C53C2211yescharacteristic
32C2×C22⋊C4×C53C2212yescomplemented
33C24×C53C2311yescharacteristic
34C24×C53C2311yescharacteristic
35C23×D53—22no—
36C23×D53—22no—
37C23×D53—28no—
38C23×D53D414yes—
39C23×D53C2312yes—
40C2×D4×C53—22no—
41C2×D4×C53—22no—
42C2×D4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53D414yescomplemented
46C22×C4×C53C2312yes—
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2312yes—
49C2×C22⋊D53—24no—
50C2×C22⋊D53—28no—
51C22×Dic53—22no—
52C22×Dic53—22no—
53C22×Dic53D414yescomplemented
54C22×Dic53C2312yes—
55C22×Dic53C2312yes—
56C22×Dic53C2312yes—
57C2×C4×D53—232no—
58C2×C4×D53—28no—
59C2×C4×D53—28no—
60C2×C4×D53—28no—
61C4⋊C4×C53—24no—
62C22⋊C4×C53—22no—
63C22⋊C4×C53—24no—
64C22⋊Dic53—22no—
65C22⋊Dic53—22no—
66C22⋊Dic53—24no—
67#11719921—24no—
68#11719921—28no—
69#11719919—24no—
70C23×C53—24no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53—21no—
74C23×C53—24no—
75C23×C53C2×D412yes—
76C23×C53C2×D412yes—
77C23×C53C2×D412yes—
78C23×C53C2411yescharacteristic
79C22×D53—216no—
80C22×D53—216no—
81C22×D53—28no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—24no—
85C22×D53—216no—
86C22×D53C2×D414yes—
87C22×D53#4118yes—
88D4×C53—44no—
89C2×C4×C53—28no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—24no—
97C2×C4×C53C2×D414yes—
98C22⋊D53—48no—
99C2×Dic53—28no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—24no—
105C2×Dic53—24no—
106C2×Dic53C2×D414yes—
107C4×D53—232no—
108C22×C53—24no—
109C22×C53—24no—
110C22×C53—24no—
111C22×C53—41no—
112C22×C53—42no—
113C22×C53—42no—
114C22×C53—42no—
115C22×C53#12014yes—
116C22×C53C22×D411yescharacteristic
117C22×C53C22×D411yescharacteristic
118C22×C53C22×D411yescharacteristic
119C22×C53C2×#4114yes—
120C2×D53—232no—
121C2×D53—24no—
122C2×D53—24no—
123C2×D53—28no—
124C2×D53—28no—
125C2×D53#12318yescomplemented
126C4×C53—24no—
127C4×C53—41no—
128C4×C53—42no—
129Dic53—24no—
130Dic53—42no—
131Dic53—42no—
132#2849—531nomaximal, Sylow
133C2×C53—24no—
134C2×C53—81no—
135C2×C53C2×#12011yescharacteristic
136C2×C53C2×#12312yes—
137C2×C53#52512yes—
138C2×C53#52512yes—
139D53—28no—
140C24×C4—531no—
141C2×#123—531no—
142C2×#123D5311yescharacteristic, complemented
143C2×#120—531no—
144#386—534no—
145#385—534no—
146#381—531no—
147#381—531no—
148#379—531no—
149C53#284911yescharacteristic, complemented, Sylow
150C25—531no—
151C22×D4—532no—
152C22×D4C2×D5311yescharacteristic
153C23×C4—531no—
154C23×C4—534no—
155C23×C4—1064no—
156C23×C4—1064no—
157C23×C4—1064no—
158C23×C4C2×D5311yescharacteristic
159#123—24no—
160#123—1064no—
161#120—1064no—
162C2×C4⋊C4—532no—
163C2×C4⋊C4C2×D5312yescomplemented
164C2×C22⋊C4—531no—
165C2×C22⋊C4—531no—
166C2×C22⋊C4—532no—
167C2×C22⋊C4—532no—
168C2×C22⋊C4—534no—
169C2×C22⋊C4C2×D5311yescharacteristic
170C2×C22⋊C4C2×D5312yescomplemented
171#95—534no—
172#95—534no—
173C24—532no—
174C24—534no—
175C24—1068no—
176C24—1062no—
177C24—1062no—
178C24C22×D5311yescharacteristic
179C24C22×D5311yescharacteristic
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—1064no—
184C2×D4—1068no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—534no—
188C22×C4—532no—
189C22×C4—532no—
190C22×C4—532no—
191C22×C4—1068no—
192C22×C4—1068no—
193C22×C4—10632no—
194C22×C4—1062no—
195C22×C4—1062no—
196C22×C4—1068no—
197C22×C4C22⋊D5314yescomplemented
198C22×C4C22×D5312yes—
199C22×C4C22×D5311yescharacteristic
200C22×C4C22×D5312yes—
201C4⋊C4—24no—
202C4⋊C4—1064no—
203C22⋊C4—22no—
204C22⋊C4—24no—
205C22⋊C4—1064no—
206C22⋊C4—1068no—
207C22⋊C4—1064no—
208C22⋊C4—1062no—
209C22⋊C4—1062no—
210C23—24no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—24no—
215C23—534no—
216C23—538no—
217C23—10616no—
218C23—10616no—
219C23—10616no—
220C23—1068no—
221C23—1062no—
222C23—1062no—
223C23—1064no—
224C23D4×D5312yes—
225C23D4×D5312yes—
226C23C2×C22⋊D5312yes—
227C23C23×D5311yescharacteristic
228D4—44no—
229D4—2128no—
230C2×C4—28no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—24no—
238C2×C4—534no—
239C2×C4—10632no—
240C2×C4—1068no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—1064no—
246C2×C4—1064no—
247C2×C4C2×C22⋊D5314yes—
248C22—24no—
249C22—24no—
250C22—24no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—538no—
256C22—10632no—
257C22—1064no—
258C22—1064no—
259C22—1068no—
260C22—1068no—
261C22#1313318514yes—
262C22C2×D4×D5311yescharacteristic
263C22C2×D4×D5311yescharacteristic
264C22C22×C22⋊D5311yescharacteristic
265C22#41×D5314yes—
266C4—24no—
267C4—41no—
268C4—42no—
269C4—1064no—
270C4—2122no—
271C4—2122no—
272C2—24no—
273C2—81no—
274C2—1068no—
275C2C2×#1313318511yescharacteristic
276C2#123×D5312yes—
277C2#1681113012yes—
278C2#1681113112yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#123×C53C2
hypercenterC2×#123D53
hypocenterC53#2849
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C23×C4×D53C2semidirect
2. C2×#13133198C2semidirect
4. #16810499C2semidirect
5. #16810494C2semidirect
6. #16810486C2semidirect
7. #16810484C2semidirect
8. #16810478C2semidirect
9. C2×#123×C53C2semidirect
19. C2×#11719919C22semidirect
22. C2×C22⋊Dic53C22semidirect
25. #13133124C22semidirect
26. #13133124C22semidirect
30. C2×C4⋊C4×C53C22semidirect
32. C2×C22⋊C4×C53C22semidirect
45. C22×C4×C53D4semidirect
53. C22×Dic53D4semidirect
125. C2×D53#123semidirect
142. C2×#123D53semidirect
149. C53#2849semidirect
163. C2×C4⋊C4C2×D53semidirect
170. C2×C22⋊C4C2×D53semidirect
197. C22×C4C22⋊D53semidirect