Finite Group Databasebeta

← #20283008

Subgroups — #20283008

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×#13133201C211yesmaximal, characteristic, complemented
3C2×#13133200C211yesmaximal, characteristic, complemented
4C2×#13133187C211yesmaximal, characteristic
5#16810501C214yesmaximal, complemented
6#16810497C214yesmaximal, complemented
7#16810484C211yesmaximal, characteristic, complemented
8#16810478C211yesmaximal, characteristic, complemented
9C53×#381C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C22×D212C2212yes—
12C22×C22⋊D53C2211yescharacteristic
13C23×Dic53C2211yescharacteristic
14C22×C4×D53—24no—
15C22×C4×D53—24no—
16C22×C4×D53—24no—
17C22×C4×D53C2214yes—
18#13133201—24no—
19#13133200—24no—
20#13133187—24no—
21C2×C4⋊Dic53C2212yescomplemented
22C2×#11719919C2212yescomplemented
23C2×C22⋊Dic53C2211yescharacteristic
24C2×#11719921C2212yes—
25C2×#11719921C2214yes—
26C2×#11719921C2212yescomplemented
27C2×#11719921C2212yescomplemented
28#13133124C2214yescomplemented
29C23×C4×C53C2211yescharacteristic
30C2×C22⋊C4×C53C2211yescharacteristic
31C2×C22⋊C4×C53C2211yescharacteristic
32#95×C53C2214yescomplemented
33C24×C53C2311yescharacteristic
34C23×D53—22no—
35C23×D53—22no—
36C23×D53—28no—
37C23×D53D414yes—
38C23×D53C2311yescharacteristic
39C23×D53C2312yes—
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53D414yescomplemented
43C22×C4×C53C2312yes—
44C22×C4×C53C2312yes—
45C22×C4×C53C2312yes—
46C2×C22⋊D53—22no—
47C2×C22⋊D53—22no—
48C2×C22⋊D53—22no—
49C22×Dic53—22no—
50C22×Dic53—22no—
51C22×Dic53D414yescomplemented
52C22×Dic53C2312yes—
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2312yes—
55C2×D212—24no—
56C2×D212—28no—
57C2×C4×D53—232no—
58C2×C4×D53—28no—
59C2×C4×D53—28no—
60C2×C4×D53—28no—
61C22⋊C4×C53—22no—
62C22⋊C4×C53—22no—
63C22⋊Dic53—22no—
64#11719921—24no—
65#11719921—28no—
66#11719921—24no—
67#11719921—24no—
68C4⋊Dic53—24no—
69#11719919—24no—
70C23×C53—24no—
71C23×C53C2×D412yes—
72C23×C53C2×D412yes—
73C23×C53C2×D412yes—
74C23×C53C2411yescharacteristic
75C22×D53—216no—
76C22×D53—216no—
77C22×D53—28no—
78C22×D53—22no—
79C22×D53—22no—
80C22×D53—24no—
81C22×D53—216no—
82C22×D53—21no—
83C22×D53—21no—
84C22×D53—21no—
85C22×D53—24no—
86C22×D53C2×D414yes—
87C22×D53#4118yes—
88C2×C4×C53—28no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—24no—
94C2×C4×C53—24no—
95C2×C4×C53C2×D414yes—
96C22⋊D53—44no—
97C2×Dic53—28no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—22no—
104C2×Dic53—24no—
105C2×Dic53C2×D414yes—
106D212—48no—
107C4×D53—232no—
108C22×C53—24no—
109C22×C53—24no—
110C22×C53—24no—
111C22×C53#12014yes—
112C22×C53C22×D411yescharacteristic
113C22×C53C22×D411yescharacteristic
114C22×C53C22×D411yescharacteristic
115C22×C53C2×#4114yes—
116C2×D53—232no—
117C2×D53—24no—
118C2×D53—24no—
119C2×D53—28no—
120C2×D53—28no—
121C2×D53—41no—
122C2×D53—42no—
123C2×D53—42no—
124C2×D53—42no—
125C2×D53#12318yescomplemented
126C4×C53—24no—
127C4×C53—42no—
128C4×C53—42no—
129Dic53—24no—
130Dic53—41no—
131Dic53—42no—
132#2849—531nomaximal, Sylow
133C2×C53—24no—
134C2×C53C2×#12011yescharacteristic
135C2×C53C2×#12312yes—
136C2×C53#52512yes—
137C2×C53#52512yes—
138D53—28no—
139D53—81no—
140C24×C4—531no—
141C2×#123—531no—
142C2×#123—531no—
143C2×#120—531no—
144#386—534no—
145#385—534no—
146#381—531no—
147#381D5311yescharacteristic, complemented
148#379—531no—
149C53#284911yescharacteristic, complemented, Sylow
150C25—531no—
151C22×D4—531no—
152C22×D4—532no—
153C23×C4—531no—
154C23×C4—534no—
155C23×C4—1064no—
156C23×C4—1064no—
157C23×C4—1064no—
158C23×C4C2×D5311yescharacteristic
159#123—1064no—
160#123—1064no—
161#120—1064no—
162C2×C4⋊C4—532no—
163C2×C4⋊C4—532no—
164C2×C22⋊C4—531no—
165C2×C22⋊C4—532no—
166C2×C22⋊C4—532no—
167C2×C22⋊C4—532no—
168C2×C22⋊C4—534no—
169C2×C22⋊C4C2×D5311yescharacteristic
170C2×C22⋊C4C2×D5311yescharacteristic
171#95—534no—
172#95C2×D5314yescomplemented
173C24—532no—
174C24—534no—
175C24—531no—
176C24—1068no—
177C24—1062no—
178C24—1062no—
179C24C22×D5311yescharacteristic
180C2×D4—1064no—
181C2×D4—1068no—
182C2×D4—1062no—
183C2×D4—1062no—
184C2×D4—1062no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—534no—
188C22×C4—532no—
189C22×C4—532no—
190C22×C4—531no—
191C22×C4—1068no—
192C22×C4—1068no—
193C22×C4—10632no—
194C22×C4—1062no—
195C22×C4—1062no—
196C22×C4—1068no—
197C22×C4D21214yescomplemented
198C22×C4C22×D5312yes—
199C22×C4C22×D5312yes—
200C22×C4C22×D5312yes—
201C4⋊C4—1064no—
202C4⋊C4—1064no—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C22⋊C4—1064no—
206C22⋊C4—1064no—
207C22⋊C4—1068no—
208C22⋊C4—1062no—
209C22⋊C4—1064no—
210C23—24no—
211C23—534no—
212C23—538no—
213C23—10616no—
214C23—10616no—
215C23—10616no—
216C23—1068no—
217C23—1062no—
218C23—1062no—
219C23—1064no—
220C23—1061no—
221C23—1061no—
222C23—1061no—
223C23—1064no—
224C23C2×D21212yes—
225C23D4×D5312yes—
226C23D4×D5312yes—
227C23C23×D5311yescharacteristic
228D4—2128no—
229D4—2124no—
230C2×C4—28no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—534no—
238C2×C4—10632no—
239C2×C4—1068no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—1061no—
243C2×C4—1061no—
244C2×C4—1062no—
245C2×C4—1061no—
246C2×C4—1064no—
247C2×C4C2×D21214yes—
248C22—24no—
249C22—24no—
250C22—24no—
251C22—538no—
252C22—10632no—
253C22—1064no—
254C22—1064no—
255C22—1068no—
256C22—1068no—
257C22—2121no—
258C22—2122no—
259C22—2122no—
260C22—2122no—
261C22#1313318714yes—
262C22C2×D4×D5311yescharacteristic
263C22C2×D4×D5311yescharacteristic
264C22C22×D21211yescharacteristic
265C22#41×D5314yes—
266C4—24no—
267C4—42no—
268C4—42no—
269C4—1064no—
270C4—2121no—
271C4—2122no—
272C2—24no—
273C2—1068no—
274C2—4241no—
275C2C2×#1313318711yescharacteristic
276C2#123×D5312yes—
277C2#1681113112yes—
278C2#1681113512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#381C2
hypercenter#381D53
hypocenterC53#2849
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C23×C4×D53C2semidirect
2. C2×#13133201C2semidirect
3. C2×#13133200C2semidirect
5. #16810501C2semidirect
6. #16810497C2semidirect
7. #16810484C2semidirect
8. #16810478C2semidirect
9. C53×#381C2semidirect
21. C2×C4⋊Dic53C22semidirect
22. C2×#11719919C22semidirect
26. C2×#11719921C22semidirect
27. C2×#11719921C22semidirect
28. #13133124C22semidirect
32. #95×C53C22semidirect
42. C22×C4×C53D4semidirect
51. C22×Dic53D4semidirect
125. C2×D53#123semidirect
147. #381D53semidirect
149. C53#2849semidirect
172. #95C2×D53semidirect
197. C22×C4D212semidirect