Finite Group Databasebeta

← #20282999

Subgroups — #20282999

Order 6784.

1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 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
2C2×#13133193C212yesmaximal, complemented
3#16810495C212yesmaximal, complemented
4#16810494C214yesmaximal, complemented
5#16810483C211yesmaximal, characteristic, complemented
6#16810480C212yesmaximal, complemented
7#16810478C212yesmaximal
8C2×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
9C24×D53C2211yescharacteristic
10C23×Dic53C2211yescharacteristic
11C22×C4×D53—24no—
12C22×C4×D53—28no—
13C22×C4×D53C2212yes—
14C22×C4×D53C2212yes—
15#13133193—28no—
16C2×C4⋊Dic53C2212yescomplemented
17C2×#11719919C2214yescomplemented
18C2×C22⋊Dic53C2212yes—
19C2×#11719921C2214yes—
20C2×#11719921C2214yes—
21#13133124C2214yescomplemented
22#13133124C2212yescomplemented
23C22×Q8×C53C2211yescharacteristic, complemented
24C23×C4×C53C2211yescharacteristic
25C22⋊Q8×C53—24no—
26C2×C4⋊C4×C53C2211yescharacteristic, complemented
27C2×C4⋊C4×C53C2212yescomplemented
28C2×C22⋊C4×C53C2212yes—
29C24×C53C2311yescharacteristic
30C23×D53—22no—
31C23×D53—22no—
32C23×D53—24no—
33C23×D53—24no—
34C23×D53Q814yescomplemented
35C23×D53C2312yes—
36C2×Q8×C53—22no—
37C2×Q8×C53—24no—
38C22×C4×C53—24no—
39C22×C4×C53D414yes—
40C22×C4×C53C2312yes—
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×Dic53—22no—
45C22×Dic53—24no—
46C22×Dic53C2314yes—
47C22×Dic53C2312yes—
48C2×C4×D53—232no—
49C2×C4×D53—24no—
50C2×C4×D53—24no—
51C2×C4×D53—24no—
52C2×C4×D53—24no—
53C2×C4×D53—24no—
54C2×C4×D53D418yes—
55C4⋊C4×C53—24no—
56C4⋊C4×C53—22no—
57C22⋊C4×C53—24no—
58C22⋊Dic53—24no—
59#11719921—28no—
60#11719921—28no—
61C4⋊Dic53—24no—
62#11719919—28no—
63C23×C53—22no—
64C23×C53—22no—
65C23×C53C2×D412yes—
66C23×C53C2×Q812yes—
67C23×C53#4112yes—
68C23×C53C2411yescharacteristic
69C22×D53—28no—
70C22×D53—216no—
71C22×D53—28no—
72C22×D53—28no—
73C22×D53—22no—
74C22×D53—22no—
75C22×D53—22no—
76C22×D53—22no—
77C22×D53—216no—
78C22×D53C2×D414yes—
79C22×D53C2×Q814yes—
80C22×D53#4114yes—
81Q8×C53—44no—
82C2×C4×C53—22no—
83C2×C4×C53—28no—
84C2×C4×C53—24no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—24no—
88C2×C4×C53C2×D414yes—
89C2×C4×C53C2×D412yes—
90C2×C4×C53C2×D412yes—
91C2×Dic53—28no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—24no—
95C2×Dic53—24no—
96C2×Dic53—24no—
97C2×Dic53C2×D414yes—
98C4×D53—216no—
99C4×D53—216no—
100C22×C53—22no—
101C22×C53—24no—
102C22×C53—22no—
103C22×C53—24no—
104C22×C53C22⋊Q814yescomplemented
105C22×C53C22×D412yes—
106C22×C53C22×D411yescharacteristic
107C22×C53C22×Q811yescharacteristic
108C22×C53C2×#4112yes—
109C22×C53C2×#4111yescharacteristic
110C2×D53—232no—
111C2×D53—24no—
112C2×D53—24no—
113C2×D53—24no—
114C2×D53—24no—
115C2×D53—24no—
116C2×D53—24no—
117C2×D53C22⋊Q818yescomplemented
118C4×C53—22no—
119C4×C53—42no—
120C4×C53—42no—
121C4×C53#12014yes—
122Dic53—24no—
123Dic53—44no—
124#2848—531nomaximal, Sylow
125C2×C53—24no—
126C2×C53C2×#12011yescharacteristic
127C2×C53C2×C22⋊Q812yes—
128C2×C53C2×C22⋊Q811yescharacteristic
129C2×C53#52511yescharacteristic
130C2×C53#52512yes—
131D53—28no—
132C24×C4—531no—
133C2×C22⋊Q8—532no—
134C2×C22⋊Q8D5311yescharacteristic, complemented
135#385—534no—
136#385—532no—
137#380—531no—
138#380—532no—
139#379—532no—
140C53#284811yescharacteristic, complemented, Sylow
141C25—531no—
142C22×Q8C2×D5311yescharacteristic, complemented
143C23×C4—532no—
144C23×C4—531no—
145C23×C4—532no—
146C23×C4—1068no—
147C23×C4—1064no—
148C23×C4C2×D5311yescharacteristic
149C22⋊Q8—24no—
150C22⋊Q8—1068no—
151C2×C4⋊C4—534no—
152C2×C4⋊C4—532no—
153C2×C4⋊C4C2×D5311yescharacteristic, complemented
154C2×C4⋊C4C2×D5312yescomplemented
155C2×C22⋊C4—532no—
156C2×C22⋊C4—534no—
157C2×C22⋊C4—534no—
158C2×C22⋊C4C2×D5312yes—
159#95—534no—
160#95—532no—
161C24—532no—
162C24—534no—
163C24—1064no—
164C24—1064no—
165C24—1062no—
166C24—1062no—
167C24C22×D5311yescharacteristic
168C2×Q8—24no—
169C2×Q8—22no—
170C22×C4—24no—
171C22×C4—532no—
172C22×C4—538no—
173C22×C4—534no—
174C22×C4—1064no—
175C22×C4—1064no—
176C22×C4—10632no—
177C22×C4—1064no—
178C22×C4—1062no—
179C22×C4—1064no—
180C22×C4—1064no—
181C22×C4—1064no—
182C22×C4C22⋊D5314yes—
183C22×C4C22×D5312yes—
184C22×C4C22×D5312yes—
185C22×C4C22×D5311yescharacteristic
186C22×C4C22×D5311yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—1068no—
190C4⋊C4—1064no—
191C22⋊C4—24no—
192C22⋊C4—1068no—
193C22⋊C4—1068no—
194C22⋊C4—1064no—
195C23—22no—
196C23—22no—
197C23—534no—
198C23—534no—
199C23—534no—
200C23—10616no—
201C23—1068no—
202C23—1068no—
203C23—10616no—
204C23—1068no—
205C23—1062no—
206C23—1062no—
207C23—1062no—
208C23—1062no—
209C23Q8×D5312yes—
210C23#1171994912yes—
211C23C2×C22⋊D5312yes—
212C23C23×D5311yescharacteristic
213Q8—44no—
214C2×C4—22no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—534no—
221C2×C4—10616no—
222C2×C4—1068no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—10616no—
226C2×C4—1064no—
227C2×C4—1064no—
228C2×C4—1064no—
229C2×C4D4×D5314yes—
230C2×C4C2×C22⋊D5312yes—
231C2×C4C2×C22⋊D5312yes—
232C22—22no—
233C22—24no—
234C22—22no—
235C22—24no—
236C22—538no—
237C22—10632no—
238C22—1064no—
239C22—1064no—
240C22—1064no—
241C22—1064no—
242C22—1064no—
243C22—1064no—
244C22#1313319314yescomplemented
245C22C2×D4×D5312yes—
246C22C22×C22⋊D5311yescharacteristic
247C22C2×Q8×D5311yescharacteristic
248C22#41×D5312yes—
249C22C2×#1171994911yescharacteristic
250C4—22no—
251C4—42no—
252C4—42no—
253C4—1064no—
254C4—2124no—
255C4#1313318514yes—
256C2—24no—
257C2—1068no—
258C2C2×#1313318511yescharacteristic
259C2C22⋊Q8×D5312yes—
260C2C2×#1313319311yescharacteristic
261C2#1681113011yescharacteristic
262C2#1681113612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C22⋊Q8×C53C2
hypercenterC2×C22⋊Q8D53
hypocenterC53#2848
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#13133193C2semidirect
3. #16810495C2semidirect
4. #16810494C2semidirect
5. #16810483C2semidirect
6. #16810480C2semidirect
8. C2×C22⋊Q8×C53C2semidirect
16. C2×C4⋊Dic53C22semidirect
17. C2×#11719919C22semidirect
21. #13133124C22semidirect
22. #13133124C22semidirect
23. C22×Q8×C53C22semidirect
26. C2×C4⋊C4×C53C22semidirect
27. C2×C4⋊C4×C53C22semidirect
34. C23×D53Q8semidirect
104. C22×C53C22⋊Q8semidirect
117. C2×D53C22⋊Q8semidirect
134. C2×C22⋊Q8D53semidirect
140. C53#2848semidirect
142. C22×Q8C2×D53semidirect
153. C2×C4⋊C4C2×D53semidirect
154. C2×C4⋊C4C2×D53semidirect
244. C22#13133193semidirect