Finite Group Databasebeta

← #20280520

Subgroups — #20280520

Order 6784.

668 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13133196C211yesmaximal, characteristic, complemented
2#16810559C212yesmaximal, complemented
3#16810546C214yesmaximal, complemented
4#16810535C212yesmaximal, complemented
5#16810496C212yesmaximal, complemented
6#16810483C211yesmaximal, characteristic, complemented
7#16810479C212yesmaximal
8C22×C22⋊C4×C53C211yesmaximal, characteristic
9C22×Dic106C2211yescharacteristic, complemented
10#13133196—24no—
11C2×C4⋊Dic53C2211yescharacteristic, complemented
12C2×C4⋊Dic53C2212yescomplemented
13C2×#11719919C2212yescomplemented
14C2×#11719919C2214yescomplemented
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2214yes—
18C2×C22⋊Dic53C2214yes—
19#13133124C2214yescomplemented
20#13133124C2212yescomplemented
21C25×C53C2211yescharacteristic
22C23×C4×C53C2211yescharacteristic
23C23×C4×C53C2211yescharacteristic
24C2×C22⋊C4×C53—24no—
25C2×C22⋊C4×C53C2212yes—
26C2×C22⋊C4×C53C2212yes—
27C24×C53—22no—
28C24×C53—22no—
29C24×C53—42no—
30C24×C53—42no—
31C24×C53Q814yescomplemented
32C24×C53C2312yes—
33C24×C53C2311yescharacteristic
34C22×C4×C53—22no—
35C22×C4×C53—42no—
36C22×C4×C53—42no—
37C22×C4×C53D414yes—
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×Dic53C2314yes—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C2×Dic106—22no—
45C2×Dic106—42no—
46C22⋊C4×C53—24no—
47C22⋊C4×C53—24no—
48C22⋊Dic53—24no—
49C22⋊Dic53—24no—
50C22⋊Dic53—44no—
51C22⋊Dic53—44no—
52C4⋊Dic53—22no—
53C4⋊Dic53—42no—
54#11719919—24no—
55#11719919—44no—
56C23×C53—22no—
57C23×C53—22no—
58C23×C53—28no—
59C23×C53—22no—
60C23×C53—22no—
61C23×C53—24no—
62C23×C53—24no—
63C23×C53—48no—
64C23×C53—44no—
65C23×C53—44no—
66C23×C53—48no—
67C23×C53C2×D412yes—
68C23×C53C2×Q814yes—
69C23×C53C2×Q812yes—
70C23×C53#4112yes—
71C23×C53C2411yescharacteristic
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—24no—
76C2×C4×C53—44no—
77C2×C4×C53—44no—
78C2×C4×C53C2×D412yes—
79C2×C4×C53C2×D412yes—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—24no—
83C2×Dic53—24no—
84C2×Dic53—24no—
85C2×Dic53—24no—
86C2×Dic53—24no—
87Dic106—44no—
88C22×C53—22no—
89C22×C53—22no—
90C22×C53—24no—
91C22×C53—24no—
92C22×C53—24no—
93C22×C53—28no—
94C22×C53—416no—
95C22×C53—42no—
96C22×C53—42no—
97C22×C53—44no—
98C22×C53C22⋊Q814yes—
99C22×C53C22×D411yescharacteristic
100C22×C53C22×Q811yescharacteristic
101C22×C53C2×#4111yescharacteristic
102C22×C53H(5,𝔽2)12yes—
103C22×C53H(5,𝔽2)12yes—
104C4×C53—22no—
105C4×C53—22no—
106C4×C53—42no—
107Dic53—42no—
108Dic53—42no—
109Dic53—44no—
110#2626—531nomaximal, Sylow
111C2×C53—24no—
112C2×C53—44no—
113C2×C53C2×C22⋊Q811yescharacteristic
114C2×C53#53411yescharacteristic
115C2×C53#53511yescharacteristic
116C2×C53#53812yes—
117C2×C53#54312yes—
118C2×C22⋊Q8—531no—
119C22×C22⋊C4D5311yescharacteristic
120#399—532no—
121#396—534no—
122#393—532no—
123#385—532no—
124#380—531no—
125#379—532no—
126C53#262611yescharacteristic, complemented, Sylow
127C25C2×D5311yescharacteristic
128C22×Q8—531no—
129C23×C4C2×D5311yescharacteristic
130C23×C4C2×D5311yescharacteristic
131C22⋊Q8—1064no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—531no—
134C2×C4⋊C4—534no—
135C2×C4⋊C4—532no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—532no—
138C2×C22⋊C4—532no—
139C2×C22⋊C4—534no—
140C2×C22⋊C4—534no—
141C2×C22⋊C4C2×D5312yes—
142C2×C22⋊C4C2×D5312yes—
143#95—532no—
144#95—534no—
145C24—22no—
146C24—22no—
147C24—42no—
148C24—42no—
149C24Dic10614yescomplemented
150C24C22×D5312yes—
151C24C22×D5311yescharacteristic
152C2×Q8—1062no—
153C2×Q8—2122no—
154C22×C4—22no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—534no—
158C22×C4—532no—
159C22×C4—532no—
160C22×C4C22⋊D5314yes—
161C22×C4C22×D5312yes—
162C22×C4C22×D5311yescharacteristic
163C22×C4C22×D5311yescharacteristic
164C4⋊C4—1064no—
165C4⋊C4—1062no—
166C4⋊C4—2124no—
167C4⋊C4—2122no—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—1064no—
171C22⋊C4—1064no—
172C22⋊C4—2124no—
173C22⋊C4—2124no—
174C23—22no—
175C23—22no—
176C23—28no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—24no—
181C23—48no—
182C23—44no—
183C23—44no—
184C23—48no—
185C23C2×Dic10614yes—
186C23C2×Dic10612yes—
187C23#1171994512yes—
188C23C2×C22⋊D5312yes—
189C23C23×D5311yescharacteristic
190Q8—2124no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—1064no—
200C2×C4—1064no—
201C2×C4—1064no—
202C2×C4—1064no—
203C2×C4—1064no—
204C2×C4C2×C22⋊D5312yes—
205C2×C4C2×C22⋊D5312yes—
206C22—22no—
207C22—22no—
208C22—24no—
209C22—24no—
210C22—24no—
211C22—28no—
212C22—416no—
213C22—42no—
214C22—42no—
215C22—44no—
216C22#1313319614yes—
217C22C22×C22⋊D5311yescharacteristic
218C22C22×Dic10611yescharacteristic
219C22C2×#1171994511yescharacteristic
220C22#1313326212yes—
221C22#1313326212yes—
222C4—22no—
223C4—22no—
224C4—42no—
225C4—2122no—
226C4—2122no—
227C4—2124no—
228C2—24no—
229C2—44no—
230C2C2×#1313319611yescharacteristic
231C2#1681118211yescharacteristic
232C2#1681119011yescharacteristic
233C2#1681121112yes—
234C2#1681124912yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC22×C22⋊C4×C53C2
hypercenterC22×C22⋊C4D53
hypocenterC53#2626
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#13133196C2semidirect
2. #16810559C2semidirect
3. #16810546C2semidirect
4. #16810535C2semidirect
5. #16810496C2semidirect
6. #16810483C2semidirect
9. C22×Dic106C22semidirect
11. C2×C4⋊Dic53C22semidirect
12. C2×C4⋊Dic53C22semidirect
13. C2×#11719919C22semidirect
14. C2×#11719919C22semidirect
19. #13133124C22semidirect
20. #13133124C22semidirect
31. C24×C53Q8semidirect
126. C53#2626semidirect
149. C24Dic106semidirect