Finite Group Databasebeta

← #20281676

Subgroups — #20281676

Order 6784.

476 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133212C212yesmaximal, complemented
2#16810560C212yesmaximal, complemented
3#16810545C212yesmaximal, complemented
4#16810540C212yesmaximal
5#16810525C211yesmaximal, characteristic
6#16810522C212yesmaximal, complemented
7#16810521C211yesmaximal, characteristic, complemented
8#16810515C212yesmaximal, complemented
9C53×#389C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11#13133212—28no—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×C4⋊Dic53C2212yes—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4⋊Dic53C2212yes—
17C2×#11719919C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×C4×Dic53C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27C2×C4⋊C4×C53C2212yes—
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2211yescharacteristic
30C2×C42×C53C2211yescharacteristic
31#95×C53C2212yes—
32C23×D53C2312yes—
33C22×C4×C53C2312yes—
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C2×D212—22no—
42C2×D212—41no—
43C2×D212—41no—
44C4⋊C4×C53—24no—
45C4⋊C4×C53—41no—
46C4⋊C4×C53D414yescomplemented
47C42×C53—41no—
48#11719921—24no—
49#11719921—24no—
50#11719921—24no—
51#11719921—24no—
52#11719921—42no—
53#11719921—42no—
54C4⋊Dic53—24no—
55C4⋊Dic53—22no—
56C4⋊Dic53—41no—
57C4⋊Dic53—42no—
58C4⋊Dic53—41no—
59#11719919—24no—
60C4×Dic53—24no—
61C23×C53C2411yescharacteristic
62C22×D53—22no—
63C22×D53—22no—
64C22×D53—22no—
65C22×D53—22no—
66C22×D53—22no—
67C22×D53—22no—
68C22×D53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53C2×D414yes—
81C2×C4×C53C2×D412yes—
82C2×C4×C53#4112yes—
83C2×C4×C53#4112yes—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93D212—44no—
94C22×C53C22×D411yescharacteristic
95C22×C53C2×#4111yescharacteristic
96C22×C53C2×#4111yescharacteristic
97C22×C53H(5,𝔽2)11yescharacteristic
98C22×C53H(5,𝔽2)11yescharacteristic
99C22×C532−1+411yescharacteristic
100C22×C532−1+411yescharacteristic
101C2×D53—42no—
102C2×D53—42no—
103C2×D53—42no—
104C2×D53—42no—
105C2×D53—42no—
106C2×D53—42no—
107C2×D53—42no—
108C4×C53—22no—
109C4×C53—22no—
110C4×C53—22no—
111C4×C53—24no—
112C4×C53—42no—
113Dic53—42no—
114Dic53—42no—
115Dic53—42no—
116#2727—531nomaximal, Sylow
117C2×C53#53711yescharacteristic
118C2×C53#53811yescharacteristic
119C2×C53#53911yescharacteristic
120C2×C53#54711yescharacteristic
121C2×C53#55011yescharacteristic
122C2×C53#55611yescharacteristic
123C2×C53#56311yescharacteristic
124D53—82no—
125C2×#126—532no—
126#399—532no—
127#396—532no—
128#394—532no—
129#390—531no—
130#389—532no—
131#389—531no—
132#389D5311yescharacteristic, complemented
133#388—532no—
134C53#272711yescharacteristic, complemented, Sylow
135C22×D4—531no—
136#126—1068no—
137C2×C4⋊C4—532no—
138C2×C4⋊C4—532no—
139C2×C4⋊C4—531no—
140C2×C4⋊C4—531no—
141C2×C4⋊C4—531no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4C2×D5312yes—
144C2×C4⋊C4C2×D5311yescharacteristic
145C2×C4⋊C4C2×D5311yescharacteristic
146C2×C22⋊C4—532no—
147C2×C22⋊C4—532no—
148C2×C22⋊C4—532no—
149C2×C22⋊C4—532no—
150C2×C22⋊C4—532no—
151C2×C22⋊C4—532no—
152C2×C42—532no—
153C2×C42C2×D5311yescharacteristic
154#95—532no—
155#95—532no—
156#95C2×D5312yes—
157C24—532no—
158C2×D4—1062no—
159C2×D4—2121no—
160C2×D4—2121no—
161C22×C4—532no—
162C22×C4—532no—
163C22×C4—532no—
164C22×C4C22×D5312yes—
165C22×C4C22×D5312yes—
166C22×C4C22×D5311yescharacteristic
167C22×C4C22×D5311yescharacteristic
168C22×C4C22×D5311yescharacteristic
169C4⋊C4—24no—
170C4⋊C4—41no—
171C4⋊C4—1064no—
172C4⋊C4—1064no—
173C4⋊C4—1062no—
174C4⋊C4—2121no—
175C4⋊C4—2121no—
176C4⋊C4—2122no—
177C4⋊C4D21214yescomplemented
178C22⋊C4—1064no—
179C22⋊C4—1064no—
180C22⋊C4—1064no—
181C22⋊C4—1064no—
182C22⋊C4—2122no—
183C22⋊C4—2122no—
184C42—41no—
185C42—1064no—
186C23—1062no—
187C23—1062no—
188C23—1062no—
189C23—1062no—
190C23—1062no—
191C23—1062no—
192C23—1062no—
193C23C23×D5311yescharacteristic
194D4—2124no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—1062no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1062no—
215C2×C4C2×D21214yes—
216C2×C4C2×D21212yes—
217C2×C4#1171994712yes—
218C2×C4#1171994912yes—
219C22—2122no—
220C22—2122no—
221C22—2122no—
222C22—2122no—
223C22—2122no—
224C22—2122no—
225C22—2122no—
226C22C22×D21211yescharacteristic
227C22C2×#1171994711yescharacteristic
228C22C2×#1171994911yescharacteristic
229C22#1313326211yescharacteristic
230C22#1313326311yescharacteristic
231C22#1313326411yescharacteristic
232C22#1313326511yescharacteristic
233C4—24no—
234C4—22no—
235C4—22no—
236C4—22no—
237C4—42no—
238C4—2122no—
239C4—2122no—
240C4—2122no—
241C2—4242no—
242C2#1681120411yescharacteristic
243C2#1681121011yescharacteristic
244C2#1681122211yescharacteristic
245C2#1681127711yescharacteristic
246C2#1681129511yescharacteristic
247C2#1681134711yescharacteristic
248C2#1681137711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#389C2
hypercenter#389D53
hypocenterC53#2727
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#13133212C2semidirect
2. #16810560C2semidirect
3. #16810545C2semidirect
6. #16810522C2semidirect
7. #16810521C2semidirect
8. #16810515C2semidirect
9. C53×#389C2semidirect
46. C4⋊C4×C53D4semidirect
132. #389D53semidirect
134. C53#2727semidirect
177. C4⋊C4D212semidirect