Finite Group Databasebeta

← #20281983

Subgroups — #20281983

Order 6784.

484 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#16810559C211yesmaximal, characteristic, complemented
2#16810548C212yesmaximal, complemented
3#16810537C212yesmaximal, complemented
4#16810526C212yesmaximal, complemented
5#16810516C212yesmaximal, complemented
6#16810506C212yesmaximal, complemented
7#16810503C212yesmaximal, complemented
8#16810484C211yesmaximal, characteristic, complemented
9C53×#392C211yesmaximal, characteristic
10C23×Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2212yes—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×#11719919C2212yes—
14C2×C22⋊Dic53C2212yes—
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×C22⋊Dic53C2211yescharacteristic
19C2×C22⋊Dic53C2212yes—
20C2×C4×Dic53C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26C22×D4×C53C2212yes—
27C22×D4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2212yes—
29C2×C22⋊C4×C53C2211yescharacteristic
30#95×C53C2211yescharacteristic
31C24×C53C2312yes—
32C24×C53C2311yescharacteristic
33C2×D4×C53—24no—
34C2×D4×C53—24no—
35C2×D4×C53—42no—
36C2×D4×C53—42no—
37C2×D4×C53—41no—
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×Dic53—22no—
42C22×Dic53—41no—
43C22×Dic53—41no—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2312yes—
47C22×Dic53C2312yes—
48C22⋊C4×C53—24no—
49C22⋊C4×C53—41no—
50C22⋊Dic53—42no—
51C22⋊Dic53—42no—
52C22⋊Dic53—42no—
53C22⋊Dic53—42no—
54C22⋊Dic53—41no—
55C22⋊Dic53—42no—
56C4⋊Dic53—24no—
57C4⋊Dic53—41no—
58#11719919—42no—
59C4×Dic53—42no—
60C23×C53—22no—
61C23×C53—21no—
62C23×C53—21no—
63C23×C53—22no—
64C23×C53—22no—
65C23×C53—22no—
66C23×C53—22no—
67C23×C53—22no—
68C23×C53—22no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53#4112yes—
72C23×C53C2411yescharacteristic
73D4×C53—48no—
74D4×C53—44no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53#4114yes—
82C2×Dic53—22no—
83C2×Dic53—22no—
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—
93C2×Dic53—22no—
94C2×Dic53—44no—
95C22×C53—22no—
96C22×C53—24no—
97C22×C53—42no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53—42no—
101C22×C53—42no—
102C22×C53—42no—
103C22×C53—42no—
104C22×C53—42no—
105C22×C53—42no—
106C22×C53—42no—
107C22×C53C2×#4111yescharacteristic
108C22×C53C2×#4112yes—
109C22×C53H(5,𝔽2)11yescharacteristic
110C22×C53H(5,𝔽2)12yes—
111C22×C53H(5,𝔽2)11yescharacteristic
112C4×C53—24no—
113C4×C53—41no—
114C4×C53—41no—
115Dic53—42no—
116Dic53—42no—
117Dic53—42no—
118Dic53—42no—
119#2752—531nomaximal, Sylow
120C2×C53—42no—
121C2×C53—82no—
122C2×C53#53811yescharacteristic
123C2×C53#54012yes—
124C2×C53#55112yes—
125C2×C53#55911yescharacteristic
126C2×C53#56011yescharacteristic
127#399—531no—
128#397—532no—
129#394—532no—
130#392D5311yescharacteristic
131#390—532no—
132#388—532no—
133#387—532no—
134#386—532no—
135#381—531no—
136C53#275211yescharacteristic, complemented, Sylow
137C22×D4C2×D5312yes—
138C22×D4C2×D5311yescharacteristic
139C23×C4—531no—
140C2×C4⋊C4—532no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—531no—
143C2×C22⋊C4—531no—
144C2×C22⋊C4—532no—
145C2×C22⋊C4—532no—
146C2×C22⋊C4—532no—
147C2×C22⋊C4—532no—
148C2×C22⋊C4—532no—
149C2×C22⋊C4C2×D5312yes—
150C2×C22⋊C4C2×D5311yescharacteristic
151C2×C42—532no—
152#95—532no—
153#95—532no—
154#95—532no—
155#95—532no—
156#95—531no—
157#95C2×D5311yescharacteristic
158C24C22×D5312yes—
159C24C22×D5311yescharacteristic
160C2×D4—24no—
161C2×D4—24no—
162C2×D4—42no—
163C2×D4—41no—
164C2×D4—42no—
165C22×C4—532no—
166C22×C4—532no—
167C22×C4—532no—
168C22×C4—532no—
169C22×C4—1062no—
170C22×C4—2121no—
171C22×C4—2121no—
172C22×C4C22×D5311yescharacteristic
173C22×C4C22×D5311yescharacteristic
174C22×C4C22×D5312yes—
175C4⋊C4—1064no—
176C4⋊C4—2122no—
177C4⋊C4—2121no—
178C22⋊C4—24no—
179C22⋊C4—41no—
180C22⋊C4—2122no—
181C22⋊C4—2122no—
182C22⋊C4—2122no—
183C22⋊C4—2121no—
184C22⋊C4—2122no—
185C22⋊C4—2122no—
186C42—2122no—
187C23—22no—
188C23—21no—
189C23—22no—
190C23—21no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23#1171994712yes—
199C23C23×D5311yescharacteristic
200D4—48no—
201D4—44no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—1062no—
209C2×C4—1062no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—2124no—
221C2×C4#1171994714yes—
222C22—24no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22C2×#1171994711yescharacteristic
235C22C2×#1171994712yes—
236C22#1313326212yes—
237C22#1313326211yescharacteristic
238C22#1313326311yescharacteristic
239C4—24no—
240C4—41no—
241C4—41no—
242C4—2122no—
243C4—2122no—
244C4—2122no—
245C4—2122no—
246C2—42no—
247C2—82no—
248C2#1681121011yescharacteristic
249C2#1681122612yes—
250C2#1681130412yes—
251C2#1681135911yescharacteristic
252C2#1681136211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2752
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #16810559C2semidirect
2. #16810548C2semidirect
3. #16810537C2semidirect
4. #16810526C2semidirect
5. #16810516C2semidirect
6. #16810506C2semidirect
7. #16810503C2semidirect
8. #16810484C2semidirect
136. C53#2752semidirect