Finite Group Databasebeta

← #20282090

Subgroups — #20282090

Order 6784.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C53⋊#401C212yesmaximal, complemented
2#16810534C212yesmaximal, complemented
3#16810532C211yesmaximal, characteristic
4#16810501C212yesmaximal, complemented
5#16810500C212yesmaximal, complemented
6#16810499C212yesmaximal, complemented
7#16810485C212yesmaximal, complemented
8#16810484C211yesmaximal, characteristic, complemented
9C53×#393C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C23×Dic53C2211yescharacteristic
13C22×C4×D53C2212yes—
14C2×C22⋊Dic53C2211yescharacteristic
15C2×#11719921C2212yes—
16C2×#11719921C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2211yescharacteristic
25#13133124C2212yes—
26C22×Q8×C53C2211yescharacteristic
27C2×C22⋊C4×C53C2212yes—
28C2×C22⋊C4×C53C2211yescharacteristic
29#95×C53C2212yes—
30#95×C53C2211yescharacteristic
31C24×C53C2311yescharacteristic
32C23×D53C2312yes—
33C2×Q8×C53—42no—
34C2×Q8×C53—41no—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2312yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C2×C22⋊D53—24no—
40C2×C22⋊D53—24no—
41C2×C22⋊D53—42no—
42C22×Dic53—22no—
43C22×Dic53—41no—
44C22×Dic53—41no—
45C22×Dic53C2312yes—
46C22×Dic53C2312yes—
47C22×Dic53C2312yes—
48C2×D212—24no—
49C2×D212—41no—
50C2×C4×D53—24no—
51C2×C4×D53—42no—
52C2×C4×D53—42no—
53C22⋊C4×C53—42no—
54C22⋊C4×C53—41no—
55C22⋊Dic53—41no—
56#11719921—42no—
57#11719921—42no—
58#11719921—42no—
59#11719921—42no—
60C23×C53—22no—
61C23×C53—21no—
62C23×C53—21no—
63C23×C53—22no—
64C23×C53#4112yes—
65C23×C53C2411yescharacteristic
66C22×D53—22no—
67C22×D53—22no—
68C22×D53—22no—
69C22×D53—22no—
70C22×D53—22no—
71C22×D53—22no—
72C22×D53#4114yes—
73Q8×C53—44no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—21no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—21no—
84C22⋊D53—48no—
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—
95D212—44no—
96C4×D53—48no—
97C22×C53—22no—
98C22×C53—24no—
99C22×C53—42no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53C2×#4112yes—
103C22×C53C2×#4111yescharacteristic
104C22×C53H(5,𝔽2)12yes—
105C22×C53H(5,𝔽2)11yescharacteristic
106C22×C53H(5,𝔽2)11yescharacteristic
107C2×D53—24no—
108C2×D53—24no—
109C2×D53—24no—
110C2×D53—42no—
111C2×D53—42no—
112C2×D53—42no—
113C2×D53—42no—
114C4×C53—42no—
115C4×C53—41no—
116C4×C53—42no—
117C4×C53—41no—
118Dic53—42no—
119Dic53—42no—
120Dic53—42no—
121#2760—531nomaximal, Sylow
122C2×C53—42no—
123C2×C53#53812yes—
124C2×C53#53811yescharacteristic
125C2×C53#55112yes—
126C2×C53#55111yescharacteristic
127C2×C53#56011yescharacteristic
128D53—44no—
129#401—532no—
130#393—532no—
131#393D5311yescharacteristic, complemented
132#392—531no—
133#386—532no—
134#386—532no—
135#386—532no—
136#381—531no—
137#381—532no—
138C53#276011yescharacteristic, complemented, Sylow
139C22×Q8C2×D5311yescharacteristic
140C22×D4—532no—
141C22×D4—531no—
142C23×C4—531no—
143C23×C4—532no—
144C2×C22⋊C4—531no—
145C2×C22⋊C4—532no—
146C2×C22⋊C4—532no—
147C2×C22⋊C4—532no—
148C2×C22⋊C4—532no—
149C2×C22⋊C4C2×D5311yescharacteristic
150C2×C22⋊C4C2×D5312yes—
151#95—532no—
152#95—532no—
153#95—532no—
154#95—532no—
155#95—532no—
156#95—532no—
157#95—531no—
158#95C2×D5311yescharacteristic
159#95C2×D5312yes—
160C24—532no—
161C24C22×D5311yescharacteristic
162C2×Q8—42no—
163C2×Q8—41no—
164C2×D4—1064no—
165C2×D4—1064no—
166C2×D4—1064no—
167C2×D4—2121no—
168C2×D4—2122no—
169C22×C4—532no—
170C22×C4—532no—
171C22×C4—532no—
172C22×C4—1064no—
173C22×C4—1062no—
174C22×C4—2122no—
175C22×C4—2122no—
176C22×C4—2121no—
177C22×C4—2121no—
178C22×C4C22×D5311yescharacteristic
179C22×C4C22×D5312yes—
180C22×C4C22×D5311yescharacteristic
181C22×C4C22×D5312yes—
182C22⋊C4—42no—
183C22⋊C4—41no—
184C22⋊C4—2122no—
185C22⋊C4—2122no—
186C22⋊C4—2122no—
187C22⋊C4—2122no—
188C22⋊C4—2121no—
189C23—22no—
190C23—21no—
191C23—22no—
192C23—21no—
193C23—534no—
194C23—1062no—
195C23—1062no—
196C23—1062no—
197C23—1062no—
198C23—1062no—
199C23—1062no—
200C23#1171994912yes—
201C23C23×D5311yescharacteristic
202Q8—44no—
203D4—2124no—
204D4—2128no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—2128no—
225C2×C4—2124no—
226C22—24no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—1064no—
232C22—1064no—
233C22—1064no—
234C22—2122no—
235C22—2122no—
236C22—2122no—
237C22—2122no—
238C22#41×D5312yes—
239C22C2×#1171994911yescharacteristic
240C22#1313326212yes—
241C22#1313326211yescharacteristic
242C22#1313326311yescharacteristic
243C4—41no—
244C4—42no—
245C4—41no—
246C4—42no—
247C4—2122no—
248C4—2122no—
249C4—2122no—
250C2—42no—
251C2—2124no—
252C2#1681120712yes—
253C2#1681121311yescharacteristic
254C2#1681130012yes—
255C2#1681130311yescharacteristic
256C2#1681136211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2760
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C53⋊#401C2semidirect
2. #16810534C2semidirect
4. #16810501C2semidirect
5. #16810500C2semidirect
6. #16810499C2semidirect
7. #16810485C2semidirect
8. #16810484C2semidirect
9. C53×#393C2semidirect
131. #393D53semidirect
138. C53#2760semidirect