Finite Group Databasebeta

← #20282790

Subgroups — #20282790

Order 6784.

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

normal subgroups only

This group has 456 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
1C2×#13133212C212yesmaximal, complemented
2#16810557C212yesmaximal, complemented
3#16810547C212yesmaximal, complemented
4#16810545C212yesmaximal, complemented
5#16810545C212yesmaximal, complemented
6#16810541C212yesmaximal
7#16810540C211yesmaximal, characteristic
8#16810492C211yesmaximal, characteristic, complemented
9C53×#398C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11#13133212—28no—
12C2×C4⋊Dic53C2212yes—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×C4×Dic53C2211yescharacteristic
25#13133124C2211yescharacteristic
26#13133124C2212yes—
27C2×C4⋊C4×C53C2212yes—
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2211yescharacteristic
30#95×C53C2212yes—
31#95×C53C2211yescharacteristic
32C23×D53C2312yes—
33C22×C4×C53C2311yescharacteristic
34C22×C4×C53C2312yes—
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C2×D212—42no—
42C2×D212—41no—
43C4⋊C4×C53—24no—
44C4⋊C4×C53—41no—
45C4⋊C4×C53—41no—
46#11719921—24no—
47#11719921—24no—
48#11719921—24no—
49#11719921—42no—
50#11719921—42no—
51#11719921—42no—
52C4⋊Dic53—24no—
53C4⋊Dic53—41no—
54#11719919—24no—
55#11719919—42no—
56#11719919—42no—
57#11719919—42no—
58C4×Dic53D414yescomplemented
59C23×C53C2411yescharacteristic
60C22×D53—22no—
61C22×D53—22no—
62C22×D53—22no—
63C22×D53—22no—
64C22×D53—22no—
65C22×D53—22no—
66C22×D53—22no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53C2×D412yes—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53C2×D414yes—
89D212—44no—
90C22×C53C22×D411yescharacteristic
91C22×C53H(5,𝔽2)11yescharacteristic
92C22×C53H(5,𝔽2)12yes—
93C22×C532−1+411yescharacteristic
94C22×C532−1+412yes—
95C2×D53—42no—
96C2×D53—42no—
97C2×D53—42no—
98C2×D53—42no—
99C2×D53—42no—
100C2×D53—42no—
101C2×D53—42no—
102C4×C53—22no—
103C4×C53—42no—
104C4×C53—42no—
105C4×C53—41no—
106C4×C53—41no—
107Dic53—24no—
108Dic53—42no—
109Dic53—42no—
110#2823—531nomaximal, Sylow
111C2×C53#53711yescharacteristic
112C2×C53#53712yes—
113C2×C53#56011yescharacteristic
114C2×C53#56311yescharacteristic
115C2×C53#56312yes—
116D53—82no—
117C2×#126—532no—
118#398—532no—
119#398D5311yescharacteristic, complemented
120#396—532no—
121#396—532no—
122#396—532no—
123#394—531no—
124#394—532no—
125#384—531no—
126C53#282311yescharacteristic, complemented, Sylow
127C22×D4—531no—
128#126—1068no—
129C2×C4⋊C4—532no—
130C2×C4⋊C4—532no—
131C2×C4⋊C4—532no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—531no—
135C2×C4⋊C4C2×D5312yes—
136C2×C4⋊C4C2×D5311yescharacteristic
137C2×C4⋊C4C2×D5311yescharacteristic
138C2×C22⋊C4—532no—
139C2×C22⋊C4—532no—
140C2×C22⋊C4—532no—
141C2×C22⋊C4—532no—
142C2×C22⋊C4—532no—
143C2×C22⋊C4—532no—
144C2×C42—531no—
145#95—531no—
146#95—532no—
147#95C2×D5312yes—
148#95C2×D5311yescharacteristic
149C24—532no—
150C2×D4—2121no—
151C2×D4—2122no—
152C22×C4—532no—
153C22×C4—532no—
154C22×C4—532no—
155C22×C4C22×D5311yescharacteristic
156C22×C4C22×D5312yes—
157C22×C4C22×D5312yes—
158C22×C4C22×D5311yescharacteristic
159C22×C4C22×D5311yescharacteristic
160C4⋊C4—24no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—1064no—
164C4⋊C4—1064no—
165C4⋊C4—2122no—
166C4⋊C4—2122no—
167C4⋊C4—2122no—
168C4⋊C4—2121no—
169C22⋊C4—1064no—
170C22⋊C4—1064no—
171C22⋊C4—1064no—
172C22⋊C4—2122no—
173C22⋊C4—2122no—
174C22⋊C4—2122no—
175C42—534no—
176C23—1062no—
177C23—1062no—
178C23—1062no—
179C23—1062no—
180C23—1062no—
181C23—1062no—
182C23—1062no—
183C23C23×D5311yescharacteristic
184D4—2124no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—534no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4D4×D5312yes—
207C22—2122no—
208C22—2122no—
209C22—2122no—
210C22—2122no—
211C22—2122no—
212C22—2122no—
213C22—2122no—
214C22C2×D4×D5311yescharacteristic
215C22#1313326211yescharacteristic
216C22#1313326312yes—
217C22#1313326411yescharacteristic
218C22#1313326512yes—
219C4—22no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4—42no—
224C4—1064no—
225C4—2122no—
226C4—2122no—
227C2—4242no—
228C2#1681120011yescharacteristic
229C2#1681120312yes—
230C2#1681136311yescharacteristic
231C2#1681137211yescharacteristic
232C2#1681137712yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2823
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#13133212C2semidirect
2. #16810557C2semidirect
3. #16810547C2semidirect
4. #16810545C2semidirect
5. #16810545C2semidirect
8. #16810492C2semidirect
9. C53×#398C2semidirect
58. C4×Dic53D4semidirect
119. #398D53semidirect
126. C53#2823semidirect