Finite Group Databasebeta

← #20282786

Subgroups — #20282786

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×#13133210C212yesmaximal, complemented
2#16810557C212yesmaximal, complemented
3#16810553C211yesmaximal, characteristic, complemented
4#16810546C212yesmaximal, complemented
5#16810546C212yesmaximal, complemented
6#16810546C212yesmaximal, complemented
7#16810537C212yesmaximal
8#16810492C211yesmaximal, characteristic, complemented
9C53×#394C211yesmaximal, characteristic
10#13133210—28no—
11C2×C4⋊Dic53C2212yes—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×C22⋊Dic53C2212yes—
20C2×C22⋊Dic53C2212yes—
21C2×C22⋊Dic53C2212yes—
22C2×C22⋊Dic53C2212yes—
23C2×C4×Dic53C2211yescharacteristic
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26#13133124C2212yes—
27C22×D4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2212yes—
30C2×C22⋊C4×C53C2212yes—
31#95×C53C2211yescharacteristic
32C24×C53C2312yes—
33C2×D4×C53—42no—
34C2×D4×C53—41no—
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C4⋊C4×C53—41no—
44C22⋊C4×C53—24no—
45C22⋊C4×C53—42no—
46C22⋊Dic53—24no—
47C22⋊Dic53—24no—
48C22⋊Dic53—42no—
49C22⋊Dic53—42no—
50C4⋊Dic53—24no—
51C4⋊Dic53—41no—
52C4⋊Dic53—41no—
53#11719919—24no—
54#11719919—24no—
55#11719919—42no—
56#11719919—42no—
57#11719919—42no—
58C4×Dic53D414yescomplemented
59C23×C53—22no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53—22no—
63C23×C53—22no—
64C23×C53—22no—
65C23×C53—22no—
66C23×C53C2411yescharacteristic
67D4×C53—44no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53C2×D412yes—
78C2×Dic53—22no—
79C2×Dic53—22no—
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×Dic53—22no—
89C2×Dic53C2×D414yes—
90C22×C53—42no—
91C22×C53—42no—
92C22×C53—42no—
93C22×C53—42no—
94C22×C53—42no—
95C22×C53—42no—
96C22×C53—42no—
97C22×C53C22×D411yescharacteristic
98C22×C53H(5,𝔽2)11yescharacteristic
99C22×C53H(5,𝔽2)12yes—
100C22×C532−1+412yes—
101C22×C532−1+411yescharacteristic
102C4×C53—22no—
103C4×C53—42no—
104C4×C53—41no—
105C4×C53—41no—
106Dic53—24no—
107Dic53—42no—
108Dic53—42no—
109Dic53—42no—
110#2823—531nomaximal, Sylow
111C2×C53—82no—
112C2×C53#53711yescharacteristic
113C2×C53#53712yes—
114C2×C53#56011yescharacteristic
115C2×C53#56311yescharacteristic
116C2×C53#56312yes—
117C2×#126—532no—
118#398—531no—
119#398—532no—
120#396—532no—
121#396—532no—
122#396—532no—
123#394—532no—
124#394D5311yescharacteristic
125#384—531no—
126C53#282311yescharacteristic, complemented, Sylow
127C22×D4C2×D5311yescharacteristic
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—532no—
135C2×C4⋊C4—531no—
136C2×C4⋊C4—531no—
137C2×C4⋊C4C2×D5311yescharacteristic
138C2×C22⋊C4—532no—
139C2×C22⋊C4—532no—
140C2×C22⋊C4—532no—
141C2×C22⋊C4—532no—
142C2×C22⋊C4C2×D5312yes—
143C2×C22⋊C4C2×D5312yes—
144C2×C42—531no—
145#95—532no—
146#95—531no—
147#95—532no—
148#95C2×D5311yescharacteristic
149C24C22×D5312yes—
150C2×D4—41no—
151C2×D4—42no—
152C22×C4—532no—
153C22×C4—532no—
154C22×C4—532no—
155C22×C4—532no—
156C22×C4C22×D5311yescharacteristic
157C22×C4C22×D5312yes—
158C22×C4C22×D5311yescharacteristic
159C22×C4C22×D5311yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—1064no—
162C4⋊C4—1064no—
163C4⋊C4—1064no—
164C4⋊C4—2122no—
165C4⋊C4—2122no—
166C4⋊C4—2122no—
167C4⋊C4—2121no—
168C4⋊C4—2121no—
169C22⋊C4—24no—
170C22⋊C4—42no—
171C22⋊C4—1064no—
172C22⋊C4—1064no—
173C22⋊C4—2122no—
174C22⋊C4—2122no—
175C42—534no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23C23×D5311yescharacteristic
184D4—44no—
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—534no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1062no—
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—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22C2×D4×D5311yescharacteristic
215C22#1313326211yescharacteristic
216C22#1313326212yes—
217C22#1313326412yes—
218C22#1313326411yescharacteristic
219C4—22no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4—1064no—
224C4—2122no—
225C4—2122no—
226C4—2122no—
227C2—82no—
228C2#1681120011yescharacteristic
229C2#1681120212yes—
230C2#1681136211yescharacteristic
231C2#1681137311yescharacteristic
232C2#1681137612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2823
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#13133210C2semidirect
2. #16810557C2semidirect
3. #16810553C2semidirect
4. #16810546C2semidirect
5. #16810546C2semidirect
6. #16810546C2semidirect
8. #16810492C2semidirect
58. C4×Dic53D4semidirect
126. C53#2823semidirect