Finite Group Databasebeta

← #20281674

Subgroups — #20281674

Order 6784.

476 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 133 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#16810559C212yesmaximal, complemented
3#16810546C212yesmaximal, complemented
4#16810537C212yesmaximal
5#16810521C211yesmaximal, characteristic, complemented
6#16810521C211yesmaximal, characteristic, complemented
7#16810518C212yesmaximal, complemented
8#16810516C212yesmaximal, complemented
9C53×#390C211yesmaximal, characteristic
10#13133210—28no—
11C2×C4⋊Dic53C2211yescharacteristic
12C2×C4⋊Dic53C2211yescharacteristic
13C2×C4⋊Dic53C2211yescharacteristic
14C2×C4⋊Dic53C2211yescharacteristic
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×C22⋊Dic53C2212yes—
20C2×C22⋊Dic53C2212yes—
21C2×C22⋊Dic53C2212yes—
22C2×C22⋊Dic53C2212yes—
23C2×C4×Dic53C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27C22×D4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2212yes—
30C2×C22⋊C4×C53C2212yes—
31C2×C42×C53C2211yescharacteristic
32C24×C53C2312yes—
33C2×D4×C53—22no—
34C2×D4×C53—41no—
35C2×D4×C53—41no—
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C4⋊C4×C53—41no—
45C22⋊C4×C53—24no—
46C22⋊C4×C53—24no—
47C42×C53—41no—
48C22⋊Dic53—24no—
49C22⋊Dic53—24no—
50C22⋊Dic53—42no—
51C22⋊Dic53—42no—
52C4⋊Dic53—22no—
53C4⋊Dic53—41no—
54C4⋊Dic53—41no—
55C4⋊Dic53D414yescomplemented
56#11719919—24no—
57#11719919—24no—
58#11719919—24no—
59#11719919—42no—
60C4×Dic53—24no—
61C23×C53—22no—
62C23×C53—22no—
63C23×C53—22no—
64C23×C53—22no—
65C23×C53—22no—
66C23×C53—22no—
67C23×C53—22no—
68C23×C53C2411yescharacteristic
69D4×C53—44no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53C2×D412yes—
80C2×C4×C53#4112yes—
81C2×C4×C53#4112yes—
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×Dic53C2×D414yes—
94C22×C53—42no—
95C22×C53—42no—
96C22×C53—42no—
97C22×C53—42no—
98C22×C53—42no—
99C22×C53—42no—
100C22×C53—42no—
101C22×C53C22×D411yescharacteristic
102C22×C53C2×#4111yescharacteristic
103C22×C53C2×#4111yescharacteristic
104C22×C53H(5,𝔽2)11yescharacteristic
105C22×C53H(5,𝔽2)11yescharacteristic
106C22×C532−1+411yescharacteristic
107C22×C532−1+411yescharacteristic
108C4×C53—22no—
109C4×C53—22no—
110C4×C53—22no—
111C4×C53—42no—
112Dic53—24no—
113Dic53—42no—
114Dic53—42no—
115Dic53—42no—
116#2727—531nomaximal, Sylow
117C2×C53—82no—
118C2×C53#53711yescharacteristic
119C2×C53#53811yescharacteristic
120C2×C53#53911yescharacteristic
121C2×C53#54711yescharacteristic
122C2×C53#55011yescharacteristic
123C2×C53#55611yescharacteristic
124C2×C53#56311yescharacteristic
125C2×#126—532no—
126#399—532no—
127#396—532no—
128#394—532no—
129#390D5311yescharacteristic
130#389—532no—
131#389—531no—
132#389—531no—
133#388—532no—
134C53#272711yescharacteristic, complemented, Sylow
135C22×D4C2×D5311yescharacteristic
136#126—1068no—
137C2×C4⋊C4—531no—
138C2×C4⋊C4—532no—
139C2×C4⋊C4—532no—
140C2×C4⋊C4—532no—
141C2×C4⋊C4—531no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4—531no—
144C2×C4⋊C4—531no—
145C2×C4⋊C4C2×D5311yescharacteristic
146C2×C22⋊C4—532no—
147C2×C22⋊C4—532no—
148C2×C22⋊C4—532no—
149C2×C22⋊C4—532no—
150C2×C22⋊C4C2×D5312yes—
151C2×C22⋊C4C2×D5312yes—
152C2×C42—532no—
153C2×C42C2×D5311yescharacteristic
154#95—532no—
155#95—532no—
156#95—532no—
157C24C22×D5312yes—
158C2×D4—22no—
159C2×D4—41no—
160C2×D4—41no—
161C22×C4—532no—
162C22×C4—532no—
163C22×C4—532no—
164C22×C4—532no—
165C22×C4C22×D5312yes—
166C22×C4C22×D5311yescharacteristic
167C22×C4C22×D5311yescharacteristic
168C22×C4C22×D5311yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—534no—
171C4⋊C4—1064no—
172C4⋊C4—1064no—
173C4⋊C4—1064no—
174C4⋊C4—1062no—
175C4⋊C4—2122no—
176C4⋊C4—2121no—
177C4⋊C4—2121no—
178C22⋊C4—24no—
179C22⋊C4—24no—
180C22⋊C4—1064no—
181C22⋊C4—1064no—
182C22⋊C4—2122no—
183C22⋊C4—2122no—
184C42—41no—
185C42—1064no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23C23×D5311yescharacteristic
194D4—44no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—534no—
205C2×C4—1062no—
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×C4—1062no—
216C2×C4#1171994512yes—
217C2×C4D4×D5312yes—
218C2×C4#1171994712yes—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22C2×D4×D5311yescharacteristic
227C22C2×#1171994711yescharacteristic
228C22C2×#1171994511yescharacteristic
229C22#1313326211yescharacteristic
230C22#1313326211yescharacteristic
231C22#1313326411yescharacteristic
232C22#1313326411yescharacteristic
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—1064no—
238C4—2122no—
239C4—2122no—
240C4—2122no—
241C2—82no—
242C2#1681120011yescharacteristic
243C2#1681121111yescharacteristic
244C2#1681121811yescharacteristic
245C2#1681127811yescharacteristic
246C2#1681129311yescharacteristic
247C2#1681134311yescharacteristic
248C2#1681137311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#390C2
hypercenter#390D53
hypocenterC53#2727
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#13133210C2semidirect
2. #16810559C2semidirect
3. #16810546C2semidirect
5. #16810521C2semidirect
6. #16810521C2semidirect
7. #16810518C2semidirect
8. #16810516C2semidirect
55. C4⋊Dic53D4semidirect
134. C53#2727semidirect