Finite Group Databasebeta

← #20281644

Subgroups — #20281644

Order 6784.

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

normal subgroups only

This group has 504 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×#13133202C212yesmaximal
2#16810556C212yesmaximal, complemented
3#16810550C212yesmaximal, complemented
4#16810549C212yesmaximal, complemented
5#16810525C211yesmaximal, characteristic
6#16810521C211yesmaximal, characteristic, complemented
7#16810497C212yesmaximal, complemented
8#16810494C212yesmaximal, complemented
9C53×#389C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C4×D53C2212yes—
12#13133202—28no—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×C4⋊Dic53C2211yescharacteristic
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2212yes—
29C2×C4⋊C4×C53C2211yescharacteristic
30C2×C42×C53C2211yescharacteristic
31#95×C53C2212yes—
32C23×D53C2312yes—
33C22×C4×C53C2312yes—
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C2×D212—41no—
42C2×D212—41no—
43C2×D212D414yes—
44C2×C4×D53—24no—
45C2×C4×D53—42no—
46C2×C4×D53—42no—
47C4⋊C4×C53—24no—
48C4⋊C4×C53—22no—
49C4⋊C4×C53—41no—
50C42×C53—41no—
51#11719921—24no—
52#11719921—24no—
53#11719921—24no—
54#11719921—24no—
55#11719921—24no—
56C4⋊Dic53—22no—
57C4⋊Dic53—41no—
58C4⋊Dic53—41no—
59#11719919—24no—
60#11719919—42no—
61C23×C53C2411yescharacteristic
62C22×D53—22no—
63C22×D53—22no—
64C22×D53—22no—
65C22×D53—22no—
66C22×D53—22no—
67C22×D53—22no—
68C22×D53C2×D414yes—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53C2×D412yes—
82C2×C4×C53#4112yes—
83C2×C4×C53#4112yes—
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—
93D212—44no—
94C4×D53—48no—
95C22×C53C22×D411yescharacteristic
96C22×C53C2×#4111yescharacteristic
97C22×C53C2×#4111yescharacteristic
98C22×C53H(5,𝔽2)11yescharacteristic
99C22×C53H(5,𝔽2)11yescharacteristic
100C22×C532−1+411yescharacteristic
101C22×C532−1+411yescharacteristic
102C2×D53—24no—
103C2×D53—24no—
104C2×D53—24no—
105C2×D53—42no—
106C2×D53—42no—
107C2×D53—42no—
108C2×D53—42no—
109C4×C53—22no—
110C4×C53—22no—
111C4×C53—22no—
112C4×C53—42no—
113C4×C53—42no—
114Dic53—42no—
115Dic53—42no—
116Dic53—42no—
117#2724—531nomaximal, Sylow
118C2×C53#53411yescharacteristic
119C2×C53#53911yescharacteristic
120C2×C53#53911yescharacteristic
121C2×C53#54711yescharacteristic
122C2×C53#54711yescharacteristic
123C2×C53#55911yescharacteristic
124C2×C53#56311yescharacteristic
125D53—44no—
126C2×#123—532no—
127#398—532no—
128#397—532no—
129#397—532no—
130#390—531no—
131#389—531no—
132#389D5311yescharacteristic, complemented
133#385—532no—
134#385—532no—
135C53#272411yescharacteristic, complemented, Sylow
136C22×D4—531no—
137C23×C4—532no—
138#123—1068no—
139C2×C4⋊C4—532no—
140C2×C4⋊C4—531no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—531no—
143C2×C4⋊C4—531no—
144C2×C4⋊C4C2×D5312yes—
145C2×C4⋊C4C2×D5311yescharacteristic
146C2×C4⋊C4C2×D5311yescharacteristic
147C2×C22⋊C4—532no—
148C2×C22⋊C4—532no—
149C2×C22⋊C4—532no—
150C2×C22⋊C4—532no—
151C2×C22⋊C4—532no—
152C2×C42C2×D5311yescharacteristic
153#95—532no—
154#95—532no—
155#95—532no—
156#95—532no—
157#95C2×D5312yes—
158C24—532no—
159C2×D4—534no—
160C2×D4—2121no—
161C2×D4—2121no—
162C22×C4—532no—
163C22×C4—532no—
164C22×C4—532no—
165C22×C4—1064no—
166C22×C4—2122no—
167C22×C4—2122no—
168C22×C4C22×D5312yes—
169C22×C4C22×D5312yes—
170C22×C4C22×D5311yescharacteristic
171C22×C4C22×D5311yescharacteristic
172C22×C4C22×D5311yescharacteristic
173C4⋊C4—22no—
174C4⋊C4—24no—
175C4⋊C4—41no—
176C4⋊C4—1064no—
177C4⋊C4—1062no—
178C4⋊C4—2122no—
179C4⋊C4—2121no—
180C4⋊C4—2121no—
181C22⋊C4—1064no—
182C22⋊C4—1064no—
183C22⋊C4—1064no—
184C22⋊C4—1064no—
185C22⋊C4—1064no—
186C42—41no—
187C23—534no—
188C23—1062no—
189C23—1062no—
190C23—1062no—
191C23—1062no—
192C23—1062no—
193C23—1062no—
194C23C23×D5311yescharacteristic
195D4—2124no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
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—2128no—
218C2×C4#1171994512yes—
219C2×C4D4×D5312yes—
220C2×C4#1171994912yes—
221C22—1064no—
222C22—1064no—
223C22—1064no—
224C22—2122no—
225C22—2122no—
226C22—2122no—
227C22—2122no—
228C22C2×D4×D5311yescharacteristic
229C22C2×#1171994511yescharacteristic
230C22C2×#1171994911yescharacteristic
231C22#1313326211yescharacteristic
232C22#1313326311yescharacteristic
233C22#1313326411yescharacteristic
234C22#1313326511yescharacteristic
235C4—22no—
236C4—22no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—2122no—
241C4—2122no—
242C4—2122no—
243C2—2124no—
244C2#1681118111yescharacteristic
245C2#1681122011yescharacteristic
246C2#1681122211yescharacteristic
247C2#1681127611yescharacteristic
248C2#1681127811yescharacteristic
249C2#1681135811yescharacteristic
250C2#1681137711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#389C2
hypercenter#389D53
hypocenterC53#2724
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. #16810556C2semidirect
3. #16810550C2semidirect
4. #16810549C2semidirect
6. #16810521C2semidirect
7. #16810497C2semidirect
8. #16810494C2semidirect
9. C53×#389C2semidirect
132. #389D53semidirect
135. C53#2724semidirect