Finite Group Databasebeta

← #20280659

Subgroups — #20280659

Order 6784.

584 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133213C211yesmaximal, characteristic
2C2×#13133208C211yesmaximal, characteristic, complemented
3#16810539C214yesmaximal, complemented
4#16810538C214yesmaximal, complemented
5#16810526C211yesmaximal, characteristic, complemented
6#16810525C211yesmaximal, characteristic, complemented
7#16810524C211yesmaximal, characteristic, complemented
8#16810469C211yesmaximal, characteristic, complemented
9C53×#390C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C22×C22⋊D53C2212yes—
13#13133213—24no—
14#13133208—24no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4⋊Dic53C2212yes—
17C2×#11719919C2212yes—
18C2×C22⋊Dic53C2214yes—
19C2×#11719921C2214yes—
20C2×#11719921C2214yes—
21C2×C4×Dic53C2211yescharacteristic
22C2×C4×Dic53C2211yescharacteristic
23#13133124C2212yes—
24#13133124C2212yes—
25C22×D4×C53C2211yescharacteristic
26C2×C4⋊C4×C53C2211yescharacteristic
27C2×C22⋊C4×C53C2214yes—
28C2×C42×C53C2211yescharacteristic
29C24×C53C2312yes—
30C23×D53C2312yes—
31C2×D4×C53—22no—
32C2×D4×C53—22no—
33C2×D4×C53—22no—
34C22×C4×C53C2312yes—
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C2×C22⋊D53—24no—
38C2×C22⋊D53—24no—
39C2×C22⋊D53—44no—
40C2×C22⋊D53—44no—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22×Dic53C2311yescharacteristic
44C22×Dic53C2311yescharacteristic
45C2×D212—22no—
46C2×D212—22no—
47C2×D212—22no—
48C4⋊C4×C53—22no—
49C22⋊C4×C53—44no—
50C42×C53—22no—
51C22⋊Dic53—44no—
52#11719921—44no—
53#11719921—44no—
54C4⋊Dic53—22no—
55C4⋊Dic53—24no—
56#11719919—24no—
57C4×Dic53—22no—
58C4×Dic53D414yes—
59C23×C53—22no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53—24no—
63C23×C53—24no—
64C23×C53C2411yescharacteristic
65C22×D53—22no—
66C22×D53—22no—
67C22×D53—22no—
68C22×D53—24no—
69C22×D53—24no—
70D4×C53—44no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—24no—
75C2×C4×C53—22no—
76C2×C4×C53C2×D412yes—
77C2×C4×C53#4112yes—
78C2×C4×C53#4112yes—
79C22⋊D53—48no—
80C22⋊D53—48no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—24no—
87C2×Dic53—22no—
88C2×Dic53C2×D412yes—
89C2×Dic53C2×D412yes—
90D212—44no—
91C22×C53—42no—
92C22×C53—42no—
93C22×C53—42no—
94C22×C53—44no—
95C22×C53—44no—
96C22×C53C22×D411yescharacteristic
97C22×C53C2×#4111yescharacteristic
98C22×C53C2×#4111yescharacteristic
99C22×C53H(5,𝔽2)12yes—
100C22×C53H(5,𝔽2)12yes—
101C2×D53—42no—
102C2×D53—42no—
103C2×D53—42no—
104C2×D53—44no—
105C2×D53—44no—
106C4×C53—42no—
107C4×C53—42no—
108C4×C53#12414yes—
109Dic53—22no—
110Dic53—22no—
111Dic53—42no—
112Dic53—42no—
113#2641—531nomaximal, Sylow
114C2×C53—82no—
115C2×C53C2×#12411yescharacteristic
116C2×C53#53511yescharacteristic
117C2×C53#53511yescharacteristic
118C2×C53#54012yes—
119C2×C53#54012yes—
120D53—82no—
121C2×#127—531no—
122C2×#125—531no—
123#394—534no—
124#394—534no—
125#390—531no—
126#390—531no—
127#390—531no—
128#390D5311yescharacteristic, complemented
129#376—531no—
130C53#264111yescharacteristic, complemented, Sylow
131C22×D4—532no—
132C22×D4—532no—
133C22×D4—531no—
134C22×D4C2×D5311yescharacteristic
135#127—1064no—
136#125—1064no—
137C2×C4⋊C4—532no—
138C2×C4⋊C4—531no—
139C2×C4⋊C4—532no—
140C2×C4⋊C4C2×D5311yescharacteristic
141C2×C22⋊C4—534no—
142C2×C22⋊C4—534no—
143C2×C22⋊C4—534no—
144C2×C22⋊C4C2×D5314yes—
145C2×C42—531no—
146C2×C42—531no—
147C2×C42C2×D5311yescharacteristic
148#95—532no—
149#95—532no—
150C24—532no—
151C24C22×D5312yes—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—1064no—
156C2×D4—1064no—
157C2×D4—1062no—
158C2×D4—1062no—
159C2×D4—1062no—
160C2×D4—2124no—
161C2×D4—2124no—
162C22×C4—531no—
163C22×C4—531no—
164C22×C4—532no—
165C22×C4—532no—
166C22×C4C22×D5312yes—
167C22×C4C22×D5312yes—
168C22×C4C22×D5311yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—1064no—
171C4⋊C4—1064no—
172C4⋊C4—1062no—
173C22⋊C4—44no—
174C22⋊C4—2124no—
175C22⋊C4—2124no—
176C22⋊C4—2124no—
177C42—22no—
178C42—534no—
179C42—1062no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—24no—
184C23—24no—
185C23—1062no—
186C23—1062no—
187C23—1062no—
188C23—1064no—
189C23—1064no—
190C23C23×D5311yescharacteristic
191D4—44no—
192D4—2128no—
193D4—2128no—
194D4—2124no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—532no—
201C2×C4—532no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—1064no—
208C2×C4—1062no—
209C2×C4#1171994512yes—
210C2×C4D4×D5312yes—
211C2×C4#1171994712yes—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—44no—
216C22—44no—
217C22—2122no—
218C22—2122no—
219C22—2122no—
220C22—2124no—
221C22—2124no—
222C22C2×D4×D5311yescharacteristic
223C22C2×#1171994711yescharacteristic
224C22C2×#1171994511yescharacteristic
225C22#1313326212yes—
226C22#1313326312yes—
227C4—42no—
228C4—42no—
229C4—1062no—
230C4—1062no—
231C4—2122no—
232C4—2122no—
233C4#1313320314yes—
234C2—82no—
235C2—4242no—
236C2C2×#1313320311yescharacteristic
237C2#1681118611yescharacteristic
238C2#1681119111yescharacteristic
239C2#1681122412yes—
240C2#1681122612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#390C2
hypercenter#390D53
hypocenterC53#2641
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#13133208C2semidirect
3. #16810539C2semidirect
4. #16810538C2semidirect
5. #16810526C2semidirect
6. #16810525C2semidirect
7. #16810524C2semidirect
8. #16810469C2semidirect
9. C53×#390C2semidirect
128. #390D53semidirect
130. C53#2641semidirect