Finite Group Databasebeta

← #20277887

Subgroups — #20277887

Order 6784.

520 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13133206C211yesmaximal, characteristic
2#16810530C211yesmaximal, characteristic, complemented
3#16810514C212yesmaximal, complemented
4#16810512C212yesmaximal, complemented
5#16810501C212yesmaximal
6#16810497C212yesmaximal, complemented
7#16810494C212yesmaximal, complemented
8#16810487C212yesmaximal, complemented
9C53×#391C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C4×D53C2212yes—
12#13133206C418yescomplemented
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719921C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×C4×Dic53C2211yescharacteristic
22C2×C4×Dic53C2211yescharacteristic
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27C22×Q8×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C42×C53C2211yescharacteristic
30#95×C53C2212yes—
31#95×C53C2212yes—
32C23×D53C2312yes—
33C2×Q8×C53—41no—
34C2×Q8×C53—41no—
35C2×Q8×C53C2×C414yescomplemented
36C22×C4×C53C2312yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C2×D212—41no—
45C2×D212—41no—
46C2×D212C2×C414yes—
47C2×C4×D53—42no—
48C2×C4×D53—42no—
49C2×C4×D53—42no—
50C4⋊C4×C53—41no—
51C42×C53—41no—
52#11719921—42no—
53#11719921—42no—
54#11719921—42no—
55#11719921C2×C418yescomplemented
56#11719921C2×C418yescomplemented
57C4⋊Dic53—41no—
58#11719919—42no—
59#11719919—42no—
60C4×Dic53—41no—
61C4×Dic53C2×C414yescomplemented
62C23×C53C2411yescharacteristic
63C22×D53—22no—
64C22×D53—22no—
65C22×D53—22no—
66C22×D53—22no—
67C22×D53—22no—
68C22×D53—22no—
69C22×D53C22×C414yes—
70Q8×C53—44no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53C22×C412yes—
84C2×C4×C53C22×C412yes—
85C2×C4×C53C22×C412yes—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53C22×C414yes—
95D212—44no—
96C4×D53—48no—
97C22×C53C23×C411yescharacteristic
98C22×C53H(5,𝔽2)11yescharacteristic
99C22×C53H(5,𝔽2)11yescharacteristic
100C22×C53H(5,𝔽2)12yes—
101C22×C532−1+412yes—
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—24no—
115Dic53—42no—
116Dic53—42no—
117#2379—531nomaximal, Sylow
118C2×C53#51811yescharacteristic
119C2×C53#52012yes—
120C2×C53H(3,𝔽4)11yescharacteristic
121C2×C53#56212yes—
122C2×C53#56311yescharacteristic
123D53—44no—
124C2×#124—531no—
125#391—531no—
126#391D5311yescharacteristic, complemented
127#388—532no—
128#388—532no—
129#386—532no—
130#385—532no—
131#385—532no—
132#382—532no—
133C53#237911yescharacteristic, complemented, Sylow
134C22×Q8C2×D5311yescharacteristic
135C22×D4—531no—
136C23×C4—532no—
137#124—538no—
138C2×C4⋊C4—532no—
139C2×C4⋊C4—532no—
140C2×C4⋊C4—531no—
141C2×C4⋊C4C2×D5311yescharacteristic
142C2×C22⋊C4—532no—
143C2×C22⋊C4—532no—
144C2×C22⋊C4—532no—
145C2×C22⋊C4—532no—
146C2×C22⋊C4—532no—
147C2×C42—531no—
148C2×C42—531no—
149C2×C42C2×D5311yescharacteristic
150#95—532no—
151#95—532no—
152#95—532no—
153#95—532no—
154#95C2×D5312yes—
155#95C2×D5312yes—
156C24—532no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8C4×D5314yescomplemented
160C2×D4—534no—
161C2×D4—2121no—
162C2×D4—2121no—
163C22×C4—532no—
164C22×C4—532no—
165C22×C4—532no—
166C22×C4—2122no—
167C22×C4—2122no—
168C22×C4—2122no—
169C22×C4C22×D5312yes—
170C22×C4C22×D5312yes—
171C22×C4C22×D5311yescharacteristic
172C22×C4C22×D5311yescharacteristic
173C22×C4C22×D5311yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—2122no—
176C4⋊C4—2122no—
177C4⋊C4—2121no—
178C22⋊C4—538no—
179C22⋊C4—538no—
180C22⋊C4—2122no—
181C22⋊C4—2122no—
182C22⋊C4—2122no—
183C42—41no—
184C42—534no—
185C42—2121no—
186C23—534no—
187C23—1062no—
188C23—1062no—
189C23—1062no—
190C23—1062no—
191C23—1062no—
192C23—1062no—
193C23C23×D5311yescharacteristic
194Q8—44no—
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—534no—
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×C4C2×C4×D5312yes—
219C2×C4C2×C4×D5312yes—
220C2×C4C2×C4×D5312yes—
221C22—1064no—
222C22—1064no—
223C22—1064no—
224C22—2122no—
225C22—2122no—
226C22—2122no—
227C22—2122no—
228C22C22×C4×D5311yescharacteristic
229C22#1313326211yescharacteristic
230C22#1313326211yescharacteristic
231C22#1313326312yes—
232C22#1313326512yes—
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—1064no—
239C4—2122no—
240C4—2122no—
241C2—2124no—
242C2#1681109611yescharacteristic
243C2#1681110512yes—
244C2#1681136511yescharacteristic
245C2#1681136912yes—
246C2#1681137211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#391C2
socleC23×C53C24
hypercenter#391D53
hypocenterC53#2379
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #16810530C2semidirect
3. #16810514C2semidirect
4. #16810512C2semidirect
6. #16810497C2semidirect
7. #16810494C2semidirect
8. #16810487C2semidirect
9. C53×#391C2semidirect
12. #13133206C4semidirect
35. C2×Q8×C53C2×C4semidirect
55. #11719921C2×C4semidirect
56. #11719921C2×C4semidirect
61. C4×Dic53C2×C4semidirect
126. #391D53semidirect
133. C53#2379semidirect
159. C2×Q8C4×D53semidirect