Finite Group Databasebeta

← #20274702

Subgroups — #20274702

Order 6784.

380 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 87 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×#13133177C211yesmaximal, characteristic
2C2×#13133131C212yesmaximal
3#16810481C212yesmaximal, complemented
4#16810390C211yesmaximal, characteristic, complemented
5C53×#342C211yesmaximal, characteristic, complemented
6C22×C22⋊D53C2211yescharacteristic
7C23×Dic53C2211yescharacteristic
8C22×C4×D53C411yescharacteristic, complemented
9#13133177—22no—
10#13133177—22no—
11C2×#11719919—22no—
12C2×#11719919C411yescharacteristic, complemented
13C2×C22⋊Dic53C2212yes—
14C2×#11719921C411yescharacteristic, complemented
15C2×C4×Dic53C411yescharacteristic, complemented
16#13133131—24no—
17#13133124—22no—
18C2×C22⋊C4×C53C2212yes—
19C2×C22⋊C4×C53C2211yescharacteristic
20C24×C53C2311yescharacteristic
21C23×D53C2×C411yescharacteristic
22C22×C4×C53—22no—
23C22×C4×C53C2×C411yescharacteristic
24C22×C4×C53C2×C411yescharacteristic
25C2×C22⋊D53—22no—
26C2×C22⋊D53—22no—
27C2×C22⋊D53D412yes—
28C2×C22⋊D53Q812yes—
29C22×Dic53—22no—
30C22×Dic53—42no—
31C22×Dic53C2×C411yescharacteristic
32C22×Dic53C2×C411yescharacteristic
33C22×Dic53C2×C411yescharacteristic
34C22×Dic53D412yes—
35C22×Dic53D412yes—
36C2×C4×D53—21no—
37C2×C4×D53—21no—
38C2×C4×D53—42no—
39C22⋊C4×C53—22no—
40C22⋊C4×C53—22no—
41C22⋊C4×C53—22no—
42C22⋊Dic53—22no—
43C22⋊Dic53—22no—
44#11719921—22no—
45#11719919—22no—
46#11719919—44no—
47C4×Dic53—21no—
48C4×Dic53—21no—
49C23×C53—41no—
50C23×C53—41no—
51C23×C53C22×C411yescharacteristic
52C23×C53C2×D412yes—
53C23×C53C2×D411yescharacteristic
54C23×C53C2×Q811yescharacteristic
55C23×C53#4112yes—
56C22×D53—21no—
57C22×D53—21no—
58C22×D53—41no—
59C22×D53—41no—
60C22×D53C4⋊C412yescomplemented
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—42no—
66C2×C4×C53—42no—
67C2×C4×C53—42no—
68C22⋊D53—24no—
69C22⋊D53—24no—
70C2×Dic53—21no—
71C2×Dic53—21no—
72C2×Dic53—22no—
73C2×Dic53—24no—
74C2×Dic53—41no—
75C2×Dic53—42no—
76C2×Dic53—42no—
77C2×Dic53—42no—
78C2×Dic53—42no—
79C2×Dic53C22⋊C411yescharacteristic
80C2×Dic53C22⋊C412yes—
81C2×Dic53C22⋊C411yescharacteristic
82C2×Dic53C4⋊C412yescomplemented
83C4×D53—42no—
84C4×D53—42no—
85C22×C53—22no—
86C22×C53—21no—
87C22×C53—22no—
88C22×C53—21no—
89C22×C53—41no—
90C22×C53—41no—
91C22×C53—42no—
92C22×C53C2×C22⋊C411yescharacteristic
93C22×C53C2×C4⋊C411yescharacteristic
94C22×C53#11711yescharacteristic
95C22×C53#12112yes—
96C22×C53C22⋊Q812yes—
97C2×D53—22no—
98C2×D53—42no—
99C2×D53—41no—
100C2×D53—41no—
101C2×D53—42no—
102C4×C53—41no—
103C4×C53—41no—
104C4×C53—82no—
105Dic53—22no—
106Dic53—21no—
107Dic53—22no—
108Dic53—82no—
109Dic53#9912yescomplemented
110#1799—531nomaximal, Sylow
111C2×C53—22no—
112C2×C53—21no—
113C2×C53—21no—
114C2×C53—22no—
115C2×C53#38011yescharacteristic
116C2×C53C2×#9911yescharacteristic
117C2×C53#41011yescharacteristic
118D53—42no—
119C2×C4×D4—531no—
120C2×#99—532no—
121#380—532no—
122#342—531no—
123#342D5311yescharacteristic, complemented
124C53#179911yescharacteristic, complemented, Sylow
125C22×D4—531no—
126C23×C4—531no—
127C23×C4—531no—
128C4×D4—1062no—
129C4×D4—1062no—
130C2×C4⋊C4—531no—
131C2×C4⋊C4—1062no—
132C2×C22⋊C4—531no—
133C2×C22⋊C4—532no—
134C2×C22⋊C4C2×D5312yes—
135C2×C22⋊C4C2×D5311yescharacteristic
136C2×C42—531no—
137#99—1064no—
138#95—1062no—
139C24—531no—
140C24C22×D5311yescharacteristic
141C2×D4—532no—
142C2×D4—532no—
143C2×D4—1062no—
144C2×D4—1062no—
145C22×C4—22no—
146C22×C4—531no—
147C22×C4—531no—
148C22×C4—532no—
149C22×C4—531no—
150C22×C4—532no—
151C22×C4—1061no—
152C22×C4—1061no—
153C22×C4—1062no—
154C22×C4—2122no—
155C22×C4—2122no—
156C22×C4C4×D5311yescharacteristic
157C22×C4C4×D5311yescharacteristic
158C4⋊C4—1062no—
159C4⋊C4—2124no—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—1062no—
164C22⋊C4—1062no—
165C22⋊C4—1062no—
166C42—1061no—
167C42—1061no—
168C23—41no—
169C23—41no—
170C23—532no—
171C23—1061no—
172C23—1061no—
173C23—2121no—
174C23—2121no—
175C23C2×C4×D5311yescharacteristic
176C23#1171994512yes—
177C23D4×D5312yes—
178C23D4×D5311yescharacteristic
179C23Q8×D5311yescharacteristic
180D4—1064no—
181D4—1064no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—531no—
190C2×C4—531no—
191C2×C4—532no—
192C2×C4—532no—
193C2×C4—1064no—
194C2×C4—1062no—
195C2×C4—1061no—
196C2×C4—1061no—
197C2×C4—2122no—
198C2×C4—2122no—
199C2×C4—2122no—
200C2×C4—2121no—
201C2×C4—2122no—
202C2×C4—2122no—
203C2×C4—2122no—
204C22—22no—
205C22—21no—
206C22—22no—
207C22—21no—
208C22—42no—
209C22—41no—
210C22—41no—
211C22—1062no—
212C22—2122no—
213C22—2122no—
214C22—2121no—
215C22—2121no—
216C22C22⋊C4×D5311yescharacteristic
217C22C4⋊C4×D5311yescharacteristic
218C22#1313317411yescharacteristic
219C22#1313319012yes—
220C22#1313319512yes—
221C4—41no—
222C4—41no—
223C4—82no—
224C4—532no—
225C4—1061no—
226C4—1062no—
227C4—1062no—
228C4—4242no—
229C2—22no—
230C2—21no—
231C2—21no—
232C2—22no—
233C2—2122no—
234C2#1681048211yescharacteristic
235C2#99×D5311yescharacteristic
236C2#1681059511yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC53×#342C2
socleC22×C53C2×C22⋊C4
hypercenter#342D53
hypocenterC53#1799
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #16810481C2semidirect
4. #16810390C2semidirect
5. C53×#342C2semidirect
8. C22×C4×D53C4semidirect
12. C2×#11719919C4semidirect
14. C2×#11719921C4semidirect
15. C2×C4×Dic53C4semidirect
60. C22×D53C4⋊C4semidirect
82. C2×Dic53C4⋊C4semidirect
109. Dic53#99semidirect
123. #342D53semidirect
124. C53#1799semidirect