Finite Group Databasebeta

← #20274701

Subgroups — #20274701

Order 6784.

380 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 95 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×D4×Dic53C211yesmaximal, characteristic
2C2×#13133132C211yesmaximal, characteristic
3#16810481C212yesmaximal, complemented
4#16810389C212yesmaximal, complemented
5C2×#99×C53C211yesmaximal, characteristic
6C23×Dic53C411yescharacteristic, complemented
7C23×Dic53C2211yescharacteristic
8D4×Dic53—22no—
9D4×Dic53—22no—
10C2×C4⋊Dic53C411yescharacteristic, complemented
11C2×#11719919—22no—
12C2×C22⋊Dic53C411yescharacteristic, complemented
13C2×C22⋊Dic53C2212yes—
14C2×C22⋊Dic53C2211yescharacteristic
15C2×C4×Dic53C411yescharacteristic, complemented
16#13133132—22no—
17#13133124—22no—
18C22×D4×C53C2211yescharacteristic
19C2×C22⋊C4×C53C2212yes—
20#99×C53—22no—
21C24×C53C2×C411yescharacteristic
22C24×C53C2311yescharacteristic
23C2×D4×C53—22no—
24C2×D4×C53—22no—
25C2×D4×C53D412yes—
26C2×D4×C53Q812yes—
27C22×C4×C53—22no—
28C22×C4×C53C2×C411yescharacteristic
29C22×Dic53—21no—
30C22×Dic53—21no—
31C22×Dic53—22no—
32C22×Dic53—42no—
33C22×Dic53—42no—
34C22×Dic53C2×C411yescharacteristic
35C22×Dic53C2×C411yescharacteristic
36C22×Dic53C2×C411yescharacteristic
37C22×Dic53C2×C411yescharacteristic
38C22×Dic53D412yes—
39C22×Dic53D412yes—
40C22⋊C4×C53—22no—
41C22⋊C4×C53—22no—
42C22⋊Dic53—22no—
43C22⋊Dic53—22no—
44C22⋊Dic53—22no—
45C22⋊Dic53—22no—
46C4⋊Dic53—22no—
47#11719919—44no—
48C4×Dic53—21no—
49C4×Dic53—21no—
50C23×C53—21no—
51C23×C53—21no—
52C23×C53—41no—
53C23×C53—41no—
54C23×C53—41no—
55C23×C53—41no—
56C23×C53C4⋊C412yescomplemented
57C23×C53C22×C411yescharacteristic
58C23×C53C2×D412yes—
59C23×C53C2×D411yescharacteristic
60C23×C53C2×Q811yescharacteristic
61C23×C53#4112yes—
62D4×C53—24no—
63D4×C53—24no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—42no—
67C2×C4×C53—42no—
68C2×C4×C53—42no—
69C2×C4×C53C4⋊C412yescomplemented
70C2×Dic53—22no—
71C2×Dic53—24no—
72C2×Dic53—21no—
73C2×Dic53—21no—
74C2×Dic53—41no—
75C2×Dic53—42no—
76C2×Dic53—42no—
77C2×Dic53—42no—
78C2×Dic53—41no—
79C2×Dic53—41no—
80C2×Dic53—42no—
81C2×Dic53—42no—
82C2×Dic53—42no—
83C2×Dic53C22⋊C411yescharacteristic
84C2×Dic53C22⋊C412yes—
85C2×Dic53C22⋊C411yescharacteristic
86C22×C53—22no—
87C22×C53—21no—
88C22×C53—22no—
89C22×C53—21no—
90C22×C53—22no—
91C22×C53—41no—
92C22×C53—41no—
93C22×C53—42no—
94C22×C53—42no—
95C22×C53—42no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53C2×C22⋊C411yescharacteristic
99C22×C53C2×C4⋊C411yescharacteristic
100C22×C53#11711yescharacteristic
101C22×C53#12112yes—
102C22×C53C22⋊Q812yes—
103C4×C53—22no—
104C4×C53—82no—
105Dic53—21no—
106Dic53—22no—
107Dic53—41no—
108Dic53—41no—
109Dic53—82no—
110Dic53#9912yescomplemented
111#1799—531nomaximal, Sylow
112C2×C53—22no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53—22no—
116C2×C53—42no—
117C2×C53#38011yescharacteristic
118C2×C53C2×#9911yescharacteristic
119C2×C53#41011yescharacteristic
120C2×C4×D4—531no—
121C2×#99—531no—
122C2×#99D5311yescharacteristic
123#380—532no—
124#342—532no—
125C53#179911yescharacteristic, complemented, Sylow
126C22×D4C2×D5311yescharacteristic
127C23×C4—531no—
128C23×C4—531no—
129C4×D4—1062no—
130C4×D4—1062no—
131C2×C4⋊C4—531no—
132C2×C4⋊C4—1062no—
133C2×C22⋊C4—531no—
134C2×C22⋊C4—531no—
135C2×C22⋊C4—532no—
136C2×C22⋊C4C2×D5312yes—
137C2×C42—531no—
138#99—22no—
139#99—1062no—
140#95—1062no—
141C24C4×D5311yescharacteristic
142C24C22×D5311yescharacteristic
143C2×D4—22no—
144C2×D4—22no—
145C2×D4Dic10612yes—
146C2×D4C22⋊D5312yes—
147C22×C4—22no—
148C22×C4—531no—
149C22×C4—532no—
150C22×C4—531no—
151C22×C4—532no—
152C22×C4—531no—
153C22×C4—531no—
154C22×C4—1061no—
155C22×C4—1061no—
156C22×C4—1062no—
157C22×C4—2122no—
158C22×C4—2122no—
159C22×C4C4×D5311yescharacteristic
160C4⋊C4—1062no—
161C4⋊C4—2124no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—1062no—
165C22⋊C4—1062no—
166C22⋊C4—1062no—
167C22⋊C4—1062no—
168C42—1061no—
169C42—1061no—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23#1171991912yescomplemented
177C23C2×Dic10611yescharacteristic
178C23C2×C4×D5311yescharacteristic
179C23D4×D5312yes—
180C23#1171994712yes—
181C23C2×C22⋊D5311yescharacteristic
182D4—24no—
183D4—24no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—531no—
190C2×C4—531no—
191C2×C4—532no—
192C2×C4—1064no—
193C2×C4—1062no—
194C2×C4—1061no—
195C2×C4—1061no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4—2121no—
199C2×C4—2122no—
200C2×C4—2122no—
201C2×C4—2122no—
202C2×C4—2122no—
203C2×C4—2121no—
204C2×C4—2121no—
205C2×C4#1171991912yescomplemented
206C22—22no—
207C22—21no—
208C22—22no—
209C22—21no—
210C22—22no—
211C22—41no—
212C22—42no—
213C22—41no—
214C22—42no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22C22⋊C4×D5311yescharacteristic
219C22C2×#1171991911yescharacteristic
220C22#1313317311yescharacteristic
221C22#1313318912yes—
222C22#1313319412yes—
223C4—22no—
224C4—82no—
225C4—532no—
226C4—1061no—
227C4—1062no—
228C4—4242no—
229C4—2121no—
230C4—2121no—
231C2—22no—
232C2—21no—
233C2—21no—
234C2—22no—
235C2—42no—
236C2#1681048111yescharacteristic
237C2#99×D5311yescharacteristic
238C2#1681059511yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC2×#99×C53C2
socleC22×C53C2×C22⋊C4
hypercenterC2×#99D53
hypocenterC53#1799
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
3. #16810481C2semidirect
4. #16810389C2semidirect
6. C23×Dic53C4semidirect
10. C2×C4⋊Dic53C4semidirect
12. C2×C22⋊Dic53C4semidirect
15. C2×C4×Dic53C4semidirect
56. C23×C53C4⋊C4semidirect
69. C2×C4×C53C4⋊C4semidirect
110. Dic53#99semidirect
125. C53#1799semidirect
176. C23#11719919semidirect
205. C2×C4#11719919semidirect