Finite Group Databasebeta

← #20274788

Subgroups — #20274788

Order 6784.

504 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 81 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×C22⋊C4×D53C211yesmaximal, characteristic
2#16810478C212yesmaximal
3#16810390C211yesmaximal, characteristic, complemented
4#16810389C212yesmaximal, complemented
5C53×#342C211yesmaximal, characteristic, complemented
6C24×D53C2211yescharacteristic
7C22×C4×D53C412yescomplemented
8C2×C22⋊Dic53C2212yes—
9C2×C22⋊Dic53C2211yescharacteristic
10C2×#11719921—24no—
11C2×#11719921C412yescomplemented
12C22⋊C4×D53—24no—
13C2×C22⋊C4×C53C2212yes—
14C2×C22⋊C4×C53C2211yescharacteristic
15C24×C53C2311yescharacteristic
16C23×D53—22no—
17C23×D53—21no—
18C23×D53—21no—
19C23×D53—44no—
20C23×D53C2×C411yescharacteristic
21C23×D53C2×C411yescharacteristic
22C23×D53D414yes—
23C22×C4×C53—22no—
24C22×C4×C53C2×C412yes—
25C22×Dic53—22no—
26C22×Dic53C2×C412yes—
27C2×C4×D53—22no—
28C2×C4×D53—22no—
29C2×C4×D53—44no—
30C22⋊C4×C53—22no—
31C22⋊C4×C53—22no—
32C22⋊C4×C53—22no—
33C22⋊Dic53—22no—
34C22⋊Dic53—22no—
35C22⋊Dic53—22no—
36#11719921—24no—
37#11719921—48no—
38C23×C53—42no—
39C23×C53C22×C411yescharacteristic
40C23×C53C2×D412yes—
41C23×C53#4112yes—
42C23×C53#4111yescharacteristic
43C23×C53#4111yescharacteristic
44C22×D53—21no—
45C22×D53—21no—
46C22×D53—24no—
47C22×D53—21no—
48C22×D53—21no—
49C22×D53—24no—
50C22×D53—24no—
51C22×D53—28no—
52C22×D53—42no—
53C22×D53—44no—
54C22×D53—48no—
55C22×D53—44no—
56C22×D53—44no—
57C22×D53C22⋊C412yes—
58C22×D53C22⋊C412yes—
59C2×C4×C53—22no—
60C2×C4×C53—42no—
61C2×C4×C53—44no—
62C2×C4×C53—42no—
63C2×Dic53—22no—
64C2×Dic53—42no—
65C2×Dic53—44no—
66C2×Dic53—42no—
67C4×D53—44no—
68C4×D53—44no—
69C22×C53—22no—
70C22×C53—24no—
71C22×C53—42no—
72C22×C53—41no—
73C22×C53—41no—
74C22×C53C2×C22⋊C411yescharacteristic
75C22×C53#11711yescharacteristic
76C22×C53#11711yescharacteristic
77C22×C53#12312yes—
78C22×C53#12312yes—
79C2×D53—24no—
80C2×D53—24no—
81C2×D53—22no—
82C2×D53—22no—
83C2×D53—22no—
84C2×D53—24no—
85C2×D53—48no—
86C2×D53—48no—
87C2×D53—42no—
88C2×D53—42no—
89C2×D53#9914yes—
90C4×C53—42no—
91C4×C53—82no—
92Dic53—42no—
93Dic53—82no—
94#1819—531nomaximal, Sylow
95C2×C53—22no—
96C2×C53—22no—
97C2×C53—21no—
98C2×C53—21no—
99C2×C53#38111yescharacteristic
100C2×C53C2×#9912yes—
101D53—24no—
102D53—42no—
103C22×C22⋊C4—531no—
104#379—532no—
105#342—532no—
106#342—531no—
107#342D5311yescharacteristic, complemented
108C53#181911yescharacteristic, complemented, Sylow
109C25—531no—
110C23×C4—532no—
111C2×C22⋊C4—531no—
112C2×C22⋊C4—532no—
113C2×C22⋊C4—532no—
114C2×C22⋊C4—1064no—
115C2×C22⋊C4—1064no—
116C2×C22⋊C4C2×D5312yes—
117C2×C22⋊C4C2×D5311yescharacteristic
118C24—531no—
119C24—534no—
120C24—531no—
121C24—1062no—
122C24—1061no—
123C24—1061no—
124C24—2124no—
125C24C22×D5311yescharacteristic
126C22×C4—22no—
127C22×C4—532no—
128C22×C4—1062no—
129C22×C4—1062no—
130C22×C4—1062no—
131C22×C4—2124no—
132C22×C4C4×D5312yes—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—1064no—
137C22⋊C4—1062no—
138C22⋊C4—1062no—
139C22⋊C4—1062no—
140C22⋊C4—2128no—
141C23—42no—
142C23—532no—
143C23—532no—
144C23—1064no—
145C23—1068no—
146C23—1061no—
147C23—1061no—
148C23—1061no—
149C23—1061no—
150C23—1064no—
151C23—1064no—
152C23—2122no—
153C23—2124no—
154C23—2128no—
155C23—2124no—
156C23—2124no—
157C23C2×C4×D5311yescharacteristic
158C23#1171994512yes—
159C23D4×D5312yes—
160C23#1171994711yescharacteristic
161C23#1171994911yescharacteristic
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—42no—
166C2×C4—1062no—
167C2×C4—2124no—
168C2×C4—2124no—
169C2×C4—2124no—
170C2×C4—2122no—
171C2×C4—2122no—
172C22—22no—
173C22—24no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—534no—
178C22—1064no—
179C22—1064no—
180C22—1062no—
181C22—1062no—
182C22—1064no—
183C22—1062no—
184C22—2128no—
185C22—2128no—
186C22—2122no—
187C22—2122no—
188C22C22⋊C4×D5311yescharacteristic
189C22#1313317411yescharacteristic
190C22#1313317511yescharacteristic
191C22#1313319912yes—
192C22#1313320212yes—
193C4—42no—
194C4—82no—
195C4—4242no—
196C4—2122no—
197C2—22no—
198C2—22no—
199C2—21no—
200C2—21no—
201C2—1064no—
202C2—2122no—
203C2#1681048511yescharacteristic
204C2#99×D5312yes—
subgroupG/H
centerC22C22⋊C4×D53
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC53×#342C2
socleC22×C53C2×C22⋊C4
hypercenter#342D53
hypocenterC53#1819
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. #16810390C2semidirect
4. #16810389C2semidirect
5. C53×#342C2semidirect
7. C22×C4×D53C4semidirect
11. C2×#11719921C4semidirect
107. #342D53semidirect
108. C53#1819semidirect