Finite Group Databasebeta

← #20290120

Subgroups — #20290120

Order 6784.

326 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 60 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
1#16811595C211yesmaximal, characteristic, complemented
2#16811566C211yesmaximal, characteristic, complemented
3#16811474C214yesmaximal, complemented
4C2×#13133188C211yesmaximal, characteristic
5C2×#11719940—21no—
6C2×C22⋊#6157—21no—
7#13133274—21no—
8#13133274C2212yescomplemented
9(C2×#6157)⋊C4—21no—
10(C2×#6157)⋊C4C2212yescomplemented
11C22×C22⋊D53—21no—
12C22×C4×D53C2211yescharacteristic
13#13133188—21no—
14#13133188—21no—
15#13133188C412yescomplemented
16#13133188C2212yes—
17C2×C4⋊Dic53C411yescharacteristic, complemented
18C2×C22⋊Dic53—21no—
19C22×D4×C53C411yescharacteristic, complemented
20C24×C53—21no—
21C23×D53D411yescharacteristic
22C22×#6157—22no—
23C2×D4×C53—21no—
24C2×D4×C53—42no—
25C2×D4×C53C2×C412yes—
26C22×C4×C53C2×C411yescharacteristic
27C2×C22⋊D53—22no—
28C2×C22⋊D53—22no—
29C2×C22⋊D53—44no—
30C22×Dic53—21no—
31C22×Dic53D411yescharacteristic
32C2×C4×D53—42no—
33C2×C4×D53C2×C411yescharacteristic
34C2×C4×D53D411yescharacteristic
35C2×C4×D53D411yescharacteristic
36C2×C4×D53C2311yescharacteristic
37C22⋊#6157—42no—
38#11719940—42no—
39C2×#88324—22no—
40C22⋊Dic53—22no—
41C22⋊Dic53—22no—
42C4⋊Dic53—21no—
43C4⋊Dic53C2×C412yes—
44C23×C53—22no—
45C23×C53—42no—
46C23×C53—42no—
47C23×C53—42no—
48C23×C53C22⋊C411yescharacteristic
49C22×D53—22no—
50C22×D53—22no—
51C22×D53—41no—
52C22×D53C22⋊C411yescharacteristic
53C22×D53C2×D411yescharacteristic
54C2×#6157—42no—
55C2×#6157—42no—
56C2×#6157—42no—
57D4×C53—42no—
58D4×C53—42no—
59C2×C4×C53—22no—
60C2×C4×C53C22⋊C411yescharacteristic
61C2×C4×C53C22×C411yescharacteristic
62C22⋊D53—44no—
63C22⋊D53—44no—
64C2×Dic53—22no—
65C2×Dic53—22no—
66C2×Dic53—42no—
67C2×Dic53C22⋊C411yescharacteristic
68C2×Dic53C2×D411yescharacteristic
69C4×D53—41no—
70C4×D53—41no—
71C4×D53—41no—
72C4×D53—41no—
73#88324—42no—
74C22×C53—21no—
75C22×C53—21no—
76C22×C53—42no—
77C22×C53—42no—
78C22×C53—42no—
79C22×C53—84no—
80C22×C53#9912yes—
81C22×C53C2×C22⋊C411yescharacteristic
82C2×D53—21no—
83C2×D53—21no—
84C2×D53—21no—
85C2×D53—21no—
86C2×D53—21no—
87C2×D53—44no—
88C2×D53AGL(1,ℤ/8)12yes—
89#6157—82no—
90C4×C53—21no—
91C4×C53—21no—
92Dic53—21no—
93Dic53—21no—
94Dic53—42no—
95#1510—531nomaximal, Sylow
96C2×C53—22no—
97C2×C53—82no—
98C2×C53C2×#9911yescharacteristic
99C2×C53#41811yescharacteristic
100C2×C53#42111yescharacteristic
101D53—22no—
102D53—41no—
103C2×#121—531no—
104#407—531no—
105#381—531no—
106#327—534no—
107C53#151011yescharacteristic, complemented, Sylow
108C22×D4—1061no—
109C22×D4#615711yescharacteristic, complemented
110C23×C4—531no—
111C2×M16—1061no—
112#121—532no—
113#121—532no—
114#121—1061no—
115#121—1061no—
116C2×C4⋊C4—531no—
117C2×C22⋊C4—1061no—
118C2×C22⋊C4—1061no—
119C22⋊C8—532no—
120C22⋊C8—1061no—
121#95—532no—
122#95—1061no—
123C24—21no—
124C24—531no—
125C2×D4—21no—
126C2×D4—42no—
127C2×D4—1062no—
128C2×D4—1062no—
129C2×D4—2124no—
130C2×D4C2×#615712yes—
131C22×C4—531no—
132C22×C4—531no—
133C22×C4—531no—
134C22×C4—531no—
135C22×C4—531no—
136C22×C4—1061no—
137C22×C4—1062no—
138C22×C4—2122no—
139C22×C4C2×#615711yescharacteristic
140M16—2122no—
141C2×C8—1062no—
142C4⋊C4—532no—
143C4⋊C4—1061no—
144C22⋊C4—1062no—
145C22⋊C4—1062no—
146C22⋊C4—2122no—
147C23—22no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23—531no—
152C23—531no—
153C23—1062no—
154C23—1062no—
155C23—2121no—
156C23C22⋊#615711yescharacteristic
157D4—42no—
158D4—42no—
159D4—2124no—
160D4—2124no—
161C2×C4—22no—
162C2×C4—531no—
163C2×C4—531no—
164C2×C4—1062no—
165C2×C4—1062no—
166C2×C4—2121no—
167C2×C4—2121no—
168C2×C4—2121no—
169C2×C4—2121no—
170C2×C4—2122no—
171C2×C4—2122no—
172C2×C4—2122no—
173C2×C4—2122no—
174C2×C4C22⋊#615711yescharacteristic
175C2×C4C22×#615711yescharacteristic
176C8—2122no—
177C22—21no—
178C22—21no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—84no—
183C22—532no—
184C22—1061no—
185C22—1061no—
186C22—1061no—
187C22—1061no—
188C22—1061no—
189C22—2124no—
190C22#1313327512yes—
191C22C2×C22⋊#615711yescharacteristic
192C4—21no—
193C4—21no—
194C4—1061no—
195C4—1061no—
196C4—4242no—
197C4—2122no—
198C2—22no—
199C2—82no—
200C2—1062no—
201C2—2121no—
202C2C2×#1313327511yescharacteristic
203C2#1681161511yescharacteristic
204C2#1681162111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22×D4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22×D4#6157
hypocenterC53#1510
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #16811595C2semidirect
2. #16811566C2semidirect
3. #16811474C2semidirect
8. #13133274C22semidirect
10. (C2×#6157)⋊C4C22semidirect
15. #13133188C4semidirect
17. C2×C4⋊Dic53C4semidirect
19. C22×D4×C53C4semidirect
107. C53#1510semidirect
109. C22×D4#6157semidirect