Finite Group Databasebeta

← #20290122

Subgroups — #20290122

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#16811475C214yesmaximal, complemented
4C2×#13133192C211yesmaximal, characteristic
5C2×#11719940—21no—
6C2×C22⋊#6157—21no—
7#13133274—21no—
8#13133274C2212yescomplemented
9(C2×#6157)⋊C4—21no—
10(C2×#6157)⋊C4C2212yescomplemented
11C22×D212—21no—
12C22×D212C411yescharacteristic, complemented
13C22×C4×D53C2211yescharacteristic
14#13133192—21no—
15#13133192—21no—
16#13133192C412yescomplemented
17#13133192C2212yes—
18C2×#11719921—21no—
19C2×C4⋊C4×C53C411yescharacteristic, complemented
20C23×D53—21no—
21C23×D53D411yescharacteristic
22C22×#6157—22no—
23C22×C4×C53—21no—
24C22×C4×C53C2×C411yescharacteristic
25C22×Dic53D411yescharacteristic
26C2×D212—22no—
27C2×D212—22no—
28C2×D212—21no—
29C2×D212—42no—
30C2×D212—44no—
31C2×D212C2×C412yes—
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—
40C4⋊C4×C53—21no—
41C4⋊C4×C53C2×C412yes—
42#11719921—22no—
43#11719921—22no—
44C23×C53C22⋊C411yescharacteristic
45C22×D53—22no—
46C22×D53—22no—
47C22×D53—22no—
48C22×D53—41no—
49C22×D53—42no—
50C22×D53—42no—
51C22×D53—42no—
52C22×D53C22⋊C411yescharacteristic
53C22×D53C2×D411yescharacteristic
54C2×#6157—42no—
55C2×#6157—42no—
56C2×#6157—42no—
57C2×C4×C53—22no—
58C2×C4×C53—22no—
59C2×C4×C53—42no—
60C2×C4×C53C22⋊C411yescharacteristic
61C2×C4×C53C22×C411yescharacteristic
62C2×Dic53—22no—
63C2×Dic53C22⋊C411yescharacteristic
64C2×Dic53C2×D411yescharacteristic
65D212—42no—
66D212—42no—
67D212—44no—
68D212—44no—
69C4×D53—41no—
70C4×D53—41no—
71C4×D53—41no—
72C4×D53—41no—
73#88324—42no—
74C22×C53—21no—
75C22×C53—21no—
76C22×C53#9912yes—
77C22×C53C2×C22⋊C411yescharacteristic
78C2×D53—21no—
79C2×D53—21no—
80C2×D53—21no—
81C2×D53—21no—
82C2×D53—21no—
83C2×D53—44no—
84C2×D53—42no—
85C2×D53—42no—
86C2×D53—42no—
87C2×D53—84no—
88C2×D53AGL(1,ℤ/8)12yes—
89#6157—82no—
90C4×C53—21no—
91C4×C53—21no—
92C4×C53—42no—
93Dic53—21no—
94Dic53—21no—
95#1510—531nomaximal, Sylow
96C2×C53—22no—
97C2×C53C2×#9911yescharacteristic
98C2×C53#41811yescharacteristic
99C2×C53#42111yescharacteristic
100D53—22no—
101D53—41no—
102D53—82no—
103C2×#121—531no—
104#407—531no—
105#381—531no—
106#327—534no—
107C53#151011yescharacteristic, complemented, Sylow
108C22×D4—531no—
109C22×D4—1061no—
110C23×C4—531no—
111C2×M16—1061no—
112#121—532no—
113#121—532no—
114#121—1061no—
115#121—1061no—
116C2×C4⋊C4#615711yescharacteristic, complemented
117C2×C22⋊C4—1061no—
118C2×C22⋊C4—1061no—
119C22⋊C8—532no—
120C22⋊C8—1061no—
121#95—532no—
122#95—1061no—
123C24—531no—
124C24—1061no—
125C2×D4—532no—
126C2×D4—1062no—
127C2×D4—1062no—
128C2×D4—1061no—
129C2×D4—2122no—
130C2×D4—2124no—
131C22×C4—21no—
132C22×C4—531no—
133C22×C4—531no—
134C22×C4—531no—
135C22×C4—531no—
136C22×C4—531no—
137C22×C4—1062no—
138C22×C4—2122no—
139C22×C4C2×#615711yescharacteristic
140M16—2122no—
141C2×C8—1062no—
142C4⋊C4—21no—
143C4⋊C4C2×#615712yes—
144C22⋊C4—1062no—
145C22⋊C4—1062no—
146C22⋊C4—2122no—
147C23—531no—
148C23—531no—
149C23—1062no—
150C23—1062no—
151C23—1062no—
152C23—2121no—
153C23—2122no—
154C23—2122no—
155C23—2122no—
156C23C22⋊#615711yescharacteristic
157D4—2122no—
158D4—2122no—
159D4—2124no—
160D4—2124no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—531no—
165C2×C4—531no—
166C2×C4—1062no—
167C2×C4—2121no—
168C2×C4—2121no—
169C2×C4—2121no—
170C2×C4—2121no—
171C2×C4—2122no—
172C2×C4—2122no—
173C2×C4—2122no—
174C2×C4C22⋊#615711yescharacteristic
175C2×C4C22×#615711yescharacteristic
176C8—2122no—
177C22—21no—
178C22—21no—
179C22—532no—
180C22—1061no—
181C22—1061no—
182C22—1061no—
183C22—1061no—
184C22—1061no—
185C22—4244no—
186C22—2124no—
187C22—2122no—
188C22—2122no—
189C22—2122no—
190C22#1313327612yes—
191C22C2×C22⋊#615711yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—1061no—
196C4—1061no—
197C4—4242no—
198C2—22no—
199C2—1062no—
200C2—4242no—
201C2—2121no—
202C2C2×#1313327611yescharacteristic
203C2#1681161511yescharacteristic
204C2#1681162211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC2×C4⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C4⋊C4#6157
hypocenterC53#1510
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #16811595C2semidirect
2. #16811566C2semidirect
3. #16811475C2semidirect
8. #13133274C22semidirect
10. (C2×#6157)⋊C4C22semidirect
12. C22×D212C4semidirect
16. #13133192C4semidirect
19. C2×C4⋊C4×C53C4semidirect
107. C53#1510semidirect
116. C2×C4⋊C4#6157semidirect