Finite Group Databasebeta

← #20290134

Subgroups — #20290134

Order 6784.

252 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 56 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#16811645C211yesmaximal, characteristic, complemented
2#16811568C211yesmaximal, characteristic, complemented
3#16811475C212yesmaximal, complemented
4#16811474C212yesmaximal, complemented
5#16811170C211yesmaximal, characteristic
6C4⋊#88324—21no—
7#13133274C2212yes—
8#13133272—21no—
9(C2×#6157)⋊C4—21no—
10(C2×#6157)⋊C4—21no—
11(C2×#6157)⋊C4C2212yes—
12C2×#11719945—21no—
13#13133216C411yescharacteristic, complemented
14#13133213C411yescharacteristic, complemented
15#13133203—21no—
16#13133192C2211yescharacteristic
17#13133190—21no—
18#13133188C2211yescharacteristic
19C4×D212C411yescharacteristic, complemented
20C4×C22⋊D53—21no—
21C42×D53C2211yescharacteristic
22C4×D4×C53C411yescharacteristic, complemented
23C22×#6157—22no—
24C2×D4×C53C2×C411yescharacteristic
25C22×C4×C53—21no—
26C2×C22⋊D53—21no—
27C2×C22⋊D53—21no—
28C2×C22⋊D53—21no—
29#11719945—44no—
30C2×D212—21no—
31C2×D212C2×C411yescharacteristic
32C2×C4×D53—21no—
33C2×C4×D53D411yescharacteristic
34C2×C4×D53D411yescharacteristic
35C2×C4×D53C2311yescharacteristic
36C2×Dic106—21no—
37C2×#88324—22no—
38C4⋊C4×C53C2×C411yescharacteristic
39C22⋊C4×C53—21no—
40C42×C53C2×C411yescharacteristic
41C22⋊Dic53—21no—
42#11719921—21no—
43#11719921—21no—
44C4⋊Dic53C2×C411yescharacteristic
45#11719919—21no—
46C4×Dic53C2×C411yescharacteristic
47C4×Dic53D411yescharacteristic
48C4×Dic53D411yescharacteristic
49C23×C53—21no—
50C22×D53—21no—
51C22×D53C2×D411yescharacteristic
52C2×#6157—42no—
53C2×#6157—42no—
54C2×#6157—42no—
55D4×C53—41no—
56C2×C4×C53—21no—
57C2×C4×C53—42no—
58C2×C4×C53C22⋊C411yescharacteristic
59C2×C4×C53C22⋊C411yescharacteristic
60C2×C4×C53C22×C411yescharacteristic
61C22⋊D53—42no—
62C22⋊D53—42no—
63C22⋊D53—42no—
64C2×Dic53—21no—
65C2×Dic53C22⋊C411yescharacteristic
66C2×Dic53C22⋊C411yescharacteristic
67C2×Dic53C2×D411yescharacteristic
68D212—41no—
69D212—42no—
70C4×D53—22no—
71C4×D53—41no—
72C4×D53—41no—
73C4×D53—42no—
74Dic106—42no—
75#88324—42no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53—41no—
79C22×C53C2×C22⋊C411yescharacteristic
80C2×D53—21no—
81C2×D53—21no—
82C2×D53—41no—
83C2×D53—41no—
84C2×D53—41no—
85C2×D53AGL(1,ℤ/8)12yes—
86#6157—82no—
87C4×C53—21no—
88C4×C53—21no—
89C4×C53—21no—
90C4×C53—41no—
91Dic53—21no—
92Dic53—21no—
93Dic53—21no—
94Dic53—41no—
95#1514—531nomaximal, Sylow
96C2×C53—81no—
97C2×C53#41011yescharacteristic
98C2×C53#41811yescharacteristic
99C2×C53#42111yescharacteristic
100D53—22no—
101D53—81no—
102#532—531no—
103#432—531no—
104#383—531no—
105#327—532no—
106#327—532no—
107C53#151411yescharacteristic, complemented, Sylow
108C2×#41—1061no—
109C4⋊Q8—531no—
110#127—531no—
111#124—1061no—
112#121—531no—
113#121—531no—
114#121—1061no—
115C4×D4—531no—
116C4×D4—1061no—
117C4×D4#615711yescharacteristic, complemented
118C2×C42—531no—
119C4⋊C8—1061no—
120C22⋊C8—532no—
121#97—1061no—
122#95—532no—
123#95—1061no—
124#95—1061no—
125#41—2124no—
126C2×Q8—1061no—
127C2×D4—531no—
128C2×D4—1061no—
129C2×D4—1061no—
130C2×D4—1061no—
131C2×D4—1061no—
132C2×D4C2×#615711yescharacteristic
133C22×C4—21no—
134C22×C4—531no—
135C22×C4—531no—
136C22×C4—531no—
137C22×C4—1061no—
138C22×C4—1062no—
139C2×C8—1062no—
140C4⋊C4—531no—
141C4⋊C4—1061no—
142C4⋊C4C2×#615711yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—1061no—
145C22⋊C4—1061no—
146C22⋊C4—1061no—
147C42—531no—
148C42—531no—
149C42—531no—
150C42C2×#615711yescharacteristic
151C23—21no—
152C23—531no—
153C23—1061no—
154Q8—2122no—
155D4—41no—
156D4—2121no—
157D4—2122no—
158D4—2122no—
159D4—2122no—
160D4—2122no—
161C2×C4—21no—
162C2×C4—42no—
163C2×C4—531no—
164C2×C4—531no—
165C2×C4—531no—
166C2×C4—1062no—
167C2×C4—1061no—
168C2×C4—2121no—
169C2×C4—2121no—
170C2×C4—2122no—
171C2×C4—2122no—
172C2×C4—2122no—
173C2×C4—2122no—
174C2×C4C22⋊#615711yescharacteristic
175C2×C4C22⋊#615711yescharacteristic
176C2×C4C22×#615711yescharacteristic
177C8—2122no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—532no—
182C22—1061no—
183C22—1061no—
184C22—2121no—
185C22—2121no—
186C22—2121no—
187C22C2×C22⋊#615711yescharacteristic
188C4—21no—
189C4—21no—
190C4—21no—
191C4—41no—
192C4—1061no—
193C4—1061no—
194C4—1061no—
195C4—4242no—
196C4—2121no—
197C2—81no—
198C2—1062no—
199C2—4241no—
200C2#1681160211yescharacteristic
201C2#1681161511yescharacteristic
202C2#1681162011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4×D4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC4×D4#6157
hypocenterC53#1514
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #16811645C2semidirect
2. #16811568C2semidirect
3. #16811475C2semidirect
4. #16811474C2semidirect
13. #13133216C4semidirect
14. #13133213C4semidirect
19. C4×D212C4semidirect
22. C4×D4×C53C4semidirect
107. C53#1514semidirect
117. C4×D4#6157semidirect