Finite Group Databasebeta

← #20285580

Subgroups — #20285580

Order 6784.

442 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 117 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#16811187C212yesmaximal, complemented
2#16810948C214yesmaximal, complemented
3#16810876C212yesmaximal, complemented
4#16810859C212yesmaximal, complemented
5#16810759C214yesmaximal, complemented
6C53×#408C211yesmaximal, characteristic, complemented
7C22×D212C2212yescomplemented
8C2×D424C2212yes—
9C2×D424C2212yes—
10#13133213—22no—
11#13133206—22no—
12#13133191C2214yes—
13#13133191C2214yescomplemented
14#13133185—22no—
15#13133185—22no—
16#13133176C2212yescomplemented
17#13133151C2212yescomplemented
18#13133149C2212yescomplemented
19#13133139C2214yescomplemented
20#13133139C2214yescomplemented
21C2×#41×C53C2211yescharacteristic
22C2×M16×C53C2211yescharacteristic, complemented
23C22×C8×C53C2211yescharacteristic, complemented
24C22⋊C8×C53C2214yescomplemented
25C23×D53—22no—
26#41×C53—41no—
27#41×C53—41no—
28C2×Q8×C53D411yescharacteristic
29C2×D4×C53D411yescharacteristic
30C2×D4×C53D412yes—
31C22×C4×C53C2312yes—
32C22×C4×C53C2311yescharacteristic
33C2×C22⋊D53—24no—
34C2×C22⋊D53—24no—
35C2×D212—22no—
36C2×D212—42no—
37C2×D212—42no—
38C2×D212C2314yes—
39M16×C53—41no—
40C2×C8×C53—41no—
41C2×C8×C53C2312yes—
42C2×C8×C53C2312yes—
43C22⋊Dic53—22no—
44#11719921—24no—
45#11719921—24no—
46C4⋊Dic53C2314yes—
47C4×Dic53—22no—
48D424—42no—
49D424—42no—
50C23×C53C2×D411yescharacteristic
51C23×C53C2×D412yes—
52C22×D53—24no—
53C22×D53—44no—
54C22×D53—84no—
55Q8×C53—41no—
56D4×C53—42no—
57D4×C53—41no—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53C2×D412yes—
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2411yescharacteristic
65C22⋊D53—84no—
66C22⋊D53—84no—
67C2×Dic53—24no—
68D212—24no—
69D212—24no—
70D212—44no—
71C8×C53—22no—
72C8×C53—22no—
73C22×C53—22no—
74C22×C53—21no—
75C22×C53—22no—
76C22×C53—24no—
77C22×C53C22×D411yescharacteristic
78C2×D53—44no—
79C2×D53—44no—
80C2×D53—44no—
81C2×D53—88no—
82C4×C53—22no—
83C4×C53—21no—
84C4×C53H(5,𝔽2)11yescharacteristic
85C4×C53H(5,𝔽2)11yescharacteristic
86Dic53—44no—
87#3199—531nomaximal, Sylow
88C2×C53—42no—
89C2×C53—41no—
90C2×C53#53411yescharacteristic
91C2×C53#57612yes—
92D53—84no—
93#535—532no—
94#482—534no—
95#469—532no—
96#466—532no—
97#447—534no—
98#408D5311yescharacteristic, complemented
99C53#319911yescharacteristic, complemented, Sylow
100C2×#41C2×D5311yescharacteristic
101C22×D4—532no—
102C2×D8—532no—
103C2×D8—532no—
104C2×M16C2×D5311yescharacteristic, complemented
105C22×C8C2×D5311yescharacteristic, complemented
106#127—1062no—
107#124—1062no—
108#121—534no—
109#121—534no—
110#120—1062no—
111#120—1062no—
112#117—532no—
113#107—532no—
114#106—532no—
115#102—534no—
116#102—534no—
117C22⋊C8C2×D5314yescomplemented
118C24—1062no—
119#41—41no—
120#41—41no—
121C2×Q8D21211yescharacteristic
122C2×D4—534no—
123C2×D4—1064no—
124C2×D4—1064no—
125C2×D4—1062no—
126C2×D4—2122no—
127C2×D4—2122no—
128C2×D4D21211yescharacteristic
129C2×D4D21212yes—
130C22×C4C22×D5312yes—
131C22×C4C22×D5311yescharacteristic
132D8—2122no—
133D8—2122no—
134M16—41no—
135C2×C8—41no—
136C2×C8C22×D5312yes—
137C2×C8C22×D5312yes—
138C4⋊C4—534no—
139C22⋊C4—1062no—
140C22⋊C4—1064no—
141C22⋊C4—1064no—
142C42—1062no—
143C23—1064no—
144C23—4244no—
145C23—2124no—
146C23C2×D21212yes—
147C23C2×D21211yescharacteristic
148Q8—41no—
149D4—42no—
150D4—41no—
151D4—1064no—
152D4—1064no—
153D4—4244no—
154D4—4244no—
155D4—2124no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—1064no—
161C2×C4C2×D21212yes—
162C2×C4C2×D21211yescharacteristic
163C2×C4C23×D5311yescharacteristic
164C8—22no—
165C8—22no—
166C22—22no—
167C22—24no—
168C22—22no—
169C22—21no—
170C22—4248no—
171C22—2124no—
172C22—2124no—
173C22—2124no—
174C22C22×D21211yescharacteristic
175C4—22no—
176C4—21no—
177C4—2124no—
178C4#1313326211yescharacteristic
179C4#1313326211yescharacteristic
180C2—42no—
181C2—41no—
182C2—4244no—
183C2#1681118511yescharacteristic
184C2#1681142812yes—
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#408C2
socleC22×C53C22×D4
hypercenter#408D53
hypocenterC53#3199
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #16811187C2semidirect
2. #16810948C2semidirect
3. #16810876C2semidirect
4. #16810859C2semidirect
5. #16810759C2semidirect
6. C53×#408C2semidirect
7. C22×D212C22semidirect
13. #13133191C22semidirect
16. #13133176C22semidirect
17. #13133151C22semidirect
18. #13133149C22semidirect
19. #13133139C22semidirect
20. #13133139C22semidirect
22. C2×M16×C53C22semidirect
23. C22×C8×C53C22semidirect
24. C22⋊C8×C53C22semidirect
98. #408D53semidirect
99. C53#3199semidirect
104. C2×M16C2×D53semidirect
105. C22×C8C2×D53semidirect
117. C22⋊C8C2×D53semidirect