Finite Group Databasebeta

← #20272742

Subgroups — #20272742

Order 6784.

116 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 116 conjugacy classes of subgroups — too many to draw. Only its 37 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#16810392C211yesmaximal, characteristic
2#16810349C211yesmaximal, characteristic
3C53×#400C211yesmaximal, characteristic
4C2×#11719917—21no—
5C22⋊#88322—21no—
6C2×C4⋊C4×C53C411yescharacteristic
7C2×C4⋊C4×C53C2211yescharacteristic
8#95×C53—21no—
9#95×C53—21no—
10#95×C53C411yescharacteristic
11C22×C4×C53—21no—
12C22×C4×C53—21no—
13C22×C4×C53C2×C411yescharacteristic
14C22×C4×C53D411yescharacteristic
15C22×C4×C53D411yescharacteristic
16C4⋊C4×C53—41no—
17C4⋊C4×C53—41no—
18#11719917—22no—
19#11719917—41no—
20C2×#88322—41no—
21C2×#88322—41no—
22C23×C53C22⋊C411yescharacteristic
23C2×C4×C53—21no—
24C2×C4×C53—21no—
25C2×C4×C53—21no—
26C2×C4×C53—21no—
27C2×C4×C53—41no—
28C2×C4×C53—41no—
29C2×C4×C53—41no—
30C2×C4×C53—41no—
31C2×C4×C53—42no—
32C2×C4×C53—41no—
33C2×C4×C53—41no—
34C2×C4×C53SD1611yescharacteristic
35C2×C4×C53Q1611yescharacteristic
36#88322—42no—
37#88322—81no—
38C22×C53—21no—
39C22×C53—21no—
40C22×C53#9911yescharacteristic
41C22×C53#10311yescharacteristic
42C22×C53#10411yescharacteristic
43C4×C53—21no—
44C4×C53—21no—
45C4×C53—41no—
46C4×C53—41no—
47C4×C53—81no—
48C4×C53—81no—
49#1353—531nomaximal, Sylow
50C2×C53—21no—
51C2×C53—21no—
52C2×C53#32811yescharacteristic
53C2×C53#35211yescharacteristic
54C2×C53#35511yescharacteristic
55#400D5311yescharacteristic
56#343—531no—
57#324—531no—
58C53#135311yescharacteristic, complemented, Sylow
59C2×M16—1061no—
60C2×C4⋊C4Dic5311yescharacteristic
61C2×C4⋊C4C2×D5311yescharacteristic
62C22⋊C8—1061no—
63#95—21no—
64#95—21no—
65#95Dic5311yescharacteristic
66C22×C4—21no—
67C22×C4—21no—
68C22×C4C2×Dic5311yescharacteristic
69C22×C4C22⋊D5311yescharacteristic
70C22×C4C22⋊D5311yescharacteristic
71M16—1062no—
72M16—2121no—
73C2×C8—2121no—
74C2×C8—2121no—
75C4⋊C4—41no—
76C4⋊C4—41no—
77C23C22⋊Dic5311yescharacteristic
78C2×C4—21no—
79C2×C4—21no—
80C2×C4—21no—
81C2×C4—21no—
82C2×C4—41no—
83C2×C4—41no—
84C2×C4—41no—
85C2×C4—41no—
86C2×C4—42no—
87C2×C4—41no—
88C2×C4—41no—
89C2×C4#1171992411yescharacteristic
90C2×C4#1171992511yescharacteristic
91C8—4241no—
92C8—2122no—
93C22—21no—
94C22—21no—
95C22#1313313211yescharacteristic
96C22#1313314111yescharacteristic
97C22#1313314411yescharacteristic
98C4—21no—
99C4—21no—
100C4—41no—
101C4—41no—
102C4—81no—
103C4—81no—
104C2—21no—
105C2—21no—
106C2#1681036011yescharacteristic
107C2#1681041311yescharacteristic
108C2#1681042011yescharacteristic
subgroupG/H
centerC22#13133132
commutatorC22×C4×C53C2×C4
FrattiniC2×C4⋊C4C2×D53
FittingC53×#400C2
socleC22×C53#99
hypercenter#400D53
hypocenterC53#1353
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #400

Complements

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

NH ≅ G/Ntype
58. C53#1353semidirect