Finite Group Databasebeta

← #20289713

Subgroups — #20289713

Order 6784.

116 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 116 conjugacy classes of subgroups — too many to draw. Only its 30 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#16811503C211yesmaximal, characteristic
2#16811469C211yesmaximal, characteristic
3#16810563C211yesmaximal, characteristic
4C2×#11719940—21no—
5C22⋊#88324—21no—
6C2×C4⋊Dic53C411yescharacteristic
7C2×#11719919C2211yescharacteristic
8#13133124—21no—
9#13133124—21no—
10#95×C53C411yescharacteristic
11C22×C4×C53—21no—
12C22×C4×C53C2×C411yescharacteristic
13C22×Dic53—21no—
14C22×Dic53D411yescharacteristic
15C22×Dic53D411yescharacteristic
16#11719940—22no—
17#11719940—41no—
18C2×#88324—41no—
19C2×#88324—41no—
20C4⋊Dic53—41no—
21#11719919—41no—
22C23×C53C22⋊C411yescharacteristic
23C2×C4×C53—21no—
24C2×C4×C53—41no—
25C2×C4×C53—41no—
26C2×C4×C53—41no—
27C2×C4×C53—41no—
28C2×Dic53—21no—
29C2×Dic53—21no—
30C2×Dic53—21no—
31C2×Dic53—41no—
32C2×Dic53—42no—
33C2×Dic53—41no—
34C2×Dic53SD1611yescharacteristic
35C2×Dic53Q1611yescharacteristic
36#88324—42no—
37#88324—81no—
38C22×C53—21no—
39C22×C53—21no—
40C22×C53#9911yescharacteristic
41C22×C53#10311yescharacteristic
42C22×C53#10411yescharacteristic
43C4×C53—41no—
44C4×C53—81no—
45Dic53—21no—
46Dic53—21no—
47Dic53—41no—
48Dic53—81no—
49#1353—531nomaximal, Sylow
50C2×C53—21no—
51C2×C53—21no—
52C2×C53#32811yescharacteristic
53C2×C53#35211yescharacteristic
54C2×C53#35511yescharacteristic
55#400—531no—
56#343—531no—
57#324—531no—
58C53#135311yescharacteristic, complemented, Sylow
59C2×M16—1061no—
60C2×C4⋊C4—531no—
61C2×C4⋊C4—531no—
62C22⋊C8—1061no—
63#95—1061no—
64#95—1061no—
65#95#615711yescharacteristic
66C22×C4—21no—
67C22×C4—531no—
68C22×C4—531no—
69C22×C4—1061no—
70C22×C4C2×#615711yescharacteristic
71M16—1062no—
72M16—2121no—
73C2×C8—2121no—
74C2×C8—2121no—
75C4⋊C4—2121no—
76C4⋊C4—2121no—
77C23C22⋊#615711yescharacteristic
78C2×C4—21no—
79C2×C4—41no—
80C2×C4—41no—
81C2×C4—41no—
82C2×C4—41no—
83C2×C4—531no—
84C2×C4—531no—
85C2×C4—1061no—
86C2×C4—1061no—
87C2×C4—1061no—
88C2×C4—2122no—
89C2×C4—2121no—
90C2×C4—2121no—
91C8—4241no—
92C8—2122no—
93C22—21no—
94C22—21no—
95C22#1313327611yescharacteristic
96C22#1313328011yescharacteristic
97C22#1313328111yescharacteristic
98C4—41no—
99C4—81no—
100C4—1061no—
101C4—1061no—
102C4—4241no—
103C4—2121no—
104C2—21no—
105C2—21no—
106C2#1681147611yescharacteristic
107C2#1681152611yescharacteristic
108C2#1681153211yescharacteristic
subgroupG/H
centerC22#13133276
commutatorC22×C4×C53C2×C4
FrattiniC22×C4C2×#6157
Fitting#95×C53C4
socleC22×C53#99
hypercenter#95#6157
hypocenterC53#1353
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #95

Complements

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

NH ≅ G/Ntype
58. C53#1353semidirect