Finite Group Databasebeta

← #20272743

Subgroups — #20272743

Order 6784.

116 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 116 conjugacy classes of subgroups — too many to draw. Only its 35 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#16810562C211yesmaximal, characteristic
2#16810392C211yesmaximal, characteristic
3C53×#324C211yesmaximal, characteristic
4C2×#11719917—21no—
5C2×C4⋊Dic53C411yescharacteristic
6#13133124—21no—
7#13133124—21no—
8#13133124C411yescharacteristic
9C2×C4⋊C4×C53C2211yescharacteristic
10C22⋊C8×C53—21no—
11C22×C4×C53C2×C411yescharacteristic
12C22×C4×C53D411yescharacteristic
13C22×C4×C53D411yescharacteristic
14C22×Dic53—21no—
15C22×Dic53—21no—
16C2×C8×C53—41no—
17C4⋊C4×C53—41no—
18C4⋊Dic53—41no—
19#11719917—22no—
20#11719917—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×C53SD1611yescharacteristic
30C2×C4×C53Q1611yescharacteristic
31C2×Dic53—41no—
32C2×Dic53—41no—
33C2×Dic53—42no—
34C2×Dic53—41no—
35C2×Dic53—41no—
36C8×C53—81no—
37#88322—42no—
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—
47Dic53—81no—
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#324D5311yescharacteristic
58C53#135311yescharacteristic, complemented, Sylow
59C2×M16—1061no—
60C2×C4⋊C4—531no—
61C2×C4⋊C4C2×D5311yescharacteristic
62C22⋊C8—21no—
63#95—531no—
64#95—1061no—
65#95—1061no—
66C22×C4—1061no—
67C22×C4—1061no—
68C22×C4C4×D5311yescharacteristic
69C22×C4D21211yescharacteristic
70C22×C4C22⋊D5311yescharacteristic
71M16—1062no—
72M16—2121no—
73C2×C8—41no—
74C2×C8—2121no—
75C4⋊C4—41no—
76C4⋊C4—2121no—
77C23#1171992111yescharacteristic
78C2×C4—21no—
79C2×C4—21no—
80C2×C4—21no—
81C2×C4—21no—
82C2×C4—41no—
83C2×C4—41no—
84C2×C4—2121no—
85C2×C4—2121no—
86C2×C4—2122no—
87C2×C4—2121no—
88C2×C4—2121no—
89C2×C4#1171991311yescharacteristic
90C2×C4Dic21211yescharacteristic
91C8—81no—
92C8—2122no—
93C22—21no—
94C22—21no—
95C22#1313313111yescharacteristic
96C22#1313314211yescharacteristic
97C22#1313314311yescharacteristic
98C4—21no—
99C4—21no—
100C4—41no—
101C4—41no—
102C4—4241no—
103C4—4241no—
104C2—21no—
105C2—21no—
106C2#1681036111yescharacteristic
107C2#1681041211yescharacteristic
108C2#1681041911yescharacteristic
subgroupG/H
centerC22#13133131
commutatorC22×C4×C53C2×C4
FrattiniC2×C4⋊C4C2×D53
FittingC53×#324C2
socleC22×C53#99
hypercenter#324D53
hypocenterC53#1353
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #324

Complements

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

NH ≅ G/Ntype
58. C53#1353semidirect