Finite Group Databasebeta

← #20272722

Subgroups — #20272722

Order 6784.

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

normal subgroups only

This group has 124 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#16810553C211yesmaximal, characteristic
2#16810392C211yesmaximal, characteristic
3C53×#324C211yesmaximal, characteristic
4C2×#11719917—21no—
5C2×C4⋊Dic53C411yescharacteristic
6C2×#11719919—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#11719919—41no—
20#11719919—41no—
21#11719917—22no—
22#11719917—41no—
23C2×#88322—41no—
24C23×C53C22⋊C411yescharacteristic
25C2×C4×C53—21no—
26C2×C4×C53—21no—
27C2×C4×C53—21no—
28C2×C4×C53—21no—
29C2×C4×C53—41no—
30C2×C4×C53—41no—
31C2×C4×C53D811yescharacteristic
32C2×C4×C53SD1611yescharacteristic
33C2×Dic53—22no—
34C2×Dic53—41no—
35C2×Dic53—41no—
36C2×Dic53—41no—
37C2×Dic53—41no—
38C2×Dic53—41no—
39C8×C53—81no—
40#88322—42no—
41C22×C53—21no—
42C22×C53—21no—
43C22×C53#9911yescharacteristic
44C22×C53#10211yescharacteristic
45C22×C53#10411yescharacteristic
46C4×C53—21no—
47C4×C53—21no—
48C4×C53—41no—
49C4×C53—41no—
50Dic53—42no—
51Dic53—81no—
52#1346—531nomaximal, Sylow
53C2×C53—21no—
54C2×C53—21no—
55C2×C53#32711yescharacteristic
56C2×C53#35211yescharacteristic
57C2×C53#35611yescharacteristic
58#398—531no—
59#343—531no—
60#324D5311yescharacteristic
61C53#134611yescharacteristic, complemented, Sylow
62C2×M16—1061no—
63C2×C4⋊C4—531no—
64C2×C4⋊C4—1061no—
65C2×C4⋊C4C2×D5311yescharacteristic
66C22⋊C8—21no—
67#95—531no—
68#95—1061no—
69C22×C4—1061no—
70C22×C4—1061no—
71C22×C4C4×D5311yescharacteristic
72C22×C4D21211yescharacteristic
73C22×C4C22⋊D5311yescharacteristic
74M16—1062no—
75M16—2121no—
76C2×C8—41no—
77C2×C8—2121no—
78C4⋊C4—41no—
79C4⋊C4—2121no—
80C4⋊C4—2121no—
81C4⋊C4—2121no—
82C23#1171992111yescharacteristic
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—41no—
88C2×C4—41no—
89C2×C4—1062no—
90C2×C4—2121no—
91C2×C4—2121no—
92C2×C4—2121no—
93C2×C4—2121no—
94C2×C4—2121no—
95C2×C4#1171991311yescharacteristic
96C2×C4D42411yescharacteristic
97C8—81no—
98C8—2122no—
99C22—21no—
100C22—21no—
101C22#1313313111yescharacteristic
102C22#1313313911yescharacteristic
103C22#1313314311yescharacteristic
104C4—21no—
105C4—21no—
106C4—41no—
107C4—41no—
108C4—4241no—
109C4—2122no—
110C2—21no—
111C2—21no—
112C2#1681035811yescharacteristic
113C2#1681041211yescharacteristic
114C2#1681042111yescharacteristic
subgroupG/H
centerC22#13133131
commutatorC22×C4×C53C2×C4
FrattiniC2×C4⋊C4C2×D53
FittingC53×#324C2
socleC22×C53#99
hypercenter#324D53
hypocenterC53#1346
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
61. C53#1346semidirect