Finite Group Databasebeta

← #20290615

Subgroups — #20290615

Order 6784.

316 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 70 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#16811595C212yesmaximal
2#16811503C214yesmaximal, complemented
3C2×C4⋊C4×D53C211yesmaximal, characteristic
4C2×#11719940C2214yes—
5#13133274—22no—
6#13133274—22no—
7C22×C4×D53C412yes—
8C22×C4×D53C2211yescharacteristic
9C2×C4⋊Dic53C411yescharacteristic
10C2×#11719919C2212yes—
11C4⋊C4×D53—22no—
12C4⋊C4×D53—22no—
13C2×C4⋊C4×C53C411yescharacteristic
14C23×D53C2×C411yescharacteristic
15C22×C4×C53C2×C411yescharacteristic
16C22×C4×C53C2×C412yes—
17C22×Dic53C2×C412yes—
18C22×Dic53C2311yescharacteristic
19C2×C4×D53—24no—
20C2×C4×D53—22no—
21C2×C4×D53—22no—
22C2×C4×D53—44no—
23C2×C4×D53D412yes—
24C2×C4×D53D412yes—
25#11719940—28no—
26C2×#88324—24no—
27C4⋊C4×C53—22no—
28C4⋊Dic53—22no—
29#11719919—24no—
30C23×C53C22×C411yescharacteristic
31C22×D53—22no—
32C22×D53—22no—
33C22×D53—22no—
34C22×D53C22⋊C411yescharacteristic
35C22×D53C22⋊C411yescharacteristic
36C2×C4×C53—22no—
37C2×C4×C53—22no—
38C2×C4×C53—42no—
39C2×C4×C53C22⋊C412yes—
40C2×Dic53—22no—
41C2×Dic53—42no—
42C2×Dic53C2×D412yes—
43C2×Dic53#4114yes—
44C4×D53—24no—
45C4×D53—24no—
46C4×D53—44no—
47C4×D53—44no—
48#88324—44no—
49C22×C53—21no—
50C22×C53—21no—
51C22×C53C2×C22⋊C411yescharacteristic
52C22×C53#11712yes—
53C2×D53—28no—
54C2×D53—21no—
55C2×D53—21no—
56C2×D53—21no—
57C2×D53—21no—
58C2×D53#10012yes—
59C2×D53#10112yes—
60C4×C53—22no—
61C4×C53—42no—
62Dic53—42no—
63Dic53#12314yes—
64#1821—531nomaximal, Sylow
65C2×C53—22no—
66C2×C53#38111yescharacteristic
67C2×C53C2×#10011yescharacteristic
68C2×C53C2×#10111yescharacteristic
69D53—22no—
70D53—22no—
71C22×C4⋊C4—531no—
72#407—532no—
73#343—534no—
74C53#182111yescharacteristic, complemented, Sylow
75C23×C4—531no—
76C23×C4—532no—
77C2×M16—534no—
78C2×C4⋊C4—532no—
79C2×C4⋊C4—531no—
80C2×C4⋊C4—1062no—
81C2×C4⋊C4—1062no—
82C2×C4⋊C4#615711yescharacteristic
83C22⋊C8—1062no—
84C22⋊C8—1062no—
85C24—531no—
86C22×C4—531no—
87C22×C4—532no—
88C22×C4—532no—
89C22×C4—532no—
90C22×C4—1064no—
91C22×C4—1062no—
92C22×C4—1062no—
93C22×C4—2124no—
94C22×C4C2×#615712yes—
95C22×C4C2×#615711yescharacteristic
96M16—1068no—
97C2×C8—1064no—
98C4⋊C4—22no—
99C4⋊C4—1064no—
100C4⋊C4—1062no—
101C23—531no—
102C23—531no—
103C23—1062no—
104C23—1062no—
105C23—1062no—
106C23C22×#615711yescharacteristic
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—532no—
111C2×C4—534no—
112C2×C4—1064no—
113C2×C4—1064no—
114C2×C4—1062no—
115C2×C4—2124no—
116C2×C4—2124no—
117C2×C4—2122no—
118C2×C4C22⋊#615712yes—
119C8—2124no—
120C22—21no—
121C22—21no—
122C22—532no—
123C22—532no—
124C22—1068no—
125C22—1061no—
126C22—1061no—
127C22—1061no—
128C22—1061no—
129C22C2×C22⋊#615711yescharacteristic
130C22#1313329612yes—
131C4—22no—
132C4—42no—
133C4—534no—
134C4—2122no—
135C2—22no—
136C2—1062no—
137C2—1062no—
138C2#1681156611yescharacteristic
139C2D53⋊#10011yescharacteristic
140C2D53⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C4⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C4⋊C4#6157
hypocenterC53#1821
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #16811503C2semidirect
74. C53#1821semidirect