Finite Group Databasebeta

← #20290994

Subgroups — #20290994

Order 6784.

270 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 54 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#16811602C212yesmaximal, complemented
2#16811527C214yesmaximal, complemented
3#16811190C211yesmaximal, characteristic
4#13133296—22no—
5#13133276C2214yescomplemented
6#13133275—22no—
7C2×#11719945C2211yescharacteristic
8#13133213C2212yes—
9#13133204C412yescomplemented
10#13133188—22no—
11#13133186—22no—
12#13133176C411yescharacteristic, complemented
13C22×D4×C53C411yescharacteristic, complemented
14C24×C53—21no—
15C2×D4×C53—21no—
16C2×D4×C53—42no—
17C2×D4×C53C2×C412yes—
18C22×C4×C53C2×C411yescharacteristic
19C2×C22⋊D53—24no—
20C2×C22⋊D53D412yes—
21#11719945—42no—
22C2×D212C2311yescharacteristic
23C2×C4×D53D412yes—
24C2×Dic106C2×C411yescharacteristic
25C22⋊#6157—24no—
26C4⋊#6157—42no—
27C4×#6157—42no—
28C22⋊Dic53—21no—
29C22⋊Dic53—24no—
30C4⋊Dic53—21no—
31C4×Dic53C2×C412yes—
32C23×C53—22no—
33C23×C53—42no—
34C23×C53—82no—
35C23×C53C22⋊C411yescharacteristic
36C22×D53C2×D412yes—
37C2×#6157—44no—
38D4×C53—42no—
39D4×C53—42no—
40C2×C4×C53—22no—
41C2×C4×C53C22⋊C411yescharacteristic
42C2×C4×C53C22×C411yescharacteristic
43C22⋊D53—42no—
44C22⋊D53—84no—
45C2×Dic53—22no—
46C2×Dic53C22⋊C412yes—
47D212—41no—
48C4×D53—22no—
49C4×D53—22no—
50Dic106—41no—
51C22×C53—21no—
52C22×C53—41no—
53C22×C53—44no—
54C22×C53—42no—
55C22×C53—84no—
56C22×C53C2×C22⋊C411yescharacteristic
57C2×D53—22no—
58C2×D53—42no—
59#6157—84no—
60C4×C53—21no—
61C4×C53#9912yes—
62Dic53—22no—
63Dic53—42no—
64#2128—531nomaximal, Sylow
65C2×C53—21no—
66C2×C53—41no—
67C2×C53—82no—
68C2×C53C2×#9911yescharacteristic
69D53—42no—
70#535—531no—
71#410—532no—
72#353—534no—
73C53#212811yescharacteristic, complemented, Sylow
74C2×#41—531no—
75C22×D4#615711yescharacteristic, complemented
76#127—532no—
77#124—532no—
78#121—1062no—
79#120—1062no—
80#117—531no—
81#117—1062no—
82#99—534no—
83#99—1062no—
84C24—21no—
85#41—2122no—
86C2×Q8—531no—
87C2×D4—21no—
88C2×D4—42no—
89C2×D4—532no—
90C2×D4—531no—
91C2×D4—1064no—
92C2×D4C2×#615712yes—
93C22×C4—532no—
94C22×C4C2×#615711yescharacteristic
95C4⋊C4—1061no—
96C4⋊C4—2122no—
97C22⋊C4—1064no—
98C22⋊C4—1061no—
99C22⋊C4—1064no—
100C42—532no—
101C42—2122no—
102C23—22no—
103C23—42no—
104C23—82no—
105C23—532no—
106C23C22⋊#615711yescharacteristic
107Q8—2121no—
108D4—42no—
109D4—42no—
110D4—4244no—
111D4—2121no—
112D4—2122no—
113C2×C4—22no—
114C2×C4—532no—
115C2×C4—1062no—
116C2×C4—1062no—
117C2×C4—1062no—
118C2×C4—2124no—
119C2×C4C22⋊#615711yescharacteristic
120C2×C4C22×#615711yescharacteristic
121C22—21no—
122C22—41no—
123C22—44no—
124C22—42no—
125C22—84no—
126C22—1062no—
127C22—2122no—
128C22C2×C22⋊#615711yescharacteristic
129C4—21no—
130C4—1062no—
131C4—4244no—
132C4—2122no—
133C4#1313327512yes—
134C2—21no—
135C2—41no—
136C2—82no—
137C2—2122no—
138C2C2×#1313327511yescharacteristic
subgroupG/H
centerC2C2×#13133275
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22×D4×C53C4
socleC2×C53C2×#99
hypercenterC22×D4#6157
hypocenterC53#2128
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #16811602C2semidirect
2. #16811527C2semidirect
5. #13133276C22semidirect
9. #13133204C4semidirect
12. #13133176C4semidirect
13. C22×D4×C53C4semidirect
73. C53#2128semidirect
75. C22×D4#6157semidirect