Finite Group Databasebeta

← #20272745

Subgroups — #20272745

Order 6784.

140 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 37 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#16810392C211yesmaximal, characteristic
2#16810349C211yesmaximal, characteristic
3C53×#395C211yesmaximal, characteristic
4C2×#11719917—21no—
5C22⋊#88322—21no—
6C22×Q8×C53C411yescharacteristic
7C2×C4⋊C4×C53—21no—
8C2×C4⋊C4×C53C2211yescharacteristic
9#95×C53—21no—
10#95×C53C411yescharacteristic
11C2×Q8×C53—42no—
12C2×Q8×C53—41no—
13C22×C4×C53—21no—
14C22×C4×C53—21no—
15C22×C4×C53C2×C411yescharacteristic
16C22×C4×C53D411yescharacteristic
17C22×C4×C53D411yescharacteristic
18C4⋊C4×C53—41no—
19C4⋊C4×C53—42no—
20#11719917—22no—
21#11719917—41no—
22C2×#88322—41no—
23C2×#88322—41no—
24C23×C53C22⋊C411yescharacteristic
25Q8×C53—42no—
26Q8×C53—42no—
27C2×C4×C53—21no—
28C2×C4×C53—21no—
29C2×C4×C53—22no—
30C2×C4×C53—22no—
31C2×C4×C53—21no—
32C2×C4×C53—21no—
33C2×C4×C53—41no—
34C2×C4×C53—41no—
35C2×C4×C53—41no—
36C2×C4×C53—41no—
37C2×C4×C53—41no—
38C2×C4×C53—41no—
39C2×C4×C53SD1611yescharacteristic
40C2×C4×C53Q1611yescharacteristic
41#88322—42no—
42#88322—81no—
43C22×C53—21no—
44C22×C53—21no—
45C22×C53#9911yescharacteristic
46C22×C53#10311yescharacteristic
47C22×C53#10411yescharacteristic
48C4×C53—21no—
49C4×C53—21no—
50C4×C53—42no—
51C4×C53—41no—
52C4×C53—42no—
53C4×C53—41no—
54#1354—531nomaximal, Sylow
55C2×C53—21no—
56C2×C53—21no—
57C2×C53#32811yescharacteristic
58C2×C53AΣL(1,𝔽16)11yescharacteristic
59C2×C53#35611yescharacteristic
60#395D5311yescharacteristic
61#343—531no—
62#324—531no—
63C53#135411yescharacteristic, complemented, Sylow
64C22×Q8Dic5311yescharacteristic
65C2×M16—1061no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2×D5311yescharacteristic
68C22⋊C8—1061no—
69#95—21no—
70#95Dic5311yescharacteristic
71C2×Q8—42no—
72C2×Q8—41no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4C2×Dic5311yescharacteristic
76C22×C4C22⋊D5311yescharacteristic
77C22×C4C22⋊D5311yescharacteristic
78M16—1062no—
79M16—2121no—
80C2×C8—2121no—
81C2×C8—2121no—
82C4⋊C4—41no—
83C4⋊C4—42no—
84C23C22⋊Dic5311yescharacteristic
85Q8—42no—
86Q8—42no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—22no—
90C2×C4—22no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4#1171992411yescharacteristic
100C2×C4#1171992511yescharacteristic
101C8—4241no—
102C8—2122no—
103C22—21no—
104C22—21no—
105C22#1313313211yescharacteristic
106C22#1313314111yescharacteristic
107C22#1313314411yescharacteristic
108C4—21no—
109C4—21no—
110C4—42no—
111C4—41no—
112C4—42no—
113C4—41no—
114C2—21no—
115C2—21no—
116C2#1681036011yescharacteristic
117C2#1681041011yescharacteristic
118C2#1681042211yescharacteristic
subgroupG/H
centerC22#13133132
commutatorC22×C4×C53C2×C4
FrattiniC2×C4⋊C4C2×D53
FittingC53×#395C2
socleC22×C53#99
hypercenter#395D53
hypocenterC53#1354
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #395

Complements

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

NH ≅ G/Ntype
63. C53#1354semidirect