Finite Group Databasebeta

← #20272744

Subgroups — #20272744

Order 6784.

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

normal subgroups only

This group has 140 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#16810543C211yesmaximal, characteristic
2#16810349C211yesmaximal, characteristic
3C53×#343C211yesmaximal, characteristic
4C22×Dic106C411yescharacteristic
5C2×#11719919—21no—
6C22⋊#88322—21no—
7#13133124—21no—
8#13133124C411yescharacteristic
9C2×M16×C53—21no—
10C2×C4⋊C4×C53C2211yescharacteristic
11C22×C4×C53C2×C411yescharacteristic
12C22×C4×C53D411yescharacteristic
13C22×C4×C53D411yescharacteristic
14C22×Dic53—21no—
15C22×Dic53—21no—
16C2×Dic106—41no—
17C2×Dic106—42no—
18M16×C53—22no—
19M16×C53—41no—
20C2×C8×C53—41no—
21C4⋊C4×C53—41no—
22#11719919—42no—
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×C53SD1611yescharacteristic
32C2×C4×C53Q1611yescharacteristic
33C2×Dic53—22no—
34C2×Dic53—22no—
35C2×Dic53—41no—
36C2×Dic53—41no—
37C2×Dic53—41no—
38C2×Dic53—41no—
39Dic106—42no—
40Dic106—42no—
41C8×C53—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—41no—
51C4×C53—41no—
52Dic53—42no—
53Dic53—42no—
54#1354—531nomaximal, Sylow
55C2×C53—21no—
56C2×C53—21no—
57C2×C53#32811yescharacteristic
58C2×C53AΣL(1,𝔽16)11yescharacteristic
59C2×C53#35611yescharacteristic
60#395—531no—
61#343D5311yescharacteristic
62#324—531no—
63C53#135411yescharacteristic, complemented, Sylow
64C22×Q8—531no—
65C2×M16—21no—
66C2×C4⋊C4—1061no—
67C2×C4⋊C4C2×D5311yescharacteristic
68C22⋊C8—1061no—
69#95—531no—
70#95—1061no—
71C2×Q8—2121no—
72C2×Q8—2122no—
73C22×C4—1061no—
74C22×C4—1061no—
75C22×C4C4×D5311yescharacteristic
76C22×C4D21211yescharacteristic
77C22×C4C22⋊D5311yescharacteristic
78M16—22no—
79M16—41no—
80C2×C8—41no—
81C2×C8—2121no—
82C4⋊C4—41no—
83C4⋊C4—2122no—
84C23#1171992111yescharacteristic
85Q8—2122no—
86Q8—2122no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—1062no—
94C2×C4—1062no—
95C2×C4—2121no—
96C2×C4—2121no—
97C2×C4—2121no—
98C2×C4—2121no—
99C2×C4#1171992311yescharacteristic
100C2×C4#1171992511yescharacteristic
101C8—42no—
102C8—4241no—
103C22—21no—
104C22—21no—
105C22#1313313111yescharacteristic
106C22#1313314011yescharacteristic
107C22#1313314511yescharacteristic
108C4—21no—
109C4—21no—
110C4—41no—
111C4—41no—
112C4—2122no—
113C4—2122no—
114C2—21no—
115C2—21no—
116C2#1681035911yescharacteristic
117C2#1681041111yescharacteristic
118C2#1681042111yescharacteristic
subgroupG/H
centerC22#13133131
commutatorC22×C4×C53C2×C4
FrattiniC2×C4⋊C4C2×D53
FittingC53×#343C2
socleC22×C53#99
hypercenter#343D53
hypocenterC53#1354
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #343

Complements

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

NH ≅ G/Ntype
63. C53#1354semidirect