Finite Group Databasebeta

← #20289708

Subgroups — #20289708

Order 6784.

104 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 30 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#16811518C211yesmaximal, characteristic
2#16811499C211yesmaximal, characteristic
3#16810865C211yesmaximal, characteristic
4C2×#11719936—21no—
5C2×#11719910—21no—
6C2×Dic212C411yescharacteristic
7C2×C8×D53C2211yescharacteristic
8#13133197—21no—
9#13133140—21no—
10#107×C53C411yescharacteristic
11C2×C4×D53D411yescharacteristic
12C2×Dic106—21no—
13C2×#88324—41no—
14#11719936—22no—
15#11719936—41no—
16C2×C8×C53C2×C411yescharacteristic
17C4⋊C4×C53—21no—
18#11719921—41no—
19C4⋊Dic53—41no—
20C2×#88322D411yescharacteristic
21Dic212—41no—
22C8×D53—41no—
23#11719910—41no—
24C22×D53SD1611yescharacteristic
25C2×C4×C53—41no—
26C2×C4×C53C22⋊C411yescharacteristic
27C2×Dic53—41no—
28C2×Dic53D811yescharacteristic
29C4×D53—21no—
30C4×D53—21no—
31Dic106—41no—
32Dic106—41no—
33#88324—42no—
34C8×C53—21no—
35#88322—21no—
36C22×C53#10211yescharacteristic
37C2×D53—21no—
38C2×D53—21no—
39C2×D53—21no—
40C4×C53—81no—
41C4×C53#9911yescharacteristic
42C4×C53#10411yescharacteristic
43Dic53—21no—
44Dic53—81no—
45#1350—531nomaximal, Sylow
46C2×C53#32711yescharacteristic
47C2×C53#35911yescharacteristic
48C2×C53#36211yescharacteristic
49D53—41no—
50#467—531no—
51C22⋊C16—531no—
52#341—531no—
53C53#135011yescharacteristic, complemented, Sylow
54C2×Q16—531no—
55C2×M16—1061no—
56C22×C8—531no—
57C22⋊Q8—1061no—
58C2×C16—1061no—
59#107#615711yescharacteristic
60#103—1061no—
61C2×Q8—1061no—
62C22×C4—531no—
63Q16—2121no—
64M16—1062no—
65M16—2121no—
66C2×C8—531no—
67C2×C8—2121no—
68C2×C8—2121no—
69C2×C8C2×#615711yescharacteristic
70C4⋊C4—21no—
71C4⋊C4—2121no—
72C22⋊C4—2121no—
73C16—2121no—
74C23—531no—
75Q8—2121no—
76Q8—2121no—
77C2×C4—41no—
78C2×C4—531no—
79C2×C4—1061no—
80C2×C4—1061no—
81C2×C4—2121no—
82C2×C4C22⋊#615711yescharacteristic
83C8—21no—
84C8—1061no—
85C8—2122no—
86C22—1061no—
87C22—1061no—
88C22—1061no—
89C22#1313327911yescharacteristic
90C4—81no—
91C4—1061no—
92C4—4241no—
93C4#1313327611yescharacteristic
94C4#1313328211yescharacteristic
95C2—2121no—
96C2#1681147511yescharacteristic
97C2#1681153711yescharacteristic
98C2#1681154211yescharacteristic
subgroupG/H
centerC22#13133279
commutatorC2×C8×C53C2×C4
FrattiniC2×C8C2×#6157
Fitting#107×C53C4
socleC22×C53#102
hypercenter#107#6157
hypocenterC53#1350
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ #107

Complements

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

NH ≅ G/Ntype
53. C53#1350semidirect