Finite Group Databasebeta

← #20274362

Subgroups — #20274362

Order 6784.

164 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 57 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#16811026C211yesmaximal, characteristic
2#16811019C211yesmaximal, characteristic
3#16810975C211yesmaximal, characteristic
4#16810974C211yesmaximal, characteristic
5#16810973C211yesmaximal, characteristic
6#16810344C211yesmaximal, characteristic
7C53×#487C211yesmaximal, characteristic
8#13133217C2211yescharacteristic
9#13133217C2211yescharacteristic
10#13133150—22no—
11#13133148—22no—
12#13133142—22no—
13#13133141—22no—
14#13133140—22no—
15C4×#88322C2211yescharacteristic
16C4×#88322C2211yescharacteristic
17C4⋊Q8×C53C2211yescharacteristic
18C4⋊Q8×C53C2211yescharacteristic
19#103×C53—22no—
20C4×C8×C53C2211yescharacteristic
21C2×Q8×C53—21no—
22C2×Q8×C53—21no—
23C2×Dic106—21no—
24C2×Dic106—21no—
25C2×C8×C53D412yes—
26C4⋊C4×C53—21no—
27C4⋊C4×C53—21no—
28C4⋊C4×C53—21no—
29C4⋊C4×C53—21no—
30C42×C53C2311yescharacteristic
31C4⋊Dic53—21no—
32C4⋊Dic53—21no—
33C4⋊Dic53—21no—
34C4⋊Dic53—21no—
35C2×#88322D412yes—
36C2×#88322D412yes—
37Q8×C53—42no—
38Q8×C53—42no—
39C2×C4×C53—41no—
40C2×C4×C53—41no—
41C2×C4×C53C2×D411yescharacteristic
42C2×C4×C53C2×D411yescharacteristic
43C2×C4×C53C2×D411yescharacteristic
44C2×Dic53—41no—
45C2×Dic53—41no—
46Dic106—42no—
47Dic106—42no—
48C8×C53—22no—
49#88322—22no—
50#88322—22no—
51C22×C53#12711yescharacteristic
52C4×C53—81no—
53C4×C53—81no—
54C4×C53#13512yes—
55C4×C53#13712yes—
56C4×C53#13712yes—
57Dic53—81no—
58Dic53—81no—
59#1718—531nomaximal, Sylow
60C2×C53#49511yescharacteristic
61C2×C53#49711yescharacteristic
62C2×C53#49711yescharacteristic
63#500—531no—
64#498—531no—
65#487—531no—
66#487—531no—
67#487—531no—
68#487D5311yescharacteristic
69#322—531no—
70C53#171811yescharacteristic, complemented, Sylow
71C4⋊Q8—531no—
72C4⋊Q8—531no—
73C4⋊Q8C2×D5311yescharacteristic
74C4⋊Q8C2×D5311yescharacteristic
75#107—1062no—
76#106—1062no—
77#103—22no—
78#103—1062no—
79#103—1062no—
80#103—1062no—
81C4×C8—531no—
82C4×C8—531no—
83C4×C8C2×D5311yescharacteristic
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—1061no—
87C2×Q8—1061no—
88C2×C8—532no—
89C2×C8—532no—
90C2×C8C22⋊D5312yes—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—1061no—
96C4⋊C4—1061no—
97C4⋊C4—1061no—
98C4⋊C4—1061no—
99C42C22×D5311yescharacteristic
100Q8—42no—
101Q8—42no—
102Q8—2122no—
103Q8—2122no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—2121no—
107C2×C4—2121no—
108C2×C4D4×D5311yescharacteristic
109C2×C4D4×D5311yescharacteristic
110C2×C4C2×C22⋊D5311yescharacteristic
111C8—22no—
112C8—1062no—
113C8—1062no—
114C22#1313321311yescharacteristic
115C4—81no—
116C4—81no—
117C4—4241no—
118C4—4241no—
119C4#1313323712yes—
120C4#1313324612yes—
121C4#1313324912yes—
122C2#1681100511yescharacteristic
123C2#1681101411yescharacteristic
124C2#1681101611yescharacteristic
subgroupG/H
centerC22#13133213
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#487C2
socleC22×C53#127
hypercenter#487D53
hypocenterC53#1718
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
70. C53#1718semidirect