Finite Group Databasebeta

← #20274007

Subgroups — #20274007

Order 6784.

230 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 45 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#16811371C211yesmaximal, characteristic, complemented
2#16811010C212yesmaximal, complemented
3#16810365C211yesmaximal, characteristic, complemented
4#16810362C212yesmaximal, complemented
5C53×#490C211yesmaximal, characteristic, complemented
6C2×#11719923—22no—
7C2×#11719922—22no—
8#13133205C2212yes—
9#13133202—22no—
10#13133195—22no—
11#13133192—22no—
12#13133128C2212yes—
13#127×C53C2211yescharacteristic
14#125×C53C2211yescharacteristic
15#102×C53—22no—
16#97×C53C2211yescharacteristic
17C2×D4×C53—21no—
18C2×D4×C53—22no—
19C2×D212—22no—
20C2×D212D412yes—
21C2×C4×D53—22no—
22C2×Dic106D412yes—
23C2×C8×C53—21no—
24C4⋊C4×C53—22no—
25C4⋊C4×C53D412yes—
26C42×C53C2311yescharacteristic
27#11719923—44no—
28#11719922—44no—
29#11719921—22no—
30#11719921—22no—
31#11719921—22no—
32#11719919—22no—
33C2×#88322—22no—
34C23×C53—41no—
35C22×D53—22no—
36D4×C53—42no—
37D4×C53—42no—
38D4×C53—42no—
39C2×C4×C53—22no—
40C2×C4×C53C2×D412yes—
41C2×C4×C53C2×D411yescharacteristic
42C2×Dic53—22no—
43D212—42no—
44D212—82no—
45C4×D53—82no—
46Dic106—42no—
47C8×C53—41no—
48#88322—24no—
49C22×C53—82no—
50C22×C53—81no—
51C22×C53#12011yescharacteristic
52C2×D53—42no—
53C2×D53—42no—
54C2×D53—42no—
55C4×C53—22no—
56C4×C53—21no—
57C4×C53—42no—
58Dic53—42no—
59#1656—531nomaximal, Sylow
60C2×C53—161no—
61C2×C53#45312yes—
62C2×C53#45411yescharacteristic
63D53—82no—
64#562—531no—
65#496—532no—
66#490D5311yescharacteristic, complemented
67#330—531no—
68#329—532no—
69C53#165611yescharacteristic, complemented, Sylow
70C2×SD16—1062no—
71C2×D8—1062no—
72#127C2×D5311yescharacteristic
73#125C2×D5311yescharacteristic
74#124—532no—
75#123—1062no—
76C22⋊Q8—1062no—
77#121—1062no—
78#102—22no—
79#97—532no—
80#97C2×D5311yescharacteristic
81C2×Q8—532no—
82C2×D4—22no—
83C2×D4—21no—
84C2×D4—532no—
85C2×D4—1062no—
86C22×C4—1062no—
87SD16—2124no—
88D8—2124no—
89C2×C8—21no—
90C2×C8—1062no—
91C4⋊C4—22no—
92C4⋊C4—1062no—
93C4⋊C4C22⋊D5312yes—
94C22⋊C4—1062no—
95C22⋊C4—1062no—
96C22⋊C4—1062no—
97C42C22×D5311yescharacteristic
98C23—41no—
99C23—1062no—
100Q8—2122no—
101D4—42no—
102D4—42no—
103D4—42no—
104D4—4242no—
105D4—2122no—
106C2×C4—22no—
107C2×C4—1062no—
108C2×C4—4242no—
109C2×C4D4×D5312yes—
110C2×C4C2×C22⋊D5311yescharacteristic
111C8—41no—
112C8—1064no—
113C22—82no—
114C22—81no—
115C22—2122no—
116C22—2122no—
117C22—2122no—
118C22#1313318511yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—2122no—
123C2—161no—
124C2—4242no—
125C2#1681078912yes—
126C2#1681079511yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#490C2
socleC22×C53#120
hypercenter#490D53
hypocenterC53#1656
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #16811371C2semidirect
2. #16811010C2semidirect
3. #16810365C2semidirect
4. #16810362C2semidirect
5. C53×#490C2semidirect
66. #490D53semidirect
69. C53#1656semidirect