Finite Group Databasebeta

← #20274010

Subgroups — #20274010

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#16811012C212yesmaximal, complemented
3#16810989C211yesmaximal, characteristic, complemented
4#16810362C212yesmaximal, complemented
5C53×#330C211yesmaximal, characteristic, complemented
6C2×#11719913—22no—
7C2×D424—22no—
8#13133214C2211yescharacteristic
9#13133205C2212yes—
10#13133202—22no—
11#13133195—22no—
12#13133192—22no—
13#13133137—22no—
14#13133128C2211yescharacteristic
15#125×C53C2211yescharacteristic
16#97×C53C2212yes—
17C2×D212—21no—
18C2×D212—22no—
19C2×D212—22no—
20C2×D212D412yes—
21C2×C4×D53—22no—
22C2×Dic106D412yes—
23C2×C8×C53—22no—
24C4⋊C4×C53—22no—
25C4⋊C4×C53D412yes—
26C42×C53C2311yescharacteristic
27#11719921—22no—
28#11719921—22no—
29#11719921—22no—
30#11719919—22no—
31C2×#88322—21no—
32D424—44no—
33#11719913—44no—
34C22×D53—22no—
35C22×D53—41no—
36C2×C4×C53—22no—
37C2×C4×C53C2×D411yescharacteristic
38C2×C4×C53C2×D412yes—
39C2×Dic53—22no—
40D212—42no—
41D212—42no—
42D212—42no—
43D212—42no—
44D212—82no—
45C4×D53—82no—
46Dic106—42no—
47C8×C53—24no—
48#88322—41no—
49C22×C53#12011yescharacteristic
50C2×D53—42no—
51C2×D53—42no—
52C2×D53—42no—
53C2×D53—82no—
54C2×D53—81no—
55C4×C53—22no—
56C4×C53—21no—
57C4×C53—42no—
58Dic53—42no—
59#1656—531nomaximal, Sylow
60C2×C53#45312yes—
61C2×C53#45411yescharacteristic
62D53—82no—
63D53—161no—
64#562—531no—
65#496—532no—
66#490—531no—
67#330D5311yescharacteristic, complemented
68#329—532no—
69C53#165611yescharacteristic, complemented, Sylow
70C2×SD16—1062no—
71C2×D8—1062no—
72#127—531no—
73#125C2×D5311yescharacteristic
74#124—532no—
75#123—1062no—
76C22⋊Q8—1062no—
77#121—1062no—
78#102—1062no—
79#97—531no—
80#97C2×D5312yes—
81C2×Q8—532no—
82C2×D4—532no—
83C2×D4—1062no—
84C2×D4—1061no—
85C2×D4—1062no—
86C22×C4—1062no—
87SD16—2124no—
88D8—2124no—
89C2×C8—22no—
90C2×C8—1061no—
91C4⋊C4—22no—
92C4⋊C4—1062no—
93C4⋊C4D21212yes—
94C22⋊C4—1062no—
95C22⋊C4—1062no—
96C22⋊C4—1062no—
97C42C22×D5311yescharacteristic
98C23—1062no—
99C23—2121no—
100Q8—2122no—
101D4—4242no—
102D4—2122no—
103D4—2122no—
104D4—2122no—
105D4—2122no—
106C2×C4—22no—
107C2×C4—1062no—
108C2×C4—4242no—
109C2×C4C2×D21211yescharacteristic
110C2×C4D4×D5312yes—
111C8—24no—
112C8—2121no—
113C22—4242no—
114C22—4241no—
115C22—2122no—
116C22—2122no—
117C22—2122no—
118C22#1313318711yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—2122no—
123C2—8481no—
124C2—4242no—
125C2#1681079212yes—
126C2#1681079811yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#330C2
socleC22×C53#120
hypercenter#330D53
hypocenterC53#1656
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
1. #16811371C2semidirect
2. #16811012C2semidirect
3. #16810989C2semidirect
4. #16810362C2semidirect
5. C53×#330C2semidirect
67. #330D53semidirect
69. C53#1656semidirect