Finite Group Databasebeta

← #41001102

Subgroups — #41001102

Order 26816.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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#27211715C211yesmaximal, characteristic, complemented
2#27211712C211yesmaximal, characteristic, complemented
3#27211703C213yesmaximal, complemented
4#27211693C213yesmaximal, complemented
5#27211691C213yesmaximal, complemented
6#27211688C213yesmaximal, complemented
7#126×C419C211yesmaximal, characteristic, complemented
8C23×D419C2211yescharacteristic
9C2×D1676C2213yes—
10C2×D1676C2213yes—
11C2×D1676C2213yescomplemented
12C2×C22⋊D419C2213yes—
13C2×C4×D419C2213yescomplemented
14Dic419⋊C4C2213yescomplemented
15(C2×D419)⋊C4C2213yes—
16(C2×D419)⋊C4C2213yes—
17(C2×D419)⋊C4C2213yescomplemented
18C4⋊C4×C419C2213yescomplemented
19C22⋊C4×C419C2213yes—
20C42×C419C2211yescharacteristic, complemented
21C22×D419—43no—
22C22×D419C2311yescharacteristic
23C22×D419C2311yescharacteristic
24C22×D419C2313yes—
25D1676—43no—
26D1676—43no—
27D1676—43no—
28C22⋊D419—43no—
29C2×Dic419C2313yes—
30C4×D419—43no—
31C23×C419C2311yescharacteristic
32C2×C4×C419C2313yes—
33C2×C4×C419C2313yes—
34C22×C419—23no—
35C22×C419C2411yescharacteristic
36C2×D419—23no—
37C2×D419—23no—
38C2×D419—23no—
39C2×D419—23no—
40C2×D419—23no—
41C2×D419—44no—
42C4×C419—23no—
43C4×C419—23no—
44Dic419—23no—
45C2×C419—41no—
46C2×C419H(5,𝔽2)13yes—
47D419—43no—
48D419—41no—
49D419—41no—
50C419#56011yescharacteristic, complemented, Sylow
51#560—4191nomaximal, Sylow
52#127—4191no—
53#126—4191no—
54#126D41911yescharacteristic, complemented
55#123—4193no—
56#121—4193no—
57#121—4193no—
58#120—4193no—
59C24—4191no—
60C2×D4—4193no—
61C2×D4—4193no—
62C2×D4—4193no—
63C2×D4—4193no—
64C22×C4—4193no—
65C4⋊C4—4193no—
66C4⋊C4C2×D41913yescomplemented
67C22⋊C4—4193no—
68C22⋊C4—4193no—
69C22⋊C4—4193no—
70C22⋊C4C2×D41913yes—
71C42C2×D41911yescharacteristic, complemented
72C23—16763no—
73C23—4191no—
74C23—4191no—
75C23—4193no—
76C23C22×D41911yescharacteristic
77D4—16763no—
78D4—16763no—
79D4—16763no—
80D4—16763no—
81C2×C4—16763no—
82C2×C4—4193no—
83C2×C4C22×D41913yes—
84C2×C4C22×D41913yes—
85C22—23no—
86C22—8383no—
87C22—8383no—
88C22—8383no—
89C22—8383no—
90C22—8383no—
91C22—16764no—
92C22C23×D41911yescharacteristic
93C4—23no—
94C4—23no—
95C4—8383no—
96C2—41no—
97C2—16761no—
98C2—16761no—
99C2—16763no—
100C2#2721176413yes—
subgroupG/H
center, FrattiniC22C23×D419
commutator, socle, cosocleC22×C419C24
Fitting#126×C419C2
hypercenter#126D419
hypocenterC419#560
LCSG ▷ C22×C419 ▷ C419
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #27211715C2semidirect
2. #27211712C2semidirect
3. #27211703C2semidirect
4. #27211693C2semidirect
5. #27211691C2semidirect
6. #27211688C2semidirect
7. #126×C419C2semidirect
11. C2×D1676C22semidirect
13. C2×C4×D419C22semidirect
14. Dic419⋊C4C22semidirect
17. (C2×D419)⋊C4C22semidirect
18. C4⋊C4×C419C22semidirect
20. C42×C419C22semidirect
50. C419#560semidirect
54. #126D419semidirect
66. C4⋊C4C2×D419semidirect
71. C42C2×D419semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#272117152
#272117122
#272117032
#272116932
#272116912
#272116882
#126×C4192
#560419