Finite Group Databasebeta

← #66945⋊C5 #13182494

Subgroups — #66945⋊C5 #13182494

Order 1760.

152 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×C4×AGL(1,𝔽11)C211yesmaximal, characteristic
2C4⋊C4×C11⋊C5C211yesmaximal, characteristic, complemented
3((C2×D11)⋊C4)⋊C5C212yesmaximal
4C4⋊(Dic11⋊C5)C211yesmaximal, characteristic, complemented
5C4×Dic11⋊C5C212yesmaximal, complemented
6C22×AGL(1,𝔽11)C2211yescharacteristic
7C2×C4×C11⋊C5C2212yes—
8C2×C4×C11⋊C5C2211yescharacteristic
9C2×Dic11⋊C5C2212yes—
10C2×Dic11⋊C5C2211yescharacteristic
11C4×AGL(1,𝔽11)C414yescomplemented
12#66945C511yesmaximal, characteristic, complemented, Hall
13C22×C11⋊C5C2311yescharacteristic
14C2×AGL(1,𝔽11)—21no—
15C2×AGL(1,𝔽11)—21no—
16C2×AGL(1,𝔽11)C2×C412yes—
17C4×C11⋊C5—22no—
18C4×C11⋊C5C2×C412yes—
19Dic11⋊C5—22no—
20Dic11⋊C5C2×C412yes—
21C2×C4×D11C2×C511yescharacteristic
22C11×C4⋊C4C2×C511yescharacteristic, complemented
23(C2×D11)⋊C4C2×C512yes—
24C4⋊Dic11C2×C511yescharacteristic, complemented
25C4×Dic11C2×C512yescomplemented
26C5×#117—111nomaximal, Hall
27C2×C11⋊C5C22×C411yescharacteristic
28C2×C11⋊C5#4111yescharacteristic
29C2×C11⋊C5#4111yescharacteristic
30AGL(1,𝔽11)—22no—
31C22×D11C22×C511yescharacteristic
32C2×C4×C11C22×C512yes—
33C2×C4×C11C22×C511yescharacteristic
34C2×Dic11C22×C512yes—
35C2×Dic11C22×C511yescharacteristic
36C4×D11C4×C514yescomplemented
37C22×C4×C5—111no—
38C5×C4⋊C4—111no—
39C5×C4⋊C4—111no—
40C5×C22⋊C4—112no—
41C42×C5—112no—
42C11⋊C5#11711yescharacteristic, complemented, Hall
43C22×C11C23×C511yescharacteristic
44C2×D11—21no—
45C2×D11—21no—
46C2×D11C2×C4×C512yes—
47C4×C11—22no—
48C4×C11C2×C4×C512yes—
49Dic11—22no—
50Dic11C2×C4×C512yes—
51C23×C5—111no—
52C2×C4×C5—114no—
53C2×C4×C5—111no—
54C2×C4×C5—112no—
55C2×C4×C5—111no—
56C2×C4×C5—112no—
57#117—111noSylow
58C2×C11C22×C4×C511yescharacteristic
59C2×C11C5×#4111yescharacteristic
60C2×C11C5×#4111yescharacteristic
61D11—22no—
62C22×C5—112no—
63C22×C5—111no—
64C22×C5—221no—
65C22×C5—221no—
66C4×C5—112no—
67C4×C5—112no—
68C4×C5—222no—
69C4×C5—222no—
70C22×C4—111no—
71C4⋊C4—111no—
72C4⋊C4AGL(1,𝔽11)11yescharacteristic, complemented
73C22⋊C4—112no—
74C42—112no—
75C11C5×#11711yescharacteristic, complemented, Sylow
76C2×C5—111no—
77C2×C5—111no—
78C2×C5—111no—
79C2×C5—222no—
80C23—111no—
81C2×C4—111no—
82C2×C4—114no—
83C2×C4—112no—
84C2×C4C2×AGL(1,𝔽11)12yes—
85C2×C4C2×AGL(1,𝔽11)11yescharacteristic
86C5—111noSylow
87C22—112no—
88C22—221no—
89C22—221no—
90C22C22×AGL(1,𝔽11)11yescharacteristic
91C4—22no—
92C4—112no—
93C4—222no—
94C4C4×AGL(1,𝔽11)12yes—
95C2—222no—
96C2C2×C4×AGL(1,𝔽11)11yescharacteristic
97C2#4341⋊C511yescharacteristic
98C2#4343⋊C511yescharacteristic
subgroupG/H
center, FrattiniC22C22×AGL(1,𝔽11)
commutatorC2×C11C22×C4×C5
FittingC11×C4⋊C4C2×C5
socle, cosocleC22×C11C23×C5
prime radicalC22×AGL(1,𝔽11)C22
hypercenterC4⋊C4AGL(1,𝔽11)
hypocenterC11C5×#117
LCSG ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×C11⋊C5C2semidirect
4. C4⋊(Dic11⋊C5)C2semidirect
5. C4×Dic11⋊C5C2semidirect
11. C4×AGL(1,𝔽11)C4semidirect
12. #66945C5semidirect
22. C11×C4⋊C4C2×C5semidirect
24. C4⋊Dic11C2×C5semidirect
25. C4×Dic11C2×C5semidirect
36. C4×D11C4×C5semidirect
42. C11⋊C5#117semidirect
72. C4⋊C4AGL(1,𝔽11)semidirect
75. C11C5×#117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×AGL(1,𝔽11)2
C4⋊C4×C11⋊C52
((C2×D11)⋊C4)⋊C52
C4⋊(Dic11⋊C5)2
C4×Dic11⋊C52
#669455
C5×#11711