Finite Group Databasebeta

← #66982⋊C5 #13182530

Subgroups — #66982⋊C5 #13182530

Order 1760.

156 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 55 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
1C22×C4×C11⋊C5C211yesmaximal, characteristic, complemented
2C2×C22⋊AGL(1,𝔽11)C211yesmaximal, characteristic
3C22⋊(Dic11⋊C5)C211yesmaximal, characteristic, complemented
4((C2×D11)⋊C4)⋊C5C212yesmaximal, complemented
5(Dic11⋊C4)⋊C5C212yesmaximal, complemented
6C23×C11⋊C5C2211yescharacteristic
7C22×AGL(1,𝔽11)C2211yescharacteristic
8C2×C4×C11⋊C5—22no—
9C2×C4×C11⋊C5C2212yes—
10C22⋊AGL(1,𝔽11)—22no—
11C2×Dic11⋊C5C2211yescharacteristic
12C2×Dic11⋊C5C2212yes—
13#66982C511yesmaximal, characteristic, complemented, Hall
14C22×C11⋊C5—22no—
15C22×C11⋊C5D412yes—
16C22×C11⋊C5C2311yescharacteristic
17C2×AGL(1,𝔽11)—21no—
18C2×AGL(1,𝔽11)—22no—
19C4×C11⋊C5—22no—
20Dic11⋊C5—21no—
21Dic11⋊C5—22no—
22C22×C4×C11C2×C511yescharacteristic, complemented
23C2×C22⋊D11C2×C511yescharacteristic
24C22⋊Dic11C2×C511yescharacteristic, complemented
25(C2×D11)⋊C4C2×C512yescomplemented
26Dic11⋊C4C2×C512yescomplemented
27C5×#123—111nomaximal, Hall
28C2×C11⋊C5—22no—
29C2×C11⋊C5C2×D411yescharacteristic
30C2×C11⋊C5#4112yes—
31AGL(1,𝔽11)—41no—
32C23×C11C22×C511yescharacteristic
33C22×D11C22×C511yescharacteristic
34C2×C4×C11—22no—
35C2×C4×C11C22×C512yes—
36C22⋊D11—22no—
37C2×Dic11C22×C511yescharacteristic
38C2×Dic11C22×C512yes—
39C2×C5×D4—111no—
40C22×C4×C5—111no—
41C5×C4⋊C4—112no—
42C5×C22⋊C4—112no—
43C5×C22⋊C4—111no—
44C11⋊C5#12311yescharacteristic, complemented, Hall
45C22×C11—22no—
46C22×C11C5×D412yes—
47C22×C11C23×C511yescharacteristic
48C2×D11—21no—
49C2×D11—22no—
50C4×C11—22no—
51Dic11—21no—
52Dic11—22no—
53C23×C5—111no—
54C23×C5—111no—
55C5×D4—222no—
56C2×C4×C5—112no—
57C2×C4×C5—111no—
58C2×C4×C5—112no—
59C2×C4×C5—222no—
60#123—111noSylow
61C2×C11—22no—
62C2×C11C2×C5×D411yescharacteristic
63C2×C11C5×#4112yes—
64D11—41no—
65C22×C5—112no—
66C22×C5—111no—
67C22×C5—222no—
68C22×C5—221no—
69C22×C5—222no—
70C4×C5—222no—
71C4×C5—221no—
72C4×C5—222no—
73C2×D4—111no—
74C22×C4AGL(1,𝔽11)11yescharacteristic, complemented
75C4⋊C4—112no—
76C22⋊C4—111no—
77C22⋊C4—112no—
78C11C5×#12311yescharacteristic, complemented, Sylow
79C2×C5—112no—
80C2×C5—111no—
81C2×C5—222no—
82C2×C5—441no—
83C23—111no—
84C23C2×AGL(1,𝔽11)11yescharacteristic
85D4—222no—
86C2×C4—22no—
87C2×C4—112no—
88C2×C4—111no—
89C2×C4C2×AGL(1,𝔽11)12yes—
90C5—111noSylow
91C22—22no—
92C22—222no—
93C22—221no—
94C22C22⋊AGL(1,𝔽11)12yes—
95C22C22×AGL(1,𝔽11)11yescharacteristic
96C4—22no—
97C4—222no—
98C4—221no—
99C2—22no—
100C2—441no—
101C2#4339⋊C512yes—
102C2C2×C22⋊AGL(1,𝔽11)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×AGL(1,𝔽11)
commutator, socle, cosocleC22×C11C23×C5
FittingC22×C4×C11C2×C5
prime radicalC2×C22⋊AGL(1,𝔽11)C2
hypercenterC22×C4AGL(1,𝔽11)
hypocenterC11C5×#123
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×C11⋊C5C2semidirect
3. C22⋊(Dic11⋊C5)C2semidirect
4. ((C2×D11)⋊C4)⋊C5C2semidirect
5. (Dic11⋊C4)⋊C5C2semidirect
13. #66982C5semidirect
22. C22×C4×C11C2×C5semidirect
24. C22⋊Dic11C2×C5semidirect
25. (C2×D11)⋊C4C2×C5semidirect
26. Dic11⋊C4C2×C5semidirect
44. C11⋊C5#123semidirect
74. C22×C4AGL(1,𝔽11)semidirect
78. C11C5×#123semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C11⋊C52
C2×C22⋊AGL(1,𝔽11)2
C22⋊(Dic11⋊C5)2
((C2×D11)⋊C4)⋊C52
(Dic11⋊C4)⋊C52
#669825
C5×#12311