Finite Group Databasebeta

← #66946⋊C5 #13182495

Subgroups — #66946⋊C5 #13182495

Order 1760.

188 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 75 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
2C2×C4×AGL(1,𝔽11)C212yesmaximal, complemented
3C4⋊C4×C11⋊C5C211yesmaximal, characteristic, complemented
4((C2×D11)⋊C4)⋊C5C212yesmaximal, complemented
5C4×Dic11⋊C5C211yesmaximal, characteristic, complemented
6C22×AGL(1,𝔽11)C2212yes—
7C2×C4×C11⋊C5C2212yes—
8C2×C4×C11⋊C5C2211yescharacteristic
9C2×Dic11⋊C5C2211yescharacteristic
10C2×Dic11⋊C5C2211yescharacteristic
11C4⋊AGL(1,𝔽11)C414yescomplemented
12C4×AGL(1,𝔽11)—24no—
13#66946C511yesmaximal, characteristic, complemented, Hall
14C22×C11⋊C5C2311yescharacteristic
15C2×AGL(1,𝔽11)—22no—
16C2×AGL(1,𝔽11)—22no—
17C2×AGL(1,𝔽11)C2×C414yescomplemented
18C4×C11⋊C5—22no—
19C4×C11⋊C5C2×C412yescomplemented
20Dic11⋊C5—21no—
21Dic11⋊C5D412yescomplemented
22C2×D44C2×C511yescharacteristic
23C2×C4×D11C2×C512yescomplemented
24C11×C4⋊C4C2×C511yescharacteristic, complemented
25(C2×D11)⋊C4C2×C512yescomplemented
26C4×Dic11C2×C511yescharacteristic, complemented
27C4×C5×D4—111nomaximal, Hall
28C2×C11⋊C5C22×C411yescharacteristic
29C2×C11⋊C5C2×D411yescharacteristic
30C2×C11⋊C5#4111yescharacteristic
31AGL(1,𝔽11)—24no—
32C22×D11C22×C512yes—
33C2×C4×C11C22×C512yes—
34C2×C4×C11C22×C511yescharacteristic
35C2×Dic11C22×C511yescharacteristic
36C2×Dic11C22×C511yescharacteristic
37D44C4×C514yescomplemented
38C4×D11—24no—
39C2×C5×D4—111no—
40C22×C4×C5—112no—
41C5×C4⋊C4—111no—
42C5×C22⋊C4—112no—
43C42×C5—111no—
44C11⋊C5C4×D411yescharacteristic, complemented, Hall
45C22×C11C23×C511yescharacteristic
46C2×D11—22no—
47C2×D11—22no—
48C2×D11C2×C4×C514yescomplemented
49C4×C11—22no—
50C4×C11C2×C4×C512yescomplemented
51Dic11—21no—
52Dic11C5×D412yescomplemented
53C23×C5—112no—
54C5×D4—114no—
55C2×C4×C5—111no—
56C2×C4×C5—111no—
57C2×C4×C5—111no—
58C2×C4×C5—112no—
59C2×C4×C5—224no—
60C4×D4—111noSylow
61C2×C11C22×C4×C511yescharacteristic
62C2×C11C2×C5×D411yescharacteristic
63C2×C11C5×#4111yescharacteristic
64D11—24no—
65C22×C5—114no—
66C22×C5—111no—
67C22×C5—222no—
68C22×C5—222no—
69C4×C5—112no—
70C4×C5—112no—
71C4×C5—222no—
72C4×C5—221no—
73C2×D4—111no—
74C22×C4—112no—
75C4⋊C4AGL(1,𝔽11)11yescharacteristic, complemented
76C22⋊C4—112no—
77C42—111no—
78C11C4×C5×D411yescharacteristic, complemented, Sylow
79C2×C5—111no—
80C2×C5—111no—
81C2×C5—111no—
82C2×C5—224no—
83C23—112no—
84D4—114no—
85C2×C4—111no—
86C2×C4—111no—
87C2×C4—224no—
88C2×C4C2×AGL(1,𝔽11)12yes—
89C2×C4C2×AGL(1,𝔽11)11yescharacteristic
90C5—111noSylow
91C22—114no—
92C22—222no—
93C22—222no—
94C22C22×AGL(1,𝔽11)11yescharacteristic
95C4—22no—
96C4—112no—
97C4—221no—
98C4C4×AGL(1,𝔽11)12yescomplemented
99C2—224no—
100C2C2×C4×AGL(1,𝔽11)11yescharacteristic
101C2D4×AGL(1,𝔽11)11yescharacteristic
102C2#4343⋊C511yescharacteristic
subgroupG/H
center, FrattiniC22C22×AGL(1,𝔽11)
commutatorC2×C11C22×C4×C5
FittingC11×C4⋊C4C2×C5
socle, cosocleC22×C11C23×C5
prime radicalC2×C4⋊AGL(1,𝔽11)C2
hypercenterC4⋊C4AGL(1,𝔽11)
hypocenterC11C4×C5×D4
LCSG ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×AGL(1,𝔽11)C2semidirect
3. C4⋊C4×C11⋊C5C2semidirect
4. ((C2×D11)⋊C4)⋊C5C2semidirect
5. C4×Dic11⋊C5C2semidirect
11. C4⋊AGL(1,𝔽11)C4semidirect
13. #66946C5semidirect
17. C2×AGL(1,𝔽11)C2×C4semidirect
19. C4×C11⋊C5C2×C4semidirect
21. Dic11⋊C5D4semidirect
23. C2×C4×D11C2×C5semidirect
24. C11×C4⋊C4C2×C5semidirect
25. (C2×D11)⋊C4C2×C5semidirect
26. C4×Dic11C2×C5semidirect
37. D44C4×C5semidirect
44. C11⋊C5C4×D4semidirect
48. C2×D11C2×C4×C5semidirect
50. C4×C11C2×C4×C5semidirect
52. Dic11C5×D4semidirect
75. C4⋊C4AGL(1,𝔽11)semidirect
78. C11C4×C5×D4semidirect
98. C4C4×AGL(1,𝔽11)semidirect

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
C2×C4×AGL(1,𝔽11)2
C4⋊C4×C11⋊C52
((C2×D11)⋊C4)⋊C52
C4×Dic11⋊C52
#669465
C4×C5×D411