Finite Group Databasebeta

← C2×C42×AGL(1,𝔽11) #72248852

Subgroups — C2×C42×AGL(1,𝔽11) #72248852

Order 3520.

996 conjugacy classes of subgroups; only the 552 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 552 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×AGL(1,𝔽11)C213yesmaximal
2C2×C42×C11⋊C5C211yesmaximal, characteristic, complemented
3C2×C4×Dic11⋊C5C213yesmaximal, complemented
4C42×AGL(1,𝔽11)C218yesmaximal, direct product factor
5C23×AGL(1,𝔽11)C2211yescharacteristic
6C22×C4×C11⋊C5C2213yes—
7C22×Dic11⋊C5C2213yes—
8C2×C4×AGL(1,𝔽11)C4124yesdirect product factor
9C2×C4×AGL(1,𝔽11)C22112yes—
10C42×C11⋊C5C2214yescomplemented
11C4×Dic11⋊C5C22112yescomplemented
12C2×C42×D11C511yesmaximal, characteristic, complemented, Hall
13C23×C11⋊C5C2311yescharacteristic
14C22×AGL(1,𝔽11)C2×C4112yes—
15C22×AGL(1,𝔽11)C2312yes—
16C2×C4×C11⋊C5C2×C4112yescomplemented
17C2×C4×C11⋊C5C2316yes—
18C2×Dic11⋊C5C2×C4112yescomplemented
19C2×Dic11⋊C5C2316yes—
20C4×AGL(1,𝔽11)C2×C4148yesdirect product factor
21C22×C4×D11C2×C513yes—
22C2×C42×C11C2×C511yescharacteristic, complemented
23C2×C4×Dic11C2×C513yescomplemented
24C42×D11C2×C518yescomplemented
25C22×C42×C5—111nomaximal, Hall
26C22×C11⋊C5C22×C416yes—
27C22×C11⋊C5C2411yescharacteristic
28C2×AGL(1,𝔽11)C42116yesdirect product factor
29C2×AGL(1,𝔽11)C22×C4112yes—
30C4×C11⋊C5C22×C4112yescomplemented
31Dic11⋊C5C22×C4112yescomplemented
32C23×D11C22×C511yescharacteristic
33C22×C4×C11C22×C513yes—
34C22×Dic11C22×C513yes—
35C2×C4×D11C4×C5124yescomplemented
36C2×C4×D11C22×C5112yes—
37C42×C11C22×C514yescomplemented
38C4×Dic11C22×C5112yescomplemented
39C23×C4×C5—113no—
40C2×C42×C5—113no—
41C2×C42×C5—118no—
42C2×C42×C5—221no—
43C2×C11⋊C5C2×C4214yescomplemented
44C2×C11⋊C5C23×C413yes—
45AGL(1,𝔽11)C2×C4218yesdirect product factor
46C23×C11C23×C511yescharacteristic
47C22×D11C2×C4×C5112yes—
48C22×D11C23×C512yes—
49C2×C4×C11C2×C4×C5112yescomplemented
50C2×C4×C11C23×C516yes—
51C2×Dic11C2×C4×C5112yescomplemented
52C2×Dic11C23×C516yes—
53C4×D11C2×C4×C5148yescomplemented
54C24×C5—111no—
55C22×C4×C5—113no—
56C22×C4×C5—1112no—
57C22×C4×C5—1124no—
58C22×C4×C5—223no—
59C42×C5—1112no—
60C42×C5—224no—
61C22×C42—111noSylow
62C11⋊C5C22×C4211yescharacteristic, complemented, Hall
63C22×C11C22×C4×C516yes—
64C22×C11C24×C511yescharacteristic
65C2×D11C42×C5116yescomplemented
66C2×D11C22×C4×C5112yes—
67C4×C11C22×C4×C5112yescomplemented
68Dic11C22×C4×C5112yescomplemented
69C23×C5—1112no—
70C23×C5—112no—
71C23×C5—221no—
72C2×C4×C5—1112no—
73C2×C4×C5—116no—
74C2×C4×C5—1148no—
75C2×C4×C5—2212no—
76C2×C4×C5—226no—
77C23×C4—113no—
78C2×C42—113no—
79C2×C42—118no—
80C2×C42AGL(1,𝔽11)11yescharacteristic, direct product factor
81C2×C11C2×C42×C514yescomplemented
82C2×C11C23×C4×C513yes—
83D11C2×C42×C518yescomplemented
84C22×C5—1112no—
85C22×C5—1116no—
86C22×C5—226no—
87C22×C5—221no—
88C4×C5—1112no—
89C4×C5—2212no—
90C24—111no—
91C22×C4—113no—
92C22×C4—1124no—
93C22×C4—1112no—
94C22×C4C2×AGL(1,𝔽11)13yes—
95C42—1112no—
96C42C2×AGL(1,𝔽11)14yesdirect product factor
97C11C22×C42×C511yescharacteristic, complemented, Sylow
98C2×C5—118no—
99C2×C5—224no—
100C2×C5—223no—
101C23—1112no—
102C23—112no—
103C23C22×AGL(1,𝔽11)11yescharacteristic
104C2×C4—1112no—
105C2×C4—116no—
106C2×C4—1148no—
107C2×C4C4×AGL(1,𝔽11)112yesdirect product factor
108C2×C4C22×AGL(1,𝔽11)16yes—
109C5—221noSylow
110C22—1116no—
111C22—1112no—
112C22C2×C4×AGL(1,𝔽11)16yes—
113C22C23×AGL(1,𝔽11)11yescharacteristic
114C4—1112no—
115C4C2×C4×AGL(1,𝔽11)112yesdirect product factor
116C2—118no—
117C2C42×AGL(1,𝔽11)14yesdirect product factor
118C2C22×C4×AGL(1,𝔽11)13yes—
subgroupG/H
center, hypercenterC2×C42AGL(1,𝔽11)
commutator, hypocenterC11C22×C42×C5
FrattiniC22C23×AGL(1,𝔽11)
FittingC2×C42×C11C2×C5
socleC23×C11C23×C5
prime radicalC23×AGL(1,𝔽11)C22
cosocleC22×C11C24×C5
LCSG ▷ C11
UCSC1 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×C42×C11⋊C5C2semidirect
3. C2×C4×Dic11⋊C5C2semidirect
4. C42×AGL(1,𝔽11)C2direct (N × H)
8. C2×C4×AGL(1,𝔽11)C4direct (N × H)
10. C42×C11⋊C5C22semidirect
11. C4×Dic11⋊C5C22semidirect
12. C2×C42×D11C5semidirect
16. C2×C4×C11⋊C5C2×C4semidirect
18. C2×Dic11⋊C5C2×C4semidirect
20. C4×AGL(1,𝔽11)C2×C4direct (N × H)
22. C2×C42×C11C2×C5semidirect
23. C2×C4×Dic11C2×C5semidirect
24. C42×D11C2×C5semidirect
28. C2×AGL(1,𝔽11)C42direct (N × H)
30. C4×C11⋊C5C22×C4semidirect
31. Dic11⋊C5C22×C4semidirect
35. C2×C4×D11C4×C5semidirect
37. C42×C11C22×C5semidirect
38. C4×Dic11C22×C5semidirect
43. C2×C11⋊C5C2×C42semidirect
45. AGL(1,𝔽11)C2×C42direct (N × H)
49. C2×C4×C11C2×C4×C5semidirect
51. C2×Dic11C2×C4×C5semidirect
53. C4×D11C2×C4×C5semidirect
62. C11⋊C5C22×C42semidirect
65. C2×D11C42×C5semidirect
67. C4×C11C22×C4×C5semidirect
68. Dic11C22×C4×C5semidirect
80. C2×C42AGL(1,𝔽11)direct (N × H)
81. C2×C11C2×C42×C5semidirect
83. D11C2×C42×C5semidirect
96. C42C2×AGL(1,𝔽11)direct (N × H)
97. C11C22×C42×C5semidirect
107. C2×C4C4×AGL(1,𝔽11)direct (N × H)
115. C4C2×C4×AGL(1,𝔽11)direct (N × H)
117. C2C42×AGL(1,𝔽11)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×AGL(1,𝔽11)2
C2×C42×C11⋊C52
C2×C4×Dic11⋊C52
C42×AGL(1,𝔽11)2
C2×C42×D115
C22×C42×C511