Finite Group Databasebeta

← AGL(1,𝔽5)×ΓL(1,𝔽8)×C11⋊C5 #37528413

Subgroups — AGL(1,𝔽5)×ΓL(1,𝔽8)×C11⋊C5 #37528413

Order 23100.

104 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 36 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
1D5×ΓL(1,𝔽8)×C11⋊C5C211yesmaximal, characteristic
2C7×AGL(1,𝔽5)×C11⋊C5C311yesmaximal, characteristic, complemented, Hall
3C5×ΓL(1,𝔽8)×C11⋊C5C411yescharacteristic, complemented, Hall
4C11×AGL(1,𝔽5)×ΓL(1,𝔽8)C511yesmaximal, characteristic, complemented
5C4×ΓL(1,𝔽8)×C11⋊C5—51nomaximal
6C7×D5×C11⋊C5C2×C311yescharacteristic
7C3×AGL(1,𝔽5)×C11⋊C5—71nomaximal, Hall
8D5×C11×ΓL(1,𝔽8)C2×C511yescharacteristic
9C2×ΓL(1,𝔽8)×C11⋊C5—51no—
10C5×AGL(1,𝔽5)×ΓL(1,𝔽8)—111nomaximal, Hall
11C5×C7×C11⋊C5C3×C411yescharacteristic, complemented, Hall
12C3×D5×C11⋊C5—71no—
13C7×C11×AGL(1,𝔽5)C3×C511yescharacteristic, complemented
14C4×C7×C11⋊C5—51no—
15C5×C11×ΓL(1,𝔽8)C4×C511yescharacteristic, complemented
16ΓL(1,𝔽8)×C11⋊C5—41no—
17ΓL(1,𝔽8)×C11⋊C5AGL(1,𝔽5)11yescharacteristic, direct product factor
18AGL(1,𝔽5)×C11⋊C5ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
19C5×D5×ΓL(1,𝔽8)—111no—
20C4×C11×ΓL(1,𝔽8)—51noHall
21C3×C5×C11⋊C5—71noHall
22C7×D5×C11C2×C3×C511yescharacteristic
23C2×C7×C11⋊C5—51no—
24C5×C7×AGL(1,𝔽5)—111noHall
25C3×C11×AGL(1,𝔽5)—71no—
26C3×C4×C11⋊C5—351no—
27D5×C11⋊C5C2×ΓL(1,𝔽8)11yescharacteristic
28C52×ΓL(1,𝔽8)—111noHall
29C2×C11×ΓL(1,𝔽8)—51no—
30AGL(1,𝔽5)×ΓL(1,𝔽8)C11⋊C511yescharacteristic, direct product factor
31C4×C5×ΓL(1,𝔽8)—551no—
32C5×C7×C11C3×C4×C511yescharacteristic, complemented
33C7×C11⋊C5—41no—
34C7×C11⋊C5C3×AGL(1,𝔽5)11yescharacteristic, complemented
35C5×C7×D5—111no—
36C3×D5×C11—71no—
37C2×C3×C11⋊C5—351no—
38C4×C7×C11—51noHall
39C3×C5×AGL(1,𝔽5)—771noHall
40C5×C11⋊C5C4×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
41C11×ΓL(1,𝔽8)C5×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
42C11×AGL(1,𝔽5)C5×ΓL(1,𝔽8)11yescharacteristic, complemented
43C4×C11⋊C5—51no—
44C2×C5×ΓL(1,𝔽8)—551no—
45D5×ΓL(1,𝔽8)C2×C11⋊C511yescharacteristic
46C52×C7—111noHall
47C3×C5×C11—71no—
48C3×C11⋊C5—71no—
49C3×C11⋊C5—281no—
50C2×C7×C11—51no—
51C3×C5×D5—771no—
52C7×AGL(1,𝔽5)C3×C11⋊C511yescharacteristic, complemented
53C4×C5×C7—551no—
54C3×C4×C11—351noHall
55D5×C11C2×C5×ΓL(1,𝔽8)11yescharacteristic
56C2×C11⋊C5—51no—
57C5×ΓL(1,𝔽8)—111no—
58C5×ΓL(1,𝔽8)—441no—
59C5×ΓL(1,𝔽8)C4×C11⋊C511yescharacteristic, complemented
60C5×AGL(1,𝔽5)—111noHall
61C4×ΓL(1,𝔽8)—51noHall
62C7×C11C3×C5×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
63C3×C52—771noHall
64C2×C5×C7—551no—
65C7×D5C2×C3×C11⋊C511yescharacteristic
66C2×C3×C11—351no—
67C3×AGL(1,𝔽5)—71no—
68C3×C4×C5—3851no—
69C5×C11C4×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
70C11⋊C5—41no—
71C11⋊C5AGL(1,𝔽5)×ΓL(1,𝔽8)11yescharacteristic, direct product factor
72C5×D5—111no—
73C4×C11—51noHall
74C2×ΓL(1,𝔽8)—51no—
75C5×C7—111no—
76C5×C7—441no—
77C5×C7C3×C4×C11⋊C511yescharacteristic, complemented
78C3×C11—71noHall
79C2×C3×C5—3851no—
80C3×D5—71no—
81C4×C7—51noHall
82C52—111noSylow
83C2×C11—51no—
84ΓL(1,𝔽8)AGL(1,𝔽5)×C11⋊C511yescharacteristic, direct product factor, Hall
85AGL(1,𝔽5)ΓL(1,𝔽8)×C11⋊C511yescharacteristic, direct product factor
86C4×C5—551no—
87C3×C5—71no—
88C3×C5—3081no—
89C3×C5—771no—
90C2×C7—51no—
91C3×C4—351noHall
92C11C5×AGL(1,𝔽5)×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
93C2×C5—551no—
94D5C2×ΓL(1,𝔽8)×C11⋊C511yescharacteristic
95C7C3×AGL(1,𝔽5)×C11⋊C511yescharacteristic, complemented, Sylow
96C2×C3—351no—
97C5—111no—
98C5—441no—
99C5C4×ΓL(1,𝔽8)×C11⋊C511yescharacteristic, complemented
100C4—51noSylow
101C3—71noSylow
102C2—51no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C7×C11C3×C4×C5
prime radicalD5×ΓL(1,𝔽8)×C11⋊C5C2
cosocleC7×D5×C11C2×C3×C5
LCSG ▷ C5×C7×C11
UCSC1

Complements

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

NH ≅ G/Ntype
2. C7×AGL(1,𝔽5)×C11⋊C5C3semidirect
3. C5×ΓL(1,𝔽8)×C11⋊C5C4semidirect
4. C11×AGL(1,𝔽5)×ΓL(1,𝔽8)C5semidirect
11. C5×C7×C11⋊C5C3×C4semidirect
13. C7×C11×AGL(1,𝔽5)C3×C5semidirect
15. C5×C11×ΓL(1,𝔽8)C4×C5semidirect
17. ΓL(1,𝔽8)×C11⋊C5AGL(1,𝔽5)direct (N × H)
18. AGL(1,𝔽5)×C11⋊C5ΓL(1,𝔽8)direct (N × H)
30. AGL(1,𝔽5)×ΓL(1,𝔽8)C11⋊C5direct (N × H)
32. C5×C7×C11C3×C4×C5semidirect
34. C7×C11⋊C5C3×AGL(1,𝔽5)semidirect
40. C5×C11⋊C5C4×ΓL(1,𝔽8)semidirect
41. C11×ΓL(1,𝔽8)C5×AGL(1,𝔽5)semidirect
42. C11×AGL(1,𝔽5)C5×ΓL(1,𝔽8)semidirect
52. C7×AGL(1,𝔽5)C3×C11⋊C5semidirect
59. C5×ΓL(1,𝔽8)C4×C11⋊C5semidirect
62. C7×C11C3×C5×AGL(1,𝔽5)semidirect
69. C5×C11C4×C5×ΓL(1,𝔽8)semidirect
71. C11⋊C5AGL(1,𝔽5)×ΓL(1,𝔽8)direct (N × H)
77. C5×C7C3×C4×C11⋊C5semidirect
84. ΓL(1,𝔽8)AGL(1,𝔽5)×C11⋊C5direct (N × H)
85. AGL(1,𝔽5)ΓL(1,𝔽8)×C11⋊C5direct (N × H)
92. C11C5×AGL(1,𝔽5)×ΓL(1,𝔽8)semidirect
95. C7C3×AGL(1,𝔽5)×C11⋊C5semidirect
99. C5C4×ΓL(1,𝔽8)×C11⋊C5semidirect

Maximal subgroups

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

subgroupindex
D5×ΓL(1,𝔽8)×C11⋊C52
C7×AGL(1,𝔽5)×C11⋊C53
C11×AGL(1,𝔽5)×ΓL(1,𝔽8)5
C4×ΓL(1,𝔽8)×C11⋊C55
C3×AGL(1,𝔽5)×C11⋊C57
C5×AGL(1,𝔽5)×ΓL(1,𝔽8)11