Finite Group Databasebeta

← C5×C13⋊C3×ΓL(1,𝔽16) #26043610

Subgroups — C5×C13⋊C3×ΓL(1,𝔽16) #26043610

Order 11700.

124 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C3×C5×D5×C13⋊C3C211yesmaximal, characteristic
2C5×C13×ΓL(1,𝔽16)C311yesmaximal, characteristic, complemented
3C5×AGL(1,𝔽5)×C13⋊C3—31nomaximal
4C3×C52×C13⋊C3C411yescharacteristic, complemented, Hall
5C13⋊C3×ΓL(1,𝔽16)C511yesmaximal, characteristic, direct product factor
6C5×Dic3×C13⋊C3—51nomaximal
7C3×C5×D5×C13C2×C311yescharacteristic
8C5×D5×C13⋊C3—21no—
9C5×D5×C13⋊C3D311yescharacteristic
10C5×C13×AGL(1,𝔽5)—31noHall
11C2×C3×C5×C13⋊C3—51no—
12C3×D5×C13⋊C3C2×C511yescharacteristic
13C3×C52×C13C3×C411yescharacteristic, complemented
14C52×C13⋊C3—21no—
15C52×C13⋊C3Dic311yescharacteristic, complemented
16C3×C5×ΓL(1,𝔽16)—131nomaximal, Hall
17C13×ΓL(1,𝔽16)C3×C511yescharacteristic, complemented
18AGL(1,𝔽5)×C13⋊C3—31no—
19C5×Dic3×C13—51no—
20C4×C5×C13⋊C3—151no—
21C5×D5×C13C3×D311yescharacteristic
22C3×C5×C13⋊C3—41no—
23C3×C5×C13⋊C3C4×C511yescharacteristic, complemented
24C3×C5×C13⋊C3AGL(1,𝔽5)11yescharacteristic, complemented
25Dic3×C13⋊C3—51noHall
26C32×C5×D5—131no—
27C2×C3×C5×C13—51no—
28C3×D5×C13C2×C3×C511yescharacteristic
29C2×C5×C13⋊C3—51no—
30C2×C5×C13⋊C3—101no—
31D5×C13⋊C3—21no—
32D5×C13⋊C3C5×D311yescharacteristic
33C52×C13C3×Dic311yescharacteristic, complemented, Hall
34C5×ΓL(1,𝔽16)C13⋊C311yescharacteristic, direct product factor
35C3×C5×AGL(1,𝔽5)—391no—
36C13×AGL(1,𝔽5)—31no—
37C4×C5×C13—151no—
38C2×C3×C13⋊C3—51no—
39C32×C52—131noHall
40C3×C5×C13—41no—
41C3×C5×C13C3×C4×C511yescharacteristic, complemented
42C3×C5×C13C3×AGL(1,𝔽5)11yescharacteristic, complemented
43C5×C13⋊C3—21no—
44C5×C13⋊C3—21no—
45C5×C13⋊C3—42no—
46C5×C13⋊C3—41no—
47C5×C13⋊C3C5×Dic311yescharacteristic, complemented
48C5×C13⋊C3ΓL(1,𝔽16)11yescharacteristic, direct product factor
49C3×ΓL(1,𝔽16)—131no—
50C3×C5×Dic3—651no—
51Dic3×C13—51no—
52C4×C13⋊C3—151no—
53C3×C5×D5—131no—
54C3×C5×D5—261no—
55C3×C5×D5C2×C13⋊C311yescharacteristic
56C2×C5×C13—51no—
57D5×C13C3×C5×D311yescharacteristic
58C3×C13⋊C3C5×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
59C5×AGL(1,𝔽5)—31noHall
60C2×C32×C5—651no—
61C32×D5—131no—
62C2×C3×C13—51no—
63C2×C13⋊C3—51no—
64C2×C13⋊C3—101no—
65C3×C52—131no—
66C3×C52—261no—
67C3×C52C4×C13⋊C311yescharacteristic, complemented
68C5×C13—41no—
69C5×C13C3×C5×Dic311yescharacteristic, complemented
70C5×C13C3×ΓL(1,𝔽16)11yescharacteristic, complemented
71ΓL(1,𝔽16)C5×C13⋊C311yescharacteristic, direct product factor
72C3×AGL(1,𝔽5)—391no—
73C3×C4×C5—1951no—
74C5×Dic3—51no—
75C4×C13—151noHall
76C5×D5D3×C13⋊C311yescharacteristic
77C32×C5—131no—
78C32×C5—131no—
79C32×C5—521no—
80C3×C13C3×C5×AGL(1,𝔽5)11yescharacteristic, complemented
81C13⋊C3—21no—
82C13⋊C3C5×ΓL(1,𝔽16)11yescharacteristic, direct product factor
83C3×Dic3—651noHall
84C2×C3×C5—51no—
85C2×C3×C5—651no—
86C2×C3×C5—1301no—
87C3×D5—131no—
88C3×D5—261no—
89C3×D5C2×C5×C13⋊C311yescharacteristic
90C2×C13—51no—
91C52Dic3×C13⋊C311yescharacteristic, complemented, Sylow
92AGL(1,𝔽5)—31no—
93C4×C5—151no—
94C2×C32—651no—
95C3×C5—41no—
96C3×C5—131no—
97C3×C5—131no—
98C3×C5—261no—
99C3×C5—261no—
100C3×C5—521no—
101C3×C5—522no—
102C3×C5C4×C5×C13⋊C311yescharacteristic, complemented
103C3×C5AGL(1,𝔽5)×C13⋊C311yescharacteristic, complemented
104C13C3×C5×ΓL(1,𝔽16)11yescharacteristic, complemented, Sylow
105C3×C4—1951no—
106Dic3—51no—
107C2×C5—51no—
108D5C5×D3×C13⋊C311yescharacteristic
109C32—131noSylow
110C2×C3—51no—
111C2×C3—651no—
112C2×C3—1301no—
113C5—41no—
114C5C5×Dic3×C13⋊C311yescharacteristic, complemented
115C5C13⋊C3×ΓL(1,𝔽16)11yescharacteristic, direct product factor
116C4—151noSylow
117C3—131no—
118C3—261no—
119C3C5×AGL(1,𝔽5)×C13⋊C311yescharacteristic, complemented
120C2—51no—
subgroupG/H
center, hypercenterC5C13⋊C3×ΓL(1,𝔽16)
commutator, hypocenterC3×C5×C13C3×C4×C5
FrattiniC1G
Fitting, socleC3×C52×C13C3×C4
prime radicalC3×C5×D5×C13⋊C3C2
cosocleC3×D5×C13C2×C3×C5
LCSG ▷ C3×C5×C13
UCSC1 ◁ C5

Complements

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

NH ≅ G/Ntype
2. C5×C13×ΓL(1,𝔽16)C3semidirect
4. C3×C52×C13⋊C3C4semidirect
5. C13⋊C3×ΓL(1,𝔽16)C5direct (N × H)
13. C3×C52×C13C3×C4semidirect
15. C52×C13⋊C3Dic3semidirect
17. C13×ΓL(1,𝔽16)C3×C5semidirect
23. C3×C5×C13⋊C3C4×C5semidirect
24. C3×C5×C13⋊C3AGL(1,𝔽5)semidirect
33. C52×C13C3×Dic3semidirect
34. C5×ΓL(1,𝔽16)C13⋊C3direct (N × H)
41. C3×C5×C13C3×C4×C5semidirect
42. C3×C5×C13C3×AGL(1,𝔽5)semidirect
47. C5×C13⋊C3C5×Dic3semidirect
48. C5×C13⋊C3ΓL(1,𝔽16)direct (N × H)
58. C3×C13⋊C3C5×AGL(1,𝔽5)semidirect
67. C3×C52C4×C13⋊C3semidirect
69. C5×C13C3×C5×Dic3semidirect
70. C5×C13C3×ΓL(1,𝔽16)semidirect
71. ΓL(1,𝔽16)C5×C13⋊C3direct (N × H)
80. C3×C13C3×C5×AGL(1,𝔽5)semidirect
82. C13⋊C3C5×ΓL(1,𝔽16)direct (N × H)
91. C52Dic3×C13⋊C3semidirect
102. C3×C5C4×C5×C13⋊C3semidirect
103. C3×C5AGL(1,𝔽5)×C13⋊C3semidirect
104. C13C3×C5×ΓL(1,𝔽16)semidirect
114. C5C5×Dic3×C13⋊C3semidirect
115. C5C13⋊C3×ΓL(1,𝔽16)direct (N × H)
119. C3C5×AGL(1,𝔽5)×C13⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×C5×D5×C13⋊C32
C5×C13×ΓL(1,𝔽16)3
C5×AGL(1,𝔽5)×C13⋊C33
C13⋊C3×ΓL(1,𝔽16)5
C5×Dic3×C13⋊C35
C3×C5×ΓL(1,𝔽16)13