Finite Group Databasebeta

← C7×C13⋊C3×ΓL(1,𝔽16) #30655421

Subgroups — C7×C13⋊C3×ΓL(1,𝔽16) #30655421

Order 16380.

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

normal subgroups only

This group has 112 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×C7×D5×C13⋊C3C211yesmaximal, characteristic
2C7×C13×ΓL(1,𝔽16)C311yesmaximal, characteristic, complemented
3C7×AGL(1,𝔽5)×C13⋊C3—31nomaximal
4C3×C5×C7×C13⋊C3C411yescharacteristic, complemented, Hall
5C7×Dic3×C13⋊C3—51nomaximal, Hall
6C7×D5×C13⋊C3—21no—
7C7×D5×C13⋊C3D311yescharacteristic
8C3×C7×D5×C13C2×C311yescharacteristic
9C13⋊C3×ΓL(1,𝔽16)C711yesmaximal, characteristic, direct product factor, Hall
10C7×C13×AGL(1,𝔽5)—31noHall
11C2×C3×C7×C13⋊C3—51no—
12C3×C5×C7×C13C3×C411yescharacteristic, complemented
13C5×C7×C13⋊C3—21no—
14C5×C7×C13⋊C3Dic311yescharacteristic, complemented
15C3×C7×ΓL(1,𝔽16)—131nomaximal, Hall
16C3×D5×C13⋊C3C2×C711yescharacteristic
17C7×Dic3×C13—51no—
18C4×C7×C13⋊C3—151no—
19C7×D5×C13C3×D311yescharacteristic
20C3×C7×C13⋊C3AGL(1,𝔽5)11yescharacteristic, complemented, Hall
21C13×ΓL(1,𝔽16)C3×C711yescharacteristic, complemented
22AGL(1,𝔽5)×C13⋊C3—31no—
23C32×C7×D5—131no—
24C3×C5×C13⋊C3C4×C711yescharacteristic, complemented, Hall
25C2×C3×C7×C13—51no—
26C2×C7×C13⋊C3—51no—
27C2×C7×C13⋊C3—101no—
28Dic3×C13⋊C3—51noHall
29C5×C7×C13C3×Dic311yescharacteristic, complemented, Hall
30C7×ΓL(1,𝔽16)C13⋊C311yescharacteristic, direct product factor
31C3×C7×AGL(1,𝔽5)—391no—
32C3×D5×C13C2×C3×C711yescharacteristic
33D5×C13⋊C3—21no—
34D5×C13⋊C3D3×C711yescharacteristic
35C4×C7×C13—151noHall
36C32×C5×C7—131noHall
37C3×C7×C13C3×AGL(1,𝔽5)11yescharacteristic, complemented
38C7×C13⋊C3—21no—
39C7×C13⋊C3ΓL(1,𝔽16)11yescharacteristic, direct product factor
40C13×AGL(1,𝔽5)—31noHall
41C3×C7×Dic3—651noHall
42C2×C3×C13⋊C3—51no—
43C3×C7×D5—131no—
44C3×C7×D5—261no—
45C3×C7×D5C2×C13⋊C311yescharacteristic
46C3×C5×C13C3×C4×C711yescharacteristic, complemented
47C5×C13⋊C3—21no—
48C5×C13⋊C3C7×Dic311yescharacteristic, complemented
49C2×C7×C13—51no—
50C3×ΓL(1,𝔽16)—131noHall
51Dic3×C13—51no—
52C4×C13⋊C3—151no—
53C7×AGL(1,𝔽5)—31noHall
54D5×C13C3×D3×C711yescharacteristic
55C2×C32×C7—651no—
56C3×C13⋊C3C7×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
57C3×C5×C7—131no—
58C3×C5×C7—261no—
59C3×C5×C7C4×C13⋊C311yescharacteristic, complemented
60C7×C13C3×ΓL(1,𝔽16)11yescharacteristic, complemented, Hall
61C32×D5—131no—
62C3×C4×C7—1951no—
63C7×Dic3—51no—
64C2×C3×C13—51no—
65C2×C13⋊C3—51no—
66C2×C13⋊C3—101no—
67C7×D5D3×C13⋊C311yescharacteristic
68C5×C13C3×C7×Dic311yescharacteristic, complemented, Hall
69C32×C7—131noHall
70ΓL(1,𝔽16)C7×C13⋊C311yescharacteristic, direct product factor
71C3×AGL(1,𝔽5)—391no—
72C4×C13—151noHall
73C32×C5—131noHall
74C2×C3×C7—51no—
75C2×C3×C7—651no—
76C2×C3×C7—1301no—
77C3×C13C3×C7×AGL(1,𝔽5)11yescharacteristic, complemented
78C13⋊C3—21no—
79C13⋊C3C7×ΓL(1,𝔽16)11yescharacteristic, direct product factor
80C3×Dic3—651noHall
81C5×C7Dic3×C13⋊C311yescharacteristic, complemented, Hall
82C3×D5—131no—
83C3×D5—261no—
84C3×D5C2×C7×C13⋊C311yescharacteristic
85C4×C7—151noHall
86C2×C13—51no—
87C3×C7—131no—
88C3×C7—261no—
89C3×C7AGL(1,𝔽5)×C13⋊C311yescharacteristic, complemented
90AGL(1,𝔽5)—31noHall
91C2×C32—651no—
92C3×C5—131no—
93C3×C5—261no—
94C3×C5C4×C7×C13⋊C311yescharacteristic, complemented
95C2×C7—51no—
96C13C3×C7×ΓL(1,𝔽16)11yescharacteristic, complemented, Sylow
97C3×C4—1951no—
98Dic3—51no—
99D5D3×C7×C13⋊C311yescharacteristic
100C32—131noSylow
101C7C13⋊C3×ΓL(1,𝔽16)11yescharacteristic, direct product factor, Sylow
102C2×C3—51no—
103C2×C3—651no—
104C2×C3—1301no—
105C5C7×Dic3×C13⋊C311yescharacteristic, complemented, Sylow
106C4—151noSylow
107C3—131no—
108C3—261no—
109C3C7×AGL(1,𝔽5)×C13⋊C311yescharacteristic, complemented
110C2—51no—
subgroupG/H
center, hypercenterC7C13⋊C3×ΓL(1,𝔽16)
commutator, hypocenterC3×C5×C13C3×C4×C7
FrattiniC1G
Fitting, socleC3×C5×C7×C13C3×C4
prime radicalC3×C7×D5×C13⋊C3C2
cosocleC3×D5×C13C2×C3×C7
LCSG ▷ C3×C5×C13
UCSC1 ◁ C7

Complements

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

NH ≅ G/Ntype
2. C7×C13×ΓL(1,𝔽16)C3semidirect
4. C3×C5×C7×C13⋊C3C4semidirect
9. C13⋊C3×ΓL(1,𝔽16)C7direct (N × H)
12. C3×C5×C7×C13C3×C4semidirect
14. C5×C7×C13⋊C3Dic3semidirect
20. C3×C7×C13⋊C3AGL(1,𝔽5)semidirect
21. C13×ΓL(1,𝔽16)C3×C7semidirect
24. C3×C5×C13⋊C3C4×C7semidirect
29. C5×C7×C13C3×Dic3semidirect
30. C7×ΓL(1,𝔽16)C13⋊C3direct (N × H)
37. C3×C7×C13C3×AGL(1,𝔽5)semidirect
39. C7×C13⋊C3ΓL(1,𝔽16)direct (N × H)
46. C3×C5×C13C3×C4×C7semidirect
48. C5×C13⋊C3C7×Dic3semidirect
56. C3×C13⋊C3C7×AGL(1,𝔽5)semidirect
59. C3×C5×C7C4×C13⋊C3semidirect
60. C7×C13C3×ΓL(1,𝔽16)semidirect
68. C5×C13C3×C7×Dic3semidirect
70. ΓL(1,𝔽16)C7×C13⋊C3direct (N × H)
77. C3×C13C3×C7×AGL(1,𝔽5)semidirect
79. C13⋊C3C7×ΓL(1,𝔽16)direct (N × H)
81. C5×C7Dic3×C13⋊C3semidirect
89. C3×C7AGL(1,𝔽5)×C13⋊C3semidirect
94. C3×C5C4×C7×C13⋊C3semidirect
96. C13C3×C7×ΓL(1,𝔽16)semidirect
101. C7C13⋊C3×ΓL(1,𝔽16)direct (N × H)
105. C5C7×Dic3×C13⋊C3semidirect
109. C3C7×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×C7×D5×C13⋊C32
C7×C13×ΓL(1,𝔽16)3
C7×AGL(1,𝔽5)×C13⋊C33
C7×Dic3×C13⋊C35
C13⋊C3×ΓL(1,𝔽16)7
C3×C7×ΓL(1,𝔽16)13