Finite Group Databasebeta

← C13×ΓL(1,𝔽8)×ΓL(1,𝔽16) #30655434

Subgroups — C13×ΓL(1,𝔽8)×ΓL(1,𝔽16) #30655434

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×D5×C13×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C7×C13×ΓL(1,𝔽16)C311yesmaximal, characteristic, complemented
3C13×AGL(1,𝔽5)×ΓL(1,𝔽8)—31nomaximal
4C3×C5×C13×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
5Dic3×C13×ΓL(1,𝔽8)—51nomaximal, Hall
6D5×C13×ΓL(1,𝔽8)—21no—
7D5×C13×ΓL(1,𝔽8)D311yescharacteristic
8C3×C7×D5×C13C2×C311yescharacteristic
9C3×C13×ΓL(1,𝔽16)—71nomaximal, Hall
10C7×C13×AGL(1,𝔽5)—31noHall
11C2×C3×C13×ΓL(1,𝔽8)—51no—
12C3×C5×C7×C13C3×C411yescharacteristic, complemented
13C5×C13×ΓL(1,𝔽8)—21no—
14C5×C13×ΓL(1,𝔽8)Dic311yescharacteristic, complemented
15ΓL(1,𝔽8)×ΓL(1,𝔽16)C1311yesmaximal, characteristic, direct product factor, Hall
16C32×D5×C13—71no—
17C7×Dic3×C13—51no—
18C4×C13×ΓL(1,𝔽8)—151no—
19C7×D5×C13C3×D311yescharacteristic
20C3×C13×ΓL(1,𝔽8)AGL(1,𝔽5)11yescharacteristic, complemented, Hall
21C13×ΓL(1,𝔽16)ΓL(1,𝔽8)11yescharacteristic, direct product factor
22C3×C13×AGL(1,𝔽5)—211no—
23C3×D5×ΓL(1,𝔽8)C2×C1311yescharacteristic
24C32×C5×C13—71noHall
25C2×C3×C7×C13—51no—
26C2×C13×ΓL(1,𝔽8)—51no—
27C2×C13×ΓL(1,𝔽8)—101no—
28C3×Dic3×C13—351noHall
29C5×C7×C13C3×Dic311yescharacteristic, complemented, Hall
30C7×ΓL(1,𝔽16)C3×C1311yescharacteristic, complemented
31AGL(1,𝔽5)×ΓL(1,𝔽8)—31no—
32C3×D5×C13—71no—
33C3×D5×C13—141no—
34C3×D5×C13C2×ΓL(1,𝔽8)11yescharacteristic
35C4×C7×C13—151noHall
36C3×C5×ΓL(1,𝔽8)C4×C1311yescharacteristic, complemented, Hall
37C3×C7×C13C3×AGL(1,𝔽5)11yescharacteristic, complemented
38C13×ΓL(1,𝔽8)—21no—
39C13×ΓL(1,𝔽8)ΓL(1,𝔽16)11yescharacteristic, direct product factor
40C13×AGL(1,𝔽5)—31noHall
41Dic3×ΓL(1,𝔽8)—51noHall
42C2×C32×C13—351no—
43C3×C7×D5C2×C3×C1311yescharacteristic
44D5×ΓL(1,𝔽8)—21no—
45D5×ΓL(1,𝔽8)D3×C1311yescharacteristic
46C3×C5×C13—71no—
47C3×C5×C13—141no—
48C3×C5×C13C4×ΓL(1,𝔽8)11yescharacteristic, complemented
49C2×C7×C13—51no—
50C3×ΓL(1,𝔽16)—71noHall
51C3×C4×C13—1051no—
52Dic3×C13—51no—
53C7×AGL(1,𝔽5)—31noHall
54D5×C13D3×ΓL(1,𝔽8)11yescharacteristic
55C2×C3×ΓL(1,𝔽8)—51no—
56C32×C13—71noHall
57C3×C5×C7C3×C4×C1311yescharacteristic, complemented
58C5×ΓL(1,𝔽8)—21no—
59C5×ΓL(1,𝔽8)Dic3×C1311yescharacteristic, complemented
60C7×C13C3×ΓL(1,𝔽16)11yescharacteristic, complemented, Hall
61C32×D5—71no—
62C7×Dic3—51no—
63C4×ΓL(1,𝔽8)—151no—
64C2×C3×C13—51no—
65C2×C3×C13—351no—
66C2×C3×C13—701no—
67C7×D5C3×D3×C1311yescharacteristic
68C5×C13Dic3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
69C3×ΓL(1,𝔽8)C13×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
70ΓL(1,𝔽16)C13×ΓL(1,𝔽8)11yescharacteristic, direct product factor
71C3×AGL(1,𝔽5)—211no—
72C4×C13—151noHall
73C32×C5—71noHall
74C2×C3×C7—51no—
75C2×ΓL(1,𝔽8)—51no—
76C2×ΓL(1,𝔽8)—101no—
77C3×C13—71no—
78C3×C13—141no—
79C3×C13AGL(1,𝔽5)×ΓL(1,𝔽8)11yescharacteristic, complemented
80C3×Dic3—351noHall
81C5×C7C3×Dic3×C1311yescharacteristic, complemented, Hall
82C3×D5—71no—
83C3×D5—141no—
84C3×D5C2×C13×ΓL(1,𝔽8)11yescharacteristic
85C4×C7—151noHall
86C2×C13—51no—
87C3×C7C3×C13×AGL(1,𝔽5)11yescharacteristic, complemented
88ΓL(1,𝔽8)—21no—
89ΓL(1,𝔽8)C13×ΓL(1,𝔽16)11yescharacteristic, direct product factor
90AGL(1,𝔽5)—31noHall
91C2×C32—351no—
92C3×C5—71no—
93C3×C5—141no—
94C3×C5C4×C13×ΓL(1,𝔽8)11yescharacteristic, complemented
95C2×C7—51no—
96C13ΓL(1,𝔽8)×ΓL(1,𝔽16)11yescharacteristic, direct product factor, Sylow
97C3×C4—1051no—
98Dic3—51no—
99D5D3×C13×ΓL(1,𝔽8)11yescharacteristic
100C32—71noSylow
101C7C3×C13×ΓL(1,𝔽16)11yescharacteristic, complemented, Sylow
102C2×C3—51no—
103C2×C3—351no—
104C2×C3—701no—
105C5Dic3×C13×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
106C4—151noSylow
107C3—71no—
108C3—141no—
109C3C13×AGL(1,𝔽5)×ΓL(1,𝔽8)11yescharacteristic, complemented
110C2—51no—
subgroupG/H
center, hypercenterC13ΓL(1,𝔽8)×ΓL(1,𝔽16)
commutator, hypocenterC3×C5×C7C3×C4×C13
FrattiniC1G
Fitting, socleC3×C5×C7×C13C3×C4
prime radicalC3×D5×C13×ΓL(1,𝔽8)C2
cosocleC3×C7×D5C2×C3×C13
LCSG ▷ C3×C5×C7
UCSC1 ◁ C13

Complements

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

NH ≅ G/Ntype
2. C7×C13×ΓL(1,𝔽16)C3semidirect
4. C3×C5×C13×ΓL(1,𝔽8)C4semidirect
12. C3×C5×C7×C13C3×C4semidirect
14. C5×C13×ΓL(1,𝔽8)Dic3semidirect
15. ΓL(1,𝔽8)×ΓL(1,𝔽16)C13direct (N × H)
20. C3×C13×ΓL(1,𝔽8)AGL(1,𝔽5)semidirect
21. C13×ΓL(1,𝔽16)ΓL(1,𝔽8)direct (N × H)
29. C5×C7×C13C3×Dic3semidirect
30. C7×ΓL(1,𝔽16)C3×C13semidirect
36. C3×C5×ΓL(1,𝔽8)C4×C13semidirect
37. C3×C7×C13C3×AGL(1,𝔽5)semidirect
39. C13×ΓL(1,𝔽8)ΓL(1,𝔽16)direct (N × H)
48. C3×C5×C13C4×ΓL(1,𝔽8)semidirect
57. C3×C5×C7C3×C4×C13semidirect
59. C5×ΓL(1,𝔽8)Dic3×C13semidirect
60. C7×C13C3×ΓL(1,𝔽16)semidirect
68. C5×C13Dic3×ΓL(1,𝔽8)semidirect
69. C3×ΓL(1,𝔽8)C13×AGL(1,𝔽5)semidirect
70. ΓL(1,𝔽16)C13×ΓL(1,𝔽8)direct (N × H)
79. C3×C13AGL(1,𝔽5)×ΓL(1,𝔽8)semidirect
81. C5×C7C3×Dic3×C13semidirect
87. C3×C7C3×C13×AGL(1,𝔽5)semidirect
89. ΓL(1,𝔽8)C13×ΓL(1,𝔽16)direct (N × H)
94. C3×C5C4×C13×ΓL(1,𝔽8)semidirect
96. C13ΓL(1,𝔽8)×ΓL(1,𝔽16)direct (N × H)
101. C7C3×C13×ΓL(1,𝔽16)semidirect
105. C5Dic3×C13×ΓL(1,𝔽8)semidirect
109. C3C13×AGL(1,𝔽5)×ΓL(1,𝔽8)semidirect

Maximal subgroups

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

subgroupindex
C3×D5×C13×ΓL(1,𝔽8)2
C7×C13×ΓL(1,𝔽16)3
C13×AGL(1,𝔽5)×ΓL(1,𝔽8)3
Dic3×C13×ΓL(1,𝔽8)5
C3×C13×ΓL(1,𝔽16)7
ΓL(1,𝔽8)×ΓL(1,𝔽16)13