Finite Group Databasebeta

← C3⋊(ΓL(1,𝔽8)⋊#63121) #30655430

Subgroups — C3⋊(ΓL(1,𝔽8)⋊#63121) #30655430

Order 16380.

112 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 32 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×ΓL(1,𝔽8)×D65C211yesmaximal, characteristic
2C3⋊(C7⋊#63121)C311yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)⋊#63121—31nomaximal
4C3×C5×C13×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
5C3⋊(ΓL(1,𝔽8)⋊#261)—51nomaximal, Hall
6ΓL(1,𝔽8)×D65—21no—
7ΓL(1,𝔽8)×D65D311yescharacteristic
8C3×C7×D65C2×C311yescharacteristic
9C3×C3⋊#63121—71nomaximal, Hall
10C7⋊#63121—31noHall
11C3×ΓL(1,𝔽8)×D13—51no—
12C3×C5×C7×C13C3×C411yescharacteristic, complemented
13C5×C13×ΓL(1,𝔽8)—21no—
14C5×C13×ΓL(1,𝔽8)Dic311yescharacteristic, complemented
15C3⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))—131nomaximal, Hall
16C32×D65—141no—
17C3⋊(C7⋊#261)—51no—
18ΓL(1,𝔽8)⋊#261—151no—
19C7×D65C3×D311yescharacteristic
20C3×C13×ΓL(1,𝔽8)AGL(1,𝔽5)11yescharacteristic, complemented, Hall
21C3⋊#63121—71no—
22C3×#63121—211no—
23C3×D5×ΓL(1,𝔽8)—131no—
24C32×C5×C13—141noHall
25C3×C7×D13—51no—
26ΓL(1,𝔽8)×D13—51no—
27ΓL(1,𝔽8)×D13—101no—
28C3×C3⋊#261—351noHall
29C5×C7×C13C3×Dic311yescharacteristic, complemented, Hall
30C3⋊(C7⋊AGL(1,𝔽5))—131no—
31ΓL(1,𝔽8)⋊AGL(1,𝔽5)—391no—
32C3×D65—141no—
33C3×D65—141no—
34C3×D65AGL(1,𝔽7)11yescharacteristic
35C7⋊#261—151noHall
36C3×C5×ΓL(1,𝔽8)#26111yescharacteristic, complemented, Hall
37C3×C7×C13C3×AGL(1,𝔽5)11yescharacteristic, complemented
38C13×ΓL(1,𝔽8)—21no—
39C13×ΓL(1,𝔽8)ΓL(1,𝔽16)11yescharacteristic, complemented
40#63121—211noHall
41C3⋊(ΓL(1,𝔽8)⋊C4)—651noHall
42C32×D13—701no—
43C3×C7×D5—131no—
44D5×ΓL(1,𝔽8)—131no—
45D5×ΓL(1,𝔽8)—261no—
46C3×C5×C13—141no—
47C3×C5×C13—141no—
48C3×C5×C13ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
49C7×D13—51no—
50C3×ΓL(1,𝔽16)—911noHall
51C3⋊#261—351no—
52C3×#261—1051no—
53C7⋊AGL(1,𝔽5)—391noHall
54D65C3⋊AGL(1,𝔽7)11yescharacteristic
55C2×C3×ΓL(1,𝔽8)—651no—
56C32×C13—141noHall
57C3×C5×C7C3×#26111yescharacteristic, complemented
58C5×ΓL(1,𝔽8)—21no—
59C5×ΓL(1,𝔽8)C3⋊#26111yescharacteristic, complemented
60C7×C13C3×ΓL(1,𝔽16)11yescharacteristic, complemented, Hall
61C32×D5—1821no—
62Dic21—651no—
63ΓL(1,𝔽8)⋊C4—1951no—
64C3×D13—51no—
65C3×D13—701no—
66C3×D13—701no—
67C7×D5—131no—
68C5×C13C3⋊(ΓL(1,𝔽8)⋊C4)11yescharacteristic, complemented, Hall
69C3×ΓL(1,𝔽8)#6312111yescharacteristic, complemented, Hall
70ΓL(1,𝔽16)—911no—
71C3×AGL(1,𝔽5)—2731no—
72#261—1051noHall
73C32×C5—141noHall
74C2×C3×C7—651no—
75C2×ΓL(1,𝔽8)—651no—
76C2×ΓL(1,𝔽8)—1301no—
77C3×C13—141no—
78C3×C13—141no—
79C3×C13ΓL(1,𝔽8)⋊AGL(1,𝔽5)11yescharacteristic, complemented
80C3×Dic3—4551noHall
81C5×C7C3×C3⋊#26111yescharacteristic, complemented, Hall
82C3×D5—131no—
83C3×D5—1821no—
84C3×D5—1821no—
85Dic7—1951noHall
86D13—51no—
87C3×C7C3×#6312111yescharacteristic, complemented
88ΓL(1,𝔽8)—21no—
89ΓL(1,𝔽8)C3⋊#6312111yescharacteristic, complemented
90AGL(1,𝔽5)—2731noHall
91C2×C32—9101no—
92C3×C5—141no—
93C3×C5—141no—
94C3×C5ΓL(1,𝔽8)⋊#26111yescharacteristic, complemented
95C2×C7—651no—
96C13C3⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))11yescharacteristic, complemented, Sylow
97C3×C4—13651no—
98Dic3—4551no—
99D5—131no—
100C32—141noSylow
101C7C3×C3⋊#6312111yescharacteristic, complemented, Sylow
102C2×C3—651no—
103C2×C3—9101no—
104C2×C3—9101no—
105C5C3⋊(ΓL(1,𝔽8)⋊#261)11yescharacteristic, complemented, Sylow
106C4—13651noSylow
107C3—141no—
108C3—141no—
109C3ΓL(1,𝔽8)⋊#6312111yescharacteristic, complemented
110C2—651no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C7×C13C3×C4
prime radicalC3×ΓL(1,𝔽8)×D65C2
cosocleC3×C7×D65C2×C3
LCSG ▷ C3×C5×C7×C13
UCSC1

Complements

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

NH ≅ G/Ntype
2. C3⋊(C7⋊#63121)C3semidirect
4. C3×C5×C13×ΓL(1,𝔽8)C4semidirect
12. C3×C5×C7×C13C3×C4semidirect
14. C5×C13×ΓL(1,𝔽8)Dic3semidirect
20. C3×C13×ΓL(1,𝔽8)AGL(1,𝔽5)semidirect
29. C5×C7×C13C3×Dic3semidirect
36. C3×C5×ΓL(1,𝔽8)#261semidirect
37. C3×C7×C13C3×AGL(1,𝔽5)semidirect
39. C13×ΓL(1,𝔽8)ΓL(1,𝔽16)semidirect
48. C3×C5×C13ΓL(1,𝔽8)⋊C4semidirect
57. C3×C5×C7C3×#261semidirect
59. C5×ΓL(1,𝔽8)C3⋊#261semidirect
60. C7×C13C3×ΓL(1,𝔽16)semidirect
68. C5×C13C3⋊(ΓL(1,𝔽8)⋊C4)semidirect
69. C3×ΓL(1,𝔽8)#63121semidirect
79. C3×C13ΓL(1,𝔽8)⋊AGL(1,𝔽5)semidirect
81. C5×C7C3×C3⋊#261semidirect
87. C3×C7C3×#63121semidirect
89. ΓL(1,𝔽8)C3⋊#63121semidirect
94. C3×C5ΓL(1,𝔽8)⋊#261semidirect
96. C13C3⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))semidirect
101. C7C3×C3⋊#63121semidirect
105. C5C3⋊(ΓL(1,𝔽8)⋊#261)semidirect
109. C3ΓL(1,𝔽8)⋊#63121semidirect

Maximal subgroups

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

subgroupindex
C3×ΓL(1,𝔽8)×D652
C3⋊(C7⋊#63121)3
ΓL(1,𝔽8)⋊#631213
C3⋊(ΓL(1,𝔽8)⋊#261)5
C3×C3⋊#631217
C3⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))13