Finite Group Databasebeta

← C3⋊(ΓL(1,𝔽8)⋊(C19⋊AGL(1,𝔽5))) #37561325

Subgroups — C3⋊(ΓL(1,𝔽8)⋊(C19⋊AGL(1,𝔽5))) #37561325

Order 23940.

112 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 38 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×C19×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C3⋊(C7⋊(C19⋊AGL(1,𝔽5)))C311yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)⋊(C19⋊AGL(1,𝔽5))—31nomaximal
4C3×C5×C19×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
5C3⋊(ΓL(1,𝔽8)⋊Dic19)—51nomaximal, Hall
6D5×C19×ΓL(1,𝔽8)—21no—
7D5×C19×ΓL(1,𝔽8)D311yescharacteristic
8C3×C7×D5×C19C2×C311yescharacteristic
9C3×C3⋊(C19⋊AGL(1,𝔽5))—71nomaximal, Hall
10C7⋊(C19⋊AGL(1,𝔽5))—31noHall
11C2×C3×C19×ΓL(1,𝔽8)—51no—
12C3×C5×C7×C19C3×C411yescharacteristic, complemented
13C5×C19×ΓL(1,𝔽8)—21no—
14C5×C19×ΓL(1,𝔽8)Dic311yescharacteristic, complemented
15C32×D5×C19—141no—
16Dic399—51no—
17ΓL(1,𝔽8)⋊Dic19—151no—
18C7×D5×C19C3×D311yescharacteristic
19C3⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))—191nomaximal, Hall
20C3×C19×ΓL(1,𝔽8)AGL(1,𝔽5)11yescharacteristic, complemented, Hall
21C3⋊(C19⋊AGL(1,𝔽5))—71no—
22C3×C19⋊AGL(1,𝔽5)—211no—
23C32×C5×C19—141noHall
24C2×C3×C7×C19—51no—
25C2×C19×ΓL(1,𝔽8)—51no—
26C2×C19×ΓL(1,𝔽8)—101no—
27C3×Dic57—351noHall
28C5×C7×C19C3×Dic311yescharacteristic, complemented, Hall
29C3×D5×ΓL(1,𝔽8)D1911yescharacteristic
30C3×D5×C19—141no—
31C3×D5×C19—141no—
32C3×D5×C19AGL(1,𝔽7)11yescharacteristic
33Dic133—151noHall
34C3⋊(C7⋊AGL(1,𝔽5))—191no—
35ΓL(1,𝔽8)⋊AGL(1,𝔽5)—571no—
36C3×C7×C19C3×AGL(1,𝔽5)11yescharacteristic, complemented
37C19×ΓL(1,𝔽8)—21no—
38C19×ΓL(1,𝔽8)ΓL(1,𝔽16)11yescharacteristic, complemented
39C19⋊AGL(1,𝔽5)—211noHall
40C2×C32×C19—351no—
41C3×C5×ΓL(1,𝔽8)Dic1911yescharacteristic, complemented, Hall
42C3×C5×C19—141no—
43C3×C5×C19—141no—
44C3×C5×C19ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
45C2×C7×C19—51no—
46C3⋊(ΓL(1,𝔽8)⋊C4)—951noHall
47Dic57—351no—
48C3×Dic19—1051no—
49C3×C7×D5C3×D1911yescharacteristic
50D5×ΓL(1,𝔽8)—21no—
51D5×ΓL(1,𝔽8)D5711yescharacteristic
52D5×C19C3⋊AGL(1,𝔽7)11yescharacteristic
53C3×ΓL(1,𝔽16)—1331noHall
54C32×C19—71noHall
55C7⋊AGL(1,𝔽5)—571noHall
56C7×C19C3×ΓL(1,𝔽16)11yescharacteristic, complemented, Hall
57C2×C3×ΓL(1,𝔽8)—51no—
58C2×C3×C19—51no—
59C2×C3×C19—351no—
60C2×C3×C19—701no—
61C3×C5×C7C3×Dic1911yescharacteristic, complemented
62C5×ΓL(1,𝔽8)—21no—
63C5×ΓL(1,𝔽8)Dic5711yescharacteristic, complemented
64C5×C19C3⋊(ΓL(1,𝔽8)⋊C4)11yescharacteristic, complemented, Hall
65C32×D5—71no—
66Dic21—951no—
67ΓL(1,𝔽8)⋊C4—2851no—
68Dic19—1051noHall
69C7×D5C3×D5711yescharacteristic
70C3×ΓL(1,𝔽8)C19⋊AGL(1,𝔽5)11yescharacteristic, complemented, Hall
71ΓL(1,𝔽16)—1331no—
72C3×AGL(1,𝔽5)—3991no—
73C3×C19—71no—
74C3×C19—141no—
75C3×C19ΓL(1,𝔽8)⋊AGL(1,𝔽5)11yescharacteristic, complemented
76C32×C5—71noHall
77C2×C3×C7—51no—
78C2×ΓL(1,𝔽8)—51no—
79C2×ΓL(1,𝔽8)—101no—
80C2×C19—51no—
81C3×Dic3—6651noHall
82C5×C7C3×Dic5711yescharacteristic, complemented, Hall
83C3×D5—141no—
84C3×D5—141no—
85C3×D5ΓL(1,𝔽8)⋊D1911yescharacteristic
86Dic7—2851noHall
87C3×C7C3×C19⋊AGL(1,𝔽5)11yescharacteristic, complemented
88ΓL(1,𝔽8)—21no—
89ΓL(1,𝔽8)C3⋊(C19⋊AGL(1,𝔽5))11yescharacteristic, complemented
90AGL(1,𝔽5)—3991noHall
91C19C3⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))11yescharacteristic, complemented, Sylow
92C2×C32—701no—
93C3×C5—141no—
94C3×C5—141no—
95C3×C5ΓL(1,𝔽8)⋊Dic1911yescharacteristic, complemented
96C2×C7—51no—
97C3×C4—19951no—
98Dic3—6651no—
99D5C3⋊(ΓL(1,𝔽8)⋊D19)11yescharacteristic
100C32—141noSylow
101C7C3×C3⋊(C19⋊AGL(1,𝔽5))11yescharacteristic, complemented, Sylow
102C2×C3—51no—
103C2×C3—701no—
104C2×C3—701no—
105C5C3⋊(ΓL(1,𝔽8)⋊Dic19)11yescharacteristic, complemented, Sylow
106C4—19951noSylow
107C3—141no—
108C3—141no—
109C3ΓL(1,𝔽8)⋊(C19⋊AGL(1,𝔽5))11yescharacteristic, complemented
110C2—51no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C7×C19C3×C4
prime radicalC3×D5×C19×ΓL(1,𝔽8)C2
cosocleC3×C7×D5×C19C2×C3
LCSG ▷ C3×C5×C7×C19
UCSC1

Complements

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

NH ≅ G/Ntype
2. C3⋊(C7⋊(C19⋊AGL(1,𝔽5)))C3semidirect
4. C3×C5×C19×ΓL(1,𝔽8)C4semidirect
12. C3×C5×C7×C19C3×C4semidirect
14. C5×C19×ΓL(1,𝔽8)Dic3semidirect
20. C3×C19×ΓL(1,𝔽8)AGL(1,𝔽5)semidirect
28. C5×C7×C19C3×Dic3semidirect
36. C3×C7×C19C3×AGL(1,𝔽5)semidirect
38. C19×ΓL(1,𝔽8)ΓL(1,𝔽16)semidirect
41. C3×C5×ΓL(1,𝔽8)Dic19semidirect
44. C3×C5×C19ΓL(1,𝔽8)⋊C4semidirect
56. C7×C19C3×ΓL(1,𝔽16)semidirect
61. C3×C5×C7C3×Dic19semidirect
63. C5×ΓL(1,𝔽8)Dic57semidirect
64. C5×C19C3⋊(ΓL(1,𝔽8)⋊C4)semidirect
70. C3×ΓL(1,𝔽8)C19⋊AGL(1,𝔽5)semidirect
75. C3×C19ΓL(1,𝔽8)⋊AGL(1,𝔽5)semidirect
82. C5×C7C3×Dic57semidirect
87. C3×C7C3×C19⋊AGL(1,𝔽5)semidirect
89. ΓL(1,𝔽8)C3⋊(C19⋊AGL(1,𝔽5))semidirect
91. C19C3⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))semidirect
95. C3×C5ΓL(1,𝔽8)⋊Dic19semidirect
101. C7C3×C3⋊(C19⋊AGL(1,𝔽5))semidirect
105. C5C3⋊(ΓL(1,𝔽8)⋊Dic19)semidirect
109. C3ΓL(1,𝔽8)⋊(C19⋊AGL(1,𝔽5))semidirect

Maximal subgroups

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

subgroupindex
C3×D5×C19×ΓL(1,𝔽8)2
C3⋊(C7⋊(C19⋊AGL(1,𝔽5)))3
ΓL(1,𝔽8)⋊(C19⋊AGL(1,𝔽5))3
C3⋊(ΓL(1,𝔽8)⋊Dic19)5
C3×C3⋊(C19⋊AGL(1,𝔽5))7
C3⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))19