Finite Group Databasebeta

← C3⋊(ΓL(1,𝔽8)⋊#10596817) #49141474

Subgroups — C3⋊(ΓL(1,𝔽8)⋊#10596817) #49141474

Order 36540.

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×C5×ΓL(1,𝔽8)×D29C211yesmaximal, characteristic
2C3⋊(C7⋊#10596817)C311yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)⋊#10596817—31nomaximal
4C3×C5×ΓL(1,𝔽8)×C29C411yescharacteristic, complemented, Hall
5C3⋊(ΓL(1,𝔽8)⋊#1181)—51nomaximal, Hall
6C5×ΓL(1,𝔽8)×D29—21no—
7C5×ΓL(1,𝔽8)×D29D311yescharacteristic
8C3×C5×C7×D29C2×C311yescharacteristic
9C3×C3⋊#10596817—71nomaximal, Hall
10C7⋊#10596817—31noHall
11C3×ΓL(1,𝔽8)×D29D511yescharacteristic
12C5×ΓL(1,𝔽8)×C29—21no—
13C5×ΓL(1,𝔽8)×C29Dic311yescharacteristic, complemented
14C3×C5×C7×C29C3×C411yescharacteristic, complemented
15C32×C5×D29—71no—
16C3⋊(C7⋊#1181)—51no—
17ΓL(1,𝔽8)⋊#1181—151no—
18C5×C7×D29C3×D311yescharacteristic
19C3×ΓL(1,𝔽8)×C29Dic511yescharacteristic, complemented, Hall
20C3⋊#10596817—71no—
21C3×#10596817—211no—
22C32×C5×C29—71noHall
23C3⋊(ΓL(1,𝔽8)⋊Dic5)—291nomaximal, Hall
24C3×C7×D29C3×D511yescharacteristic
25ΓL(1,𝔽8)×D29—21no—
26ΓL(1,𝔽8)×D29D1511yescharacteristic
27C3×C3⋊#1181—351noHall
28C5×C7×C29C3×Dic311yescharacteristic, complemented, Hall
29C3×C5×D29—71no—
30C3×C5×D29—141no—
31C3×C5×D29AGL(1,𝔽7)11yescharacteristic
32C7⋊#1181—151noHall
33C2×C3×C5×ΓL(1,𝔽8)—291no—
34C3×C7×C29C3×Dic511yescharacteristic, complemented
35ΓL(1,𝔽8)×C29—21no—
36ΓL(1,𝔽8)×C29Dic1511yescharacteristic, complemented
37#10596817—211noHall
38C32×D29—71no—
39C3×C5×C29—71no—
40C3×C5×C29—141no—
41C3×C5×C29ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
42Dic105—291no—
43ΓL(1,𝔽8)⋊Dic5—871no—
44C7×D29C3×D1511yescharacteristic
45C3⋊#1181—351no—
46C3×#1181—1051no—
47C3×C5×ΓL(1,𝔽8)#118111yescharacteristic, complemented, Hall
48C5×D29C3⋊AGL(1,𝔽7)11yescharacteristic
49C32×C29—71noHall
50C3⋊(ΓL(1,𝔽8)⋊C4)—1451noHall
51C2×C3×C5×C7—291no—
52C2×C5×ΓL(1,𝔽8)—291no—
53C2×C5×ΓL(1,𝔽8)—581no—
54C7×C29C3×Dic1511yescharacteristic, complemented, Hall
55C3×Dic15—2031noHall
56C3×D29—71no—
57C3×D29—141no—
58C3×D29C5⋊AGL(1,𝔽7)11yescharacteristic
59C5×C29C3⋊(ΓL(1,𝔽8)⋊C4)11yescharacteristic, complemented, Hall
60Dic35—871noHall
61C2×C3×ΓL(1,𝔽8)—291no—
62#1181—1051noHall
63C3×C5×C7C3×#118111yescharacteristic, complemented
64C5×ΓL(1,𝔽8)—21no—
65C5×ΓL(1,𝔽8)C3⋊#118111yescharacteristic, complemented
66C2×C32×C5—2031no—
67C3×C29—71no—
68C3×C29—141no—
69C3×C29ΓL(1,𝔽8)⋊Dic511yescharacteristic, complemented
70Dic21—1451no—
71ΓL(1,𝔽8)⋊C4—4351no—
72C2×C5×C7—291no—
73C3×ΓL(1,𝔽8)#1059681711yescharacteristic, complemented, Hall
74Dic15—2031no—
75C3×Dic5—6091no—
76D29C3⋊(C5⋊AGL(1,𝔽7))11yescharacteristic
77C32×C5—71noHall
78C2×C3×C7—291no—
79C2×ΓL(1,𝔽8)—291no—
80C2×ΓL(1,𝔽8)—581no—
81C3×Dic3—10151noHall
82C5×C7C3×C3⋊#118111yescharacteristic, complemented, Hall
83C2×C3×C5—291no—
84C2×C3×C5—4061no—
85C2×C3×C5—2031no—
86C29C3⋊(ΓL(1,𝔽8)⋊Dic5)11yescharacteristic, complemented, Sylow
87Dic7—4351noHall
88C3×C7C3×#1059681711yescharacteristic, complemented
89ΓL(1,𝔽8)—21no—
90ΓL(1,𝔽8)C3⋊#1059681711yescharacteristic, complemented
91Dic5—6091noHall
92C2×C32—2031no—
93C3×C5—71no—
94C3×C5—141no—
95C3×C5ΓL(1,𝔽8)⋊#118111yescharacteristic, complemented
96C2×C7—291no—
97C3×C4—30451no—
98Dic3—10151no—
99C2×C5—291no—
100C32—71noSylow
101C7C3×C3⋊#1059681711yescharacteristic, complemented, Sylow
102C2×C3—291no—
103C2×C3—4061no—
104C2×C3—2031no—
105C5C3⋊(ΓL(1,𝔽8)⋊#1181)11yescharacteristic, complemented, Sylow
106C4—30451noSylow
107C3—71no—
108C3—141no—
109C3ΓL(1,𝔽8)⋊#1059681711yescharacteristic, complemented
110C2—291no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C7×C29C3×C4
prime radicalC3×C5×ΓL(1,𝔽8)×D29C2
cosocleC3×C5×C7×D29C2×C3
LCSG ▷ C3×C5×C7×C29
UCSC1

Complements

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

NH ≅ G/Ntype
2. C3⋊(C7⋊#10596817)C3semidirect
4. C3×C5×ΓL(1,𝔽8)×C29C4semidirect
13. C5×ΓL(1,𝔽8)×C29Dic3semidirect
14. C3×C5×C7×C29C3×C4semidirect
19. C3×ΓL(1,𝔽8)×C29Dic5semidirect
28. C5×C7×C29C3×Dic3semidirect
34. C3×C7×C29C3×Dic5semidirect
36. ΓL(1,𝔽8)×C29Dic15semidirect
41. C3×C5×C29ΓL(1,𝔽8)⋊C4semidirect
47. C3×C5×ΓL(1,𝔽8)#1181semidirect
54. C7×C29C3×Dic15semidirect
59. C5×C29C3⋊(ΓL(1,𝔽8)⋊C4)semidirect
63. C3×C5×C7C3×#1181semidirect
65. C5×ΓL(1,𝔽8)C3⋊#1181semidirect
69. C3×C29ΓL(1,𝔽8)⋊Dic5semidirect
73. C3×ΓL(1,𝔽8)#10596817semidirect
82. C5×C7C3×C3⋊#1181semidirect
86. C29C3⋊(ΓL(1,𝔽8)⋊Dic5)semidirect
88. C3×C7C3×#10596817semidirect
90. ΓL(1,𝔽8)C3⋊#10596817semidirect
95. C3×C5ΓL(1,𝔽8)⋊#1181semidirect
101. C7C3×C3⋊#10596817semidirect
105. C5C3⋊(ΓL(1,𝔽8)⋊#1181)semidirect
109. C3ΓL(1,𝔽8)⋊#10596817semidirect

Maximal subgroups

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

subgroupindex
C3×C5×ΓL(1,𝔽8)×D292
C3⋊(C7⋊#10596817)3
ΓL(1,𝔽8)⋊#105968173
C3⋊(ΓL(1,𝔽8)⋊#1181)5
C3×C3⋊#105968177
C3⋊(ΓL(1,𝔽8)⋊Dic5)29