Finite Group Databasebeta

← H(3,𝔽3)×3−1+2×C37 #41004684

Subgroups — H(3,𝔽3)×3−1+2×C37 #41004684

Order 26973.

1244 conjugacy classes of subgroups; only the 518 normal subgroups are shown.

normal subgroups only

This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 518 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
1C32×3−1+2×C37C314yesmaximal, complemented
2C32×H(3,𝔽3)×C37C311yesmaximal, characteristic
3C37×C9⋊H(3,𝔽3)C3124yesmaximal, complemented
4C37×#6891C318yesmaximal, complemented
5C9×H(3,𝔽3)×C37C313yesmaximal, complemented
6C34×C37C3214yes—
7C3×3−1+2×C37—336no—
8C3×3−1+2×C37C3218yes—
9C3×3−1+2×C37C3211yescharacteristic
10C3×3−1+2×C37C32124yescomplemented
11C3×H(3,𝔽3)×C37—39no—
12C3×H(3,𝔽3)×C37C3218yes—
13C3×H(3,𝔽3)×C37C3211yescharacteristic
14C32×C9×C37C32112yescomplemented
15C37×C32⋊C9C32124yescomplemented
16C37×C32⋊C9C32148yescomplemented
17C33×C37—312no—
18C33×C37—312no—
19C33×C37—98no—
20C33×C37C3318yes—
21C33×C37C3311yescharacteristic
22C33×C37C3314yes—
233−1+2×C37—324no—
243−1+2×C37—372no—
253−1+2×C37H(3,𝔽3)19yesdirect product factor
26H(3,𝔽3)×C37—324no—
27H(3,𝔽3)×C373−1+219yesdirect product factor
28C3×C9×C37—336no—
29C3×C9×C37C33124yes—
30C3×C9×C37C3313yes—
31H(3,𝔽3)×3−1+2C3711yesmaximal, characteristic, direct product factor, Sylow
32C32×C37—34no—
33C32×C37—31no—
34C32×C37—32no—
35C32×C37—38no—
36C32×C37—38no—
37C32×C37—38no—
38C32×C37—316no—
39C32×C37—936no—
40C32×C37C3×H(3,𝔽3)13yes—
41C32×C37C3×3−1+2112yescomplemented
42C32×C37C3411yescharacteristic
43C9×C37—324no—
44C9×C37C3×H(3,𝔽3)19yescomplemented
45C32×3−1+2C3×C3714yescomplemented
46C32×H(3,𝔽3)C3×C3711yescharacteristic
47C9⋊H(3,𝔽3)C3×C37124yescomplemented
48#6891C3×C3718yescomplemented
49C9×H(3,𝔽3)C3×C3713yescomplemented
50C3×C37—33no—
51C3×C37—312no—
52C3×C37—98no—
53C3×C37C32×H(3,𝔽3)11yescharacteristic
54C3×C37C32×3−1+211yescharacteristic
55C3×C373−1+412yes—
56C34C32×C3714yes—
57C3×3−1+2—336no—
58C3×3−1+2C32×C3718yes—
59C3×3−1+2C32×C3711yescharacteristic
60C3×3−1+2C32×C37124yescomplemented
61C3×H(3,𝔽3)—39no—
62C3×H(3,𝔽3)C32×C3718yes—
63C3×H(3,𝔽3)C32×C3711yescharacteristic
64C32×C9C32×C37112yescomplemented
65C32⋊C9C32×C37124yescomplemented
66C32⋊C9C32×C37148yescomplemented
67C37H(3,𝔽3)×3−1+211yescharacteristic, direct product factor, Sylow
68C33—312no—
69C33—312no—
70C33—98no—
71C33C33×C3718yes—
72C33C33×C3711yescharacteristic
73C33C33×C3714yes—
743−1+2—324no—
753−1+2—372no—
763−1+2H(3,𝔽3)×C3719yesdirect product factor
77H(3,𝔽3)—324no—
78H(3,𝔽3)3−1+2×C3719yesdirect product factor
79C3×C9—336no—
80C3×C9C33×C37124yes—
81C3×C9C33×C3713yes—
82C32—34no—
83C32—31no—
84C32—32no—
85C32—38no—
86C32—38no—
87C32—38no—
88C32—316no—
89C32—936no—
90C32C3×H(3,𝔽3)×C3713yes—
91C32C3×3−1+2×C37112yescomplemented
92C32C34×C3711yescharacteristic
93C9—324no—
94C9C3×H(3,𝔽3)×C3719yescomplemented
95C3—33no—
96C3—312no—
97C3—98no—
98C3C32×H(3,𝔽3)×C3711yescharacteristic
99C3C32×3−1+2×C3711yescharacteristic
100C3C37×3−1+412yes—
subgroupG/H
center, socleC32×C37C34
commutator, Frattini, cosocleC32C34×C37
prime radicalC32×H(3,𝔽3)×C37C3
LCSG ▷ C32 ▷ C1
UCSC1 ◁ C32×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C32×3−1+2×C37C3semidirect
3. C37×C9⋊H(3,𝔽3)C3semidirect
4. C37×#6891C3semidirect
5. C9×H(3,𝔽3)×C37C3semidirect
10. C3×3−1+2×C37C32semidirect
14. C32×C9×C37C32semidirect
15. C37×C32⋊C9C32semidirect
16. C37×C32⋊C9C32semidirect
25. 3−1+2×C37H(3,𝔽3)direct (N × H)
27. H(3,𝔽3)×C373−1+2direct (N × H)
31. H(3,𝔽3)×3−1+2C37direct (N × H)
41. C32×C37C3×3−1+2semidirect
44. C9×C37C3×H(3,𝔽3)semidirect
45. C32×3−1+2C3×C37semidirect
47. C9⋊H(3,𝔽3)C3×C37semidirect
48. #6891C3×C37semidirect
49. C9×H(3,𝔽3)C3×C37semidirect
60. C3×3−1+2C32×C37semidirect
64. C32×C9C32×C37semidirect
65. C32⋊C9C32×C37semidirect
66. C32⋊C9C32×C37semidirect
67. C37H(3,𝔽3)×3−1+2direct (N × H)
76. 3−1+2H(3,𝔽3)×C37direct (N × H)
78. H(3,𝔽3)3−1+2×C37direct (N × H)
91. C32C3×3−1+2×C37semidirect
94. C9C3×H(3,𝔽3)×C37semidirect

Maximal subgroups

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

subgroupindex
C32×3−1+2×C373
C32×H(3,𝔽3)×C373
C37×C9⋊H(3,𝔽3)3
C37×#68913
C9×H(3,𝔽3)×C373
H(3,𝔽3)×3−1+237