Finite Group Databasebeta

← C3×C53×C9⋊H(3,𝔽3) #50311752

Subgroups — C3×C53×C9⋊H(3,𝔽3) #50311752

Order 38637.

1328 conjugacy classes of subgroups; only the 548 normal subgroups are shown.

normal subgroups only

This group has 1328 conjugacy classes of subgroups — too many to draw. Only its 548 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×C53C313yesmaximal, complemented
2C32×H(3,𝔽3)×C53C311yesmaximal, characteristic
3C33×C9×C53C311yesmaximal, characteristic, complemented
4C53×C9⋊H(3,𝔽3)C3127yesmaximal, direct product factor
5C3×C53×C32⋊C9C316yesmaximal, complemented
6C3×C53×C32⋊C9C312yesmaximal, complemented
7C34×C53C3213yes—
8C34×C53C3211yescharacteristic
9C3×3−1+2×C53—327no—
10C3×3−1+2×C53C32127yescomplemented
11C3×H(3,𝔽3)×C53—39no—
12C3×H(3,𝔽3)×C53C3219yes—
13C32×C9×C53—39no—
14C32×C9×C53C3212yes—
15C32×C9×C53C3216yes—
16C32×C9×C53C3211yescharacteristic
17C32×C9×C53C3219yescomplemented
18C53×C32⋊C9C32154yescomplemented
19C53×C32⋊C9C32118yescomplemented
20C33×C53—33no—
21C33×C53—36no—
22C33×C53—39no—
23C33×C53—39no—
24C33×C53—318no—
25C33×C533−1+219yescomplemented
26C33×C53C3319yes—
27C33×C53C3313yes—
28C33×C53C3311yescharacteristic
293−1+2×C53—381no—
30H(3,𝔽3)×C53—327no—
31C3×C9×C53—36no—
32C3×C9×C53—318no—
33C3×C9×C53—327no—
34C3×C9×C53H(3,𝔽3)19yescomplemented
35C3×C9×C53C3316yes—
36C3×C9×C53C33118yes—
37C3×C9×C53C3313yes—
38C3×C9⋊H(3,𝔽3)C5311yesmaximal, characteristic, direct product factor, Sylow
39C32×C53—33no—
40C32×C53—36no—
41C32×C53—327no—
42C32×C53—39no—
43C32×C53—39no—
44C32×C53—318no—
45C32×C53—927no—
46C32×C53C3×H(3,𝔽3)13yes—
47C32×C53C3×3−1+213yes—
48C32×C53C3×3−1+219yescomplemented
49C32×C53#73716yes—
50C32×C53C3411yescharacteristic
51C9×C53—36no—
52C9×C53—318no—
53C9×C53C3×H(3,𝔽3)19yescomplemented
54C32×3−1+2C3×C5313yescomplemented
55C32×H(3,𝔽3)C3×C5311yescharacteristic
56C33×C9C3×C5311yescharacteristic, complemented
57C9⋊H(3,𝔽3)C3×C53127yesdirect product factor
58C3×C32⋊C9C3×C5316yescomplemented
59C3×C32⋊C9C3×C5312yescomplemented
60C3×C53—39no—
61C3×C53—99no—
62C3×C53C9⋊H(3,𝔽3)19yesdirect product factor
63C3×C53C32×H(3,𝔽3)11yescharacteristic
64C3×C53C32×3−1+211yescharacteristic
65C3×C53C3×#73712yes—
66C34C32×C5313yes—
67C34C32×C5311yescharacteristic
68C3×3−1+2—327no—
69C3×3−1+2C32×C53127yescomplemented
70C3×H(3,𝔽3)—39no—
71C3×H(3,𝔽3)C32×C5319yes—
72C32×C9—39no—
73C32×C9C32×C5312yes—
74C32×C9C32×C5316yes—
75C32×C9C32×C5311yescharacteristic
76C32×C9C32×C5319yescomplemented
77C32⋊C9C32×C53154yescomplemented
78C32⋊C9C32×C53118yescomplemented
79C53C3×C9⋊H(3,𝔽3)11yescharacteristic, direct product factor, Sylow
80C33—33no—
81C33—36no—
82C33—39no—
83C33—39no—
84C33—318no—
85C333−1+2×C5319yescomplemented
86C33C33×C5319yes—
87C33C33×C5313yes—
88C33C33×C5311yescharacteristic
893−1+2—381no—
90H(3,𝔽3)—327no—
91C3×C9—36no—
92C3×C9—318no—
93C3×C9—327no—
94C3×C9H(3,𝔽3)×C5319yescomplemented
95C3×C9C33×C5316yes—
96C3×C9C33×C53118yes—
97C3×C9C33×C5313yes—
98C32—327no—
99C32—33no—
100C32—36no—
101C32—39no—
102C32—39no—
103C32—318no—
104C32—927no—
105C32C3×H(3,𝔽3)×C5313yes—
106C32C3×3−1+2×C5313yes—
107C32C3×3−1+2×C5319yescomplemented
108C32C53×#73716yes—
109C32C34×C5311yescharacteristic
110C9—36no—
111C9—318no—
112C9C3×H(3,𝔽3)×C5319yescomplemented
113C3—39no—
114C3—99no—
115C3C53×C9⋊H(3,𝔽3)19yesdirect product factor
116C3C32×H(3,𝔽3)×C5311yescharacteristic
117C3C32×3−1+2×C5311yescharacteristic
118C3C3×C53×#73712yes—
subgroupG/H
center, socleC33×C53C33
commutator, Frattini, cosocleC32C34×C53
prime radicalC32×H(3,𝔽3)×C53C3
LCSG ▷ C32 ▷ C1
UCSC1 ◁ C33×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C32×3−1+2×C53C3semidirect
3. C33×C9×C53C3semidirect
4. C53×C9⋊H(3,𝔽3)C3direct (N × H)
5. C3×C53×C32⋊C9C3semidirect
6. C3×C53×C32⋊C9C3semidirect
10. C3×3−1+2×C53C32semidirect
17. C32×C9×C53C32semidirect
18. C53×C32⋊C9C32semidirect
19. C53×C32⋊C9C32semidirect
25. C33×C533−1+2semidirect
34. C3×C9×C53H(3,𝔽3)semidirect
38. C3×C9⋊H(3,𝔽3)C53direct (N × H)
48. C32×C53C3×3−1+2semidirect
53. C9×C53C3×H(3,𝔽3)semidirect
54. C32×3−1+2C3×C53semidirect
56. C33×C9C3×C53semidirect
57. C9⋊H(3,𝔽3)C3×C53direct (N × H)
58. C3×C32⋊C9C3×C53semidirect
59. C3×C32⋊C9C3×C53semidirect
62. C3×C53C9⋊H(3,𝔽3)direct (N × H)
69. C3×3−1+2C32×C53semidirect
76. C32×C9C32×C53semidirect
77. C32⋊C9C32×C53semidirect
78. C32⋊C9C32×C53semidirect
79. C53C3×C9⋊H(3,𝔽3)direct (N × H)
85. C333−1+2×C53semidirect
94. C3×C9H(3,𝔽3)×C53semidirect
107. C32C3×3−1+2×C53semidirect
112. C9C3×H(3,𝔽3)×C53semidirect
115. C3C53×C9⋊H(3,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C32×3−1+2×C533
C32×H(3,𝔽3)×C533
C33×C9×C533
C53×C9⋊H(3,𝔽3)3
C3×C53×C32⋊C93
C3×C9⋊H(3,𝔽3)53