Finite Group Databasebeta

← C3×#23782921 #44535403

Subgroups — C3×#23782921 #44535403

Order 31347.

5588 conjugacy classes of subgroups; only the 2884 normal subgroups are shown.

normal subgroups only

This group has 5588 conjugacy classes of subgroups — too many to draw. Only its 2884 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
1#23782921C3181yesmaximal, direct product factor
2C3×#16812526C3116yesmaximal, complemented
3C3×#16812525C318yesmaximal, complemented
4C3×#16812524C312yesmaximal, complemented
5C32×#11923956C314yesmaximal, complemented
6C3×3−1+2×C43⋊C3C318yesmaximal, complemented
7C3×H(3,𝔽3)×C43⋊C3C311yesmaximal, characteristic
8C3×C43×#737C311yesmaximal, characteristic, complemented
9C33×C43⋊C3C3214yes—
10#16812526C321432yescomplemented
11#16812525C321216yescomplemented
12#16812524C32154yescomplemented
13C3×#11923958C32116yescomplemented
14C3×#11923957C32124yescomplemented
15C3×#11923956C3218yes—
16C3×#11923956C3211yescharacteristic
17C3×#11923956C3218yescomplemented
18C3×#11923956C32116yescomplemented
19C3×#11923956C321108yescomplemented
203−1+2×C43⋊C3C321216yescomplemented
21C3×C32⋊(C43⋊C3)C3218yes—
22H(3,𝔽3)×C43⋊C3C32127yes—
23C32×C43⋊C9C3218yescomplemented
24C3×C9×C43⋊C3C32124yescomplemented
25C43×#737C32127yescomplemented
26C3×3−1+2×C43C3218yescomplemented
27C3×H(3,𝔽3)×C43C3211yescharacteristic
28C32×C9×C43C3214yescomplemented
29C32×C43⋊C3—912no—
30C32×C43⋊C3C3318yes—
31C32×C43⋊C3C3311yescharacteristic
32C32×C43⋊C3C33136yes—
33#11923958C331144yescomplemented
34#11923957C331216yescomplemented
35#11923956C33172yes—
36#11923956C3319yes—
37#11923956C33172yescomplemented
38#11923956C331144yescomplemented
39C32⋊(C43⋊C3)C33172yes—
40C3×C43⋊C9C33116yes—
41C3×C43⋊C9C3312yes—
42C3×C43⋊C9C33172yescomplemented
43C9×C43⋊C3C331216yescomplemented
44C33×C43C3314yes—
453−1+2×C43C33172yescomplemented
46H(3,𝔽3)×C43C3319yes—
47C3×C9×C43C3311yescharacteristic
48C3×C9×C43C3318yes—
49C3×C9×C43C33136yescomplemented
50C3×3−1+4—431nomaximal, Sylow
51C32×C43—312no—
52C32×C43C34112yes—
53C32×C43C3411yescharacteristic
54C3×C43⋊C3—33no—
55C3×C43⋊C3—324no—
56C3×C43⋊C3—936no—
57C3×C43⋊C3C34124yes—
58C3×C43⋊C3C3413yes—
59C9×C43C3413yes—
60C9×C43C34124yes—
61C43⋊C9C34148yes—
62C43⋊C9C3416yes—
633−1+4—4381no—
64C3×#737—4316no—
65C3×#737—432no—
66C3×#737—438no—
67C3×#737C43⋊C311yescharacteristic, complemented
68C32×3−1+2—438no—
69C32×3−1+2—434no—
70C32×H(3,𝔽3)—431no—
71C3×C43—312no—
72C3×C433−1+413yescomplemented
73C3×C43C3511yescharacteristic
74C43⋊C3—33no—
75C43⋊C3—324no—
76C34—434no—
77#737—43432no—
78#737—4354no—
79#737—43216no—
80#737C3×C43⋊C3127yescomplemented
81C3×3−1+2—4316no—
82C3×3−1+2—43216no—
83C3×3−1+2—438no—
84C3×3−1+2—431no—
85C3×3−1+2—4316no—
86C3×3−1+2—438no—
87C3×3−1+2—4324no—
88C3×3−1+2—43108no—
89C3×3−1+2C3×C43⋊C318yescomplemented
90C3×H(3,𝔽3)—438no—
91C3×H(3,𝔽3)—4327no—
92C3×H(3,𝔽3)C3×C43⋊C311yescharacteristic
93C32×C9—438no—
94C32×C9—4324no—
95C32×C9C3×C43⋊C314yescomplemented
96C43C3×3−1+411yescharacteristic, complemented, Sylow
97C33—438no—
98C33—431no—
99C33—4336no—
100C33—38712no—
101C33C32×C43⋊C314yes—
1023−1+2—43144no—
1033−1+2—4372no—
1043−1+2—439no—
1053−1+2—43144no—
1063−1+2—4372no—
1073−1+2—43216no—
1083−1+2C32×C43⋊C3172yescomplemented
109H(3,𝔽3)—4372no—
110H(3,𝔽3)C32×C43⋊C319yes—
111C3×C9—4372no—
112C3×C9—432no—
113C3×C9—4316no—
114C3×C9—43216no—
115C3×C9C32×C43⋊C318yes—
116C3×C9C32×C43⋊C311yescharacteristic
117C3×C9C32×C43⋊C3136yescomplemented
118C32—312no—
119C32—4324no—
120C32—433no—
121C32—12924no—
122C32—1293no—
123C32—38736no—
124C32C33×C43⋊C3112yes—
125C32C33×C43⋊C311yescharacteristic
126C9—4348no—
127C9—436no—
128C9C33×C43⋊C3124yes—
129C9C33×C43⋊C313yes—
130C3—312no—
131C3—12924no—
132C3—1293no—
133C3#2378292113yesdirect product factor
134C3C34×C43⋊C311yescharacteristic
subgroupG/H
centerC32C33×C43⋊C3
commutator, cosocleC3×C43C35
FrattiniC3C34×C43⋊C3
FittingC3×C43×#737C3
socleC32×C43C34
prime radicalC3×H(3,𝔽3)×C43⋊C3C3
hypercenterC3×#737C43⋊C3
hypocenterC43C3×3−1+4
LCSG ▷ C3×C43 ▷ C43
UCSC1 ◁ C32 ◁ C3×#737

Complements

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

NH ≅ G/Ntype
1. #23782921C3direct (N × H)
2. C3×#16812526C3semidirect
3. C3×#16812525C3semidirect
4. C3×#16812524C3semidirect
5. C32×#11923956C3semidirect
6. C3×3−1+2×C43⋊C3C3semidirect
8. C3×C43×#737C3semidirect
10. #16812526C32semidirect
11. #16812525C32semidirect
12. #16812524C32semidirect
13. C3×#11923958C32semidirect
14. C3×#11923957C32semidirect
17. C3×#11923956C32semidirect
18. C3×#11923956C32semidirect
19. C3×#11923956C32semidirect
20. 3−1+2×C43⋊C3C32semidirect
23. C32×C43⋊C9C32semidirect
24. C3×C9×C43⋊C3C32semidirect
25. C43×#737C32semidirect
26. C3×3−1+2×C43C32semidirect
28. C32×C9×C43C32semidirect
33. #11923958C33semidirect
34. #11923957C33semidirect
37. #11923956C33semidirect
38. #11923956C33semidirect
42. C3×C43⋊C9C33semidirect
43. C9×C43⋊C3C33semidirect
45. 3−1+2×C43C33semidirect
49. C3×C9×C43C33semidirect
67. C3×#737C43⋊C3semidirect
72. C3×C433−1+4semidirect
80. #737C3×C43⋊C3semidirect
89. C3×3−1+2C3×C43⋊C3semidirect
95. C32×C9C3×C43⋊C3semidirect
96. C43C3×3−1+4semidirect
108. 3−1+2C32×C43⋊C3semidirect
117. C3×C9C32×C43⋊C3semidirect
133. C3#23782921direct (N × H)

Maximal subgroups

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

subgroupindex
#237829213
C3×#168125263
C3×#168125253
C3×#168125243
C32×#119239563
C3×3−1+2×C43⋊C33
C3×H(3,𝔽3)×C43⋊C33
C3×C43×#7373
C3×3−1+443