Finite Group Databasebeta

← C3×#22582678 #41008256

Subgroups — C3×#22582678 #41008256

Order 26973.

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#22582678C3181yesmaximal, direct product factor
2C3×#15688532C3116yesmaximal, complemented
3C3×#15688531C318yesmaximal, complemented
4C3×#15688530C312yesmaximal, complemented
5C32×#11758605C314yesmaximal, complemented
6C3×3−1+2×C37⋊C3C318yesmaximal, complemented
7C3×H(3,𝔽3)×C37⋊C3C311yesmaximal, characteristic
8C3×C37×#737C311yesmaximal, characteristic, complemented
9C33×C37⋊C3C3214yes—
10#15688532C321432yescomplemented
11#15688531C321216yescomplemented
12#15688530C32154yescomplemented
13C3×#11758608C32116yescomplemented
14C3×#11758605C3218yes—
15C3×#11758605C3211yescharacteristic
16C3×#11758605C321108yescomplemented
17C3×#11758605C3218yescomplemented
18C3×#11758605C32116yescomplemented
19C3×#11758606C32124yescomplemented
203−1+2×C37⋊C3C321216yescomplemented
21C3×C32⋊(C37⋊C3)C3218yes—
22H(3,𝔽3)×C37⋊C3C32127yes—
23C32×#66524C3218yescomplemented
24C3×C9×C37⋊C3C32124yescomplemented
25C37×#737C32127yescomplemented
26C3×3−1+2×C37C3218yescomplemented
27C3×H(3,𝔽3)×C37C3211yescharacteristic
28C32×C9×C37C3214yescomplemented
29C33×C37C3314yes—
30C32×C37⋊C3—912no—
31C32×C37⋊C3C3318yes—
32C32×C37⋊C3C3311yescharacteristic
33C32×C37⋊C3C33136yes—
343−1+2×C37C33172yescomplemented
35H(3,𝔽3)×C37C3319yes—
36C3×C9×C37C3311yescharacteristic
37C3×C9×C37C3318yes—
38C3×C9×C37C33136yescomplemented
39C32⋊(C37⋊C3)C33172yes—
40#11758608C331144yescomplemented
41C3×#66524C33116yes—
42C3×#66524C3312yes—
43C3×#66524C33172yescomplemented
44#11758606C331216yescomplemented
45#11758605C33172yes—
46#11758605C3319yes—
47#11758605C33172yescomplemented
48#11758605C331144yescomplemented
49C9×C37⋊C3C331216yescomplemented
50C3×3−1+4—371nomaximal, Sylow
51C32×C37—312no—
52C32×C37C34112yes—
53C32×C37C3411yescharacteristic
54C3×C37⋊C3—33no—
55C3×C37⋊C3—324no—
56C3×C37⋊C3—936no—
57C3×C37⋊C3C34124yes—
58C3×C37⋊C3C3413yes—
59C9×C37C34124yes—
60C9×C37C3413yes—
61#66524C34148yes—
62#66524C3416yes—
633−1+4—3781no—
64C3×#737—3716no—
65C3×#737—372no—
66C3×#737—378no—
67C3×#737C37⋊C311yescharacteristic, complemented
68C32×3−1+2—378no—
69C32×3−1+2—374no—
70C32×H(3,𝔽3)—371no—
71C3×C37—312no—
72C3×C373−1+413yescomplemented
73C3×C37C3511yescharacteristic
74C37⋊C3—33no—
75C37⋊C3—324no—
76C34—374no—
77#737—37432no—
78#737—3754no—
79#737—37216no—
80#737C3×C37⋊C3127yescomplemented
81C3×3−1+2—3716no—
82C3×3−1+2—37216no—
83C3×3−1+2—378no—
84C3×3−1+2—371no—
85C3×3−1+2—3724no—
86C3×3−1+2—3716no—
87C3×3−1+2—378no—
88C3×3−1+2—37108no—
89C3×3−1+2C3×C37⋊C318yescomplemented
90C3×H(3,𝔽3)—378no—
91C3×H(3,𝔽3)—3727no—
92C3×H(3,𝔽3)C3×C37⋊C311yescharacteristic
93C32×C9—378no—
94C32×C9—3724no—
95C32×C9C3×C37⋊C314yescomplemented
96C37C3×3−1+411yescharacteristic, complemented, Sylow
97C33—3736no—
98C33—378no—
99C33—371no—
100C33—33312no—
101C33C32×C37⋊C314yes—
1023−1+2—37144no—
1033−1+2—3772no—
1043−1+2—379no—
1053−1+2—37144no—
1063−1+2—3772no—
1073−1+2—37216no—
1083−1+2C32×C37⋊C3172yescomplemented
109H(3,𝔽3)—3772no—
110H(3,𝔽3)C32×C37⋊C319yes—
111C3×C9—3772no—
112C3×C9—3716no—
113C3×C9—372no—
114C3×C9—37216no—
115C3×C9C32×C37⋊C318yes—
116C3×C9C32×C37⋊C311yescharacteristic
117C3×C9C32×C37⋊C3136yescomplemented
118C32—312no—
119C32—3724no—
120C32—373no—
121C32—33336no—
122C32—1113no—
123C32—11124no—
124C32C33×C37⋊C3112yes—
125C32C33×C37⋊C311yescharacteristic
126C9—376no—
127C9—3748no—
128C9C33×C37⋊C3124yes—
129C9C33×C37⋊C313yes—
130C3—312no—
131C3—1113no—
132C3—11124no—
133C3#2258267813yesdirect product factor
134C3C34×C37⋊C311yescharacteristic
subgroupG/H
centerC32C33×C37⋊C3
commutator, cosocleC3×C37C35
FrattiniC3C34×C37⋊C3
FittingC3×C37×#737C3
socleC32×C37C34
prime radicalC3×H(3,𝔽3)×C37⋊C3C3
hypercenterC3×#737C37⋊C3
hypocenterC37C3×3−1+4
LCSG ▷ C3×C37 ▷ C37
UCSC1 ◁ C32 ◁ C3×#737

Complements

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

NH ≅ G/Ntype
1. #22582678C3direct (N × H)
2. C3×#15688532C3semidirect
3. C3×#15688531C3semidirect
4. C3×#15688530C3semidirect
5. C32×#11758605C3semidirect
6. C3×3−1+2×C37⋊C3C3semidirect
8. C3×C37×#737C3semidirect
10. #15688532C32semidirect
11. #15688531C32semidirect
12. #15688530C32semidirect
13. C3×#11758608C32semidirect
16. C3×#11758605C32semidirect
17. C3×#11758605C32semidirect
18. C3×#11758605C32semidirect
19. C3×#11758606C32semidirect
20. 3−1+2×C37⋊C3C32semidirect
23. C32×#66524C32semidirect
24. C3×C9×C37⋊C3C32semidirect
25. C37×#737C32semidirect
26. C3×3−1+2×C37C32semidirect
28. C32×C9×C37C32semidirect
34. 3−1+2×C37C33semidirect
38. C3×C9×C37C33semidirect
40. #11758608C33semidirect
43. C3×#66524C33semidirect
44. #11758606C33semidirect
47. #11758605C33semidirect
48. #11758605C33semidirect
49. C9×C37⋊C3C33semidirect
67. C3×#737C37⋊C3semidirect
72. C3×C373−1+4semidirect
80. #737C3×C37⋊C3semidirect
89. C3×3−1+2C3×C37⋊C3semidirect
95. C32×C9C3×C37⋊C3semidirect
96. C37C3×3−1+4semidirect
108. 3−1+2C32×C37⋊C3semidirect
117. C3×C9C32×C37⋊C3semidirect
133. C3#22582678direct (N × H)

Maximal subgroups

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

subgroupindex
#225826783
C3×#156885323
C3×#156885313
C3×#156885303
C32×#117586053
C3×3−1+2×C37⋊C33
C3×H(3,𝔽3)×C37⋊C33
C3×C37×#7373
C3×3−1+437