Finite Group Databasebeta

← C3×#737×C37⋊C3 #41008248

Subgroups — C3×#737×C37⋊C3 #41008248

Order 26973.

5804 conjugacy classes of subgroups; only the 2902 normal subgroups are shown.

normal subgroups only

This group has 5804 conjugacy classes of subgroups — too many to draw. Only its 2902 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×#15688532C3116yesmaximal, complemented
2C3×#15688531C318yesmaximal, complemented
3C3×#15688530C312yesmaximal, complemented
4#737×C37⋊C3C3181yesmaximal, direct product factor
5C3×3−1+2×C37⋊C3C318yesmaximal, complemented
6C3×H(3,𝔽3)×C37⋊C3C311yesmaximal, characteristic
7C32×C9×C37⋊C3C314yesmaximal, complemented
8C3×C37×#737C311yesmaximal, characteristic, complemented
9C33×C37⋊C3C3214yes—
10#15688532C321432yescomplemented
11#15688531C321216yescomplemented
12#15688530C32154yescomplemented
13C3×#11758608C32116yescomplemented
14C3×#11758605C32124yescomplemented
15C3×#11758606C32124yescomplemented
163−1+2×C37⋊C3C321216yescomplemented
17C3×C32⋊(C37⋊C3)C3218yes—
18H(3,𝔽3)×C37⋊C3C32127yes—
19C32×#66524C3218yescomplemented
20C3×C9×C37⋊C3C3211yescharacteristic
21C3×C9×C37⋊C3C3218yes—
22C3×C9×C37⋊C3C321108yescomplemented
23C3×C9×C37⋊C3C3218yescomplemented
24C3×C9×C37⋊C3C32116yescomplemented
25C37×#737C32127yescomplemented
26C3×3−1+2×C37C3218yescomplemented
27C3×H(3,𝔽3)×C37C3211yescharacteristic
28C32×C9×C37C3214yescomplemented
29C33×C37C3314yes—
30C32×C37⋊C3—336no—
31C32×C37⋊C3C33136yes—
32C32×C37⋊C3C3318yes—
33C32×C37⋊C3C3311yescharacteristic
343−1+2×C37C33172yescomplemented
35H(3,𝔽3)×C37C3319yes—
36C3×C9×C37C3318yes—
37C3×C9×C37C3311yescharacteristic
38C3×C9×C37C33136yescomplemented
39C32⋊(C37⋊C3)C33172yes—
40#11758608C331144yescomplemented
41C3×#66524C3312yes—
42C3×#66524C33116yes—
43C3×#66524C33172yescomplemented
44#11758606C331216yescomplemented
45#11758605C331216yescomplemented
46C9×C37⋊C3C3319yes—
47C9×C37⋊C3C33172yes—
48C9×C37⋊C3C33172yescomplemented
49C9×C37⋊C3C331144yescomplemented
50C32×#737—371nomaximal, Sylow
51C32×C37—312no—
52C32×C37C34112yes—
53C32×C37C3411yescharacteristic
54C3×C37⋊C3—324no—
55C3×C37⋊C3—3108no—
56C3×C37⋊C3#73719yesdirect product factor
57C3×C37⋊C3C34124yes—
58C3×C37⋊C3C3413yes—
59C9×C37C3413yes—
60C9×C37C34124yes—
61#66524C3416yes—
62#66524C34148yes—
63C3×#737—3781no—
64C3×#737—3716no—
65C3×#737—372no—
66C3×#737—378no—
67C3×#737C37⋊C311yescharacteristic, direct product factor
68C32×3−1+2—378no—
69C32×H(3,𝔽3)—371no—
70C33×C9—374no—
71C3×C37—312no—
72C3×C37C3×#73713yescomplemented
73C3×C37C3511yescharacteristic
74C37⋊C3—324no—
75C37⋊C3C3×#73719yesdirect product factor
76C34—374no—
77#737—37432no—
78#737—3754no—
79#737—37216no—
80#737C3×C37⋊C3127yesdirect product factor
81C3×3−1+2—3716no—
82C3×3−1+2—37216no—
83C3×3−1+2—3724no—
84C3×3−1+2—3724no—
85C3×3−1+2C3×C37⋊C318yescomplemented
86C3×H(3,𝔽3)—378no—
87C3×H(3,𝔽3)—3727no—
88C3×H(3,𝔽3)C3×C37⋊C311yescharacteristic
89C32×C9—378no—
90C32×C9—378no—
91C32×C9—3716no—
92C32×C9—371no—
93C32×C9—378no—
94C32×C9—37108no—
95C32×C9C3×C37⋊C314yescomplemented
96C37C32×#73711yescharacteristic, complemented, Sylow
97C33—371no—
98C33—3736no—
99C33—378no—
100C33—11136no—
101C33C32×C37⋊C314yes—
1023−1+2—37144no—
1033−1+2—37216no—
1043−1+2—37216no—
1053−1+2C32×C37⋊C3172yescomplemented
106H(3,𝔽3)—3772no—
107H(3,𝔽3)C32×C37⋊C319yes—
108C3×C9—372no—
109C3×C9—3716no—
110C3×C9—3772no—
111C3×C9—379no—
112C3×C9—3772no—
113C3×C9—37144no—
114C3×C9—3772no—
115C3×C9C32×C37⋊C318yes—
116C3×C9C32×C37⋊C311yescharacteristic
117C3×C9C32×C37⋊C3136yescomplemented
118C32—312no—
119C32—379no—
120C32—373no—
121C32—3724no—
122C32—111108no—
123C32—11124no—
124C32C33×C37⋊C3112yes—
125C32C33×C37⋊C311yescharacteristic
126C9—376no—
127C9—3748no—
128C9C33×C37⋊C313yes—
129C9C33×C37⋊C3124yes—
130C3—312no—
131C3—379no—
132C3—11124no—
133C3#737×C37⋊C313yesdirect product factor
134C3C34×C37⋊C311yescharacteristic
subgroupG/H
centerC3×C9C32×C37⋊C3
commutator, cosocleC3×C37C35
FrattiniC3C34×C37⋊C3
FittingC3×C37×#737C3
socleC32×C37C34
prime radicalC3×H(3,𝔽3)×C37⋊C3C3
hypercenterC3×#737C37⋊C3
hypocenterC37C32×#737
LCSG ▷ C3×C37 ▷ C37
UCSC1 ◁ C3×C9 ◁ C3×#737

Complements

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

NH ≅ G/Ntype
1. C3×#15688532C3semidirect
2. C3×#15688531C3semidirect
3. C3×#15688530C3semidirect
4. #737×C37⋊C3C3direct (N × H)
5. C3×3−1+2×C37⋊C3C3semidirect
7. C32×C9×C37⋊C3C3semidirect
8. C3×C37×#737C3semidirect
10. #15688532C32semidirect
11. #15688531C32semidirect
12. #15688530C32semidirect
13. C3×#11758608C32semidirect
14. C3×#11758605C32semidirect
15. C3×#11758606C32semidirect
16. 3−1+2×C37⋊C3C32semidirect
19. C32×#66524C32semidirect
22. C3×C9×C37⋊C3C32semidirect
23. C3×C9×C37⋊C3C32semidirect
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
45. #11758605C33semidirect
48. C9×C37⋊C3C33semidirect
49. C9×C37⋊C3C33semidirect
56. C3×C37⋊C3#737direct (N × H)
67. C3×#737C37⋊C3direct (N × H)
72. C3×C37C3×#737semidirect
75. C37⋊C3C3×#737direct (N × H)
80. #737C3×C37⋊C3direct (N × H)
85. C3×3−1+2C3×C37⋊C3semidirect
95. C32×C9C3×C37⋊C3semidirect
96. C37C32×#737semidirect
105. 3−1+2C32×C37⋊C3semidirect
117. C3×C9C32×C37⋊C3semidirect
133. C3#737×C37⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#156885323
C3×#156885313
C3×#156885303
#737×C37⋊C33
C3×3−1+2×C37⋊C33
C3×H(3,𝔽3)×C37⋊C33
C32×C9×C37⋊C33
C3×C37×#7373
C32×#73737