Finite Group Databasebeta

← C37×#10624934 #41004723

Subgroups — C37×#10624934 #41004723

Order 26973.

980 conjugacy classes of subgroups; only the 470 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 470 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
1C37×3−1+4C313yesmaximal, complemented
2C32×3−1+2×C37C311yesmaximal, characteristic, complemented
3C37×#6913C3118yesmaximal, complemented
4C37×#6912C312yesmaximal, complemented
5C37×#6911C311yesmaximal, characteristic, complemented
6C37×#6910C313yesmaximal, complemented
7C37×#6908C319yesmaximal, complemented
8C3×C37×AΣL(1,𝔽27)C313yesmaximal, complemented
9C34×C37C3211yescharacteristic
10C37×#737—318no—
11C37×#737C32127yescomplemented
12C3×3−1+2×C37—39no—
13C3×3−1+2×C37—39no—
14C3×3−1+2×C37C3216yes—
15C3×3−1+2×C37C3211yescharacteristic
16C3×3−1+2×C37C3211yescharacteristic
17C3×3−1+2×C37C3211yescharacteristic
18C3×3−1+2×C37C3216yescomplemented
19C3×H(3,𝔽3)×C37C3213yes—
20C32×C9×C37C3213yescomplemented
21C37×#733C32118yescomplemented
22C37×#732C3219yescomplemented
23C37×#731C32127yescomplemented
24C37×AΣL(1,𝔽27)C3219yescomplemented
25C37×AΣL(1,𝔽27)C32118yescomplemented
26C33×C37—33no—
27C33×C37—33no—
28C33×C37—93no—
29C33×C37C3311yescharacteristic
30C33×C37C3312yes—
31C33×C37C3311yescharacteristic
323−1+2×C37—36no—
333−1+2×C37—33no—
343−1+2×C37—33no—
353−1+2×C37—36no—
363−1+2×C37—318no—
373−1+2×C37—354no—
383−1+2×C37H(3,𝔽3)19yes—
393−1+2×C37C33118yes—
40H(3,𝔽3)×C37—36no—
41H(3,𝔽3)×C37C3319yes—
42C3×C9×C37—36no—
43C3×C9×C37—39no—
44C3×C9×C37—327no—
45C3×C9×C37C3313yes—
46C3×C9×C37C3313yes—
47C3×C9×C37C3313yes—
48#10624934C3711yesmaximal, characteristic, direct product factor, Sylow
49C32×C37—33no—
50C32×C37—31no—
51C32×C37—39no—
52C32×C37—32no—
53C32×C37—91no—
54C32×C37—99no—
55C32×C37—92no—
56C32×C37—273no—
57C32×C37C3×H(3,𝔽3)13yes—
58C32×C37C3411yescharacteristic
59C9×C37—318no—
60C9×C37—33no—
61C9×C37—33no—
62C9×C37C3×H(3,𝔽3)19yes—
633−1+4C3×C3713yescomplemented
64C32×3−1+2C3×C3711yescharacteristic, complemented
65#6913C3×C37118yescomplemented
66#6912C3×C3712yescomplemented
67#6911C3×C3711yescharacteristic, complemented
68#6910C3×C3713yescomplemented
69#6908C3×C3719yescomplemented
70C3×AΣL(1,𝔽27)C3×C3713yescomplemented
71C3×C37—33no—
72C3×C37—31no—
73C3×C37—33no—
74C3×C37—92no—
75C3×C37—99no—
76C3×C37C32×H(3,𝔽3)11yescharacteristic
77C34C32×C3711yescharacteristic
78#737—318no—
79#737C32×C37127yescomplemented
80C3×3−1+2—39no—
81C3×3−1+2—39no—
82C3×3−1+2C32×C3716yes—
83C3×3−1+2C32×C3711yescharacteristic
84C3×3−1+2C32×C3711yescharacteristic
85C3×3−1+2C32×C3711yescharacteristic
86C3×3−1+2C32×C3716yescomplemented
87C3×H(3,𝔽3)C32×C3713yes—
88C32×C9C32×C3713yescomplemented
89#733C32×C37118yescomplemented
90#732C32×C3719yescomplemented
91#731C32×C37127yescomplemented
92AΣL(1,𝔽27)C32×C37118yescomplemented
93AΣL(1,𝔽27)C32×C3719yescomplemented
94C37#1062493411yescharacteristic, direct product factor, Sylow
95C33—33no—
96C33—33no—
97C33—93no—
98C33C33×C3711yescharacteristic
99C33C33×C3712yes—
100C33C33×C3711yescharacteristic
1013−1+2—36no—
1023−1+2—33no—
1033−1+2—33no—
1043−1+2—36no—
1053−1+2—318no—
1063−1+2—354no—
1073−1+2H(3,𝔽3)×C3719yes—
1083−1+2C33×C37118yes—
109H(3,𝔽3)—36no—
110H(3,𝔽3)C33×C3719yes—
111C3×C9—36no—
112C3×C9—39no—
113C3×C9—327no—
114C3×C9C33×C3713yes—
115C3×C9C33×C3713yes—
116C3×C9C33×C3713yes—
117C32—33no—
118C32—31no—
119C32—39no—
120C32—32no—
121C32—91no—
122C32—99no—
123C32—92no—
124C32—273no—
125C32C3×H(3,𝔽3)×C3713yes—
126C32C34×C3711yescharacteristic
127C9—318no—
128C9—33no—
129C9—33no—
130C9C3×H(3,𝔽3)×C3719yes—
131C3—31no—
132C3—33no—
133C3—33no—
134C3—92no—
135C3—99no—
136C3C32×H(3,𝔽3)×C3711yescharacteristic
subgroupG/H
center, socleC3×C37C32×H(3,𝔽3)
commutator, Frattini, cosocleC32C34×C37
LCSG ▷ C32 ▷ C3 ▷ C1
UCSC1 ◁ C3×C37 ◁ C3×3−1+2×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C37×3−1+4C3semidirect
2. C32×3−1+2×C37C3semidirect
3. C37×#6913C3semidirect
4. C37×#6912C3semidirect
5. C37×#6911C3semidirect
6. C37×#6910C3semidirect
7. C37×#6908C3semidirect
8. C3×C37×AΣL(1,𝔽27)C3semidirect
11. C37×#737C32semidirect
18. C3×3−1+2×C37C32semidirect
20. C32×C9×C37C32semidirect
21. C37×#733C32semidirect
22. C37×#732C32semidirect
23. C37×#731C32semidirect
24. C37×AΣL(1,𝔽27)C32semidirect
25. C37×AΣL(1,𝔽27)C32semidirect
48. #10624934C37direct (N × H)
63. 3−1+4C3×C37semidirect
64. C32×3−1+2C3×C37semidirect
65. #6913C3×C37semidirect
66. #6912C3×C37semidirect
67. #6911C3×C37semidirect
68. #6910C3×C37semidirect
69. #6908C3×C37semidirect
70. C3×AΣL(1,𝔽27)C3×C37semidirect
79. #737C32×C37semidirect
86. C3×3−1+2C32×C37semidirect
88. C32×C9C32×C37semidirect
89. #733C32×C37semidirect
90. #732C32×C37semidirect
91. #731C32×C37semidirect
92. AΣL(1,𝔽27)C32×C37semidirect
93. AΣL(1,𝔽27)C32×C37semidirect
94. C37#10624934direct (N × H)

Maximal subgroups

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

subgroupindex
C37×3−1+43
C32×3−1+2×C373
C37×#69133
C37×#69123
C37×#69113
C37×#69103
C37×#69083
C3×C37×AΣL(1,𝔽27)3
#1062493437