Finite Group Databasebeta

← C3×C37×#6909 #41004713

Subgroups — C3×C37×#6909 #41004713

Order 26973.

1196 conjugacy classes of subgroups; only the 476 normal subgroups are shown.

normal subgroups only

This group has 1196 conjugacy classes of subgroups — too many to draw. Only its 476 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×C37C312yesmaximal, complemented
2C32×H(3,𝔽3)×C37C311yesmaximal, characteristic, complemented
3C32×H(3,𝔽3)×C37C311yesmaximal, characteristic, complemented
4C37×#6909C3127yesmaximal, direct product factor
5C3×C37×AΣL(1,𝔽27)C319yesmaximal, complemented
6C34×C37—31no—
7C34×C37C3211yescharacteristic
8C34×C37C3213yescomplemented
9C3×3−1+2×C37—312no—
10C3×3−1+2×C37—32no—
11C3×3−1+2×C37—32no—
12C3×3−1+2×C37C3216yescomplemented
13C3×3−1+2×C37C32118yescomplemented
14C3×H(3,𝔽3)×C37—36no—
15C3×H(3,𝔽3)×C37—36no—
16C3×H(3,𝔽3)×C37—32no—
17C3×H(3,𝔽3)×C37—31no—
18C3×H(3,𝔽3)×C37—32no—
19C3×H(3,𝔽3)×C37C3213yes—
20C3×H(3,𝔽3)×C37C3219yescomplemented
21C3×H(3,𝔽3)×C37C3219yescomplemented
22C32×C9×C37—32no—
23C37×AΣL(1,𝔽27)C32181yescomplemented
24C33×C37—36no—
25C33×C37—39no—
26C33×C37—33no—
27C33×C37—33no—
28C33×C37—93no—
29C33×C37—99no—
30C33×C37—99no—
31C33×C37H(3,𝔽3)12yes—
32C33×C37H(3,𝔽3)11yescharacteristic
33C33×C37H(3,𝔽3)16yes—
34C33×C37C3311yescharacteristic
35C33×C37C3313yes—
36C33×C37C3319yescomplemented
373−1+2×C37—318no—
383−1+2×C37—312no—
393−1+2×C37—318no—
403−1+2×C37C33118yescomplemented
41H(3,𝔽3)×C37—36no—
42H(3,𝔽3)×C37—318no—
43H(3,𝔽3)×C37—39no—
44H(3,𝔽3)×C37—318no—
45H(3,𝔽3)×C37C3319yes—
46C3×C9×C37—318no—
47C3×C9×C37—36no—
48C3×#6909C3711yesmaximal, characteristic, direct product factor, Sylow
49C32×C37—33no—
50C32×C37—33no—
51C32×C37—36no—
52C32×C37—39no—
53C32×C37—39no—
54C32×C37—96no—
55C32×C37—93no—
56C32×C37—99no—
57C32×C37—918no—
58C32×C37—927no—
59C32×C37—99no—
60C32×C37—99no—
61C32×C37C3×H(3,𝔽3)16yes—
62C32×C37C3×H(3,𝔽3)12yes—
63C32×C37C3×H(3,𝔽3)13yes—
64C32×C37C3×H(3,𝔽3)11yescharacteristic
65C32×C37C3411yescharacteristic
66C9×C37—318no—
67C32×3−1+2C3×C3712yescomplemented
68C32×H(3,𝔽3)C3×C3711yescharacteristic, complemented
69C32×H(3,𝔽3)C3×C3711yescharacteristic, complemented
70#6909C3×C37127yesdirect product factor
71C3×AΣL(1,𝔽27)C3×C3719yescomplemented
72C3×C37—33no—
73C3×C37—32no—
74C3×C37—31no—
75C3×C37—36no—
76C3×C37—99no—
77C3×C37—99no—
78C3×C37#690913yesdirect product factor
79C3×C37C32×H(3,𝔽3)11yescharacteristic
80C34—31no—
81C34C32×C3711yescharacteristic
82C34C32×C3713yescomplemented
83C3×3−1+2—32no—
84C3×3−1+2—32no—
85C3×3−1+2—312no—
86C3×3−1+2C32×C3716yescomplemented
87C3×3−1+2C32×C37118yescomplemented
88C3×H(3,𝔽3)—36no—
89C3×H(3,𝔽3)—32no—
90C3×H(3,𝔽3)—31no—
91C3×H(3,𝔽3)—36no—
92C3×H(3,𝔽3)—32no—
93C3×H(3,𝔽3)C32×C3713yes—
94C3×H(3,𝔽3)C32×C3719yescomplemented
95C3×H(3,𝔽3)C32×C3719yescomplemented
96C32×C9—32no—
97AΣL(1,𝔽27)C32×C37181yescomplemented
98C37C3×#690911yescharacteristic, direct product factor, Sylow
99C33—36no—
100C33—33no—
101C33—33no—
102C33—39no—
103C33—93no—
104C33—99no—
105C33—99no—
106C33H(3,𝔽3)×C3712yes—
107C33H(3,𝔽3)×C3711yescharacteristic
108C33H(3,𝔽3)×C3716yes—
109C33C33×C3711yescharacteristic
110C33C33×C3713yes—
111C33C33×C3719yescomplemented
1123−1+2—318no—
1133−1+2—312no—
1143−1+2—318no—
1153−1+2C33×C37118yescomplemented
116H(3,𝔽3)—36no—
117H(3,𝔽3)—318no—
118H(3,𝔽3)—39no—
119H(3,𝔽3)—318no—
120H(3,𝔽3)C33×C3719yes—
121C3×C9—318no—
122C3×C9—36no—
123C32—33no—
124C32—33no—
125C32—36no—
126C32—39no—
127C32—39no—
128C32—96no—
129C32—93no—
130C32—99no—
131C32—927no—
132C32—918no—
133C32—99no—
134C32—99no—
135C32C3×H(3,𝔽3)×C3716yes—
136C32C3×H(3,𝔽3)×C3712yes—
137C32C3×H(3,𝔽3)×C3713yes—
138C32C3×H(3,𝔽3)×C3711yescharacteristic
139C32C34×C3711yescharacteristic
140C9—318no—
141C3—33no—
142C3—32no—
143C3—31no—
144C3—36no—
145C3—99no—
146C3—99no—
147C3C37×#690913yesdirect product factor
148C3C32×H(3,𝔽3)×C3711yescharacteristic
subgroupG/H
center, socleC32×C37C3×H(3,𝔽3)
commutator, Frattini, cosocleC32C34×C37
LCSG ▷ C32 ▷ C3 ▷ C1
UCSC1 ◁ C32×C37 ◁ C34×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C32×3−1+2×C37C3semidirect
2. C32×H(3,𝔽3)×C37C3semidirect
3. C32×H(3,𝔽3)×C37C3semidirect
4. C37×#6909C3direct (N × H)
5. C3×C37×AΣL(1,𝔽27)C3semidirect
8. C34×C37C32semidirect
12. C3×3−1+2×C37C32semidirect
13. C3×3−1+2×C37C32semidirect
20. C3×H(3,𝔽3)×C37C32semidirect
21. C3×H(3,𝔽3)×C37C32semidirect
23. C37×AΣL(1,𝔽27)C32semidirect
36. C33×C37C33semidirect
40. 3−1+2×C37C33semidirect
48. C3×#6909C37direct (N × H)
67. C32×3−1+2C3×C37semidirect
68. C32×H(3,𝔽3)C3×C37semidirect
69. C32×H(3,𝔽3)C3×C37semidirect
70. #6909C3×C37direct (N × H)
71. C3×AΣL(1,𝔽27)C3×C37semidirect
78. C3×C37#6909direct (N × H)
82. C34C32×C37semidirect
86. C3×3−1+2C32×C37semidirect
87. C3×3−1+2C32×C37semidirect
94. C3×H(3,𝔽3)C32×C37semidirect
95. C3×H(3,𝔽3)C32×C37semidirect
97. AΣL(1,𝔽27)C32×C37semidirect
98. C37C3×#6909direct (N × H)
111. C33C33×C37semidirect
115. 3−1+2C33×C37semidirect
147. C3C37×#6909direct (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×C373
C32×H(3,𝔽3)×C373
C37×#69093
C3×C37×AΣL(1,𝔽27)3
C3×#690937