Finite Group Databasebeta

← C37×#10624896 #41004685

Subgroups — C37×#10624896 #41004685

Order 26973.

1082 conjugacy classes of subgroups; only the 482 normal subgroups are shown.

normal subgroups only

This group has 1082 conjugacy classes of subgroups — too many to draw. Only its 482 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×C37×#737C313yesmaximal, complemented
2C32×3−1+2×C37C311yesmaximal, characteristic, complemented
3C37×#6893C316yesmaximal, complemented
4C37×C9⋊H(3,𝔽3)C316yesmaximal, complemented
5C37×C9⋊H(3,𝔽3)C3118yesmaximal, complemented
6C37×#6891C311yesmaximal, characteristic, complemented
7C37×C32⋊H(3,𝔽3)C311yesmaximal, characteristic
8C9×H(3,𝔽3)×C37C313yesmaximal, complemented
9C3×C37×C32⋊C9C311yesmaximal, characteristic, complemented
10C34×C37C3211yescharacteristic
11C37×#737—327no—
12C3×3−1+2×C37—39no—
13C3×3−1+2×C37C3213yes—
14C3×3−1+2×C37C3216yes—
15C3×3−1+2×C37C3212yes—
16C3×3−1+2×C37C3211yescharacteristic
17C3×3−1+2×C37C32118yescomplemented
18C3×H(3,𝔽3)×C37C3213yes—
19C3×H(3,𝔽3)×C37C3213yes—
20C3×H(3,𝔽3)×C37C3216yes—
21C32×C9×C37C3213yes—
22C32×C9×C37C3213yes—
23C32×C9×C37C3219yescomplemented
24C37×C32⋊C9C3211yescharacteristic
25C37×C32⋊C9C3216yes—
26C37×C32⋊C9C3212yes—
27C37×C32⋊C9C3213yes—
28C37×C32⋊C9C3216yes—
29C37×C32⋊C9C32118yescomplemented
30C37×C32⋊C9C32136yescomplemented
31C33×C37—33no—
32C33×C37—33no—
33C33×C37—92no—
34C33×C37C3312yes—
35C33×C37C3319yes—
36C33×C37C3311yescharacteristic
37C33×C37C3311yescharacteristic
383−1+2×C37—36no—
393−1+2×C37—39no—
403−1+2×C37—318no—
413−1+2×C37—354no—
423−1+2×C37H(3,𝔽3)19yescomplemented
43H(3,𝔽3)×C37—39no—
44H(3,𝔽3)×C37—39no—
45H(3,𝔽3)×C37—96no—
46C3×C9×C37—39no—
47C3×C9×C37—39no—
48C3×C9×C37—327no—
49C3×C9×C37C3313yes—
50C3×C9×C37C3316yes—
51C3×C9×C37C3319yes—
52C3×C9×C37C3316yes—
53C3×C9×C37C3313yes—
54#10624896C3711yesmaximal, characteristic, direct product factor, Sylow
55C32×C37—31no—
56C32×C37—31no—
57C32×C37—32no—
58C32×C37—32no—
59C32×C37—39no—
60C32×C37—32no—
61C32×C37—39no—
62C32×C37—32no—
63C32×C37—32no—
64C32×C37—318no—
65C32×C37—32no—
66C32×C37—99no—
67C32×C37C3×H(3,𝔽3)13yes—
68C32×C37#73713yescomplemented
69C32×C37C3411yescharacteristic
70C9×C37—33no—
71C9×C37—36no—
72C9×C37—36no—
73C9×C37—39no—
74C9×C37C3×H(3,𝔽3)19yes—
75C3×#737C3×C3713yescomplemented
76C32×3−1+2C3×C3711yescharacteristic, complemented
77#6893C3×C3716yescomplemented
78C9⋊H(3,𝔽3)C3×C3716yescomplemented
79C9⋊H(3,𝔽3)C3×C37118yescomplemented
80#6891C3×C3711yescharacteristic, complemented
81C32⋊H(3,𝔽3)C3×C3711yescharacteristic
82C9×H(3,𝔽3)C3×C3713yescomplemented
83C3×C32⋊C9C3×C3711yescharacteristic, complemented
84C3×C37—33no—
85C3×C37—33no—
86C3×C37—99no—
87C3×C37—92no—
88C3×C37C32×H(3,𝔽3)11yescharacteristic
89C3×C37C3×#73711yescharacteristic
90C3×C373−1+412yes—
91C34C32×C3711yescharacteristic
92#737—327no—
93C3×3−1+2—39no—
94C3×3−1+2C32×C3716yes—
95C3×3−1+2C32×C3712yes—
96C3×3−1+2C32×C3713yes—
97C3×3−1+2C32×C3711yescharacteristic
98C3×3−1+2C32×C37118yescomplemented
99C3×H(3,𝔽3)C32×C3713yes—
100C3×H(3,𝔽3)C32×C3713yes—
101C3×H(3,𝔽3)C32×C3716yes—
102C32×C9C32×C3713yes—
103C32×C9C32×C3713yes—
104C32×C9C32×C3719yescomplemented
105C32⋊C9C32×C3711yescharacteristic
106C32⋊C9C32×C3716yes—
107C32⋊C9C32×C3712yes—
108C32⋊C9C32×C3713yes—
109C32⋊C9C32×C3716yes—
110C32⋊C9C32×C37118yescomplemented
111C32⋊C9C32×C37136yescomplemented
112C37#1062489611yescharacteristic, direct product factor, Sylow
113C33—33no—
114C33—33no—
115C33—92no—
116C33C33×C3712yes—
117C33C33×C3719yes—
118C33C33×C3711yescharacteristic
119C33C33×C3711yescharacteristic
1203−1+2—36no—
1213−1+2—39no—
1223−1+2—318no—
1233−1+2—354no—
1243−1+2H(3,𝔽3)×C3719yescomplemented
125H(3,𝔽3)—39no—
126H(3,𝔽3)—39no—
127H(3,𝔽3)—96no—
128C3×C9—39no—
129C3×C9—39no—
130C3×C9—327no—
131C3×C9C33×C3713yes—
132C3×C9C33×C3716yes—
133C3×C9C33×C3719yes—
134C3×C9C33×C3716yes—
135C3×C9C33×C3713yes—
136C32—31no—
137C32—31no—
138C32—32no—
139C32—32no—
140C32—39no—
141C32—32no—
142C32—39no—
143C32—32no—
144C32—32no—
145C32—318no—
146C32—32no—
147C32—99no—
148C32C3×H(3,𝔽3)×C3713yes—
149C32C37×#73713yescomplemented
150C32C34×C3711yescharacteristic
151C9—33no—
152C9—36no—
153C9—36no—
154C9—39no—
155C9C3×H(3,𝔽3)×C3719yes—
156C3—33no—
157C3—33no—
158C3—92no—
159C3—99no—
160C3C32×H(3,𝔽3)×C3711yescharacteristic
161C3C3×C37×#73711yescharacteristic
162C3C37×3−1+412yes—
subgroupG/H
center, socleC32×C37C34
commutator, Frattini, cosocleC32C34×C37
prime radicalC37×C32⋊H(3,𝔽3)C3
LCSG ▷ C32 ▷ C1
UCSC1 ◁ C32×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C37×#737C3semidirect
2. C32×3−1+2×C37C3semidirect
3. C37×#6893C3semidirect
4. C37×C9⋊H(3,𝔽3)C3semidirect
5. C37×C9⋊H(3,𝔽3)C3semidirect
6. C37×#6891C3semidirect
8. C9×H(3,𝔽3)×C37C3semidirect
9. C3×C37×C32⋊C9C3semidirect
17. C3×3−1+2×C37C32semidirect
23. C32×C9×C37C32semidirect
29. C37×C32⋊C9C32semidirect
30. C37×C32⋊C9C32semidirect
42. 3−1+2×C37H(3,𝔽3)semidirect
54. #10624896C37direct (N × H)
68. C32×C37#737semidirect
75. C3×#737C3×C37semidirect
76. C32×3−1+2C3×C37semidirect
77. #6893C3×C37semidirect
78. C9⋊H(3,𝔽3)C3×C37semidirect
79. C9⋊H(3,𝔽3)C3×C37semidirect
80. #6891C3×C37semidirect
82. C9×H(3,𝔽3)C3×C37semidirect
83. C3×C32⋊C9C3×C37semidirect
98. C3×3−1+2C32×C37semidirect
104. C32×C9C32×C37semidirect
110. C32⋊C9C32×C37semidirect
111. C32⋊C9C32×C37semidirect
112. C37#10624896direct (N × H)
124. 3−1+2H(3,𝔽3)×C37semidirect
149. C32C37×#737semidirect

Maximal subgroups

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

subgroupindex
C3×C37×#7373
C32×3−1+2×C373
C37×#68933
C37×C9⋊H(3,𝔽3)3
C37×#68913
C37×C32⋊H(3,𝔽3)3
C9×H(3,𝔽3)×C373
C3×C37×C32⋊C93
#1062489637