Finite Group Databasebeta

← #44533720

Subgroups — #44533720

Order 31347.

560 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C43⋊H(5,𝔽3)C311yesmaximal, characteristic, complemented
2#23782891C311yesmaximal, characteristic, complemented
3#23782891C312yesmaximal, complemented
4#23782883C312yesmaximal, complemented
5#23782881C312yesmaximal, complemented
6C3×#16812508C311yesmaximal, characteristic, complemented
7C3×#16812506C311yesmaximal, characteristic, complemented
8(C3×C43⋊C3)⋊H(3,𝔽3)C312yesmaximal, complemented
9C43×#6911C311yesmaximal, characteristic, complemented
10C33×C43⋊C3C3211yescharacteristic, complemented
11C3×#11923958C3212yescomplemented
12C3×#11923958C3211yescharacteristic, complemented
13C3×C32⋊(C43⋊C3)—33no—
14C3×C32⋊(C43⋊C3)—36no—
15C3×C32⋊(C43⋊C3)C3212yes—
16C3×C32⋊(C43⋊C3)C3211yescharacteristic
17C3×C32⋊(C43⋊C3)C3212yescomplemented
18H(3,𝔽3)×C43⋊C3—36no—
19H(3,𝔽3)×C43⋊C3—33no—
20#16812512—36no—
21#16812512—33no—
22#16812508—36no—
23#16812508—33no—
24#16812506—36no—
25#16812506—33no—
26C3×3−1+2×C43C3211yescharacteristic, complemented
27C3×H(3,𝔽3)×C43C3211yescharacteristic
28C3×H(3,𝔽3)×C43C3212yescomplemented
29C43×#732—33no—
30C32×C43⋊C3—32no—
31C32×C43⋊C3—39no—
32C32×C43⋊C3—31no—
33C32×C43⋊C3—31no—
34C32×C43⋊C3—31no—
35C32×C43⋊C3—32no—
36C32×C43⋊C3—93no—
37C32×C43⋊C3H(3,𝔽3)16yescomplemented
38#11923958—32no—
39#11923958—32no—
40#11923958—32no—
41#11923958—31no—
42#11923958—32no—
43C32⋊(C43⋊C3)—318no—
44C32⋊(C43⋊C3)—32no—
45C32⋊(C43⋊C3)—32no—
46C32⋊(C43⋊C3)—32no—
47C32⋊(C43⋊C3)—32no—
48C32⋊(C43⋊C3)—32no—
49C32⋊(C43⋊C3)—32no—
50C32⋊(C43⋊C3)—32no—
51C32⋊(C43⋊C3)—918no—
52C32⋊(C43⋊C3)H(3,𝔽3)13yescomplemented
53C3×C43⋊C9—32no—
54C3×C43⋊C9—31no—
55C33×C43—31no—
56C33×C43—32no—
57C33×C43C3311yescharacteristic
583−1+2×C43—31no—
593−1+2×C43—32no—
60H(3,𝔽3)×C43—32no—
61H(3,𝔽3)×C43—32no—
62H(3,𝔽3)×C43—32no—
63H(3,𝔽3)×C43—32no—
64H(3,𝔽3)×C43H(3,𝔽3)13yescomplemented
65C3×C9×C43—31no—
66UT(4,𝔽3)—431nomaximal, Sylow
67C32×C43—33no—
68C32×C43—91no—
69C32×C43—96no—
70C32×C43—92no—
71C32×C43—271no—
72C32×C43C3×H(3,𝔽3)11yescharacteristic
73C32×C43C3×H(3,𝔽3)12yes—
74C32×C43C3×H(3,𝔽3)11yescharacteristic
75C3×C43⋊C3—36no—
76C3×C43⋊C3—36no—
77C3×C43⋊C3—33no—
78C3×C43⋊C3—94no—
79C3×C43⋊C3—91no—
80C3×C43⋊C3—93no—
81C3×C43⋊C3—93no—
82C3×C43⋊C3—96no—
83C3×C43⋊C3—96no—
84C3×C43⋊C3—92no—
85C3×C43⋊C3—271no—
86C3×C43⋊C3—279no—
87C9×C43—91no—
88C43⋊C9—92no—
89C43⋊C9—91no—
90H(5,𝔽3)—431no—
91#6911—431no—
92#6911—432no—
93#6911C43⋊C311yescharacteristic, complemented
94#6909—432no—
95#6909—432no—
96C3×AΣL(1,𝔽27)—431no—
97C3×AΣL(1,𝔽27)—431no—
98C32⋊H(3,𝔽3)—432no—
99C3×C43—32no—
100C3×C43—31no—
101C3×C43—31no—
102C3×C43—93no—
103C3×C43—272no—
104C3×C43C32⋊H(3,𝔽3)11yescharacteristic
105C43⋊C3—93no—
106C43⋊C3—96no—
107C43⋊C3—93no—
108C43⋊C3—272no—
109C34—431no—
110C3×3−1+2—432no—
111C3×3−1+2—431no—
112C3×3−1+2C3×C43⋊C311yescharacteristic, complemented
113C3×H(3,𝔽3)—431no—
114C3×H(3,𝔽3)—432no—
115C3×H(3,𝔽3)—432no—
116C3×H(3,𝔽3)—1296no—
117C3×H(3,𝔽3)—1293no—
118C3×H(3,𝔽3)—1293no—
119C3×H(3,𝔽3)—1296no—
120C3×H(3,𝔽3)C3×C43⋊C311yescharacteristic
121C3×H(3,𝔽3)C3×C43⋊C312yescomplemented
122#732—33no—
123#732—1296no—
124#732—1293no—
125AΣL(1,𝔽27)—1296no—
126AΣL(1,𝔽27)—1293no—
127AΣL(1,𝔽27)—1296no—
128AΣL(1,𝔽27)—1293no—
129C43UT(4,𝔽3)11yescharacteristic, complemented, Sylow
130C33—31no—
131C33—32no—
132C33—436no—
133C33—1299no—
134C33—1292no—
135C33—1291no—
136C33—1291no—
137C33—1292no—
138C33—1291no—
139C33—3873no—
140C33C32×C43⋊C311yescharacteristic
1413−1+2—31no—
1423−1+2—32no—
1433−1+2—1291no—
1443−1+2—1292no—
1453−1+2—1292no—
1463−1+2—1292no—
1473−1+2—1292no—
148H(3,𝔽3)—32no—
149H(3,𝔽3)—32no—
150H(3,𝔽3)—32no—
151H(3,𝔽3)—32no—
152H(3,𝔽3)—433no—
153H(3,𝔽3)—12918no—
154H(3,𝔽3)—1292no—
155H(3,𝔽3)—1292no—
156H(3,𝔽3)—1292no—
157H(3,𝔽3)—1292no—
158H(3,𝔽3)—1292no—
159H(3,𝔽3)—1292no—
160H(3,𝔽3)—1292no—
161H(3,𝔽3)—38718no—
162H(3,𝔽3)C32⋊(C43⋊C3)13yescomplemented
163C3×C9—31no—
164C3×C9—1292no—
165C3×C9—1291no—
166C32—33no—
167C32—91no—
168C32—96no—
169C32—92no—
170C32—271no—
171C32—1296no—
172C32—1293no—
173C32—1296no—
174C32—3876no—
175C32—3873no—
176C32—3873no—
177C32—3874no—
178C32—3876no—
179C32—3871no—
180C32—3872no—
181C32—11611no—
182C32—11619no—
183C32H(3,𝔽3)×C43⋊C311yescharacteristic
184C32H(3,𝔽3)×C43⋊C312yes—
185C32C3×C32⋊(C43⋊C3)11yescharacteristic
186C9—91no—
187C9—3872no—
188C9—3871no—
189C3—32no—
190C3—31no—
191C3—31no—
192C3—93no—
193C3—272no—
194C3—3873no—
195C3—3876no—
196C3—3873no—
197C3—11612no—
198C3(C3×C43⋊C3)⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC3(C3×C43⋊C3)⋊H(3,𝔽3)
commutator, cosocleC33×C43C33
FrattiniC33C32×C43⋊C3
FittingC43×#6911C3
socleC3×C43C32⋊H(3,𝔽3)
hypercenter#6911C43⋊C3
hypocenterC43UT(4,𝔽3)
LCSG ▷ C33×C43 ▷ C3×C43 ▷ C43
UCSC1 ◁ C3 ◁ C33 ◁ #6911

Complements

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

NH ≅ G/Ntype
1. C43⋊H(5,𝔽3)C3semidirect
2. #23782891C3semidirect
3. #23782891C3semidirect
4. #23782883C3semidirect
5. #23782881C3semidirect
6. C3×#16812508C3semidirect
7. C3×#16812506C3semidirect
8. (C3×C43⋊C3)⋊H(3,𝔽3)C3semidirect
9. C43×#6911C3semidirect
10. C33×C43⋊C3C32semidirect
11. C3×#11923958C32semidirect
12. C3×#11923958C32semidirect
17. C3×C32⋊(C43⋊C3)C32semidirect
26. C3×3−1+2×C43C32semidirect
28. C3×H(3,𝔽3)×C43C32semidirect
37. C32×C43⋊C3H(3,𝔽3)semidirect
52. C32⋊(C43⋊C3)H(3,𝔽3)semidirect
64. H(3,𝔽3)×C43H(3,𝔽3)semidirect
93. #6911C43⋊C3semidirect
112. C3×3−1+2C3×C43⋊C3semidirect
121. C3×H(3,𝔽3)C3×C43⋊C3semidirect
129. C43UT(4,𝔽3)semidirect
162. H(3,𝔽3)C32⋊(C43⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C43⋊H(5,𝔽3)3
#237828913
#237828833
#237828813
C3×#168125083
C3×#168125063
(C3×C43⋊C3)⋊H(3,𝔽3)3
C43×#69113
UT(4,𝔽3)43