Finite Group Databasebeta

← #44535347

Subgroups — #44535347

Order 31347.

978 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 978 conjugacy classes of subgroups — too many to draw. Only its 268 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
1#23782923C312yesmaximal, complemented
2C43⋊H(5,𝔽3)C311yesmaximal, characteristic
3C3×#16812526C311yesmaximal, characteristic, complemented
4#23782888C312yesmaximal, complemented
5#23782887C312yesmaximal, complemented
6#23782886C311yesmaximal, characteristic, complemented
7#23782885C319yesmaximal, complemented
8#23782884C319yesmaximal, complemented
9C3×#16812511C312yesmaximal, complemented
10C3×#16812509C311yesmaximal, characteristic, complemented
11#731×C43⋊C3C319yesmaximal, complemented
12C43×#6910C311yesmaximal, characteristic, complemented
13#16812526—36no—
14#16812526—33no—
15#16812526—36no—
16#16812526—36no—
17#16812525—36no—
18C3×#11923958C3212yescomplemented
19C3×#11923958C3212yescomplemented
20C3×#11923958C3211yescharacteristic, complemented
21C3×#11923957C3213yescomplemented
22C3×#11923956C3213yescomplemented
233−1+2×C43⋊C3C32118yescomplemented
24C3×C32⋊(C43⋊C3)—33no—
25C3×C32⋊(C43⋊C3)C3212yes—
26C3×C32⋊(C43⋊C3)C3211yescharacteristic
27H(3,𝔽3)×C43⋊C3—36no—
28H(3,𝔽3)×C43⋊C3C3219yes—
29C32×C43⋊C9C3211yescharacteristic, complemented
30C3×C9×C43⋊C3C3213yescomplemented
31#16812511C3216yescomplemented
32#16812511C32118yescomplemented
33#16812511C3216yescomplemented
34#16812511C3216yescomplemented
35#16812510C32118yescomplemented
36#16812509C3213yescomplemented
37#16812509C3219yescomplemented
38#16812509C3213yescomplemented
39#16812509C3213yescomplemented
40C3×3−1+2×C43C3212yescomplemented
41C3×3−1+2×C43C3211yescharacteristic, complemented
42C3×H(3,𝔽3)×C43C3211yescharacteristic
43C43×#731C3219yescomplemented
44C32×C43⋊C3—32no—
45C32×C43⋊C3—39no—
46C32×C43⋊C3C3313yes—
47#11923958—32no—
48#11923958—32no—
49#11923958—32no—
50#11923958—32no—
51#11923958—32no—
52#11923958—31no—
53#11923958C3316yes—
54#11923958C3316yes—
55#11923957—318no—
56#11923957—39no—
57#11923956—318no—
58#11923956—39no—
59C32⋊(C43⋊C3)—32no—
60C32⋊(C43⋊C3)—32no—
61C32⋊(C43⋊C3)—318no—
62C32⋊(C43⋊C3)H(3,𝔽3)16yes—
63C32⋊(C43⋊C3)H(3,𝔽3)13yes—
64C32⋊(C43⋊C3)C3316yes—
65C3×C43⋊C9—32no—
66C3×C43⋊C9—31no—
67C3×C43⋊C9—32no—
68C3×C43⋊C9—32no—
69C3×C43⋊C9C3311yescharacteristic
70C3×C43⋊C9C3313yes—
71C3×C43⋊C9C3311yescharacteristic
72C3×C43⋊C9C3311yescharacteristic
73C9×C43⋊C3—39no—
74C9×C43⋊C3—318no—
75C33×C43—31no—
76C33×C43C3311yescharacteristic
773−1+2×C43—32no—
783−1+2×C43—32no—
793−1+2×C43—32no—
803−1+2×C43—31no—
813−1+2×C43C3316yes—
82H(3,𝔽3)×C43—32no—
83H(3,𝔽3)×C43C3313yes—
84C3×C9×C43—32no—
85C3×C9×C43C3313yes—
86#10624931—431nomaximal, Sylow
87C32×C43—33no—
88C32×C43—91no—
89C32×C43—271no—
90C32×C43C3×H(3,𝔽3)12yes—
91C32×C43C3×H(3,𝔽3)11yescharacteristic
92C32×C43C3411yescharacteristic
93C3×C43⋊C3—36no—
94C3×C43⋊C3—93no—
95C3×C43⋊C3—279no—
96C3×C43⋊C3—272no—
97C3×C43⋊C3C3×H(3,𝔽3)19yes—
98C9×C43—36no—
99C9×C43—33no—
100C43⋊C9—31no—
101C43⋊C9—36no—
102C43⋊C9—36no—
103C43⋊C9—33no—
104C43⋊C9—31no—
105C43⋊C9—31no—
1063−1+4—432no—
107H(5,𝔽3)—431no—
108C3×#737—431no—
109#6910—432no—
110#6910—432no—
111#6910—431no—
112#6910—439no—
113#6910—439no—
114#6910C43⋊C311yescharacteristic, complemented
115C3×#731—432no—
116C3×#731—431no—
117C3×#731—439no—
118C3×C43—31no—
119C3×C43—32no—
120C3×C43—31no—
121C3×C43—93no—
122C3×C43C32×H(3,𝔽3)11yescharacteristic
123C43⋊C3—39no—
124C43⋊C3—96no—
125#737—1296no—
126#737—1296no—
127#737—1293no—
128#737—1296no—
129#737—1296no—
130C3×3−1+2—431no—
131C3×3−1+2—433no—
132C3×3−1+2—433no—
133C3×3−1+2—432no—
134C3×3−1+2—432no—
135C3×3−1+2—4318no—
136C3×3−1+2C3×C43⋊C312yescomplemented
137C3×3−1+2C3×C43⋊C311yescharacteristic, complemented
138C3×H(3,𝔽3)—431no—
139C3×H(3,𝔽3)—432no—
140C3×H(3,𝔽3)—439no—
141C3×H(3,𝔽3)—1296no—
142C3×H(3,𝔽3)—1293no—
143C3×H(3,𝔽3)C3×C43⋊C311yescharacteristic
144C32×C9—431no—
145C32×C9—433no—
146#731—436no—
147#731—4318no—
148#731—436no—
149#731—436no—
150#731—433no—
151#731—439no—
152#731—433no—
153#731—433no—
154#731—4318no—
155#731C3×C43⋊C319yescomplemented
156C43#1062493111yescharacteristic, complemented, Sylow
157C33—31no—
158C33—433no—
159C33—1299no—
160C33—1292no—
161C33C32×C43⋊C311yescharacteristic
1623−1+2—32no—
1633−1+2—32no—
1643−1+2—32no—
1653−1+2—31no—
1663−1+2—436no—
1673−1+2—436no—
1683−1+2—12918no—
1693−1+2—1299no—
1703−1+2—12918no—
1713−1+2—1299no—
1723−1+2—1292no—
1733−1+2—1291no—
1743−1+2—1292no—
1753−1+2—1292no—
1763−1+2—1292no—
1773−1+2—1292no—
1783−1+2C32×C43⋊C316yes—
179H(3,𝔽3)—32no—
180H(3,𝔽3)—436no—
181H(3,𝔽3)—433no—
182H(3,𝔽3)—436no—
183H(3,𝔽3)—12918no—
184H(3,𝔽3)—1292no—
185H(3,𝔽3)—1292no—
186H(3,𝔽3)C32×C43⋊C313yes—
187C3×C9—32no—
188C3×C9—431no—
189C3×C9—431no—
190C3×C9—431no—
191C3×C9—433no—
192C3×C9—1299no—
193C3×C9—12918no—
194C3×C9—1292no—
195C3×C9—1291no—
196C3×C9—1292no—
197C3×C9—1292no—
198C3×C9C32×C43⋊C313yes—
199C32—33no—
200C32—91no—
201C32—271no—
202C32—439no—
203C32—1296no—
204C32—3873no—
205C32—11612no—
206C32—11619no—
207C32H(3,𝔽3)×C43⋊C312yes—
208C32H(3,𝔽3)×C43⋊C311yescharacteristic
209C32C33×C43⋊C311yescharacteristic
210C9—36no—
211C9—33no—
212C9—1291no—
213C9—1293no—
214C9—1291no—
215C9—1291no—
216C9—1296no—
217C9—1296no—
218C3—31no—
219C3—31no—
220C3—32no—
221C3—93no—
222C3—1299no—
223C3—3876no—
224C3C3×H(3,𝔽3)×C43⋊C311yescharacteristic
subgroupG/H
centerC3C3×H(3,𝔽3)×C43⋊C3
commutator, cosocleC32×C43C34
FrattiniC32C33×C43⋊C3
FittingC43×#6910C3
socleC3×C43C32×H(3,𝔽3)
prime radicalC43⋊H(5,𝔽3)C3
hypercenter#6910C43⋊C3
hypocenterC43#10624931
LCSG ▷ C32×C43 ▷ C3×C43 ▷ C43
UCSC1 ◁ C3 ◁ C33 ◁ #6910

Complements

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

NH ≅ G/Ntype
1. #23782923C3semidirect
3. C3×#16812526C3semidirect
4. #23782888C3semidirect
5. #23782887C3semidirect
6. #23782886C3semidirect
7. #23782885C3semidirect
8. #23782884C3semidirect
9. C3×#16812511C3semidirect
10. C3×#16812509C3semidirect
11. #731×C43⋊C3C3semidirect
12. C43×#6910C3semidirect
18. C3×#11923958C32semidirect
19. C3×#11923958C32semidirect
20. C3×#11923958C32semidirect
21. C3×#11923957C32semidirect
22. C3×#11923956C32semidirect
23. 3−1+2×C43⋊C3C32semidirect
29. C32×C43⋊C9C32semidirect
30. C3×C9×C43⋊C3C32semidirect
31. #16812511C32semidirect
32. #16812511C32semidirect
33. #16812511C32semidirect
34. #16812511C32semidirect
35. #16812510C32semidirect
36. #16812509C32semidirect
37. #16812509C32semidirect
38. #16812509C32semidirect
39. #16812509C32semidirect
40. C3×3−1+2×C43C32semidirect
41. C3×3−1+2×C43C32semidirect
43. C43×#731C32semidirect
114. #6910C43⋊C3semidirect
136. C3×3−1+2C3×C43⋊C3semidirect
137. C3×3−1+2C3×C43⋊C3semidirect
155. #731C3×C43⋊C3semidirect
156. C43#10624931semidirect

Maximal subgroups

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

subgroupindex
#237829233
C43⋊H(5,𝔽3)3
C3×#168125263
#237828883
#237828873
#237828863
#237828853
#237828843
C3×#168125113
C3×#168125093
#731×C43⋊C33
C43×#69103
#1062493143