Finite Group Databasebeta

← #36382949

Subgroups — #36382949

Order 22599.

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#21419437C312yesmaximal, complemented
2C31⋊H(5,𝔽3)C311yesmaximal, characteristic
3C3×#14578913C311yesmaximal, characteristic, complemented
4#21419402C312yesmaximal, complemented
5#21419401C312yesmaximal, complemented
6#21419400C311yesmaximal, characteristic, complemented
7#21419399C319yesmaximal, complemented
8#21419398C319yesmaximal, complemented
9C3×#14578898C312yesmaximal, complemented
10C3×#14578896C311yesmaximal, characteristic, complemented
11#731×C31⋊C3C319yesmaximal, complemented
12C31×#6910C311yesmaximal, characteristic, complemented
13#14578913—36no—
14#14578913—33no—
15#14578913—36no—
16#14578913—36no—
17#14578912—36no—
18C3×#11719689C3212yescomplemented
19C3×#11719689C3212yescomplemented
20C3×#11719689C3211yescharacteristic, complemented
21C3×#11719686C3213yescomplemented
22C3×#11719687C3213yescomplemented
233−1+2×C31⋊C3C32118yescomplemented
24C3×C32⋊(C31⋊C3)—33no—
25C3×C32⋊(C31⋊C3)C3212yes—
26C3×C32⋊(C31⋊C3)C3211yescharacteristic
27H(3,𝔽3)×C31⋊C3—36no—
28H(3,𝔽3)×C31⋊C3C3219yes—
29C32×C31⋊C9C3211yescharacteristic, complemented
30C3×C9×C31⋊C3C3213yescomplemented
31#14578898C3216yescomplemented
32#14578898C32118yescomplemented
33#14578898C3216yescomplemented
34#14578898C3216yescomplemented
35#14578897C32118yescomplemented
36#14578896C3213yescomplemented
37#14578896C3219yescomplemented
38#14578896C3213yescomplemented
39#14578896C3213yescomplemented
40C3×3−1+2×C31C3212yescomplemented
41C3×3−1+2×C31C3211yescharacteristic, complemented
42C3×H(3,𝔽3)×C31C3211yescharacteristic
43C31×#731C3219yescomplemented
44C33×C31—31no—
45C33×C31C3311yescharacteristic
46C32×C31⋊C3—32no—
47C32×C31⋊C3—39no—
48C32×C31⋊C3C3313yes—
493−1+2×C31—32no—
503−1+2×C31—32no—
513−1+2×C31—31no—
523−1+2×C31—32no—
533−1+2×C31C3316yes—
54H(3,𝔽3)×C31—32no—
55H(3,𝔽3)×C31C3313yes—
56C3×C9×C31—32no—
57C3×C9×C31C3313yes—
58C32⋊(C31⋊C3)—32no—
59C32⋊(C31⋊C3)—32no—
60C32⋊(C31⋊C3)—318no—
61C32⋊(C31⋊C3)H(3,𝔽3)16yes—
62C32⋊(C31⋊C3)H(3,𝔽3)13yes—
63C32⋊(C31⋊C3)C3316yes—
64#11719689—32no—
65#11719689—32no—
66#11719689—32no—
67#11719689—32no—
68#11719689—31no—
69#11719689—32no—
70#11719689C3316yes—
71#11719689C3316yes—
72C3×C31⋊C9—31no—
73C3×C31⋊C9—32no—
74C3×C31⋊C9—32no—
75C3×C31⋊C9—32no—
76C3×C31⋊C9C3311yescharacteristic
77C3×C31⋊C9C3313yes—
78C3×C31⋊C9C3311yescharacteristic
79C3×C31⋊C9C3311yescharacteristic
80#11719687—318no—
81#11719687—39no—
82#11719686—318no—
83#11719686—39no—
84C9×C31⋊C3—39no—
85C9×C31⋊C3—318no—
86#10624931—311nomaximal, Sylow
87C32×C31—33no—
88C32×C31—91no—
89C32×C31—271no—
90C32×C31C3×H(3,𝔽3)11yescharacteristic
91C32×C31C3×H(3,𝔽3)12yes—
92C32×C31C3411yescharacteristic
93C3×C31⋊C3—36no—
94C3×C31⋊C3—93no—
95C3×C31⋊C3—279no—
96C3×C31⋊C3—272no—
97C3×C31⋊C3C3×H(3,𝔽3)19yes—
98C9×C31—36no—
99C9×C31—33no—
100C31⋊C9—31no—
101C31⋊C9—36no—
102C31⋊C9—36no—
103C31⋊C9—33no—
104C31⋊C9—31no—
105C31⋊C9—31no—
1063−1+4—312no—
107H(5,𝔽3)—311no—
108C3×#737—311no—
109#6910—312no—
110#6910—312no—
111#6910—311no—
112#6910—319no—
113#6910—319no—
114#6910C31⋊C311yescharacteristic, complemented
115C3×#731—312no—
116C3×#731—311no—
117C3×#731—319no—
118C3×C31—31no—
119C3×C31—32no—
120C3×C31—31no—
121C3×C31—93no—
122C3×C31C32×H(3,𝔽3)11yescharacteristic
123C31⋊C3—39no—
124C31⋊C3—96no—
125#737—936no—
126#737—936no—
127#737—933no—
128#737—936no—
129#737—936no—
130C3×3−1+2—311no—
131C3×3−1+2—313no—
132C3×3−1+2—313no—
133C3×3−1+2—312no—
134C3×3−1+2—312no—
135C3×3−1+2—3118no—
136C3×3−1+2C3×C31⋊C312yescomplemented
137C3×3−1+2C3×C31⋊C311yescharacteristic, complemented
138C3×H(3,𝔽3)—311no—
139C3×H(3,𝔽3)—312no—
140C3×H(3,𝔽3)—319no—
141C3×H(3,𝔽3)—936no—
142C3×H(3,𝔽3)—933no—
143C3×H(3,𝔽3)C3×C31⋊C311yescharacteristic
144C32×C9—311no—
145C32×C9—313no—
146#731—316no—
147#731—3118no—
148#731—316no—
149#731—316no—
150#731—313no—
151#731—319no—
152#731—313no—
153#731—313no—
154#731—3118no—
155#731C3×C31⋊C319yescomplemented
156C31#1062493111yescharacteristic, complemented, Sylow
157C33—31no—
158C33—313no—
159C33—939no—
160C33—932no—
161C33C32×C31⋊C311yescharacteristic
1623−1+2—32no—
1633−1+2—32no—
1643−1+2—32no—
1653−1+2—31no—
1663−1+2—316no—
1673−1+2—316no—
1683−1+2—9318no—
1693−1+2—939no—
1703−1+2—9318no—
1713−1+2—939no—
1723−1+2—932no—
1733−1+2—931no—
1743−1+2—932no—
1753−1+2—932no—
1763−1+2—932no—
1773−1+2—932no—
1783−1+2C32×C31⋊C316yes—
179H(3,𝔽3)—32no—
180H(3,𝔽3)—316no—
181H(3,𝔽3)—313no—
182H(3,𝔽3)—316no—
183H(3,𝔽3)—9318no—
184H(3,𝔽3)—932no—
185H(3,𝔽3)—932no—
186H(3,𝔽3)C32×C31⋊C313yes—
187C3×C9—32no—
188C3×C9—311no—
189C3×C9—313no—
190C3×C9—311no—
191C3×C9—311no—
192C3×C9—939no—
193C3×C9—9318no—
194C3×C9—932no—
195C3×C9—931no—
196C3×C9—932no—
197C3×C9—932no—
198C3×C9C32×C31⋊C313yes—
199C32—33no—
200C32—91no—
201C32—2793no—
202C32—271no—
203C32—319no—
204C32—8372no—
205C32—8379no—
206C32—936no—
207C32H(3,𝔽3)×C31⋊C312yes—
208C32H(3,𝔽3)×C31⋊C311yescharacteristic
209C32C33×C31⋊C311yescharacteristic
210C9—36no—
211C9—33no—
212C9—931no—
213C9—933no—
214C9—931no—
215C9—931no—
216C9—936no—
217C9—936no—
218C3—31no—
219C3—31no—
220C3—32no—
221C3—93no—
222C3—2796no—
223C3—939no—
224C3C3×H(3,𝔽3)×C31⋊C311yescharacteristic
subgroupG/H
centerC3C3×H(3,𝔽3)×C31⋊C3
commutator, cosocleC32×C31C34
FrattiniC32C33×C31⋊C3
FittingC31×#6910C3
socleC3×C31C32×H(3,𝔽3)
prime radicalC31⋊H(5,𝔽3)C3
hypercenter#6910C31⋊C3
hypocenterC31#10624931
LCSG ▷ C32×C31 ▷ C3×C31 ▷ C31
UCSC1 ◁ C3 ◁ C33 ◁ #6910

Complements

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

NH ≅ G/Ntype
1. #21419437C3semidirect
3. C3×#14578913C3semidirect
4. #21419402C3semidirect
5. #21419401C3semidirect
6. #21419400C3semidirect
7. #21419399C3semidirect
8. #21419398C3semidirect
9. C3×#14578898C3semidirect
10. C3×#14578896C3semidirect
11. #731×C31⋊C3C3semidirect
12. C31×#6910C3semidirect
18. C3×#11719689C32semidirect
19. C3×#11719689C32semidirect
20. C3×#11719689C32semidirect
21. C3×#11719686C32semidirect
22. C3×#11719687C32semidirect
23. 3−1+2×C31⋊C3C32semidirect
29. C32×C31⋊C9C32semidirect
30. C3×C9×C31⋊C3C32semidirect
31. #14578898C32semidirect
32. #14578898C32semidirect
33. #14578898C32semidirect
34. #14578898C32semidirect
35. #14578897C32semidirect
36. #14578896C32semidirect
37. #14578896C32semidirect
38. #14578896C32semidirect
39. #14578896C32semidirect
40. C3×3−1+2×C31C32semidirect
41. C3×3−1+2×C31C32semidirect
43. C31×#731C32semidirect
114. #6910C31⋊C3semidirect
136. C3×3−1+2C3×C31⋊C3semidirect
137. C3×3−1+2C3×C31⋊C3semidirect
155. #731C3×C31⋊C3semidirect
156. C31#10624931semidirect

Maximal subgroups

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

subgroupindex
#214194373
C31⋊H(5,𝔽3)3
C3×#145789133
#214194023
#214194013
#214194003
#214193993
#214193983
C3×#145788983
C3×#145788963
#731×C31⋊C33
C31×#69103
#1062493131