Finite Group Databasebeta

← #36382963

Subgroups — #36382963

Order 22599.

980 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 286 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
2C3×3−1+2×C31⋊C3C311yesmaximal, characteristic, complemented
3#21419414C319yesmaximal, complemented
4#21419413C319yesmaximal, complemented
5#21419409C312yesmaximal, complemented
6#21419405C311yesmaximal, characteristic, complemented
7#21419402C312yesmaximal, complemented
8#21419400C311yesmaximal, characteristic, complemented
9#21419388C319yesmaximal, complemented
10C3×#14578895C312yesmaximal, complemented
11C3×#14578893C311yesmaximal, characteristic, complemented
12C31×3−1+4C311yesmaximal, characteristic, complemented
13C33×C31⋊C3C3211yescharacteristic
14#14578913—36no—
15#14578913—36no—
16#14578912C32118yescomplemented
17C3×#11719689C3212yes—
18C3×#11719689C3212yes—
19C3×#11719689C3211yescharacteristic
20C3×#11719689C3211yescharacteristic
21C3×#11719686C3213yescomplemented
22C3×#11719687C3213yescomplemented
233−1+2×C31⋊C3—33no—
243−1+2×C31⋊C3—36no—
253−1+2×C31⋊C3—36no—
26C3×C32⋊(C31⋊C3)C3212yes—
27C3×C9×C31⋊C3C3213yescomplemented
28#14578901C32118yescomplemented
29#14578899C3219yescomplemented
30#14578898C32118yescomplemented
31#14578896C3219yescomplemented
32#14578895C3216yescomplemented
33#14578895C3216yescomplemented
34#14578895C3216yescomplemented
35#14578893C3213yescomplemented
36#14578893C3213yescomplemented
37#14578893C3213yescomplemented
38C31×#737—36no—
39C31×#737C3219yescomplemented
40C3×3−1+2×C31—33no—
41C3×3−1+2×C31C3212yes—
42C3×3−1+2×C31C3211yescharacteristic
43C3×H(3,𝔽3)×C31C3211yescharacteristic
44C33×C31—31no—
45C33×C31C3311yescharacteristic
46C32×C31⋊C3—31no—
47C32×C31⋊C3—32no—
48C32×C31⋊C3—32no—
49C32×C31⋊C3—93no—
50C32×C31⋊C3C3311yescharacteristic
51C32×C31⋊C3C3311yescharacteristic
52C32×C31⋊C3C3311yescharacteristic
533−1+2×C31—32no—
543−1+2×C31—32no—
553−1+2×C31—318no—
563−1+2×C31H(3,𝔽3)13yes—
573−1+2×C31H(3,𝔽3)16yes—
583−1+2×C31C3316yes—
59H(3,𝔽3)×C31—32no—
60H(3,𝔽3)×C31C3313yes—
61C3×C9×C31—32no—
62C3×C9×C31—39no—
63C3×C9×C31C3313yes—
64C32⋊(C31⋊C3)—32no—
65C32⋊(C31⋊C3)—32no—
66C32⋊(C31⋊C3)C3316yes—
67#11719689—32no—
68#11719689—32no—
69#11719689—31no—
70#11719689—31no—
71#11719689—32no—
72#11719689—32no—
73#11719689—32no—
74#11719689—32no—
75#11719689C3316yes—
76#11719689C3316yes—
77C3×C31⋊C9—32no—
78C3×C31⋊C9—32no—
79C3×C31⋊C9C3313yes—
80C3×C31⋊C9C3313yes—
81#11719687—39no—
82#11719687—318no—
83#11719686—39no—
84#11719686—318no—
85C9×C31⋊C3—39no—
86C9×C31⋊C3—318no—
87#10624934—311nomaximal, Sylow
88C32×C31—33no—
89C32×C31—91no—
90C32×C31—271no—
91C32×C31C3×H(3,𝔽3)11yescharacteristic
92C32×C31C3×H(3,𝔽3)12yes—
93C32×C31C3411yescharacteristic
94C3×C31⋊C3—31no—
95C3×C31⋊C3—33no—
96C3×C31⋊C3—36no—
97C3×C31⋊C3—31no—
98C3×C31⋊C3—31no—
99C3×C31⋊C3—96no—
100C3×C31⋊C3—93no—
101C3×C31⋊C3—91no—
102C3×C31⋊C3—91no—
103C3×C31⋊C3—272no—
104C9×C31—36no—
105C9×C31C3×H(3,𝔽3)19yes—
106C31⋊C9—36no—
107C31⋊C9—36no—
108C31⋊C9—33no—
109C31⋊C9—33no—
1103−1+4—312no—
1113−1+4C31⋊C311yescharacteristic, complemented
112C32×3−1+2—311no—
113#6913—319no—
114#6913—319no—
115#6912—312no—
116#6911—311no—
117#6910—312no—
118#6910—311no—
119#6908—319no—
120C3×AΣL(1,𝔽27)—312no—
121C3×AΣL(1,𝔽27)—311no—
122C3×C31—31no—
123C3×C31—32no—
124C3×C31—31no—
125C3×C31—93no—
126C3×C31C32×H(3,𝔽3)11yescharacteristic
127C31⋊C3—33no—
128C31⋊C3—91no—
129C31⋊C3—91no—
130C31⋊C3—96no—
131C34—311no—
132#737—36no—
133#737—3118no—
134#737—936no—
135#737—936no—
136#737C3×C31⋊C319yescomplemented
137C3×3−1+2—33no—
138C3×3−1+2—311no—
139C3×3−1+2—311no—
140C3×3−1+2—313no—
141C3×3−1+2—313no—
142C3×3−1+2—312no—
143C3×3−1+2—312no—
144C3×3−1+2—933no—
145C3×3−1+2—936no—
146C3×3−1+2—936no—
147C3×3−1+2C3×C31⋊C312yes—
148C3×3−1+2C3×C31⋊C311yescharacteristic
149C3×H(3,𝔽3)—312no—
150C3×H(3,𝔽3)C3×C31⋊C311yescharacteristic
151C32×C9—313no—
152#733—3118no—
153#732—319no—
154#731—3118no—
155#731—319no—
156AΣL(1,𝔽27)—316no—
157AΣL(1,𝔽27)—316no—
158AΣL(1,𝔽27)—316no—
159AΣL(1,𝔽27)—313no—
160AΣL(1,𝔽27)—313no—
161AΣL(1,𝔽27)—313no—
162C31#1062493411yescharacteristic, complemented, Sylow
163C33—31no—
164C33—2793no—
165C33—311no—
166C33—311no—
167C33—311no—
168C33—932no—
169C33—931no—
170C33—932no—
171C33C32×C31⋊C311yescharacteristic
1723−1+2—32no—
1733−1+2—32no—
1743−1+2—318no—
1753−1+2—316no—
1763−1+2—316no—
1773−1+2—932no—
1783−1+2—932no—
1793−1+2—931no—
1803−1+2—931no—
1813−1+2—932no—
1823−1+2—932no—
1833−1+2—932no—
1843−1+2—932no—
1853−1+2—939no—
1863−1+2—9318no—
1873−1+2—939no—
1883−1+2—9318no—
1893−1+2C32⋊(C31⋊C3)13yes—
1903−1+2C32⋊(C31⋊C3)16yes—
1913−1+2C32×C31⋊C316yes—
192H(3,𝔽3)—32no—
193H(3,𝔽3)—316no—
194H(3,𝔽3)—932no—
195H(3,𝔽3)—932no—
196H(3,𝔽3)C32×C31⋊C313yes—
197C3×C9—32no—
198C3×C9—39no—
199C3×C9—313no—
200C3×C9—313no—
201C3×C9—932no—
202C3×C9—932no—
203C3×C9—939no—
204C3×C9—9318no—
205C3×C9C32×C31⋊C313yes—
206C32—33no—
207C32—91no—
208C32—2791no—
209C32—2791no—
210C32—2793no—
211C32—2796no—
212C32—271no—
213C32—8372no—
214C32—933no—
215C32—936no—
216C32—931no—
217C32—931no—
218C32—931no—
219C32C3×C32⋊(C31⋊C3)12yes—
220C32C3×C32⋊(C31⋊C3)11yescharacteristic
221C32C33×C31⋊C311yescharacteristic
222C9—36no—
223C9—933no—
224C9—933no—
225C9—936no—
226C9—936no—
227C9C3×C32⋊(C31⋊C3)19yes—
228C3—31no—
229C3—32no—
230C3—31no—
231C3—93no—
232C3—2791no—
233C3—2791no—
234C3—2796no—
235C3—933no—
236C3C32×C32⋊(C31⋊C3)11yescharacteristic
subgroupG/H
centerC3C32×C32⋊(C31⋊C3)
commutator, cosocleC32×C31C34
FrattiniC32C33×C31⋊C3
FittingC31×3−1+4C3
socleC3×C31C32×H(3,𝔽3)
hypercenter3−1+4C31⋊C3
hypocenterC31#10624934
LCSG ▷ C32×C31 ▷ C3×C31 ▷ C31
UCSC1 ◁ C3 ◁ C3×3−1+2 ◁ 3−1+4

Complements

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

NH ≅ G/Ntype
1. #21419437C3semidirect
2. C3×3−1+2×C31⋊C3C3semidirect
3. #21419414C3semidirect
4. #21419413C3semidirect
5. #21419409C3semidirect
6. #21419405C3semidirect
7. #21419402C3semidirect
8. #21419400C3semidirect
9. #21419388C3semidirect
10. C3×#14578895C3semidirect
11. C3×#14578893C3semidirect
12. C31×3−1+4C3semidirect
16. #14578912C32semidirect
21. C3×#11719686C32semidirect
22. C3×#11719687C32semidirect
27. C3×C9×C31⋊C3C32semidirect
28. #14578901C32semidirect
29. #14578899C32semidirect
30. #14578898C32semidirect
31. #14578896C32semidirect
32. #14578895C32semidirect
33. #14578895C32semidirect
34. #14578895C32semidirect
35. #14578893C32semidirect
36. #14578893C32semidirect
37. #14578893C32semidirect
39. C31×#737C32semidirect
111. 3−1+4C31⋊C3semidirect
136. #737C3×C31⋊C3semidirect
162. C31#10624934semidirect

Maximal subgroups

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

subgroupindex
#214194373
C3×3−1+2×C31⋊C33
#214194143
#214194133
#214194093
#214194053
#214194023
#214194003
#214193883
C3×#145788953
C3×#145788933
C31×3−1+43
#1062493431