Finite Group Databasebeta

← #36382962

Subgroups — #36382962

Order 22599.

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

normal subgroups only

This group has 980 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#21419438C311yesmaximal, characteristic, complemented
2#21419437C312yesmaximal, complemented
3C32×#11719689C311yesmaximal, characteristic, complemented
4#21419412C319yesmaximal, complemented
5#21419411C319yesmaximal, complemented
6#21419409C312yesmaximal, complemented
7#21419405C311yesmaximal, characteristic, complemented
8#21419402C312yesmaximal, complemented
9#21419400C311yesmaximal, characteristic, complemented
10#21419387C319yesmaximal, complemented
11C3×#14578894C312yesmaximal, complemented
12C3×C31×AΣL(1,𝔽27)C311yesmaximal, characteristic, complemented
13#14578913—36no—
14#14578913C32118yescomplemented
15#14578912—36no—
16#14578911—36no—
17#14578911C3219yescomplemented
18C3×#11719689—36no—
19C3×#11719689—33no—
20C3×#11719689—33no—
21C3×#11719689C3212yes—
22C3×#11719689C3212yes—
23C3×#11719689C3211yescharacteristic
24C3×#11719689C3211yescharacteristic
25C3×#11719689C3211yescharacteristic
26C3×#11719689C3213yescomplemented
27C3×#11719689C3213yescomplemented
283−1+2×C31⋊C3—36no—
29C3×C32⋊(C31⋊C3)C3212yes—
30C32×C31⋊C9C3213yescomplemented
31#14578901C32118yescomplemented
32#14578899C3219yescomplemented
33#14578898C32118yescomplemented
34#14578896C3219yescomplemented
35#14578894C3216yescomplemented
36#14578894C3216yescomplemented
37#14578894C3216yescomplemented
38C34×C31C3211yescharacteristic
39C3×3−1+2×C31C3212yes—
40C3×H(3,𝔽3)×C31C3211yescharacteristic
41C31×AΣL(1,𝔽27)C3213yescomplemented
42C31×AΣL(1,𝔽27)C3213yescomplemented
43C31×AΣL(1,𝔽27)C3213yescomplemented
44C33×C31—32no—
45C33×C31—31no—
46C33×C31—31no—
47C33×C31—93no—
48C33×C31C3311yescharacteristic
49C33×C31C3311yescharacteristic
50C33×C31C3311yescharacteristic
51C33×C31C3311yescharacteristic
52C32×C31⋊C3—32no—
533−1+2×C31—32no—
543−1+2×C31—32no—
553−1+2×C31C3316yes—
56H(3,𝔽3)×C31—32no—
57H(3,𝔽3)×C31C3313yes—
58C3×C9×C31—32no—
59C32⋊(C31⋊C3)—32no—
60C32⋊(C31⋊C3)—32no—
61C32⋊(C31⋊C3)C3316yes—
62#11719689—32no—
63#11719689—32no—
64#11719689—32no—
65#11719689—32no—
66#11719689—31no—
67#11719689—31no—
68#11719689—32no—
69#11719689—32no—
70#11719689—39no—
71#11719689—39no—
72#11719689—318no—
73#11719689H(3,𝔽3)16yes—
74#11719689H(3,𝔽3)13yes—
75#11719689C3316yes—
76#11719689C3316yes—
77C3×C31⋊C9—32no—
78C3×C31⋊C9—32no—
79C3×C31⋊C9—39no—
80C3×C31⋊C9—39no—
81C3×C31⋊C9C3313yes—
82C3×C31⋊C9C3313yes—
83C3×C31⋊C9C3313yes—
84#11719687—318no—
85#11719686—318no—
86C9×C31⋊C3—318no—
87#10624934—311nomaximal, Sylow
88C32×C31—33no—
89C32×C31—31no—
90C32×C31—33no—
91C32×C31—31no—
92C32×C31—31no—
93C32×C31—91no—
94C32×C31—96no—
95C32×C31—93no—
96C32×C31—91no—
97C32×C31—91no—
98C32×C31—271no—
99C32×C31C3×H(3,𝔽3)12yes—
100C32×C31C3×H(3,𝔽3)11yescharacteristic
101C32×C31C3411yescharacteristic
102C3×C31⋊C3—36no—
103C3×C31⋊C3—272no—
104C9×C31—36no—
105C31⋊C9—36no—
106C31⋊C9—36no—
107C31⋊C9—33no—
108C31⋊C9—33no—
109C31⋊C9C3×H(3,𝔽3)19yes—
1103−1+4—312no—
1113−1+4—311no—
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)C31⋊C311yescharacteristic, complemented
122C3×C31—32no—
123C3×C31—33no—
124C3×C31—31no—
125C3×C31—31no—
126C3×C31—91no—
127C3×C31—91no—
128C3×C31—93no—
129C3×C31C32×H(3,𝔽3)11yescharacteristic
130C31⋊C3—96no—
131C34C3×C31⋊C311yescharacteristic
132#737—3118no—
133#737—319no—
134#737—936no—
135#737—936no—
136#737—936no—
137C3×3−1+2—311no—
138C3×3−1+2—313no—
139C3×3−1+2—313no—
140C3×3−1+2—311no—
141C3×3−1+2—311no—
142C3×3−1+2—312no—
143C3×3−1+2—312no—
144C3×3−1+2—936no—
145C3×3−1+2—933no—
146C3×3−1+2—936no—
147C3×3−1+2—933no—
148C3×3−1+2C3×C31⋊C312yes—
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)C3×C31⋊C313yescomplemented
160AΣL(1,𝔽27)C3×C31⋊C313yescomplemented
161AΣL(1,𝔽27)C3×C31⋊C313yescomplemented
162C31#1062493411yescharacteristic, complemented, Sylow
163C33—31no—
164C33—32no—
165C33—31no—
166C33—93no—
167C33—932no—
168C33C32×C31⋊C311yescharacteristic
169C33C32×C31⋊C311yescharacteristic
170C33C32×C31⋊C311yescharacteristic
171C33C32×C31⋊C311yescharacteristic
1723−1+2—32no—
1733−1+2—32no—
1743−1+2—313no—
1753−1+2—316no—
1763−1+2—316no—
1773−1+2—316no—
1783−1+2—9318no—
1793−1+2—9318no—
1803−1+2—939no—
1813−1+2—939no—
1823−1+2—9318no—
1833−1+2—931no—
1843−1+2—931no—
1853−1+2—932no—
1863−1+2—932no—
1873−1+2—932no—
1883−1+2—932no—
1893−1+2—932no—
1903−1+2—932no—
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—313no—
199C3×C9—313no—
200C3×C9—313no—
201C3×C9—9318no—
202C3×C9—939no—
203C3×C9—939no—
204C3×C9—932no—
205C3×C9—932no—
206C32—33no—
207C32—31no—
208C32—33no—
209C32—31no—
210C32—31no—
211C32—91no—
212C32—93no—
213C32—96no—
214C32—91no—
215C32—91no—
216C32—271no—
217C32—8372no—
218C32—936no—
219C32H(3,𝔽3)×C31⋊C311yescharacteristic
220C32H(3,𝔽3)×C31⋊C312yes—
221C32C33×C31⋊C311yescharacteristic
222C9—36no—
223C9—319no—
224C9—933no—
225C9—933no—
226C9—936no—
227C9—936no—
228C3—31no—
229C3—33no—
230C3—31no—
231C3—32no—
232C3—91no—
233C3—91no—
234C3—93no—
235C3—2796no—
236C3C3×H(3,𝔽3)×C31⋊C311yescharacteristic
subgroupG/H
centerC3C3×H(3,𝔽3)×C31⋊C3
commutator, cosocleC32×C31C34
FrattiniC32C33×C31⋊C3
FittingC3×C31×AΣL(1,𝔽27)C3
socleC3×C31C32×H(3,𝔽3)
hypercenterC3×AΣL(1,𝔽27)C31⋊C3
hypocenterC31#10624934
LCSG ▷ C32×C31 ▷ C3×C31 ▷ C31
UCSC1 ◁ C3 ◁ C33 ◁ C3×AΣL(1,𝔽27)

Complements

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

NH ≅ G/Ntype
1. #21419438C3semidirect
2. #21419437C3semidirect
3. C32×#11719689C3semidirect
4. #21419412C3semidirect
5. #21419411C3semidirect
6. #21419409C3semidirect
7. #21419405C3semidirect
8. #21419402C3semidirect
9. #21419400C3semidirect
10. #21419387C3semidirect
11. C3×#14578894C3semidirect
12. C3×C31×AΣL(1,𝔽27)C3semidirect
14. #14578913C32semidirect
17. #14578911C32semidirect
26. C3×#11719689C32semidirect
27. C3×#11719689C32semidirect
30. C32×C31⋊C9C32semidirect
31. #14578901C32semidirect
32. #14578899C32semidirect
33. #14578898C32semidirect
34. #14578896C32semidirect
35. #14578894C32semidirect
36. #14578894C32semidirect
37. #14578894C32semidirect
41. C31×AΣL(1,𝔽27)C32semidirect
42. C31×AΣL(1,𝔽27)C32semidirect
43. C31×AΣL(1,𝔽27)C32semidirect
121. C3×AΣL(1,𝔽27)C31⋊C3semidirect
159. AΣL(1,𝔽27)C3×C31⋊C3semidirect
160. AΣL(1,𝔽27)C3×C31⋊C3semidirect
161. AΣL(1,𝔽27)C3×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
#214194383
#214194373
C32×#117196893
#214194123
#214194113
#214194093
#214194053
#214194023
#214194003
#214193873
C3×#145788943
C3×C31×AΣL(1,𝔽27)3
#1062493431