Finite Group Databasebeta

← #36382980

Subgroups — #36382980

Order 22599.

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

normal subgroups only

This group has 948 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#21419435C313yesmaximal, complemented
2C3×#14578912C311yesmaximal, characteristic, complemented
3#21419418C316yesmaximal, complemented
4#21419414C316yesmaximal, complemented
5#21419411C316yesmaximal, complemented
6#21419399C313yesmaximal, complemented
7#21419393C313yesmaximal, complemented
8#21419392C314yesmaximal, complemented
9#21419391C312yesmaximal, complemented
10#21419389C312yesmaximal, complemented
11#733×C31⋊C3C312yesmaximal, complemented
12#732×C31⋊C3C311yesmaximal, characteristic, complemented
13C31×#6908C311yesmaximal, characteristic, complemented
14#14578913—312no—
15#14578913C32112yescomplemented
16#14578912—32no—
17#14578912—31no—
18#14578912—32no—
19#14578912—34no—
20#14578912—36no—
21#14578912C3216yescomplemented
22#14578911C3216yescomplemented
23C3×#11719689C3212yescomplemented
24C3×#11719686C3211yescharacteristic
25C3×#11719686C3212yescomplemented
26C3×#11719687—33no—
27C3×#11719687C3211yescharacteristic
28C3×#11719687C3212yescomplemented
293−1+2×C31⋊C3—36no—
303−1+2×C31⋊C3C3216yes—
31C3×C32⋊(C31⋊C3)C3211yescharacteristic
32H(3,𝔽3)×C31⋊C3C3213yes—
33C3×C9×C31⋊C3C3211yescharacteristic
34C3×C9×C31⋊C3C3212yescomplemented
35#14578902C3214yescomplemented
36#14578901C32112yescomplemented
37#14578900C3212yes—
38#14578899C3216yes—
39#14578898C32112yescomplemented
40#14578897C3216yescomplemented
41#14578896C3216yescomplemented
42#14578895C32112yescomplemented
43#14578894C3216yescomplemented
44#14578893C3216yescomplemented
45C31×#737C3213yescomplemented
46C32×C9×C31C3211yescharacteristic, complemented
47C31×#733C3212yescomplemented
48C31×#732C3211yescharacteristic
49C31×#731C3213yescomplemented
50C31×AΣL(1,𝔽27)C3213yescomplemented
51C33×C31C3311yescharacteristic
52C32×C31⋊C3—33no—
53C32×C31⋊C3C3312yes—
54C32×C31⋊C3C3311yescharacteristic
553−1+2×C31—36no—
563−1+2×C31C3316yes—
57H(3,𝔽3)×C31C3313yes—
58C3×C9×C31—32no—
59C3×C9×C31—31no—
60C3×C9×C31—33no—
61C3×C9×C31C3311yescharacteristic
62C3×C9×C31C3311yescharacteristic
63C3×C9×C31C3311yescharacteristic
64C32⋊(C31⋊C3)—36no—
65C32⋊(C31⋊C3)—32no—
66C32⋊(C31⋊C3)—31no—
67C32⋊(C31⋊C3)C3316yes—
68#11719689—312no—
69#11719689—34no—
70#11719689—32no—
71#11719689C33112yes—
72C3×C31⋊C9—36no—
73C3×C31⋊C9C3312yes—
74C3×C31⋊C9C3312yes—
75C3×C31⋊C9C3312yes—
76#11719687—36no—
77#11719687—36no—
78#11719687—34no—
79#11719687—32no—
80#11719687—312no—
81#11719687H(3,𝔽3)12yes—
82#11719687H(3,𝔽3)11yescharacteristic
83#11719687H(3,𝔽3)12yes—
84#11719687H(3,𝔽3)14yes—
85#11719686—36no—
86#11719686—32no—
87#11719686—31no—
88#11719686—36no—
89#11719686—312no—
90#11719686—34no—
91#11719686—32no—
92C9×C31⋊C3—32no—
93C9×C31⋊C3—31no—
94C9×C31⋊C3—36no—
95C9×C31⋊C3—36no—
96C9×C31⋊C3—34no—
97C9×C31⋊C3—32no—
98C9×C31⋊C3—312no—
99#10624936—311nomaximal, Sylow
100C32×C31—33no—
101C32×C31—31no—
102C32×C31—91no—
103C32×C31C3411yescharacteristic
104C3×C31⋊C3—32no—
105C3×C31⋊C3—36no—
106C3×C31⋊C3—91no—
107C3×C31⋊C3—92no—
108C3×C31⋊C3—273no—
109C3×C31⋊C3C3×H(3,𝔽3)12yes—
110C3×C31⋊C3C3×H(3,𝔽3)11yescharacteristic
111C9×C31—31no—
112C9×C31—36no—
113C9×C31—31no—
114C9×C31C3×H(3,𝔽3)12yes—
115C9×C31C3×H(3,𝔽3)11yescharacteristic
116C31⋊C9—312no—
117C31⋊C9—32no—
118C31⋊C9—32no—
119C31⋊C9C3×H(3,𝔽3)14yes—
120C31⋊C9C3×H(3,𝔽3)12yes—
1213−1+4—313no—
122C3×#737—311no—
123#6913—316no—
124#6913—316no—
125#6913—316no—
126#6910—313no—
127#6909—313no—
128#6908—312no—
129#6908—314no—
130#6908—312no—
131#6908C31⋊C311yescharacteristic, complemented
132C3×#733—312no—
133C3×#732—311no—
134C3×C31—31no—
135C3×C31—93no—
136C3×C31—91no—
137C3×C31C32×H(3,𝔽3)11yescharacteristic
138C31⋊C3—31no—
139C31⋊C3—32no—
140C31⋊C3—92no—
141C31⋊C3—96no—
142#737—3112no—
143#737—316no—
144#737—316no—
145#737—931no—
146#737—932no—
147#737—932no—
148#737—934no—
149#737—936no—
150#737—9312no—
151#737C3×C31⋊C313yescomplemented
152C3×3−1+2—312no—
153C3×3−1+2—312no—
154C3×3−1+2—311no—
155C3×3−1+2—311no—
156C3×3−1+2—312no—
157C3×3−1+2—316no—
158C3×3−1+2—933no—
159C3×3−1+2—936no—
160C3×H(3,𝔽3)—313no—
161C3×H(3,𝔽3)—311no—
162C32×C9—311no—
163C32×C9—312no—
164C32×C9C3×C31⋊C311yescharacteristic, complemented
165#733—3112no—
166#733—314no—
167#733C3×C31⋊C312yescomplemented
168#732—316no—
169#732—312no—
170#732C3×C31⋊C311yescharacteristic
171#731—3112no—
172#731—316no—
173#731—316no—
174#731C3×C31⋊C313yescomplemented
175AΣL(1,𝔽27)—316no—
176AΣL(1,𝔽27)—3112no—
177AΣL(1,𝔽27)—316no—
178AΣL(1,𝔽27)C3×C31⋊C313yescomplemented
179C31#1062493611yescharacteristic, complemented, Sylow
180C33—311no—
181C33—312no—
182C33—933no—
183C33C32×C31⋊C311yescharacteristic
1843−1+2—36no—
1853−1+2—311no—
1863−1+2—312no—
1873−1+2—312no—
1883−1+2—314no—
1893−1+2—3112no—
1903−1+2—934no—
1913−1+2—932no—
1923−1+2—9312no—
1933−1+2—9312no—
1943−1+2—932no—
1953−1+2—931no—
1963−1+2—936no—
1973−1+2—936no—
1983−1+2—934no—
1993−1+2—932no—
2003−1+2—9312no—
2013−1+2—936no—
2023−1+2—932no—
2033−1+2—934no—
2043−1+2—936no—
2053−1+2C32×C31⋊C316yes—
206H(3,𝔽3)—316no—
207H(3,𝔽3)—932no—
208H(3,𝔽3)—931no—
209H(3,𝔽3)—936no—
210H(3,𝔽3)C32×C31⋊C313yes—
211C3×C9—31no—
212C3×C9—32no—
213C3×C9—33no—
214C3×C9—312no—
215C3×C9—312no—
216C3×C9—312no—
217C3×C9—934no—
218C3×C9—932no—
219C3×C9—9312no—
220C3×C9—932no—
221C3×C9—931no—
222C3×C9—936no—
223C3×C9—936no—
224C3×C9—936no—
225C3×C9C32×C31⋊C311yescharacteristic
226C3×C9C32×C31⋊C311yescharacteristic
227C3×C9C32×C31⋊C311yescharacteristic
228C32—31no—
229C32—33no—
230C32—91no—
231C32—2792no—
232C32—2791no—
233C32—312no—
234C32—311no—
235C32—8373no—
236C32—936no—
237C32—932no—
238C32C33×C31⋊C311yescharacteristic
239C9—31no—
240C9—31no—
241C9—36no—
242C9—314no—
243C9—312no—
244C9—932no—
245C9—932no—
246C9—9312no—
247C9H(3,𝔽3)×C31⋊C311yescharacteristic
248C9H(3,𝔽3)×C31⋊C312yes—
249C3—31no—
250C3—91no—
251C3—93no—
252C3—2792no—
253C3—2796no—
254C3—931no—
255C3—932no—
256C3C3×H(3,𝔽3)×C31⋊C311yescharacteristic
subgroupG/H
centerC3C3×H(3,𝔽3)×C31⋊C3
commutator, cosocleC32×C31C34
FrattiniC32C33×C31⋊C3
FittingC31×#6908C3
socleC3×C31C32×H(3,𝔽3)
hypercenter#6908C31⋊C3
hypocenterC31#10624936
LCSG ▷ C32×C31 ▷ C3×C31 ▷ C31
UCSC1 ◁ C3 ◁ C3×C9 ◁ #6908

Complements

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

NH ≅ G/Ntype
1. #21419435C3semidirect
2. C3×#14578912C3semidirect
3. #21419418C3semidirect
4. #21419414C3semidirect
5. #21419411C3semidirect
6. #21419399C3semidirect
7. #21419393C3semidirect
8. #21419392C3semidirect
9. #21419391C3semidirect
10. #21419389C3semidirect
11. #733×C31⋊C3C3semidirect
12. #732×C31⋊C3C3semidirect
13. C31×#6908C3semidirect
15. #14578913C32semidirect
21. #14578912C32semidirect
22. #14578911C32semidirect
23. C3×#11719689C32semidirect
25. C3×#11719686C32semidirect
28. C3×#11719687C32semidirect
34. C3×C9×C31⋊C3C32semidirect
35. #14578902C32semidirect
36. #14578901C32semidirect
39. #14578898C32semidirect
40. #14578897C32semidirect
41. #14578896C32semidirect
42. #14578895C32semidirect
43. #14578894C32semidirect
44. #14578893C32semidirect
45. C31×#737C32semidirect
46. C32×C9×C31C32semidirect
47. C31×#733C32semidirect
49. C31×#731C32semidirect
50. C31×AΣL(1,𝔽27)C32semidirect
131. #6908C31⋊C3semidirect
151. #737C3×C31⋊C3semidirect
164. C32×C9C3×C31⋊C3semidirect
167. #733C3×C31⋊C3semidirect
174. #731C3×C31⋊C3semidirect
178. AΣL(1,𝔽27)C3×C31⋊C3semidirect
179. C31#10624936semidirect

Maximal subgroups

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

subgroupindex
#214194353
C3×#145789123
#214194183
#214194143
#214194113
#214193993
#214193933
#214193923
#214193913
#214193893
#733×C31⋊C33
#732×C31⋊C33
C31×#69083
#1062493631