Finite Group Databasebeta

← #36382971

Subgroups — #36382971

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#21419436C313yesmaximal, complemented
2C3×#14578912C311yesmaximal, characteristic, complemented
3#21419421C314yesmaximal, complemented
4#21419416C312yesmaximal, complemented
5#21419413C316yesmaximal, complemented
6#21419411C316yesmaximal, complemented
7#21419408C312yesmaximal, complemented
8#21419404C311yesmaximal, characteristic, complemented
9#21419393C313yesmaximal, complemented
10#21419391C312yesmaximal, complemented
11#21419390C316yesmaximal, complemented
12#731×C31⋊C3C313yesmaximal, complemented
13C31×#6908C311yesmaximal, characteristic, complemented
14#14578913—312no—
15#14578913C32112yescomplemented
16#14578912—36no—
17#14578912—33no—
18#14578912—36no—
19#14578912C3216yescomplemented
20#14578911C3216yescomplemented
21C3×#11719689C3212yescomplemented
22C3×#11719686—33no—
23C3×#11719686C3211yescharacteristic
24C3×#11719686C3212yescomplemented
25C3×#11719687C3211yescharacteristic
26C3×#11719687C3212yescomplemented
273−1+2×C31⋊C3—36no—
283−1+2×C31⋊C3C3216yes—
29C3×C32⋊(C31⋊C3)C3211yescharacteristic
30H(3,𝔽3)×C31⋊C3C3213yes—
31C3×C9×C31⋊C3C3211yescharacteristic
32C3×C9×C31⋊C3C3212yescomplemented
33#14578902C3214yescomplemented
34#14578901C32112yescomplemented
35#14578900C3212yes—
36#14578899C3216yes—
37#14578898C32112yescomplemented
38#14578897C3216yescomplemented
39#14578896C3216yescomplemented
40#14578895C32112yescomplemented
41#14578894C3216yescomplemented
42#14578893C3216yescomplemented
43C31×#737C3213yescomplemented
44C32×C9×C31C3211yescharacteristic, complemented
45C31×#733C3212yescomplemented
46C31×#732C3211yescharacteristic
47C31×#731C3213yescomplemented
48C31×AΣL(1,𝔽27)C3213yescomplemented
49C33×C31C3311yescharacteristic
50C32×C31⋊C3—33no—
51C32×C31⋊C3C3312yes—
52C32×C31⋊C3C3311yescharacteristic
533−1+2×C31—36no—
543−1+2×C31C3316yes—
55H(3,𝔽3)×C31C3313yes—
56C3×C9×C31—32no—
57C3×C9×C31—31no—
58C3×C9×C31—33no—
59C3×C9×C31C3311yescharacteristic
60C3×C9×C31C3311yescharacteristic
61C3×C9×C31C3311yescharacteristic
62C32⋊(C31⋊C3)—36no—
63C32⋊(C31⋊C3)—33no—
64C32⋊(C31⋊C3)C3316yes—
65#11719689—312no—
66#11719689—36no—
67#11719689C33112yes—
68C3×C31⋊C9—36no—
69C3×C31⋊C9C3312yes—
70C3×C31⋊C9C3312yes—
71C3×C31⋊C9C3312yes—
72#11719687—36no—
73#11719687—33no—
74#11719687—36no—
75#11719687—312no—
76#11719687—36no—
77#11719686—36no—
78#11719686—36no—
79#11719686—34no—
80#11719686—32no—
81#11719686—312no—
82#11719686H(3,𝔽3)16yes—
83#11719686H(3,𝔽3)13yes—
84C9×C31⋊C3—33no—
85C9×C31⋊C3—36no—
86C9×C31⋊C3—36no—
87C9×C31⋊C3—36no—
88C9×C31⋊C3—312no—
89#10624935—311nomaximal, Sylow
90C32×C31—33no—
91C32×C31—31no—
92C32×C31—91no—
93C32×C31C3411yescharacteristic
94C3×C31⋊C3—32no—
95C3×C31⋊C3—36no—
96C3×C31⋊C3—91no—
97C3×C31⋊C3—92no—
98C3×C31⋊C3—273no—
99C3×C31⋊C3C3×H(3,𝔽3)13yes—
100C9×C31—31no—
101C9×C31—36no—
102C9×C31—31no—
103C9×C31C3×H(3,𝔽3)12yes—
104C9×C31C3×H(3,𝔽3)11yescharacteristic
105C31⋊C9—312no—
106C31⋊C9—32no—
107C31⋊C9—32no—
108C31⋊C9C3×H(3,𝔽3)16yes—
1093−1+4—313no—
110C3×#737—311no—
111#6913—316no—
112#6913—314no—
113#6913—312no—
114#6913—316no—
115#6912—312no—
116#6911—311no—
117#6909—313no—
118#6908—312no—
119#6908—316no—
120#6908C31⋊C311yescharacteristic, complemented
121C3×#731—313no—
122C3×C31—31no—
123C3×C31—93no—
124C3×C31—91no—
125C3×C31C32×H(3,𝔽3)11yescharacteristic
126C31⋊C3—33no—
127C31⋊C3—92no—
128C31⋊C3—96no—
129#737—3112no—
130#737—316no—
131#737—316no—
132#737—933no—
133#737—936no—
134#737—936no—
135#737—9312no—
136#737C3×C31⋊C313yescomplemented
137C3×3−1+2—312no—
138C3×3−1+2—311no—
139C3×3−1+2—312no—
140C3×3−1+2—311no—
141C3×3−1+2—312no—
142C3×3−1+2—316no—
143C3×3−1+2—933no—
144C3×3−1+2—936no—
145C3×H(3,𝔽3)—313no—
146C3×H(3,𝔽3)—311no—
147C32×C9—311no—
148C32×C9—312no—
149C32×C9C3×C31⋊C311yescharacteristic, complemented
150#733—3112no—
151#733—314no—
152#733C3×C31⋊C312yescomplemented
153#732—316no—
154#732—312no—
155#732C3×C31⋊C311yescharacteristic
156#731—3112no—
157#731—316no—
158#731—316no—
159#731C3×C31⋊C313yescomplemented
160AΣL(1,𝔽27)—316no—
161AΣL(1,𝔽27)—3112no—
162AΣL(1,𝔽27)—316no—
163AΣL(1,𝔽27)C3×C31⋊C313yescomplemented
164C31#1062493511yescharacteristic, complemented, Sylow
165C33—311no—
166C33—312no—
167C33—933no—
168C33C32×C31⋊C311yescharacteristic
1693−1+2—36no—
1703−1+2—313no—
1713−1+2—316no—
1723−1+2—3112no—
1733−1+2—9312no—
1743−1+2—936no—
1753−1+2—9312no—
1763−1+2—936no—
1773−1+2—933no—
1783−1+2—936no—
1793−1+2—936no—
1803−1+2—9312no—
1813−1+2—932no—
1823−1+2—934no—
1833−1+2—936no—
1843−1+2—936no—
1853−1+2C32×C31⋊C316yes—
186H(3,𝔽3)—316no—
187H(3,𝔽3)—933no—
188H(3,𝔽3)—936no—
189H(3,𝔽3)C32×C31⋊C313yes—
190C3×C9—31no—
191C3×C9—32no—
192C3×C9—33no—
193C3×C9—312no—
194C3×C9—312no—
195C3×C9—312no—
196C3×C9—936no—
197C3×C9—9312no—
198C3×C9—933no—
199C3×C9—936no—
200C3×C9—936no—
201C3×C9—936no—
202C3×C9C32×C31⋊C311yescharacteristic
203C3×C9C32×C31⋊C311yescharacteristic
204C3×C9C32×C31⋊C311yescharacteristic
205C32—31no—
206C32—33no—
207C32—91no—
208C32—2792no—
209C32—2791no—
210C32—313no—
211C32—8373no—
212C32—936no—
213C32—932no—
214C32C33×C31⋊C311yescharacteristic
215C9—31no—
216C9—31no—
217C9—36no—
218C9—316no—
219C9—932no—
220C9—932no—
221C9—9312no—
222C9H(3,𝔽3)×C31⋊C311yescharacteristic
223C9H(3,𝔽3)×C31⋊C312yes—
224C3—31no—
225C3—91no—
226C3—93no—
227C3—2792no—
228C3—2796no—
229C3—933no—
230C3C3×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#10624935
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. #21419436C3semidirect
2. C3×#14578912C3semidirect
3. #21419421C3semidirect
4. #21419416C3semidirect
5. #21419413C3semidirect
6. #21419411C3semidirect
7. #21419408C3semidirect
8. #21419404C3semidirect
9. #21419393C3semidirect
10. #21419391C3semidirect
11. #21419390C3semidirect
12. #731×C31⋊C3C3semidirect
13. C31×#6908C3semidirect
15. #14578913C32semidirect
19. #14578912C32semidirect
20. #14578911C32semidirect
21. C3×#11719689C32semidirect
24. C3×#11719686C32semidirect
26. C3×#11719687C32semidirect
32. C3×C9×C31⋊C3C32semidirect
33. #14578902C32semidirect
34. #14578901C32semidirect
37. #14578898C32semidirect
38. #14578897C32semidirect
39. #14578896C32semidirect
40. #14578895C32semidirect
41. #14578894C32semidirect
42. #14578893C32semidirect
43. C31×#737C32semidirect
44. C32×C9×C31C32semidirect
45. C31×#733C32semidirect
47. C31×#731C32semidirect
48. C31×AΣL(1,𝔽27)C32semidirect
120. #6908C31⋊C3semidirect
136. #737C3×C31⋊C3semidirect
149. C32×C9C3×C31⋊C3semidirect
152. #733C3×C31⋊C3semidirect
159. #731C3×C31⋊C3semidirect
163. AΣL(1,𝔽27)C3×C31⋊C3semidirect
164. C31#10624935semidirect

Maximal subgroups

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

subgroupindex
#214194363
C3×#145789123
#214194213
#214194163
#214194133
#214194113
#214194083
#214194043
#214193933
#214193913
#214193903
#731×C31⋊C33
C31×#69083
#1062493531