Finite Group Databasebeta

← C31⋊C3×#6911 #36382910

Subgroups — C31⋊C3×#6911 #36382910

Order 22599.

1304 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 1304 conjugacy classes of subgroups — too many to draw. Only its 271 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
1C3×3−1+2×C31⋊C3C311yesmaximal, characteristic, complemented
2C3×H(3,𝔽3)×C31⋊C3C313yesmaximal, complemented
3#21419406C312yesmaximal, complemented
4#21419405C316yesmaximal, complemented
5#21419404C319yesmaximal, complemented
6#21419403C319yesmaximal, complemented
7#732×C31⋊C3C319yesmaximal, complemented
8C31×#6911C311yesmaximal, characteristic, complemented
9C33×C31⋊C3—33no—
10C33×C31⋊C3C3211yescharacteristic
11C3×#11719689C3212yescomplemented
12C3×#11719686C3213yescomplemented
13C3×#11719687C3213yescomplemented
143−1+2×C31⋊C3—33no—
153−1+2×C31⋊C3—36no—
16C3×C32⋊(C31⋊C3)—36no—
17C3×C32⋊(C31⋊C3)C3216yescomplemented
18H(3,𝔽3)×C31⋊C3—36no—
19H(3,𝔽3)×C31⋊C3—36no—
20H(3,𝔽3)×C31⋊C3—36no—
21H(3,𝔽3)×C31⋊C3C3219yescomplemented
22H(3,𝔽3)×C31⋊C3C3219yescomplemented
23H(3,𝔽3)×C31⋊C3C3219yescomplemented
24C3×C9×C31⋊C3C3213yescomplemented
25#14578900C32118yescomplemented
26#14578899C32154yescomplemented
27C3×3−1+2×C31C3211yescharacteristic, complemented
28C3×H(3,𝔽3)×C31C3213yescomplemented
29C31×#732C3219yescomplemented
30C33×C31—33no—
31C33×C31C3311yescharacteristic
32C32×C31⋊C3—36no—
33C32×C31⋊C3—39no—
34C32×C31⋊C3—39no—
35C32×C31⋊C3—39no—
36C32×C31⋊C3—93no—
37C32×C31⋊C3—927no—
38C32×C31⋊C3H(3,𝔽3)13yes—
39C32×C31⋊C3H(3,𝔽3)16yes—
40C32×C31⋊C3C3311yescharacteristic
41C32×C31⋊C3C3311yescharacteristic
42C32×C31⋊C3C3311yescharacteristic
433−1+2×C31—33no—
44H(3,𝔽3)×C31—36no—
45H(3,𝔽3)×C31C3319yescomplemented
46C3×C9×C31C3313yescomplemented
47C32⋊(C31⋊C3)—312no—
48C32⋊(C31⋊C3)—318no—
49C32⋊(C31⋊C3)—318no—
50C32⋊(C31⋊C3)—318no—
51C32⋊(C31⋊C3)C33118yescomplemented
52#11719689—36no—
53C3×C31⋊C9C3316yescomplemented
54#11719687—39no—
55#11719686—39no—
56C9×C31⋊C3—36no—
57C9×C31⋊C3—33no—
58C3×#6911—311nomaximal, Sylow
59C32×C31—39no—
60C32×C31—91no—
61C32×C31—99no—
62C32×C31C3×H(3,𝔽3)13yes—
63C32×C31C3411yescharacteristic
64C3×C31⋊C3—39no—
65C3×C31⋊C3—33no—
66C3×C31⋊C3—318no—
67C3×C31⋊C3—96no—
68C3×C31⋊C3—91no—
69C3×C31⋊C3—91no—
70C3×C31⋊C3—927no—
71C3×C31⋊C3—927no—
72C3×C31⋊C3—918no—
73C3×C31⋊C3—927no—
74C3×C31⋊C3C3×H(3,𝔽3)12yes—
75C3×C31⋊C3C3×H(3,𝔽3)13yes—
76C3×C31⋊C3C3×H(3,𝔽3)13yes—
77C3×C31⋊C3C3×H(3,𝔽3)11yescharacteristic
78C9×C31—33no—
79C31⋊C9—36no—
80C32×3−1+2—311no—
81C32×H(3,𝔽3)—313no—
82#6911—312no—
83#6911—316no—
84#6911—319no—
85#6911—319no—
86#6911C31⋊C311yescharacteristic, direct product factor
87C3×#732—319no—
88C3×C31—33no—
89C3×C31—31no—
90C3×C31—99no—
91C3×C31C32×H(3,𝔽3)11yescharacteristic
92C31⋊C3—33no—
93C31⋊C3—33no—
94C31⋊C3—32no—
95C31⋊C3—918no—
96C31⋊C3#691113yesdirect product factor
97C34—311no—
98C34—933no—
99C3×3−1+2—312no—
100C3×3−1+2—313no—
101C3×3−1+2—313no—
102C3×3−1+2—936no—
103C3×3−1+2—933no—
104C3×3−1+2C3×C31⋊C311yescharacteristic, complemented
105C3×H(3,𝔽3)—316no—
106C3×H(3,𝔽3)—319no—
107C3×H(3,𝔽3)—319no—
108C3×H(3,𝔽3)—319no—
109C3×H(3,𝔽3)—936no—
110C3×H(3,𝔽3)—936no—
111C3×H(3,𝔽3)—936no—
112C3×H(3,𝔽3)—936no—
113C3×H(3,𝔽3)C3×C31⋊C313yescomplemented
114C32×C9—313no—
115#732—3154no—
116#732—3118no—
117#732C3×C31⋊C319yescomplemented
118C31C3×#691111yescharacteristic, complemented, Sylow
119C33—33no—
120C33—27927no—
121C33—2793no—
122C33—316no—
123C33—311no—
124C33—311no—
125C33—313no—
126C33—311no—
127C33—939no—
128C33—939no—
129C33—939no—
130C33—936no—
131C33C32×C31⋊C311yescharacteristic
1323−1+2—33no—
1333−1+2—939no—
1343−1+2—939no—
1353−1+2—936no—
136H(3,𝔽3)—36no—
137H(3,𝔽3)—3118no—
138H(3,𝔽3)—9318no—
139H(3,𝔽3)—9318no—
140H(3,𝔽3)—9318no—
141H(3,𝔽3)—9312no—
142H(3,𝔽3)C32×C31⋊C319yescomplemented
143C3×C9—316no—
144C3×C9—933no—
145C3×C9—936no—
146C3×C9C32×C31⋊C313yescomplemented
147C32—39no—
148C32—91no—
149C32—99no—
150C32—27918no—
151C32—2796no—
152C32—2791no—
153C32—2791no—
154C32—27927no—
155C32—27927no—
156C32—27927no—
157C32—311no—
158C32—312no—
159C32—313no—
160C32—313no—
161C32—939no—
162C32—933no—
163C32—9318no—
164C32H(3,𝔽3)×C31⋊C313yes—
165C32C33×C31⋊C311yescharacteristic
166C9—33no—
167C9—936no—
168C3—33no—
169C3—31no—
170C3—99no—
171C3—27918no—
172C3—313no—
173C3—933no—
174C3—933no—
175C3—932no—
176C3C3×H(3,𝔽3)×C31⋊C311yescharacteristic
subgroupG/H
centerC3C3×H(3,𝔽3)×C31⋊C3
commutator, cosocleC32×C31C34
FrattiniC32C33×C31⋊C3
FittingC31×#6911C3
socleC3×C31C32×H(3,𝔽3)
hypercenter#6911C31⋊C3
hypocenterC31C3×#6911
LCSG ▷ C32×C31 ▷ C3×C31 ▷ C31
UCSC1 ◁ C3 ◁ C33 ◁ #6911

Complements

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

NH ≅ G/Ntype
1. C3×3−1+2×C31⋊C3C3semidirect
2. C3×H(3,𝔽3)×C31⋊C3C3semidirect
3. #21419406C3semidirect
4. #21419405C3semidirect
5. #21419404C3semidirect
6. #21419403C3semidirect
7. #732×C31⋊C3C3semidirect
8. C31×#6911C3semidirect
11. C3×#11719689C32semidirect
12. C3×#11719686C32semidirect
13. C3×#11719687C32semidirect
17. C3×C32⋊(C31⋊C3)C32semidirect
21. H(3,𝔽3)×C31⋊C3C32semidirect
22. H(3,𝔽3)×C31⋊C3C32semidirect
23. H(3,𝔽3)×C31⋊C3C32semidirect
24. C3×C9×C31⋊C3C32semidirect
25. #14578900C32semidirect
26. #14578899C32semidirect
27. C3×3−1+2×C31C32semidirect
28. C3×H(3,𝔽3)×C31C32semidirect
29. C31×#732C32semidirect
45. H(3,𝔽3)×C31C33semidirect
46. C3×C9×C31C33semidirect
51. C32⋊(C31⋊C3)C33semidirect
53. C3×C31⋊C9C33semidirect
86. #6911C31⋊C3direct (N × H)
96. C31⋊C3#6911direct (N × H)
104. C3×3−1+2C3×C31⋊C3semidirect
113. C3×H(3,𝔽3)C3×C31⋊C3semidirect
117. #732C3×C31⋊C3semidirect
118. C31C3×#6911semidirect
142. H(3,𝔽3)C32×C31⋊C3semidirect
146. C3×C9C32×C31⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×3−1+2×C31⋊C33
C3×H(3,𝔽3)×C31⋊C33
#214194063
#214194053
#214194043
#214194033
#732×C31⋊C33
C31×#69113
C3×#691131