Finite Group Databasebeta

← #36382168

Subgroups — #36382168

Order 22599.

200 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 46 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#21419419C312yesmaximal, complemented
2#21419406C311yesmaximal, characteristic, complemented
3#21419281C312yesmaximal, complemented
4#21419271C312yesmaximal, complemented
5#21419271C312yesmaximal, complemented
6#21419265C311yesmaximal, characteristic, complemented
7#21419261C311yesmaximal, characteristic, complemented
8#21419260C311yesmaximal, characteristic, complemented
9C31×#6900C311yesmaximal, characteristic, complemented
10#14578912—32no—
11C3×C32⋊(C31⋊C3)—31no—
12#14578902C3212yescomplemented
13#14578902C3212yescomplemented
14#14578900—32no—
15#14578900C3211yescharacteristic
16#14578900C3211yescharacteristic
17#14578900C3211yescharacteristic
18#14578900C3212yescomplemented
19#14578897—32no—
20#14578894—32no—
21C3×3−1+2×C31—31no—
22C3×3−1+2×C31C3211yescharacteristic
23C31×C9⋊C9—32no—
24C31×C9⋊C9C3212yescomplemented
25C92×C31C3211yescharacteristic, complemented
26C33×C31H(3,𝔽3)11yescharacteristic
27C32×C31⋊C3—91no—
283−1+2×C31—91no—
293−1+2×C31—93no—
30C3×C9×C31—31no—
31C3×C9×C31—32no—
32C3×C9×C31H(3,𝔽3)12yes—
33C3×C9×C31C3311yescharacteristic
34C32⋊(C31⋊C3)—32no—
35C32⋊(C31⋊C3)—31no—
36C32⋊(C31⋊C3)—31no—
37C32⋊(C31⋊C3)—31no—
38C32⋊(C31⋊C3)—92no—
39#11719689—32no—
40#11719689—32no—
41#11719687—92no—
42#11719686—92no—
43C9×C31⋊C3—92no—
44#10624851—311nomaximal, Sylow
45C32×C31—31no—
46C32×C31—91no—
47C32×C31C3×H(3,𝔽3)11yescharacteristic
48C3×C31⋊C3—92no—
49C3×C31⋊C3—91no—
50C3×C31⋊C3—91no—
51C3×C31⋊C3—91no—
52C3×C31⋊C3—273no—
53C9×C31—33no—
54C9×C31—32no—
55C9×C31—31no—
56C9×C31—92no—
57C31⋊C9—92no—
58C31⋊C9—92no—
59#6913—312no—
60#6911—311no—
61#6900C31⋊C311yescharacteristic, complemented
62#6883—312no—
63#6881—312no—
64#6881—312no—
65#6879—311no—
66#6878—311no—
67#6878—311no—
68C3×C31—31no—
69C3×C31—91no—
70C3×C31#691111yescharacteristic
71C31⋊C3—272no—
72C31⋊C3—271no—
73C31⋊C3—271no—
74C31⋊C3—271no—
75#737—932no—
76C3×3−1+2—31no—
77C3×3−1+2C3×C31⋊C311yescharacteristic
78C3×H(3,𝔽3)—931no—
79#733—312no—
80#733—312no—
81#732—312no—
82#732—311no—
83#732—311no—
84#732—311no—
85#732—932no—
86#731—932no—
87AΣL(1,𝔽27)—932no—
88C9⋊C9—32no—
89C9⋊C9C3×C31⋊C312yescomplemented
90C92C3×C31⋊C311yescharacteristic, complemented
91C31#1062485111yescharacteristic, complemented, Sylow
92C33—2791no—
93C33C32⋊(C31⋊C3)11yescharacteristic
943−1+2—91no—
953−1+2—93no—
963−1+2—2792no—
973−1+2—2792no—
983−1+2—932no—
993−1+2—932no—
100H(3,𝔽3)—2792no—
101H(3,𝔽3)—932no—
102H(3,𝔽3)—931no—
103H(3,𝔽3)—931no—
104H(3,𝔽3)—931no—
105C3×C9—31no—
106C3×C9—32no—
107C3×C9—2792no—
108C3×C9C32⋊(C31⋊C3)12yes—
109C3×C9C32×C31⋊C311yescharacteristic
110C32—31no—
111C32—91no—
112C32—2792no—
113C32—2791no—
114C32—2791no—
115C32—2791no—
116C32—8373no—
117C32C3×C32⋊(C31⋊C3)11yescharacteristic
118C9—33no—
119C9—32no—
120C9—31no—
121C9—92no—
122C9—2792no—
123C9—2792no—
124C3—31no—
125C3—91no—
126C3—8371no—
127C3—8371no—
128C3—8371no—
129C3—8372no—
130C3#2141940611yescharacteristic
subgroupG/H
centerC3#21419406
commutator, cosocleC3×C9×C31C33
FrattiniC3×C9C32×C31⋊C3
FittingC31×#6900C3
socleC3×C31#6911
hypercenter#6900C31⋊C3
hypocenterC31#10624851
LCSG ▷ C3×C9×C31 ▷ C32×C31 ▷ C3×C31 ▷ C31
UCSC1 ◁ C3 ◁ C32 ◁ C3×3−1+2 ◁ #6900

Complements

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

NH ≅ G/Ntype
1. #21419419C3semidirect
2. #21419406C3semidirect
3. #21419281C3semidirect
4. #21419271C3semidirect
5. #21419271C3semidirect
6. #21419265C3semidirect
7. #21419261C3semidirect
8. #21419260C3semidirect
9. C31×#6900C3semidirect
12. #14578902C32semidirect
13. #14578902C32semidirect
18. #14578900C32semidirect
24. C31×C9⋊C9C32semidirect
25. C92×C31C32semidirect
61. #6900C31⋊C3semidirect
89. C9⋊C9C3×C31⋊C3semidirect
90. C92C3×C31⋊C3semidirect
91. C31#10624851semidirect

Maximal subgroups

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

subgroupindex
#214194193
#214194063
#214192813
#214192713
#214192653
#214192613
#214192603
C31×#69003
#1062485131