Finite Group Databasebeta

← #36382209

Subgroups — #36382209

Order 22599.

178 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 43 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#21419421C312yesmaximal, complemented
2#21419408C311yesmaximal, characteristic, complemented
3#21419353C311yesmaximal, characteristic, complemented
4#21419279C312yesmaximal, complemented
5#21419277C311yesmaximal, characteristic, complemented
6#21419274C312yesmaximal, complemented
7#21419272C311yesmaximal, characteristic, complemented
8#21419260C312yesmaximal, complemented
9C31×#6880C311yesmaximal, characteristic, complemented
10#14578913—32no—
11C3×#11719687C3211yescharacteristic
123−1+2×C31⋊C3—31no—
13#14578902C3212yes—
14#14578902C3212yes—
15#14578901—32no—
16#14578900C3212yes—
17#14578898—32no—
18#14578895—32no—
19#14578887C3211yescharacteristic
20#14578887C3211yescharacteristic
21(C31⋊C9)⋊C9—32no—
22(C3×C31⋊C3)⋊C9—31no—
23C31×#733C3211yescharacteristic
24C31×#733C3212yes—
25C92×C31C3211yescharacteristic
26C32×C31⋊C3H(3,𝔽3)11yescharacteristic
273−1+2×C31—32no—
283−1+2×C31—31no—
29C3×C9×C31—31no—
30C3×C9×C31C3311yescharacteristic
31C32⋊(C31⋊C3)—32no—
32#11719689—32no—
33#11719689—32no—
34C3×C31⋊C9—31no—
35C3×C31⋊C9—31no—
36C3×C31⋊C9H(3,𝔽3)12yes—
37#11719687—91no—
38#11719687—92no—
39#11719687—91no—
40#11719686—92no—
41#11719686—91no—
42C9×C31⋊C3—92no—
43C9×C31⋊C3—91no—
44#10624854—311nomaximal, Sylow
45C32×C31C3×H(3,𝔽3)11yescharacteristic
46C3×C31⋊C3—31no—
47C3×C31⋊C3—91no—
48C3×C31⋊C3—92no—
49C9×C31—31no—
50C9×C31—91no—
51C9×C31—92no—
52C9×C31—91no—
53C31⋊C9—32no—
54C31⋊C9—91no—
55C31⋊C9—91no—
56C31⋊C9—92no—
57C31⋊C9—92no—
58#6913—312no—
59#6912—311no—
60#6899—311no—
61#6883—312no—
62#6882—312no—
63#6882—311no—
64#6881—311no—
65#6880C31⋊C311yescharacteristic, complemented
66#6878—312no—
67C3×C31—31no—
68C3×C31#691111yescharacteristic
69C31⋊C3—91no—
70C31⋊C3—272no—
71#737—932no—
72C3×3−1+2—311no—
73C3×3−1+2—931no—
74#733—312no—
75#733—312no—
76#733—932no—
77#733C3×C31⋊C311yescharacteristic
78#733C3×C31⋊C312yes—
79#732—312no—
80#731—932no—
81AΣL(1,𝔽27)—932no—
82C9⋊C9—311no—
83C9⋊C9—311no—
84C9⋊C9—932no—
85C32⋊C9—931no—
86C92C3×C31⋊C311yescharacteristic
87C31#1062485411yescharacteristic, complemented, Sylow
88C33—311no—
893−1+2—32no—
903−1+2—31no—
913−1+2—2791no—
923−1+2—2791no—
933−1+2—2791no—
943−1+2—2792no—
953−1+2—2792no—
963−1+2—932no—
973−1+2—932no—
98H(3,𝔽3)—932no—
99C3×C9—31no—
100C3×C9—2791no—
101C3×C9—2792no—
102C3×C9—312no—
103C3×C9—931no—
104C3×C9—931no—
105C3×C9C32×C31⋊C311yescharacteristic
106C32—2791no—
107C32—2792no—
108C32—931no—
109C32H(3,𝔽3)×C31⋊C311yescharacteristic
110C9—31no—
111C9—91no—
112C9—91no—
113C9—92no—
114C9—2791no—
115C9—2791no—
116C9—2792no—
117C9—2792no—
118C9—932no—
119C3—31no—
120C3—2791no—
121C3—8372no—
122C3#2141940411yescharacteristic
subgroupG/H
centerC3#21419404
commutator, cosocleC3×C9×C31C33
FrattiniC3×C9C32×C31⋊C3
FittingC31×#6880C3
socleC3×C31#6911
hypercenter#6880C31⋊C3
hypocenterC31#10624854
LCSG ▷ C3×C9×C31 ▷ C32×C31 ▷ C3×C31 ▷ C31
UCSC1 ◁ C3 ◁ C32 ◁ C3×C9 ◁ #6880

Complements

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

NH ≅ G/Ntype
1. #21419421C3semidirect
2. #21419408C3semidirect
3. #21419353C3semidirect
4. #21419279C3semidirect
5. #21419277C3semidirect
6. #21419274C3semidirect
7. #21419272C3semidirect
8. #21419260C3semidirect
9. C31×#6880C3semidirect
65. #6880C31⋊C3semidirect
87. C31#10624854semidirect

Maximal subgroups

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

subgroupindex
#214194213
#214194083
#214193533
#214192793
#214192773
#214192743
#214192723
#214192603
C31×#68803
#1062485431