Finite Group Databasebeta

← #36382183

Subgroups — #36382183

Order 22599.

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

normal subgroups only

This group has 188 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#21419420C312yesmaximal, complemented
2#21419407C311yesmaximal, characteristic, complemented
3#21419360C311yesmaximal, characteristic, complemented
4#21419280C312yesmaximal, complemented
5#21419278C311yesmaximal, characteristic, complemented
6#21419275C312yesmaximal, complemented
7#21419273C311yesmaximal, characteristic, complemented
8#21419261C312yesmaximal, complemented
9C31×#6880C311yesmaximal, characteristic, complemented
10#14578913—32no—
11C3×#11719686—31no—
12C3×#11719686C3211yescharacteristic
133−1+2×C31⋊C3—31no—
14#14578902C3212yes—
15#14578902C3212yes—
16#14578901—32no—
17#14578900C3212yes—
18#14578898—32no—
19#14578895—32no—
20#14578886—32no—
21#14578886C3211yescharacteristic
22#14578886C3211yescharacteristic
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—92no—
38#11719687—91no—
39#11719686—91no—
40#11719686—92no—
41#11719686—93no—
42#11719686—91no—
43C9×C31⋊C3—92no—
44C9×C31⋊C3—91no—
45#10624852—311nomaximal, Sylow
46C32×C31C3×H(3,𝔽3)11yescharacteristic
47C3×C31⋊C3—31no—
48C3×C31⋊C3—91no—
49C3×C31⋊C3—92no—
50C9×C31—33no—
51C9×C31—31no—
52C9×C31—92no—
53C9×C31—91no—
54C31⋊C9—32no—
55C31⋊C9—91no—
56C31⋊C9—91no—
57C31⋊C9—92no—
58C31⋊C9—92no—
59#6913—312no—
60#6912—311no—
61#6900—311no—
62#6883—312no—
63#6882—312no—
64#6882—311no—
65#6881—311no—
66#6880C31⋊C311yescharacteristic, complemented
67#6878—312no—
68C3×C31—31no—
69C3×C31#691111yescharacteristic
70C31⋊C3—91no—
71C31⋊C3—272no—
72#737—932no—
73C3×3−1+2—311no—
74C3×3−1+2—931no—
75C3×3−1+2—931no—
76#733—312no—
77#733—312no—
78#733—932no—
79#733C3×C31⋊C311yescharacteristic
80#733C3×C31⋊C312yes—
81#732—312no—
82#731—932no—
83AΣL(1,𝔽27)—932no—
84C9⋊C9—311no—
85C9⋊C9—311no—
86C9⋊C9—932no—
87C92C3×C31⋊C311yescharacteristic
88C31#1062485211yescharacteristic, complemented, Sylow
89C33—311no—
903−1+2—32no—
913−1+2—31no—
923−1+2—2793no—
933−1+2—2791no—
943−1+2—2791no—
953−1+2—2791no—
963−1+2—2792no—
973−1+2—2792no—
983−1+2—932no—
993−1+2—932no—
100H(3,𝔽3)—932no—
101C3×C9—31no—
102C3×C9—2791no—
103C3×C9—2792no—
104C3×C9—312no—
105C3×C9—931no—
106C3×C9—931no—
107C3×C9C32×C31⋊C311yescharacteristic
108C32—2791no—
109C32—2792no—
110C32—931no—
111C32H(3,𝔽3)×C31⋊C311yescharacteristic
112C9—33no—
113C9—31no—
114C9—91no—
115C9—92no—
116C9—2791no—
117C9—2791no—
118C9—2792no—
119C9—2792no—
120C9—932no—
121C3—31no—
122C3—2791no—
123C3—8372no—
124C3#2141940311yescharacteristic
subgroupG/H
centerC3#21419403
commutator, cosocleC3×C9×C31C33
FrattiniC3×C9C32×C31⋊C3
FittingC31×#6880C3
socleC3×C31#6911
hypercenter#6880C31⋊C3
hypocenterC31#10624852
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. #21419420C3semidirect
2. #21419407C3semidirect
3. #21419360C3semidirect
4. #21419280C3semidirect
5. #21419278C3semidirect
6. #21419275C3semidirect
7. #21419273C3semidirect
8. #21419261C3semidirect
9. C31×#6880C3semidirect
66. #6880C31⋊C3semidirect
88. C31#10624852semidirect

Maximal subgroups

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

subgroupindex
#214194203
#214194073
#214193603
#214192803
#214192783
#214192753
#214192733
#214192613
C31×#68803
#1062485231