Finite Group Databasebeta

← #36382139

Subgroups — #36382139

Order 22599.

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

normal subgroups only

This group has 186 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#733×C31⋊C3C311yesmaximal, characteristic
2#733×C31⋊C3C311yesmaximal, characteristic
3#733×C31⋊C3C311yesmaximal, characteristic
4#21419295C311yesmaximal, characteristic
5#21419278C311yesmaximal, characteristic, complemented
6#21419277C311yesmaximal, characteristic, complemented
7#21419276C311yesmaximal, characteristic, complemented
8#21419276C311yesmaximal, characteristic, complemented
9#21419275C311yesmaximal, characteristic, complemented
10#21419275C311yesmaximal, characteristic, complemented
11#21419274C311yesmaximal, characteristic, complemented
12#21419274C311yesmaximal, characteristic, complemented
13C31×#6882C311yesmaximal, characteristic, complemented
143−1+2×C31⋊C3—31no—
153−1+2×C31⋊C3—31no—
163−1+2×C31⋊C3—31no—
17C3×C9×C31⋊C3C3211yescharacteristic
18#14578902C3211yescharacteristic
19#14578902C3211yescharacteristic
20#14578902C3211yescharacteristic
21#14578902C3211yescharacteristic
22#14578902C3211yescharacteristic
23#14578902C3211yescharacteristic
24#14578901—31no—
25#14578901—31no—
26#14578901—31no—
27#14578901—31no—
28#14578901—31no—
29#14578901—31no—
30#14578887C3211yescharacteristic
31#14578886C3211yescharacteristic
32(C31⋊C9)⋊C9—31no—
33(C3×C31⋊C3)⋊C9—31no—
34C9×C31⋊C9—31no—
35C31×#733C3211yescharacteristic
36C31×#733C3211yescharacteristic
37C31×#733C3211yescharacteristic
38C31×C9⋊C9C3211yescharacteristic
39C32×C31⋊C3H(3,𝔽3)11yescharacteristic
403−1+2×C31—31no—
413−1+2×C31—31no—
423−1+2×C31—31no—
43C3×C9×C31—31no—
44C3×C9×C31C3311yescharacteristic
45#11719689—31no—
46#11719689—31no—
47#11719689—31no—
48#11719689—31no—
49#11719689—31no—
50#11719689—31no—
51C3×C31⋊C9—31no—
52C3×C31⋊C9—31no—
53C3×C31⋊C9H(3,𝔽3)11yescharacteristic
54C3×C31⋊C9H(3,𝔽3)11yescharacteristic
55#11719687—91no—
56#11719687—91no—
57#11719687—91no—
58#11719686—91no—
59#11719686—91no—
60#11719686—91no—
61C9×C31⋊C3—31no—
62C9×C31⋊C3—31no—
63C9×C31⋊C3—31no—
64C9×C31⋊C3—91no—
65C9×C31⋊C3—91no—
66C9×C31⋊C3—91no—
67#10624847—311nomaximal, Sylow
68C32×C31C3×H(3,𝔽3)11yescharacteristic
69C3×C31⋊C3—91no—
70C3×C31⋊C3#73211yescharacteristic
71C3×C31⋊C3#73211yescharacteristic
72C3×C31⋊C3#73211yescharacteristic
73C9×C31—31no—
74C9×C31—91no—
75C9×C31—91no—
76C9×C31—91no—
77C9×C31—91no—
78C31⋊C9—31no—
79C31⋊C9—31no—
80C31⋊C9—91no—
81C31⋊C9—91no—
82C31⋊C9—91no—
83C31⋊C9—91no—
84C31⋊C9—91no—
85C31⋊C9—91no—
86C31⋊C9—91no—
87C31⋊C9—91no—
88C3×#733—311no—
89C3×#733—311no—
90C3×#733—311no—
91#6887—311no—
92#6882—311no—
93#6882—311no—
94#6882—311no—
95#6882—311no—
96#6882—311no—
97#6882—311no—
98#6882—311no—
99#6882—311no—
100#6882C31⋊C311yescharacteristic, complemented
101C3×C31—31no—
102C3×C31C3×#73211yescharacteristic
103C31⋊C3—31no—
104C31⋊C3—31no—
105C31⋊C3—31no—
106C3×3−1+2—931no—
107C3×3−1+2—931no—
108C3×3−1+2—931no—
109C32×C9—311no—
110#733—311no—
111#733—311no—
112#733—311no—
113#733—311no—
114#733—311no—
115#733—311no—
116#733—931no—
117#733—931no—
118#733—931no—
119#733—931no—
120#733—931no—
121#733—931no—
122#733C3×C31⋊C311yescharacteristic
123#733C3×C31⋊C311yescharacteristic
124#733C3×C31⋊C311yescharacteristic
125C9⋊C9—311no—
126C9⋊C9—311no—
127C9⋊C9—931no—
128C9⋊C9C3×C31⋊C311yescharacteristic
129C32⋊C9—931no—
130C92—931no—
131C31#1062484711yescharacteristic, complemented, Sylow
132C33—311no—
1333−1+2—31no—
1343−1+2—31no—
1353−1+2—31no—
1363−1+2—2791no—
1373−1+2—2791no—
1383−1+2—2791no—
1393−1+2—2791no—
1403−1+2—2791no—
1413−1+2—2791no—
1423−1+2—931no—
1433−1+2—931no—
1443−1+2—931no—
1453−1+2—931no—
1463−1+2—931no—
1473−1+2—931no—
148C3×C9—31no—
149C3×C9—2791no—
150C3×C9—2791no—
151C3×C9—2791no—
152C3×C9—311no—
153C3×C9—311no—
154C3×C9—931no—
155C3×C9—931no—
156C3×C9—931no—
157C3×C9—931no—
158C3×C9—931no—
159C3×C9C32×C31⋊C311yescharacteristic
160C32—2791no—
161C32—311no—
162C32—311no—
163C32—311no—
164C32H(3,𝔽3)×C31⋊C311yescharacteristic
165C9—31no—
166C9—91no—
167C9—91no—
168C9—91no—
169C9—91no—
170C9—2791no—
171C9—2791no—
172C9—2791no—
173C9—2791no—
174C9—2791no—
175C9—2791no—
176C9—2791no—
177C9—2791no—
178C9—931no—
179C9—931no—
180C3—31no—
181C3—931no—
182C3—931no—
183C3—931no—
184C3#732×C31⋊C311yescharacteristic
subgroupG/H
centerC3#732×C31⋊C3
commutator, cosocleC3×C9×C31C33
FrattiniC3×C9C32×C31⋊C3
FittingC31×#6882C3
socleC3×C31C3×#732
prime radicalC32×C31⋊C3H(3,𝔽3)
hypercenter#6882C31⋊C3
hypocenterC31#10624847
LCSG ▷ C3×C9×C31 ▷ C32×C31 ▷ C3×C31 ▷ C31
UCSC1 ◁ C3 ◁ C32 ◁ C3×C9 ◁ #6882

Complements

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

NH ≅ G/Ntype
5. #21419278C3semidirect
6. #21419277C3semidirect
7. #21419276C3semidirect
8. #21419276C3semidirect
9. #21419275C3semidirect
10. #21419275C3semidirect
11. #21419274C3semidirect
12. #21419274C3semidirect
13. C31×#6882C3semidirect
100. #6882C31⋊C3semidirect
131. C31#10624847semidirect

Maximal subgroups

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

subgroupindex
#733×C31⋊C33
#214192953
#214192783
#214192773
#214192763
#214192753
#214192743
C31×#68823
#1062484731