Finite Group Databasebeta

← #36382204

Subgroups — #36382204

Order 22599.

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

normal subgroups only

This group has 190 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#21419419C311yesmaximal, characteristic, complemented
2#21419406C311yesmaximal, characteristic, complemented
3#21419352C311yesmaximal, characteristic, complemented
4#21419280C311yesmaximal, characteristic, complemented
5#21419279C311yesmaximal, characteristic, complemented
6#21419273C311yesmaximal, characteristic, complemented
7#21419272C311yesmaximal, characteristic, complemented
8#21419270C311yesmaximal, characteristic, complemented
9#21419269C311yesmaximal, characteristic, complemented
10#21419264C311yesmaximal, characteristic, complemented
11#21419263C311yesmaximal, characteristic, complemented
12#21419262C311yesmaximal, characteristic, complemented
13C31×#6913C311yesmaximal, characteristic, complemented
14#14578912—31no—
15C3×C32⋊(C31⋊C3)—31no—
16#14578902C3211yescharacteristic, complemented
17#14578902C3211yescharacteristic, complemented
18#14578900—31no—
19#14578900—31no—
20#14578900C3211yescharacteristic
21#14578900C3211yescharacteristic
22#14578900C3211yescharacteristic
23#14578900C3211yescharacteristic, complemented
24#14578897—31no—
25#14578894—31no—
26#14578887C3211yescharacteristic, complemented
27#14578886C3211yescharacteristic, complemented
28(C31⋊C9)⋊C9—31no—
29(C31⋊C9)⋊C9—31no—
30C32⋊(C31⋊C9)—31no—
31C9×C31⋊C9C3211yescharacteristic, complemented
32C31×#737—31no—
33C3×3−1+2×C31C3211yescharacteristic
34C31×#733C3211yescharacteristic, complemented
35C31×#733C3211yescharacteristic, complemented
36C31×#732C3211yescharacteristic, complemented
37C31×#731—31no—
38C31×AΣL(1,𝔽27)—31no—
39C33×C31H(3,𝔽3)11yescharacteristic
40C32×C31⋊C3—91no—
413−1+2×C31—31no—
423−1+2×C31—31no—
433−1+2×C31—91no—
443−1+2×C31—92no—
45H(3,𝔽3)×C31—31no—
46C3×C9×C31—91no—
47C3×C9×C31H(3,𝔽3)11yescharacteristic
48C3×C9×C31H(3,𝔽3)11yescharacteristic
49C3×C9×C31C3311yescharacteristic
50C32⋊(C31⋊C3)—31no—
51C32⋊(C31⋊C3)—31no—
52C32⋊(C31⋊C3)—31no—
53C32⋊(C31⋊C3)—31no—
54C32⋊(C31⋊C3)—92no—
55#11719689—31no—
56#11719689—31no—
57C3×C31⋊C9—31no—
58C3×C31⋊C9—31no—
59C3×C31⋊C9—31no—
60#11719687—91no—
61#11719686—91no—
62C9×C31⋊C3—91no—
63#10624853—311nomaximal, Sylow
64C32×C31—31no—
65C32×C31—91no—
66C32×C31—91no—
67C32×C31C3×H(3,𝔽3)11yescharacteristic
68C3×C31⋊C3—91no—
69C3×C31⋊C3—91no—
70C3×C31⋊C3—91no—
71C3×C31⋊C3—91no—
72C3×C31⋊C3—273no—
73C9×C31—31no—
74C9×C31—31no—
75C9×C31—31no—
76C9×C31—91no—
77C9×C31—91no—
78C31⋊C9—91no—
79C31⋊C9—91no—
80C31⋊C9—91no—
81C31⋊C9—91no—
82C31⋊C9—91no—
83#6913—311no—
84#6913C31⋊C311yescharacteristic, complemented
85#6911—311no—
86#6899—311no—
87#6883—311no—
88#6883—311no—
89#6881—311no—
90#6881—311no—
91#6881—311no—
92#6881—311no—
93#6879—311no—
94#6878—311no—
95#6878—311no—
96C3×C31—31no—
97C3×C31—91no—
98C3×C31—271no—
99C3×C31#691111yescharacteristic
100C31⋊C3—271no—
101C31⋊C3—271no—
102C31⋊C3—271no—
103C31⋊C3—271no—
104#737—31no—
105#737—931no—
106C3×3−1+2C3×C31⋊C311yescharacteristic
107C3×H(3,𝔽3)—931no—
108#733—311no—
109#733—311no—
110#733C3×C31⋊C311yescharacteristic, complemented
111#733C3×C31⋊C311yescharacteristic, complemented
112#732—311no—
113#732—311no—
114#732—311no—
115#732—311no—
116#732—931no—
117#732—931no—
118#732C3×C31⋊C311yescharacteristic, complemented
119#731—31no—
120#731—931no—
121AΣL(1,𝔽27)—31no—
122AΣL(1,𝔽27)—931no—
123C9⋊C9—311no—
124C9⋊C9—311no—
125C9⋊C9—931no—
126C9⋊C9—931no—
127C32⋊C9—931no—
128C92—311no—
129C31#1062485311yescharacteristic, complemented, Sylow
130C33—2791no—
131C33C32⋊(C31⋊C3)11yescharacteristic
1323−1+2—31no—
1333−1+2—31no—
1343−1+2—91no—
1353−1+2—92no—
1363−1+2—2791no—
1373−1+2—2791no—
1383−1+2—931no—
1393−1+2—931no—
140H(3,𝔽3)—31no—
141H(3,𝔽3)—2792no—
142H(3,𝔽3)—931no—
143H(3,𝔽3)—931no—
144H(3,𝔽3)—931no—
145H(3,𝔽3)—931no—
146C3×C9—91no—
147C3×C9—2791no—
148C3×C9—931no—
149C3×C9—931no—
150C3×C9—931no—
151C3×C9C32⋊(C31⋊C3)11yescharacteristic
152C3×C9C32⋊(C31⋊C3)11yescharacteristic
153C3×C9C32×C31⋊C311yescharacteristic
154C32—31no—
155C32—91no—
156C32—91no—
157C32—2791no—
158C32—2791no—
159C32—2791no—
160C32—2791no—
161C32—8373no—
162C32C3×C32⋊(C31⋊C3)11yescharacteristic
163C9—31no—
164C9—31no—
165C9—31no—
166C9—91no—
167C9—91no—
168C9—2791no—
169C9—2791no—
170C9—2791no—
171C9—2791no—
172C9—2791no—
173C3—31no—
174C3—91no—
175C3—271no—
176C3—8371no—
177C3—8371no—
178C3—8371no—
179C3—8371no—
180C3#2141940511yescharacteristic
subgroupG/H
centerC3#21419405
commutator, cosocleC3×C9×C31C33
FrattiniC3×C9C32×C31⋊C3
FittingC31×#6913C3
socleC3×C31#6911
hypercenter#6913C31⋊C3
hypocenterC31#10624853
LCSG ▷ C3×C9×C31 ▷ C32×C31 ▷ C3×C31 ▷ C31
UCSC1 ◁ C3 ◁ C32 ◁ C3×3−1+2 ◁ #6913

Complements

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

NH ≅ G/Ntype
1. #21419419C3semidirect
2. #21419406C3semidirect
3. #21419352C3semidirect
4. #21419280C3semidirect
5. #21419279C3semidirect
6. #21419273C3semidirect
7. #21419272C3semidirect
8. #21419270C3semidirect
9. #21419269C3semidirect
10. #21419264C3semidirect
11. #21419263C3semidirect
12. #21419262C3semidirect
13. C31×#6913C3semidirect
16. #14578902C32semidirect
17. #14578902C32semidirect
23. #14578900C32semidirect
26. #14578887C32semidirect
27. #14578886C32semidirect
31. C9×C31⋊C9C32semidirect
34. C31×#733C32semidirect
35. C31×#733C32semidirect
36. C31×#732C32semidirect
84. #6913C31⋊C3semidirect
110. #733C3×C31⋊C3semidirect
111. #733C3×C31⋊C3semidirect
118. #732C3×C31⋊C3semidirect
129. C31#10624853semidirect

Maximal subgroups

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

subgroupindex
#214194193
#214194063
#214193523
#214192803
#214192793
#214192733
#214192723
#214192703
#214192693
#214192643
#214192633
#214192623
C31×#69133
#1062485331