Finite Group Databasebeta

← #36382166

Subgroups — #36382166

Order 22599.

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

normal subgroups only

This group has 200 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#21419420C311yesmaximal, characteristic, complemented
2#21419415C311yesmaximal, characteristic, complemented
3#21419403C311yesmaximal, characteristic, complemented
4#21419363C311yesmaximal, characteristic, complemented
5#21419281C311yesmaximal, characteristic, complemented
6#21419279C311yesmaximal, characteristic, complemented
7#21419271C311yesmaximal, characteristic, complemented
8#21419269C311yesmaximal, characteristic, complemented
9#21419269C311yesmaximal, characteristic, complemented
10#21419264C311yesmaximal, characteristic, complemented
11#21419263C311yesmaximal, characteristic, complemented
12#21419259C311yesmaximal, characteristic, complemented
13C31×#6881C311yesmaximal, characteristic, complemented
14#14578913—31no—
15#14578911—31no—
16C3×#11719686—31no—
17C3×#11719686C3211yescharacteristic
18H(3,𝔽3)×C31⋊C3—31no—
19#14578902C3211yescharacteristic, complemented
20#14578902C3211yescharacteristic, complemented
21#14578902C3211yescharacteristic, complemented
22#14578900C3211yescharacteristic
23#14578900C3211yescharacteristic
24#14578900C3211yescharacteristic, complemented
25#14578899—31no—
26#14578899—31no—
27#14578898—31no—
28#14578896—31no—
29#14578895—31no—
30#14578893—31no—
31#14578887C3211yescharacteristic, complemented
32(C31⋊C9)⋊C9—31no—
33(C31⋊C9)⋊C9—31no—
34C9×C31⋊C9C3211yescharacteristic, complemented
35C31×#733C3211yescharacteristic, complemented
36C31×#732C3211yescharacteristic
37C31×#732C3211yescharacteristic, complemented
38C31×C9⋊C9C3211yescharacteristic, complemented
39C32×C31⋊C3—91no—
40C32×C31⋊C3H(3,𝔽3)11yescharacteristic
413−1+2×C31—31no—
42H(3,𝔽3)×C31—31no—
43H(3,𝔽3)×C31—31no—
44C3×C9×C31—31no—
45C3×C9×C31C3311yescharacteristic
46C32⋊(C31⋊C3)—31no—
47C32⋊(C31⋊C3)—31no—
48C32⋊(C31⋊C3)—31no—
49C32⋊(C31⋊C3)—92no—
50#11719689—31no—
51#11719689—31no—
52#11719689—31no—
53#11719689—92no—
54C3×C31⋊C9—31no—
55C3×C31⋊C9—31no—
56C3×C31⋊C9—91no—
57C3×C31⋊C9H(3,𝔽3)11yescharacteristic
58C3×C31⋊C9H(3,𝔽3)11yescharacteristic
59#11719687—91no—
60#11719686—91no—
61#11719686—93no—
62#11719686—91no—
63C9×C31⋊C3—91no—
64#10624851—311nomaximal, Sylow
65C32×C31—91no—
66C32×C31—91no—
67C32×C31C3×H(3,𝔽3)11yescharacteristic
68C3×C31⋊C3—31no—
69C3×C31⋊C3—91no—
70C3×C31⋊C3—91no—
71C3×C31⋊C3—91no—
72C3×C31⋊C3—91no—
73C3×C31⋊C3—273no—
74C9×C31—31no—
75C9×C31—91no—
76C9×C31—91no—
77C31⋊C9—33no—
78C31⋊C9—31no—
79C31⋊C9—31no—
80C31⋊C9—91no—
81C31⋊C9—91no—
82C31⋊C9—91no—
83C31⋊C9—91no—
84#6913—311no—
85#6913—311no—
86#6911—311no—
87#6900—311no—
88#6883—311no—
89#6883—311no—
90#6881—311no—
91#6881—311no—
92#6881—311no—
93#6881C31⋊C311yescharacteristic, complemented
94#6879—311no—
95#6878—311no—
96#6878—311no—
97C3×C31—31no—
98C3×C31—271no—
99C3×C31—271no—
100C3×C31#691111yescharacteristic
101C31⋊C3—91no—
102C31⋊C3—271no—
103C31⋊C3—271no—
104C31⋊C3—271no—
105#737—931no—
106#737—931no—
107C3×3−1+2—311no—
108C3×3−1+2—931no—
109C3×H(3,𝔽3)—931no—
110#733—311no—
111#733—311no—
112#733—311no—
113#733C3×C31⋊C311yescharacteristic, complemented
114#732—311no—
115#732—311no—
116#732—311no—
117#732—931no—
118#732—931no—
119#732C3×C31⋊C311yescharacteristic
120#732C3×C31⋊C311yescharacteristic, complemented
121#731—931no—
122#731—931no—
123AΣL(1,𝔽27)—931no—
124AΣL(1,𝔽27)—931no—
125C9⋊C9—311no—
126C9⋊C9—931no—
127C9⋊C9—931no—
128C9⋊C9C3×C31⋊C311yescharacteristic, complemented
129C92—311no—
130C31#1062485111yescharacteristic, complemented, Sylow
131C33—2791no—
132C33—311no—
1333−1+2—31no—
1343−1+2—2791no—
1353−1+2—2793no—
1363−1+2—2791no—
1373−1+2—2791no—
1383−1+2—2792no—
1393−1+2—931no—
1403−1+2—931no—
1413−1+2—931no—
142H(3,𝔽3)—31no—
143H(3,𝔽3)—31no—
144H(3,𝔽3)—2792no—
145H(3,𝔽3)—931no—
146H(3,𝔽3)—931no—
147H(3,𝔽3)—931no—
148C3×C9—31no—
149C3×C9—2791no—
150C3×C9—2791no—
151C3×C9—311no—
152C3×C9—311no—
153C3×C9—931no—
154C3×C9—931no—
155C3×C9C32×C31⋊C311yescharacteristic
156C32—91no—
157C32—91no—
158C32—2791no—
159C32—2791no—
160C32—2791no—
161C32—2791no—
162C32—8373no—
163C32—931no—
164C32H(3,𝔽3)×C31⋊C311yescharacteristic
165C9—31no—
166C9—91no—
167C9—91no—
168C9—2791no—
169C9—2791no—
170C9—2791no—
171C9—2791no—
172C9—933no—
173C9—931no—
174C9—931no—
175C3—31no—
176C3—2791no—
177C3—271no—
178C3—271no—
179C3—8371no—
180C3—8371no—
181C3—8371no—
182C3#2141940311yescharacteristic
subgroupG/H
centerC3#21419403
commutator, cosocleC3×C9×C31C33
FrattiniC3×C9C32×C31⋊C3
FittingC31×#6881C3
socleC3×C31#6911
hypercenter#6881C31⋊C3
hypocenterC31#10624851
LCSG ▷ C3×C9×C31 ▷ C32×C31 ▷ C3×C31 ▷ C31
UCSC1 ◁ C3 ◁ C32 ◁ C3×C9 ◁ #6881

Complements

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

NH ≅ G/Ntype
1. #21419420C3semidirect
2. #21419415C3semidirect
3. #21419403C3semidirect
4. #21419363C3semidirect
5. #21419281C3semidirect
6. #21419279C3semidirect
7. #21419271C3semidirect
8. #21419269C3semidirect
9. #21419269C3semidirect
10. #21419264C3semidirect
11. #21419263C3semidirect
12. #21419259C3semidirect
13. C31×#6881C3semidirect
19. #14578902C32semidirect
20. #14578902C32semidirect
21. #14578902C32semidirect
24. #14578900C32semidirect
31. #14578887C32semidirect
34. C9×C31⋊C9C32semidirect
35. C31×#733C32semidirect
37. C31×#732C32semidirect
38. C31×C9⋊C9C32semidirect
93. #6881C31⋊C3semidirect
113. #733C3×C31⋊C3semidirect
120. #732C3×C31⋊C3semidirect
128. C9⋊C9C3×C31⋊C3semidirect
130. C31#10624851semidirect

Maximal subgroups

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

subgroupindex
#214194203
#214194153
#214194033
#214193633
#214192813
#214192793
#214192713
#214192693
#214192643
#214192633
#214192593
C31×#68813
#1062485131