Finite Group Databasebeta

← #36382133

Subgroups — #36382133

Order 22599.

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

normal subgroups only

This group has 210 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, complemented
2#732×C31⋊C3C311yesmaximal, characteristic, complemented
3#732×C31⋊C3C311yesmaximal, characteristic, complemented
4#21419295C311yesmaximal, characteristic, complemented
5#21419273C311yesmaximal, characteristic, complemented
6#21419272C311yesmaximal, characteristic, complemented
7#21419271C311yesmaximal, characteristic, complemented
8#21419271C311yesmaximal, characteristic, complemented
9#21419270C311yesmaximal, characteristic, complemented
10#21419270C311yesmaximal, characteristic, complemented
11#21419269C311yesmaximal, characteristic, complemented
12#21419269C311yesmaximal, characteristic, complemented
13C31×#6881C311yesmaximal, characteristic, complemented
143−1+2×C31⋊C3—31no—
15H(3,𝔽3)×C31⋊C3—31no—
16H(3,𝔽3)×C31⋊C3—31no—
17C3×C9×C31⋊C3C3211yescharacteristic
18#14578902C3211yescharacteristic, complemented
19#14578902C3211yescharacteristic, complemented
20#14578901—31no—
21#14578901—31no—
22#14578900C3211yescharacteristic, complemented
23#14578900C3211yescharacteristic, complemented
24#14578900C3211yescharacteristic, complemented
25#14578900C3211yescharacteristic, complemented
26#14578899—31no—
27#14578899—31no—
28#14578899—31no—
29#14578899—31no—
30#14578887C3211yescharacteristic, complemented
31#14578886C3211yescharacteristic, complemented
32(C31⋊C9)⋊C9—31no—
33(C3×C31⋊C3)⋊C9—31no—
34C9×C31⋊C9—31no—
35C31×#733C3211yescharacteristic, complemented
36C31×#732C3211yescharacteristic, complemented
37C31×#732C3211yescharacteristic, complemented
38C31×C9⋊C9C3211yescharacteristic, complemented
39C32×C31⋊C3—91no—
40C32×C31⋊C3—91no—
41C32×C31⋊C3H(3,𝔽3)11yescharacteristic
423−1+2×C31—31no—
43H(3,𝔽3)×C31—31no—
44H(3,𝔽3)×C31—31no—
45C3×C9×C31—31no—
46C3×C9×C31C3311yescharacteristic
47C32⋊(C31⋊C3)—31no—
48C32⋊(C31⋊C3)—31no—
49C32⋊(C31⋊C3)—31no—
50C32⋊(C31⋊C3)—31no—
51C32⋊(C31⋊C3)—91no—
52C32⋊(C31⋊C3)—91no—
53C32⋊(C31⋊C3)—91no—
54C32⋊(C31⋊C3)—91no—
55#11719689—31no—
56#11719689—31no—
57C3×C31⋊C9—31no—
58C3×C31⋊C9—31no—
59C3×C31⋊C9H(3,𝔽3)11yescharacteristic
60C3×C31⋊C9H(3,𝔽3)11yescharacteristic
61#11719687—91no—
62#11719686—91no—
63C9×C31⋊C3—31no—
64C9×C31⋊C3—31no—
65C9×C31⋊C3—31no—
66C9×C31⋊C3—91no—
67#10624846—311nomaximal, Sylow
68C32×C31—91no—
69C32×C31—91no—
70C32×C31C3×H(3,𝔽3)11yescharacteristic
71C3×C31⋊C3—91no—
72C3×C31⋊C3—91no—
73C3×C31⋊C3—91no—
74C3×C31⋊C3—91no—
75C3×C31⋊C3—91no—
76C3×C31⋊C3—273no—
77C3×C31⋊C3—273no—
78C3×C31⋊C3#73211yescharacteristic
79C3×C31⋊C3#73211yescharacteristic
80C3×C31⋊C3#73211yescharacteristic
81C9×C31—31no—
82C9×C31—91no—
83C9×C31—91no—
84C31⋊C9—31no—
85C31⋊C9—31no—
86C31⋊C9—91no—
87C31⋊C9—91no—
88C31⋊C9—91no—
89C31⋊C9—91no—
90C3×#733—311no—
91C3×#732—311no—
92C3×#732—311no—
93#6887—311no—
94#6881—311no—
95#6881—311no—
96#6881—311no—
97#6881—311no—
98#6881—311no—
99#6881—311no—
100#6881—311no—
101#6881—311no—
102#6881C31⋊C311yescharacteristic, complemented
103C3×C31—31no—
104C3×C31—271no—
105C3×C31—271no—
106C3×C31C3×#73211yescharacteristic
107C31⋊C3—31no—
108C31⋊C3—31no—
109C31⋊C3—31no—
110C31⋊C3—271no—
111C31⋊C3—271no—
112C31⋊C3—271no—
113C31⋊C3—271no—
114C3×3−1+2—931no—
115C3×H(3,𝔽3)—931no—
116C3×H(3,𝔽3)—931no—
117C32×C9—311no—
118#733—311no—
119#733—311no—
120#733—931no—
121#733—931no—
122#733C3×C31⋊C311yescharacteristic, complemented
123#732—311no—
124#732—311no—
125#732—311no—
126#732—311no—
127#732—931no—
128#732—931no—
129#732—931no—
130#732—931no—
131#732C3×C31⋊C311yescharacteristic, complemented
132#732C3×C31⋊C311yescharacteristic, complemented
133C9⋊C9—311no—
134C9⋊C9—311no—
135C9⋊C9—931no—
136C9⋊C9C3×C31⋊C311yescharacteristic, complemented
137C32⋊C9—931no—
138C92—931no—
139C31#1062484611yescharacteristic, complemented, Sylow
140C33—2791no—
141C33—2791no—
142C33—311no—
1433−1+2—31no—
1443−1+2—2791no—
1453−1+2—2791no—
1463−1+2—931no—
1473−1+2—931no—
148H(3,𝔽3)—31no—
149H(3,𝔽3)—31no—
150H(3,𝔽3)—2791no—
151H(3,𝔽3)—2791no—
152H(3,𝔽3)—2791no—
153H(3,𝔽3)—2791no—
154H(3,𝔽3)—931no—
155H(3,𝔽3)—931no—
156H(3,𝔽3)—931no—
157H(3,𝔽3)—931no—
158C3×C9—31no—
159C3×C9—2791no—
160C3×C9—311no—
161C3×C9—311no—
162C3×C9—931no—
163C3×C9—931no—
164C3×C9—931no—
165C3×C9—931no—
166C3×C9—931no—
167C3×C9C32×C31⋊C311yescharacteristic
168C32—91no—
169C32—91no—
170C32—2791no—
171C32—2791no—
172C32—2791no—
173C32—2791no—
174C32—2791no—
175C32—311no—
176C32—311no—
177C32—311no—
178C32—8373no—
179C32—8373no—
180C32H(3,𝔽3)×C31⋊C311yescharacteristic
181C9—31no—
182C9—91no—
183C9—91no—
184C9—2791no—
185C9—2791no—
186C9—2791no—
187C9—2791no—
188C9—931no—
189C9—931no—
190C3—31no—
191C3—271no—
192C3—271no—
193C3—8371no—
194C3—8371no—
195C3—8371no—
196C3—8371no—
197C3—931no—
198C3—931no—
199C3—931no—
200C3#732×C31⋊C311yescharacteristic
subgroupG/H
centerC3#732×C31⋊C3
commutator, cosocleC3×C9×C31C33
FrattiniC3×C9C32×C31⋊C3
FittingC31×#6881C3
socleC3×C31C3×#732
hypercenter#6881C31⋊C3
hypocenterC31#10624846
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. #733×C31⋊C3C3semidirect
2. #732×C31⋊C3C3semidirect
3. #732×C31⋊C3C3semidirect
4. #21419295C3semidirect
5. #21419273C3semidirect
6. #21419272C3semidirect
7. #21419271C3semidirect
8. #21419271C3semidirect
9. #21419270C3semidirect
10. #21419270C3semidirect
11. #21419269C3semidirect
12. #21419269C3semidirect
13. C31×#6881C3semidirect
18. #14578902C32semidirect
19. #14578902C32semidirect
22. #14578900C32semidirect
23. #14578900C32semidirect
24. #14578900C32semidirect
25. #14578900C32semidirect
30. #14578887C32semidirect
31. #14578886C32semidirect
35. C31×#733C32semidirect
36. C31×#732C32semidirect
37. C31×#732C32semidirect
38. C31×C9⋊C9C32semidirect
102. #6881C31⋊C3semidirect
122. #733C3×C31⋊C3semidirect
131. #732C3×C31⋊C3semidirect
132. #732C3×C31⋊C3semidirect
136. C9⋊C9C3×C31⋊C3semidirect
139. C31#10624846semidirect

Maximal subgroups

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

subgroupindex
#733×C31⋊C33
#732×C31⋊C33
#214192953
#214192733
#214192723
#214192713
#214192703
#214192693
C31×#68813
#1062484631