Finite Group Databasebeta

← #36381321

Subgroups — #36381321

Order 22599.

560 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C31⋊H(5,𝔽3)C311yesmaximal, characteristic, complemented
2#21419406C312yesmaximal, complemented
3#21419405C312yesmaximal, complemented
4#21419397C312yesmaximal, complemented
5#21419396C311yesmaximal, characteristic, complemented
6C3×#14578895C312yesmaximal, complemented
7(C3×C31⋊C3)⋊H(3,𝔽3)C311yesmaximal, characteristic, complemented
8(C3×C31⋊C3)⋊H(3,𝔽3)C311yesmaximal, characteristic, complemented
9C31×#6909C311yesmaximal, characteristic, complemented
10C33×C31⋊C3C3211yescharacteristic, complemented
11C3×#11719689C3212yescomplemented
12C3×C32⋊(C31⋊C3)—33no—
13C3×C32⋊(C31⋊C3)—33no—
14C3×C32⋊(C31⋊C3)—33no—
15C3×C32⋊(C31⋊C3)C3211yescharacteristic
16C3×C32⋊(C31⋊C3)C3212yes—
17C3×C32⋊(C31⋊C3)C3211yescharacteristic, complemented
18C3×C32⋊(C31⋊C3)C3212yescomplemented
19H(3,𝔽3)×C31⋊C3—33no—
20H(3,𝔽3)×C31⋊C3—36no—
21#14578900—36no—
22#14578899—36no—
23#14578895—36no—
24#14578895—36no—
25#14578894—33no—
26C3×3−1+2×C31C3212yescomplemented
27C3×H(3,𝔽3)×C31C3211yescharacteristic
28C3×H(3,𝔽3)×C31C3211yescharacteristic, complemented
29C31×AΣL(1,𝔽27)—33no—
30C33×C31—31no—
31C33×C31H(3,𝔽3)13yescomplemented
32C33×C31C3311yescharacteristic
33C32×C31⋊C3—31no—
34C32×C31⋊C3—31no—
35C32×C31⋊C3—39no—
36C32×C31⋊C3—32no—
37C32×C31⋊C3—32no—
38C32×C31⋊C3—31no—
39C32×C31⋊C3—32no—
40C32×C31⋊C3—93no—
41C32×C31⋊C3H(3,𝔽3)13yescomplemented
423−1+2×C31—32no—
433−1+2×C31—32no—
443−1+2×C31—32no—
45H(3,𝔽3)×C31—32no—
46H(3,𝔽3)×C31—31no—
47H(3,𝔽3)×C31—31no—
48H(3,𝔽3)×C31—32no—
49C3×C9×C31—32no—
50C32⋊(C31⋊C3)—318no—
51C32⋊(C31⋊C3)—32no—
52C32⋊(C31⋊C3)—32no—
53C32⋊(C31⋊C3)—31no—
54C32⋊(C31⋊C3)—32no—
55C32⋊(C31⋊C3)—31no—
56C32⋊(C31⋊C3)—32no—
57C32⋊(C31⋊C3)—32no—
58C32⋊(C31⋊C3)—32no—
59C32⋊(C31⋊C3)—32no—
60C32⋊(C31⋊C3)—99no—
61C32⋊(C31⋊C3)—99no—
62C32⋊(C31⋊C3)H(3,𝔽3)16yescomplemented
63#11719689—32no—
64#11719689—32no—
65#11719689—32no—
66C3×C31⋊C9—32no—
67UT(4,𝔽3)—311nomaximal, Sylow
68C32×C31—33no—
69C32×C31—91no—
70C32×C31—93no—
71C32×C31—93no—
72C32×C31—91no—
73C32×C31C3×H(3,𝔽3)12yes—
74C32×C31C3×H(3,𝔽3)11yescharacteristic
75C32×C31C3×H(3,𝔽3)11yescharacteristic
76C3×C31⋊C3—33no—
77C3×C31⋊C3—33no—
78C3×C31⋊C3—33no—
79C3×C31⋊C3—36no—
80C3×C31⋊C3—92no—
81C3×C31⋊C3—92no—
82C3×C31⋊C3—91no—
83C3×C31⋊C3—96no—
84C3×C31⋊C3—93no—
85C3×C31⋊C3—93no—
86C3×C31⋊C3—96no—
87C3×C31⋊C3—91no—
88C3×C31⋊C3—92no—
89C3×C31⋊C3—272no—
90C3×C31⋊C3—279no—
91C9×C31—92no—
92C31⋊C9—92no—
93H(5,𝔽3)—311no—
94#6911—312no—
95#6911—312no—
96#6909—312no—
97#6909—311no—
98#6909C31⋊C311yescharacteristic, complemented
99C3×AΣL(1,𝔽27)—312no—
100C32⋊H(3,𝔽3)—311no—
101C32⋊H(3,𝔽3)—311no—
102C3×C31—31no—
103C3×C31—31no—
104C3×C31—32no—
105C3×C31—93no—
106C3×C31—271no—
107C3×C31C32⋊H(3,𝔽3)11yescharacteristic
108C31⋊C3—93no—
109C31⋊C3—93no—
110C31⋊C3—96no—
111C31⋊C3—271no—
112C31⋊C3—272no—
113C34—311no—
114C3×3−1+2—312no—
115C3×3−1+2C3×C31⋊C312yescomplemented
116C3×H(3,𝔽3)—312no—
117C3×H(3,𝔽3)—311no—
118C3×H(3,𝔽3)—312no—
119C3×H(3,𝔽3)—311no—
120C3×H(3,𝔽3)—936no—
121C3×H(3,𝔽3)—933no—
122C3×H(3,𝔽3)—933no—
123C3×H(3,𝔽3)—933no—
124C3×H(3,𝔽3)—933no—
125C3×H(3,𝔽3)C3×C31⋊C311yescharacteristic
126C3×H(3,𝔽3)C3×C31⋊C311yescharacteristic, complemented
127#732—936no—
128#732—936no—
129AΣL(1,𝔽27)—33no—
130AΣL(1,𝔽27)—936no—
131AΣL(1,𝔽27)—936no—
132AΣL(1,𝔽27)—933no—
133C31UT(4,𝔽3)11yescharacteristic, complemented, Sylow
134C33—31no—
135C33—2793no—
136C33—313no—
137C33—939no—
138C33—932no—
139C33—931no—
140C33—932no—
141C33—931no—
142C33—931no—
143C33—932no—
144C33C32⋊(C31⋊C3)13yescomplemented
145C33C32×C31⋊C311yescharacteristic
1463−1+2—32no—
1473−1+2—32no—
1483−1+2—32no—
1493−1+2—932no—
1503−1+2—932no—
1513−1+2—932no—
152H(3,𝔽3)—32no—
153H(3,𝔽3)—31no—
154H(3,𝔽3)—32no—
155H(3,𝔽3)—31no—
156H(3,𝔽3)—2799no—
157H(3,𝔽3)—2799no—
158H(3,𝔽3)—316no—
159H(3,𝔽3)—9318no—
160H(3,𝔽3)—932no—
161H(3,𝔽3)—932no—
162H(3,𝔽3)—932no—
163H(3,𝔽3)—932no—
164H(3,𝔽3)—931no—
165H(3,𝔽3)—932no—
166H(3,𝔽3)—932no—
167H(3,𝔽3)—932no—
168H(3,𝔽3)—931no—
169C3×C9—32no—
170C3×C9—932no—
171C32—33no—
172C32—91no—
173C32—93no—
174C32—93no—
175C32—91no—
176C32—2793no—
177C32—2796no—
178C32—2792no—
179C32—2792no—
180C32—2793no—
181C32—2796no—
182C32—2791no—
183C32—2792no—
184C32—2791no—
185C32—8372no—
186C32—8379no—
187C32—933no—
188C32—933no—
189C32—936no—
190C32—933no—
191C32H(3,𝔽3)×C31⋊C312yes—
192C32H(3,𝔽3)×C31⋊C311yescharacteristic
193C32C3×C32⋊(C31⋊C3)11yescharacteristic
194C9—92no—
195C9—2792no—
196C3—31no—
197C3—31no—
198C3—32no—
199C3—93no—
200C3—2793no—
201C3—2793no—
202C3—2796no—
203C3—271no—
204C3—8371no—
205C3—8372no—
206C3(C3×C31⋊C3)⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC3(C3×C31⋊C3)⋊H(3,𝔽3)
commutator, cosocleC33×C31C33
FrattiniC33C32×C31⋊C3
FittingC31×#6909C3
socleC3×C31C32⋊H(3,𝔽3)
hypercenter#6909C31⋊C3
hypocenterC31UT(4,𝔽3)
LCSG ▷ C33×C31 ▷ C3×C31 ▷ C31
UCSC1 ◁ C3 ◁ C33 ◁ #6909

Complements

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

NH ≅ G/Ntype
1. C31⋊H(5,𝔽3)C3semidirect
2. #21419406C3semidirect
3. #21419405C3semidirect
4. #21419397C3semidirect
5. #21419396C3semidirect
6. C3×#14578895C3semidirect
7. (C3×C31⋊C3)⋊H(3,𝔽3)C3semidirect
8. (C3×C31⋊C3)⋊H(3,𝔽3)C3semidirect
9. C31×#6909C3semidirect
10. C33×C31⋊C3C32semidirect
11. C3×#11719689C32semidirect
17. C3×C32⋊(C31⋊C3)C32semidirect
18. C3×C32⋊(C31⋊C3)C32semidirect
26. C3×3−1+2×C31C32semidirect
28. C3×H(3,𝔽3)×C31C32semidirect
31. C33×C31H(3,𝔽3)semidirect
41. C32×C31⋊C3H(3,𝔽3)semidirect
62. C32⋊(C31⋊C3)H(3,𝔽3)semidirect
98. #6909C31⋊C3semidirect
115. C3×3−1+2C3×C31⋊C3semidirect
126. C3×H(3,𝔽3)C3×C31⋊C3semidirect
133. C31UT(4,𝔽3)semidirect
144. C33C32⋊(C31⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C31⋊H(5,𝔽3)3
#214194063
#214194053
#214193973
#214193963
C3×#145788953
(C3×C31⋊C3)⋊H(3,𝔽3)3
C31×#69093
UT(4,𝔽3)31