Finite Group Databasebeta

← C22×D5×AGL(1,ℤ/9) #72247331

Subgroups — C22×D5×AGL(1,ℤ/9) #72247331

Order 2160.

836 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×C5⋊AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
2C22×C5×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
3C22×D5×3−1+2C211yesmaximal, characteristic, complemented
4C2×D5×AGL(1,ℤ/9)C2112yesmaximal, direct product factor
5C22×C3×D3×D5—31nomaximal
6C22×D5×D9C311yesmaximal, characteristic, complemented
7C22×C5×3−1+2C2211yescharacteristic, complemented
8C2×C5⋊AGL(1,ℤ/9)C2216yescomplemented
9C2×C5×AGL(1,ℤ/9)C2216yescomplemented
10C2×D5×3−1+2C2216yescomplemented
11D5×AGL(1,ℤ/9)C22116yesdirect product factor
12C23×AGL(1,ℤ/9)—51nomaximal, Hall
13C22×C3×D15—31no—
14C22×C3×C5×D3—31no—
15C22×C32×D5D311yescharacteristic
16C2×C3×D3×D5—312no—
17C22×D45C2×C311yescharacteristic, complemented
18C22×C5×D9C2×C311yescharacteristic, complemented
19C22×C9×D5—21no—
20C22×C9×D5C2×C311yescharacteristic, complemented
21C2×D5×D9C2×C3112yescomplemented
22C2×C5×3−1+2C2313yescomplemented
23C5⋊AGL(1,ℤ/9)C2314yescomplemented
24C5×AGL(1,ℤ/9)C2314yescomplemented
25D5×3−1+2C2314yescomplemented
26C23×C3×D5—91no—
27C22×D3×D5—31no—
28C23×3−1+2—51no—
29C22×AGL(1,ℤ/9)—51no—
30C22×AGL(1,ℤ/9)—512no—
31C22×AGL(1,ℤ/9)D511yescharacteristic, direct product factor
32C22×C32×C5C2×D311yescharacteristic
33C2×C3×D15—36no—
34C2×C3×C5×D3—36no—
35C2×C32×D5C2×D316yes—
36C3×D3×D5—316no—
37C22×C5×C9—21no—
38C22×C5×C9C22×C311yescharacteristic, complemented
39C2×D45C22×C316yescomplemented
40C2×C5×D9C22×C316yescomplemented
41C2×C9×D5—26no—
42C2×C9×D5C22×C316yescomplemented
43D5×D9C22×C3116yescomplemented
44C23×C3×D3—151no—
45C23×D9—51no—
46C5×3−1+2C2411yescharacteristic, complemented, Hall
47C23×C3×C5—91no—
48C22×D15—31no—
49C22×C5×D3—31no—
50C22×C3×D5—61no—
51C22×C3×D5—912no—
52C22×C3×D5—91no—
53C22×C3×D5C3×D311yescharacteristic
54C2×D3×D5—312no—
55C22×3−1+2—56no—
56C22×3−1+2C2×D511yescharacteristic, complemented
57C2×AGL(1,ℤ/9)—56no—
58C2×AGL(1,ℤ/9)—516no—
59C2×AGL(1,ℤ/9)C2×D516yesdirect product factor
60C2×C32×C5C22×D313yes—
61C3×D15—34no—
62C3×C5×D3—34no—
63C32×D5C22×D314yes—
64C2×C5×C9—23no—
65C2×C5×C9C23×C313yescomplemented
66D45C23×C314yescomplemented
67C9×D5—24no—
68C9×D5C23×C314yescomplemented
69C5×D9C23×C314yescomplemented
70C23×D5—91noHall
71C23×C32—51no—
72C22×C3×D3—31no—
73C22×C3×D3—1512no—
74C22×C3×D3—151no—
75C23×C9—51no—
76C23×C9—101no—
77C22×D9—51no—
78C22×D9—512no—
79C22×D9C3×D511yescharacteristic, complemented
80C22×C3×C5—61no—
81C22×C3×C5—96no—
82C22×C3×C5C2×C3×D311yescharacteristic
83C2×D15—36no—
84C2×C5×D3—36no—
85C2×C3×D5—66no—
86C2×C3×D5—916no—
87C2×C3×D5—96no—
88C2×C3×D5C2×C3×D316yes—
89D3×D5—316no—
90C2×3−1+2—54no—
91C2×3−1+2C22×D513yescomplemented
92AGL(1,ℤ/9)—54no—
93AGL(1,ℤ/9)C22×D514yesdirect product factor
94C24×C3—451no—
95C23×D3—151no—
96C32×C5C23×D311yescharacteristic
97C5×C9—21no—
98C5×C9C24×C311yescharacteristic, complemented
99C23×C5—91no—
100C22×D5—91no—
101C22×D5—912no—
102C22×D5AGL(1,ℤ/9)11yescharacteristic, direct product factor
103C22×C32—56no—
104C22×C32D3×D511yescharacteristic
105C2×C3×D3—36no—
106C2×C3×D3—1516no—
107C2×C3×D3—156no—
108C22×C9—21no—
109C22×C9—56no—
110C22×C9—106no—
111C22×C9C2×C3×D511yescharacteristic, complemented
112C2×D9—56no—
113C2×D9—516no—
114C2×D9C2×C3×D516yescomplemented
115C2×C3×C5—63no—
116C2×C3×C5—94no—
117C2×C3×C5C22×C3×D313yes—
118D15—34no—
119C3×D5—64no—
120C3×D5—94no—
121C3×D5C22×C3×D314yes—
122C5×D3—34no—
1233−1+2C23×D511yescharacteristic, complemented, Sylow
124C23×C3—51no—
125C23×C3—91no—
126C23×C3—301no—
127C23×C3—451no—
128C23×C3—4512no—
129C22×D3—31no—
130C22×D3—1512no—
131C22×D3—151no—
132C22×C5—96no—
133C22×C5C2×AGL(1,ℤ/9)11yescharacteristic, complemented
134C2×D5—96no—
135C2×D5—916no—
136C2×D5C2×AGL(1,ℤ/9)16yesdirect product factor
137C2×C32—54no—
138C2×C32C2×D3×D513yes—
139C3×D3—34no—
140C3×D3—154no—
141C2×C9—23no—
142C2×C9—54no—
143C2×C9—104no—
144C2×C9C22×C3×D513yescomplemented
145D9—54no—
146D9C22×C3×D514yescomplemented
147C24—451noSylow
148C3×C5—61no—
149C3×C5C23×C3×D311yescharacteristic
150C22×C3—56no—
151C22×C3—61no—
152C22×C3—96no—
153C22×C3—306no—
154C22×C3—456no—
155C22×C3—4516no—
156C22×C3C3×D3×D511yescharacteristic
157C2×D3—36no—
158C2×D3—1516no—
159C2×D3—156no—
160C2×C5—94no—
161C2×C5C22×AGL(1,ℤ/9)13yescomplemented
162D5—94no—
163D5C22×AGL(1,ℤ/9)14yesdirect product factor
164C32C22×D3×D511yescharacteristic
165C9—21no—
166C9C23×C3×D511yescharacteristic, complemented
167C23—51no—
168C23—91no—
169C23—4512no—
170C23—451no—
171C2×C3—54no—
172C2×C3—63no—
173C2×C3—94no—
174C2×C3—304no—
175C2×C3—454no—
176C2×C3C2×C3×D3×D513yes—
177D3—34no—
178D3—154no—
179C5C23×AGL(1,ℤ/9)11yescharacteristic, complemented, Sylow
180C22—56no—
181C22—96no—
182C22—4516no—
183C22—456no—
184C22D5×AGL(1,ℤ/9)11yescharacteristic, direct product factor
185C3—61no—
186C3C22×C3×D3×D511yescharacteristic
187C2—54no—
188C2—94no—
189C2—454no—
190C2C2×D5×AGL(1,ℤ/9)13yesdirect product factor
subgroupG/H
center, hypercenterC22D5×AGL(1,ℤ/9)
commutator, cosocle, hypocenterC5×C9C24×C3
FrattiniC3C22×C3×D3×D5
FittingC22×C5×3−1+2C22
socleC22×C3×C5C2×C3×D3
LCSG ▷ C5×C9
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C5⋊AGL(1,ℤ/9)C2semidirect
2. C22×C5×AGL(1,ℤ/9)C2semidirect
3. C22×D5×3−1+2C2semidirect
4. C2×D5×AGL(1,ℤ/9)C2direct (N × H)
6. C22×D5×D9C3semidirect
7. C22×C5×3−1+2C22semidirect
8. C2×C5⋊AGL(1,ℤ/9)C22semidirect
9. C2×C5×AGL(1,ℤ/9)C22semidirect
10. C2×D5×3−1+2C22semidirect
11. D5×AGL(1,ℤ/9)C22direct (N × H)
17. C22×D45C2×C3semidirect
18. C22×C5×D9C2×C3semidirect
20. C22×C9×D5C2×C3semidirect
21. C2×D5×D9C2×C3semidirect
22. C2×C5×3−1+2C23semidirect
23. C5⋊AGL(1,ℤ/9)C23semidirect
24. C5×AGL(1,ℤ/9)C23semidirect
25. D5×3−1+2C23semidirect
31. C22×AGL(1,ℤ/9)D5direct (N × H)
38. C22×C5×C9C22×C3semidirect
39. C2×D45C22×C3semidirect
40. C2×C5×D9C22×C3semidirect
42. C2×C9×D5C22×C3semidirect
43. D5×D9C22×C3semidirect
46. C5×3−1+2C24semidirect
56. C22×3−1+2C2×D5semidirect
59. C2×AGL(1,ℤ/9)C2×D5direct (N × H)
65. C2×C5×C9C23×C3semidirect
66. D45C23×C3semidirect
68. C9×D5C23×C3semidirect
69. C5×D9C23×C3semidirect
79. C22×D9C3×D5semidirect
91. C2×3−1+2C22×D5semidirect
93. AGL(1,ℤ/9)C22×D5direct (N × H)
98. C5×C9C24×C3semidirect
102. C22×D5AGL(1,ℤ/9)direct (N × H)
111. C22×C9C2×C3×D5semidirect
114. C2×D9C2×C3×D5semidirect
123. 3−1+2C23×D5semidirect
133. C22×C5C2×AGL(1,ℤ/9)semidirect
136. C2×D5C2×AGL(1,ℤ/9)direct (N × H)
144. C2×C9C22×C3×D5semidirect
146. D9C22×C3×D5semidirect
161. C2×C5C22×AGL(1,ℤ/9)semidirect
163. D5C22×AGL(1,ℤ/9)direct (N × H)
166. C9C23×C3×D5semidirect
179. C5C23×AGL(1,ℤ/9)semidirect
184. C22D5×AGL(1,ℤ/9)direct (N × H)
190. C2C2×D5×AGL(1,ℤ/9)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5⋊AGL(1,ℤ/9)2
C22×C5×AGL(1,ℤ/9)2
C22×D5×3−1+22
C2×D5×AGL(1,ℤ/9)2
C22×C3×D3×D53
C22×D5×D93
C23×AGL(1,ℤ/9)5