Finite Group Databasebeta

← A4×C5⋊AGL(1,ℤ/9) #72247486

Subgroups — A4×C5⋊AGL(1,ℤ/9) #72247486

Order 3240.

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

normal subgroups only

This group has 234 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
1C5×A4×3−1+2C211yesmaximal, characteristic, complemented
2C3×A4×D15—31nomaximal
3C22×C5⋊AGL(1,ℤ/9)C311yesmaximal, characteristic, complemented
4C5⋊(D9⋊A4)C312yesmaximal, complemented
5A4×D45C311yesmaximal, characteristic, complemented
6C3×C5⋊AGL(1,ℤ/9)—41nomaximal
7A4×AGL(1,ℤ/9)—51nomaximal, Hall
8C32×C5×A4D311yescharacteristic
9C22×C5×3−1+2C2×C311yescharacteristic, complemented
10C2×C5⋊AGL(1,ℤ/9)—31no—
11C2×C5⋊AGL(1,ℤ/9)—31no—
12C5×#1094—21no—
13C5×C9⋊A4—22no—
14C5×C9⋊A4C2×C312yescomplemented
15C5×C9×A4—21no—
16C5×C9×A4C2×C311yescharacteristic, complemented
17C3×C5×3−1+2—41noHall
18C22×C3×D15—31no—
19A4×D15—31no—
20A4×D15—32no—
21C3×D5×A4—91no—
22C22×D45C3211yescharacteristic, complemented
23A4×3−1+2D511yescharacteristic, complemented
24C32×D15—121no—
25C2×C5×3−1+2—31no—
26C5⋊AGL(1,ℤ/9)—31no—
27C5⋊AGL(1,ℤ/9)—42no—
28C5⋊AGL(1,ℤ/9)A411yescharacteristic, direct product factor
29C3×D45—41no—
30C3×D3×A4—151no—
31C22×AGL(1,ℤ/9)—51no—
32A4×D9—51no—
33D9⋊A4—52no—
34C22×C32×C5C3×D311yescharacteristic
35C2×C3×D15—91no—
36C2×C3×D15—91no—
37C3×C5×A4—31no—
38C3×C5×A4—61no—
39C3×C5×A4C3×D311yescharacteristic
40C3×C5×A4C3×D312yes—
41C22×C5×C9—21no—
42C22×C5×C9C2×C3211yescharacteristic, complemented
43C2×D45—31no—
44C2×D45—31no—
45C5×C22⋊C9—22no—
46C5×C22⋊C9—21no—
47C3×AGL(1,ℤ/9)—201no—
48C33×C5—41no—
49C5×3−1+2—42no—
50C5×3−1+2—82no—
51C5×3−1+2—81no—
52C5×3−1+2C2×A411yescharacteristic, complemented
53C3×C5×C9—41no—
54C3×C5×C9—81no—
55C22×D15—31no—
56C22×C3×D5—91no—
57D5×A4—92no—
58D5×A4—91no—
59C32×A4D1511yescharacteristic
60C22×3−1+2C3×D511yescharacteristic, complemented
61C2×AGL(1,ℤ/9)—151no—
62C2×AGL(1,ℤ/9)—151no—
63#1094—21no—
64C9⋊A4—22no—
65C9⋊A4C3×D512yescomplemented
66C9×A4—21no—
67C9×A4C3×D511yescharacteristic, complemented
68C2×C32×C5—31no—
69C3×D15—31no—
70C3×D15—91no—
71C3×D15—121no—
72C3×D15—122no—
73C32×D5—361no—
74C2×C5×C9—31no—
75C2×C5×C9—61no—
76D45—31no—
77D45C3×A411yescharacteristic, complemented
78C3×3−1+2—41noSylow
79C22×C3×D3—151no—
80C2×C3×A4—451no—
81D3×A4—152no—
82D3×A4—151no—
83C22×D9—51no—
84C22×C3×C5—31no—
85C22×C3×C5C32×D311yescharacteristic
86C2×D15—91no—
87C2×D15—91no—
88C2×C3×D5—271no—
89C2×C3×D5—271no—
90C5×A4—21no—
91C5×A4—32no—
92C5×A4AGL(1,ℤ/9)11yescharacteristic, complemented
93C32×D3—601no—
94C2×3−1+2—31no—
95AGL(1,ℤ/9)—51no—
96AGL(1,ℤ/9)—151no—
97AGL(1,ℤ/9)—202no—
98C3×D9—201no—
99C32×C5—41no—
100C32×C5—42no—
101C32×C5—121no—
102C32×C5—241no—
103C32×C5D3×A411yescharacteristic
104C5×C9—21no—
105C5×C9—81no—
106C5×C9—82no—
107C5×C9C2×C3×A411yescharacteristic, complemented
108C22×D5—91noHall
109C22×C32C3×D1511yescharacteristic
110C2×C3×D3—451no—
111C2×C3×D3—451no—
112C3×A4—31no—
113C3×A4—61no—
114C3×A4C3×D1511yescharacteristic
115C3×A4C3×D1512yes—
116C22×C9—21no—
117C22×C9C32×D511yescharacteristic, complemented
118C2×D9—151no—
119C2×D9—151no—
120C22⋊C9—22no—
121C22⋊C9—21no—
122C2×C3×C5—31no—
123C2×C3×C5—91no—
124D15—31no—
125D15—91no—
126C3×D5—91no—
127C3×D5—271no—
128C3×D5—361no—
129C3×D5—362no—
130C33—41no—
1313−1+2—42no—
1323−1+2—82no—
1333−1+2—81no—
1343−1+2D5×A411yescharacteristic, complemented
135C3×C9—41no—
136C3×C9—81no—
137C23×C3—451no—
138C22×D3—151no—
139C2×A4—452no—
140C2×A4—451no—
141C22×C5C3×AGL(1,ℤ/9)11yescharacteristic, complemented
142C2×D5—271no—
143C2×D5—271no—
144C2×C32—31no—
145C2×C32—1801no—
146C3×D3—151no—
147C3×D3—451no—
148C3×D3—601no—
149C3×D3—602no—
150C2×C9—31no—
151C2×C9—61no—
152D9—51no—
153D9—151no—
154C3×C5—31no—
155C3×C5—41no—
156C3×C5—81no—
157C3×C5—122no—
158C3×C5C3×D3×A411yescharacteristic
159C22×C3—31no—
160C22×C3—1351no—
161C22×C3—1351no—
162C22×C3C32×D1511yescharacteristic
163C2×D3—451no—
164C2×D3—451no—
165A4—21no—
166A4—32no—
167A4C5⋊AGL(1,ℤ/9)11yescharacteristic, direct product factor
168C2×C5—31no—
169D5—91no—
170D5—271no—
171C32—41no—
172C32—42no—
173C32—121no—
174C32—241no—
175C32A4×D1511yescharacteristic
176C9—21no—
177C9—81no—
178C9—82no—
179C9C3×D5×A411yescharacteristic, complemented
180C23—451noSylow
181C2×C3—31no—
182C2×C3—91no—
183C2×C3—451no—
184C2×C3—1351no—
185C2×C3—1801no—
186C2×C3—1802no—
187D3—151no—
188D3—451no—
189C5A4×AGL(1,ℤ/9)11yescharacteristic, complemented, Sylow
190C22—1351no—
191C22—1351no—
192C22C3×C5⋊AGL(1,ℤ/9)11yescharacteristic, complemented
193C3—31no—
194C3—41no—
195C3—81no—
196C3—122no—
197C3C3×A4×D1511yescharacteristic
198C2—31no—
199C2—451no—
200C2—1351no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22×C5×C9C2×C32
FrattiniC3C3×A4×D15
FittingC22×C5×3−1+2C2×C3
socleC22×C3×C5C32×D3
LCSG ▷ C22×C5×C9
UCSC1

Complements

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

NH ≅ G/Ntype
1. C5×A4×3−1+2C2semidirect
3. C22×C5⋊AGL(1,ℤ/9)C3semidirect
4. C5⋊(D9⋊A4)C3semidirect
5. A4×D45C3semidirect
9. C22×C5×3−1+2C2×C3semidirect
14. C5×C9⋊A4C2×C3semidirect
16. C5×C9×A4C2×C3semidirect
22. C22×D45C32semidirect
23. A4×3−1+2D5semidirect
28. C5⋊AGL(1,ℤ/9)A4direct (N × H)
42. C22×C5×C9C2×C32semidirect
52. C5×3−1+2C2×A4semidirect
60. C22×3−1+2C3×D5semidirect
65. C9⋊A4C3×D5semidirect
67. C9×A4C3×D5semidirect
77. D45C3×A4semidirect
92. C5×A4AGL(1,ℤ/9)semidirect
107. C5×C9C2×C3×A4semidirect
117. C22×C9C32×D5semidirect
134. 3−1+2D5×A4semidirect
141. C22×C5C3×AGL(1,ℤ/9)semidirect
167. A4C5⋊AGL(1,ℤ/9)direct (N × H)
179. C9C3×D5×A4semidirect
189. C5A4×AGL(1,ℤ/9)semidirect
192. C22C3×C5⋊AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
C5×A4×3−1+22
C3×A4×D153
C22×C5⋊AGL(1,ℤ/9)3
C5⋊(D9⋊A4)3
A4×D453
C3×C5⋊AGL(1,ℤ/9)4
A4×AGL(1,ℤ/9)5