Finite Group Databasebeta

← AGL(1,ℤ/9)×AGL(1,𝔽23) #72275868

Subgroups — AGL(1,ℤ/9)×AGL(1,𝔽23) #72275868

Order 27324.

128 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 39 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#72275867C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/9)×C23⋊C11C211yesmaximal, characteristic, complemented
33−1+2×AGL(1,𝔽23)C211yesmaximal, characteristic, complemented
43−1+2×C23⋊C11C2211yescharacteristic, complemented, Hall
5D23×AGL(1,ℤ/9)C1111yesmaximal, characteristic, complemented, Hall
6C3×D3×AGL(1,𝔽23)—31nomaximal
7D9×AGL(1,𝔽23)C311yesmaximal, characteristic, complemented
8C3×D3×C23⋊C11—31no—
9C3×C3⋊AGL(1,𝔽23)—31no—
10C32×AGL(1,𝔽23)D311yescharacteristic
11C9⋊AGL(1,𝔽23)C2×C311yescharacteristic, complemented
12D9×C23⋊C11C2×C311yescharacteristic, complemented
13C9×AGL(1,𝔽23)—21no—
14C9×AGL(1,𝔽23)C2×C311yescharacteristic, complemented
15C2×C3×AGL(1,𝔽23)—91no—
16D3×AGL(1,𝔽23)—31no—
17C32×C23⋊C11C2×D311yescharacteristic
18C9×C23⋊C11—21no—
19C9×C23⋊C11C22×C311yescharacteristic, complemented
20C2×C3×C23⋊C11—91no—
21C3⋊AGL(1,𝔽23)—31no—
22D3×C23⋊C11—31no—
23C3×AGL(1,𝔽23)—31no—
24C3×AGL(1,𝔽23)—91no—
25C3×AGL(1,𝔽23)C3×D311yescharacteristic
26C23⋊AGL(1,ℤ/9)C2×C1111yescharacteristic, complemented
27C23×AGL(1,ℤ/9)C2×C1111yescharacteristic, complemented
283−1+2×D23C2×C1111yescharacteristic, complemented
29C2×C11×AGL(1,ℤ/9)—231nomaximal, Hall
30C2×AGL(1,𝔽23)—91noHall
31C3×D3×D23—31no—
32D9×D23C3×C1111yescharacteristic, complemented
33C3×C23⋊C11—31no—
34C3×C23⋊C11C2×C3×D311yescharacteristic
35C23×3−1+2C22×C1111yescharacteristic, complemented, Hall
36C2×C11×3−1+2—231no—
37C11×AGL(1,ℤ/9)—231no—
38C11×AGL(1,ℤ/9)—231no—
39C2×C23⋊C11—91no—
40AGL(1,𝔽23)—91no—
41AGL(1,𝔽23)AGL(1,ℤ/9)11yescharacteristic, direct product factor
42C3×D69—31no—
43C3×D3×C23—31no—
44C32×D23D3×C1111yescharacteristic
45D207C2×C3×C1111yescharacteristic, complemented
46C9×D23—21no—
47C9×D23C2×C3×C1111yescharacteristic, complemented
48D9×C23C2×C3×C1111yescharacteristic, complemented
49C2×C3×D3×C11—691no—
50C2×C11×D9—231no—
51C11×3−1+2—231noHall
52C2×C3×D23—91no—
53D3×D23—31no—
54C23⋊C11C2×AGL(1,ℤ/9)11yescharacteristic, complemented, Hall
55C32×C23C2×D3×C1111yescharacteristic
56C9×C23—21no—
57C9×C23C22×C3×C1111yescharacteristic, complemented
58C2×C32×C11—231no—
59C3×D3×C11—691no—
60C3×D3×C11—691no—
61C2×C9×C11—231no—
62C2×C9×C11—461no—
63C11×D9—231no—
64C11×D9—231no—
65C2×C3×C23—91no—
66D69—31no—
67C3×D23—31no—
68C3×D23—91no—
69C3×D23C3×D3×C1111yescharacteristic
70D3×C23—31no—
71C22×C3×C11—2071no—
72C2×D3×C11—691no—
73C2×AGL(1,ℤ/9)—231noHall
74C32×C11—231no—
75C9×C11—231no—
76C9×C11—461no—
77C2×D23—91noHall
78C3×C23—31no—
79C3×C23C2×C3×D3×C1111yescharacteristic
80C2×C3×C11—231no—
81C2×C3×C11—691no—
82C2×C3×C11—2071no—
83C2×C3×C11—2071no—
84D3×C11—691no—
85D3×C11—691no—
86C2×3−1+2—231no—
87AGL(1,ℤ/9)—231no—
88AGL(1,ℤ/9)AGL(1,𝔽23)11yescharacteristic, direct product factor
89C2×C23—91no—
90D23—91no—
91D23C11×AGL(1,ℤ/9)11yescharacteristic, complemented
92C22×C11—2071noHall
93C2×C3×D3—691no—
94C2×D9—231no—
95C3×C11—231no—
96C3×C11—691no—
973−1+2C2×AGL(1,𝔽23)11yescharacteristic, complemented, Sylow
98C23C2×C11×AGL(1,ℤ/9)11yescharacteristic, complemented, Sylow
99C2×C11—231no—
100C2×C11—2071no—
101C2×C11—2071no—
102C2×C32—231no—
103C3×D3—31no—
104C3×D3—691no—
105C2×C9—231no—
106C2×C9—461no—
107D9—231no—
108D9C3×AGL(1,𝔽23)11yescharacteristic, complemented
109C22×C3—2071no—
110C2×D3—691no—
111C11—231noSylow
112C32D3×AGL(1,𝔽23)11yescharacteristic
113C9—21no—
114C9C2×C3×AGL(1,𝔽23)11yescharacteristic, complemented
115C2×C3—91no—
116C2×C3—231no—
117C2×C3—691no—
118C2×C3—2071no—
119D3—31no—
120D3—691no—
121C22—2071noSylow
122C3—31no—
123C3C3×D3×AGL(1,𝔽23)11yescharacteristic
124C2—91no—
125C2—231no—
126C2—2071no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC9×C23C22×C3×C11
FrattiniC3C3×D3×AGL(1,𝔽23)
FittingC23×3−1+2C22×C11
socleC3×C23C2×C3×D3×C11
LCSG ▷ C9×C23
UCSC1

Complements

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

NH ≅ G/Ntype
1. #72275867C2semidirect
2. AGL(1,ℤ/9)×C23⋊C11C2semidirect
3. 3−1+2×AGL(1,𝔽23)C2semidirect
4. 3−1+2×C23⋊C11C22semidirect
5. D23×AGL(1,ℤ/9)C11semidirect
7. D9×AGL(1,𝔽23)C3semidirect
11. C9⋊AGL(1,𝔽23)C2×C3semidirect
12. D9×C23⋊C11C2×C3semidirect
14. C9×AGL(1,𝔽23)C2×C3semidirect
19. C9×C23⋊C11C22×C3semidirect
26. C23⋊AGL(1,ℤ/9)C2×C11semidirect
27. C23×AGL(1,ℤ/9)C2×C11semidirect
28. 3−1+2×D23C2×C11semidirect
32. D9×D23C3×C11semidirect
35. C23×3−1+2C22×C11semidirect
41. AGL(1,𝔽23)AGL(1,ℤ/9)direct (N × H)
45. D207C2×C3×C11semidirect
47. C9×D23C2×C3×C11semidirect
48. D9×C23C2×C3×C11semidirect
54. C23⋊C11C2×AGL(1,ℤ/9)semidirect
57. C9×C23C22×C3×C11semidirect
88. AGL(1,ℤ/9)AGL(1,𝔽23)direct (N × H)
91. D23C11×AGL(1,ℤ/9)semidirect
97. 3−1+2C2×AGL(1,𝔽23)semidirect
98. C23C2×C11×AGL(1,ℤ/9)semidirect
108. D9C3×AGL(1,𝔽23)semidirect
114. C9C2×C3×AGL(1,𝔽23)semidirect

Maximal subgroups

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

subgroupindex
#722758672
AGL(1,ℤ/9)×C23⋊C112
3−1+2×AGL(1,𝔽23)2
D23×AGL(1,ℤ/9)11
C3×D3×AGL(1,𝔽23)3
D9×AGL(1,𝔽23)3
C2×C11×AGL(1,ℤ/9)23