Finite Group Databasebeta

← C2×C3×C5×AGL(1,𝔽9) #72216990

Subgroups — C2×C3×C5×AGL(1,𝔽9) #72216990

Order 2160.

116 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 116 conjugacy classes of subgroups — too many to draw. Only its 52 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
1C2×C3×C5×#157C211yesmaximal, characteristic
2C3×C5×AGL(1,𝔽9)C212yesmaximal, direct product factor
3C2×C5×AGL(1,𝔽9)C311yesmaximal, characteristic, direct product factor
4C2×C3×C5×C3⋊D3C411yescharacteristic
5C3×C5×#157C411yescharacteristic
6C3×C5×#157C2211yescharacteristic
7C2×C3×AGL(1,𝔽9)C511yesmaximal, characteristic, direct product factor, Hall
8C2×C5×#157C2×C311yescharacteristic
9C5×AGL(1,𝔽9)C2×C312yesdirect product factor
10C2×C33×C5C811yescharacteristic, complemented
11C3×C5×C3⋊D3C811yescharacteristic, complemented
12C3×C5×C3⋊D3C2×C411yescharacteristic
13C2×C3×C5×C8—91nomaximal
14C2×C3×#157C2×C511yescharacteristic
15C3×AGL(1,𝔽9)C2×C512yesdirect product factor
16C2×C5×C3⋊D3C3×C411yescharacteristic
17C2×C3×C5×D3—121no—
18C5×#157C3×C411yescharacteristic
19C5×#157C22×C311yescharacteristic
20C2×AGL(1,𝔽9)C3×C511yescharacteristic, direct product factor
21C33×C5C2×C811yescharacteristic, complemented, Hall
22C2×C3×C4×C5—91no—
23C3×C5×C8—92no—
24C2×C3×C3⋊D3C4×C511yescharacteristic
25C3×#157C4×C511yescharacteristic
26C3×#157C22×C511yescharacteristic
27C2×C32×C5—41no—
28C2×C32×C5—81no—
29C2×C32×C5C3×C811yescharacteristic, complemented
30C5×C3⋊D3C3×C811yescharacteristic, complemented
31C5×C3⋊D3C2×C3×C411yescharacteristic
32C3×C5×D3—121no—
33C3×C5×D3—121no—
34C2×C5×C8—91noHall
35C2×#157C2×C3×C511yescharacteristic
36AGL(1,𝔽9)C2×C3×C512yesdirect product factor
37C22×C3×C5—91no—
38C2×C5×D3—121no—
39C3×C4×C5—91no—
40C3×C4×C5—91no—
41C2×C33C5×C811yescharacteristic, complemented
42C3×C3⋊D3C5×C811yescharacteristic, complemented
43C3×C3⋊D3C2×C4×C511yescharacteristic
44C2×C3×C8—91no—
45C32×C5—41no—
46C32×C5—81no—
47C32×C5C2×C3×C811yescharacteristic, complemented
48C2×C4×C5—91no—
49C5×C8—92no—
50C2×C3⋊D3C3×C4×C511yescharacteristic
51C2×C3×D3—121no—
52#157C3×C4×C511yescharacteristic
53#157C22×C3×C511yescharacteristic
54C2×C3×C5—41no—
55C2×C3×C5—81no—
56C2×C3×C5—91no—
57C2×C3×C5—91no—
58C2×C3×C5AGL(1,𝔽9)11yescharacteristic, direct product factor
59C5×D3—121no—
60C5×D3—121no—
61C33C2×C5×C811yescharacteristic, complemented, Sylow
62C2×C3×C4—91no—
63C3×C8—92no—
64C22×C5—91no—
65C4×C5—91no—
66C4×C5—91no—
67C2×C32—41no—
68C2×C32—81no—
69C2×C32C3×C5×C811yescharacteristic, complemented
70C3⋊D3C3×C5×C811yescharacteristic, complemented
71C3⋊D3C2×C3×C4×C511yescharacteristic
72C3×D3—121no—
73C3×D3—121no—
74C2×C8—91noSylow
75C3×C5—41no—
76C3×C5—81no—
77C3×C5C2×AGL(1,𝔽9)11yescharacteristic, direct product factor
78C22×C3—91no—
79C2×D3—121no—
80C3×C4—91no—
81C3×C4—91no—
82C2×C5—91no—
83C2×C5—91no—
84C2×C5C3×AGL(1,𝔽9)11yescharacteristic, direct product factor
85C32—41no—
86C32—81no—
87C32C2×C3×C5×C811yescharacteristic, complemented
88C2×C4—91no—
89C8—92no—
90C2×C3—41no—
91C2×C3—81no—
92C2×C3—91no—
93C2×C3—91no—
94C2×C3C5×AGL(1,𝔽9)11yescharacteristic, direct product factor
95D3—121no—
96D3—121no—
97C5C2×C3×AGL(1,𝔽9)11yescharacteristic, direct product factor, Sylow
98C22—91no—
99C4—91no—
100C4—91no—
101C3—41no—
102C3—81no—
103C3C2×C5×AGL(1,𝔽9)11yescharacteristic, direct product factor
104C2—91no—
105C2—91no—
106C2C3×C5×AGL(1,𝔽9)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5AGL(1,𝔽9)
commutator, hypocenterC32C2×C3×C5×C8
FrattiniC1G
Fitting, socleC2×C33×C5C8
prime radicalC2×C3×C5×C3⋊D3C4
cosocle#157C22×C3×C5
LCSG ▷ C32
UCSC1 ◁ C2×C3×C5

Complements

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

NH ≅ G/Ntype
2. C3×C5×AGL(1,𝔽9)C2direct (N × H)
3. C2×C5×AGL(1,𝔽9)C3direct (N × H)
7. C2×C3×AGL(1,𝔽9)C5direct (N × H)
9. C5×AGL(1,𝔽9)C2×C3direct (N × H)
10. C2×C33×C5C8semidirect
11. C3×C5×C3⋊D3C8semidirect
15. C3×AGL(1,𝔽9)C2×C5direct (N × H)
20. C2×AGL(1,𝔽9)C3×C5direct (N × H)
21. C33×C5C2×C8semidirect
29. C2×C32×C5C3×C8semidirect
30. C5×C3⋊D3C3×C8semidirect
36. AGL(1,𝔽9)C2×C3×C5direct (N × H)
41. C2×C33C5×C8semidirect
42. C3×C3⋊D3C5×C8semidirect
47. C32×C5C2×C3×C8semidirect
58. C2×C3×C5AGL(1,𝔽9)direct (N × H)
61. C33C2×C5×C8semidirect
69. C2×C32C3×C5×C8semidirect
70. C3⋊D3C3×C5×C8semidirect
77. C3×C5C2×AGL(1,𝔽9)direct (N × H)
84. C2×C5C3×AGL(1,𝔽9)direct (N × H)
87. C32C2×C3×C5×C8semidirect
94. C2×C3C5×AGL(1,𝔽9)direct (N × H)
97. C5C2×C3×AGL(1,𝔽9)direct (N × H)
103. C3C2×C5×AGL(1,𝔽9)direct (N × H)
106. C2C3×C5×AGL(1,𝔽9)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1572
C3×C5×AGL(1,𝔽9)2
C2×C5×AGL(1,𝔽9)3
C2×C3×AGL(1,𝔽9)5
C2×C3×C5×C89