Finite Group Databasebeta

← C23×C4×C5×C9×C17 #72201752

Subgroups — C23×C4×C5×C9×C17 #72201752

Order 24480.

1416 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C5×C9×C17C211yesmaximal, characteristic
2C22×C4×C5×C9×C17C2114yesmaximal, direct product factor
3C23×C3×C4×C5×C17C311yesmaximal, characteristic
4C23×C5×C9×C17C418yesdirect product factor
5C23×C5×C9×C17C2217yes—
6C2×C4×C5×C9×C17C22128yesdirect product factor
7C23×C4×C9×C17C511yesmaximal, characteristic, direct product factor, Hall
8C24×C3×C5×C17C2×C311yescharacteristic
9C22×C3×C4×C5×C17C2×C3114yes—
10C23×C4×C5×C17C911yescharacteristic, direct product factor, Hall
11C24×C9×C17C2×C511yescharacteristic
12C22×C4×C9×C17C2×C5114yesdirect product factor
13C23×C3×C5×C17C3×C418yes—
14C23×C3×C5×C17C22×C317yes—
15C2×C3×C4×C5×C17C22×C3128yes—
16C22×C5×C9×C17C2×C4128yesdirect product factor
17C22×C5×C9×C17C2317yes—
18C4×C5×C9×C17C2318yesdirect product factor
19C23×C3×C4×C17C3×C511yescharacteristic
20C2×C5×C9×C17C22×C4114yesdirect product factor
21C2×C5×C9×C17C2411yescharacteristic
22C23×C4×C5×C9C1711yesmaximal, characteristic, direct product factor, Hall
23C24×C5×C17C2×C911yescharacteristic
24C22×C4×C5×C17C2×C9114yesdirect product factor
25C23×C9×C17C4×C518yesdirect product factor
26C23×C9×C17C22×C517yes—
27C2×C4×C9×C17C22×C5128yesdirect product factor
28C22×C3×C5×C17C2×C3×C4128yes—
29C22×C3×C5×C17C23×C317yes—
30C3×C4×C5×C17C23×C318yes—
31C24×C3×C17C2×C3×C511yescharacteristic
32C22×C3×C4×C17C2×C3×C5114yes—
33C5×C9×C17C23×C411yescharacteristic, direct product factor, Hall
34C24×C5×C9C2×C1711yescharacteristic
35C22×C4×C5×C9C2×C17114yesdirect product factor
36C23×C5×C17C4×C918yesdirect product factor
37C23×C5×C17C22×C917yes—
38C2×C4×C5×C17C22×C9128yesdirect product factor
39C22×C9×C17C2×C4×C5128yesdirect product factor
40C22×C9×C17C23×C517yes—
41C4×C9×C17C23×C518yesdirect product factor
42C23×C4×C17C5×C911yescharacteristic, direct product factor, Hall
43C2×C3×C5×C17C22×C3×C4114yes—
44C2×C3×C5×C17C24×C311yescharacteristic
45C23×C3×C4×C5C3×C1711yescharacteristic
46C23×C3×C17C3×C4×C518yes—
47C23×C3×C17C22×C3×C517yes—
48C2×C3×C4×C17C22×C3×C5128yes—
49C23×C5×C9C4×C1718yesdirect product factor
50C23×C5×C9C22×C1717yes—
51C2×C4×C5×C9C22×C17128yesdirect product factor
52C22×C5×C17C2×C4×C9128yesdirect product factor
53C22×C5×C17C23×C917yes—
54C4×C5×C17C23×C918yesdirect product factor
55C2×C9×C17C22×C4×C5114yesdirect product factor
56C2×C9×C17C24×C511yescharacteristic
57C23×C4×C9C5×C1711yescharacteristic, direct product factor, Hall
58C24×C17C2×C5×C911yescharacteristic
59C22×C4×C17C2×C5×C9114yesdirect product factor
60C3×C5×C17C23×C3×C411yescharacteristic
61C24×C3×C5C2×C3×C1711yescharacteristic
62C22×C3×C4×C5C2×C3×C17114yes—
63C22×C3×C17C2×C3×C4×C5128yes—
64C22×C3×C17C23×C3×C517yes—
65C3×C4×C17C23×C3×C518yes—
66C22×C5×C9C2×C4×C17128yesdirect product factor
67C22×C5×C9C23×C1717yes—
68C4×C5×C9C23×C1718yesdirect product factor
69C2×C5×C17C22×C4×C9114yesdirect product factor
70C2×C5×C17C24×C911yescharacteristic
71C23×C4×C5C9×C1711yescharacteristic, direct product factor, Hall
72C9×C17C23×C4×C511yescharacteristic, direct product factor, Hall
73C24×C9C2×C5×C1711yescharacteristic
74C22×C4×C9C2×C5×C17114yesdirect product factor
75C23×C17C4×C5×C918yesdirect product factor
76C23×C17C22×C5×C917yes—
77C2×C4×C17C22×C5×C9128yesdirect product factor
78C23×C3×C5C3×C4×C1718yes—
79C23×C3×C5C22×C3×C1717yes—
80C2×C3×C4×C5C22×C3×C17128yes—
81C2×C3×C17C22×C3×C4×C5114yes—
82C2×C3×C17C24×C3×C511yescharacteristic
83C23×C3×C4C3×C5×C1711yescharacteristic
84C2×C5×C9C22×C4×C17114yesdirect product factor
85C2×C5×C9C24×C1711yescharacteristic
86C5×C17C23×C4×C911yescharacteristic, direct product factor, Hall
87C24×C5C2×C9×C1711yescharacteristic
88C22×C4×C5C2×C9×C17114yesdirect product factor
89C23×C9C4×C5×C1718yesdirect product factor
90C23×C9C22×C5×C1717yes—
91C2×C4×C9C22×C5×C17128yesdirect product factor
92C22×C17C2×C4×C5×C9128yesdirect product factor
93C22×C17C23×C5×C917yes—
94C4×C17C23×C5×C918yesdirect product factor
95C22×C3×C5C2×C3×C4×C17128yes—
96C22×C3×C5C23×C3×C1717yes—
97C3×C4×C5C23×C3×C1718yes—
98C3×C17C23×C3×C4×C511yescharacteristic
99C24×C3C2×C3×C5×C1711yescharacteristic
100C22×C3×C4C2×C3×C5×C17114yes—
101C5×C9C23×C4×C1711yescharacteristic, direct product factor, Hall
102C23×C5C4×C9×C1718yesdirect product factor
103C23×C5C22×C9×C1717yes—
104C2×C4×C5C22×C9×C17128yesdirect product factor
105C22×C9C2×C4×C5×C17128yesdirect product factor
106C22×C9C23×C5×C1717yes—
107C4×C9C23×C5×C1718yesdirect product factor
108C2×C17C22×C4×C5×C9114yesdirect product factor
109C2×C17C24×C5×C911yescharacteristic
110C23×C4C5×C9×C1711yescharacteristic, direct product factor, Sylow
111C2×C3×C5C22×C3×C4×C17114yes—
112C2×C3×C5C24×C3×C1711yescharacteristic
113C23×C3C3×C4×C5×C1718yes—
114C23×C3C22×C3×C5×C1717yes—
115C2×C3×C4C22×C3×C5×C17128yes—
116C22×C5C2×C4×C9×C17128yesdirect product factor
117C22×C5C23×C9×C1717yes—
118C4×C5C23×C9×C1718yesdirect product factor
119C2×C9C22×C4×C5×C17114yesdirect product factor
120C2×C9C24×C5×C1711yescharacteristic
121C17C23×C4×C5×C911yescharacteristic, direct product factor, Sylow
122C24C2×C5×C9×C1711yescharacteristic
123C22×C4C2×C5×C9×C17114yesdirect product factor
124C3×C5C23×C3×C4×C1711yescharacteristic
125C22×C3C2×C3×C4×C5×C17128yes—
126C22×C3C23×C3×C5×C1717yes—
127C3×C4C23×C3×C5×C1718yes—
128C2×C5C22×C4×C9×C17114yesdirect product factor
129C2×C5C24×C9×C1711yescharacteristic
130C9C23×C4×C5×C1711yescharacteristic, direct product factor, Sylow
131C23C4×C5×C9×C1718yesdirect product factor
132C23C22×C5×C9×C1717yes—
133C2×C4C22×C5×C9×C17128yesdirect product factor
134C2×C3C22×C3×C4×C5×C17114yes—
135C2×C3C24×C3×C5×C1711yescharacteristic
136C5C23×C4×C9×C1711yescharacteristic, direct product factor, Sylow
137C22C2×C4×C5×C9×C17128yesdirect product factor
138C22C23×C5×C9×C1717yes—
139C4C23×C5×C9×C1718yesdirect product factor
140C3C23×C3×C4×C5×C1711yescharacteristic
141C2C22×C4×C5×C9×C17114yesdirect product factor
142C2C24×C5×C9×C1711yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2×C3C24×C3×C5×C17
socle, prime radicalC24×C3×C5×C17C2×C3
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C4×C5×C9×C17C2direct (N × H)
4. C23×C5×C9×C17C4direct (N × H)
6. C2×C4×C5×C9×C17C22direct (N × H)
7. C23×C4×C9×C17C5direct (N × H)
10. C23×C4×C5×C17C9direct (N × H)
12. C22×C4×C9×C17C2×C5direct (N × H)
16. C22×C5×C9×C17C2×C4direct (N × H)
18. C4×C5×C9×C17C23direct (N × H)
20. C2×C5×C9×C17C22×C4direct (N × H)
22. C23×C4×C5×C9C17direct (N × H)
24. C22×C4×C5×C17C2×C9direct (N × H)
25. C23×C9×C17C4×C5direct (N × H)
27. C2×C4×C9×C17C22×C5direct (N × H)
33. C5×C9×C17C23×C4direct (N × H)
35. C22×C4×C5×C9C2×C17direct (N × H)
36. C23×C5×C17C4×C9direct (N × H)
38. C2×C4×C5×C17C22×C9direct (N × H)
39. C22×C9×C17C2×C4×C5direct (N × H)
41. C4×C9×C17C23×C5direct (N × H)
42. C23×C4×C17C5×C9direct (N × H)
49. C23×C5×C9C4×C17direct (N × H)
51. C2×C4×C5×C9C22×C17direct (N × H)
52. C22×C5×C17C2×C4×C9direct (N × H)
54. C4×C5×C17C23×C9direct (N × H)
55. C2×C9×C17C22×C4×C5direct (N × H)
57. C23×C4×C9C5×C17direct (N × H)
59. C22×C4×C17C2×C5×C9direct (N × H)
66. C22×C5×C9C2×C4×C17direct (N × H)
68. C4×C5×C9C23×C17direct (N × H)
69. C2×C5×C17C22×C4×C9direct (N × H)
71. C23×C4×C5C9×C17direct (N × H)
72. C9×C17C23×C4×C5direct (N × H)
74. C22×C4×C9C2×C5×C17direct (N × H)
75. C23×C17C4×C5×C9direct (N × H)
77. C2×C4×C17C22×C5×C9direct (N × H)
84. C2×C5×C9C22×C4×C17direct (N × H)
86. C5×C17C23×C4×C9direct (N × H)
88. C22×C4×C5C2×C9×C17direct (N × H)
89. C23×C9C4×C5×C17direct (N × H)
91. C2×C4×C9C22×C5×C17direct (N × H)
92. C22×C17C2×C4×C5×C9direct (N × H)
94. C4×C17C23×C5×C9direct (N × H)
101. C5×C9C23×C4×C17direct (N × H)
102. C23×C5C4×C9×C17direct (N × H)
104. C2×C4×C5C22×C9×C17direct (N × H)
105. C22×C9C2×C4×C5×C17direct (N × H)
107. C4×C9C23×C5×C17direct (N × H)
108. C2×C17C22×C4×C5×C9direct (N × H)
110. C23×C4C5×C9×C17direct (N × H)
116. C22×C5C2×C4×C9×C17direct (N × H)
118. C4×C5C23×C9×C17direct (N × H)
119. C2×C9C22×C4×C5×C17direct (N × H)
121. C17C23×C4×C5×C9direct (N × H)
123. C22×C4C2×C5×C9×C17direct (N × H)
128. C2×C5C22×C4×C9×C17direct (N × H)
130. C9C23×C4×C5×C17direct (N × H)
131. C23C4×C5×C9×C17direct (N × H)
133. C2×C4C22×C5×C9×C17direct (N × H)
136. C5C23×C4×C9×C17direct (N × H)
137. C22C2×C4×C5×C9×C17direct (N × H)
139. C4C23×C5×C9×C17direct (N × H)
141. C2C22×C4×C5×C9×C17direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C5×C9×C172
C22×C4×C5×C9×C172
C23×C3×C4×C5×C173
C23×C4×C9×C175
C23×C4×C5×C917