Finite Group Databasebeta

← C26×C3×C5×C27 #72203112

Subgroups — C26×C3×C5×C27 #72203112

Order 25920.

79100 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C25×C3×C5×C27C2163yesmaximal, direct product factor
2C26×C3×C5×C9C311yesmaximal, characteristic
3C26×C5×C27C313yesmaximal, direct product factor
4C24×C3×C5×C27C221651yesdirect product factor
5C26×C3×C27C511yesmaximal, characteristic, direct product factor, Hall
6C25×C3×C5×C9C2×C3163yes—
7C25×C5×C27C2×C31189yesdirect product factor
8C23×C3×C5×C27C2311395yesdirect product factor
9C26×C5×C9C912yes—
10C26×C5×C9C3211yescharacteristic
11C26×C32×C5C911yescharacteristic
12C25×C3×C27C2×C5163yesdirect product factor
13C24×C3×C5×C9C22×C31651yes—
14C24×C5×C27C22×C311953yesdirect product factor
15C26×C3×C9C3×C511yescharacteristic
16C26×C27C3×C513yesdirect product factor
17C22×C3×C5×C27C241651yesdirect product factor
18C25×C32×C5C2×C9163yes—
19C25×C5×C9C2×C91126yes—
20C25×C5×C9C2×C32163yes—
21C24×C3×C27C22×C51651yesdirect product factor
22C23×C3×C5×C9C23×C311395yes—
23C23×C5×C27C23×C314185yesdirect product factor
24C26×C3×C5C2713yesdirect product factor
25C26×C3×C5C3×C911yescharacteristic
26C25×C3×C9C2×C3×C5163yes—
27C25×C27C2×C3×C51189yesdirect product factor
28C2×C3×C5×C27C25163yesdirect product factor
29C24×C32×C5C22×C91651yes—
30C24×C5×C9C22×C911302yes—
31C24×C5×C9C22×C321651yes—
32C23×C3×C27C23×C511395yesdirect product factor
33C26×C32C5×C911yescharacteristic
34C26×C9C5×C912yes—
35C26×C9C32×C511yescharacteristic
36C22×C3×C5×C9C24×C31651yes—
37C22×C5×C27C24×C311953yesdirect product factor
38C25×C3×C5C2×C271189yesdirect product factor
39C25×C3×C5C2×C3×C9163yes—
40C24×C3×C9C22×C3×C51651yes—
41C24×C27C22×C3×C511953yesdirect product factor
42C3×C5×C27C2611yescharacteristic, direct product factor, Hall
43C23×C32×C5C23×C911395yes—
44C23×C5×C9C23×C912790yes—
45C23×C5×C9C23×C3211395yes—
46C22×C3×C27C24×C51651yesdirect product factor
47C26×C5C3×C2711yescharacteristic, direct product factor, Hall
48C25×C32C2×C5×C9163yes—
49C25×C9C2×C5×C91126yes—
50C25×C9C2×C32×C5163yes—
51C2×C3×C5×C9C25×C3163yes—
52C2×C5×C27C25×C31189yesdirect product factor
53C24×C3×C5C22×C2711953yesdirect product factor
54C24×C3×C5C22×C3×C91651yes—
55C23×C3×C9C23×C3×C511395yes—
56C23×C27C23×C3×C514185yesdirect product factor
57C26×C3C5×C2713yesdirect product factor
58C26×C3C3×C5×C911yescharacteristic
59C22×C32×C5C24×C91651yes—
60C22×C5×C9C24×C911302yes—
61C22×C5×C9C24×C321651yes—
62C2×C3×C27C25×C5163yesdirect product factor
63C25×C5C2×C3×C27163yesdirect product factor
64C24×C32C22×C5×C91651yes—
65C24×C9C22×C5×C911302yes—
66C24×C9C22×C32×C51651yes—
67C3×C5×C9C26×C311yescharacteristic
68C5×C27C26×C313yesdirect product factor
69C23×C3×C5C23×C2714185yesdirect product factor
70C23×C3×C5C23×C3×C911395yes—
71C22×C3×C9C24×C3×C51651yes—
72C22×C27C24×C3×C511953yesdirect product factor
73C25×C3C2×C5×C271189yesdirect product factor
74C25×C3C2×C3×C5×C9163yes—
75C2×C32×C5C25×C9163yes—
76C2×C5×C9C25×C91126yes—
77C2×C5×C9C25×C32163yes—
78C3×C27C26×C511yescharacteristic, direct product factor, Sylow
79C24×C5C22×C3×C271651yesdirect product factor
80C23×C32C23×C5×C911395yes—
81C23×C9C23×C5×C912790yes—
82C23×C9C23×C32×C511395yes—
83C26C3×C5×C2711yescharacteristic, direct product factor, Sylow
84C22×C3×C5C24×C2711953yesdirect product factor
85C22×C3×C5C24×C3×C91651yes—
86C2×C3×C9C25×C3×C5163yes—
87C2×C27C25×C3×C51189yesdirect product factor
88C24×C3C22×C5×C2711953yesdirect product factor
89C24×C3C22×C3×C5×C91651yes—
90C32×C5C26×C911yescharacteristic
91C5×C9C26×C912yes—
92C5×C9C26×C3211yescharacteristic
93C23×C5C23×C3×C2711395yesdirect product factor
94C22×C32C24×C5×C91651yes—
95C22×C9C24×C5×C911302yes—
96C22×C9C24×C32×C51651yes—
97C25C2×C3×C5×C27163yesdirect product factor
98C2×C3×C5C25×C271189yesdirect product factor
99C2×C3×C5C25×C3×C9163yes—
100C3×C9C26×C3×C511yescharacteristic
101C27C26×C3×C513yesdirect product factor
102C23×C3C23×C5×C2714185yesdirect product factor
103C23×C3C23×C3×C5×C911395yes—
104C22×C5C24×C3×C271651yesdirect product factor
105C2×C32C25×C5×C9163yes—
106C2×C9C25×C5×C91126yes—
107C2×C9C25×C32×C5163yes—
108C24C22×C3×C5×C271651yesdirect product factor
109C3×C5C26×C2713yesdirect product factor
110C3×C5C26×C3×C911yescharacteristic
111C22×C3C24×C5×C2711953yesdirect product factor
112C22×C3C24×C3×C5×C91651yes—
113C2×C5C25×C3×C27163yesdirect product factor
114C32C26×C5×C911yescharacteristic
115C9C26×C32×C511yescharacteristic
116C9C26×C5×C912yes—
117C23C23×C3×C5×C2711395yesdirect product factor
118C2×C3C25×C5×C271189yesdirect product factor
119C2×C3C25×C3×C5×C9163yes—
120C5C26×C3×C2711yescharacteristic, direct product factor, Sylow
121C22C24×C3×C5×C271651yesdirect product factor
122C3C26×C5×C2713yesdirect product factor
123C3C26×C3×C5×C911yescharacteristic
124C2C25×C3×C5×C27163yesdirect product factor
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC9C26×C32×C5
socle, prime radicalC26×C32×C5C9
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
1. C25×C3×C5×C27C2direct (N × H)
3. C26×C5×C27C3direct (N × H)
4. C24×C3×C5×C27C22direct (N × H)
5. C26×C3×C27C5direct (N × H)
7. C25×C5×C27C2×C3direct (N × H)
8. C23×C3×C5×C27C23direct (N × H)
12. C25×C3×C27C2×C5direct (N × H)
14. C24×C5×C27C22×C3direct (N × H)
16. C26×C27C3×C5direct (N × H)
17. C22×C3×C5×C27C24direct (N × H)
21. C24×C3×C27C22×C5direct (N × H)
23. C23×C5×C27C23×C3direct (N × H)
24. C26×C3×C5C27direct (N × H)
27. C25×C27C2×C3×C5direct (N × H)
28. C2×C3×C5×C27C25direct (N × H)
32. C23×C3×C27C23×C5direct (N × H)
37. C22×C5×C27C24×C3direct (N × H)
38. C25×C3×C5C2×C27direct (N × H)
41. C24×C27C22×C3×C5direct (N × H)
42. C3×C5×C27C26direct (N × H)
46. C22×C3×C27C24×C5direct (N × H)
47. C26×C5C3×C27direct (N × H)
52. C2×C5×C27C25×C3direct (N × H)
53. C24×C3×C5C22×C27direct (N × H)
56. C23×C27C23×C3×C5direct (N × H)
57. C26×C3C5×C27direct (N × H)
62. C2×C3×C27C25×C5direct (N × H)
63. C25×C5C2×C3×C27direct (N × H)
68. C5×C27C26×C3direct (N × H)
69. C23×C3×C5C23×C27direct (N × H)
72. C22×C27C24×C3×C5direct (N × H)
73. C25×C3C2×C5×C27direct (N × H)
78. C3×C27C26×C5direct (N × H)
79. C24×C5C22×C3×C27direct (N × H)
83. C26C3×C5×C27direct (N × H)
84. C22×C3×C5C24×C27direct (N × H)
87. C2×C27C25×C3×C5direct (N × H)
88. C24×C3C22×C5×C27direct (N × H)
93. C23×C5C23×C3×C27direct (N × H)
97. C25C2×C3×C5×C27direct (N × H)
98. C2×C3×C5C25×C27direct (N × H)
101. C27C26×C3×C5direct (N × H)
102. C23×C3C23×C5×C27direct (N × H)
104. C22×C5C24×C3×C27direct (N × H)
108. C24C22×C3×C5×C27direct (N × H)
109. C3×C5C26×C27direct (N × H)
111. C22×C3C24×C5×C27direct (N × H)
113. C2×C5C25×C3×C27direct (N × H)
117. C23C23×C3×C5×C27direct (N × H)
118. C2×C3C25×C5×C27direct (N × H)
120. C5C26×C3×C27direct (N × H)
121. C22C24×C3×C5×C27direct (N × H)
122. C3C26×C5×C27direct (N × H)
124. C2C25×C3×C5×C27direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C3×C5×C272
C26×C3×C5×C93
C26×C5×C273
C26×C3×C275