Finite Group Databasebeta

← C22×C53×#585 #27270612

Subgroups — C22×C53×#585 #27270612

Order 13568.

60366 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
1C2×C53×#585C2196yesmaximal, direct product factor
2C22×2−1+4×C53C216yesmaximal, complemented
3C22×H(5,𝔽2)×C53C2110yesmaximal, complemented
4C23×#41×C53C2115yesmaximal, complemented
5C53×#585C2211024yesdirect product factor
6C2×2−1+4×C53C221288yescomplemented
7C2×H(5,𝔽2)×C53C221480yescomplemented
8C22×#41×C53C22120yescomplemented
9C22×#41×C53C22160yescomplemented
10C22×#41×C53C221720yescomplemented
11C23×Q8×C53C22115yescomplemented
12C23×D4×C53C22145yescomplemented
13C24×C4×C53C22115yescomplemented
14C25×C53C23115yes—
152−1+4×C53C2311536yescomplemented
16H(5,𝔽2)×C53C2312560yescomplemented
17C2×#41×C53C231480yescomplemented
18C2×#41×C53C2311440yescomplemented
19C2×#41×C53C2313840yescomplemented
20C22×Q8×C53C23120yes—
21C22×Q8×C53C231360yescomplemented
22C22×D4×C53C23160yes—
23C22×D4×C53C2311080yescomplemented
24C23×C4×C53C23145yes—
25C23×C4×C53C23115yes—
26C23×C4×C53C231360yescomplemented
27C24×C53—460no—
28C24×C53C24115yes—
29C24×C53C241180yes—
30#41×C53C2411280yescomplemented
31#41×C53C2413840yescomplemented
32C2×Q8×C53C241240yes—
33C2×Q8×C53C241960yescomplemented
34C2×D4×C53C241720yes—
35C2×D4×C53C2412880yescomplemented
36C22×C4×C53C24115yes—
37C22×C4×C53C241180yes—
38C22×C4×C53C241540yes—
39C22×C4×C53C2411yescharacteristic
40C22×C4×C53C241960yescomplemented
41C23×C53—260no—
42C23×C53—4360no—
43C23×C53C25190yes—
44C23×C53C251240yes—
45C23×C53C2511yescharacteristic
46Q8×C53C251320yes—
47D4×C53C251960yes—
48C2×C4×C53C2516yes—
49C2×C4×C53C25190yes—
50C2×C4×C53C251240yes—
51C2×C4×C53C251720yes—
52C22×#585C5311yesmaximal, characteristic, direct product factor, Sylow
53C22×C53—2180no—
54C22×C53—4240no—
55C22×C53#58514yesdirect product factor
56C22×C53C26160yes—
57C22×C53C2613yes—
58C4×C53C2614yes—
59C4×C53C26160yes—
60C2×#585C2×C53196yesdirect product factor
61C22×2−1+4C2×C5316yescomplemented
62C22×H(5,𝔽2)C2×C53110yescomplemented
63C23×#41C2×C53115yescomplemented
64C2×C53—260no—
65C2×C53C2×#58516yesdirect product factor
66C2×C53C2711yescharacteristic
67#585C22×C5311024yesdirect product factor
68C2×2−1+4C22×C531288yescomplemented
69C2×H(5,𝔽2)C22×C531480yescomplemented
70C22×#41C22×C53120yescomplemented
71C22×#41C22×C53160yescomplemented
72C22×#41C22×C531720yescomplemented
73C23×Q8C22×C53115yescomplemented
74C23×D4C22×C53145yescomplemented
75C24×C4C22×C53115yescomplemented
76C53C22×#58511yescharacteristic, direct product factor, Sylow
77C25C23×C53115yes—
782−1+4C23×C5311536yescomplemented
79H(5,𝔽2)C23×C5312560yescomplemented
80C2×#41C23×C531480yescomplemented
81C2×#41C23×C5311440yescomplemented
82C2×#41C23×C5313840yescomplemented
83C22×Q8C23×C53120yes—
84C22×Q8C23×C531360yescomplemented
85C22×D4C23×C53160yes—
86C22×D4C23×C5311080yescomplemented
87C23×C4C23×C53115yes—
88C23×C4C23×C53145yes—
89C23×C4C23×C531360yescomplemented
90C24—460no—
91C24C24×C53115yes—
92C24C24×C531180yes—
93#41C24×C5311280yescomplemented
94#41C24×C5313840yescomplemented
95C2×Q8C24×C531240yes—
96C2×Q8C24×C531960yescomplemented
97C2×D4C24×C531720yes—
98C2×D4C24×C5312880yescomplemented
99C22×C4C24×C53115yes—
100C22×C4C24×C531180yes—
101C22×C4C24×C531540yes—
102C22×C4C24×C5311yescharacteristic
103C22×C4C24×C531960yescomplemented
104C23—260no—
105C23—4360no—
106C23C25×C53190yes—
107C23C25×C531240yes—
108C23C25×C5311yescharacteristic
109Q8C25×C531320yes—
110D4C25×C531960yes—
111C2×C4C25×C5316yes—
112C2×C4C25×C53190yes—
113C2×C4C25×C531240yes—
114C2×C4C25×C531720yes—
115C22—2180no—
116C22—4240no—
117C22C53×#58514yesdirect product factor
118C22C26×C53160yes—
119C22C26×C5313yes—
120C4C26×C5314yes—
121C4C26×C53160yes—
122C2—260no—
123C2C2×C53×#58516yesdirect product factor
124C2C27×C5311yescharacteristic
subgroupG/H
centerC22×C4×C53C24
commutator, Frattini, cosocleC2C27×C53
socleC23×C53C25
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C53×#585C2direct (N × H)
2. C22×2−1+4×C53C2semidirect
3. C22×H(5,𝔽2)×C53C2semidirect
4. C23×#41×C53C2semidirect
5. C53×#585C22direct (N × H)
6. C2×2−1+4×C53C22semidirect
7. C2×H(5,𝔽2)×C53C22semidirect
8. C22×#41×C53C22semidirect
9. C22×#41×C53C22semidirect
10. C22×#41×C53C22semidirect
11. C23×Q8×C53C22semidirect
12. C23×D4×C53C22semidirect
13. C24×C4×C53C22semidirect
15. 2−1+4×C53C23semidirect
16. H(5,𝔽2)×C53C23semidirect
17. C2×#41×C53C23semidirect
18. C2×#41×C53C23semidirect
19. C2×#41×C53C23semidirect
21. C22×Q8×C53C23semidirect
23. C22×D4×C53C23semidirect
26. C23×C4×C53C23semidirect
30. #41×C53C24semidirect
31. #41×C53C24semidirect
33. C2×Q8×C53C24semidirect
35. C2×D4×C53C24semidirect
40. C22×C4×C53C24semidirect
52. C22×#585C53direct (N × H)
55. C22×C53#585direct (N × H)
60. C2×#585C2×C53direct (N × H)
61. C22×2−1+4C2×C53semidirect
62. C22×H(5,𝔽2)C2×C53semidirect
63. C23×#41C2×C53semidirect
65. C2×C53C2×#585direct (N × H)
67. #585C22×C53direct (N × H)
68. C2×2−1+4C22×C53semidirect
69. C2×H(5,𝔽2)C22×C53semidirect
70. C22×#41C22×C53semidirect
71. C22×#41C22×C53semidirect
72. C22×#41C22×C53semidirect
73. C23×Q8C22×C53semidirect
74. C23×D4C22×C53semidirect
75. C24×C4C22×C53semidirect
76. C53C22×#585direct (N × H)
78. 2−1+4C23×C53semidirect
79. H(5,𝔽2)C23×C53semidirect
80. C2×#41C23×C53semidirect
81. C2×#41C23×C53semidirect
82. C2×#41C23×C53semidirect
84. C22×Q8C23×C53semidirect
86. C22×D4C23×C53semidirect
89. C23×C4C23×C53semidirect
93. #41C24×C53semidirect
94. #41C24×C53semidirect
96. C2×Q8C24×C53semidirect
98. C2×D4C24×C53semidirect
103. C22×C4C24×C53semidirect
117. C22C53×#585direct (N × H)
123. C2C2×C53×#585direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#5852
C22×2−1+4×C532
C22×H(5,𝔽2)×C532
C23×#41×C532
C22×#58553