Finite Group Databasebeta

← C23×H(5,𝔽2)×C53 #27270610

Subgroups — C23×H(5,𝔽2)×C53 #27270610

Order 13568.

65726 conjugacy classes of subgroups; only the 58526 normal subgroups are shown.

normal subgroups only

This group has 65726 conjugacy classes of subgroups — too many to draw. Only its 58526 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
1C22×H(5,𝔽2)×C53C21112yesmaximal, direct product factor
2C23×#41×C53C216yesmaximal, complemented
3C24×D4×C53C219yesmaximal, complemented
4C26×C53C2216yescomplemented
5C2×H(5,𝔽2)×C53C2211792yesdirect product factor
6C22×#41×C53C221336yescomplemented
7C23×Q8×C53C2212yescomplemented
8C23×D4×C53C22118yescomplemented
9C23×D4×C53C221504yescomplemented
10C24×C4×C53C2219yescomplemented
11C25×C53—448no—
12C25×C53C2319yes—
13C25×C53C231168yescomplemented
14H(5,𝔽2)×C53C2314096yesdirect product factor
15C2×#41×C53C2312688yescomplemented
16C22×Q8×C53C23156yescomplemented
17C22×D4×C53C231504yescomplemented
18C22×D4×C53C2314032yescomplemented
19C23×C4×C53C2316yes—
20C23×C4×C53C231252yescomplemented
21C24×C53—272no—
22C24×C53—4672no—
23C24×C53C241126yes—
24C24×C53C2411yescharacteristic
25C24×C53C241672yescomplemented
26#41×C53C2413072yescomplemented
27C2×Q8×C53C241224yescomplemented
28C2×D4×C53C2412016yescomplemented
29C2×D4×C53C2414608yescomplemented
30C22×C4×C53C24184yes—
31C22×C4×C53C2411008yescomplemented
32C23×C53—2504no—
33C23×C53—41344no—
34C23×C53H(5,𝔽2)18yesdirect product factor
35C23×C53C251252yes—
36C23×C53C2517yes—
37C23×C53C251384yescomplemented
38Q8×C53C251128yescomplemented
39D4×C53C2511152yescomplemented
40C2×C4×C53C251168yes—
41C2×C4×C53C251576yescomplemented
42C23×H(5,𝔽2)C5311yesmaximal, characteristic, direct product factor, Sylow
43C22×C53—2504no—
44C22×C53—4384no—
45C22×C53C2×H(5,𝔽2)128yesdirect product factor
46C22×C53C26172yes—
47C22×C53C2617yes—
48C4×C53C26148yes—
49C22×H(5,𝔽2)C2×C531112yesdirect product factor
50C23×#41C2×C5316yescomplemented
51C24×D4C2×C5319yescomplemented
52C2×C53—272no—
53C2×C53C22×H(5,𝔽2)114yesdirect product factor
54C2×C53C2711yescharacteristic
55C26C22×C5316yescomplemented
56C2×H(5,𝔽2)C22×C5311792yesdirect product factor
57C22×#41C22×C531336yescomplemented
58C23×Q8C22×C5312yescomplemented
59C23×D4C22×C53118yescomplemented
60C23×D4C22×C531504yescomplemented
61C24×C4C22×C5319yescomplemented
62C53C23×H(5,𝔽2)11yescharacteristic, direct product factor, Sylow
63C25—448no—
64C25C23×C5319yes—
65C25C23×C531168yescomplemented
66H(5,𝔽2)C23×C5314096yesdirect product factor
67C2×#41C23×C5312688yescomplemented
68C22×Q8C23×C53156yescomplemented
69C22×D4C23×C531504yescomplemented
70C22×D4C23×C5314032yescomplemented
71C23×C4C23×C5316yes—
72C23×C4C23×C531252yescomplemented
73C24—272no—
74C24—4672no—
75C24C24×C531126yes—
76C24C24×C5311yescharacteristic
77C24C24×C531672yescomplemented
78#41C24×C5313072yescomplemented
79C2×Q8C24×C531224yescomplemented
80C2×D4C24×C5312016yescomplemented
81C2×D4C24×C5314608yescomplemented
82C22×C4C24×C53184yes—
83C22×C4C24×C5311008yescomplemented
84C23—2504no—
85C23—41344no—
86C23H(5,𝔽2)×C5318yesdirect product factor
87C23C25×C531252yes—
88C23C25×C5317yes—
89C23C25×C531384yescomplemented
90Q8C25×C531128yescomplemented
91D4C25×C5311152yescomplemented
92C2×C4C25×C531168yes—
93C2×C4C25×C531576yescomplemented
94C22—2504no—
95C22—4384no—
96C22C2×H(5,𝔽2)×C53128yesdirect product factor
97C22C26×C53172yes—
98C22C26×C5317yes—
99C4C26×C53148yes—
100C2—272no—
101C2C22×H(5,𝔽2)×C53114yesdirect product factor
102C2C27×C5311yescharacteristic
subgroupG/H
center, socleC24×C53C24
commutator, Frattini, cosocleC2C27×C53
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C24×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×H(5,𝔽2)×C53C2direct (N × H)
2. C23×#41×C53C2semidirect
3. C24×D4×C53C2semidirect
4. C26×C53C22semidirect
5. C2×H(5,𝔽2)×C53C22direct (N × H)
6. C22×#41×C53C22semidirect
7. C23×Q8×C53C22semidirect
8. C23×D4×C53C22semidirect
9. C23×D4×C53C22semidirect
10. C24×C4×C53C22semidirect
13. C25×C53C23semidirect
14. H(5,𝔽2)×C53C23direct (N × H)
15. C2×#41×C53C23semidirect
16. C22×Q8×C53C23semidirect
17. C22×D4×C53C23semidirect
18. C22×D4×C53C23semidirect
20. C23×C4×C53C23semidirect
25. C24×C53C24semidirect
26. #41×C53C24semidirect
27. C2×Q8×C53C24semidirect
28. C2×D4×C53C24semidirect
29. C2×D4×C53C24semidirect
31. C22×C4×C53C24semidirect
34. C23×C53H(5,𝔽2)direct (N × H)
37. C23×C53C25semidirect
38. Q8×C53C25semidirect
39. D4×C53C25semidirect
41. C2×C4×C53C25semidirect
42. C23×H(5,𝔽2)C53direct (N × H)
45. C22×C53C2×H(5,𝔽2)direct (N × H)
49. C22×H(5,𝔽2)C2×C53direct (N × H)
50. C23×#41C2×C53semidirect
51. C24×D4C2×C53semidirect
53. C2×C53C22×H(5,𝔽2)direct (N × H)
55. C26C22×C53semidirect
56. C2×H(5,𝔽2)C22×C53direct (N × H)
57. C22×#41C22×C53semidirect
58. C23×Q8C22×C53semidirect
59. C23×D4C22×C53semidirect
60. C23×D4C22×C53semidirect
61. C24×C4C22×C53semidirect
62. C53C23×H(5,𝔽2)direct (N × H)
65. C25C23×C53semidirect
66. H(5,𝔽2)C23×C53direct (N × H)
67. C2×#41C23×C53semidirect
68. C22×Q8C23×C53semidirect
69. C22×D4C23×C53semidirect
70. C22×D4C23×C53semidirect
72. C23×C4C23×C53semidirect
77. C24C24×C53semidirect
78. #41C24×C53semidirect
79. C2×Q8C24×C53semidirect
80. C2×D4C24×C53semidirect
81. C2×D4C24×C53semidirect
83. C22×C4C24×C53semidirect
86. C23H(5,𝔽2)×C53direct (N × H)
89. C23C25×C53semidirect
90. Q8C25×C53semidirect
91. D4C25×C53semidirect
93. C2×C4C25×C53semidirect
96. C22C2×H(5,𝔽2)×C53direct (N × H)
101. C2C22×H(5,𝔽2)×C53direct (N × H)

Maximal subgroups

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

subgroupindex
C22×H(5,𝔽2)×C532
C23×#41×C532
C24×D4×C532
C23×H(5,𝔽2)53