Finite Group Databasebeta

← C23×C13×H(5,𝔽2) #15749437

Subgroups — C23×C13×H(5,𝔽2) #15749437

Order 3328.

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×C13×H(5,𝔽2)C21112yesmaximal, direct product factor
2C23×C13×#41C216yesmaximal, complemented
3C24×D4×C13C219yesmaximal, complemented
4C26×C13C2216yescomplemented
5C2×C13×H(5,𝔽2)C2211792yesdirect product factor
6C22×C13×#41C221336yescomplemented
7C23×Q8×C13C2212yescomplemented
8C23×D4×C13C22118yescomplemented
9C23×D4×C13C221504yescomplemented
10C24×C4×C13C2219yescomplemented
11C25×C13—448no—
12C25×C13C2319yes—
13C25×C13C231168yescomplemented
14C13×H(5,𝔽2)C2314096yesdirect product factor
15C2×C13×#41C2312688yescomplemented
16C22×Q8×C13C23156yescomplemented
17C22×D4×C13C231504yescomplemented
18C22×D4×C13C2314032yescomplemented
19C23×C4×C13C2316yes—
20C23×C4×C13C231252yescomplemented
21C23×H(5,𝔽2)C1311yesmaximal, characteristic, direct product factor, Sylow
22C24×C13—272no—
23C24×C13—4672no—
24C24×C13C241126yes—
25C24×C13C2411yescharacteristic
26C24×C13C241672yescomplemented
27C13×#41C2413072yescomplemented
28C2×Q8×C13C241224yescomplemented
29C2×D4×C13C2412016yescomplemented
30C2×D4×C13C2414608yescomplemented
31C22×C4×C13C24184yes—
32C22×C4×C13C2411008yescomplemented
33C22×H(5,𝔽2)C2×C131112yesdirect product factor
34C23×#41C2×C1316yescomplemented
35C24×D4C2×C1319yescomplemented
36C23×C13—2504no—
37C23×C13—41344no—
38C23×C13H(5,𝔽2)18yesdirect product factor
39C23×C13C251252yes—
40C23×C13C2517yes—
41C23×C13C251384yescomplemented
42Q8×C13C251128yescomplemented
43D4×C13C2511152yescomplemented
44C2×C4×C13C251168yes—
45C2×C4×C13C251576yescomplemented
46C26C22×C1316yescomplemented
47C2×H(5,𝔽2)C22×C1311792yesdirect product factor
48C22×#41C22×C131336yescomplemented
49C23×Q8C22×C1312yescomplemented
50C23×D4C22×C13118yescomplemented
51C23×D4C22×C131504yescomplemented
52C24×C4C22×C1319yescomplemented
53C22×C13—2504no—
54C22×C13—4384no—
55C22×C13C2×H(5,𝔽2)128yesdirect product factor
56C22×C13C26172yes—
57C22×C13C2617yes—
58C4×C13C26148yes—
59C25—448no—
60C25C23×C1319yes—
61C25C23×C131168yescomplemented
62H(5,𝔽2)C23×C1314096yesdirect product factor
63C2×#41C23×C1312688yescomplemented
64C22×Q8C23×C13156yescomplemented
65C22×D4C23×C131504yescomplemented
66C22×D4C23×C1314032yescomplemented
67C23×C4C23×C1316yes—
68C23×C4C23×C131252yescomplemented
69C2×C13—272no—
70C2×C13C22×H(5,𝔽2)114yesdirect product factor
71C2×C13C2711yescharacteristic
72C24—272no—
73C24—4672no—
74C24C24×C131126yes—
75C24C24×C1311yescharacteristic
76C24C24×C131672yescomplemented
77#41C24×C1313072yescomplemented
78C2×Q8C24×C131224yescomplemented
79C2×D4C24×C1312016yescomplemented
80C2×D4C24×C1314608yescomplemented
81C22×C4C24×C13184yes—
82C22×C4C24×C1311008yescomplemented
83C13C23×H(5,𝔽2)11yescharacteristic, direct product factor, Sylow
84C23—2504no—
85C23—41344no—
86C23C13×H(5,𝔽2)18yesdirect product factor
87C23C25×C131252yes—
88C23C25×C1317yes—
89C23C25×C131384yescomplemented
90Q8C25×C131128yescomplemented
91D4C25×C1311152yescomplemented
92C2×C4C25×C131168yes—
93C2×C4C25×C131576yescomplemented
94C22—2504no—
95C22—4384no—
96C22C2×C13×H(5,𝔽2)128yesdirect product factor
97C22C26×C13172yes—
98C22C26×C1317yes—
99C4C26×C13148yes—
100C2—272no—
101C2C22×C13×H(5,𝔽2)114yesdirect product factor
102C2C27×C1311yescharacteristic
subgroupG/H
center, socleC24×C13C24
commutator, Frattini, cosocleC2C27×C13
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C24×C13 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C13×H(5,𝔽2)C2direct (N × H)
2. C23×C13×#41C2semidirect
3. C24×D4×C13C2semidirect
4. C26×C13C22semidirect
5. C2×C13×H(5,𝔽2)C22direct (N × H)
6. C22×C13×#41C22semidirect
7. C23×Q8×C13C22semidirect
8. C23×D4×C13C22semidirect
9. C23×D4×C13C22semidirect
10. C24×C4×C13C22semidirect
13. C25×C13C23semidirect
14. C13×H(5,𝔽2)C23direct (N × H)
15. C2×C13×#41C23semidirect
16. C22×Q8×C13C23semidirect
17. C22×D4×C13C23semidirect
18. C22×D4×C13C23semidirect
20. C23×C4×C13C23semidirect
21. C23×H(5,𝔽2)C13direct (N × H)
26. C24×C13C24semidirect
27. C13×#41C24semidirect
28. C2×Q8×C13C24semidirect
29. C2×D4×C13C24semidirect
30. C2×D4×C13C24semidirect
32. C22×C4×C13C24semidirect
33. C22×H(5,𝔽2)C2×C13direct (N × H)
34. C23×#41C2×C13semidirect
35. C24×D4C2×C13semidirect
38. C23×C13H(5,𝔽2)direct (N × H)
41. C23×C13C25semidirect
42. Q8×C13C25semidirect
43. D4×C13C25semidirect
45. C2×C4×C13C25semidirect
46. C26C22×C13semidirect
47. C2×H(5,𝔽2)C22×C13direct (N × H)
48. C22×#41C22×C13semidirect
49. C23×Q8C22×C13semidirect
50. C23×D4C22×C13semidirect
51. C23×D4C22×C13semidirect
52. C24×C4C22×C13semidirect
55. C22×C13C2×H(5,𝔽2)direct (N × H)
61. C25C23×C13semidirect
62. H(5,𝔽2)C23×C13direct (N × H)
63. C2×#41C23×C13semidirect
64. C22×Q8C23×C13semidirect
65. C22×D4C23×C13semidirect
66. C22×D4C23×C13semidirect
68. C23×C4C23×C13semidirect
70. C2×C13C22×H(5,𝔽2)direct (N × H)
76. C24C24×C13semidirect
77. #41C24×C13semidirect
78. C2×Q8C24×C13semidirect
79. C2×D4C24×C13semidirect
80. C2×D4C24×C13semidirect
82. C22×C4C24×C13semidirect
83. C13C23×H(5,𝔽2)direct (N × H)
86. C23C13×H(5,𝔽2)direct (N × H)
89. C23C25×C13semidirect
90. Q8C25×C13semidirect
91. D4C25×C13semidirect
93. C2×C4C25×C13semidirect
96. C22C2×C13×H(5,𝔽2)direct (N × H)
101. C2C22×C13×H(5,𝔽2)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C13×H(5,𝔽2)2
C23×C13×#412
C24×D4×C132
C23×H(5,𝔽2)13