Finite Group Databasebeta

← C23×H(5,𝔽2)×C61 #29566531

Subgroups — C23×H(5,𝔽2)×C61 #29566531

Order 15616.

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)×C61C21112yesmaximal, direct product factor
2C23×#41×C61C216yesmaximal, complemented
3C24×D4×C61C219yesmaximal, complemented
4C26×C61C2216yescomplemented
5C2×H(5,𝔽2)×C61C2211792yesdirect product factor
6C22×#41×C61C221336yescomplemented
7C23×Q8×C61C2212yescomplemented
8C23×D4×C61C22118yescomplemented
9C23×D4×C61C221504yescomplemented
10C24×C4×C61C2219yescomplemented
11C25×C61—448no—
12C25×C61C2319yes—
13C25×C61C231168yescomplemented
14H(5,𝔽2)×C61C2314096yesdirect product factor
15C2×#41×C61C2312688yescomplemented
16C22×Q8×C61C23156yescomplemented
17C22×D4×C61C231504yescomplemented
18C22×D4×C61C2314032yescomplemented
19C23×C4×C61C2316yes—
20C23×C4×C61C231252yescomplemented
21C24×C61—272no—
22C24×C61—4672no—
23C24×C61C241126yes—
24C24×C61C2411yescharacteristic
25C24×C61C241672yescomplemented
26#41×C61C2413072yescomplemented
27C2×Q8×C61C241224yescomplemented
28C2×D4×C61C2412016yescomplemented
29C2×D4×C61C2414608yescomplemented
30C22×C4×C61C24184yes—
31C22×C4×C61C2411008yescomplemented
32C23×C61—2504no—
33C23×C61—41344no—
34C23×C61H(5,𝔽2)18yesdirect product factor
35C23×C61C251252yes—
36C23×C61C2517yes—
37C23×C61C251384yescomplemented
38Q8×C61C251128yescomplemented
39D4×C61C2511152yescomplemented
40C2×C4×C61C251168yes—
41C2×C4×C61C251576yescomplemented
42C23×H(5,𝔽2)C6111yesmaximal, characteristic, direct product factor, Sylow
43C22×C61—2504no—
44C22×C61—4384no—
45C22×C61C2×H(5,𝔽2)128yesdirect product factor
46C22×C61C26172yes—
47C22×C61C2617yes—
48C4×C61C26148yes—
49C22×H(5,𝔽2)C2×C611112yesdirect product factor
50C23×#41C2×C6116yescomplemented
51C24×D4C2×C6119yescomplemented
52C2×C61—272no—
53C2×C61C22×H(5,𝔽2)114yesdirect product factor
54C2×C61C2711yescharacteristic
55C26C22×C6116yescomplemented
56C2×H(5,𝔽2)C22×C6111792yesdirect product factor
57C22×#41C22×C611336yescomplemented
58C23×Q8C22×C6112yescomplemented
59C23×D4C22×C61118yescomplemented
60C23×D4C22×C611504yescomplemented
61C24×C4C22×C6119yescomplemented
62C61C23×H(5,𝔽2)11yescharacteristic, direct product factor, Sylow
63C25—448no—
64C25C23×C6119yes—
65C25C23×C611168yescomplemented
66H(5,𝔽2)C23×C6114096yesdirect product factor
67C2×#41C23×C6112688yescomplemented
68C22×Q8C23×C61156yescomplemented
69C22×D4C23×C611504yescomplemented
70C22×D4C23×C6114032yescomplemented
71C23×C4C23×C6116yes—
72C23×C4C23×C611252yescomplemented
73C24—272no—
74C24—4672no—
75C24C24×C611126yes—
76C24C24×C6111yescharacteristic
77C24C24×C611672yescomplemented
78#41C24×C6113072yescomplemented
79C2×Q8C24×C611224yescomplemented
80C2×D4C24×C6112016yescomplemented
81C2×D4C24×C6114608yescomplemented
82C22×C4C24×C61184yes—
83C22×C4C24×C6111008yescomplemented
84C23—2504no—
85C23—41344no—
86C23H(5,𝔽2)×C6118yesdirect product factor
87C23C25×C611252yes—
88C23C25×C6117yes—
89C23C25×C611384yescomplemented
90Q8C25×C611128yescomplemented
91D4C25×C6111152yescomplemented
92C2×C4C25×C611168yes—
93C2×C4C25×C611576yescomplemented
94C22—2504no—
95C22—4384no—
96C22C2×H(5,𝔽2)×C61128yesdirect product factor
97C22C26×C61172yes—
98C22C26×C6117yes—
99C4C26×C61148yes—
100C2—272no—
101C2C22×H(5,𝔽2)×C61114yesdirect product factor
102C2C27×C6111yescharacteristic
subgroupG/H
center, socleC24×C61C24
commutator, Frattini, cosocleC2C27×C61
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C24×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×H(5,𝔽2)×C61C2direct (N × H)
2. C23×#41×C61C2semidirect
3. C24×D4×C61C2semidirect
4. C26×C61C22semidirect
5. C2×H(5,𝔽2)×C61C22direct (N × H)
6. C22×#41×C61C22semidirect
7. C23×Q8×C61C22semidirect
8. C23×D4×C61C22semidirect
9. C23×D4×C61C22semidirect
10. C24×C4×C61C22semidirect
13. C25×C61C23semidirect
14. H(5,𝔽2)×C61C23direct (N × H)
15. C2×#41×C61C23semidirect
16. C22×Q8×C61C23semidirect
17. C22×D4×C61C23semidirect
18. C22×D4×C61C23semidirect
20. C23×C4×C61C23semidirect
25. C24×C61C24semidirect
26. #41×C61C24semidirect
27. C2×Q8×C61C24semidirect
28. C2×D4×C61C24semidirect
29. C2×D4×C61C24semidirect
31. C22×C4×C61C24semidirect
34. C23×C61H(5,𝔽2)direct (N × H)
37. C23×C61C25semidirect
38. Q8×C61C25semidirect
39. D4×C61C25semidirect
41. C2×C4×C61C25semidirect
42. C23×H(5,𝔽2)C61direct (N × H)
45. C22×C61C2×H(5,𝔽2)direct (N × H)
49. C22×H(5,𝔽2)C2×C61direct (N × H)
50. C23×#41C2×C61semidirect
51. C24×D4C2×C61semidirect
53. C2×C61C22×H(5,𝔽2)direct (N × H)
55. C26C22×C61semidirect
56. C2×H(5,𝔽2)C22×C61direct (N × H)
57. C22×#41C22×C61semidirect
58. C23×Q8C22×C61semidirect
59. C23×D4C22×C61semidirect
60. C23×D4C22×C61semidirect
61. C24×C4C22×C61semidirect
62. C61C23×H(5,𝔽2)direct (N × H)
65. C25C23×C61semidirect
66. H(5,𝔽2)C23×C61direct (N × H)
67. C2×#41C23×C61semidirect
68. C22×Q8C23×C61semidirect
69. C22×D4C23×C61semidirect
70. C22×D4C23×C61semidirect
72. C23×C4C23×C61semidirect
77. C24C24×C61semidirect
78. #41C24×C61semidirect
79. C2×Q8C24×C61semidirect
80. C2×D4C24×C61semidirect
81. C2×D4C24×C61semidirect
83. C22×C4C24×C61semidirect
86. C23H(5,𝔽2)×C61direct (N × H)
89. C23C25×C61semidirect
90. Q8C25×C61semidirect
91. D4C25×C61semidirect
93. C2×C4C25×C61semidirect
96. C22C2×H(5,𝔽2)×C61direct (N × H)
101. C2C22×H(5,𝔽2)×C61direct (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)×C612
C23×#41×C612
C24×D4×C612
C23×H(5,𝔽2)61