Finite Group Databasebeta

← C23×H(5,𝔽2)×C79 #34174787

Subgroups — C23×H(5,𝔽2)×C79 #34174787

Order 20224.

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)×C79C21112yesmaximal, direct product factor
2C23×#41×C79C216yesmaximal, complemented
3C24×D4×C79C219yesmaximal, complemented
4C26×C79C2216yescomplemented
5C2×H(5,𝔽2)×C79C2211792yesdirect product factor
6C22×#41×C79C221336yescomplemented
7C23×Q8×C79C2212yescomplemented
8C23×D4×C79C22118yescomplemented
9C23×D4×C79C221504yescomplemented
10C24×C4×C79C2219yescomplemented
11C25×C79—448no—
12C25×C79C2319yes—
13C25×C79C231168yescomplemented
14H(5,𝔽2)×C79C2314096yesdirect product factor
15C2×#41×C79C2312688yescomplemented
16C22×Q8×C79C23156yescomplemented
17C22×D4×C79C231504yescomplemented
18C22×D4×C79C2314032yescomplemented
19C23×C4×C79C2316yes—
20C23×C4×C79C231252yescomplemented
21C24×C79—272no—
22C24×C79—4672no—
23C24×C79C241126yes—
24C24×C79C2411yescharacteristic
25C24×C79C241672yescomplemented
26#41×C79C2413072yescomplemented
27C2×Q8×C79C241224yescomplemented
28C2×D4×C79C2412016yescomplemented
29C2×D4×C79C2414608yescomplemented
30C22×C4×C79C24184yes—
31C22×C4×C79C2411008yescomplemented
32C23×C79—2504no—
33C23×C79—41344no—
34C23×C79H(5,𝔽2)18yesdirect product factor
35C23×C79C251252yes—
36C23×C79C2517yes—
37C23×C79C251384yescomplemented
38Q8×C79C251128yescomplemented
39D4×C79C2511152yescomplemented
40C2×C4×C79C251168yes—
41C2×C4×C79C251576yescomplemented
42C22×C79—2504no—
43C22×C79—4384no—
44C22×C79C2×H(5,𝔽2)128yesdirect product factor
45C22×C79C26172yes—
46C22×C79C2617yes—
47C4×C79C26148yes—
48C23×H(5,𝔽2)C7911yesmaximal, characteristic, direct product factor, Sylow
49C2×C79—272no—
50C2×C79C22×H(5,𝔽2)114yesdirect product factor
51C2×C79C2711yescharacteristic
52C22×H(5,𝔽2)C2×C791112yesdirect product factor
53C23×#41C2×C7916yescomplemented
54C24×D4C2×C7919yescomplemented
55C79C23×H(5,𝔽2)11yescharacteristic, direct product factor, Sylow
56C26C22×C7916yescomplemented
57C2×H(5,𝔽2)C22×C7911792yesdirect product factor
58C22×#41C22×C791336yescomplemented
59C23×Q8C22×C7912yescomplemented
60C23×D4C22×C79118yescomplemented
61C23×D4C22×C791504yescomplemented
62C24×C4C22×C7919yescomplemented
63C25—448no—
64C25C23×C7919yes—
65C25C23×C791168yescomplemented
66H(5,𝔽2)C23×C7914096yesdirect product factor
67C2×#41C23×C7912688yescomplemented
68C22×Q8C23×C79156yescomplemented
69C22×D4C23×C791504yescomplemented
70C22×D4C23×C7914032yescomplemented
71C23×C4C23×C7916yes—
72C23×C4C23×C791252yescomplemented
73C24—272no—
74C24—4672no—
75C24C24×C791126yes—
76C24C24×C7911yescharacteristic
77C24C24×C791672yescomplemented
78#41C24×C7913072yescomplemented
79C2×Q8C24×C791224yescomplemented
80C2×D4C24×C7912016yescomplemented
81C2×D4C24×C7914608yescomplemented
82C22×C4C24×C79184yes—
83C22×C4C24×C7911008yescomplemented
84C23—2504no—
85C23—41344no—
86C23H(5,𝔽2)×C7918yesdirect product factor
87C23C25×C791252yes—
88C23C25×C7917yes—
89C23C25×C791384yescomplemented
90Q8C25×C791128yescomplemented
91D4C25×C7911152yescomplemented
92C2×C4C25×C791168yes—
93C2×C4C25×C791576yescomplemented
94C22—2504no—
95C22—4384no—
96C22C2×H(5,𝔽2)×C79128yesdirect product factor
97C22C26×C79172yes—
98C22C26×C7917yes—
99C4C26×C79148yes—
100C2—272no—
101C2C22×H(5,𝔽2)×C79114yesdirect product factor
102C2C27×C7911yescharacteristic
subgroupG/H
center, socleC24×C79C24
commutator, Frattini, cosocleC2C27×C79
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C24×C79 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×H(5,𝔽2)×C79C2direct (N × H)
2. C23×#41×C79C2semidirect
3. C24×D4×C79C2semidirect
4. C26×C79C22semidirect
5. C2×H(5,𝔽2)×C79C22direct (N × H)
6. C22×#41×C79C22semidirect
7. C23×Q8×C79C22semidirect
8. C23×D4×C79C22semidirect
9. C23×D4×C79C22semidirect
10. C24×C4×C79C22semidirect
13. C25×C79C23semidirect
14. H(5,𝔽2)×C79C23direct (N × H)
15. C2×#41×C79C23semidirect
16. C22×Q8×C79C23semidirect
17. C22×D4×C79C23semidirect
18. C22×D4×C79C23semidirect
20. C23×C4×C79C23semidirect
25. C24×C79C24semidirect
26. #41×C79C24semidirect
27. C2×Q8×C79C24semidirect
28. C2×D4×C79C24semidirect
29. C2×D4×C79C24semidirect
31. C22×C4×C79C24semidirect
34. C23×C79H(5,𝔽2)direct (N × H)
37. C23×C79C25semidirect
38. Q8×C79C25semidirect
39. D4×C79C25semidirect
41. C2×C4×C79C25semidirect
44. C22×C79C2×H(5,𝔽2)direct (N × H)
48. C23×H(5,𝔽2)C79direct (N × H)
50. C2×C79C22×H(5,𝔽2)direct (N × H)
52. C22×H(5,𝔽2)C2×C79direct (N × H)
53. C23×#41C2×C79semidirect
54. C24×D4C2×C79semidirect
55. C79C23×H(5,𝔽2)direct (N × H)
56. C26C22×C79semidirect
57. C2×H(5,𝔽2)C22×C79direct (N × H)
58. C22×#41C22×C79semidirect
59. C23×Q8C22×C79semidirect
60. C23×D4C22×C79semidirect
61. C23×D4C22×C79semidirect
62. C24×C4C22×C79semidirect
65. C25C23×C79semidirect
66. H(5,𝔽2)C23×C79direct (N × H)
67. C2×#41C23×C79semidirect
68. C22×Q8C23×C79semidirect
69. C22×D4C23×C79semidirect
70. C22×D4C23×C79semidirect
72. C23×C4C23×C79semidirect
77. C24C24×C79semidirect
78. #41C24×C79semidirect
79. C2×Q8C24×C79semidirect
80. C2×D4C24×C79semidirect
81. C2×D4C24×C79semidirect
83. C22×C4C24×C79semidirect
86. C23H(5,𝔽2)×C79direct (N × H)
89. C23C25×C79semidirect
90. Q8C25×C79semidirect
91. D4C25×C79semidirect
93. C2×C4C25×C79semidirect
96. C22C2×H(5,𝔽2)×C79direct (N × H)
101. C2C22×H(5,𝔽2)×C79direct (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)×C792
C23×#41×C792
C24×D4×C792
C23×H(5,𝔽2)79