Finite Group Databasebeta

← C24×#41×C53 #27270609

Subgroups — C24×#41×C53 #27270609

Order 13568.

65470 conjugacy classes of subgroups; only the 59038 normal subgroups are shown.

normal subgroups only

This group has 65470 conjugacy classes of subgroups — too many to draw. Only its 59038 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
1C23×#41×C53C21120yesmaximal, direct product factor
2C24×Q8×C53C211yesmaximal, characteristic, complemented
3C24×D4×C53C213yesmaximal, complemented
4C25×C4×C53C213yesmaximal, complemented
5C26×C53C2213yes—
6C22×#41×C53C2212240yesdirect product factor
7C23×Q8×C53C22160yescomplemented
8C23×D4×C53C221180yescomplemented
9C24×C4×C53C2213yes—
10C24×C4×C53C2211yescharacteristic
11C24×C4×C53C221180yescomplemented
12C25×C53—248no—
13C25×C53C23190yes—
14C25×C53C2311yescharacteristic
15C2×#41×C53C2317680yesdirect product factor
16C22×Q8×C53C231560yescomplemented
17C22×D4×C53C2311680yescomplemented
18C23×C4×C53C23190yes—
19C23×C4×C53C23130yes—
20C23×C4×C53C2311680yescomplemented
21C24×C53—2720no—
22C24×C53#41116yesdirect product factor
23C24×C53C241420yes—
24C24×C53C24115yes—
25#41×C53C2414096yesdirect product factor
26C2×Q8×C53C241960yescomplemented
27C2×D4×C53C2412880yescomplemented
28C22×C4×C53C241140yes—
29C22×C4×C53C241420yes—
30C22×C4×C53C2412880yescomplemented
31C23×C53—21680no—
32C23×C53C2×#411120yesdirect product factor
33C23×C53C251360yes—
34C23×C53C25135yes—
35Q8×C53C251256yescomplemented
36D4×C53C251768yescomplemented
37C2×C4×C53C251120yes—
38C2×C4×C53C251360yes—
39C2×C4×C53C251768yescomplemented
40C24×#41C5311yesmaximal, characteristic, direct product factor, Sylow
41C22×C53—2720no—
42C22×C53C22×#411140yesdirect product factor
43C22×C53C26148yes—
44C22×C53C26115yes—
45C4×C53C26116yes—
46C4×C53C26148yes—
47C23×#41C2×C531120yesdirect product factor
48C24×Q8C2×C5311yescharacteristic, complemented
49C24×D4C2×C5313yescomplemented
50C25×C4C2×C5313yescomplemented
51C2×C53—248no—
52C2×C53C23×#41130yesdirect product factor
53C2×C53C2711yescharacteristic
54C26C22×C5313yes—
55C22×#41C22×C5312240yesdirect product factor
56C23×Q8C22×C53160yescomplemented
57C23×D4C22×C531180yescomplemented
58C24×C4C22×C5311yescharacteristic
59C24×C4C22×C5313yes—
60C24×C4C22×C531180yescomplemented
61C53C24×#4111yescharacteristic, direct product factor, Sylow
62C25—248no—
63C25C23×C53190yes—
64C25C23×C5311yescharacteristic
65C2×#41C23×C5317680yesdirect product factor
66C22×Q8C23×C531560yescomplemented
67C22×D4C23×C5311680yescomplemented
68C23×C4C23×C53130yes—
69C23×C4C23×C53190yes—
70C23×C4C23×C5311680yescomplemented
71C24—2720no—
72C24#41×C53116yesdirect product factor
73C24C24×C531420yes—
74C24C24×C53115yes—
75#41C24×C5314096yesdirect product factor
76C2×Q8C24×C531960yescomplemented
77C2×D4C24×C5312880yescomplemented
78C22×C4C24×C531140yes—
79C22×C4C24×C531420yes—
80C22×C4C24×C5312880yescomplemented
81C23—21680no—
82C23C2×#41×C531120yesdirect product factor
83C23C25×C531360yes—
84C23C25×C53135yes—
85Q8C25×C531256yescomplemented
86D4C25×C531768yescomplemented
87C2×C4C25×C531120yes—
88C2×C4C25×C531360yes—
89C2×C4C25×C531768yescomplemented
90C22—2720no—
91C22C22×#41×C531140yesdirect product factor
92C22C26×C53148yes—
93C22C26×C53115yes—
94C4C26×C53116yes—
95C4C26×C53148yes—
96C2—248no—
97C2C23×#41×C53130yesdirect product factor
98C2C27×C5311yescharacteristic
subgroupG/H
centerC24×C4×C53C22
commutator, Frattini, cosocleC2C27×C53
socleC25×C53C23
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C24×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#41×C53C2direct (N × H)
2. C24×Q8×C53C2semidirect
3. C24×D4×C53C2semidirect
4. C25×C4×C53C2semidirect
6. C22×#41×C53C22direct (N × H)
7. C23×Q8×C53C22semidirect
8. C23×D4×C53C22semidirect
11. C24×C4×C53C22semidirect
15. C2×#41×C53C23direct (N × H)
16. C22×Q8×C53C23semidirect
17. C22×D4×C53C23semidirect
20. C23×C4×C53C23semidirect
22. C24×C53#41direct (N × H)
25. #41×C53C24direct (N × H)
26. C2×Q8×C53C24semidirect
27. C2×D4×C53C24semidirect
30. C22×C4×C53C24semidirect
32. C23×C53C2×#41direct (N × H)
35. Q8×C53C25semidirect
36. D4×C53C25semidirect
39. C2×C4×C53C25semidirect
40. C24×#41C53direct (N × H)
42. C22×C53C22×#41direct (N × H)
47. C23×#41C2×C53direct (N × H)
48. C24×Q8C2×C53semidirect
49. C24×D4C2×C53semidirect
50. C25×C4C2×C53semidirect
52. C2×C53C23×#41direct (N × H)
55. C22×#41C22×C53direct (N × H)
56. C23×Q8C22×C53semidirect
57. C23×D4C22×C53semidirect
60. C24×C4C22×C53semidirect
61. C53C24×#41direct (N × H)
65. C2×#41C23×C53direct (N × H)
66. C22×Q8C23×C53semidirect
67. C22×D4C23×C53semidirect
70. C23×C4C23×C53semidirect
72. C24#41×C53direct (N × H)
75. #41C24×C53direct (N × H)
76. C2×Q8C24×C53semidirect
77. C2×D4C24×C53semidirect
80. C22×C4C24×C53semidirect
82. C23C2×#41×C53direct (N × H)
85. Q8C25×C53semidirect
86. D4C25×C53semidirect
89. C2×C4C25×C53semidirect
91. C22C22×#41×C53direct (N × H)
97. C2C23×#41×C53direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#41×C532
C24×Q8×C532
C24×D4×C532
C25×C4×C532
C24×#4153