Finite Group Databasebeta

← C2×H(5,𝔽2)×C13⋊C3 #72276212

Subgroups — C2×H(5,𝔽2)×C13⋊C3 #72276212

Order 2496.

1796 conjugacy classes of subgroups; only the 1131 normal subgroups are shown.

normal subgroups only

This group has 1796 conjugacy classes of subgroups — too many to draw. Only its 1131 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
1H(5,𝔽2)×C13⋊C3C2116yesmaximal, direct product factor
2C2×#41×C13⋊C3C216yesmaximal, complemented
3C22×D4×C13⋊C3C219yesmaximal, complemented
4C2×C13×H(5,𝔽2)C311yesmaximal, characteristic, complemented, Hall
5C24×C13⋊C3C2216yescomplemented
6#41×C13⋊C3C22148yescomplemented
7C2×Q8×C13⋊C3C2212yescomplemented
8C2×D4×C13⋊C3C22118yescomplemented
9C2×D4×C13⋊C3C22172yescomplemented
10C22×C4×C13⋊C3C2219yescomplemented
11C13×H(5,𝔽2)C2×C3116yescomplemented
12C2×C13×#41C2×C316yescomplemented
13C22×D4×C13C2×C319yescomplemented
14C23×C13⋊C3—412no—
15C23×C13⋊C3C2319yes—
16C23×C13⋊C3C23124yescomplemented
17Q8×C13⋊C3C2318yescomplemented
18D4×C13⋊C3C23172yescomplemented
19C2×C4×C13⋊C3C2316yes—
20C2×C4×C13⋊C3C23136yescomplemented
21C24×C13C22×C316yescomplemented
22C13×#41C22×C3148yescomplemented
23C2×Q8×C13C22×C312yescomplemented
24C2×D4×C13C22×C3118yescomplemented
25C2×D4×C13C22×C3172yescomplemented
26C22×C4×C13C22×C319yescomplemented
27C2×C3×H(5,𝔽2)—131nomaximal, Hall
28C22×C13⋊C3—218no—
29C22×C13⋊C3—424no—
30C22×C13⋊C3C24118yes—
31C22×C13⋊C3C2411yescharacteristic
32C4×C13⋊C3C24112yes—
33C23×C13—412no—
34C23×C13C23×C319yes—
35C23×C13C23×C3124yescomplemented
36Q8×C13C23×C318yescomplemented
37D4×C13C23×C3172yescomplemented
38C2×C4×C13C23×C316yes—
39C2×C4×C13C23×C3136yescomplemented
40C3×H(5,𝔽2)—1316no—
41C2×C3×#41—136no—
42C22×C3×D4—139no—
43C2×C13⋊C3—218no—
44C2×C13⋊C3H(5,𝔽2)12yesdirect product factor
45C2×C13⋊C3C2511yescharacteristic
46C2×H(5,𝔽2)C13⋊C311yescharacteristic, direct product factor, Sylow
47C22×C13—218no—
48C22×C13—424no—
49C22×C13C24×C3118yes—
50C22×C13C24×C311yescharacteristic
51C4×C13C24×C3112yes—
52C24×C3—136no—
53C3×#41—1348no—
54C2×C3×Q8—132no—
55C2×C3×D4—1318no—
56C2×C3×D4—1372no—
57C22×C3×C4—139no—
58C13⋊C3C2×H(5,𝔽2)11yescharacteristic, direct product factor, Hall
59H(5,𝔽2)C2×C13⋊C3116yesdirect product factor
60C2×#41C2×C13⋊C316yescomplemented
61C22×D4C2×C13⋊C319yescomplemented
62C2×C13—218no—
63C2×C13C3×H(5,𝔽2)12yescomplemented
64C2×C13C25×C311yescharacteristic
65C23×C3—139no—
66C23×C3—1324no—
67C23×C3—5212no—
68C3×Q8—138no—
69C3×D4—1372no—
70C2×C3×C4—136no—
71C2×C3×C4—1336no—
72C24C22×C13⋊C316yescomplemented
73#41C22×C13⋊C3148yescomplemented
74C2×Q8C22×C13⋊C312yescomplemented
75C2×D4C22×C13⋊C3118yescomplemented
76C2×D4C22×C13⋊C3172yescomplemented
77C22×C4C22×C13⋊C319yescomplemented
78C13C2×C3×H(5,𝔽2)11yescharacteristic, complemented, Sylow
79C22×C3—1318no—
80C22×C3—131no—
81C22×C3—2618no—
82C22×C3—5224no—
83C3×C4—1312no—
84C23—412no—
85C23C23×C13⋊C319yes—
86C23C23×C13⋊C3124yescomplemented
87Q8C23×C13⋊C318yescomplemented
88D4C23×C13⋊C3172yescomplemented
89C2×C4C23×C13⋊C316yes—
90C2×C4C23×C13⋊C3136yescomplemented
91C2×C3—132no—
92C2×C3—131no—
93C2×C3—2618no—
94C22—218no—
95C22—424no—
96C22C24×C13⋊C3118yes—
97C22C24×C13⋊C311yescharacteristic
98C4C24×C13⋊C3112yes—
99C3—131noSylow
100C2—218no—
101C2H(5,𝔽2)×C13⋊C312yesdirect product factor
102C2C25×C13⋊C311yescharacteristic
subgroupG/H
centerC22C24×C13⋊C3
commutator, cosocleC2×C13C25×C3
FrattiniC2C25×C13⋊C3
FittingC2×C13×H(5,𝔽2)C3
socleC22×C13C24×C3
hypercenterC2×H(5,𝔽2)C13⋊C3
hypocenterC13C2×C3×H(5,𝔽2)
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. H(5,𝔽2)×C13⋊C3C2direct (N × H)
2. C2×#41×C13⋊C3C2semidirect
3. C22×D4×C13⋊C3C2semidirect
4. C2×C13×H(5,𝔽2)C3semidirect
5. C24×C13⋊C3C22semidirect
6. #41×C13⋊C3C22semidirect
7. C2×Q8×C13⋊C3C22semidirect
8. C2×D4×C13⋊C3C22semidirect
9. C2×D4×C13⋊C3C22semidirect
10. C22×C4×C13⋊C3C22semidirect
11. C13×H(5,𝔽2)C2×C3semidirect
12. C2×C13×#41C2×C3semidirect
13. C22×D4×C13C2×C3semidirect
16. C23×C13⋊C3C23semidirect
17. Q8×C13⋊C3C23semidirect
18. D4×C13⋊C3C23semidirect
20. C2×C4×C13⋊C3C23semidirect
21. C24×C13C22×C3semidirect
22. C13×#41C22×C3semidirect
23. C2×Q8×C13C22×C3semidirect
24. C2×D4×C13C22×C3semidirect
25. C2×D4×C13C22×C3semidirect
26. C22×C4×C13C22×C3semidirect
35. C23×C13C23×C3semidirect
36. Q8×C13C23×C3semidirect
37. D4×C13C23×C3semidirect
39. C2×C4×C13C23×C3semidirect
44. C2×C13⋊C3H(5,𝔽2)direct (N × H)
46. C2×H(5,𝔽2)C13⋊C3direct (N × H)
58. C13⋊C3C2×H(5,𝔽2)direct (N × H)
59. H(5,𝔽2)C2×C13⋊C3direct (N × H)
60. C2×#41C2×C13⋊C3semidirect
61. C22×D4C2×C13⋊C3semidirect
63. C2×C13C3×H(5,𝔽2)semidirect
72. C24C22×C13⋊C3semidirect
73. #41C22×C13⋊C3semidirect
74. C2×Q8C22×C13⋊C3semidirect
75. C2×D4C22×C13⋊C3semidirect
76. C2×D4C22×C13⋊C3semidirect
77. C22×C4C22×C13⋊C3semidirect
78. C13C2×C3×H(5,𝔽2)semidirect
86. C23C23×C13⋊C3semidirect
87. Q8C23×C13⋊C3semidirect
88. D4C23×C13⋊C3semidirect
90. C2×C4C23×C13⋊C3semidirect
101. C2H(5,𝔽2)×C13⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
H(5,𝔽2)×C13⋊C32
C2×#41×C13⋊C32
C22×D4×C13⋊C32
C2×C13×H(5,𝔽2)3
C2×C3×H(5,𝔽2)13