Finite Group Databasebeta

← C3×#88256 #11929281

Subgroups — C3×#88256 #11929281

Order 1248.

332 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C3×C13×#41C211yesmaximal, characteristic, complemented
2C3×#6131C212yesmaximal, complemented
3C3×D4×D13C216yesmaximal, complemented
4C3×#6127C213yesmaximal, complemented
5C2×C3×D52C213yesmaximal, complemented
6#88256C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×D13C2216yescomplemented
8C3×Q8×C13C2211yescharacteristic, complemented
9C3×D4×C13C2213yescomplemented
10C2×C3×C4×C13C2213yescomplemented
11C3×C22⋊D13C2216yescomplemented
12C3×D52C2213yescomplemented
13C3×D52C2216yescomplemented
14C3×C4×D13C2216yescomplemented
15C3×Dic26C2211yescharacteristic, complemented
16C13×#41C2×C311yescharacteristic, complemented
17#6131C2×C312yescomplemented
18D4×D13C2×C316yescomplemented
19#6127C2×C313yescomplemented
20C2×D52C2×C313yescomplemented
21C22×C3×C13C2313yes—
22C2×C3×D13—46no—
23C2×C3×D13C2316yes—
24C3×C4×C13C2311yescharacteristic
25C3×C4×C13C2313yes—
26C3×Dic13C2312yes—
27C22×D13C22×C316yescomplemented
28Q8×C13C22×C311yescharacteristic, complemented
29D4×C13C22×C313yescomplemented
30C2×C4×C13C22×C313yescomplemented
31C22⋊D13C22×C316yescomplemented
32D52C22×C313yescomplemented
33D52C22×C316yescomplemented
34C4×D13C22×C316yescomplemented
35Dic26C22×C311yescharacteristic, complemented
36C3×H(5,𝔽2)—131nomaximal, Hall
37C2×C3×C13—23no—
38C2×C3×C13C2411yescharacteristic
39C3×D13—26no—
40C22×C13C23×C313yes—
41C2×D13—46no—
42C2×D13C23×C316yes—
43C4×C13C23×C311yescharacteristic
44C4×C13C23×C313yes—
45Dic13C23×C312yes—
46C3×#41—132no—
47C3×#41—133no—
48C3×#41D1311yescharacteristic, complemented
49C2×C3×D4—136no—
50C2×C3×D4—133no—
51C3×C13H(5,𝔽2)11yescharacteristic, complemented, Hall
52H(5,𝔽2)—131noSylow
53C2×C13—23no—
54C2×C13C24×C311yescharacteristic
55D13—26no—
56C23×C3—136no—
57C3×Q8—131no—
58C3×Q8C2×D1311yescharacteristic, complemented
59C3×D4—136no—
60C3×D4—136no—
61C3×D4—133no—
62C3×D4C2×D1313yescomplemented
63C2×C3×C4—136no—
64C2×C3×C4C2×D1313yescomplemented
65#41—132no—
66#41—133no—
67#41C3×D1311yescharacteristic, complemented
68C2×D4—136no—
69C2×D4—133no—
70C13C3×H(5,𝔽2)11yescharacteristic, complemented, Sylow
71C22×C3—136no—
72C22×C3—526no—
73C22×C3C22×D1313yes—
74C3×C4—132no—
75C3×C4C22×D1313yes—
76C3×C4C22×D1311yescharacteristic
77C23—136no—
78Q8—131no—
79Q8C2×C3×D1311yescharacteristic, complemented
80D4—136no—
81D4—136no—
82D4—133no—
83D4C2×C3×D1313yescomplemented
84C2×C4—136no—
85C2×C4C2×C3×D1313yescomplemented
86C2×C3—23no—
87C2×C3—266no—
88C2×C3C23×D1311yescharacteristic
89C22—136no—
90C22—526no—
91C22C22×C3×D1313yes—
92C4—132no—
93C4C22×C3×D1313yes—
94C4C22×C3×D1311yescharacteristic
95C3#8825611yescharacteristic, direct product factor, Sylow
96C2—23no—
97C2—266no—
98C2C23×C3×D1311yescharacteristic
subgroupG/H
centerC2×C3C23×D13
commutator, cosocleC2×C13C24×C3
FrattiniC2C23×C3×D13
FittingC3×C13×#41C2
socleC2×C3×C13C24
hypercenterC3×#41D13
hypocenterC13C3×H(5,𝔽2)
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C2×C3 ◁ C3×#41

Complements

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

NH ≅ G/Ntype
1. C3×C13×#41C2semidirect
2. C3×#6131C2semidirect
3. C3×D4×D13C2semidirect
4. C3×#6127C2semidirect
5. C2×C3×D52C2semidirect
6. #88256C3direct (N × H)
7. C22×C3×D13C22semidirect
8. C3×Q8×C13C22semidirect
9. C3×D4×C13C22semidirect
10. C2×C3×C4×C13C22semidirect
11. C3×C22⋊D13C22semidirect
12. C3×D52C22semidirect
13. C3×D52C22semidirect
14. C3×C4×D13C22semidirect
15. C3×Dic26C22semidirect
16. C13×#41C2×C3semidirect
17. #6131C2×C3semidirect
18. D4×D13C2×C3semidirect
19. #6127C2×C3semidirect
20. C2×D52C2×C3semidirect
27. C22×D13C22×C3semidirect
28. Q8×C13C22×C3semidirect
29. D4×C13C22×C3semidirect
30. C2×C4×C13C22×C3semidirect
31. C22⋊D13C22×C3semidirect
32. D52C22×C3semidirect
33. D52C22×C3semidirect
34. C4×D13C22×C3semidirect
35. Dic26C22×C3semidirect
48. C3×#41D13semidirect
51. C3×C13H(5,𝔽2)semidirect
58. C3×Q8C2×D13semidirect
62. C3×D4C2×D13semidirect
64. C2×C3×C4C2×D13semidirect
67. #41C3×D13semidirect
70. C13C3×H(5,𝔽2)semidirect
79. Q8C2×C3×D13semidirect
83. D4C2×C3×D13semidirect
85. C2×C4C2×C3×D13semidirect
95. C3#88256direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C13×#412
C3×#61312
C3×D4×D132
C3×#61272
C2×C3×D522
#882563
C3×H(5,𝔽2)13