Finite Group Databasebeta

← C23×C42×D67 #31762379

Subgroups — C23×C42×D67 #31762379

Order 17152.

24030 conjugacy classes of subgroups; only the 13515 normal subgroups are shown.

normal subgroups only

This group has 24030 conjugacy classes of subgroups — too many to draw. Only its 13515 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
1C24×C4×D67C213yesmaximal
2C23×C4×Dic67C213yesmaximal, complemented
3C22×C42×D67C2156yesmaximal, direct product factor
4C23×C42×C67C211yesmaximal, characteristic, complemented
5C25×D67C2211yescharacteristic
6C24×Dic67C2213yes—
7C23×C4×D67C4196yesdirect product factor
8C23×C4×D67C22184yes—
9C22×C4×Dic67C22184yescomplemented
10C2×C42×D67C221448yesdirect product factor
11C24×C4×C67C2213yes—
12C22×C42×C67C22128yescomplemented
13C24×D67C2×C4148yes—
14C24×D67C23114yes—
15C23×Dic67C2×C4148yescomplemented
16C23×Dic67C23142yes—
17C22×C4×D67C2×C411344yesdirect product factor
18C22×C4×D67C231336yes—
19C2×C4×Dic67C231336yescomplemented
20C42×D67C231512yesdirect product factor
21C25×C67C2311yescharacteristic
22C23×C4×C67C2×C4148yescomplemented
23C23×C4×C67C23142yes—
24C2×C42×C67C231112yescomplemented
25C23×D67C421256yesdirect product factor
26C23×D67C22×C41336yes—
27C23×D67C24128yes—
28C22×Dic67C22×C41336yescomplemented
29C22×Dic67C24184yes—
30C2×C4×D67C22×C412688yesdirect product factor
31C2×C4×D67C241192yes—
32C4×Dic67C241192yescomplemented
33C24×C67C22×C4124yes—
34C24×C67C2417yes—
35C22×C4×C67C22×C41336yescomplemented
36C22×C4×C67C24184yes—
37C42×C67C24164yescomplemented
38C23×C67C2×C42164yescomplemented
39C23×C67C23×C4184yes—
40C23×C67C2517yes—
41C22×D67C2×C421896yesdirect product factor
42C22×D67C23×C41336yes—
43C22×D67C2518yes—
44C2×C4×C67C23×C41336yescomplemented
45C2×C4×C67C25124yes—
46C2×Dic67C23×C41336yescomplemented
47C2×Dic67C25124yes—
48C4×D67C23×C41768yesdirect product factor
49C22×C67C22×C421112yescomplemented
50C22×C67C24×C4142yes—
51C22×C67C2611yescharacteristic
52C2×D67C22×C421448yesdirect product factor
53C2×D67C24×C4148yes—
54C4×C67C24×C4148yescomplemented
55Dic67C24×C4148yescomplemented
56C24×C42—671nomaximal, Sylow
57C2×C67C23×C42128yescomplemented
58C2×C67C25×C413yes—
59D67C23×C42132yesdirect product factor
60C25×C4—673no—
61C23×C42—6756no—
62C23×C42—673no—
63C23×C42D6711yescharacteristic, direct product factor
64C67C24×C4211yescharacteristic, complemented, Sylow
65C26—671no—
66C24×C4—673no—
67C24×C4—6796no—
68C24×C4—6784no—
69C24×C4C2×D6713yes—
70C22×C42—6784no—
71C22×C42—67448no—
72C22×C42C2×D67128yesdirect product factor
73C25—6748no—
74C25—6714no—
75C25C22×D6711yescharacteristic
76C23×C4—6748no—
77C23×C4—6742no—
78C23×C4—671344no—
79C23×C4—67336no—
80C23×C4C4×D67148yesdirect product factor
81C23×C4C22×D67142yes—
82C2×C42—67336no—
83C2×C42—67512no—
84C2×C42C22×D671112yesdirect product factor
85C24—67256no—
86C24—67336no—
87C24—6728no—
88C24C2×C4×D67124yes—
89C24C23×D6717yes—
90C22×C4—67336no—
91C22×C4—6784no—
92C22×C4—672688no—
93C22×C4—67192no—
94C22×C4C2×C4×D671336yesdirect product factor
95C22×C4C23×D67184yes—
96C42—67192no—
97C42C23×D67164yesdirect product factor
98C23—67896no—
99C23—67336no—
100C23—678no—
101C23C42×D67164yesdirect product factor
102C23C22×C4×D67184yes—
103C23C24×D6717yes—
104C2×C4—67336no—
105C2×C4—6724no—
106C2×C4—67768no—
107C2×C4C22×C4×D671336yesdirect product factor
108C2×C4C24×D67124yes—
109C22—67448no—
110C22—6748no—
111C22C2×C42×D671112yesdirect product factor
112C22C23×C4×D67142yes—
113C22C25×D6711yescharacteristic
114C4—6748no—
115C4C23×C4×D67148yesdirect product factor
116C2—6732no—
117C2C22×C42×D67128yesdirect product factor
118C2C24×C4×D6713yes—
subgroupG/H
center, hypercenterC23×C42D67
commutator, hypocenterC67C24×C42
FrattiniC22C25×D67
FittingC23×C42×C67C2
socleC25×C67C23
prime radicalC25×D67C22
cosocleC22×C67C26
LCSG ▷ C67
UCSC1 ◁ C23×C42

Complements

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

NH ≅ G/Ntype
2. C23×C4×Dic67C2semidirect
3. C22×C42×D67C2direct (N × H)
4. C23×C42×C67C2semidirect
7. C23×C4×D67C4direct (N × H)
9. C22×C4×Dic67C22semidirect
10. C2×C42×D67C22direct (N × H)
12. C22×C42×C67C22semidirect
15. C23×Dic67C2×C4semidirect
17. C22×C4×D67C2×C4direct (N × H)
19. C2×C4×Dic67C23semidirect
20. C42×D67C23direct (N × H)
22. C23×C4×C67C2×C4semidirect
24. C2×C42×C67C23semidirect
25. C23×D67C42direct (N × H)
28. C22×Dic67C22×C4semidirect
30. C2×C4×D67C22×C4direct (N × H)
32. C4×Dic67C24semidirect
35. C22×C4×C67C22×C4semidirect
37. C42×C67C24semidirect
38. C23×C67C2×C42semidirect
41. C22×D67C2×C42direct (N × H)
44. C2×C4×C67C23×C4semidirect
46. C2×Dic67C23×C4semidirect
48. C4×D67C23×C4direct (N × H)
49. C22×C67C22×C42semidirect
52. C2×D67C22×C42direct (N × H)
54. C4×C67C24×C4semidirect
55. Dic67C24×C4semidirect
57. C2×C67C23×C42semidirect
59. D67C23×C42direct (N × H)
63. C23×C42D67direct (N × H)
64. C67C24×C42semidirect
72. C22×C42C2×D67direct (N × H)
80. C23×C4C4×D67direct (N × H)
84. C2×C42C22×D67direct (N × H)
94. C22×C4C2×C4×D67direct (N × H)
97. C42C23×D67direct (N × H)
101. C23C42×D67direct (N × H)
107. C2×C4C22×C4×D67direct (N × H)
111. C22C2×C42×D67direct (N × H)
115. C4C23×C4×D67direct (N × H)
117. C2C22×C42×D67direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C4×D672
C23×C4×Dic672
C22×C42×D672
C23×C42×C672
C24×C4267