Finite Group Databasebeta

← C23×D4×D251 #44565156

Subgroups — C23×D4×D251 #44565156

Order 32128.

10552 conjugacy classes of subgroups; only the 3557 normal subgroups are shown.

normal subgroups only

This group has 10552 conjugacy classes of subgroups — too many to draw. Only its 3557 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
1C25×D251C212yesmaximal, complemented
2C23×D1004C211yesmaximal, characteristic, complemented
3C23×C22⋊D251C212yesmaximal, complemented
4C22×D4×D251C2156yesmaximal, direct product factor
5C23×C4×D251C211yesmaximal, characteristic, complemented
6C23×D4×C251C211yesmaximal, characteristic, complemented
7C24×D251—232no—
8C24×D251C2211yescharacteristic
9C24×D251C2212yescomplemented
10C24×D251C22156yescomplemented
11C22×D1004C22128yescomplemented
12C22×C22⋊D251C22156yescomplemented
13C2×D4×D251C221448yesdirect product factor
14C23×Dic251C2211yescharacteristic, complemented
15C22×C4×D251C22128yescomplemented
16C25×C251C2212yescomplemented
17C22×D4×C251C22128yescomplemented
18C23×C4×C251C2211yescharacteristic, complemented
19C23×D251—2448no—
20C23×D251—216no—
21C23×D251D4116yesdirect product factor
22C23×D251C23114yes—
23C23×D251C23128yescomplemented
24C23×D251C231224yescomplemented
25C2×D1004C231112yescomplemented
26C2×C22⋊D251C231224yescomplemented
27D4×D251C231512yesdirect product factor
28C22×Dic251C23114yescomplemented
29C2×C4×D251C231112yescomplemented
30C24×C251—216no—
31C24×C251C2311yescharacteristic
32C24×C251C23128yescomplemented
33C2×D4×C251C231112yescomplemented
34C22×C4×C251C23114yescomplemented
35C22×D251—2112no—
36C22×D251—2896no—
37C22×D251C2×D41112yesdirect product factor
38C22×D251C24128yes—
39C22×D251C241128yescomplemented
40C22×D251C24156yescomplemented
41D1004C24164yescomplemented
42C22⋊D251C241128yescomplemented
43C2×Dic251C24128yescomplemented
44C4×D251C24164yescomplemented
45C23×C251—2112no—
46C23×C251C2×D418yescomplemented
47C23×C251C2417yes—
48C23×C251C24156yescomplemented
49D4×C251C24164yescomplemented
50C2×C4×C251C24128yescomplemented
51C22×C251—2112no—
52C22×C251C22×D4128yescomplemented
53C22×C251C2517yes—
54C22×C251C25116yescomplemented
55C2×D251—2112no—
56C2×D251—2256no—
57C2×D251C22×D41112yesdirect product factor
58C2×D251C2518yes—
59C2×D251C25116yescomplemented
60C4×C251C2518yescomplemented
61Dic251C2518yescomplemented
62C2×C251—216no—
63C2×C251C23×D4114yescomplemented
64C2×C251C2611yescharacteristic
65D251—216no—
66D251C23×D4116yesdirect product factor
67C251C24×D411yescharacteristic, complemented, Sylow
68C24×D4—2511nomaximal, Sylow
69C26—2512no—
70C23×D4—2512no—
71C23×D4—25156no—
72C23×D4—2511no—
73C23×D4D25111yescharacteristic, direct product factor
74C24×C4—2511no—
75C25—50232no—
76C25—2512no—
77C25—25156no—
78C25—2511no—
79C25C2×D25112yescomplemented
80C22×D4—25156no—
81C22×D4—251448no—
82C22×D4—25128no—
83C22×D4C2×D251128yesdirect product factor
84C23×C4—2511no—
85C23×C4—25128no—
86C23×C4C2×D25111yescharacteristic, complemented
87C24—216no—
88C24—502448no—
89C24—50216no—
90C24—25128no—
91C24—251224no—
92C24—25116no—
93C24—25114no—
94C24C22×D25111yescharacteristic
95C24C22×D251128yescomplemented
96C2×D4—251224no—
97C2×D4—251512no—
98C2×D4—251112no—
99C2×D4C22×D2511112yesdirect product factor
100C22×C4—25114no—
101C22×C4—251112no—
102C22×C4C22×D251114yescomplemented
103C23—2112no—
104C23—502896no—
105C23—502112no—
106C23—25156no—
107C23—251128no—
108C23—251112no—
109C23—25128no—
110C23D4×D25118yesdirect product factor
111C23C23×D25117yes—
112C23C23×D251156yescomplemented
113D4—251128no—
114D4—25164no—
115D4C23×D251164yesdirect product factor
116C2×C4—25128no—
117C2×C4—25164no—
118C2×C4C23×D251128yescomplemented
119C22—2112no—
120C22—502256no—
121C22—502112no—
122C22—25116no—
123C22—251112no—
124C22—2518no—
125C22C2×D4×D251128yesdirect product factor
126C22C24×D25117yes—
127C22C24×D251116yescomplemented
128C4—2518no—
129C4C24×D25118yescomplemented
130C2—216no—
131C2—50216no—
132C2—25116no—
133C2C22×D4×D251114yesdirect product factor
134C2C25×D25111yescharacteristic
subgroupG/H
centerC24C22×D251
commutator, cosocleC2×C251C26
FrattiniC2C25×D251
FittingC23×D4×C251C2
socleC24×C251C23
hypercenterC23×D4D251
hypocenterC251C24×D4
LCSG ▷ C2×C251 ▷ C251
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C25×D251C2semidirect
2. C23×D1004C2semidirect
3. C23×C22⋊D251C2semidirect
4. C22×D4×D251C2direct (N × H)
5. C23×C4×D251C2semidirect
6. C23×D4×C251C2semidirect
9. C24×D251C22semidirect
10. C24×D251C22semidirect
11. C22×D1004C22semidirect
12. C22×C22⋊D251C22semidirect
13. C2×D4×D251C22direct (N × H)
14. C23×Dic251C22semidirect
15. C22×C4×D251C22semidirect
16. C25×C251C22semidirect
17. C22×D4×C251C22semidirect
18. C23×C4×C251C22semidirect
21. C23×D251D4direct (N × H)
23. C23×D251C23semidirect
24. C23×D251C23semidirect
25. C2×D1004C23semidirect
26. C2×C22⋊D251C23semidirect
27. D4×D251C23direct (N × H)
28. C22×Dic251C23semidirect
29. C2×C4×D251C23semidirect
32. C24×C251C23semidirect
33. C2×D4×C251C23semidirect
34. C22×C4×C251C23semidirect
37. C22×D251C2×D4direct (N × H)
39. C22×D251C24semidirect
40. C22×D251C24semidirect
41. D1004C24semidirect
42. C22⋊D251C24semidirect
43. C2×Dic251C24semidirect
44. C4×D251C24semidirect
46. C23×C251C2×D4semidirect
48. C23×C251C24semidirect
49. D4×C251C24semidirect
50. C2×C4×C251C24semidirect
52. C22×C251C22×D4semidirect
54. C22×C251C25semidirect
57. C2×D251C22×D4direct (N × H)
59. C2×D251C25semidirect
60. C4×C251C25semidirect
61. Dic251C25semidirect
63. C2×C251C23×D4semidirect
66. D251C23×D4direct (N × H)
67. C251C24×D4semidirect
73. C23×D4D251direct (N × H)
79. C25C2×D251semidirect
83. C22×D4C2×D251direct (N × H)
86. C23×C4C2×D251semidirect
95. C24C22×D251semidirect
99. C2×D4C22×D251direct (N × H)
102. C22×C4C22×D251semidirect
110. C23D4×D251direct (N × H)
112. C23C23×D251semidirect
115. D4C23×D251direct (N × H)
118. C2×C4C23×D251semidirect
125. C22C2×D4×D251direct (N × H)
127. C22C24×D251semidirect
129. C4C24×D251semidirect
133. C2C22×D4×D251direct (N × H)

Maximal subgroups

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

subgroupindex
C25×D2512
C23×D10042
C23×C22⋊D2512
C22×D4×D2512
C23×C4×D2512
C23×D4×C2512
C24×D4251