Finite Group Databasebeta

← C24×C22⋊C4×C23 #19210265

Subgroups — C24×C22⋊C4×C23 #19210265

Order 5888.

43678 conjugacy classes of subgroups; only the 15454 normal subgroups are shown.

normal subgroups only

This group has 43678 conjugacy classes of subgroups — too many to draw. Only its 15454 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
1C27×C23C211yesmaximal, characteristic
2C25×C4×C23C212yesmaximal, complemented
3C23×C22⋊C4×C23C2160yesmaximal, direct product factor
4C26×C23—232no—
5C26×C23C4132yescomplemented
6C26×C23C22130yes—
7C26×C23C2211yescharacteristic
8C24×C4×C23—232no—
9C24×C4×C23C22160yescomplemented
10C22×C22⋊C4×C23C221560yesdirect product factor
11C25×C23—2512no—
12C25×C23—2480no—
13C25×C23C2×C4116yes—
14C25×C23C2×C41480yescomplemented
15C25×C23D4132yes—
16C25×C23C231140yes—
17C25×C23C23115yes—
18C23×C4×C23—2480no—
19C23×C4×C23C231280yescomplemented
20C2×C22⋊C4×C23C231960yesdirect product factor
21C24×C23—23840no—
22C24×C23—21120no—
23C24×C23C22⋊C41256yesdirect product factor
24C24×C23C22×C41120yes—
25C24×C23C22×C411120yescomplemented
26C24×C23C2×D41240yes—
27C24×C23C241120yes—
28C24×C23C24135yes—
29C22×C4×C23—21120no—
30C22×C4×C23C241240yescomplemented
31C22⋊C4×C23C241256yesdirect product factor
32C24×C22⋊C4C2311yesmaximal, characteristic, direct product factor, Sylow
33C23×C23—24480no—
34C23×C23—2480no—
35C23×C23C2×C22⋊C41960yesdirect product factor
36C23×C23C23×C41140yes—
37C23×C23C23×C41480yescomplemented
38C23×C23C22×D41280yes—
39C23×C23C25116yes—
40C23×C23C25115yes—
41C2×C4×C23—2480no—
42C2×C4×C23C25132yescomplemented
43C27C2×C2311yescharacteristic
44C25×C4C2×C2312yescomplemented
45C23×C22⋊C4C2×C23160yesdirect product factor
46C22×C23—232no—
47C22×C23—2960no—
48C22×C23C22×C22⋊C41560yesdirect product factor
49C22×C23C24×C4130yes—
50C22×C23C24×C4132yescomplemented
51C22×C23C23×D4160yes—
52C22×C23C2611yescharacteristic
53C4×C23—232no—
54C26—232no—
55C26C4×C23132yescomplemented
56C26C22×C23130yes—
57C26C22×C2311yescharacteristic
58C24×C4—232no—
59C24×C4C22×C23160yescomplemented
60C22×C22⋊C4C22×C231560yesdirect product factor
61C2×C23—232no—
62C2×C23C23×C22⋊C4160yesdirect product factor
63C2×C23C25×C411yescharacteristic
64C2×C23C24×D412yes—
65C25—2512no—
66C25—2480no—
67C25C2×C4×C23116yes—
68C25C2×C4×C231480yescomplemented
69C25D4×C23132yes—
70C25C23×C231140yes—
71C25C23×C23115yes—
72C23×C4—2480no—
73C23×C4C23×C231280yescomplemented
74C2×C22⋊C4C23×C231960yesdirect product factor
75C23C24×C22⋊C411yescharacteristic, direct product factor, Sylow
76C24—23840no—
77C24—21120no—
78C24C22⋊C4×C231256yesdirect product factor
79C24C22×C4×C231120yes—
80C24C22×C4×C2311120yescomplemented
81C24C2×D4×C231240yes—
82C24C24×C231120yes—
83C24C24×C23135yes—
84C22×C4—21120no—
85C22×C4C24×C231240yescomplemented
86C22⋊C4C24×C231256yesdirect product factor
87C23—2480no—
88C23—24480no—
89C23C2×C22⋊C4×C231960yesdirect product factor
90C23C23×C4×C231140yes—
91C23C23×C4×C231480yescomplemented
92C23C22×D4×C231280yes—
93C23C25×C23116yes—
94C23C25×C23115yes—
95C2×C4—2480no—
96C2×C4C25×C23132yescomplemented
97C22—232no—
98C22—2960no—
99C22C22×C22⋊C4×C231560yesdirect product factor
100C22C24×C4×C23130yes—
101C22C24×C4×C23132yescomplemented
102C22C23×D4×C23160yes—
103C22C26×C2311yescharacteristic
104C4—232no—
105C2—232no—
106C2C23×C22⋊C4×C23160yesdirect product factor
107C2C25×C4×C2311yescharacteristic
108C2C24×D4×C2312yes—
subgroupG/H
center, socleC26×C23C22
commutatorC2C25×C4×C23
Frattini, cosocleC22C26×C23
prime radicalC27×C23C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C26×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C25×C4×C23C2semidirect
3. C23×C22⋊C4×C23C2direct (N × H)
5. C26×C23C4semidirect
9. C24×C4×C23C22semidirect
10. C22×C22⋊C4×C23C22direct (N × H)
14. C25×C23C2×C4semidirect
19. C23×C4×C23C23semidirect
20. C2×C22⋊C4×C23C23direct (N × H)
23. C24×C23C22⋊C4direct (N × H)
25. C24×C23C22×C4semidirect
30. C22×C4×C23C24semidirect
31. C22⋊C4×C23C24direct (N × H)
32. C24×C22⋊C4C23direct (N × H)
35. C23×C23C2×C22⋊C4direct (N × H)
37. C23×C23C23×C4semidirect
42. C2×C4×C23C25semidirect
44. C25×C4C2×C23semidirect
45. C23×C22⋊C4C2×C23direct (N × H)
48. C22×C23C22×C22⋊C4direct (N × H)
50. C22×C23C24×C4semidirect
55. C26C4×C23semidirect
59. C24×C4C22×C23semidirect
60. C22×C22⋊C4C22×C23direct (N × H)
62. C2×C23C23×C22⋊C4direct (N × H)
68. C25C2×C4×C23semidirect
73. C23×C4C23×C23semidirect
74. C2×C22⋊C4C23×C23direct (N × H)
75. C23C24×C22⋊C4direct (N × H)
78. C24C22⋊C4×C23direct (N × H)
80. C24C22×C4×C23semidirect
85. C22×C4C24×C23semidirect
86. C22⋊C4C24×C23direct (N × H)
89. C23C2×C22⋊C4×C23direct (N × H)
91. C23C23×C4×C23semidirect
96. C2×C4C25×C23semidirect
99. C22C22×C22⋊C4×C23direct (N × H)
101. C22C24×C4×C23semidirect
106. C2C23×C22⋊C4×C23direct (N × H)

Maximal subgroups

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

subgroupindex
C27×C232
C25×C4×C232
C23×C22⋊C4×C232
C24×C22⋊C423