Finite Group Databasebeta

← C2×C4×D4×C421 #41002487

Subgroups — C2×C4×D4×C421 #41002487

Order 26944.

426 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 274 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
1C22×D4×C421C211yesmaximal, characteristic
2C23×C4×C421C212yesmaximal, complemented
3C4×D4×C421C218yesmaximal, direct product factor
4C2×C4⋊C4×C421C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×C421C212yesmaximal, complemented
6C2×C42×C421C211yesmaximal, characteristic, complemented
7C24×C421C2212yes—
8C2×D4×C421C418yesdirect product factor
9C2×D4×C421C2214yes—
10C22×C4×C421—28no—
11C22×C4×C421C2211yescharacteristic
12C22×C4×C421C2212yes—
13C22×C4×C421C2211yescharacteristic
14C22×C4×C421C2211yescharacteristic
15C22×C4×C421C2218yescomplemented
16C4⋊C4×C421C2214yescomplemented
17C22⋊C4×C421C2218yescomplemented
18C42×C421C2214yescomplemented
19C23×C421—24no—
20C23×C421—24no—
21C23×C421C2×C418yescomplemented
22C23×C421C2314yes—
23C23×C421C2311yescharacteristic
24D4×C421C2×C4116yesdirect product factor
25C2×C4×C421—24no—
26C2×C4×C421—22no—
27C2×C4×C421—216no—
28C2×C4×C421C2×C414yescomplemented
29C2×C4×C421D414yesdirect product factor
30C2×C4×C421C2312yes—
31C2×C4×C421C2314yes—
32C2×C4×C421C2312yes—
33C2×C4×C421C2312yes—
34C22×C421—216no—
35C22×C421—24no—
36C22×C421—24no—
37C22×C421C22×C412yes—
38C22×C421C22×C418yescomplemented
39C22×C421C2×D412yes—
40C22×C421#4112yes—
41C22×C421C2411yescharacteristic
42C4×C421—24no—
43C4×C421—22no—
44C4×C421C22×C414yescomplemented
45C4×C421C2×D414yesdirect product factor
46C2×C421—28no—
47C2×C421C4×D414yesdirect product factor
48C2×C421C23×C411yescharacteristic
49C2×C421C22×D411yescharacteristic
50C2×C421C2×#4111yescharacteristic
51C421C2×C4×D411yescharacteristic, direct product factor, Sylow
52C2×C4×D4C42111yesmaximal, characteristic, direct product factor, Sylow
53C22×D4C2×C42111yescharacteristic
54C23×C4C2×C42112yescomplemented
55C4×D4C2×C42118yesdirect product factor
56C2×C4⋊C4C2×C42111yescharacteristic, complemented
57C2×C22⋊C4C2×C42112yescomplemented
58C2×C42C2×C42111yescharacteristic, complemented
59C24C22×C42112yes—
60C2×D4C4×C42118yesdirect product factor
61C2×D4C22×C42114yes—
62C22×C4—28no—
63C22×C4C22×C42111yescharacteristic
64C22×C4C22×C42112yes—
65C22×C4C22×C42111yescharacteristic
66C22×C4C22×C42111yescharacteristic
67C22×C4C22×C42118yescomplemented
68C4⋊C4C22×C42114yescomplemented
69C22⋊C4C22×C42118yescomplemented
70C42C22×C42114yescomplemented
71C23—24no—
72C23—24no—
73C23C2×C4×C42118yescomplemented
74C23C23×C42114yes—
75C23C23×C42111yescharacteristic
76D4C2×C4×C421116yesdirect product factor
77C2×C4—24no—
78C2×C4—22no—
79C2×C4—216no—
80C2×C4C2×C4×C42114yescomplemented
81C2×C4D4×C42114yesdirect product factor
82C2×C4C23×C42112yes—
83C2×C4C23×C42114yes—
84C2×C4C23×C42112yes—
85C2×C4C23×C42112yes—
86C22—216no—
87C22—24no—
88C22—24no—
89C22C22×C4×C42112yes—
90C22C22×C4×C42118yescomplemented
91C22C2×D4×C42112yes—
92C22#41×C42112yes—
93C22C24×C42111yescharacteristic
94C4—24no—
95C4—22no—
96C4C22×C4×C42114yescomplemented
97C4C2×D4×C42114yesdirect product factor
98C2—28no—
99C2C4×D4×C42114yesdirect product factor
100C2C23×C4×C42111yescharacteristic
101C2C22×D4×C42111yescharacteristic
102C2C2×#41×C42111yescharacteristic
subgroupG/H
centerC22×C4×C421C22
commutatorC2C23×C4×C421
Frattini, cosocleC22C24×C421
socleC23×C421C23
prime radicalC22×D4×C421C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C421 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C4×C421C2semidirect
3. C4×D4×C421C2direct (N × H)
4. C2×C4⋊C4×C421C2semidirect
5. C2×C22⋊C4×C421C2semidirect
6. C2×C42×C421C2semidirect
8. C2×D4×C421C4direct (N × H)
15. C22×C4×C421C22semidirect
16. C4⋊C4×C421C22semidirect
17. C22⋊C4×C421C22semidirect
18. C42×C421C22semidirect
21. C23×C421C2×C4semidirect
24. D4×C421C2×C4direct (N × H)
28. C2×C4×C421C2×C4semidirect
29. C2×C4×C421D4direct (N × H)
38. C22×C421C22×C4semidirect
44. C4×C421C22×C4semidirect
45. C4×C421C2×D4direct (N × H)
47. C2×C421C4×D4direct (N × H)
51. C421C2×C4×D4direct (N × H)
52. C2×C4×D4C421direct (N × H)
54. C23×C4C2×C421semidirect
55. C4×D4C2×C421direct (N × H)
56. C2×C4⋊C4C2×C421semidirect
57. C2×C22⋊C4C2×C421semidirect
58. C2×C42C2×C421semidirect
60. C2×D4C4×C421direct (N × H)
67. C22×C4C22×C421semidirect
68. C4⋊C4C22×C421semidirect
69. C22⋊C4C22×C421semidirect
70. C42C22×C421semidirect
73. C23C2×C4×C421semidirect
76. D4C2×C4×C421direct (N × H)
80. C2×C4C2×C4×C421semidirect
81. C2×C4D4×C421direct (N × H)
90. C22C22×C4×C421semidirect
96. C4C22×C4×C421semidirect
97. C4C2×D4×C421direct (N × H)
99. C2C4×D4×C421direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×C4212
C23×C4×C4212
C4×D4×C4212
C2×C4⋊C4×C4212
C2×C22⋊C4×C4212
C2×C42×C4212
C2×C4×D4421