Finite Group Databasebeta

← C23×D52⋊C2 #13107919

Subgroups — C23×D52⋊C2 #13107919

Order 1600.

2228 conjugacy classes of subgroups; only the 441 normal subgroups are shown.

normal subgroups only

This group has 2228 conjugacy classes of subgroups — too many to draw. Only its 441 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
1C23×D52C212yesmaximal, complemented
2C23×#1044C211yesmaximal, characteristic, complemented
3C22×D52⋊C2C2128yesmaximal, direct product factor
4C23×C5⋊D5C2211yescharacteristic
5C23×C5×D5—22no—
6C22×D52—214no—
7C22×D52C22128yescomplemented
8C22×#1044C22114yescomplemented
9C2×D52⋊C2C221112yesdirect product factor
10C23×C52D411yescharacteristic, complemented
11C22×C5⋊D5D417yescomplemented
12C22×C5⋊D5C2317yes—
13C22×C5×D5—228no—
14C2×D52—284no—
15C2×D52C23156yescomplemented
16C2×#1044C23128yescomplemented
17D52⋊C2C23164yesdirect product factor
18C24×D5—102no—
19C23×AGL(1,𝔽5)—101no—
20C22×C52C2×D417yescomplemented
21C2×C5⋊D5C2×D4121yescomplemented
22C2×C5⋊D5C2417yes—
23C2×C5×D5—256no—
24D52—256no—
25D52C24116yescomplemented
26#1044C2418yescomplemented
27C24×C5—102no—
28C23×D5—22no—
29C23×D5—1028no—
30C23×D5—101no—
31C23×D5—102no—
32C23×D5—1028no—
33C22×AGL(1,𝔽5)—1014no—
34C23×D4—251nomaximal, Sylow
35C2×C52C22×D417yescomplemented
36C5⋊D5C22×D417yescomplemented
37C5⋊D5C2511yescharacteristic
38C5×D5—216no—
39C23×C5—22no—
40C23×C5—21no—
41C23×C5—1028no—
42C22×D5—228no—
43C22×D5—1056no—
44C22×D5—107no—
45C22×D5—1014no—
46C22×D5—107no—
47C22×D5—1014no—
48C22×D5—10168no—
49C2×AGL(1,𝔽5)—1028no—
50C25—252no—
51C22×D4—2528no—
52C23×C4—251no—
53C52C23×D411yescharacteristic, complemented, Sylow
54C22×C5—214no—
55C22×C5—27no—
56C22×C5—1056no—
57C2×D5—256no—
58C2×D5—1016no—
59C2×D5—1021no—
60C2×D5—1042no—
61C2×D5—107no—
62C2×D5—1014no—
63C2×D5—10112no—
64AGL(1,𝔽5)—108no—
65C24—102no—
66C24—2528no—
67C24—251no—
68C24—5014no—
69C2×D4—25112no—
70C22×C4—2514no—
71C2×C5—214no—
72C2×C5—27no—
73C2×C5—1016no—
74D5—216no—
75D5—107no—
76D5—1014no—
77D5—101no—
78D5—102no—
79C23—1028no—
80C23—2556no—
81C23—257no—
82C23—257no—
83C23—5084no—
84C23D52⋊C211yescharacteristic, direct product factor
85D4—2564no—
86C2×C4—2528no—
87C5—22no—
88C5—21no—
89C22—1056no—
90C22—2516no—
91C22—257no—
92C22—2521no—
93C22—5056no—
94C22C2×D52⋊C217yesdirect product factor
95C4—258no—
96C2—1016no—
97C2—251no—
98C2—257no—
99C2C22×D52⋊C217yesdirect product factor
subgroupG/H
center, hypercenterC23D52⋊C2
commutator, cosocleC5⋊D5C25
FrattiniC1G
Fitting, socleC23×C52D4
hypocenterC52C23×D4
LCSG ▷ C5⋊D5 ▷ C52
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×D52C2semidirect
2. C23×#1044C2semidirect
3. C22×D52⋊C2C2direct (N × H)
7. C22×D52C22semidirect
8. C22×#1044C22semidirect
9. C2×D52⋊C2C22direct (N × H)
10. C23×C52D4semidirect
11. C22×C5⋊D5D4semidirect
15. C2×D52C23semidirect
16. C2×#1044C23semidirect
17. D52⋊C2C23direct (N × H)
20. C22×C52C2×D4semidirect
21. C2×C5⋊D5C2×D4semidirect
25. D52C24semidirect
26. #1044C24semidirect
35. C2×C52C22×D4semidirect
36. C5⋊D5C22×D4semidirect
53. C52C23×D4semidirect
84. C23D52⋊C2direct (N × H)
94. C22C2×D52⋊C2direct (N × H)
99. C2C22×D52⋊C2direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D522
C23×#10442
C22×D52⋊C22
C23×D425