Finite Group Databasebeta

← C2×#26045838 #37560253

Subgroups — C2×#26045838 #37560253

Order 23872.

898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 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
1#26045838C2116yesmaximal, direct product factor
2C2×#20238611C212yesmaximal, complemented
3C2×#20238610C213yesmaximal, complemented
4C22×D1492C213yesmaximal, complemented
5C2×D4×D373C216yesmaximal, complemented
6C2×#41×C373C211yesmaximal, characteristic, complemented
7C23×D373C2216yescomplemented
8#20238611C22116yescomplemented
9#20238610C22124yescomplemented
10C2×Dic746C2211yescharacteristic, complemented
11C2×D1492C2216yescomplemented
12C2×D1492C2213yescomplemented
13C2×D1492C22124yescomplemented
14C2×C22⋊D373C2216yescomplemented
15D4×D373C22148yescomplemented
16C2×C4×D373C2216yescomplemented
17#41×C373C2218yescomplemented
18C2×Q8×C373C2211yescharacteristic, complemented
19C2×D4×C373C2213yescomplemented
20C22×C4×C373C2213yescomplemented
21C22×D373—412no—
22C22×D373C2316yes—
23C22×D373C23124yescomplemented
24Dic746C2314yescomplemented
25D1492C23124yescomplemented
26D1492C23112yescomplemented
27C22⋊D373C23124yescomplemented
28C2×Dic373C2312yes—
29C4×D373C23124yescomplemented
30C23×C373C2313yes—
31Q8×C373C2314yescomplemented
32D4×C373C23112yescomplemented
33C2×C4×C373C2313yes—
34C2×C4×C373C2311yescharacteristic
35C2×C4×C373C23112yescomplemented
36C22×C373—26no—
37C22×C373C2416yes—
38C22×C373C2411yescharacteristic
39C2×D373—212no—
40C2×D373—424no—
41C2×D373C24112yes—
42C4×C373C2412yes—
43C4×C373C2416yes—
44Dic373C2414yes—
45C2×C373—26no—
46C2×C373H(5,𝔽2)12yescomplemented
47C2×C373C2511yescharacteristic
48D373—212no—
49C373C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
50C2×H(5,𝔽2)—3731nomaximal, Sylow
51H(5,𝔽2)—37316no—
52C2×#41—3732no—
53C2×#41—3733no—
54C2×#41D37311yescharacteristic, complemented
55C22×D4—3736no—
56C22×D4—3733no—
57C24—3736no—
58#41—37316no—
59#41—37324no—
60#41C2×D37318yescomplemented
61C2×Q8—3731no—
62C2×Q8C2×D37311yescharacteristic, complemented
63C2×D4—3736no—
64C2×D4—37348no—
65C2×D4—3736no—
66C2×D4—3733no—
67C2×D4—37324no—
68C2×D4C2×D37313yescomplemented
69C22×C4—3736no—
70C22×C4C2×D37313yescomplemented
71C23—37324no—
72C23—3736no—
73C23—149212no—
74C23C22×D37313yes—
75Q8—3734no—
76Q8C22×D37314yescomplemented
77D4—37324no—
78D4—37324no—
79D4—37312no—
80D4C22×D373112yescomplemented
81C2×C4—3732no—
82C2×C4—37324no—
83C2×C4C22×D37313yes—
84C2×C4C22×D37311yescharacteristic
85C2×C4C22×D373112yescomplemented
86C22—26no—
87C22—37312no—
88C22—149224no—
89C22—74612no—
90C22C23×D37316yes—
91C22C23×D37311yescharacteristic
92C4—3734no—
93C4C23×D37316yes—
94C4C23×D37312yes—
95C2—26no—
96C2—74612no—
97C2#2604583812yesdirect product factor
98C2C24×D37311yescharacteristic
subgroupG/H
centerC22C23×D373
commutator, cosocleC2×C373C25
FrattiniC2C24×D373
FittingC2×#41×C373C2
socleC22×C373C24
hypercenterC2×#41D373
hypocenterC373C2×H(5,𝔽2)
LCSG ▷ C2×C373 ▷ C373
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #26045838C2direct (N × H)
2. C2×#20238611C2semidirect
3. C2×#20238610C2semidirect
4. C22×D1492C2semidirect
5. C2×D4×D373C2semidirect
6. C2×#41×C373C2semidirect
7. C23×D373C22semidirect
8. #20238611C22semidirect
9. #20238610C22semidirect
10. C2×Dic746C22semidirect
11. C2×D1492C22semidirect
12. C2×D1492C22semidirect
13. C2×D1492C22semidirect
14. C2×C22⋊D373C22semidirect
15. D4×D373C22semidirect
16. C2×C4×D373C22semidirect
17. #41×C373C22semidirect
18. C2×Q8×C373C22semidirect
19. C2×D4×C373C22semidirect
20. C22×C4×C373C22semidirect
23. C22×D373C23semidirect
24. Dic746C23semidirect
25. D1492C23semidirect
26. D1492C23semidirect
27. C22⋊D373C23semidirect
29. C4×D373C23semidirect
31. Q8×C373C23semidirect
32. D4×C373C23semidirect
35. C2×C4×C373C23semidirect
46. C2×C373H(5,𝔽2)semidirect
49. C373C2×H(5,𝔽2)semidirect
54. C2×#41D373semidirect
60. #41C2×D373semidirect
62. C2×Q8C2×D373semidirect
68. C2×D4C2×D373semidirect
70. C22×C4C2×D373semidirect
76. Q8C22×D373semidirect
80. D4C22×D373semidirect
85. C2×C4C22×D373semidirect
97. C2#26045838direct (N × H)

Maximal subgroups

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

subgroupindex
#260458382
C2×#202386112
C2×#202386102
C22×D14922
C2×D4×D3732
C2×#41×C3732
C2×H(5,𝔽2)373