Finite Group Databasebeta

← C2×#27133887 #37589182

Subgroups — C2×#27133887 #37589182

Order 24512.

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#27133887C2116yesmaximal, direct product factor
2C2×#20259458C212yesmaximal, complemented
3C2×#20259457C213yesmaximal, complemented
4C22×D1532C213yesmaximal, complemented
5C2×D4×D383C216yesmaximal, complemented
6C2×#41×C383C211yesmaximal, characteristic, complemented
7C23×D383C2216yescomplemented
8#20259458C22116yescomplemented
9#20259457C22124yescomplemented
10C2×Dic766C2211yescharacteristic, complemented
11C2×D1532C2216yescomplemented
12C2×D1532C2213yescomplemented
13C2×D1532C22124yescomplemented
14C2×C22⋊D383C2216yescomplemented
15D4×D383C22148yescomplemented
16C2×C4×D383C2216yescomplemented
17#41×C383C2218yescomplemented
18C2×Q8×C383C2211yescharacteristic, complemented
19C2×D4×C383C2213yescomplemented
20C22×C4×C383C2213yescomplemented
21C22×D383—412no—
22C22×D383C2316yes—
23C22×D383C23124yescomplemented
24Dic766C2314yescomplemented
25D1532C23124yescomplemented
26D1532C23112yescomplemented
27C22⋊D383C23124yescomplemented
28C2×Dic383C2312yes—
29C4×D383C23124yescomplemented
30C23×C383C2313yes—
31Q8×C383C2314yescomplemented
32D4×C383C23112yescomplemented
33C2×C4×C383C2311yescharacteristic
34C2×C4×C383C2313yes—
35C2×C4×C383C23112yescomplemented
36C22×C383—26no—
37C22×C383C2416yes—
38C22×C383C2411yescharacteristic
39C2×D383—212no—
40C2×D383—424no—
41C2×D383C24112yes—
42C4×C383C2412yes—
43C4×C383C2416yes—
44Dic383C2414yes—
45C2×C383—26no—
46C2×C383H(5,𝔽2)12yescomplemented
47C2×C383C2511yescharacteristic
48D383—212no—
49C383C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
50C2×H(5,𝔽2)—3831nomaximal, Sylow
51H(5,𝔽2)—38316no—
52C2×#41—3832no—
53C2×#41—3833no—
54C2×#41D38311yescharacteristic, complemented
55C22×D4—3836no—
56C22×D4—3833no—
57C24—3836no—
58#41—38316no—
59#41—38324no—
60#41C2×D38318yescomplemented
61C2×Q8—3831no—
62C2×Q8C2×D38311yescharacteristic, complemented
63C2×D4—3836no—
64C2×D4—38348no—
65C2×D4—3836no—
66C2×D4—3833no—
67C2×D4—38324no—
68C2×D4C2×D38313yescomplemented
69C22×C4—3836no—
70C22×C4C2×D38313yescomplemented
71C23—38324no—
72C23—3836no—
73C23—153212no—
74C23C22×D38313yes—
75Q8—3834no—
76Q8C22×D38314yescomplemented
77D4—38324no—
78D4—38324no—
79D4—38312no—
80D4C22×D383112yescomplemented
81C2×C4—3832no—
82C2×C4—38324no—
83C2×C4C22×D38313yes—
84C2×C4C22×D38311yescharacteristic
85C2×C4C22×D383112yescomplemented
86C22—26no—
87C22—38312no—
88C22—153224no—
89C22—76612no—
90C22C23×D38316yes—
91C22C23×D38311yescharacteristic
92C4—3834no—
93C4C23×D38316yes—
94C4C23×D38312yes—
95C2—26no—
96C2—76612no—
97C2#2713388712yesdirect product factor
98C2C24×D38311yescharacteristic
subgroupG/H
centerC22C23×D383
commutator, cosocleC2×C383C25
FrattiniC2C24×D383
FittingC2×#41×C383C2
socleC22×C383C24
hypercenterC2×#41D383
hypocenterC383C2×H(5,𝔽2)
LCSG ▷ C2×C383 ▷ C383
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #27133887C2direct (N × H)
2. C2×#20259458C2semidirect
3. C2×#20259457C2semidirect
4. C22×D1532C2semidirect
5. C2×D4×D383C2semidirect
6. C2×#41×C383C2semidirect
7. C23×D383C22semidirect
8. #20259458C22semidirect
9. #20259457C22semidirect
10. C2×Dic766C22semidirect
11. C2×D1532C22semidirect
12. C2×D1532C22semidirect
13. C2×D1532C22semidirect
14. C2×C22⋊D383C22semidirect
15. D4×D383C22semidirect
16. C2×C4×D383C22semidirect
17. #41×C383C22semidirect
18. C2×Q8×C383C22semidirect
19. C2×D4×C383C22semidirect
20. C22×C4×C383C22semidirect
23. C22×D383C23semidirect
24. Dic766C23semidirect
25. D1532C23semidirect
26. D1532C23semidirect
27. C22⋊D383C23semidirect
29. C4×D383C23semidirect
31. Q8×C383C23semidirect
32. D4×C383C23semidirect
35. C2×C4×C383C23semidirect
46. C2×C383H(5,𝔽2)semidirect
49. C383C2×H(5,𝔽2)semidirect
54. C2×#41D383semidirect
60. #41C2×D383semidirect
62. C2×Q8C2×D383semidirect
68. C2×D4C2×D383semidirect
70. C22×C4C2×D383semidirect
76. Q8C22×D383semidirect
80. D4C22×D383semidirect
85. C2×C4C22×D383semidirect
97. C2#27133887direct (N × H)

Maximal subgroups

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

subgroupindex
#271338872
C2×#202594582
C2×#202594572
C22×D15322
C2×D4×D3832
C2×#41×C3832
C2×H(5,𝔽2)383