Finite Group Databasebeta

← H(3,𝔽4)×D163 #35228655

Subgroups — H(3,𝔽4)×D163 #35228655

Order 20864.

1408 conjugacy classes of subgroups; only the 459 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 459 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#23782408C216yesmaximal, complemented
2#23782407C219yesmaximal, complemented
3#124×D163C219yesmaximal, complemented
4#120×D163C216yesmaximal, complemented
5H(3,𝔽4)×C163C211yesmaximal, characteristic, complemented
6C24×D163C2212yescomplemented
7C2×Q8×D163C2213yescomplemented
8C2×D4×D163C2219yes—
9#19104724C2219yescomplemented
10#19104723C2219yescomplemented
11#19104722C2219yescomplemented
12#19104721C22136yescomplemented
13#19104705C22118yescomplemented
14#19104704C2216yescomplemented
15#19104703C22118yescomplemented
16C22⋊C4×D163C22118yescomplemented
17C42×D163C2213yescomplemented
18#124×C163C2219yescomplemented
19#120×C163C2216yescomplemented
20C23×D163—412no—
21C23×D163C2316yes—
22C23×D163C2316yescomplemented
23C2×Dic326C2319yescomplemented
24Q8×D163—46no—
25C2×D652C2319yes—
26C2×C22⋊D163C23118yes—
27D4×D163—418no—
28C2×C4×D163C2319yes—
29C22⋊Dic163C23118yescomplemented
30(C2×D163)⋊C4C23136yescomplemented
31C4×Dic163C2319yescomplemented
32C24×C163C2312yescomplemented
33C2×Q8×C163C2313yescomplemented
34C2×D4×C163C2319yes—
35C22⋊C4×C163C23118yescomplemented
36C42×C163C2313yescomplemented
37C22×D163—236no—
38C22×D163—418no—
39C22×D163—432no—
40C22×D163C2416yes—
41C22×D163C2411yescharacteristic
42Dic326—49no—
43D652—49no—
44C22⋊D163—418no—
45C2×Dic163C2419yes—
46C4×D163—218no—
47C23×C163—46no—
48C23×C163C2416yes—
49Q8×C163—43no—
50D4×C163—49no—
51C2×C4×C163C2419yes—
52C22×C163—218no—
53C22×C163—48no—
54C22×C163C2511yescharacteristic
55C2×D163—218no—
56C2×D163—424no—
57C2×D163—424no—
58C2×D163H(5,𝔽2)16yes—
59C4×C163—29no—
60Dic163—29no—
61C2×C163—46no—
62C2×C163C2×H(5,𝔽2)13yes—
63D163—46no—
64D163H(3,𝔽4)14yesdirect product factor
65C163C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
66C2×H(3,𝔽4)—1631nomaximal, Sylow
67H(3,𝔽4)—1639no—
68H(3,𝔽4)—1636no—
69H(3,𝔽4)D16311yescharacteristic, direct product factor
70C2×#124—1639no—
71C2×#120—1636no—
72C25—1632no—
73C22×Q8—1633no—
74C22×D4—1639no—
75#124—1639no—
76#124—1639no—
77#124—16336no—
78#124—1639no—
79#124C2×D16319yescomplemented
80#120—1636no—
81#120—16318no—
82#120—16318no—
83#120C2×D16316yescomplemented
84C2×C22⋊C4—16318no—
85C2×C42—1633no—
86C24—65212no—
87C24—1636no—
88C24—1636no—
89C24C22×D16312yescomplemented
90C2×Q8—6526no—
91C2×Q8—1639no—
92C2×Q8C22×D16313yescomplemented
93C2×D4—65218no—
94C2×D4—16318no—
95C2×D4—1639no—
96C2×D4C22×D16319yes—
97C22×C4—1639no—
98C22⋊C4—16318no—
99C22⋊C4—16336no—
100C22⋊C4C22×D163118yescomplemented
101C42—1639no—
102C42C22×D16313yescomplemented
103C23—46no—
104C23—32636no—
105C23—65218no—
106C23—65232no—
107C23—1636no—
108C23—1631no—
109C23C23×D16316yes—
110Q8—43no—
111Q8—6529no—
112D4—49no—
113D4—6529no—
114D4—65218no—
115C2×C4—32618no—
116C2×C4—1639no—
117C2×C4C23×D16319yes—
118C22—218no—
119C22—48no—
120C22—32618no—
121C22—65224no—
122C22—65224no—
123C22—1636no—
124C22C24×D16311yescharacteristic
125C4—29no—
126C4—3269no—
127C2—46no—
128C2—6526no—
129C2—1634no—
130C2H(5,𝔽2)×D16313yes—
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
FittingH(3,𝔽4)×C163C2
hypercenterH(3,𝔽4)D163
hypocenterC163C2×H(3,𝔽4)
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #23782408C2semidirect
2. #23782407C2semidirect
3. #124×D163C2semidirect
4. #120×D163C2semidirect
5. H(3,𝔽4)×C163C2semidirect
6. C24×D163C22semidirect
7. C2×Q8×D163C22semidirect
9. #19104724C22semidirect
10. #19104723C22semidirect
11. #19104722C22semidirect
12. #19104721C22semidirect
13. #19104705C22semidirect
14. #19104704C22semidirect
15. #19104703C22semidirect
16. C22⋊C4×D163C22semidirect
17. C42×D163C22semidirect
18. #124×C163C22semidirect
19. #120×C163C22semidirect
22. C23×D163C23semidirect
23. C2×Dic326C23semidirect
29. C22⋊Dic163C23semidirect
30. (C2×D163)⋊C4C23semidirect
31. C4×Dic163C23semidirect
32. C24×C163C23semidirect
33. C2×Q8×C163C23semidirect
35. C22⋊C4×C163C23semidirect
36. C42×C163C23semidirect
64. D163H(3,𝔽4)direct (N × H)
65. C163C2×H(3,𝔽4)semidirect
69. H(3,𝔽4)D163direct (N × H)
79. #124C2×D163semidirect
83. #120C2×D163semidirect
89. C24C22×D163semidirect
92. C2×Q8C22×D163semidirect
100. C22⋊C4C22×D163semidirect
102. C42C22×D163semidirect

Maximal subgroups

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

subgroupindex
#237824082
#237824072
#124×D1632
#120×D1632
H(3,𝔽4)×C1632
C2×H(3,𝔽4)163