Finite Group Databasebeta

← H(3,𝔽4)×D173 #36367682

Subgroups — H(3,𝔽4)×D173 #36367682

Order 22144.

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#26012234C216yesmaximal, complemented
2#26012233C219yesmaximal, complemented
3#124×D173C219yesmaximal, complemented
4#120×D173C216yesmaximal, complemented
5H(3,𝔽4)×C173C211yesmaximal, characteristic, complemented
6C24×D173C2212yescomplemented
7C2×Q8×D173C2213yescomplemented
8C2×D4×D173C2219yes—
9#19149668C2219yescomplemented
10#19149667C2219yescomplemented
11#19149666C2219yescomplemented
12#19149665C22136yescomplemented
13#19149649C22118yescomplemented
14#19149648C2216yescomplemented
15#19149647C22118yescomplemented
16C22⋊C4×D173C22118yescomplemented
17C42×D173C2213yescomplemented
18#124×C173C2219yescomplemented
19#120×C173C2216yescomplemented
20C23×D173—412no—
21C23×D173C2316yes—
22C23×D173C2316yescomplemented
23C2×Dic346C2319yescomplemented
24Q8×D173—46no—
25C2×D692C2319yes—
26C2×C22⋊D173C23118yes—
27D4×D173—418no—
28C2×C4×D173C2319yes—
29C22⋊Dic173C23118yescomplemented
30#14583803C23136yescomplemented
31C4×Dic173C2319yescomplemented
32C24×C173C2312yescomplemented
33C2×Q8×C173C2313yescomplemented
34C2×D4×C173C2319yes—
35C22⋊C4×C173C23118yescomplemented
36C42×C173C2313yescomplemented
37C22×D173—236no—
38C22×D173—418no—
39C22×D173—432no—
40C22×D173C2416yes—
41C22×D173C2411yescharacteristic
42Dic346—49no—
43D692—49no—
44C22⋊D173—418no—
45C2×Dic173C2419yes—
46C4×D173—218no—
47C23×C173—46no—
48C23×C173C2416yes—
49Q8×C173—43no—
50D4×C173—49no—
51C2×C4×C173C2419yes—
52C22×C173—218no—
53C22×C173—48no—
54C22×C173C2511yescharacteristic
55C2×D173—218no—
56C2×D173—424no—
57C2×D173—424no—
58C2×D173H(5,𝔽2)16yes—
59C4×C173—29no—
60Dic173—29no—
61C2×C173—46no—
62C2×C173C2×H(5,𝔽2)13yes—
63D173—46no—
64D173H(3,𝔽4)14yesdirect product factor
65C173C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
66C2×H(3,𝔽4)—1731nomaximal, Sylow
67H(3,𝔽4)—1739no—
68H(3,𝔽4)—1736no—
69H(3,𝔽4)D17311yescharacteristic, direct product factor
70C2×#124—1739no—
71C2×#120—1736no—
72C25—1732no—
73C22×Q8—1733no—
74C22×D4—1739no—
75#124—1739no—
76#124—1739no—
77#124—17336no—
78#124—1739no—
79#124C2×D17319yescomplemented
80#120—1736no—
81#120—17318no—
82#120—17318no—
83#120C2×D17316yescomplemented
84C2×C22⋊C4—17318no—
85C2×C42—1733no—
86C24—1736no—
87C24—1736no—
88C24—69212no—
89C24C22×D17312yescomplemented
90C2×Q8—1739no—
91C2×Q8—6926no—
92C2×Q8C22×D17313yescomplemented
93C2×D4—17318no—
94C2×D4—1739no—
95C2×D4—69218no—
96C2×D4C22×D17319yes—
97C22×C4—1739no—
98C22⋊C4—17318no—
99C22⋊C4—17336no—
100C22⋊C4C22×D173118yescomplemented
101C42—1739no—
102C42C22×D17313yescomplemented
103C23—46no—
104C23—34636no—
105C23—1736no—
106C23—1731no—
107C23—69218no—
108C23—69232no—
109C23C23×D17316yes—
110Q8—43no—
111Q8—6929no—
112D4—49no—
113D4—6929no—
114D4—69218no—
115C2×C4—34618no—
116C2×C4—1739no—
117C2×C4C23×D17319yes—
118C22—218no—
119C22—48no—
120C22—34618no—
121C22—1736no—
122C22—69224no—
123C22—69224no—
124C22C24×D17311yescharacteristic
125C4—29no—
126C4—3469no—
127C2—46no—
128C2—1734no—
129C2—6926no—
130C2H(5,𝔽2)×D17313yes—
subgroupG/H
center, FrattiniC22C24×D173
commutator, socle, cosocleC22×C173C25
FittingH(3,𝔽4)×C173C2
hypercenterH(3,𝔽4)D173
hypocenterC173C2×H(3,𝔽4)
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #26012234C2semidirect
2. #26012233C2semidirect
3. #124×D173C2semidirect
4. #120×D173C2semidirect
5. H(3,𝔽4)×C173C2semidirect
6. C24×D173C22semidirect
7. C2×Q8×D173C22semidirect
9. #19149668C22semidirect
10. #19149667C22semidirect
11. #19149666C22semidirect
12. #19149665C22semidirect
13. #19149649C22semidirect
14. #19149648C22semidirect
15. #19149647C22semidirect
16. C22⋊C4×D173C22semidirect
17. C42×D173C22semidirect
18. #124×C173C22semidirect
19. #120×C173C22semidirect
22. C23×D173C23semidirect
23. C2×Dic346C23semidirect
29. C22⋊Dic173C23semidirect
30. #14583803C23semidirect
31. C4×Dic173C23semidirect
32. C24×C173C23semidirect
33. C2×Q8×C173C23semidirect
35. C22⋊C4×C173C23semidirect
36. C42×C173C23semidirect
64. D173H(3,𝔽4)direct (N × H)
65. C173C2×H(3,𝔽4)semidirect
69. H(3,𝔽4)D173direct (N × H)
79. #124C2×D173semidirect
83. #120C2×D173semidirect
89. C24C22×D173semidirect
92. C2×Q8C22×D173semidirect
100. C22⋊C4C22×D173semidirect
102. C42C22×D173semidirect

Maximal subgroups

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

subgroupindex
#260122342
#260122332
#124×D1732
#120×D1732
H(3,𝔽4)×C1732
C2×H(3,𝔽4)173