Finite Group Databasebeta

← H(3,𝔽4)×D157 #34114255

Subgroups — H(3,𝔽4)×D157 #34114255

Order 20096.

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#23758016C216yesmaximal, complemented
2#23758015C219yesmaximal, complemented
3#124×D157C219yesmaximal, complemented
4#120×D157C216yesmaximal, complemented
5H(3,𝔽4)×C157C211yesmaximal, characteristic, complemented
6C24×D157C2212yescomplemented
7C2×Q8×D157C2213yescomplemented
8C2×D4×D157C2219yes—
9#19097760C2219yescomplemented
10#19097759C2219yescomplemented
11#19097758C2219yescomplemented
12#19097757C22136yescomplemented
13#19097741C22118yescomplemented
14#19097740C2216yescomplemented
15#19097739C22118yescomplemented
16C22⋊C4×D157C22118yescomplemented
17C42×D157C2213yescomplemented
18#124×C157C2219yescomplemented
19#120×C157C2216yescomplemented
20C23×D157—412no—
21C23×D157C2316yes—
22C23×D157C2316yescomplemented
23C2×Dic314C2319yescomplemented
24Q8×D157—46no—
25C2×D628C2319yes—
26C2×C22⋊D157C23118yes—
27D4×D157—418no—
28C2×C4×D157C2319yes—
29C22⋊Dic157C23118yescomplemented
30#14578931C23136yescomplemented
31C4×Dic157C2319yescomplemented
32C24×C157C2312yescomplemented
33C2×Q8×C157C2313yescomplemented
34C2×D4×C157C2319yes—
35C22⋊C4×C157C23118yescomplemented
36C42×C157C2313yescomplemented
37C22×D157—236no—
38C22×D157—418no—
39C22×D157—432no—
40C22×D157C2416yes—
41C22×D157C2411yescharacteristic
42Dic314—49no—
43D628—49no—
44C22⋊D157—418no—
45C2×Dic157C2419yes—
46C4×D157—218no—
47C23×C157—46no—
48C23×C157C2416yes—
49Q8×C157—43no—
50D4×C157—49no—
51C2×C4×C157C2419yes—
52C22×C157—218no—
53C22×C157—48no—
54C22×C157C2511yescharacteristic
55C2×D157—218no—
56C2×D157—424no—
57C2×D157—424no—
58C2×D157H(5,𝔽2)16yes—
59C4×C157—29no—
60Dic157—29no—
61C2×C157—46no—
62C2×C157C2×H(5,𝔽2)13yes—
63D157—46no—
64D157H(3,𝔽4)14yesdirect product factor
65C157C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
66C2×H(3,𝔽4)—1571nomaximal, Sylow
67H(3,𝔽4)—1579no—
68H(3,𝔽4)—1576no—
69H(3,𝔽4)D15711yescharacteristic, direct product factor
70C2×#124—1579no—
71C2×#120—1576no—
72C25—1572no—
73C22×Q8—1573no—
74C22×D4—1579no—
75#124—1579no—
76#124—1579no—
77#124—15736no—
78#124—1579no—
79#124C2×D15719yescomplemented
80#120—1576no—
81#120—15718no—
82#120—15718no—
83#120C2×D15716yescomplemented
84C2×C22⋊C4—15718no—
85C2×C42—1573no—
86C24—62812no—
87C24—1576no—
88C24—1576no—
89C24C22×D15712yescomplemented
90C2×Q8—6286no—
91C2×Q8—1579no—
92C2×Q8C22×D15713yescomplemented
93C2×D4—62818no—
94C2×D4—15718no—
95C2×D4—1579no—
96C2×D4C22×D15719yes—
97C22×C4—1579no—
98C22⋊C4—15718no—
99C22⋊C4—15736no—
100C22⋊C4C22×D157118yescomplemented
101C42—1579no—
102C42C22×D15713yescomplemented
103C23—46no—
104C23—31436no—
105C23—62818no—
106C23—62832no—
107C23—1576no—
108C23—1571no—
109C23C23×D15716yes—
110Q8—43no—
111Q8—6289no—
112D4—49no—
113D4—6289no—
114D4—62818no—
115C2×C4—31418no—
116C2×C4—1579no—
117C2×C4C23×D15719yes—
118C22—218no—
119C22—48no—
120C22—31418no—
121C22—62824no—
122C22—62824no—
123C22—1576no—
124C22C24×D15711yescharacteristic
125C4—29no—
126C4—3149no—
127C2—46no—
128C2—6286no—
129C2—1574no—
130C2H(5,𝔽2)×D15713yes—
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
FittingH(3,𝔽4)×C157C2
hypercenterH(3,𝔽4)D157
hypocenterC157C2×H(3,𝔽4)
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #23758016C2semidirect
2. #23758015C2semidirect
3. #124×D157C2semidirect
4. #120×D157C2semidirect
5. H(3,𝔽4)×C157C2semidirect
6. C24×D157C22semidirect
7. C2×Q8×D157C22semidirect
9. #19097760C22semidirect
10. #19097759C22semidirect
11. #19097758C22semidirect
12. #19097757C22semidirect
13. #19097741C22semidirect
14. #19097740C22semidirect
15. #19097739C22semidirect
16. C22⋊C4×D157C22semidirect
17. C42×D157C22semidirect
18. #124×C157C22semidirect
19. #120×C157C22semidirect
22. C23×D157C23semidirect
23. C2×Dic314C23semidirect
29. C22⋊Dic157C23semidirect
30. #14578931C23semidirect
31. C4×Dic157C23semidirect
32. C24×C157C23semidirect
33. C2×Q8×C157C23semidirect
35. C22⋊C4×C157C23semidirect
36. C42×C157C23semidirect
64. D157H(3,𝔽4)direct (N × H)
65. C157C2×H(3,𝔽4)semidirect
69. H(3,𝔽4)D157direct (N × H)
79. #124C2×D157semidirect
83. #120C2×D157semidirect
89. C24C22×D157semidirect
92. C2×Q8C22×D157semidirect
100. C22⋊C4C22×D157semidirect
102. C42C22×D157semidirect

Maximal subgroups

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

subgroupindex
#237580162
#237580152
#124×D1572
#120×D1572
H(3,𝔽4)×C1572
C2×H(3,𝔽4)157