Finite Group Databasebeta

← H(3,𝔽4)×D127 #30652849

Subgroups — H(3,𝔽4)×D127 #30652849

Order 16256.

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#22550619C216yesmaximal, complemented
2#22550618C219yesmaximal, complemented
3#124×D127C219yesmaximal, complemented
4#120×D127C216yesmaximal, complemented
5H(3,𝔽4)×C127C211yesmaximal, characteristic, complemented
6C24×D127C2212yescomplemented
7C2×Q8×D127C2213yescomplemented
8C2×D4×D127C2219yes—
9#16861198C2219yescomplemented
10#16861197C2219yescomplemented
11#16861196C2219yescomplemented
12#16861195C22136yescomplemented
13#16861179C22118yescomplemented
14#16861178C2216yescomplemented
15#16861177C22118yescomplemented
16C22⋊C4×D127C22118yescomplemented
17C42×D127C2213yescomplemented
18#124×C127C2219yescomplemented
19#120×C127C2216yescomplemented
20C23×D127—412no—
21C23×D127C2316yes—
22C23×D127C2316yescomplemented
23C2×Dic254C2319yescomplemented
24Q8×D127—46no—
25C2×D508C2319yes—
26C2×C22⋊D127C23118yes—
27D4×D127—418no—
28C2×C4×D127C2319yes—
29C22⋊Dic127C23118yescomplemented
30(C2×D127)⋊C4C23136yescomplemented
31C4×Dic127C2319yescomplemented
32C24×C127C2312yescomplemented
33C2×Q8×C127C2313yescomplemented
34C2×D4×C127C2319yes—
35C22⋊C4×C127C23118yescomplemented
36C42×C127C2313yescomplemented
37C22×D127—236no—
38C22×D127—418no—
39C22×D127—432no—
40C22×D127C2416yes—
41C22×D127C2411yescharacteristic
42Dic254—49no—
43D508—49no—
44C22⋊D127—418no—
45C2×Dic127C2419yes—
46C4×D127—218no—
47C23×C127—46no—
48C23×C127C2416yes—
49Q8×C127—43no—
50D4×C127—49no—
51C2×C4×C127C2419yes—
52C22×C127—218no—
53C22×C127—48no—
54C22×C127C2511yescharacteristic
55C2×D127—218no—
56C2×D127—424no—
57C2×D127—424no—
58C2×D127H(5,𝔽2)16yes—
59C4×C127—29no—
60Dic127—29no—
61C2×C127—46no—
62C2×C127C2×H(5,𝔽2)13yes—
63D127—46no—
64D127H(3,𝔽4)14yesdirect product factor
65C2×H(3,𝔽4)—1271nomaximal, Sylow
66C127C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
67H(3,𝔽4)—1279no—
68H(3,𝔽4)—1276no—
69H(3,𝔽4)D12711yescharacteristic, direct product factor
70C2×#124—1279no—
71C2×#120—1276no—
72C25—1272no—
73C22×Q8—1273no—
74C22×D4—1279no—
75#124—1279no—
76#124—1279no—
77#124—12736no—
78#124—1279no—
79#124C2×D12719yescomplemented
80#120—1276no—
81#120—12718no—
82#120—12718no—
83#120C2×D12716yescomplemented
84C2×C22⋊C4—12718no—
85C2×C42—1273no—
86C24—1276no—
87C24—1276no—
88C24—50812no—
89C24C22×D12712yescomplemented
90C2×Q8—1279no—
91C2×Q8—5086no—
92C2×Q8C22×D12713yescomplemented
93C2×D4—12718no—
94C2×D4—1279no—
95C2×D4—50818no—
96C2×D4C22×D12719yes—
97C22×C4—1279no—
98C22⋊C4—12718no—
99C22⋊C4—12736no—
100C22⋊C4C22×D127118yescomplemented
101C42—1279no—
102C42C22×D12713yescomplemented
103C23—46no—
104C23—1271no—
105C23—1276no—
106C23—50818no—
107C23—50832no—
108C23—25436no—
109C23C23×D12716yes—
110Q8—43no—
111Q8—5089no—
112D4—49no—
113D4—5089no—
114D4—50818no—
115C2×C4—1279no—
116C2×C4—25418no—
117C2×C4C23×D12719yes—
118C22—218no—
119C22—48no—
120C22—1276no—
121C22—50824no—
122C22—50824no—
123C22—25418no—
124C22C24×D12711yescharacteristic
125C4—29no—
126C4—2549no—
127C2—46no—
128C2—1274no—
129C2—5086no—
130C2H(5,𝔽2)×D12713yes—
subgroupG/H
center, FrattiniC22C24×D127
commutator, socle, cosocleC22×C127C25
FittingH(3,𝔽4)×C127C2
hypercenterH(3,𝔽4)D127
hypocenterC127C2×H(3,𝔽4)
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #22550619C2semidirect
2. #22550618C2semidirect
3. #124×D127C2semidirect
4. #120×D127C2semidirect
5. H(3,𝔽4)×C127C2semidirect
6. C24×D127C22semidirect
7. C2×Q8×D127C22semidirect
9. #16861198C22semidirect
10. #16861197C22semidirect
11. #16861196C22semidirect
12. #16861195C22semidirect
13. #16861179C22semidirect
14. #16861178C22semidirect
15. #16861177C22semidirect
16. C22⋊C4×D127C22semidirect
17. C42×D127C22semidirect
18. #124×C127C22semidirect
19. #120×C127C22semidirect
22. C23×D127C23semidirect
23. C2×Dic254C23semidirect
29. C22⋊Dic127C23semidirect
30. (C2×D127)⋊C4C23semidirect
31. C4×Dic127C23semidirect
32. C24×C127C23semidirect
33. C2×Q8×C127C23semidirect
35. C22⋊C4×C127C23semidirect
36. C42×C127C23semidirect
64. D127H(3,𝔽4)direct (N × H)
66. C127C2×H(3,𝔽4)semidirect
69. H(3,𝔽4)D127direct (N × H)
79. #124C2×D127semidirect
83. #120C2×D127semidirect
89. C24C22×D127semidirect
92. C2×Q8C22×D127semidirect
100. C22⋊C4C22×D127semidirect
102. C42C22×D127semidirect

Maximal subgroups

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

subgroupindex
#225506192
#225506182
#124×D1272
#120×D1272
H(3,𝔽4)×C1272
C2×H(3,𝔽4)127