Finite Group Databasebeta

← H(3,𝔽4)×D113 #28404693

Subgroups — H(3,𝔽4)×D113 #28404693

Order 14464.

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#20299341C216yesmaximal, complemented
2#20299340C219yesmaximal, complemented
3#124×D113C219yesmaximal, complemented
4#120×D113C216yesmaximal, complemented
5H(3,𝔽4)×C113C211yesmaximal, characteristic, complemented
6C24×D113C2212yescomplemented
7C2×Q8×D113C2213yescomplemented
8C2×D4×D113C2219yes—
9#16813611C2219yescomplemented
10#16813610C2219yescomplemented
11#16813609C2219yescomplemented
12#16813608C22136yescomplemented
13#16813592C22118yescomplemented
14#16813591C2216yescomplemented
15#16813590C22118yescomplemented
16C22⋊C4×D113C22118yescomplemented
17C42×D113C2213yescomplemented
18#124×C113C2219yescomplemented
19#120×C113C2216yescomplemented
20C23×D113—412no—
21C23×D113C2316yes—
22C23×D113C2316yescomplemented
23C2×Dic226C2319yescomplemented
24Q8×D113—46no—
25C2×D452C2319yes—
26C2×C22⋊D113C23118yes—
27D4×D113—418no—
28C2×C4×D113C2319yes—
29C22⋊Dic113C23118yescomplemented
30#14268624C23136yescomplemented
31C4×Dic113C2319yescomplemented
32C24×C113C2312yescomplemented
33C2×Q8×C113C2313yescomplemented
34C2×D4×C113C2319yes—
35C22⋊C4×C113C23118yescomplemented
36C42×C113C2313yescomplemented
37C23×C113—46no—
38C23×C113C2416yes—
39C22×D113—236no—
40C22×D113—418no—
41C22×D113—432no—
42C22×D113C2416yes—
43C22×D113C2411yescharacteristic
44Q8×C113—43no—
45D4×C113—49no—
46C2×C4×C113C2419yes—
47C22⋊D113—418no—
48C2×Dic113C2419yes—
49D452—49no—
50C4×D113—218no—
51Dic226—49no—
52C22×C113—218no—
53C22×C113—48no—
54C22×C113C2511yescharacteristic
55C2×D113—218no—
56C2×D113—424no—
57C2×D113—424no—
58C2×D113H(5,𝔽2)16yes—
59C4×C113—29no—
60Dic113—29no—
61C2×C113—46no—
62C2×C113C2×H(5,𝔽2)13yes—
63D113—46no—
64D113H(3,𝔽4)14yesdirect product factor
65C2×H(3,𝔽4)—1131nomaximal, Sylow
66C113C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
67H(3,𝔽4)—1139no—
68H(3,𝔽4)—1136no—
69H(3,𝔽4)D11311yescharacteristic, direct product factor
70C2×#124—1139no—
71C2×#120—1136no—
72C25—1132no—
73C22×Q8—1133no—
74C22×D4—1139no—
75#124—1139no—
76#124—1139no—
77#124—11336no—
78#124—1139no—
79#124C2×D11319yescomplemented
80#120—1136no—
81#120—11318no—
82#120—11318no—
83#120C2×D11316yescomplemented
84C2×C22⋊C4—11318no—
85C2×C42—1133no—
86C24—1136no—
87C24—1136no—
88C24—45212no—
89C24C22×D11312yescomplemented
90C2×Q8—1139no—
91C2×Q8—4526no—
92C2×Q8C22×D11313yescomplemented
93C2×D4—11318no—
94C2×D4—1139no—
95C2×D4—45218no—
96C2×D4C22×D11319yes—
97C22×C4—1139no—
98C22⋊C4—11318no—
99C22⋊C4—11336no—
100C22⋊C4C22×D113118yescomplemented
101C42—1139no—
102C42C22×D11313yescomplemented
103C23—46no—
104C23—1131no—
105C23—1136no—
106C23—45218no—
107C23—45232no—
108C23—22636no—
109C23C23×D11316yes—
110Q8—43no—
111Q8—4529no—
112D4—49no—
113D4—4529no—
114D4—45218no—
115C2×C4—1139no—
116C2×C4—22618no—
117C2×C4C23×D11319yes—
118C22—218no—
119C22—48no—
120C22—1136no—
121C22—45224no—
122C22—45224no—
123C22—22618no—
124C22C24×D11311yescharacteristic
125C4—29no—
126C4—2269no—
127C2—46no—
128C2—1134no—
129C2—4526no—
130C2H(5,𝔽2)×D11313yes—
subgroupG/H
center, FrattiniC22C24×D113
commutator, socle, cosocleC22×C113C25
FittingH(3,𝔽4)×C113C2
hypercenterH(3,𝔽4)D113
hypocenterC113C2×H(3,𝔽4)
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #20299341C2semidirect
2. #20299340C2semidirect
3. #124×D113C2semidirect
4. #120×D113C2semidirect
5. H(3,𝔽4)×C113C2semidirect
6. C24×D113C22semidirect
7. C2×Q8×D113C22semidirect
9. #16813611C22semidirect
10. #16813610C22semidirect
11. #16813609C22semidirect
12. #16813608C22semidirect
13. #16813592C22semidirect
14. #16813591C22semidirect
15. #16813590C22semidirect
16. C22⋊C4×D113C22semidirect
17. C42×D113C22semidirect
18. #124×C113C22semidirect
19. #120×C113C22semidirect
22. C23×D113C23semidirect
23. C2×Dic226C23semidirect
29. C22⋊Dic113C23semidirect
30. #14268624C23semidirect
31. C4×Dic113C23semidirect
32. C24×C113C23semidirect
33. C2×Q8×C113C23semidirect
35. C22⋊C4×C113C23semidirect
36. C42×C113C23semidirect
64. D113H(3,𝔽4)direct (N × H)
66. C113C2×H(3,𝔽4)semidirect
69. H(3,𝔽4)D113direct (N × H)
79. #124C2×D113semidirect
83. #120C2×D113semidirect
89. C24C22×D113semidirect
92. C2×Q8C22×D113semidirect
100. C22⋊C4C22×D113semidirect
102. C42C22×D113semidirect

Maximal subgroups

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

subgroupindex
#202993412
#202993402
#124×D1132
#120×D1132
H(3,𝔽4)×C1132
C2×H(3,𝔽4)113