Finite Group Databasebeta

← H(3,𝔽4)×D101 #27185251

Subgroups — H(3,𝔽4)×D101 #27185251

Order 12928.

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#20265920C216yesmaximal, complemented
2#20265919C219yesmaximal, complemented
3#124×D101C219yesmaximal, complemented
4#120×D101C216yesmaximal, complemented
5H(3,𝔽4)×C101C211yesmaximal, characteristic, complemented
6C24×D101C2212yescomplemented
7C2×Q8×D101C2213yescomplemented
8C2×D4×D101C2219yes—
9#15692244C2219yescomplemented
10#15692243C2219yescomplemented
11#15692242C2219yescomplemented
12#15692241C22136yescomplemented
13#15692225C22118yescomplemented
14#15692224C2216yescomplemented
15#15692223C22118yescomplemented
16C22⋊C4×D101C22118yescomplemented
17C42×D101C2213yescomplemented
18#124×C101C2219yescomplemented
19#120×C101C2216yescomplemented
20C23×D101—412no—
21C23×D101C2316yes—
22C23×D101C2316yescomplemented
23C2×Dic202C2319yescomplemented
24Q8×D101—46no—
25C2×D404C2319yes—
26C2×C22⋊D101C23118yes—
27D4×D101—418no—
28C2×C4×D101C2319yes—
29C22⋊Dic101C23118yescomplemented
30#13108051C23136yescomplemented
31C4×Dic101C2319yescomplemented
32C24×C101C2312yescomplemented
33C2×Q8×C101C2313yescomplemented
34C2×D4×C101C2319yes—
35C22⋊C4×C101C23118yescomplemented
36C42×C101C2313yescomplemented
37C23×C101—46no—
38C23×C101C2416yes—
39C22×D101—236no—
40C22×D101—418no—
41C22×D101—432no—
42C22×D101C2416yes—
43C22×D101C2411yescharacteristic
44Q8×C101—43no—
45D4×C101—49no—
46C2×C4×C101C2419yes—
47C22⋊D101—418no—
48C2×Dic101C2419yes—
49D404—49no—
50C4×D101—218no—
51Dic202—49no—
52C22×C101—218no—
53C22×C101—48no—
54C22×C101C2511yescharacteristic
55C2×D101—218no—
56C2×D101—424no—
57C2×D101—424no—
58C2×D101H(5,𝔽2)16yes—
59C4×C101—29no—
60Dic101—29no—
61C2×C101—46no—
62C2×C101C2×H(5,𝔽2)13yes—
63D101—46no—
64D101H(3,𝔽4)14yesdirect product factor
65C2×H(3,𝔽4)—1011nomaximal, Sylow
66C101C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
67H(3,𝔽4)—1019no—
68H(3,𝔽4)—1016no—
69H(3,𝔽4)D10111yescharacteristic, direct product factor
70C2×#124—1019no—
71C2×#120—1016no—
72C25—1012no—
73C22×Q8—1013no—
74C22×D4—1019no—
75#124—1019no—
76#124—1019no—
77#124—10136no—
78#124—1019no—
79#124C2×D10119yescomplemented
80#120—1016no—
81#120—10118no—
82#120—10118no—
83#120C2×D10116yescomplemented
84C2×C22⋊C4—10118no—
85C2×C42—1013no—
86C24—1016no—
87C24—1016no—
88C24—40412no—
89C24C22×D10112yescomplemented
90C2×Q8—1019no—
91C2×Q8—4046no—
92C2×Q8C22×D10113yescomplemented
93C2×D4—10118no—
94C2×D4—1019no—
95C2×D4—40418no—
96C2×D4C22×D10119yes—
97C22×C4—1019no—
98C22⋊C4—10118no—
99C22⋊C4—10136no—
100C22⋊C4C22×D101118yescomplemented
101C42—1019no—
102C42C22×D10113yescomplemented
103C23—46no—
104C23—1011no—
105C23—1016no—
106C23—40418no—
107C23—40432no—
108C23—20236no—
109C23C23×D10116yes—
110Q8—43no—
111Q8—4049no—
112D4—49no—
113D4—4049no—
114D4—40418no—
115C2×C4—1019no—
116C2×C4—20218no—
117C2×C4C23×D10119yes—
118C22—218no—
119C22—48no—
120C22—1016no—
121C22—40424no—
122C22—40424no—
123C22—20218no—
124C22C24×D10111yescharacteristic
125C4—29no—
126C4—2029no—
127C2—46no—
128C2—1014no—
129C2—4046no—
130C2H(5,𝔽2)×D10113yes—
subgroupG/H
center, FrattiniC22C24×D101
commutator, socle, cosocleC22×C101C25
FittingH(3,𝔽4)×C101C2
hypercenterH(3,𝔽4)D101
hypocenterC101C2×H(3,𝔽4)
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #20265920C2semidirect
2. #20265919C2semidirect
3. #124×D101C2semidirect
4. #120×D101C2semidirect
5. H(3,𝔽4)×C101C2semidirect
6. C24×D101C22semidirect
7. C2×Q8×D101C22semidirect
9. #15692244C22semidirect
10. #15692243C22semidirect
11. #15692242C22semidirect
12. #15692241C22semidirect
13. #15692225C22semidirect
14. #15692224C22semidirect
15. #15692223C22semidirect
16. C22⋊C4×D101C22semidirect
17. C42×D101C22semidirect
18. #124×C101C22semidirect
19. #120×C101C22semidirect
22. C23×D101C23semidirect
23. C2×Dic202C23semidirect
29. C22⋊Dic101C23semidirect
30. #13108051C23semidirect
31. C4×Dic101C23semidirect
32. C24×C101C23semidirect
33. C2×Q8×C101C23semidirect
35. C22⋊C4×C101C23semidirect
36. C42×C101C23semidirect
64. D101H(3,𝔽4)direct (N × H)
66. C101C2×H(3,𝔽4)semidirect
69. H(3,𝔽4)D101direct (N × H)
79. #124C2×D101semidirect
83. #120C2×D101semidirect
89. C24C22×D101semidirect
92. C2×Q8C22×D101semidirect
100. C22⋊C4C22×D101semidirect
102. C42C22×D101semidirect

Maximal subgroups

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

subgroupindex
#202659202
#202659192
#124×D1012
#120×D1012
H(3,𝔽4)×C1012
C2×H(3,𝔽4)101