Finite Group Databasebeta

← H(3,𝔽4)×D109 #28377151

Subgroups — H(3,𝔽4)×D109 #28377151

Order 13952.

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#20295769C216yesmaximal, complemented
2#20295768C219yesmaximal, complemented
3#124×D109C219yesmaximal, complemented
4#120×D109C216yesmaximal, complemented
5H(3,𝔽4)×C109C211yesmaximal, characteristic, complemented
6C24×D109C2212yescomplemented
7C2×Q8×D109C2213yescomplemented
8C2×D4×D109C2219yes—
9#16812688C2219yescomplemented
10#16812687C2219yescomplemented
11#16812686C2219yescomplemented
12#16812685C22136yescomplemented
13#16812669C22118yescomplemented
14#16812668C2216yescomplemented
15#16812667C22118yescomplemented
16C22⋊C4×D109C22118yescomplemented
17C42×D109C2213yescomplemented
18#124×C109C2219yescomplemented
19#120×C109C2216yescomplemented
20C23×D109—412no—
21C23×D109C2316yes—
22C23×D109C2316yescomplemented
23C2×Dic218C2319yescomplemented
24Q8×D109—46no—
25C2×D436C2319yes—
26C2×C22⋊D109C23118yes—
27D4×D109—418no—
28C2×C4×D109C2319yes—
29C22⋊Dic109C23118yescomplemented
30#13182006C23136yescomplemented
31C4×Dic109C2319yescomplemented
32C24×C109C2312yescomplemented
33C2×Q8×C109C2313yescomplemented
34C2×D4×C109C2319yes—
35C22⋊C4×C109C23118yescomplemented
36C42×C109C2313yescomplemented
37C23×C109—46no—
38C23×C109C2416yes—
39C22×D109—236no—
40C22×D109—418no—
41C22×D109—432no—
42C22×D109C2416yes—
43C22×D109C2411yescharacteristic
44Q8×C109—43no—
45D4×C109—49no—
46C2×C4×C109C2419yes—
47C22⋊D109—418no—
48C2×Dic109C2419yes—
49D436—49no—
50C4×D109—218no—
51Dic218—49no—
52C22×C109—218no—
53C22×C109—48no—
54C22×C109C2511yescharacteristic
55C2×D109—218no—
56C2×D109—424no—
57C2×D109—424no—
58C2×D109H(5,𝔽2)16yes—
59C4×C109—29no—
60Dic109—29no—
61C2×C109—46no—
62C2×C109C2×H(5,𝔽2)13yes—
63D109—46no—
64D109H(3,𝔽4)14yesdirect product factor
65C2×H(3,𝔽4)—1091nomaximal, Sylow
66C109C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
67H(3,𝔽4)—1099no—
68H(3,𝔽4)—1096no—
69H(3,𝔽4)D10911yescharacteristic, direct product factor
70C2×#124—1099no—
71C2×#120—1096no—
72C25—1092no—
73C22×Q8—1093no—
74C22×D4—1099no—
75#124—1099no—
76#124—1099no—
77#124—10936no—
78#124—1099no—
79#124C2×D10919yescomplemented
80#120—1096no—
81#120—10918no—
82#120—10918no—
83#120C2×D10916yescomplemented
84C2×C22⋊C4—10918no—
85C2×C42—1093no—
86C24—1096no—
87C24—1096no—
88C24—43612no—
89C24C22×D10912yescomplemented
90C2×Q8—1099no—
91C2×Q8—4366no—
92C2×Q8C22×D10913yescomplemented
93C2×D4—10918no—
94C2×D4—1099no—
95C2×D4—43618no—
96C2×D4C22×D10919yes—
97C22×C4—1099no—
98C22⋊C4—10918no—
99C22⋊C4—10936no—
100C22⋊C4C22×D109118yescomplemented
101C42—1099no—
102C42C22×D10913yescomplemented
103C23—46no—
104C23—1091no—
105C23—1096no—
106C23—43618no—
107C23—43632no—
108C23—21836no—
109C23C23×D10916yes—
110Q8—43no—
111Q8—4369no—
112D4—49no—
113D4—4369no—
114D4—43618no—
115C2×C4—1099no—
116C2×C4—21818no—
117C2×C4C23×D10919yes—
118C22—218no—
119C22—48no—
120C22—1096no—
121C22—43624no—
122C22—43624no—
123C22—21818no—
124C22C24×D10911yescharacteristic
125C4—29no—
126C4—2189no—
127C2—46no—
128C2—1094no—
129C2—4366no—
130C2H(5,𝔽2)×D10913yes—
subgroupG/H
center, FrattiniC22C24×D109
commutator, socle, cosocleC22×C109C25
FittingH(3,𝔽4)×C109C2
hypercenterH(3,𝔽4)D109
hypocenterC109C2×H(3,𝔽4)
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #20295769C2semidirect
2. #20295768C2semidirect
3. #124×D109C2semidirect
4. #120×D109C2semidirect
5. H(3,𝔽4)×C109C2semidirect
6. C24×D109C22semidirect
7. C2×Q8×D109C22semidirect
9. #16812688C22semidirect
10. #16812687C22semidirect
11. #16812686C22semidirect
12. #16812685C22semidirect
13. #16812669C22semidirect
14. #16812668C22semidirect
15. #16812667C22semidirect
16. C22⋊C4×D109C22semidirect
17. C42×D109C22semidirect
18. #124×C109C22semidirect
19. #120×C109C22semidirect
22. C23×D109C23semidirect
23. C2×Dic218C23semidirect
29. C22⋊Dic109C23semidirect
30. #13182006C23semidirect
31. C4×Dic109C23semidirect
32. C24×C109C23semidirect
33. C2×Q8×C109C23semidirect
35. C22⋊C4×C109C23semidirect
36. C42×C109C23semidirect
64. D109H(3,𝔽4)direct (N × H)
66. C109C2×H(3,𝔽4)semidirect
69. H(3,𝔽4)D109direct (N × H)
79. #124C2×D109semidirect
83. #120C2×D109semidirect
89. C24C22×D109semidirect
92. C2×Q8C22×D109semidirect
100. C22⋊C4C22×D109semidirect
102. C42C22×D109semidirect

Maximal subgroups

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

subgroupindex
#202957692
#202957682
#124×D1092
#120×D1092
H(3,𝔽4)×C1092
C2×H(3,𝔽4)109