Finite Group Databasebeta

← H(3,𝔽4)×D97 #27155855

Subgroups — H(3,𝔽4)×D97 #27155855

Order 12416.

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#20261478C216yesmaximal, complemented
2#20261477C219yesmaximal, complemented
3#124×D97C219yesmaximal, complemented
4#120×D97C216yesmaximal, complemented
5H(3,𝔽4)×C97C211yesmaximal, characteristic, complemented
6C24×D97C2212yescomplemented
7C2×Q8×D97C2213yescomplemented
8C2×D4×D97C2219yes—
9#15690799C2219yescomplemented
10#15690798C2219yescomplemented
11#15690797C2219yescomplemented
12#15690796C22136yescomplemented
13#15690780C22118yescomplemented
14#15690779C2216yescomplemented
15#15690778C22118yescomplemented
16C22⋊C4×D97C22118yescomplemented
17C42×D97C2213yescomplemented
18#124×C97C2219yescomplemented
19#120×C97C2216yescomplemented
20C23×D97—412no—
21C23×D97C2316yes—
22C23×D97C2316yescomplemented
23C2×Dic194C2319yescomplemented
24Q8×D97—46no—
25C2×D388C2319yes—
26C2×C22⋊D97C23118yes—
27D4×D97—418no—
28C2×C4×D97C2319yes—
29C22⋊Dic97C23118yescomplemented
30#13095488C23136yescomplemented
31C4×Dic97C2319yescomplemented
32C24×C97C2312yescomplemented
33C2×Q8×C97C2313yescomplemented
34C2×D4×C97C2319yes—
35C22⋊C4×C97C23118yescomplemented
36C42×C97C2313yescomplemented
37C23×C97—46no—
38C23×C97C2416yes—
39C22×D97—236no—
40C22×D97—418no—
41C22×D97—432no—
42C22×D97C2416yes—
43C22×D97C2411yescharacteristic
44Q8×C97—43no—
45D4×C97—49no—
46C2×C4×C97C2419yes—
47C22⋊D97—418no—
48C2×Dic97C2419yes—
49D388—49no—
50C4×D97—218no—
51Dic194—49no—
52C22×C97—218no—
53C22×C97—48no—
54C22×C97C2511yescharacteristic
55C2×D97—218no—
56C2×D97—424no—
57C2×D97—424no—
58C2×D97H(5,𝔽2)16yes—
59C4×C97—29no—
60Dic97—29no—
61C2×C97—46no—
62C2×C97C2×H(5,𝔽2)13yes—
63D97—46no—
64D97H(3,𝔽4)14yesdirect product factor
65C2×H(3,𝔽4)—971nomaximal, Sylow
66C97C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
67H(3,𝔽4)—979no—
68H(3,𝔽4)—976no—
69H(3,𝔽4)D9711yescharacteristic, direct product factor
70C2×#124—979no—
71C2×#120—976no—
72C25—972no—
73C22×Q8—973no—
74C22×D4—979no—
75#124—979no—
76#124—979no—
77#124—9736no—
78#124—979no—
79#124C2×D9719yescomplemented
80#120—976no—
81#120—9718no—
82#120—9718no—
83#120C2×D9716yescomplemented
84C2×C22⋊C4—9718no—
85C2×C42—973no—
86C24—976no—
87C24—976no—
88C24—38812no—
89C24C22×D9712yescomplemented
90C2×Q8—979no—
91C2×Q8—3886no—
92C2×Q8C22×D9713yescomplemented
93C2×D4—9718no—
94C2×D4—979no—
95C2×D4—38818no—
96C2×D4C22×D9719yes—
97C22×C4—979no—
98C22⋊C4—9718no—
99C22⋊C4—9736no—
100C22⋊C4C22×D97118yescomplemented
101C42—979no—
102C42C22×D9713yescomplemented
103C23—46no—
104C23—971no—
105C23—976no—
106C23—38818no—
107C23—38832no—
108C23—19436no—
109C23C23×D9716yes—
110Q8—43no—
111Q8—3889no—
112D4—49no—
113D4—3889no—
114D4—38818no—
115C2×C4—979no—
116C2×C4—19418no—
117C2×C4C23×D9719yes—
118C22—218no—
119C22—48no—
120C22—976no—
121C22—38824no—
122C22—38824no—
123C22—19418no—
124C22C24×D9711yescharacteristic
125C4—29no—
126C4—1949no—
127C2—46no—
128C2—974no—
129C2—3886no—
130C2H(5,𝔽2)×D9713yes—
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
FittingH(3,𝔽4)×C97C2
hypercenterH(3,𝔽4)D97
hypocenterC97C2×H(3,𝔽4)
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #20261478C2semidirect
2. #20261477C2semidirect
3. #124×D97C2semidirect
4. #120×D97C2semidirect
5. H(3,𝔽4)×C97C2semidirect
6. C24×D97C22semidirect
7. C2×Q8×D97C22semidirect
9. #15690799C22semidirect
10. #15690798C22semidirect
11. #15690797C22semidirect
12. #15690796C22semidirect
13. #15690780C22semidirect
14. #15690779C22semidirect
15. #15690778C22semidirect
16. C22⋊C4×D97C22semidirect
17. C42×D97C22semidirect
18. #124×C97C22semidirect
19. #120×C97C22semidirect
22. C23×D97C23semidirect
23. C2×Dic194C23semidirect
29. C22⋊Dic97C23semidirect
30. #13095488C23semidirect
31. C4×Dic97C23semidirect
32. C24×C97C23semidirect
33. C2×Q8×C97C23semidirect
35. C22⋊C4×C97C23semidirect
36. C42×C97C23semidirect
64. D97H(3,𝔽4)direct (N × H)
66. C97C2×H(3,𝔽4)semidirect
69. H(3,𝔽4)D97direct (N × H)
79. #124C2×D97semidirect
83. #120C2×D97semidirect
89. C24C22×D97semidirect
92. C2×Q8C22×D97semidirect
100. C22⋊C4C22×D97semidirect
102. C42C22×D97semidirect

Maximal subgroups

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

subgroupindex
#202614782
#202614772
#124×D972
#120×D972
H(3,𝔽4)×C972
C2×H(3,𝔽4)97