Finite Group Databasebeta

← H(3,𝔽3)×C32⋊(C37⋊C3) #41007795

Subgroups — H(3,𝔽3)×C32⋊(C37⋊C3) #41007795

Order 26973.

2102 conjugacy classes of subgroups; only the 316 normal subgroups are shown.

normal subgroups only

This group has 2102 conjugacy classes of subgroups — too many to draw. Only its 316 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
1C32×C32⋊(C37⋊C3)C314yesmaximal, complemented
2C3×H(3,𝔽3)×C37⋊C3C313yesmaximal, complemented
3#22582550C318yesmaximal, complemented
4(C3×C37⋊C3)⋊H(3,𝔽3)C3124yesmaximal, complemented
5C32×H(3,𝔽3)×C37C311yesmaximal, characteristic, complemented
6C33×C37⋊C3C32112yescomplemented
7C3×C32⋊(C37⋊C3)—336no—
8C3×C32⋊(C37⋊C3)C3211yescharacteristic
9C3×C32⋊(C37⋊C3)C32124yescomplemented
10C3×C32⋊(C37⋊C3)C32124yescomplemented
11C3×C32⋊(C37⋊C3)C3218yescomplemented
12C3×C32⋊(C37⋊C3)C32148yescomplemented
13H(3,𝔽3)×C37⋊C3—327no—
14C34×C37C3214yescomplemented
15C3×H(3,𝔽3)×C37—39no—
16C3×H(3,𝔽3)×C37C3211yescharacteristic
17C3×H(3,𝔽3)×C37C3218yescomplemented
18C33×C37—312no—
19C33×C37—312no—
20C33×C37—98no—
21C33×C37C3318yes—
22C33×C37C3314yes—
23C33×C37C3311yescharacteristic
24C32×C37⋊C3—336no—
25C32×C37⋊C3—336no—
26C32×C37⋊C3—924no—
27C32×C37⋊C3C33124yes—
28C32×C37⋊C3C3313yes—
29H(3,𝔽3)×C37—324no—
30H(3,𝔽3)×C37H(3,𝔽3)19yescomplemented
31C32⋊(C37⋊C3)—372no—
32C32⋊(C37⋊C3)—324no—
33C32⋊(C37⋊C3)—372no—
34C32⋊(C37⋊C3)—948no—
35C32⋊(C37⋊C3)H(3,𝔽3)19yesdirect product factor
36H(3,𝔽3)2—371nomaximal, Sylow
37C32×C37—34no—
38C32×C37—31no—
39C32×C37—38no—
40C32×C37—32no—
41C32×C37—38no—
42C32×C37—38no—
43C32×C37—316no—
44C32×C37—936no—
45C32×C37C3×H(3,𝔽3)13yescomplemented
46C32×C37C3×H(3,𝔽3)112yescomplemented
47C32×C37C3411yescharacteristic
48C3×C37⋊C3—33no—
49C3×C37⋊C3—36no—
50C3×C37⋊C3—324no—
51C3×C37⋊C3—324no—
52C3×C37⋊C3—348no—
53C3×C37⋊C3—9108no—
54C3×C37⋊C3C3×H(3,𝔽3)19yescomplemented
55C32×H(3,𝔽3)—374no—
56C32×H(3,𝔽3)—373no—
57C32×H(3,𝔽3)C37⋊C311yescharacteristic, complemented
58C32⋊H(3,𝔽3)—378no—
59C32⋊H(3,𝔽3)—3724no—
60C3×C37—312no—
61C3×C37—33no—
62C3×C37—98no—
63C3×C37C32×H(3,𝔽3)11yescharacteristic
64C3×C37C32×H(3,𝔽3)11yescharacteristic
65C3×C37H(5,𝔽3)12yes—
66C37⋊C3—39no—
67C37⋊C3—924no—
68C34—3712no—
69C34C3×C37⋊C314yescomplemented
70C3×H(3,𝔽3)—39no—
71C3×H(3,𝔽3)—371no—
72C3×H(3,𝔽3)—3748no—
73C3×H(3,𝔽3)—3724no—
74C3×H(3,𝔽3)—378no—
75C3×H(3,𝔽3)—3724no—
76C3×H(3,𝔽3)—11136no—
77C3×H(3,𝔽3)—11127no—
78C3×H(3,𝔽3)C3×C37⋊C311yescharacteristic
79C3×H(3,𝔽3)C3×C37⋊C318yescomplemented
80C37H(3,𝔽3)211yescharacteristic, complemented, Sylow
81C33—312no—
82C33—312no—
83C33—98no—
84C33—373no—
85C33—3724no—
86C33—33324no—
87C33—11136no—
88C33—11136no—
89C33C32×C37⋊C318yes—
90C33C32×C37⋊C311yescharacteristic
91C33C32×C37⋊C314yes—
92H(3,𝔽3)—324no—
93H(3,𝔽3)—379no—
94H(3,𝔽3)—33348no—
95H(3,𝔽3)—11172no—
96H(3,𝔽3)—11124no—
97H(3,𝔽3)—11172no—
98H(3,𝔽3)C32⋊(C37⋊C3)19yesdirect product factor
99C32—34no—
100C32—31no—
101C32—32no—
102C32—38no—
103C32—38no—
104C32—38no—
105C32—316no—
106C32—936no—
107C32—379no—
108C32—333108no—
109C32—11124no—
110C32—11124no—
111C32—11148no—
112C32—1113no—
113C32—1116no—
114C32H(3,𝔽3)×C37⋊C313yescomplemented
115C32C3×C32⋊(C37⋊C3)112yescomplemented
116C32C33×C37⋊C311yescharacteristic
117C3—33no—
118C3—312no—
119C3—98no—
120C3—33324no—
121C3—1119no—
122C3C3×H(3,𝔽3)×C37⋊C311yescharacteristic
123C3C32×C32⋊(C37⋊C3)11yescharacteristic
124C3C37⋊H(5,𝔽3)12yes—
subgroupG/H
center, FrattiniC32C33×C37⋊C3
commutator, socle, cosocleC32×C37C34
FittingC32×H(3,𝔽3)×C37C3
hypercenterC32×H(3,𝔽3)C37⋊C3
hypocenterC37H(3,𝔽3)2
LCSG ▷ C32×C37 ▷ C37
UCSC1 ◁ C32 ◁ C32×H(3,𝔽3)

Complements

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

NH ≅ G/Ntype
1. C32×C32⋊(C37⋊C3)C3semidirect
2. C3×H(3,𝔽3)×C37⋊C3C3semidirect
3. #22582550C3semidirect
4. (C3×C37⋊C3)⋊H(3,𝔽3)C3semidirect
5. C32×H(3,𝔽3)×C37C3semidirect
6. C33×C37⋊C3C32semidirect
9. C3×C32⋊(C37⋊C3)C32semidirect
10. C3×C32⋊(C37⋊C3)C32semidirect
11. C3×C32⋊(C37⋊C3)C32semidirect
12. C3×C32⋊(C37⋊C3)C32semidirect
14. C34×C37C32semidirect
17. C3×H(3,𝔽3)×C37C32semidirect
30. H(3,𝔽3)×C37H(3,𝔽3)semidirect
35. C32⋊(C37⋊C3)H(3,𝔽3)direct (N × H)
45. C32×C37C3×H(3,𝔽3)semidirect
46. C32×C37C3×H(3,𝔽3)semidirect
54. C3×C37⋊C3C3×H(3,𝔽3)semidirect
57. C32×H(3,𝔽3)C37⋊C3semidirect
69. C34C3×C37⋊C3semidirect
79. C3×H(3,𝔽3)C3×C37⋊C3semidirect
80. C37H(3,𝔽3)2semidirect
98. H(3,𝔽3)C32⋊(C37⋊C3)direct (N × H)
114. C32H(3,𝔽3)×C37⋊C3semidirect
115. C32C3×C32⋊(C37⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C32×C32⋊(C37⋊C3)3
C3×H(3,𝔽3)×C37⋊C33
#225825503
(C3×C37⋊C3)⋊H(3,𝔽3)3
C32×H(3,𝔽3)×C373
H(3,𝔽3)237