Finite Group Databasebeta

← C37⋊C3×#6911 #41008160

Subgroups — C37⋊C3×#6911 #41008160

Order 26973.

1304 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 1304 conjugacy classes of subgroups — too many to draw. Only its 271 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
1C3×3−1+2×C37⋊C3C311yesmaximal, characteristic, complemented
2C3×H(3,𝔽3)×C37⋊C3C313yesmaximal, complemented
3#22582648C312yesmaximal, complemented
4#22582647C316yesmaximal, complemented
5#22582646C319yesmaximal, complemented
6#22582645C319yesmaximal, complemented
7#732×C37⋊C3C319yesmaximal, complemented
8C37×#6911C311yesmaximal, characteristic, complemented
9C33×C37⋊C3—33no—
10C33×C37⋊C3C3211yescharacteristic
11C3×#11758608C3212yescomplemented
12C3×#11758605C3213yescomplemented
13C3×#11758606C3213yescomplemented
143−1+2×C37⋊C3—33no—
153−1+2×C37⋊C3—36no—
16C3×C32⋊(C37⋊C3)—36no—
17C3×C32⋊(C37⋊C3)C3216yescomplemented
18H(3,𝔽3)×C37⋊C3—36no—
19H(3,𝔽3)×C37⋊C3—36no—
20H(3,𝔽3)×C37⋊C3—36no—
21H(3,𝔽3)×C37⋊C3C3219yescomplemented
22H(3,𝔽3)×C37⋊C3C3219yescomplemented
23H(3,𝔽3)×C37⋊C3C3219yescomplemented
24C3×C9×C37⋊C3C3213yescomplemented
25#15688519C32118yescomplemented
26#15688518C32154yescomplemented
27C3×3−1+2×C37C3211yescharacteristic, complemented
28C3×H(3,𝔽3)×C37C3213yescomplemented
29C37×#732C3219yescomplemented
30C33×C37—33no—
31C33×C37C3311yescharacteristic
32C32×C37⋊C3—36no—
33C32×C37⋊C3—39no—
34C32×C37⋊C3—39no—
35C32×C37⋊C3—39no—
36C32×C37⋊C3—93no—
37C32×C37⋊C3—927no—
38C32×C37⋊C3H(3,𝔽3)13yes—
39C32×C37⋊C3H(3,𝔽3)16yes—
40C32×C37⋊C3C3311yescharacteristic
41C32×C37⋊C3C3311yescharacteristic
42C32×C37⋊C3C3311yescharacteristic
433−1+2×C37—33no—
44H(3,𝔽3)×C37—36no—
45H(3,𝔽3)×C37C3319yescomplemented
46C3×C9×C37C3313yescomplemented
47C32⋊(C37⋊C3)—312no—
48C32⋊(C37⋊C3)—318no—
49C32⋊(C37⋊C3)—318no—
50C32⋊(C37⋊C3)—318no—
51C32⋊(C37⋊C3)C33118yescomplemented
52#11758608—36no—
53C3×#66524C3316yescomplemented
54#11758606—39no—
55#11758605—39no—
56C9×C37⋊C3—36no—
57C9×C37⋊C3—33no—
58C3×#6911—371nomaximal, Sylow
59C32×C37—39no—
60C32×C37—91no—
61C32×C37—99no—
62C32×C37C3×H(3,𝔽3)13yes—
63C32×C37C3411yescharacteristic
64C3×C37⋊C3—39no—
65C3×C37⋊C3—33no—
66C3×C37⋊C3—318no—
67C3×C37⋊C3—96no—
68C3×C37⋊C3—91no—
69C3×C37⋊C3—91no—
70C3×C37⋊C3—927no—
71C3×C37⋊C3—927no—
72C3×C37⋊C3—918no—
73C3×C37⋊C3—927no—
74C3×C37⋊C3C3×H(3,𝔽3)12yes—
75C3×C37⋊C3C3×H(3,𝔽3)13yes—
76C3×C37⋊C3C3×H(3,𝔽3)13yes—
77C3×C37⋊C3C3×H(3,𝔽3)11yescharacteristic
78C9×C37—33no—
79#66524—36no—
80C32×3−1+2—371no—
81C32×H(3,𝔽3)—373no—
82#6911—372no—
83#6911—376no—
84#6911—379no—
85#6911—379no—
86#6911C37⋊C311yescharacteristic, direct product factor
87C3×#732—379no—
88C3×C37—33no—
89C3×C37—31no—
90C3×C37—99no—
91C3×C37C32×H(3,𝔽3)11yescharacteristic
92C37⋊C3—33no—
93C37⋊C3—33no—
94C37⋊C3—32no—
95C37⋊C3—918no—
96C37⋊C3#691113yesdirect product factor
97C34—371no—
98C34—1113no—
99C3×3−1+2—372no—
100C3×3−1+2—373no—
101C3×3−1+2—373no—
102C3×3−1+2—1116no—
103C3×3−1+2—1113no—
104C3×3−1+2C3×C37⋊C311yescharacteristic, complemented
105C3×H(3,𝔽3)—376no—
106C3×H(3,𝔽3)—379no—
107C3×H(3,𝔽3)—379no—
108C3×H(3,𝔽3)—379no—
109C3×H(3,𝔽3)—1116no—
110C3×H(3,𝔽3)—1116no—
111C3×H(3,𝔽3)—1116no—
112C3×H(3,𝔽3)—1116no—
113C3×H(3,𝔽3)C3×C37⋊C313yescomplemented
114C32×C9—373no—
115#732—3754no—
116#732—3718no—
117#732C3×C37⋊C319yescomplemented
118C37C3×#691111yescharacteristic, complemented, Sylow
119C33—33no—
120C33—376no—
121C33—371no—
122C33—371no—
123C33—373no—
124C33—371no—
125C33—33327no—
126C33—3333no—
127C33—1119no—
128C33—1119no—
129C33—1119no—
130C33—1116no—
131C33C32×C37⋊C311yescharacteristic
1323−1+2—33no—
1333−1+2—1119no—
1343−1+2—1119no—
1353−1+2—1116no—
136H(3,𝔽3)—36no—
137H(3,𝔽3)—3718no—
138H(3,𝔽3)—11118no—
139H(3,𝔽3)—11118no—
140H(3,𝔽3)—11118no—
141H(3,𝔽3)—11112no—
142H(3,𝔽3)C32×C37⋊C319yescomplemented
143C3×C9—376no—
144C3×C9—1113no—
145C3×C9—1116no—
146C3×C9C32×C37⋊C313yescomplemented
147C32—39no—
148C32—91no—
149C32—99no—
150C32—371no—
151C32—372no—
152C32—373no—
153C32—373no—
154C32—33318no—
155C32—3336no—
156C32—3331no—
157C32—3331no—
158C32—33327no—
159C32—33327no—
160C32—33327no—
161C32—1119no—
162C32—1113no—
163C32—11118no—
164C32H(3,𝔽3)×C37⋊C313yes—
165C32C33×C37⋊C311yescharacteristic
166C9—33no—
167C9—1116no—
168C3—33no—
169C3—31no—
170C3—99no—
171C3—373no—
172C3—33318no—
173C3—1113no—
174C3—1113no—
175C3—1112no—
176C3C3×H(3,𝔽3)×C37⋊C311yescharacteristic
subgroupG/H
centerC3C3×H(3,𝔽3)×C37⋊C3
commutator, cosocleC32×C37C34
FrattiniC32C33×C37⋊C3
FittingC37×#6911C3
socleC3×C37C32×H(3,𝔽3)
hypercenter#6911C37⋊C3
hypocenterC37C3×#6911
LCSG ▷ C32×C37 ▷ C3×C37 ▷ C37
UCSC1 ◁ C3 ◁ C33 ◁ #6911

Complements

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

NH ≅ G/Ntype
1. C3×3−1+2×C37⋊C3C3semidirect
2. C3×H(3,𝔽3)×C37⋊C3C3semidirect
3. #22582648C3semidirect
4. #22582647C3semidirect
5. #22582646C3semidirect
6. #22582645C3semidirect
7. #732×C37⋊C3C3semidirect
8. C37×#6911C3semidirect
11. C3×#11758608C32semidirect
12. C3×#11758605C32semidirect
13. C3×#11758606C32semidirect
17. C3×C32⋊(C37⋊C3)C32semidirect
21. H(3,𝔽3)×C37⋊C3C32semidirect
22. H(3,𝔽3)×C37⋊C3C32semidirect
23. H(3,𝔽3)×C37⋊C3C32semidirect
24. C3×C9×C37⋊C3C32semidirect
25. #15688519C32semidirect
26. #15688518C32semidirect
27. C3×3−1+2×C37C32semidirect
28. C3×H(3,𝔽3)×C37C32semidirect
29. C37×#732C32semidirect
45. H(3,𝔽3)×C37C33semidirect
46. C3×C9×C37C33semidirect
51. C32⋊(C37⋊C3)C33semidirect
53. C3×#66524C33semidirect
86. #6911C37⋊C3direct (N × H)
96. C37⋊C3#6911direct (N × H)
104. C3×3−1+2C3×C37⋊C3semidirect
113. C3×H(3,𝔽3)C3×C37⋊C3semidirect
117. #732C3×C37⋊C3semidirect
118. C37C3×#6911semidirect
142. H(3,𝔽3)C32×C37⋊C3semidirect
146. C3×C9C32×C37⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×3−1+2×C37⋊C33
C3×H(3,𝔽3)×C37⋊C33
#225826483
#225826473
#225826463
#225826453
#732×C37⋊C33
C37×#69113
C3×#691137