Finite Group Databasebeta

← #41006572

Subgroups — #41006572

Order 26973.

560 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C37⋊H(5,𝔽3)C311yesmaximal, characteristic, complemented
2#22582647C311yesmaximal, characteristic, complemented
3#22582647C312yesmaximal, complemented
4#22582639C312yesmaximal, complemented
5#22582637C312yesmaximal, complemented
6C3×#15688514C311yesmaximal, characteristic, complemented
7C3×#15688512C311yesmaximal, characteristic, complemented
8(C3×C37⋊C3)⋊H(3,𝔽3)C312yesmaximal, complemented
9C37×#6911C311yesmaximal, characteristic, complemented
10C33×C37⋊C3C3211yescharacteristic, complemented
11C3×#11758608C3212yescomplemented
12C3×#11758608C3211yescharacteristic, complemented
13C3×C32⋊(C37⋊C3)—33no—
14C3×C32⋊(C37⋊C3)—36no—
15C3×C32⋊(C37⋊C3)C3212yes—
16C3×C32⋊(C37⋊C3)C3211yescharacteristic
17C3×C32⋊(C37⋊C3)C3212yescomplemented
18H(3,𝔽3)×C37⋊C3—36no—
19H(3,𝔽3)×C37⋊C3—33no—
20#15688518—33no—
21#15688518—36no—
22#15688514—36no—
23#15688514—33no—
24#15688512—36no—
25#15688512—33no—
26C3×3−1+2×C37C3211yescharacteristic, complemented
27C3×H(3,𝔽3)×C37C3211yescharacteristic
28C3×H(3,𝔽3)×C37C3212yescomplemented
29C37×#732—33no—
30C33×C37—31no—
31C33×C37—32no—
32C33×C37C3311yescharacteristic
33C32×C37⋊C3—32no—
34C32×C37⋊C3—39no—
35C32×C37⋊C3—31no—
36C32×C37⋊C3—31no—
37C32×C37⋊C3—31no—
38C32×C37⋊C3—32no—
39C32×C37⋊C3—93no—
40C32×C37⋊C3H(3,𝔽3)16yescomplemented
413−1+2×C37—31no—
423−1+2×C37—32no—
43H(3,𝔽3)×C37—32no—
44H(3,𝔽3)×C37—32no—
45H(3,𝔽3)×C37—32no—
46H(3,𝔽3)×C37—32no—
47H(3,𝔽3)×C37H(3,𝔽3)13yescomplemented
48C3×C9×C37—31no—
49C32⋊(C37⋊C3)—318no—
50C32⋊(C37⋊C3)—32no—
51C32⋊(C37⋊C3)—32no—
52C32⋊(C37⋊C3)—32no—
53C32⋊(C37⋊C3)—32no—
54C32⋊(C37⋊C3)—32no—
55C32⋊(C37⋊C3)—32no—
56C32⋊(C37⋊C3)—32no—
57C32⋊(C37⋊C3)—918no—
58C32⋊(C37⋊C3)H(3,𝔽3)13yescomplemented
59#11758608—32no—
60#11758608—32no—
61#11758608—32no—
62#11758608—31no—
63#11758608—32no—
64C3×#66524—32no—
65C3×#66524—31no—
66UT(4,𝔽3)—371nomaximal, Sylow
67C32×C37—33no—
68C32×C37—91no—
69C32×C37—96no—
70C32×C37—92no—
71C32×C37—271no—
72C32×C37C3×H(3,𝔽3)11yescharacteristic
73C32×C37C3×H(3,𝔽3)12yes—
74C32×C37C3×H(3,𝔽3)11yescharacteristic
75C3×C37⋊C3—36no—
76C3×C37⋊C3—36no—
77C3×C37⋊C3—33no—
78C3×C37⋊C3—94no—
79C3×C37⋊C3—91no—
80C3×C37⋊C3—93no—
81C3×C37⋊C3—93no—
82C3×C37⋊C3—96no—
83C3×C37⋊C3—96no—
84C3×C37⋊C3—92no—
85C3×C37⋊C3—271no—
86C3×C37⋊C3—279no—
87C9×C37—91no—
88#66524—92no—
89#66524—91no—
90H(5,𝔽3)—371no—
91#6911—371no—
92#6911—372no—
93#6911C37⋊C311yescharacteristic, complemented
94#6909—372no—
95#6909—372no—
96C3×AΣL(1,𝔽27)—371no—
97C3×AΣL(1,𝔽27)—371no—
98C32⋊H(3,𝔽3)—372no—
99C3×C37—32no—
100C3×C37—31no—
101C3×C37—31no—
102C3×C37—93no—
103C3×C37—272no—
104C3×C37C32⋊H(3,𝔽3)11yescharacteristic
105C37⋊C3—93no—
106C37⋊C3—96no—
107C37⋊C3—93no—
108C37⋊C3—272no—
109C34—371no—
110C3×3−1+2—372no—
111C3×3−1+2—371no—
112C3×3−1+2C3×C37⋊C311yescharacteristic, complemented
113C3×H(3,𝔽3)—371no—
114C3×H(3,𝔽3)—372no—
115C3×H(3,𝔽3)—372no—
116C3×H(3,𝔽3)—1116no—
117C3×H(3,𝔽3)—1113no—
118C3×H(3,𝔽3)—1116no—
119C3×H(3,𝔽3)—1113no—
120C3×H(3,𝔽3)C3×C37⋊C311yescharacteristic
121C3×H(3,𝔽3)C3×C37⋊C312yescomplemented
122#732—33no—
123#732—1116no—
124#732—1113no—
125AΣL(1,𝔽27)—1116no—
126AΣL(1,𝔽27)—1113no—
127AΣL(1,𝔽27)—1116no—
128AΣL(1,𝔽27)—1113no—
129C37UT(4,𝔽3)11yescharacteristic, complemented, Sylow
130C33—31no—
131C33—32no—
132C33—376no—
133C33—3333no—
134C33—1119no—
135C33—1111no—
136C33—1112no—
137C33—1111no—
138C33—1112no—
139C33—1111no—
140C33C32×C37⋊C311yescharacteristic
1413−1+2—31no—
1423−1+2—32no—
1433−1+2—1111no—
1443−1+2—1112no—
1453−1+2—1112no—
1463−1+2—1112no—
1473−1+2—1112no—
148H(3,𝔽3)—32no—
149H(3,𝔽3)—32no—
150H(3,𝔽3)—32no—
151H(3,𝔽3)—32no—
152H(3,𝔽3)—373no—
153H(3,𝔽3)—33318no—
154H(3,𝔽3)—11118no—
155H(3,𝔽3)—1112no—
156H(3,𝔽3)—1112no—
157H(3,𝔽3)—1112no—
158H(3,𝔽3)—1112no—
159H(3,𝔽3)—1112no—
160H(3,𝔽3)—1112no—
161H(3,𝔽3)—1112no—
162H(3,𝔽3)C32⋊(C37⋊C3)13yescomplemented
163C3×C9—31no—
164C3×C9—1112no—
165C3×C9—1111no—
166C32—33no—
167C32—91no—
168C32—96no—
169C32—92no—
170C32—271no—
171C32—3336no—
172C32—3333no—
173C32—3333no—
174C32—3334no—
175C32—3336no—
176C32—3331no—
177C32—3332no—
178C32—1116no—
179C32—1113no—
180C32—1116no—
181C32—9991no—
182C32—9999no—
183C32H(3,𝔽3)×C37⋊C311yescharacteristic
184C32H(3,𝔽3)×C37⋊C312yes—
185C32C3×C32⋊(C37⋊C3)11yescharacteristic
186C9—91no—
187C9—3332no—
188C9—3331no—
189C3—32no—
190C3—31no—
191C3—31no—
192C3—93no—
193C3—272no—
194C3—3333no—
195C3—3336no—
196C3—3333no—
197C3—9992no—
198C3(C3×C37⋊C3)⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC3(C3×C37⋊C3)⋊H(3,𝔽3)
commutator, cosocleC33×C37C33
FrattiniC33C32×C37⋊C3
FittingC37×#6911C3
socleC3×C37C32⋊H(3,𝔽3)
hypercenter#6911C37⋊C3
hypocenterC37UT(4,𝔽3)
LCSG ▷ C33×C37 ▷ C3×C37 ▷ C37
UCSC1 ◁ C3 ◁ C33 ◁ #6911

Complements

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

NH ≅ G/Ntype
1. C37⋊H(5,𝔽3)C3semidirect
2. #22582647C3semidirect
3. #22582647C3semidirect
4. #22582639C3semidirect
5. #22582637C3semidirect
6. C3×#15688514C3semidirect
7. C3×#15688512C3semidirect
8. (C3×C37⋊C3)⋊H(3,𝔽3)C3semidirect
9. C37×#6911C3semidirect
10. C33×C37⋊C3C32semidirect
11. C3×#11758608C32semidirect
12. C3×#11758608C32semidirect
17. C3×C32⋊(C37⋊C3)C32semidirect
26. C3×3−1+2×C37C32semidirect
28. C3×H(3,𝔽3)×C37C32semidirect
40. C32×C37⋊C3H(3,𝔽3)semidirect
47. H(3,𝔽3)×C37H(3,𝔽3)semidirect
58. C32⋊(C37⋊C3)H(3,𝔽3)semidirect
93. #6911C37⋊C3semidirect
112. C3×3−1+2C3×C37⋊C3semidirect
121. C3×H(3,𝔽3)C3×C37⋊C3semidirect
129. C37UT(4,𝔽3)semidirect
162. H(3,𝔽3)C32⋊(C37⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C37⋊H(5,𝔽3)3
#225826473
#225826393
#225826373
C3×#156885143
C3×#156885123
(C3×C37⋊C3)⋊H(3,𝔽3)3
C37×#69113
UT(4,𝔽3)37