Finite Group Databasebeta

← #41008199

Subgroups — #41008199

Order 26973.

978 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 978 conjugacy classes of subgroups — too many to draw. Only its 268 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#22582679C312yesmaximal, complemented
2C37⋊H(5,𝔽3)C311yesmaximal, characteristic
3C3×#15688532C311yesmaximal, characteristic, complemented
4#22582644C312yesmaximal, complemented
5#22582643C312yesmaximal, complemented
6#22582642C311yesmaximal, characteristic, complemented
7#22582641C319yesmaximal, complemented
8#22582640C319yesmaximal, complemented
9C3×#15688517C312yesmaximal, complemented
10C3×#15688515C311yesmaximal, characteristic, complemented
11#731×C37⋊C3C319yesmaximal, complemented
12C37×#6910C311yesmaximal, characteristic, complemented
13#15688532—36no—
14#15688532—33no—
15#15688532—36no—
16#15688532—36no—
17#15688531—36no—
18C3×#11758608C3212yescomplemented
19C3×#11758608C3212yescomplemented
20C3×#11758608C3211yescharacteristic, complemented
21C3×#11758605C3213yescomplemented
22C3×#11758606C3213yescomplemented
233−1+2×C37⋊C3C32118yescomplemented
24C3×C32⋊(C37⋊C3)—33no—
25C3×C32⋊(C37⋊C3)C3212yes—
26C3×C32⋊(C37⋊C3)C3211yescharacteristic
27H(3,𝔽3)×C37⋊C3—36no—
28H(3,𝔽3)×C37⋊C3C3219yes—
29C32×#66524C3211yescharacteristic, complemented
30C3×C9×C37⋊C3C3213yescomplemented
31#15688517C3216yescomplemented
32#15688517C32118yescomplemented
33#15688517C3216yescomplemented
34#15688517C3216yescomplemented
35#15688516C32118yescomplemented
36#15688515C3213yescomplemented
37#15688515C3219yescomplemented
38#15688515C3213yescomplemented
39#15688515C3213yescomplemented
40C3×3−1+2×C37C3212yescomplemented
41C3×3−1+2×C37C3211yescharacteristic, complemented
42C3×H(3,𝔽3)×C37C3211yescharacteristic
43C37×#731C3219yescomplemented
44C33×C37—31no—
45C33×C37C3311yescharacteristic
46C32×C37⋊C3—32no—
47C32×C37⋊C3—39no—
48C32×C37⋊C3C3313yes—
493−1+2×C37—32no—
503−1+2×C37—32no—
513−1+2×C37—31no—
523−1+2×C37—32no—
533−1+2×C37C3316yes—
54H(3,𝔽3)×C37—32no—
55H(3,𝔽3)×C37C3313yes—
56C3×C9×C37—32no—
57C3×C9×C37C3313yes—
58C32⋊(C37⋊C3)—32no—
59C32⋊(C37⋊C3)—32no—
60C32⋊(C37⋊C3)—318no—
61C32⋊(C37⋊C3)H(3,𝔽3)16yes—
62C32⋊(C37⋊C3)H(3,𝔽3)13yes—
63C32⋊(C37⋊C3)C3316yes—
64#11758608—32no—
65#11758608—32no—
66#11758608—32no—
67#11758608—32no—
68#11758608—31no—
69#11758608—32no—
70#11758608C3316yes—
71#11758608C3316yes—
72C3×#66524—31no—
73C3×#66524—32no—
74C3×#66524—32no—
75C3×#66524—32no—
76C3×#66524C3311yescharacteristic
77C3×#66524C3313yes—
78C3×#66524C3311yescharacteristic
79C3×#66524C3311yescharacteristic
80#11758606—318no—
81#11758606—39no—
82#11758605—318no—
83#11758605—39no—
84C9×C37⋊C3—39no—
85C9×C37⋊C3—318no—
86#10624931—371nomaximal, Sylow
87C32×C37—33no—
88C32×C37—91no—
89C32×C37—271no—
90C32×C37C3×H(3,𝔽3)11yescharacteristic
91C32×C37C3×H(3,𝔽3)12yes—
92C32×C37C3411yescharacteristic
93C3×C37⋊C3—36no—
94C3×C37⋊C3—93no—
95C3×C37⋊C3—279no—
96C3×C37⋊C3—272no—
97C3×C37⋊C3C3×H(3,𝔽3)19yes—
98C9×C37—36no—
99C9×C37—33no—
100#66524—31no—
101#66524—36no—
102#66524—36no—
103#66524—33no—
104#66524—31no—
105#66524—31no—
1063−1+4—372no—
107H(5,𝔽3)—371no—
108C3×#737—371no—
109#6910—372no—
110#6910—372no—
111#6910—371no—
112#6910—379no—
113#6910—379no—
114#6910C37⋊C311yescharacteristic, complemented
115C3×#731—372no—
116C3×#731—371no—
117C3×#731—379no—
118C3×C37—31no—
119C3×C37—32no—
120C3×C37—31no—
121C3×C37—93no—
122C3×C37C32×H(3,𝔽3)11yescharacteristic
123C37⋊C3—39no—
124C37⋊C3—96no—
125#737—1116no—
126#737—1116no—
127#737—1113no—
128#737—1116no—
129#737—1116no—
130C3×3−1+2—371no—
131C3×3−1+2—373no—
132C3×3−1+2—373no—
133C3×3−1+2—372no—
134C3×3−1+2—372no—
135C3×3−1+2—3718no—
136C3×3−1+2C3×C37⋊C312yescomplemented
137C3×3−1+2C3×C37⋊C311yescharacteristic, complemented
138C3×H(3,𝔽3)—371no—
139C3×H(3,𝔽3)—372no—
140C3×H(3,𝔽3)—379no—
141C3×H(3,𝔽3)—1116no—
142C3×H(3,𝔽3)—1113no—
143C3×H(3,𝔽3)C3×C37⋊C311yescharacteristic
144C32×C9—371no—
145C32×C9—373no—
146#731—376no—
147#731—3718no—
148#731—376no—
149#731—376no—
150#731—373no—
151#731—379no—
152#731—373no—
153#731—373no—
154#731—3718no—
155#731C3×C37⋊C319yescomplemented
156C37#1062493111yescharacteristic, complemented, Sylow
157C33—31no—
158C33—373no—
159C33—1119no—
160C33—1112no—
161C33C32×C37⋊C311yescharacteristic
1623−1+2—32no—
1633−1+2—32no—
1643−1+2—32no—
1653−1+2—31no—
1663−1+2—376no—
1673−1+2—376no—
1683−1+2—11118no—
1693−1+2—1119no—
1703−1+2—11118no—
1713−1+2—1119no—
1723−1+2—1112no—
1733−1+2—1111no—
1743−1+2—1112no—
1753−1+2—1112no—
1763−1+2—1112no—
1773−1+2—1112no—
1783−1+2C32×C37⋊C316yes—
179H(3,𝔽3)—32no—
180H(3,𝔽3)—376no—
181H(3,𝔽3)—373no—
182H(3,𝔽3)—376no—
183H(3,𝔽3)—11118no—
184H(3,𝔽3)—1112no—
185H(3,𝔽3)—1112no—
186H(3,𝔽3)C32×C37⋊C313yes—
187C3×C9—32no—
188C3×C9—371no—
189C3×C9—373no—
190C3×C9—371no—
191C3×C9—371no—
192C3×C9—1119no—
193C3×C9—11118no—
194C3×C9—1112no—
195C3×C9—1111no—
196C3×C9—1112no—
197C3×C9—1112no—
198C3×C9C32×C37⋊C313yes—
199C32—33no—
200C32—91no—
201C32—271no—
202C32—379no—
203C32—3333no—
204C32—1116no—
205C32—9992no—
206C32—9999no—
207C32H(3,𝔽3)×C37⋊C312yes—
208C32H(3,𝔽3)×C37⋊C311yescharacteristic
209C32C33×C37⋊C311yescharacteristic
210C9—36no—
211C9—33no—
212C9—1111no—
213C9—1113no—
214C9—1111no—
215C9—1111no—
216C9—1116no—
217C9—1116no—
218C3—31no—
219C3—31no—
220C3—32no—
221C3—93no—
222C3—3336no—
223C3—1119no—
224C3C3×H(3,𝔽3)×C37⋊C311yescharacteristic
subgroupG/H
centerC3C3×H(3,𝔽3)×C37⋊C3
commutator, cosocleC32×C37C34
FrattiniC32C33×C37⋊C3
FittingC37×#6910C3
socleC3×C37C32×H(3,𝔽3)
prime radicalC37⋊H(5,𝔽3)C3
hypercenter#6910C37⋊C3
hypocenterC37#10624931
LCSG ▷ C32×C37 ▷ C3×C37 ▷ C37
UCSC1 ◁ C3 ◁ C33 ◁ #6910

Complements

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

NH ≅ G/Ntype
1. #22582679C3semidirect
3. C3×#15688532C3semidirect
4. #22582644C3semidirect
5. #22582643C3semidirect
6. #22582642C3semidirect
7. #22582641C3semidirect
8. #22582640C3semidirect
9. C3×#15688517C3semidirect
10. C3×#15688515C3semidirect
11. #731×C37⋊C3C3semidirect
12. C37×#6910C3semidirect
18. C3×#11758608C32semidirect
19. C3×#11758608C32semidirect
20. C3×#11758608C32semidirect
21. C3×#11758605C32semidirect
22. C3×#11758606C32semidirect
23. 3−1+2×C37⋊C3C32semidirect
29. C32×#66524C32semidirect
30. C3×C9×C37⋊C3C32semidirect
31. #15688517C32semidirect
32. #15688517C32semidirect
33. #15688517C32semidirect
34. #15688517C32semidirect
35. #15688516C32semidirect
36. #15688515C32semidirect
37. #15688515C32semidirect
38. #15688515C32semidirect
39. #15688515C32semidirect
40. C3×3−1+2×C37C32semidirect
41. C3×3−1+2×C37C32semidirect
43. C37×#731C32semidirect
114. #6910C37⋊C3semidirect
136. C3×3−1+2C3×C37⋊C3semidirect
137. C3×3−1+2C3×C37⋊C3semidirect
155. #731C3×C37⋊C3semidirect
156. C37#10624931semidirect

Maximal subgroups

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

subgroupindex
#225826793
C37⋊H(5,𝔽3)3
C3×#156885323
#225826443
#225826433
#225826423
#225826413
#225826403
C3×#156885173
C3×#156885153
#731×C37⋊C33
C37×#69103
#1062493137