Finite Group Databasebeta

← #41008213

Subgroups — #41008213

Order 26973.

980 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 286 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
2C3×3−1+2×C37⋊C3C311yesmaximal, characteristic, complemented
3#22582656C319yesmaximal, complemented
4#22582655C319yesmaximal, complemented
5#22582651C312yesmaximal, complemented
6#22582647C311yesmaximal, characteristic, complemented
7#22582644C312yesmaximal, complemented
8#22582642C311yesmaximal, characteristic, complemented
9#22582630C319yesmaximal, complemented
10C3×#15688514C312yesmaximal, complemented
11C3×#15688512C311yesmaximal, characteristic, complemented
12C37×3−1+4C311yesmaximal, characteristic, complemented
13C33×C37⋊C3C3211yescharacteristic
14#15688532—36no—
15#15688532—36no—
16#15688531C32118yescomplemented
17C3×#11758608C3212yes—
18C3×#11758608C3212yes—
19C3×#11758608C3211yescharacteristic
20C3×#11758608C3211yescharacteristic
21C3×#11758605C3213yescomplemented
22C3×#11758606C3213yescomplemented
233−1+2×C37⋊C3—33no—
243−1+2×C37⋊C3—36no—
253−1+2×C37⋊C3—36no—
26C3×C32⋊(C37⋊C3)C3212yes—
27C3×C9×C37⋊C3C3213yescomplemented
28#15688520C32118yescomplemented
29#15688518C3219yescomplemented
30#15688517C32118yescomplemented
31#15688515C3219yescomplemented
32#15688514C3216yescomplemented
33#15688514C3216yescomplemented
34#15688514C3216yescomplemented
35#15688512C3213yescomplemented
36#15688512C3213yescomplemented
37#15688512C3213yescomplemented
38C37×#737—36no—
39C37×#737C3219yescomplemented
40C3×3−1+2×C37—33no—
41C3×3−1+2×C37C3212yes—
42C3×3−1+2×C37C3211yescharacteristic
43C3×H(3,𝔽3)×C37C3211yescharacteristic
44C33×C37—31no—
45C33×C37C3311yescharacteristic
46C32×C37⋊C3—31no—
47C32×C37⋊C3—32no—
48C32×C37⋊C3—32no—
49C32×C37⋊C3—93no—
50C32×C37⋊C3C3311yescharacteristic
51C32×C37⋊C3C3311yescharacteristic
52C32×C37⋊C3C3311yescharacteristic
533−1+2×C37—32no—
543−1+2×C37—32no—
553−1+2×C37—318no—
563−1+2×C37H(3,𝔽3)13yes—
573−1+2×C37H(3,𝔽3)16yes—
583−1+2×C37C3316yes—
59H(3,𝔽3)×C37—32no—
60H(3,𝔽3)×C37C3313yes—
61C3×C9×C37—32no—
62C3×C9×C37—39no—
63C3×C9×C37C3313yes—
64C32⋊(C37⋊C3)—32no—
65C32⋊(C37⋊C3)—32no—
66C32⋊(C37⋊C3)C3316yes—
67#11758608—32no—
68#11758608—32no—
69#11758608—31no—
70#11758608—31no—
71#11758608—32no—
72#11758608—32no—
73#11758608—32no—
74#11758608—32no—
75#11758608C3316yes—
76#11758608C3316yes—
77C3×#66524—32no—
78C3×#66524—32no—
79C3×#66524C3313yes—
80C3×#66524C3313yes—
81#11758606—39no—
82#11758606—318no—
83#11758605—39no—
84#11758605—318no—
85C9×C37⋊C3—39no—
86C9×C37⋊C3—318no—
87#10624934—371nomaximal, Sylow
88C32×C37—33no—
89C32×C37—91no—
90C32×C37—271no—
91C32×C37C3×H(3,𝔽3)11yescharacteristic
92C32×C37C3×H(3,𝔽3)12yes—
93C32×C37C3411yescharacteristic
94C3×C37⋊C3—31no—
95C3×C37⋊C3—33no—
96C3×C37⋊C3—36no—
97C3×C37⋊C3—31no—
98C3×C37⋊C3—31no—
99C3×C37⋊C3—96no—
100C3×C37⋊C3—93no—
101C3×C37⋊C3—91no—
102C3×C37⋊C3—91no—
103C3×C37⋊C3—272no—
104C9×C37—36no—
105C9×C37C3×H(3,𝔽3)19yes—
106#66524—36no—
107#66524—36no—
108#66524—33no—
109#66524—33no—
1103−1+4—372no—
1113−1+4C37⋊C311yescharacteristic, complemented
112C32×3−1+2—371no—
113#6913—379no—
114#6913—379no—
115#6912—372no—
116#6911—371no—
117#6910—372no—
118#6910—371no—
119#6908—379no—
120C3×AΣL(1,𝔽27)—372no—
121C3×AΣL(1,𝔽27)—371no—
122C3×C37—31no—
123C3×C37—32no—
124C3×C37—31no—
125C3×C37—93no—
126C3×C37C32×H(3,𝔽3)11yescharacteristic
127C37⋊C3—33no—
128C37⋊C3—91no—
129C37⋊C3—91no—
130C37⋊C3—96no—
131C34—371no—
132#737—36no—
133#737—3718no—
134#737—1116no—
135#737—1116no—
136#737C3×C37⋊C319yescomplemented
137C3×3−1+2—33no—
138C3×3−1+2—371no—
139C3×3−1+2—371no—
140C3×3−1+2—373no—
141C3×3−1+2—373no—
142C3×3−1+2—372no—
143C3×3−1+2—372no—
144C3×3−1+2—1113no—
145C3×3−1+2—1116no—
146C3×3−1+2—1116no—
147C3×3−1+2C3×C37⋊C312yes—
148C3×3−1+2C3×C37⋊C311yescharacteristic
149C3×H(3,𝔽3)—372no—
150C3×H(3,𝔽3)C3×C37⋊C311yescharacteristic
151C32×C9—373no—
152#733—3718no—
153#732—379no—
154#731—3718no—
155#731—379no—
156AΣL(1,𝔽27)—376no—
157AΣL(1,𝔽27)—376no—
158AΣL(1,𝔽27)—376no—
159AΣL(1,𝔽27)—373no—
160AΣL(1,𝔽27)—373no—
161AΣL(1,𝔽27)—373no—
162C37#1062493411yescharacteristic, complemented, Sylow
163C33—31no—
164C33—371no—
165C33—371no—
166C33—371no—
167C33—3333no—
168C33—1112no—
169C33—1111no—
170C33—1112no—
171C33C32×C37⋊C311yescharacteristic
1723−1+2—32no—
1733−1+2—32no—
1743−1+2—318no—
1753−1+2—376no—
1763−1+2—376no—
1773−1+2—1112no—
1783−1+2—1112no—
1793−1+2—1111no—
1803−1+2—1111no—
1813−1+2—1112no—
1823−1+2—1112no—
1833−1+2—1112no—
1843−1+2—1112no—
1853−1+2—1119no—
1863−1+2—11118no—
1873−1+2—1119no—
1883−1+2—11118no—
1893−1+2C32⋊(C37⋊C3)13yes—
1903−1+2C32⋊(C37⋊C3)16yes—
1913−1+2C32×C37⋊C316yes—
192H(3,𝔽3)—32no—
193H(3,𝔽3)—376no—
194H(3,𝔽3)—1112no—
195H(3,𝔽3)—1112no—
196H(3,𝔽3)C32×C37⋊C313yes—
197C3×C9—32no—
198C3×C9—39no—
199C3×C9—373no—
200C3×C9—373no—
201C3×C9—1112no—
202C3×C9—1112no—
203C3×C9—1119no—
204C3×C9—11118no—
205C3×C9C32×C37⋊C313yes—
206C32—33no—
207C32—91no—
208C32—271no—
209C32—3331no—
210C32—3331no—
211C32—3333no—
212C32—3336no—
213C32—1113no—
214C32—1116no—
215C32—1111no—
216C32—1111no—
217C32—1111no—
218C32—9992no—
219C32C3×C32⋊(C37⋊C3)12yes—
220C32C3×C32⋊(C37⋊C3)11yescharacteristic
221C32C33×C37⋊C311yescharacteristic
222C9—36no—
223C9—1113no—
224C9—1113no—
225C9—1116no—
226C9—1116no—
227C9C3×C32⋊(C37⋊C3)19yes—
228C3—31no—
229C3—32no—
230C3—31no—
231C3—93no—
232C3—3331no—
233C3—3331no—
234C3—3336no—
235C3—1113no—
236C3C32×C32⋊(C37⋊C3)11yescharacteristic
subgroupG/H
centerC3C32×C32⋊(C37⋊C3)
commutator, cosocleC32×C37C34
FrattiniC32C33×C37⋊C3
FittingC37×3−1+4C3
socleC3×C37C32×H(3,𝔽3)
hypercenter3−1+4C37⋊C3
hypocenterC37#10624934
LCSG ▷ C32×C37 ▷ C3×C37 ▷ C37
UCSC1 ◁ C3 ◁ C3×3−1+2 ◁ 3−1+4

Complements

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

NH ≅ G/Ntype
1. #22582679C3semidirect
2. C3×3−1+2×C37⋊C3C3semidirect
3. #22582656C3semidirect
4. #22582655C3semidirect
5. #22582651C3semidirect
6. #22582647C3semidirect
7. #22582644C3semidirect
8. #22582642C3semidirect
9. #22582630C3semidirect
10. C3×#15688514C3semidirect
11. C3×#15688512C3semidirect
12. C37×3−1+4C3semidirect
16. #15688531C32semidirect
21. C3×#11758605C32semidirect
22. C3×#11758606C32semidirect
27. C3×C9×C37⋊C3C32semidirect
28. #15688520C32semidirect
29. #15688518C32semidirect
30. #15688517C32semidirect
31. #15688515C32semidirect
32. #15688514C32semidirect
33. #15688514C32semidirect
34. #15688514C32semidirect
35. #15688512C32semidirect
36. #15688512C32semidirect
37. #15688512C32semidirect
39. C37×#737C32semidirect
111. 3−1+4C37⋊C3semidirect
136. #737C3×C37⋊C3semidirect
162. C37#10624934semidirect

Maximal subgroups

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

subgroupindex
#225826793
C3×3−1+2×C37⋊C33
#225826563
#225826553
#225826513
#225826473
#225826443
#225826423
#225826303
C3×#156885143
C3×#156885123
C37×3−1+43
#1062493437