Finite Group Databasebeta

← #41008212

Subgroups — #41008212

Order 26973.

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

normal subgroups only

This group has 980 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#22582680C311yesmaximal, characteristic, complemented
2#22582679C312yesmaximal, complemented
3C32×#11758608C311yesmaximal, characteristic, complemented
4#22582654C319yesmaximal, complemented
5#22582653C319yesmaximal, complemented
6#22582651C312yesmaximal, complemented
7#22582647C311yesmaximal, characteristic, complemented
8#22582644C312yesmaximal, complemented
9#22582642C311yesmaximal, characteristic, complemented
10#22582629C319yesmaximal, complemented
11C3×#15688513C312yesmaximal, complemented
12C3×C37×AΣL(1,𝔽27)C311yesmaximal, characteristic, complemented
13#15688532—36no—
14#15688532C32118yescomplemented
15#15688531—36no—
16#15688530—36no—
17#15688530C3219yescomplemented
18C3×#11758608—36no—
19C3×#11758608—33no—
20C3×#11758608—33no—
21C3×#11758608C3212yes—
22C3×#11758608C3212yes—
23C3×#11758608C3211yescharacteristic
24C3×#11758608C3211yescharacteristic
25C3×#11758608C3211yescharacteristic
26C3×#11758608C3213yescomplemented
27C3×#11758608C3213yescomplemented
283−1+2×C37⋊C3—36no—
29C3×C32⋊(C37⋊C3)C3212yes—
30C32×#66524C3213yescomplemented
31#15688520C32118yescomplemented
32#15688518C3219yescomplemented
33#15688517C32118yescomplemented
34#15688515C3219yescomplemented
35#15688513C3216yescomplemented
36#15688513C3216yescomplemented
37#15688513C3216yescomplemented
38C34×C37C3211yescharacteristic
39C3×3−1+2×C37C3212yes—
40C3×H(3,𝔽3)×C37C3211yescharacteristic
41C37×AΣL(1,𝔽27)C3213yescomplemented
42C37×AΣL(1,𝔽27)C3213yescomplemented
43C37×AΣL(1,𝔽27)C3213yescomplemented
44C33×C37—32no—
45C33×C37—31no—
46C33×C37—31no—
47C33×C37—93no—
48C33×C37C3311yescharacteristic
49C33×C37C3311yescharacteristic
50C33×C37C3311yescharacteristic
51C33×C37C3311yescharacteristic
52C32×C37⋊C3—32no—
533−1+2×C37—32no—
543−1+2×C37—32no—
553−1+2×C37C3316yes—
56H(3,𝔽3)×C37—32no—
57H(3,𝔽3)×C37C3313yes—
58C3×C9×C37—32no—
59C32⋊(C37⋊C3)—32no—
60C32⋊(C37⋊C3)—32no—
61C32⋊(C37⋊C3)C3316yes—
62#11758608—32no—
63#11758608—32no—
64#11758608—32no—
65#11758608—32no—
66#11758608—31no—
67#11758608—31no—
68#11758608—32no—
69#11758608—32no—
70#11758608—39no—
71#11758608—39no—
72#11758608—318no—
73#11758608H(3,𝔽3)16yes—
74#11758608H(3,𝔽3)13yes—
75#11758608C3316yes—
76#11758608C3316yes—
77C3×#66524—32no—
78C3×#66524—32no—
79C3×#66524—39no—
80C3×#66524—39no—
81C3×#66524C3313yes—
82C3×#66524C3313yes—
83C3×#66524C3313yes—
84#11758606—318no—
85#11758605—318no—
86C9×C37⋊C3—318no—
87#10624934—371nomaximal, Sylow
88C32×C37—33no—
89C32×C37—31no—
90C32×C37—33no—
91C32×C37—31no—
92C32×C37—31no—
93C32×C37—91no—
94C32×C37—96no—
95C32×C37—93no—
96C32×C37—91no—
97C32×C37—91no—
98C32×C37—271no—
99C32×C37C3×H(3,𝔽3)12yes—
100C32×C37C3×H(3,𝔽3)11yescharacteristic
101C32×C37C3411yescharacteristic
102C3×C37⋊C3—36no—
103C3×C37⋊C3—272no—
104C9×C37—36no—
105#66524—36no—
106#66524—36no—
107#66524—33no—
108#66524—33no—
109#66524C3×H(3,𝔽3)19yes—
1103−1+4—372no—
1113−1+4—371no—
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)C37⋊C311yescharacteristic, complemented
122C3×C37—32no—
123C3×C37—33no—
124C3×C37—31no—
125C3×C37—31no—
126C3×C37—91no—
127C3×C37—91no—
128C3×C37—93no—
129C3×C37C32×H(3,𝔽3)11yescharacteristic
130C37⋊C3—96no—
131C34C3×C37⋊C311yescharacteristic
132#737—3718no—
133#737—379no—
134#737—1116no—
135#737—1116no—
136#737—1116no—
137C3×3−1+2—371no—
138C3×3−1+2—373no—
139C3×3−1+2—373no—
140C3×3−1+2—371no—
141C3×3−1+2—371no—
142C3×3−1+2—372no—
143C3×3−1+2—372no—
144C3×3−1+2—1116no—
145C3×3−1+2—1113no—
146C3×3−1+2—1116no—
147C3×3−1+2—1113no—
148C3×3−1+2C3×C37⋊C312yes—
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)C3×C37⋊C313yescomplemented
160AΣL(1,𝔽27)C3×C37⋊C313yescomplemented
161AΣL(1,𝔽27)C3×C37⋊C313yescomplemented
162C37#1062493411yescharacteristic, complemented, Sylow
163C33—31no—
164C33—32no—
165C33—31no—
166C33—93no—
167C33—1112no—
168C33C32×C37⋊C311yescharacteristic
169C33C32×C37⋊C311yescharacteristic
170C33C32×C37⋊C311yescharacteristic
171C33C32×C37⋊C311yescharacteristic
1723−1+2—32no—
1733−1+2—32no—
1743−1+2—373no—
1753−1+2—376no—
1763−1+2—376no—
1773−1+2—376no—
1783−1+2—11118no—
1793−1+2—11118no—
1803−1+2—1119no—
1813−1+2—1119no—
1823−1+2—11118no—
1833−1+2—1111no—
1843−1+2—1111no—
1853−1+2—1112no—
1863−1+2—1112no—
1873−1+2—1112no—
1883−1+2—1112no—
1893−1+2—1112no—
1903−1+2—1112no—
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—373no—
199C3×C9—373no—
200C3×C9—373no—
201C3×C9—11118no—
202C3×C9—1119no—
203C3×C9—1119no—
204C3×C9—1112no—
205C3×C9—1112no—
206C32—33no—
207C32—31no—
208C32—33no—
209C32—31no—
210C32—31no—
211C32—91no—
212C32—93no—
213C32—96no—
214C32—91no—
215C32—91no—
216C32—271no—
217C32—1116no—
218C32—9992no—
219C32H(3,𝔽3)×C37⋊C311yescharacteristic
220C32H(3,𝔽3)×C37⋊C312yes—
221C32C33×C37⋊C311yescharacteristic
222C9—36no—
223C9—379no—
224C9—1113no—
225C9—1113no—
226C9—1116no—
227C9—1116no—
228C3—31no—
229C3—33no—
230C3—31no—
231C3—32no—
232C3—91no—
233C3—91no—
234C3—93no—
235C3—3336no—
236C3C3×H(3,𝔽3)×C37⋊C311yescharacteristic
subgroupG/H
centerC3C3×H(3,𝔽3)×C37⋊C3
commutator, cosocleC32×C37C34
FrattiniC32C33×C37⋊C3
FittingC3×C37×AΣL(1,𝔽27)C3
socleC3×C37C32×H(3,𝔽3)
hypercenterC3×AΣL(1,𝔽27)C37⋊C3
hypocenterC37#10624934
LCSG ▷ C32×C37 ▷ C3×C37 ▷ C37
UCSC1 ◁ C3 ◁ C33 ◁ C3×AΣL(1,𝔽27)

Complements

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

NH ≅ G/Ntype
1. #22582680C3semidirect
2. #22582679C3semidirect
3. C32×#11758608C3semidirect
4. #22582654C3semidirect
5. #22582653C3semidirect
6. #22582651C3semidirect
7. #22582647C3semidirect
8. #22582644C3semidirect
9. #22582642C3semidirect
10. #22582629C3semidirect
11. C3×#15688513C3semidirect
12. C3×C37×AΣL(1,𝔽27)C3semidirect
14. #15688532C32semidirect
17. #15688530C32semidirect
26. C3×#11758608C32semidirect
27. C3×#11758608C32semidirect
30. C32×#66524C32semidirect
31. #15688520C32semidirect
32. #15688518C32semidirect
33. #15688517C32semidirect
34. #15688515C32semidirect
35. #15688513C32semidirect
36. #15688513C32semidirect
37. #15688513C32semidirect
41. C37×AΣL(1,𝔽27)C32semidirect
42. C37×AΣL(1,𝔽27)C32semidirect
43. C37×AΣL(1,𝔽27)C32semidirect
121. C3×AΣL(1,𝔽27)C37⋊C3semidirect
159. AΣL(1,𝔽27)C3×C37⋊C3semidirect
160. AΣL(1,𝔽27)C3×C37⋊C3semidirect
161. AΣL(1,𝔽27)C3×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
#225826803
#225826793
C32×#117586083
#225826543
#225826533
#225826513
#225826473
#225826443
#225826423
#225826293
C3×#156885133
C3×C37×AΣL(1,𝔽27)3
#1062493437