Finite Group Databasebeta

← #41008230

Subgroups — #41008230

Order 26973.

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

normal subgroups only

This group has 948 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#22582677C313yesmaximal, complemented
2C3×#15688531C311yesmaximal, characteristic, complemented
3#22582660C316yesmaximal, complemented
4#22582656C316yesmaximal, complemented
5#22582653C316yesmaximal, complemented
6#22582641C313yesmaximal, complemented
7#22582635C313yesmaximal, complemented
8#22582634C314yesmaximal, complemented
9#22582633C312yesmaximal, complemented
10#22582631C312yesmaximal, complemented
11#733×C37⋊C3C312yesmaximal, complemented
12#732×C37⋊C3C311yesmaximal, characteristic, complemented
13C37×#6908C311yesmaximal, characteristic, complemented
14#15688532—312no—
15#15688532C32112yescomplemented
16#15688531—32no—
17#15688531—31no—
18#15688531—32no—
19#15688531—34no—
20#15688531—36no—
21#15688531C3216yescomplemented
22#15688530C3216yescomplemented
23C3×#11758608C3212yescomplemented
24C3×#11758605C3211yescharacteristic
25C3×#11758605C3212yescomplemented
26C3×#11758606—33no—
27C3×#11758606C3211yescharacteristic
28C3×#11758606C3212yescomplemented
293−1+2×C37⋊C3—36no—
303−1+2×C37⋊C3C3216yes—
31C3×C32⋊(C37⋊C3)C3211yescharacteristic
32H(3,𝔽3)×C37⋊C3C3213yes—
33C3×C9×C37⋊C3C3211yescharacteristic
34C3×C9×C37⋊C3C3212yescomplemented
35#15688521C3214yescomplemented
36#15688520C32112yescomplemented
37#15688519C3212yes—
38#15688518C3216yes—
39#15688517C32112yescomplemented
40#15688516C3216yescomplemented
41#15688515C3216yescomplemented
42#15688514C32112yescomplemented
43#15688513C3216yescomplemented
44#15688512C3216yescomplemented
45C37×#737C3213yescomplemented
46C32×C9×C37C3211yescharacteristic, complemented
47C37×#733C3212yescomplemented
48C37×#732C3211yescharacteristic
49C37×#731C3213yescomplemented
50C37×AΣL(1,𝔽27)C3213yescomplemented
51C33×C37C3311yescharacteristic
52C32×C37⋊C3—33no—
53C32×C37⋊C3C3312yes—
54C32×C37⋊C3C3311yescharacteristic
553−1+2×C37—36no—
563−1+2×C37C3316yes—
57H(3,𝔽3)×C37C3313yes—
58C3×C9×C37—32no—
59C3×C9×C37—31no—
60C3×C9×C37—33no—
61C3×C9×C37C3311yescharacteristic
62C3×C9×C37C3311yescharacteristic
63C3×C9×C37C3311yescharacteristic
64C32⋊(C37⋊C3)—36no—
65C32⋊(C37⋊C3)—32no—
66C32⋊(C37⋊C3)—31no—
67C32⋊(C37⋊C3)C3316yes—
68#11758608—312no—
69#11758608—34no—
70#11758608—32no—
71#11758608C33112yes—
72C3×#66524—36no—
73C3×#66524C3312yes—
74C3×#66524C3312yes—
75C3×#66524C3312yes—
76#11758606—36no—
77#11758606—36no—
78#11758606—34no—
79#11758606—32no—
80#11758606—312no—
81#11758606H(3,𝔽3)12yes—
82#11758606H(3,𝔽3)11yescharacteristic
83#11758606H(3,𝔽3)12yes—
84#11758606H(3,𝔽3)14yes—
85#11758605—36no—
86#11758605—32no—
87#11758605—31no—
88#11758605—36no—
89#11758605—312no—
90#11758605—34no—
91#11758605—32no—
92C9×C37⋊C3—32no—
93C9×C37⋊C3—31no—
94C9×C37⋊C3—36no—
95C9×C37⋊C3—36no—
96C9×C37⋊C3—34no—
97C9×C37⋊C3—32no—
98C9×C37⋊C3—312no—
99#10624936—371nomaximal, Sylow
100C32×C37—33no—
101C32×C37—31no—
102C32×C37—91no—
103C32×C37C3411yescharacteristic
104C3×C37⋊C3—32no—
105C3×C37⋊C3—36no—
106C3×C37⋊C3—91no—
107C3×C37⋊C3—92no—
108C3×C37⋊C3—273no—
109C3×C37⋊C3C3×H(3,𝔽3)12yes—
110C3×C37⋊C3C3×H(3,𝔽3)11yescharacteristic
111C9×C37—31no—
112C9×C37—36no—
113C9×C37—31no—
114C9×C37C3×H(3,𝔽3)12yes—
115C9×C37C3×H(3,𝔽3)11yescharacteristic
116#66524—312no—
117#66524—32no—
118#66524—32no—
119#66524C3×H(3,𝔽3)14yes—
120#66524C3×H(3,𝔽3)12yes—
1213−1+4—373no—
122C3×#737—371no—
123#6913—376no—
124#6913—376no—
125#6913—376no—
126#6910—373no—
127#6909—373no—
128#6908—372no—
129#6908—374no—
130#6908—372no—
131#6908C37⋊C311yescharacteristic, complemented
132C3×#733—372no—
133C3×#732—371no—
134C3×C37—31no—
135C3×C37—93no—
136C3×C37—91no—
137C3×C37C32×H(3,𝔽3)11yescharacteristic
138C37⋊C3—31no—
139C37⋊C3—32no—
140C37⋊C3—92no—
141C37⋊C3—96no—
142#737—3712no—
143#737—376no—
144#737—376no—
145#737—1111no—
146#737—1112no—
147#737—1112no—
148#737—1114no—
149#737—1116no—
150#737—11112no—
151#737C3×C37⋊C313yescomplemented
152C3×3−1+2—372no—
153C3×3−1+2—372no—
154C3×3−1+2—371no—
155C3×3−1+2—371no—
156C3×3−1+2—372no—
157C3×3−1+2—376no—
158C3×3−1+2—1113no—
159C3×3−1+2—1116no—
160C3×H(3,𝔽3)—373no—
161C3×H(3,𝔽3)—371no—
162C32×C9—371no—
163C32×C9—372no—
164C32×C9C3×C37⋊C311yescharacteristic, complemented
165#733—3712no—
166#733—374no—
167#733C3×C37⋊C312yescomplemented
168#732—376no—
169#732—372no—
170#732C3×C37⋊C311yescharacteristic
171#731—3712no—
172#731—376no—
173#731—376no—
174#731C3×C37⋊C313yescomplemented
175AΣL(1,𝔽27)—376no—
176AΣL(1,𝔽27)—3712no—
177AΣL(1,𝔽27)—376no—
178AΣL(1,𝔽27)C3×C37⋊C313yescomplemented
179C37#1062493611yescharacteristic, complemented, Sylow
180C33—371no—
181C33—372no—
182C33—1113no—
183C33C32×C37⋊C311yescharacteristic
1843−1+2—36no—
1853−1+2—371no—
1863−1+2—372no—
1873−1+2—372no—
1883−1+2—374no—
1893−1+2—3712no—
1903−1+2—1114no—
1913−1+2—1112no—
1923−1+2—11112no—
1933−1+2—11112no—
1943−1+2—1112no—
1953−1+2—1111no—
1963−1+2—1116no—
1973−1+2—1116no—
1983−1+2—1114no—
1993−1+2—1112no—
2003−1+2—11112no—
2013−1+2—1116no—
2023−1+2—1112no—
2033−1+2—1114no—
2043−1+2—1116no—
2053−1+2C32×C37⋊C316yes—
206H(3,𝔽3)—376no—
207H(3,𝔽3)—1112no—
208H(3,𝔽3)—1111no—
209H(3,𝔽3)—1116no—
210H(3,𝔽3)C32×C37⋊C313yes—
211C3×C9—31no—
212C3×C9—32no—
213C3×C9—33no—
214C3×C9—372no—
215C3×C9—372no—
216C3×C9—372no—
217C3×C9—1114no—
218C3×C9—1112no—
219C3×C9—11112no—
220C3×C9—1112no—
221C3×C9—1111no—
222C3×C9—1116no—
223C3×C9—1116no—
224C3×C9—1116no—
225C3×C9C32×C37⋊C311yescharacteristic
226C3×C9C32×C37⋊C311yescharacteristic
227C3×C9C32×C37⋊C311yescharacteristic
228C32—31no—
229C32—33no—
230C32—91no—
231C32—372no—
232C32—371no—
233C32—3332no—
234C32—3331no—
235C32—1116no—
236C32—1112no—
237C32—9993no—
238C32C33×C37⋊C311yescharacteristic
239C9—31no—
240C9—31no—
241C9—36no—
242C9—374no—
243C9—372no—
244C9—1112no—
245C9—1112no—
246C9—11112no—
247C9H(3,𝔽3)×C37⋊C311yescharacteristic
248C9H(3,𝔽3)×C37⋊C312yes—
249C3—31no—
250C3—91no—
251C3—93no—
252C3—3332no—
253C3—3336no—
254C3—1111no—
255C3—1112no—
256C3C3×H(3,𝔽3)×C37⋊C311yescharacteristic
subgroupG/H
centerC3C3×H(3,𝔽3)×C37⋊C3
commutator, cosocleC32×C37C34
FrattiniC32C33×C37⋊C3
FittingC37×#6908C3
socleC3×C37C32×H(3,𝔽3)
hypercenter#6908C37⋊C3
hypocenterC37#10624936
LCSG ▷ C32×C37 ▷ C3×C37 ▷ C37
UCSC1 ◁ C3 ◁ C3×C9 ◁ #6908

Complements

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

NH ≅ G/Ntype
1. #22582677C3semidirect
2. C3×#15688531C3semidirect
3. #22582660C3semidirect
4. #22582656C3semidirect
5. #22582653C3semidirect
6. #22582641C3semidirect
7. #22582635C3semidirect
8. #22582634C3semidirect
9. #22582633C3semidirect
10. #22582631C3semidirect
11. #733×C37⋊C3C3semidirect
12. #732×C37⋊C3C3semidirect
13. C37×#6908C3semidirect
15. #15688532C32semidirect
21. #15688531C32semidirect
22. #15688530C32semidirect
23. C3×#11758608C32semidirect
25. C3×#11758605C32semidirect
28. C3×#11758606C32semidirect
34. C3×C9×C37⋊C3C32semidirect
35. #15688521C32semidirect
36. #15688520C32semidirect
39. #15688517C32semidirect
40. #15688516C32semidirect
41. #15688515C32semidirect
42. #15688514C32semidirect
43. #15688513C32semidirect
44. #15688512C32semidirect
45. C37×#737C32semidirect
46. C32×C9×C37C32semidirect
47. C37×#733C32semidirect
49. C37×#731C32semidirect
50. C37×AΣL(1,𝔽27)C32semidirect
131. #6908C37⋊C3semidirect
151. #737C3×C37⋊C3semidirect
164. C32×C9C3×C37⋊C3semidirect
167. #733C3×C37⋊C3semidirect
174. #731C3×C37⋊C3semidirect
178. AΣL(1,𝔽27)C3×C37⋊C3semidirect
179. C37#10624936semidirect

Maximal subgroups

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

subgroupindex
#225826773
C3×#156885313
#225826603
#225826563
#225826533
#225826413
#225826353
#225826343
#225826333
#225826313
#733×C37⋊C33
#732×C37⋊C33
C37×#69083
#1062493637