Finite Group Databasebeta

← C37×#10624750 #41004539

Subgroups — C37×#10624750 #41004539

Order 26973.

414 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C37×3−1+4C311yesmaximal, characteristic, complemented
2C37×#6912C311yesmaximal, characteristic, complemented
3C37×#6912C311yesmaximal, characteristic, complemented
4C37×#6912C311yesmaximal, characteristic, complemented
5C37×#6910C311yesmaximal, characteristic, complemented
6C37×#6910C311yesmaximal, characteristic, complemented
7C37×#6909C311yesmaximal, characteristic, complemented
8C37×#6909C311yesmaximal, characteristic, complemented
9C37×#6909C311yesmaximal, characteristic, complemented
10C3×C37×#733C311yesmaximal, characteristic, complemented
11C3×C37×#731C311yesmaximal, characteristic, complemented
12C37×#6893C311yesmaximal, characteristic, complemented
13C37×#6893C311yesmaximal, characteristic, complemented
14C37×#737—33no—
15C37×#737—33no—
16C37×#737—33no—
17C3×3−1+2×C37—33no—
18C3×3−1+2×C37C3211yescharacteristic
19C3×3−1+2×C37C3211yescharacteristic
20C3×3−1+2×C37C3211yescharacteristic
21C3×3−1+2×C37C3211yescharacteristic
22C3×3−1+2×C37C3211yescharacteristic
23C3×3−1+2×C37C3211yescharacteristic
24C3×3−1+2×C37C3211yescharacteristic
25C3×3−1+2×C37C3211yescharacteristic
26C3×3−1+2×C37C3211yescharacteristic
27C3×H(3,𝔽3)×C37C3211yescharacteristic
28C3×H(3,𝔽3)×C37C3211yescharacteristic
29C3×H(3,𝔽3)×C37C3211yescharacteristic
30C32×C9×C37C3211yescharacteristic
31C37×#733—33no—
32C37×#733—33no—
33C37×#733—33no—
34C37×#733—33no—
35C37×#731—33no—
36C37×#731—33no—
37C37×#731—33no—
38C37×AΣL(1,𝔽27)—33no—
39C37×AΣL(1,𝔽27)—33no—
40C37×AΣL(1,𝔽27)—33no—
41C37×C32⋊C9—33no—
42C37×C32⋊C9—33no—
43C33×C37—31no—
44C33×C37—31no—
45C33×C37H(3,𝔽3)13yes—
46C33×C37C3311yescharacteristic
473−1+2×C37—318no—
483−1+2×C37—31no—
493−1+2×C37—31no—
503−1+2×C37—31no—
513−1+2×C37—31no—
523−1+2×C37—31no—
533−1+2×C37—31no—
543−1+2×C37—31no—
553−1+2×C37—31no—
563−1+2×C37—31no—
573−1+2×C37—31no—
583−1+2×C37—31no—
593−1+2×C37—31no—
603−1+2×C37—31no—
613−1+2×C37—31no—
623−1+2×C37—31no—
633−1+2×C37—31no—
643−1+2×C37—31no—
653−1+2×C37—31no—
663−1+2×C37—31no—
673−1+2×C37—31no—
683−1+2×C37—31no—
693−1+2×C37—31no—
703−1+2×C37—31no—
713−1+2×C37—31no—
723−1+2×C37—31no—
733−1+2×C37H(3,𝔽3)13yes—
743−1+2×C37H(3,𝔽3)13yes—
75H(3,𝔽3)×C37—31no—
76H(3,𝔽3)×C37—31no—
77H(3,𝔽3)×C37—31no—
78H(3,𝔽3)×C37—31no—
79H(3,𝔽3)×C37—31no—
80H(3,𝔽3)×C37—31no—
81H(3,𝔽3)×C37—31no—
82H(3,𝔽3)×C37—31no—
83H(3,𝔽3)×C37—31no—
84C3×C9×C37—39no—
85C3×C9×C37—31no—
86C3×C9×C37—31no—
87C3×C9×C37—31no—
88C3×C9×C37—31no—
89C3×C9×C37—31no—
90C3×C9×C37—31no—
91C3×C9×C37—31no—
92C3×C9×C37—31no—
93C3×C9×C37—31no—
94C3×C9×C37—31no—
95C3×C9×C37—31no—
96C3×C9×C37—31no—
97C3×C9×C37H(3,𝔽3)13yes—
98#10624750C3711yesmaximal, characteristic, direct product factor, Sylow
99C32×C37—33no—
100C32×C37—91no—
101C32×C37—93no—
102C32×C37—93no—
103C32×C37—91no—
104C32×C37—91no—
105C32×C37—271no—
106C32×C37C3×H(3,𝔽3)11yescharacteristic
107C32×C37C3×H(3,𝔽3)11yescharacteristic
108C32×C37C3×H(3,𝔽3)11yescharacteristic
109C32×C37C3×H(3,𝔽3)11yescharacteristic
110C9×C37—33no—
111C9×C37—33no—
112C9×C37—33no—
113C9×C37—91no—
114C9×C37—91no—
115C9×C37—91no—
116C9×C37—91no—
117C9×C37—91no—
118C9×C37—91no—
119C9×C37—91no—
1203−1+4C3×C3711yescharacteristic, complemented
121#6912C3×C3711yescharacteristic, complemented
122#6912C3×C3711yescharacteristic, complemented
123#6912C3×C3711yescharacteristic, complemented
124#6910C3×C3711yescharacteristic, complemented
125#6910C3×C3711yescharacteristic, complemented
126#6909C3×C3711yescharacteristic, complemented
127#6909C3×C3711yescharacteristic, complemented
128#6909C3×C3711yescharacteristic, complemented
129C3×#733C3×C3711yescharacteristic, complemented
130C3×#731C3×C3711yescharacteristic, complemented
131#6893C3×C3711yescharacteristic, complemented
132#6893C3×C3711yescharacteristic, complemented
133C3×C37—31no—
134C3×C37—31no—
135C3×C37—31no—
136C3×C37—31no—
137C3×C37—93no—
138C3×C37—271no—
139C3×C37—271no—
140C3×C37C32⋊H(3,𝔽3)11yescharacteristic
141#737—33no—
142#737—33no—
143#737—33no—
144C3×3−1+2—33no—
145C3×3−1+2C32×C3711yescharacteristic
146C3×3−1+2C32×C3711yescharacteristic
147C3×3−1+2C32×C3711yescharacteristic
148C3×3−1+2C32×C3711yescharacteristic
149C3×3−1+2C32×C3711yescharacteristic
150C3×3−1+2C32×C3711yescharacteristic
151C3×3−1+2C32×C3711yescharacteristic
152C3×3−1+2C32×C3711yescharacteristic
153C3×3−1+2C32×C3711yescharacteristic
154C3×H(3,𝔽3)C32×C3711yescharacteristic
155C3×H(3,𝔽3)C32×C3711yescharacteristic
156C3×H(3,𝔽3)C32×C3711yescharacteristic
157C32×C9C32×C3711yescharacteristic
158#733—33no—
159#733—33no—
160#733—33no—
161#733—33no—
162#731—33no—
163#731—33no—
164#731—33no—
165AΣL(1,𝔽27)—33no—
166AΣL(1,𝔽27)—33no—
167AΣL(1,𝔽27)—33no—
168C32⋊C9—33no—
169C32⋊C9—33no—
170C37#1062475011yescharacteristic, direct product factor, Sylow
171C33—31no—
172C33—31no—
173C33H(3,𝔽3)×C3713yes—
174C33C33×C3711yescharacteristic
1753−1+2—318no—
1763−1+2—31no—
1773−1+2—31no—
1783−1+2—31no—
1793−1+2—31no—
1803−1+2—31no—
1813−1+2—31no—
1823−1+2—31no—
1833−1+2—31no—
1843−1+2—31no—
1853−1+2—31no—
1863−1+2—31no—
1873−1+2—31no—
1883−1+2—31no—
1893−1+2—31no—
1903−1+2—31no—
1913−1+2—31no—
1923−1+2—31no—
1933−1+2—31no—
1943−1+2—31no—
1953−1+2—31no—
1963−1+2—31no—
1973−1+2—31no—
1983−1+2—31no—
1993−1+2—31no—
2003−1+2—31no—
2013−1+2H(3,𝔽3)×C3713yes—
2023−1+2H(3,𝔽3)×C3713yes—
203H(3,𝔽3)—31no—
204H(3,𝔽3)—31no—
205H(3,𝔽3)—31no—
206H(3,𝔽3)—31no—
207H(3,𝔽3)—31no—
208H(3,𝔽3)—31no—
209H(3,𝔽3)—31no—
210H(3,𝔽3)—31no—
211H(3,𝔽3)—31no—
212C3×C9—39no—
213C3×C9—31no—
214C3×C9—31no—
215C3×C9—31no—
216C3×C9—31no—
217C3×C9—31no—
218C3×C9—31no—
219C3×C9—31no—
220C3×C9—31no—
221C3×C9—31no—
222C3×C9—31no—
223C3×C9—31no—
224C3×C9—31no—
225C3×C9H(3,𝔽3)×C3713yes—
226C32—33no—
227C32—91no—
228C32—93no—
229C32—93no—
230C32—91no—
231C32—91no—
232C32—271no—
233C32C3×H(3,𝔽3)×C3711yescharacteristic
234C32C3×H(3,𝔽3)×C3711yescharacteristic
235C32C3×H(3,𝔽3)×C3711yescharacteristic
236C32C3×H(3,𝔽3)×C3711yescharacteristic
237C9—33no—
238C9—33no—
239C9—33no—
240C9—91no—
241C9—91no—
242C9—91no—
243C9—91no—
244C9—91no—
245C9—91no—
246C9—91no—
247C3—31no—
248C3—31no—
249C3—31no—
250C3—31no—
251C3—93no—
252C3—271no—
253C3—271no—
254C3C37×C32⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
center, socleC3×C37C32⋊H(3,𝔽3)
commutator, Frattini, cosocleC33C33×C37
LCSG ▷ C33 ▷ C3 ▷ C1
UCSC1 ◁ C3×C37 ◁ C33×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C37×3−1+4C3semidirect
2. C37×#6912C3semidirect
3. C37×#6912C3semidirect
4. C37×#6912C3semidirect
5. C37×#6910C3semidirect
6. C37×#6910C3semidirect
7. C37×#6909C3semidirect
8. C37×#6909C3semidirect
9. C37×#6909C3semidirect
10. C3×C37×#733C3semidirect
11. C3×C37×#731C3semidirect
12. C37×#6893C3semidirect
13. C37×#6893C3semidirect
98. #10624750C37direct (N × H)
120. 3−1+4C3×C37semidirect
121. #6912C3×C37semidirect
122. #6912C3×C37semidirect
123. #6912C3×C37semidirect
124. #6910C3×C37semidirect
125. #6910C3×C37semidirect
126. #6909C3×C37semidirect
127. #6909C3×C37semidirect
128. #6909C3×C37semidirect
129. C3×#733C3×C37semidirect
130. C3×#731C3×C37semidirect
131. #6893C3×C37semidirect
132. #6893C3×C37semidirect
170. C37#10624750direct (N × H)

Maximal subgroups

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

subgroupindex
C37×3−1+43
C37×#69123
C37×#69103
C37×#69093
C3×C37×#7333
C3×C37×#7313
C37×#68933
#1062475037