Finite Group Databasebeta

← C37⋊C3×#6910 #41008154

Subgroups — C37⋊C3×#6910 #41008154

Order 26973.

1088 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 271 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
1C3×3−1+2×C37⋊C3C312yesmaximal, complemented
2C3×3−1+2×C37⋊C3C311yesmaximal, characteristic, complemented
3C3×H(3,𝔽3)×C37⋊C3C311yesmaximal, characteristic
4#22582644C314yesmaximal, complemented
5#22582643C312yesmaximal, complemented
6#22582642C312yesmaximal, complemented
7#22582641C319yesmaximal, complemented
8#22582640C319yesmaximal, complemented
9#731×C37⋊C3C319yesmaximal, complemented
10C37×#6910C311yesmaximal, characteristic, complemented
11C33×C37⋊C3—31no—
12C33×C37⋊C3C3211yescharacteristic
13C3×#11758608C3214yescomplemented
14C3×#11758608C3212yescomplemented
15C3×#11758605—32no—
16C3×#11758605C3213yescomplemented
17C3×#11758606—32no—
18C3×#11758606C3213yescomplemented
193−1+2×C37⋊C3—32no—
203−1+2×C37⋊C3—32no—
213−1+2×C37⋊C3—32no—
223−1+2×C37⋊C3—32no—
233−1+2×C37⋊C3—32no—
243−1+2×C37⋊C3—32no—
253−1+2×C37⋊C3—32no—
263−1+2×C37⋊C3—31no—
273−1+2×C37⋊C3—32no—
283−1+2×C37⋊C3—32no—
293−1+2×C37⋊C3—32no—
303−1+2×C37⋊C3C3216yescomplemented
313−1+2×C37⋊C3C3216yescomplemented
323−1+2×C37⋊C3C3216yescomplemented
33C3×C32⋊(C37⋊C3)—32no—
34C3×C32⋊(C37⋊C3)C3212yes—
35H(3,𝔽3)×C37⋊C3—32no—
36H(3,𝔽3)×C37⋊C3—32no—
37H(3,𝔽3)×C37⋊C3—32no—
38H(3,𝔽3)×C37⋊C3C3213yes—
39H(3,𝔽3)×C37⋊C3C3213yes—
40H(3,𝔽3)×C37⋊C3C3213yes—
41C3×C9×C37⋊C3—32no—
42C3×C9×C37⋊C3C3213yescomplemented
43#15688517C32136yescomplemented
44#15688516C32118yescomplemented
45#15688515C32118yescomplemented
46C3×3−1+2×C37C3212yescomplemented
47C3×3−1+2×C37C3211yescharacteristic, complemented
48C3×H(3,𝔽3)×C37C3211yescharacteristic
49C37×#731C3219yescomplemented
50C33×C37—31no—
51C33×C37C3311yescharacteristic
52C32×C37⋊C3—32no—
53C32×C37⋊C3—33no—
54C32×C37⋊C3—33no—
55C32×C37⋊C3—33no—
56C32×C37⋊C3—93no—
57C32×C37⋊C3—99no—
58C32×C37⋊C3H(3,𝔽3)12yes—
59C32×C37⋊C3H(3,𝔽3)11yescharacteristic
60C32×C37⋊C3H(3,𝔽3)12yes—
61C32×C37⋊C3H(3,𝔽3)12yes—
62C32×C37⋊C3H(3,𝔽3)12yes—
63C32×C37⋊C3C3311yescharacteristic
64C32×C37⋊C3C3311yescharacteristic
65C32×C37⋊C3C3311yescharacteristic
663−1+2×C37—32no—
673−1+2×C37—32no—
683−1+2×C37—32no—
693−1+2×C37—31no—
703−1+2×C37C3316yescomplemented
71H(3,𝔽3)×C37—32no—
72H(3,𝔽3)×C37C3313yes—
73C3×C9×C37—32no—
74C3×C9×C37C3313yescomplemented
75C32⋊(C37⋊C3)—34no—
76C32⋊(C37⋊C3)—36no—
77C32⋊(C37⋊C3)—36no—
78C32⋊(C37⋊C3)—36no—
79C32⋊(C37⋊C3)C3316yes—
80#11758608—34no—
81#11758608—34no—
82#11758608—34no—
83#11758608—32no—
84#11758608C33112yescomplemented
85C3×#66524—34no—
86C3×#66524C3316yescomplemented
87#11758606—36no—
88#11758606—33no—
89#11758606—36no—
90#11758606—36no—
91#11758606—36no—
92#11758605—36no—
93#11758605—33no—
94#11758605—36no—
95#11758605—36no—
96#11758605—36no—
97C9×C37⋊C3—36no—
98C9×C37⋊C3—36no—
99C9×C37⋊C3—36no—
100C9×C37⋊C3—36no—
101C9×C37⋊C3—33no—
102C3×#6910—371nomaximal, Sylow
103C32×C37—33no—
104C32×C37—91no—
105C32×C37—93no—
106C32×C37C3×H(3,𝔽3)12yes—
107C32×C37C3×H(3,𝔽3)11yescharacteristic
108C32×C37C3411yescharacteristic
109C3×C37⋊C3—33no—
110C3×C37⋊C3—36no—
111C3×C37⋊C3—33no—
112C3×C37⋊C3—36no—
113C3×C37⋊C3—92no—
114C3×C37⋊C3—92no—
115C3×C37⋊C3—92no—
116C3×C37⋊C3—91no—
117C3×C37⋊C3—91no—
118C3×C37⋊C3—99no—
119C3×C37⋊C3—99no—
120C3×C37⋊C3—96no—
121C3×C37⋊C3—99no—
122C3×C37⋊C3C3×H(3,𝔽3)12yes—
123C3×C37⋊C3C3×H(3,𝔽3)12yes—
124C3×C37⋊C3C3×H(3,𝔽3)12yes—
125C3×C37⋊C3C3×H(3,𝔽3)11yescharacteristic
126C3×C37⋊C3C3×H(3,𝔽3)11yescharacteristic
127C3×C37⋊C3C3×H(3,𝔽3)11yescharacteristic
128C9×C37—36no—
129C9×C37—33no—
130#66524—312no—
131#66524—36no—
132C32×3−1+2—371no—
133C32×3−1+2—372no—
134C32×H(3,𝔽3)—371no—
135#6910—372no—
136#6910—374no—
137#6910—372no—
138#6910—379no—
139#6910—379no—
140#6910C37⋊C311yescharacteristic, direct product factor
141C3×#731—379no—
142C3×C37—31no—
143C3×C37—32no—
144C3×C37—31no—
145C3×C37—93no—
146C3×C37C32×H(3,𝔽3)11yescharacteristic
147C37⋊C3—31no—
148C37⋊C3—31no—
149C37⋊C3—32no—
150C37⋊C3—32no—
151C37⋊C3—32no—
152C37⋊C3—96no—
153C37⋊C3#691013yesdirect product factor
154C34—371no—
155C34—1111no—
156C3×3−1+2—372no—
157C3×3−1+2—373no—
158C3×3−1+2—373no—
159C3×3−1+2—374no—
160C3×3−1+2—376no—
161C3×3−1+2—376no—
162C3×3−1+2—376no—
163C3×3−1+2—1112no—
164C3×3−1+2—1112no—
165C3×3−1+2—1112no—
166C3×3−1+2—1112no—
167C3×3−1+2—1112no—
168C3×3−1+2—1112no—
169C3×3−1+2—1111no—
170C3×3−1+2—1112no—
171C3×3−1+2—1112no—
172C3×3−1+2—1112no—
173C3×3−1+2—1112no—
174C3×3−1+2—1112no—
175C3×3−1+2—1112no—
176C3×3−1+2C3×C37⋊C312yescomplemented
177C3×3−1+2C3×C37⋊C311yescharacteristic, complemented
178C3×H(3,𝔽3)—372no—
179C3×H(3,𝔽3)—373no—
180C3×H(3,𝔽3)—373no—
181C3×H(3,𝔽3)—373no—
182C3×H(3,𝔽3)—1112no—
183C3×H(3,𝔽3)—1112no—
184C3×H(3,𝔽3)—1112no—
185C3×H(3,𝔽3)—1112no—
186C3×H(3,𝔽3)C3×C37⋊C311yescharacteristic
187C32×C9—373no—
188C32×C9—1112no—
189#731—3736no—
190#731—3718no—
191#731—3718no—
192#731C3×C37⋊C319yescomplemented
193C37C3×#691011yescharacteristic, complemented, Sylow
194C33—31no—
195C33—372no—
196C33—372no—
197C33—372no—
198C33—371no—
199C33—371no—
200C33—372no—
201C33—371no—
202C33—371no—
203C33—3339no—
204C33—3333no—
205C33—1113no—
206C33—1113no—
207C33—1113no—
208C33—1112no—
209C33C32×C37⋊C311yescharacteristic
2103−1+2—32no—
2113−1+2—32no—
2123−1+2—32no—
2133−1+2—31no—
2143−1+2—3712no—
2153−1+2—1116no—
2163−1+2—1116no—
2173−1+2—1116no—
2183−1+2—1116no—
2193−1+2—1113no—
2203−1+2—1116no—
2213−1+2—1116no—
2223−1+2—1116no—
2233−1+2—1116no—
2243−1+2—1113no—
2253−1+2—1114no—
2263−1+2—1112no—
2273−1+2—1114no—
2283−1+2—1114no—
2293−1+2C32×C37⋊C316yescomplemented
230H(3,𝔽3)—32no—
231H(3,𝔽3)—376no—
232H(3,𝔽3)—1116no—
233H(3,𝔽3)—1116no—
234H(3,𝔽3)—1116no—
235H(3,𝔽3)—1114no—
236H(3,𝔽3)C32×C37⋊C313yes—
237C3×C9—32no—
238C3×C9—376no—
239C3×C9—1113no—
240C3×C9—1116no—
241C3×C9—1116no—
242C3×C9—1116no—
243C3×C9—1116no—
244C3×C9—1114no—
245C3×C9C32×C37⋊C313yescomplemented
246C32—33no—
247C32—91no—
248C32—93no—
249C32—371no—
250C32—372no—
251C32—372no—
252C32—372no—
253C32—371no—
254C32—371no—
255C32—3336no—
256C32—3332no—
257C32—3332no—
258C32—3332no—
259C32—3331no—
260C32—3331no—
261C32—3339no—
262C32—3339no—
263C32—3339no—
264C32—1113no—
265C32—1116no—
266C32—1113no—
267C32—1116no—
268C32H(3,𝔽3)×C37⋊C312yes—
269C32H(3,𝔽3)×C37⋊C311yescharacteristic
270C32C33×C37⋊C311yescharacteristic
271C9—36no—
272C9—33no—
273C9—1116no—
274C9—11112no—
275C3—31no—
276C3—32no—
277C3—31no—
278C3—93no—
279C3—373no—
280C3—3336no—
281C3—1111no—
282C3—1111no—
283C3—1112no—
284C3—1112no—
285C3—1112no—
286C3C3×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 radicalC3×H(3,𝔽3)×C37⋊C3C3
hypercenter#6910C37⋊C3
hypocenterC37C3×#6910
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. C3×3−1+2×C37⋊C3C3semidirect
2. C3×3−1+2×C37⋊C3C3semidirect
4. #22582644C3semidirect
5. #22582643C3semidirect
6. #22582642C3semidirect
7. #22582641C3semidirect
8. #22582640C3semidirect
9. #731×C37⋊C3C3semidirect
10. C37×#6910C3semidirect
13. C3×#11758608C32semidirect
14. C3×#11758608C32semidirect
16. C3×#11758605C32semidirect
18. C3×#11758606C32semidirect
30. 3−1+2×C37⋊C3C32semidirect
31. 3−1+2×C37⋊C3C32semidirect
32. 3−1+2×C37⋊C3C32semidirect
42. C3×C9×C37⋊C3C32semidirect
43. #15688517C32semidirect
44. #15688516C32semidirect
45. #15688515C32semidirect
46. C3×3−1+2×C37C32semidirect
47. C3×3−1+2×C37C32semidirect
49. C37×#731C32semidirect
70. 3−1+2×C37C33semidirect
74. C3×C9×C37C33semidirect
84. #11758608C33semidirect
86. C3×#66524C33semidirect
140. #6910C37⋊C3direct (N × H)
153. C37⋊C3#6910direct (N × H)
176. C3×3−1+2C3×C37⋊C3semidirect
177. C3×3−1+2C3×C37⋊C3semidirect
192. #731C3×C37⋊C3semidirect
193. C37C3×#6910semidirect
229. 3−1+2C32×C37⋊C3semidirect
245. C3×C9C32×C37⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×3−1+2×C37⋊C33
C3×H(3,𝔽3)×C37⋊C33
#225826443
#225826433
#225826423
#225826413
#225826403
#731×C37⋊C33
C37×#69103
C3×#691037