Finite Group Databasebeta

← AGL(1,𝔽5)×#4183 #72247229

Subgroups — AGL(1,𝔽5)×#4183 #72247229

Order 3240.

320 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 38 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
1D5×#4183C211yesmaximal, characteristic
2#13108191C211yesmaximal, characteristic, complemented
3AGL(1,𝔽5)×AΣL(1,𝔽27)C211yesmaximal, characteristic, complemented
4C3×C3⋊D3×AGL(1,𝔽5)—31nomaximal
5AGL(1,𝔽5)×AGL(1,ℤ/9)—31nomaximal
6AGL(1,𝔽5)×AGL(1,ℤ/9)—31nomaximal
7AGL(1,𝔽5)×(C3⋊D3)⋊C3—31nomaximal
8C5⋊#4183C411yescharacteristic, complemented
9C5×#4183C411yescharacteristic, complemented
10D5×AΣL(1,𝔽27)C2211yescharacteristic
11C4×#4183—51nomaximal, Hall
12C3×D5×C3⋊D3—31no—
13C3×C3⋊ΓL(1,𝔽16)—31no—
14C33×AGL(1,𝔽5)D311yescharacteristic, complemented
15D5×AGL(1,ℤ/9)—31no—
16D5×AGL(1,ℤ/9)—31no—
17D5×(C3⋊D3)⋊C3—31no—
183−1+2⋊AGL(1,𝔽5)—31no—
193−1+2⋊AGL(1,𝔽5)—31no—
20(C3⋊ΓL(1,𝔽16))⋊C3—31no—
21AGL(1,𝔽5)×3−1+2D311yescharacteristic, complemented
22AGL(1,𝔽5)×3−1+2D311yescharacteristic, complemented
23AGL(1,𝔽5)×H(3,𝔽3)D311yescharacteristic, complemented
24C5×AΣL(1,𝔽27)C2×C411yescharacteristic, complemented, Hall
25C3⋊D3×AGL(1,𝔽5)—31no—
26C3⋊D3×AGL(1,𝔽5)—31no—
27C3×D3×AGL(1,𝔽5)—91no—
28C3×D3×AGL(1,𝔽5)—91no—
29C3×D3×AGL(1,𝔽5)—91no—
30C3×D3×AGL(1,𝔽5)—91no—
31D9×AGL(1,𝔽5)—31no—
32D9×AGL(1,𝔽5)—31no—
33C2×#4183—51no—
34C4×AΣL(1,𝔽27)—51no—
35#66330—51no—
36C3×C3⋊D15—31no—
37C3×C5×C3⋊D3—31no—
38C33×D5C2×D311yescharacteristic
39C5⋊AGL(1,ℤ/9)—31no—
40C5⋊AGL(1,ℤ/9)—31no—
41C5⋊((C3⋊D3)⋊C3)—31no—
42C5×AGL(1,ℤ/9)—31no—
43C5×AGL(1,ℤ/9)—31no—
44C5×(C3⋊D3)⋊C3—31no—
45D5×3−1+2C2×D311yescharacteristic
46D5×3−1+2C2×D311yescharacteristic
47D5×H(3,𝔽3)C2×D311yescharacteristic
48C3×C4×C3⋊D3—151no—
49C4×AGL(1,ℤ/9)—151no—
50C4×AGL(1,ℤ/9)—151no—
51C4×(C3⋊D3)⋊C3—151no—
52D5×C3⋊D3—31no—
53D5×C3⋊D3—31no—
54C3×D3×D5—91no—
55C3×D3×D5—91no—
56C3×D3×D5—91no—
57C3×D3×D5—91no—
58C3⋊ΓL(1,𝔽16)—31no—
59C3⋊ΓL(1,𝔽16)—31no—
60C3×ΓL(1,𝔽16)—91no—
61C3×ΓL(1,𝔽16)—91no—
62C3×ΓL(1,𝔽16)—91no—
63C3×ΓL(1,𝔽16)—91no—
64C32×AGL(1,𝔽5)—31no—
65C32×AGL(1,𝔽5)—31no—
66C32×AGL(1,𝔽5)—31no—
67C32×AGL(1,𝔽5)—61no—
68C32×AGL(1,𝔽5)—61no—
69C32×AGL(1,𝔽5)C3⋊D311yescharacteristic
70D5×D9—31no—
71D5×D9—31no—
72C9⋊AGL(1,𝔽5)—31no—
73C9⋊AGL(1,𝔽5)—31no—
74C9×AGL(1,𝔽5)—31no—
75C9×AGL(1,𝔽5)—61no—
76C2×AΣL(1,𝔽27)—51no—
77#4183—51no—
78#4183AGL(1,𝔽5)11yescharacteristic, direct product factor
79C33×C5C4×D311yescharacteristic, complemented
80C5×3−1+2C4×D311yescharacteristic, complemented
81C5×3−1+2C4×D311yescharacteristic, complemented
82C5×H(3,𝔽3)C4×D311yescharacteristic, complemented
83C2×C3×AGL(1,𝔽5)—271no—
84D3×AGL(1,𝔽5)—91no—
85D3×AGL(1,𝔽5)—91no—
86D3×AGL(1,𝔽5)—91no—
87D3×AGL(1,𝔽5)—91no—
88D3×AGL(1,𝔽5)—271no—
89C2×C3×C3⋊D3—151no—
90C33×C4—51no—
91C3×C3⋊Dic3—151no—
92C2×AGL(1,ℤ/9)—151no—
93C2×AGL(1,ℤ/9)—151no—
94C2×(C3⋊D3)⋊C3—151no—
95C4×3−1+2—51no—
96C4×3−1+2—51no—
97C4×H(3,𝔽3)—51no—
983−1+2⋊C4—151no—
993−1+2⋊C4—151no—
100(C3⋊Dic3)⋊C3—151no—
101C3⋊D15—31no—
102C3⋊D15—31no—
103C5×C3⋊D3—31no—
104C5×C3⋊D3—31no—
105C3×D15—91no—
106C3×D15—91no—
107C3×D15—91no—
108C3×D15—91no—
109C3×C5×D3—91no—
110C3×C5×D3—91no—
111C3×C5×D3—91no—
112C3×C5×D3—91no—
113C32×D5—31no—
114C32×D5—31no—
115C32×D5—31no—
116C32×D5—61no—
117C32×D5—61no—
118C32×D5C2×C3⋊D311yescharacteristic
119D45—31no—
120D45—31no—
121C9×D5—31no—
122C9×D5—61no—
123C5×D9—31no—
124C5×D9—31no—
125AΣL(1,𝔽27)C2×AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
126C4×C3⋊D3—151no—
127C4×C3⋊D3—151no—
128C3×C4×D3—451no—
129C3×C4×D3—451no—
130C3×C4×D3—451no—
131C3×C4×D3—451no—
132C4×D9—151no—
133C4×D9—151no—
134C2×C3×D5—271no—
135D3×D5—91no—
136D3×D5—91no—
137D3×D5—91no—
138D3×D5—91no—
139D3×D5—271no—
140ΓL(1,𝔽16)—91no—
141ΓL(1,𝔽16)—91no—
142ΓL(1,𝔽16)—91no—
143ΓL(1,𝔽16)—91no—
144ΓL(1,𝔽16)—271no—
145C3×AGL(1,𝔽5)—31no—
146C3×AGL(1,𝔽5)—31no—
147C3×AGL(1,𝔽5)—61no—
148C3×AGL(1,𝔽5)—61no—
149C3×AGL(1,𝔽5)—181no—
150C3×AGL(1,𝔽5)—271no—
151C3×AGL(1,𝔽5)#27211yescharacteristic
152C2×C33—51no—
153C3×C3⋊D3—31no—
154C3×C3⋊D3—151no—
155C2×3−1+2—51no—
156C2×3−1+2—51no—
157C2×H(3,𝔽3)—51no—
158AGL(1,ℤ/9)—31no—
159AGL(1,ℤ/9)—31no—
160AGL(1,ℤ/9)—151no—
161AGL(1,ℤ/9)—151no—
162(C3⋊D3)⋊C3—31no—
163(C3⋊D3)⋊C3—151no—
164C32×C5—31no—
165C32×C5—31no—
166C32×C5—31no—
167C32×C5—61no—
168C32×C5—61no—
169C32×C5C4×C3⋊D311yescharacteristic
170C5×C9—31no—
171C5×C9—61no—
172C2×AGL(1,𝔽5)—271noHall
173C2×C3⋊D3—151no—
174C2×C3⋊D3—151no—
175C2×C3×D3—451no—
176C2×C3×D3—451no—
177C2×C3×D3—451no—
178C2×C3×D3—451no—
179C32×C4—51no—
180C32×C4—151no—
181C32×C4—301no—
182C32×C4—301no—
183C32×C4—301no—
184C32×C4—301no—
185C3⋊Dic3—151no—
186C3⋊Dic3—151no—
187C3×Dic3—451no—
188C3×Dic3—451no—
189C3×Dic3—451no—
190C3×Dic3—451no—
191C2×D9—151no—
192C2×D9—151no—
193C4×C9—301no—
194C4×C9—301no—
195Dic9—151no—
196Dic9—151no—
197C2×C3×C5—271no—
198D15—91no—
199D15—91no—
200D15—91no—
201D15—91no—
202D15—271no—
203C3×D5—31no—
204C3×D5—31no—
205C3×D5—61no—
206C3×D5—61no—
207C3×D5—181no—
208C3×D5—271no—
209C3×D5C2×#27211yescharacteristic
210C5×D3—91no—
211C5×D3—91no—
212C5×D3—91no—
213C5×D3—91no—
214C5×D3—271no—
215C33D3×AGL(1,𝔽5)11yescharacteristic, complemented
2163−1+2D3×AGL(1,𝔽5)11yescharacteristic, complemented
2173−1+2D3×AGL(1,𝔽5)11yescharacteristic, complemented
218H(3,𝔽3)D3×AGL(1,𝔽5)11yescharacteristic, complemented
219C2×C3×C4—1351no—
220C4×D3—451no—
221C4×D3—451no—
222C4×D3—451no—
223C4×D3—451no—
224C4×D3—1351no—
225C2×D5—271no—
226AGL(1,𝔽5)—271no—
227AGL(1,𝔽5)#418311yescharacteristic, direct product factor
228C2×C32—51no—
229C2×C32—151no—
230C2×C32—301no—
231C2×C32—301no—
232C2×C32—301no—
233C2×C32—301no—
234C3⋊D3—31no—
235C3⋊D3—31no—
236C3⋊D3—151no—
237C3⋊D3—151no—
238C3×D3—91no—
239C3×D3—91no—
240C3×D3—91no—
241C3×D3—91no—
242C3×D3—451no—
243C3×D3—451no—
244C3×D3—451no—
245C3×D3—451no—
246C2×C9—301no—
247C2×C9—301no—
248D9—31no—
249D9—31no—
250D9—151no—
251D9—151no—
252C3×C5—31no—
253C3×C5—31no—
254C3×C5—61no—
255C3×C5—61no—
256C3×C5—181no—
257C3×C5C4×#27211yescharacteristic
258C22×C3—1351no—
259C2×D3—451no—
260C2×D3—451no—
261C2×D3—451no—
262C2×D3—451no—
263C2×D3—1351no—
264C3×C4—51no—
265C3×C4—151no—
266C3×C4—301no—
267C3×C4—301no—
268C3×C4—301no—
269C3×C4—901no—
270C3×C4—1351no—
271Dic3—451no—
272Dic3—451no—
273Dic3—451no—
274Dic3—451no—
275Dic3—1351no—
276C2×C5—271no—
277D5—271no—
278D5C2×#418311yescharacteristic
279C32—31no—
280C32—61no—
281C32—61no—
282C32—61no—
283C32—61no—
284C32C3⋊D3×AGL(1,𝔽5)11yescharacteristic
285C9—61no—
286C9—61no—
287C2×C4—1351noSylow
288C2×C3—51no—
289C2×C3—151no—
290C2×C3—271no—
291C2×C3—301no—
292C2×C3—301no—
293C2×C3—301no—
294C2×C3—901no—
295C2×C3—1351no—
296D3—91no—
297D3—91no—
298D3—91no—
299D3—91no—
300D3—271no—
301D3—451no—
302D3—451no—
303D3—451no—
304D3—451no—
305D3—1351no—
306C5C4×#418311yescharacteristic, complemented, Sylow
307C22—1351no—
308C4—51no—
309C4—1351no—
310C3—31no—
311C3—61no—
312C3—61no—
313C3—61no—
314C3—181no—
315C3AGL(1,𝔽5)×#27211yescharacteristic
316C2—51no—
317C2—271no—
318C2—1351no—
subgroupG/H
center, hypercenterC1G
commutator, Fitting, hypocenterC5×AΣL(1,𝔽27)C2×C4
FrattiniC32C3⋊D3×AGL(1,𝔽5)
socleC3×C5C4×#272
prime radicalD5×#4183C2
cosocleD5×AΣL(1,𝔽27)C22
LCSG ▷ C5×AΣL(1,𝔽27)
UCSC1

Complements

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

NH ≅ G/Ntype
2. #13108191C2semidirect
3. AGL(1,𝔽5)×AΣL(1,𝔽27)C2semidirect
8. C5⋊#4183C4semidirect
9. C5×#4183C4semidirect
14. C33×AGL(1,𝔽5)D3semidirect
21. AGL(1,𝔽5)×3−1+2D3semidirect
22. AGL(1,𝔽5)×3−1+2D3semidirect
23. AGL(1,𝔽5)×H(3,𝔽3)D3semidirect
24. C5×AΣL(1,𝔽27)C2×C4semidirect
78. #4183AGL(1,𝔽5)direct (N × H)
79. C33×C5C4×D3semidirect
80. C5×3−1+2C4×D3semidirect
81. C5×3−1+2C4×D3semidirect
82. C5×H(3,𝔽3)C4×D3semidirect
125. AΣL(1,𝔽27)C2×AGL(1,𝔽5)semidirect
215. C33D3×AGL(1,𝔽5)semidirect
216. 3−1+2D3×AGL(1,𝔽5)semidirect
217. 3−1+2D3×AGL(1,𝔽5)semidirect
218. H(3,𝔽3)D3×AGL(1,𝔽5)semidirect
227. AGL(1,𝔽5)#4183direct (N × H)
306. C5C4×#4183semidirect

Maximal subgroups

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

subgroupindex
D5×#41832
#131081912
AGL(1,𝔽5)×AΣL(1,𝔽27)2
C3×C3⋊D3×AGL(1,𝔽5)3
AGL(1,𝔽5)×AGL(1,ℤ/9)3
AGL(1,𝔽5)×(C3⋊D3)⋊C33
C4×#41835