Finite Group Databasebeta

← A42×#1043 #72229853

Subgroups — A42×#1043 #72229853

Order 14400.

1624 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 1624 conjugacy classes of subgroups — too many to draw. Only its 110 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×A4×#1043—42nomaximal
2C22×A4×#1043C312yesmaximal, complemented
3C2×A4×#1043—32no—
4C2×A4×#1043—32no—
5A42×C5⋊D5C211yesmaximal, characteristic
6C22×A4×C5⋊D5C2×C312yes—
7C52×A42C411yescharacteristic, complemented
8A42×AGL(1,𝔽5)—56nomaximal
9C22⋊A4×#1043C312yesmaximal, complemented
10C22⋊A4×C5⋊D5C2×C312yes—
11C3×A4×C5⋊D5—42no—
12C24×#1043C3211yescharacteristic, complemented, Hall
13D5×A42—56no—
14C52×C22⋊A4C3×C412yescomplemented
15C22×C52×A4C3×C412yescomplemented
16C22×C3×#1043—42no—
17C2×A4×C5⋊D5—32no—
18C2×A4×C5⋊D5—32no—
19A4×#1043—32no—
20A4×#1043—44no—
21A4×#1043—122no—
22A4×#1043A412yesdirect product factor
23AGL(1,𝔽5)×C22⋊A4—512no—
24C22×A4×AGL(1,𝔽5)—512no—
25C3×C52×A4—42no—
26C32×#1043—161no—
27C24×C5⋊D5C2×C3211yescharacteristic
28C23×#1043—32no—
29C23×#1043—32no—
30C23×#1043—91no—
31C23×#1043—91no—
32C5×A42AGL(1,𝔽5)16yescomplemented
33C3×A4×AGL(1,𝔽5)—2012no—
34C22×C3×C5⋊D5—42no—
35C2×C52×A4—32no—
36C2×C3×#1043—122no—
37C2×C3×#1043—122no—
38A4×C5⋊D5—32no—
39A4×C5⋊D5—44no—
40A4×C5⋊D5—122no—
41A4×C5⋊D5C2×A412yes—
42C4×A42—251noHall
43D5×C22⋊A4—512no—
44C22×D5×A4—512no—
45C2×A4×AGL(1,𝔽5)—1512no—
46C2×A4×AGL(1,𝔽5)—1512no—
47C32×C5⋊D5—161no—
48C24×C52C32×C411yescharacteristic, complemented
49C23×C5⋊D5—32no—
50C23×C5⋊D5—32no—
51C23×C5⋊D5—91no—
52C23×C5⋊D5—91no—
53C22×#1043—32no—
54C22×#1043—32no—
55C22×#1043—91no—
56C22×#1043—92no—
57C22×#1043—92no—
58C22×#1043—92no—
59C22×#1043—92no—
60C22×#1043—94no—
61C22×#1043—91no—
62C22×#1043C3×A412yescomplemented
63C3×D5×A4—2012no—
64C24×AGL(1,𝔽5)—56no—
65C22×C3×C52—42no—
66C2×C3×C5⋊D5—122no—
67C2×C3×C5⋊D5—122no—
68C52×A4—44no—
69C52×A4—122no—
70C52×A4C4×A412yescomplemented
71C3×#1043—42no—
72C3×#1043—122no—
73C3×#1043—162no—
74C2×A42—251no—
75C5×C22⋊A4C3×AGL(1,𝔽5)112yescomplemented
76C22×C5×A4C3×AGL(1,𝔽5)112yescomplemented
77C22×C3×AGL(1,𝔽5)—2012no—
78C2×D5×A4—1512no—
79C2×D5×A4—1512no—
80A4×AGL(1,𝔽5)—512no—
81A4×AGL(1,𝔽5)—1512no—
82A4×AGL(1,𝔽5)—2024no—
83A4×AGL(1,𝔽5)—6012no—
84C32×C52—161noHall
85C23×C52—32no—
86C23×C52—91no—
87C22×C5⋊D5—32no—
88C22×C5⋊D5—32no—
89C22×C5⋊D5—91no—
90C22×C5⋊D5—92no—
91C22×C5⋊D5—92no—
92C22×C5⋊D5—92no—
93C22×C5⋊D5—92no—
94C22×C5⋊D5—94no—
95C22×C5⋊D5—91no—
96C22×C5⋊D5C2×C3×A412yes—
97C2×#1043—32no—
98C2×#1043—32no—
99C2×#1043—91no—
100C2×#1043—92no—
101C2×#1043—94no—
102C2×#1043—91no—
103C2×#1043—92no—
104C2×#1043—92no—
105C4×C22⋊A4—252no—
106C22×C4×A4—252no—
107C3×C5×A4—412no—
108C32×AGL(1,𝔽5)—806no—
109C24×D5—56no—
110C23×AGL(1,𝔽5)—1512no—
111C23×AGL(1,𝔽5)—1512no—
112C23×AGL(1,𝔽5)—456no—
113C23×AGL(1,𝔽5)—456no—
114C2×C3×C52—122no—
115C3×C5⋊D5—42no—
116C3×C5⋊D5—122no—
117C3×C5⋊D5—162no—
118A42#104311yescharacteristic, direct product factor
119C3×C4×A4—1002no—
120C22×C3×D5—2012no—
121C2×C5×A4—312no—
122C2×C3×AGL(1,𝔽5)—6012no—
123C2×C3×AGL(1,𝔽5)—6012no—
124D5×A4—512no—
125D5×A4—1512no—
126D5×A4—2024no—
127D5×A4—6012no—
128C22×C52—32no—
129C22×C52—91no—
130C22×C52—92no—
131C22×C52C3×C4×A412yescomplemented
132C2×C5⋊D5—32no—
133C2×C5⋊D5—32no—
134C2×C5⋊D5—91no—
135C2×C5⋊D5—92no—
136C2×C5⋊D5—94no—
137C2×C5⋊D5—91no—
138C2×C5⋊D5—92no—
139C2×C5⋊D5—92no—
140#1043—32no—
141#1043—91no—
142#1043A4211yescharacteristic, direct product factor
143C2×C22⋊A4—252no—
144C23×A4—252no—
145C2×C4×A4—752no—
146C2×C4×A4—752no—
147C32×D5—806no—
148C24×C5C32×AGL(1,𝔽5)16yescomplemented
149C23×D5—1512no—
150C23×D5—1512no—
151C23×D5—456no—
152C23×D5—456no—
153C22×AGL(1,𝔽5)—512no—
154C22×AGL(1,𝔽5)—1512no—
155C22×AGL(1,𝔽5)—1512no—
156C22×AGL(1,𝔽5)—456no—
157C22×AGL(1,𝔽5)—4512no—
158C22×AGL(1,𝔽5)—4512no—
159C22×AGL(1,𝔽5)—4512no—
160C22×AGL(1,𝔽5)—4512no—
161C22×AGL(1,𝔽5)—4524no—
162C22×AGL(1,𝔽5)—456no—
163C3×C52—42no—
164C3×C52—162no—
165C2×C3×A4—1002no—
166C24×C4—251noSylow
167C22×C3×C5—412no—
168C2×C3×D5—6012no—
169C2×C3×D5—6012no—
170C5×A4—424no—
171C5×A4—1212no—
172C5×A4A4×AGL(1,𝔽5)112yescomplemented
173C3×AGL(1,𝔽5)—2012no—
174C3×AGL(1,𝔽5)—6012no—
175C3×AGL(1,𝔽5)—8012no—
176C2×C52—32no—
177C2×C52—91no—
178C5⋊D5—32no—
179C5⋊D5—91no—
180C5⋊D5C2×A4211yescharacteristic
181C22⋊A4C3×#104312yescomplemented
182C22×A4—752no—
183C22×A4—752no—
184C22×A4C3×#104312yescomplemented
185C22×C3×C4—1002no—
186C4×A4—252no—
187C4×A4—3002no—
188C4×A4—752no—
189C4×A4—1004no—
190C32×C5—166no—
191C23×C5—312no—
192C23×C5—96no—
193C22×D5—512no—
194C22×D5—1512no—
195C22×D5—1512no—
196C22×D5—456no—
197C22×D5—4512no—
198C22×D5—4512no—
199C22×D5—4512no—
200C22×D5—4512no—
201C22×D5—4524no—
202C22×D5—456no—
203C2×AGL(1,𝔽5)—1512no—
204C2×AGL(1,𝔽5)—1512no—
205C2×AGL(1,𝔽5)—456no—
206C2×AGL(1,𝔽5)—456no—
207C2×AGL(1,𝔽5)—4512no—
208C2×AGL(1,𝔽5)—4524no—
209C2×AGL(1,𝔽5)—4512no—
210C2×AGL(1,𝔽5)—4512no—
211C3×A4—42no—
212C32×C4—4001no—
213C25—251no—
214C23×C4—752no—
215C23×C4—752no—
216C23×C4—2251no—
217C23×C4—2251no—
218C2×C3×C5—1212no—
219C3×D5—2012no—
220C3×D5—6012no—
221C3×D5—8012no—
222C52C4×A4211yescharacteristic, complemented, Sylow
223C23×C3—1002no—
224C2×A4—32no—
225C2×A4—252no—
226C2×A4—3002no—
227C2×A4—752no—
228C2×A4—1004no—
229C2×C3×C4—3002no—
230C2×C3×C4—3002no—
231C22×C5—312no—
232C22×C5—96no—
233C22×C5—912no—
234C22×C5C3×A4×AGL(1,𝔽5)112yescomplemented
235C2×D5—1512no—
236C2×D5—1512no—
237C2×D5—456no—
238C2×D5—456no—
239C2×D5—4512no—
240C2×D5—4524no—
241C2×D5—4512no—
242C2×D5—4512no—
243AGL(1,𝔽5)—56no—
244AGL(1,𝔽5)—1512no—
245AGL(1,𝔽5)—456no—
246C2×C32—4001no—
247C24—752no—
248C24—752no—
249C24—2251no—
250C24—2251no—
251C24C32×#104311yescharacteristic, complemented
252C22×C4—252no—
253C22×C4—752no—
254C22×C4—752no—
255C22×C4—2251no—
256C22×C4—2252no—
257C22×C4—2252no—
258C22×C4—2252no—
259C22×C4—2252no—
260C22×C4—2254no—
261C22×C4—2251no—
262C3×C5—412no—
263C3×C5—1612no—
264C22×C3—42no—
265C22×C3—3002no—
266C22×C3—3002no—
267A4—44no—
268A4—122no—
269A4A4×#104312yesdirect product factor
270C3×C4—3002no—
271C3×C4—1002no—
272C3×C4—4002no—
273C2×C5—312no—
274C2×C5—96no—
275D5—56no—
276D5—1512no—
277D5—456no—
278C32—161noSylow
279C23—32no—
280C23—91no—
281C23—252no—
282C23—752no—
283C23—752no—
284C23—2252no—
285C23—2251no—
286C23—2252no—
287C23—2252no—
288C23—2252no—
289C23—2251no—
290C23—2254no—
291C2×C4—752no—
292C2×C4—752no—
293C2×C4—2251no—
294C2×C4—2251no—
295C2×C4—2252no—
296C2×C4—2254no—
297C2×C4—2252no—
298C2×C4—2252no—
299C2×C3—122no—
300C2×C3—3002no—
301C2×C3—1002no—
302C2×C3—4002no—
303C5A42×AGL(1,𝔽5)16yescomplemented
304C22—32no—
305C22—91no—
306C22—92no—
307C22—752no—
308C22—752no—
309C22—2252no—
310C22—2252no—
311C22—2251no—
312C22—2251no—
313C22—2254no—
314C22—2252no—
315C22C3×A4×#104312yescomplemented
316C4—251no—
317C4—752no—
318C4—2251no—
319C3—42no—
320C3—162no—
321C2—32no—
322C2—91no—
323C2—251no—
324C2—752no—
325C2—2251no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC24×C52C32×C4
prime radicalA42×C5⋊D5C2
cosocleC24×C5⋊D5C2×C32
LCSG ▷ C24×C52
UCSC1

Complements

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

NH ≅ G/Ntype
2. C22×A4×#1043C3semidirect
7. C52×A42C4semidirect
9. C22⋊A4×#1043C3semidirect
12. C24×#1043C32semidirect
14. C52×C22⋊A4C3×C4semidirect
15. C22×C52×A4C3×C4semidirect
22. A4×#1043A4direct (N × H)
32. C5×A42AGL(1,𝔽5)semidirect
48. C24×C52C32×C4semidirect
62. C22×#1043C3×A4semidirect
70. C52×A4C4×A4semidirect
75. C5×C22⋊A4C3×AGL(1,𝔽5)semidirect
76. C22×C5×A4C3×AGL(1,𝔽5)semidirect
118. A42#1043direct (N × H)
131. C22×C52C3×C4×A4semidirect
142. #1043A42direct (N × H)
148. C24×C5C32×AGL(1,𝔽5)semidirect
172. C5×A4A4×AGL(1,𝔽5)semidirect
181. C22⋊A4C3×#1043semidirect
184. C22×A4C3×#1043semidirect
222. C52C4×A42semidirect
234. C22×C5C3×A4×AGL(1,𝔽5)semidirect
251. C24C32×#1043semidirect
269. A4A4×#1043direct (N × H)
303. C5A42×AGL(1,𝔽5)semidirect
315. C22C3×A4×#1043semidirect

Maximal subgroups

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

subgroupindex
C3×A4×#10434
C22×A4×#10433
A42×C5⋊D52
A42×AGL(1,𝔽5)5
C22⋊A4×#10433