Finite Group Databasebeta

← A4×#4003 #14512559

Subgroups — A4×#4003 #14512559

Order 1920.

524 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C2×C4×D5×A4C211yesmaximal, characteristic
2C2×A4×#170C212yesmaximal, complemented
3C22×#4003C311yesmaximal, characteristic, complemented, Hall
4C22×D5×A4C411yescharacteristic
5C2×C4×C5×A4C411yescharacteristic
6C2×A4×Dic5C2211yescharacteristic
7C4×D5×A4—22no—
8A4×#170—22no—
9C3×#4003—41nomaximal
10A4×C22⋊C8—51nomaximal, Hall
11C23×C4×D5C2×C311yescharacteristic
12C23×#170C2×C312yescomplemented
13C2×#4003—32no—
14C2×#4003—31no—
15C2×#4003—31no—
16C22×C5×A4C2×C411yescharacteristic
17C2×D5×A4—21no—
18C2×D5×A4—21no—
19C2×D5×A4C812yescomplemented
20C2×C3×C4×D5—41no—
21C4×C5×A4—21no—
22C2×C3×#170—42no—
23A4×Dic5D412yes—
24C22×C4×A4—51no—
25C2×C8×A4—52no—
26C24×D5C3×C411yescharacteristic
27C23×C4×C5C3×C411yescharacteristic
28C23×Dic5C22×C311yescharacteristic
29C22×C4×D5—22no—
30C22×C4×D5—31no—
31C22×C4×D5—31no—
32C22×C4×D5—31no—
33C22×C4×D5—31no—
34C22×C4×D5—62no—
35C22×#170—22no—
36C22×#170—32no—
37C22×#170—32no—
38C22×#170—62no—
39#4003—32no—
40#4003—31no—
41#4003—32no—
42#4003A411yescharacteristic, direct product factor
43C22×C22⋊C8—51noSylow
44C22×C3×D5—41no—
45C2×C5×A4C22⋊C411yescharacteristic
46C2×C5×A4C2×C811yescharacteristic
47C2×C5×A4M1611yescharacteristic
48D5×A4—22no—
49C2×C3×C4×C5—41no—
50C2×C3×Dic5—41no—
51C3×C4×D5—82no—
52C3×#170—82no—
53C23×A4—51no—
54C2×C4×A4—51no—
55C2×C4×A4—102no—
56C2×C4×A4AGL(1,𝔽5)11yescharacteristic
57C8×A4—102no—
58C3×C22⋊C8—201no—
59C24×C5C2×C3×C411yescharacteristic
60C23×D5—21no—
61C23×D5—21no—
62C23×D5—31no—
63C23×D5—31no—
64C23×D5—32no—
65C23×D5—61no—
66C23×D5—61no—
67C23×D5C3×C812yescomplemented
68C22×C4×C5—21no—
69C22×C4×C5—31no—
70C22×C4×C5—31no—
71C22×C4×C5—61no—
72C22×Dic5—31no—
73C22×Dic5—32no—
74C22×Dic5—31no—
75C22×Dic5C3×D412yes—
76C2×C4×D5—32no—
77C2×C4×D5—31no—
78C2×C4×D5—31no—
79C2×C4×D5—31no—
80C2×C4×D5—62no—
81C2×C4×D5—62no—
82C2×C4×D5—62no—
83C2×C4×D5—62no—
84C2×C4×D5—62no—
85C2×C4×D5—62no—
86C2×C4×D5—62no—
87C2×C4×D5—62no—
88C2×C4×D5C2×A411yescharacteristic
89C2×#170—32no—
90C2×#170—62no—
91C2×#170—62no—
92C2×#170—62no—
93C2×#170—62no—
94C2×#170C2×A412yescomplemented
95C24×C4—51no—
96C23×C8—52no—
97C2×C22⋊C8—151no—
98C2×C22⋊C8—152no—
99C2×C22⋊C8—151no—
100C22×C3×C5—41no—
101C2×C3×D5—42no—
102C2×C3×D5—81no—
103C2×C3×D5—81no—
104C5×A4C22⋊C811yescharacteristic, complemented
105C3×C4×C5—81no—
106C3×Dic5—42no—
107C22×A4—52no—
108C22×A4—101no—
109C22×A4—101no—
110C22×A4C2×AGL(1,𝔽5)11yescharacteristic
111C22×C3×C4—201no—
112C4×A4—21no—
113C4×A4—52no—
114C2×C3×C8—202no—
115C23×C5—31no—
116C23×C5—31no—
117C23×C5—31no—
118C23×C5—31no—
119C23×C5C3×C22⋊C411yescharacteristic
120C23×C5C2×C3×C811yescharacteristic
121C23×C5C3×M1611yescharacteristic
122C22×D5—22no—
123C22×D5—32no—
124C22×D5—31no—
125C22×D5—32no—
126C22×D5—32no—
127C22×D5—32no—
128C22×D5—61no—
129C22×D5—61no—
130C22×D5—61no—
131C22×D5—61no—
132C22×D5—64no—
133C22×D5—61no—
134C22×D5—61no—
135C22×D5—62no—
136C22×D5—62no—
137C22×D5—61no—
138C22×D5—61no—
139C22×D5—62no—
140C22×D5C4×A411yescharacteristic
141C2×C4×C5—31no—
142C2×C4×C5—61no—
143C2×C4×C5—61no—
144C2×C4×C5—61no—
145C2×C4×C5—61no—
146C2×C4×C5C4×A411yescharacteristic
147C2×Dic5—32no—
148C2×Dic5—31no—
149C2×Dic5—32no—
150C2×Dic5—32no—
151C2×Dic5—32no—
152C2×Dic5C22×A411yescharacteristic
153C4×D5—22no—
154C4×D5—62no—
155C4×D5—62no—
156C4×D5—64no—
157C4×D5—62no—
158#170—22no—
159#170—62no—
160C25—51no—
161C23×C4—51no—
162C23×C4—102no—
163C23×C4—151no—
164C23×C4—151no—
165C23×C4—151no—
166C23×C4—151no—
167C23×C4—302no—
168C23×C4C3×AGL(1,𝔽5)11yescharacteristic
169C22×C8—102no—
170C22×C8—152no—
171C22×C8—152no—
172C22×C8—302no—
173C22⋊C8—51no—
174C22⋊C8—152no—
175C22⋊C8—151no—
176C22⋊C8—152no—
177C2×C3×C5—41no—
178C2×C3×C5—41no—
179C2×C3×C5—41no—
180C3×D5—82no—
181C23×C3—201no—
182C2×A4—102no—
183C2×A4D5⋊C811yescharacteristic
184C2×A4#70011yescharacteristic
185C2×A4C22⋊AGL(1,𝔽5)11yescharacteristic
186C2×C3×C4—41no—
187C2×C3×C4—201no—
188C2×C3×C4—402no—
189C3×C8—402no—
190C22×C5—31no—
191C22×C5—31no—
192C22×C5—31no—
193C22×C5—32no—
194C22×C5—31no—
195C22×C5—31no—
196C22×C5—31no—
197C22×C5—31no—
198C22×C5—31no—
199C22×C5—31no—
200C22×C5C3×C22⋊C811yescharacteristic, complemented
201C22×C5C2×C4×A411yescharacteristic
202C2×D5—21no—
203C2×D5—21no—
204C2×D5—32no—
205C2×D5—62no—
206C2×D5—62no—
207C2×D5—62no—
208C2×D5—62no—
209C2×D5—62no—
210C2×D5—62no—
211C2×D5—61no—
212C2×D5—61no—
213C2×D5—62no—
214C2×D5—62no—
215C2×D5C8×A412yescomplemented
216C4×C5—21no—
217C4×C5—61no—
218Dic5—32no—
219Dic5D4×A412yes—
220C24—52no—
221C24—101no—
222C24—101no—
223C24—151no—
224C24—151no—
225C24—152no—
226C24—301no—
227C24—301no—
228C24C2×C3×AGL(1,𝔽5)11yescharacteristic
229C22×C4—21no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—51no—
233C22×C4—52no—
234C22×C4—61no—
235C22×C4—151no—
236C22×C4—151no—
237C22×C4—152no—
238C22×C4—151no—
239C22×C4—152no—
240C22×C4—151no—
241C22×C4—151no—
242C22×C4—302no—
243C22×C4—302no—
244C22×C4—302no—
245C22×C4—302no—
246C22×C4—302no—
247C22×C4—302no—
248C22×C4—302no—
249C22×C4—302no—
250C2×C8—52no—
251C2×C8—152no—
252C2×C8—302no—
253C2×C8—302no—
254C2×C8—302no—
255C2×C8—302no—
256C3×C5—41noHall
257C22×C3—41no—
258C22×C3—202no—
259C22×C3—401no—
260C22×C3—401no—
261A4#400311yescharacteristic, direct product factor
262C3×C4—81no—
263C3×C4—202no—
264C2×C5—31no—
265C2×C5—31no—
266C2×C5—31no—
267C2×C5—31no—
268C2×C5A4×C22⋊C411yescharacteristic
269C2×C5C2×C8×A411yescharacteristic
270C2×C5A4×M1611yescharacteristic
271D5—22no—
272D5—62no—
273C23—31no—
274C23—31no—
275C23—31no—
276C23—31no—
277C23—51no—
278C23—102no—
279C23—152no—
280C23—152no—
281C23—152no—
282C23—152no—
283C23—151no—
284C23—301no—
285C23—301no—
286C23—301no—
287C23—301no—
288C23—302no—
289C23—301no—
290C23—301no—
291C23—302no—
292C23—302no—
293C23—301no—
294C23—301no—
295C23—304no—
296C23C3×D5⋊C811yescharacteristic
297C23C3×#70011yescharacteristic
298C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
299C2×C4—31no—
300C2×C4—51no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—102no—
306C2×C4—152no—
307C2×C4—152no—
308C2×C4—152no—
309C2×C4—151no—
310C2×C4—152no—
311C2×C4—302no—
312C2×C4—302no—
313C2×C4—302no—
314C2×C4—304no—
315C2×C4A4×AGL(1,𝔽5)11yescharacteristic
316C8—102no—
317C8—302no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—402no—
322C5A4×C22⋊C811yescharacteristic, complemented, Sylow
323C22—31no—
324C22—31no—
325C22—31no—
326C22—32no—
327C22—31no—
328C22—31no—
329C22—31no—
330C22—31no—
331C22—31no—
332C22—31no—
333C22—52no—
334C22—101no—
335C22—101no—
336C22—152no—
337C22—302no—
338C22—302no—
339C22—302no—
340C22—302no—
341C22—302no—
342C22—302no—
343C22—302no—
344C22—302no—
345C22—301no—
346C22—301no—
347C22C3×#400311yescharacteristic, complemented
348C22C2×A4×AGL(1,𝔽5)11yescharacteristic
349C4—21no—
350C4—52no—
351C4—61no—
352C4—152no—
353C3—41noSylow
354C2—31no—
355C2—31no—
356C2—31no—
357C2—31no—
358C2—102no—
359C2—302no—
360C2A4×D5⋊C811yescharacteristic
361C2A4×#70011yescharacteristic
362C2A4×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×AGL(1,𝔽5)
commutatorC23×C5C2×C3×C8
FittingC23×C4×C5C3×C4
socleC24×C5C2×C3×C4
prime radicalC22×D5×A4C4
cosocleC23×Dic5C22×C3
hypercenterC2×C4A4×AGL(1,𝔽5)
hypocenterC22×C5C3×C22⋊C8
LCSG ▷ C23×C5 ▷ C22×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×A4×#170C2semidirect
3. C22×#4003C3semidirect
12. C23×#170C2×C3semidirect
19. C2×D5×A4C8semidirect
42. #4003A4direct (N × H)
67. C23×D5C3×C8semidirect
94. C2×#170C2×A4semidirect
104. C5×A4C22⋊C8semidirect
200. C22×C5C3×C22⋊C8semidirect
215. C2×D5C8×A4semidirect
261. A4#4003direct (N × H)
322. C5A4×C22⋊C8semidirect
347. C22C3×#4003semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D5×A42
C2×A4×#1702
C22×#40033
C3×#40034
A4×C22⋊C85