Finite Group Databasebeta

← A4×#3999 #14512556

Subgroups — A4×#3999 #14512556

Order 1920.

544 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 48 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×A4×D20C211yesmaximal, characteristic
2A4×C22⋊AGL(1,𝔽5)C212yesmaximal, complemented
3C22×#3999C311yesmaximal, characteristic, complemented, Hall
4C22×D5×A4C411yescharacteristic, complemented
5C22×D5×A4C2211yescharacteristic
6C2×A4×AGL(1,𝔽5)—22no—
7C2×C4×C5×A4C411yescharacteristic, complemented
8A4×D20—22no—
9C3×#3999—41nomaximal
10A4×#99—51nomaximal, Hall
11C23×D20C2×C311yescharacteristic
12C22×C22⋊AGL(1,𝔽5)C2×C312yescomplemented
13C2×#3999—32no—
14C2×#3999—31no—
15C2×#3999—31no—
16C22×C5×A4C2×C411yescharacteristic
17C2×D5×A4—21no—
18C2×D5×A4—41no—
19C2×D5×A4—41no—
20C2×D5×A4D412yes—
21A4×AGL(1,𝔽5)—42no—
22C2×C3×D20—41no—
23C4×C5×A4—21no—
24C3×C22⋊AGL(1,𝔽5)—42no—
25C2×D4×A4—51no—
26A4×C22⋊C4—52no—
27C24×D5C3×C411yescharacteristic, complemented
28C24×D5C22×C311yescharacteristic
29C23×AGL(1,𝔽5)—22no—
30C23×C4×C5C3×C411yescharacteristic, complemented
31C22×D20—22no—
32C22×D20—31no—
33C22×D20—31no—
34C22×D20—31no—
35C22×D20—31no—
36C22×D20—62no—
37C2×C22⋊AGL(1,𝔽5)—32no—
38C2×C22⋊AGL(1,𝔽5)—32no—
39C2×C22⋊AGL(1,𝔽5)—62no—
40#3999—32no—
41#3999—31no—
42#3999—32no—
43#3999A411yescharacteristic, direct product factor
44C22×#99—51noSylow
45C22×C3×D5—41no—
46C22×C3×D5—41no—
47C2×C5×A4—21no—
48C2×C5×A4C22⋊C411yescharacteristic
49C2×C3×AGL(1,𝔽5)—82no—
50D5×A4—22no—
51D5×A4—41no—
52C2×C3×C4×C5—41no—
53C3×D20—82no—
54C23×A4—51no—
55C23×A4—51no—
56D4×A4—102no—
57C2×C4×A4—102no—
58C2×C4×A4AGL(1,𝔽5)11yescharacteristic, complemented
59C3×#99—201no—
60C24×C5C2×C3×C411yescharacteristic
61C23×D5—21no—
62C23×D5—31no—
63C23×D5—32no—
64C23×D5—31no—
65C23×D5—31no—
66C23×D5—31no—
67C23×D5—41no—
68C23×D5—41no—
69C23×D5—61no—
70C23×D5—121no—
71C23×D5—121no—
72C23×D5C3×D412yes—
73C22×AGL(1,𝔽5)—42no—
74C22×AGL(1,𝔽5)—62no—
75C22×AGL(1,𝔽5)—62no—
76C22×AGL(1,𝔽5)—122no—
77C22×C4×C5—21no—
78C22×C4×C5—31no—
79C22×C4×C5—31no—
80C22×C4×C5—61no—
81C2×D20—32no—
82C2×D20—31no—
83C2×D20—31no—
84C2×D20—31no—
85C2×D20—62no—
86C2×D20—62no—
87C2×D20—62no—
88C2×D20—62no—
89C2×D20—62no—
90C2×D20—62no—
91C2×D20—62no—
92C2×D20—62no—
93C2×D20C2×A411yescharacteristic
94C22⋊AGL(1,𝔽5)—32no—
95C22⋊AGL(1,𝔽5)—62no—
96C22⋊AGL(1,𝔽5)—62no—
97C22⋊AGL(1,𝔽5)C2×A412yescomplemented
98C23×D4—51no—
99C22×C22⋊C4—52no—
100C2×#99—151no—
101C2×#99—152no—
102C2×#99—151no—
103C22×C3×C5—41no—
104C2×C3×D5—42no—
105C2×C3×D5—81no—
106C2×C3×D5—161no—
107C2×C3×D5—161no—
108C5×A4#9911yescharacteristic, complemented
109C3×AGL(1,𝔽5)—162no—
110C3×C4×C5—81no—
111C22×A4—52no—
112C22×A4—101no—
113C22×A4—201no—
114C22×A4—201no—
115C22×A4C2×AGL(1,𝔽5)11yescharacteristic
116C2×C3×D4—201no—
117C4×A4—21no—
118C4×A4—202no—
119C3×C22⋊C4—202no—
120C23×C5—21no—
121C23×C5—31no—
122C23×C5—31no—
123C23×C5—61no—
124C23×C5C3×C22⋊C411yescharacteristic
125C22×D5—22no—
126C22×D5—32no—
127C22×D5—31no—
128C22×D5—32no—
129C22×D5—32no—
130C22×D5—32no—
131C22×D5—31no—
132C22×D5—41no—
133C22×D5—61no—
134C22×D5—61no—
135C22×D5—62no—
136C22×D5—61no—
137C22×D5—61no—
138C22×D5—121no—
139C22×D5—121no—
140C22×D5—121no—
141C22×D5—121no—
142C22×D5—122no—
143C22×D5—121no—
144C22×D5—122no—
145C22×D5—121no—
146C22×D5—122no—
147C22×D5—122no—
148C22×D5—121no—
149C22×D5—121no—
150C22×D5—121no—
151C22×D5C4×A411yescharacteristic, complemented
152C22×D5C22×A411yescharacteristic
153C2×AGL(1,𝔽5)—22no—
154C2×AGL(1,𝔽5)—62no—
155C2×AGL(1,𝔽5)—122no—
156C2×AGL(1,𝔽5)—122no—
157C2×AGL(1,𝔽5)—122no—
158C2×AGL(1,𝔽5)—122no—
159C2×C4×C5—31no—
160C2×C4×C5—61no—
161C2×C4×C5—61no—
162C2×C4×C5—61no—
163C2×C4×C5—61no—
164C2×C4×C5C4×A411yescharacteristic, complemented
165D20—22no—
166D20—62no—
167D20—62no—
168D20—64no—
169D20—62no—
170C25—51no—
171C25—51no—
172C22×D4—102no—
173C22×D4—151no—
174C22×D4—151no—
175C22×D4—151no—
176C22×D4—151no—
177C22×D4—302no—
178C23×C4—102no—
179C23×C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
180C2×C22⋊C4—152no—
181C2×C22⋊C4—152no—
182C2×C22⋊C4—302no—
183#99—51no—
184#99—152no—
185#99—151no—
186#99—152no—
187C2×C3×C5—41no—
188C2×C3×C5—81no—
189C3×D5—82no—
190C3×D5—161no—
191C23×C3—201no—
192C23×C3—201no—
193C2×A4—21no—
194C2×A4—102no—
195C2×A4—201no—
196C2×A4C22⋊AGL(1,𝔽5)11yescharacteristic
197C3×D4—402no—
198C2×C3×C4—41no—
199C2×C3×C4—402no—
200C22×C5—31no—
201C22×C5—31no—
202C22×C5—31no—
203C22×C5—61no—
204C22×C5—61no—
205C22×C5—61no—
206C22×C5—61no—
207C22×C5C3×#9911yescharacteristic, complemented
208C22×C5C2×C4×A411yescharacteristic
209C2×D5—21no—
210C2×D5—32no—
211C2×D5—41no—
212C2×D5—41no—
213C2×D5—62no—
214C2×D5—62no—
215C2×D5—62no—
216C2×D5—62no—
217C2×D5—61no—
218C2×D5—122no—
219C2×D5—121no—
220C2×D5—121no—
221C2×D5—122no—
222C2×D5—121no—
223C2×D5—122no—
224C2×D5—121no—
225C2×D5—121no—
226C2×D5—121no—
227C2×D5—122no—
228C2×D5D4×A412yes—
229AGL(1,𝔽5)—42no—
230AGL(1,𝔽5)—122no—
231C4×C5—21no—
232C4×C5—61no—
233C24—52no—
234C24—101no—
235C24—151no—
236C24—151no—
237C24—152no—
238C24—151no—
239C24—151no—
240C24—201no—
241C24—201no—
242C24—301no—
243C24—601no—
244C24—601no—
245C24C2×C3×AGL(1,𝔽5)11yescharacteristic
246C2×D4—51no—
247C2×D4—151no—
248C2×D4—151no—
249C2×D4—152no—
250C2×D4—151no—
251C2×D4—302no—
252C2×D4—302no—
253C2×D4—302no—
254C2×D4—302no—
255C2×D4—302no—
256C2×D4—302no—
257C2×D4—302no—
258C2×D4—302no—
259C22×C4—21no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—61no—
263C22×C4—202no—
264C22×C4—302no—
265C22×C4—302no—
266C22×C4—602no—
267C22⋊C4—52no—
268C22⋊C4—152no—
269C22⋊C4—302no—
270C22⋊C4—302no—
271C3×C5—41noHall
272C22×C3—41no—
273C22×C3—202no—
274C22×C3—401no—
275C22×C3—801no—
276C22×C3—801no—
277A4#399911yescharacteristic, direct product factor
278C3×C4—81no—
279C3×C4—802no—
280C2×C5—21no—
281C2×C5—31no—
282C2×C5—31no—
283C2×C5—61no—
284C2×C5A4×C22⋊C411yescharacteristic
285D5—22no—
286D5—41no—
287D5—62no—
288D5—121no—
289C23—21no—
290C23—31no—
291C23—31no—
292C23—51no—
293C23—51no—
294C23—61no—
295C23—102no—
296C23—152no—
297C23—152no—
298C23—152no—
299C23—152no—
300C23—151no—
301C23—151no—
302C23—201no—
303C23—301no—
304C23—301no—
305C23—302no—
306C23—301no—
307C23—301no—
308C23—601no—
309C23—601no—
310C23—601no—
311C23—601no—
312C23—601no—
313C23—601no—
314C23—602no—
315C23—601no—
316C23—602no—
317C23—601no—
318C23—601no—
319C23—602no—
320C23—602no—
321C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
322D4—102no—
323D4—302no—
324D4—302no—
325D4—302no—
326D4—304no—
327C2×C4—31no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—102no—
333C2×C4—302no—
334C2×C4—602no—
335C2×C4—602no—
336C2×C4—602no—
337C2×C4—602no—
338C2×C4A4×AGL(1,𝔽5)11yescharacteristic, complemented
339C2×C3—41no—
340C2×C3—81no—
341C2×C3—402no—
342C2×C3—801no—
343C5A4×#9911yescharacteristic, complemented, Sylow
344C22—31no—
345C22—31no—
346C22—31no—
347C22—52no—
348C22—61no—
349C22—61no—
350C22—61no—
351C22—61no—
352C22—101no—
353C22—152no—
354C22—201no—
355C22—201no—
356C22—302no—
357C22—302no—
358C22—302no—
359C22—302no—
360C22—301no—
361C22—601no—
362C22—601no—
363C22—601no—
364C22—601no—
365C22—601no—
366C22—601no—
367C22—602no—
368C22—602no—
369C22—602no—
370C22—602no—
371C22C3×#399911yescharacteristic, complemented
372C22C2×A4×AGL(1,𝔽5)11yescharacteristic
373C4—21no—
374C4—61no—
375C4—202no—
376C4—602no—
377C3—41noSylow
378C2—21no—
379C2—31no—
380C2—31no—
381C2—61no—
382C2—102no—
383C2—201no—
384C2—302no—
385C2—601no—
386C2A4×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2A4×C22⋊AGL(1,𝔽5)
commutatorC24×C5C2×C3×C4
FrattiniC22C2×A4×AGL(1,𝔽5)
FittingC23×C4×C5C3×C4
socleC23×C5C3×C22⋊C4
prime radicalC2×A4×D20C2
cosocleC24×D5C22×C3
hypercenterC2×C4A4×AGL(1,𝔽5)
hypocenterC22×C5C3×#99
LCSG ▷ C24×C5 ▷ C23×C5 ▷ C22×C5
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. A4×C22⋊AGL(1,𝔽5)C2semidirect
3. C22×#3999C3semidirect
4. C22×D5×A4C4semidirect
7. C2×C4×C5×A4C4semidirect
12. C22×C22⋊AGL(1,𝔽5)C2×C3semidirect
27. C24×D5C3×C4semidirect
30. C23×C4×C5C3×C4semidirect
43. #3999A4direct (N × H)
58. C2×C4×A4AGL(1,𝔽5)semidirect
97. C22⋊AGL(1,𝔽5)C2×A4semidirect
108. C5×A4#99semidirect
151. C22×D5C4×A4semidirect
164. C2×C4×C5C4×A4semidirect
179. C23×C4C3×AGL(1,𝔽5)semidirect
207. C22×C5C3×#99semidirect
277. A4#3999direct (N × H)
338. C2×C4A4×AGL(1,𝔽5)semidirect
343. C5A4×#99semidirect
371. C22C3×#3999semidirect

Maximal subgroups

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

subgroupindex
C2×A4×D202
A4×C22⋊AGL(1,𝔽5)2
C22×#39993
C3×#39994
A4×#995