Finite Group Databasebeta

← A4×#4023 #14513982

Subgroups — A4×#4023 #14513982

Order 1920.

880 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C22×A4×Dic5C211yesmaximal, characteristic
2C5×A4×C22⋊C4C211yesmaximal, characteristic
3A4×C22⋊Dic5C211yesmaximal, characteristic
4A4×#683C212yesmaximal, complemented
5C4×A4×Dic5C212yesmaximal, complemented
6C22×#4023C311yesmaximal, characteristic, complemented, Hall
7C23×C5×A4C2211yescharacteristic
8C2×C4×C5×A4C2212yes—
9C2×A4×Dic5C414yescomplemented
10C2×A4×Dic5C2212yes—
11C2×A4×Dic5C2211yescharacteristic
12C2×A4×Dic5C2211yescharacteristic
13C3×#4023—41nomaximal
14A4×#117—51nomaximal, Hall
15C24×Dic5C2×C311yescharacteristic
16C22×C5×C22⋊C4C2×C311yescharacteristic
17C22×C22⋊Dic5C2×C311yescharacteristic
18C22×#683C2×C312yescomplemented
19C22×C4×Dic5C2×C312yescomplemented
20C2×#4023—31no—
21C2×#4023—31no—
22C2×#4023—31no—
23C2×#4023—31no—
24C2×#4023—32no—
25C2×#4023—32no—
26C22×C5×A4—22no—
27C22×C5×A4C2×C412yes—
28C22×C5×A4C2311yescharacteristic
29C22×C3×Dic5—41no—
30C4×C5×A4—22no—
31A4×Dic5—22no—
32A4×Dic5C2×C412yes—
33A4×Dic5C2×C412yes—
34C3×C5×C22⋊C4—41no—
35C3×C22⋊Dic5—41no—
36C3×#683—42no—
37C3×C4×Dic5—42no—
38C22×C4×A4—51no—
39A4×C4⋊C4—52no—
40A4×C22⋊C4—51no—
41A4×C22⋊C4D511yescharacteristic
42C42×A4—52no—
43C25×C5C22×C311yescharacteristic
44C23×C4×C5C22×C312yes—
45C23×Dic5—31no—
46C23×Dic5—31no—
47C23×Dic5—31no—
48C23×Dic5—31no—
49C23×Dic5—34no—
50C23×Dic5C3×C414yescomplemented
51C23×Dic5C22×C312yes—
52C23×Dic5C22×C311yescharacteristic
53C23×Dic5C22×C311yescharacteristic
54C2×C5×C22⋊C4—31no—
55C2×C5×C22⋊C4—31no—
56C2×C5×C22⋊C4—32no—
57C2×C22⋊Dic5—31no—
58C2×C22⋊Dic5—31no—
59C2×C22⋊Dic5—32no—
60C2×#683—32no—
61C2×#683—32no—
62C2×#683—32no—
63C2×#683—32no—
64C2×C4×Dic5—32no—
65C2×C4×Dic5—32no—
66C2×C4×Dic5—32no—
67C2×C4×Dic5—32no—
68#4023—32no—
69#4023—32no—
70#4023—34no—
71#4023—34no—
72#4023—32no—
73#4023—31no—
74#4023—31no—
75#4023—31no—
76#4023—32no—
77#4023—32no—
78#4023A411yescharacteristic, direct product factor
79C22×#117—51noSylow
80C23×C3×C5—41no—
81C2×C5×A4—22no—
82C2×C5×A4C22×C411yescharacteristic
83C2×C5×A4#4112yes—
84C2×C3×C4×C5—42no—
85C2×C3×Dic5—44no—
86C2×C3×Dic5—41no—
87C2×C3×Dic5—41no—
88C2×C3×Dic5—42no—
89C23×A4C2×D511yescharacteristic
90C2×C4×A4—54no—
91C2×C4×A4—51no—
92C2×C4×A4—51no—
93C2×C4×A4—52no—
94C2×C4×A4C2×D512yes—
95C3×#117—201no—
96C24×C5—22no—
97C24×C5—31no—
98C24×C5—32no—
99C24×C5—31no—
100C24×C5—62no—
101C24×C5C2×C3×C412yes—
102C24×C5C23×C311yescharacteristic
103C22×C4×C5—22no—
104C22×C4×C5—32no—
105C22×C4×C5—32no—
106C22×C4×C5—62no—
107C22×Dic5—22no—
108C22×Dic5—34no—
109C22×Dic5—34no—
110C22×Dic5—34no—
111C22×Dic5—32no—
112C22×Dic5—31no—
113C22×Dic5—34no—
114C22×Dic5—34no—
115C22×Dic5—31no—
116C22×Dic5—31no—
117C22×Dic5—31no—
118C22×Dic5—31no—
119C22×Dic5—34no—
120C22×Dic5—34no—
121C22×Dic5—34no—
122C22×Dic5—32no—
123C22×Dic5—32no—
124C22×Dic5—31no—
125C22×Dic5—31no—
126C22×Dic5—32no—
127C22×Dic5—32no—
128C22×Dic5—62no—
129C22×Dic5C2×C3×C412yes—
130C22×Dic5C2×C3×C412yes—
131C22×Dic5C2×A411yescharacteristic
132C5×C22⋊C4—32no—
133C5×C22⋊C4—31no—
134C5×C22⋊C4—32no—
135C5×C22⋊C4C2×A411yescharacteristic
136C22⋊Dic5—32no—
137C22⋊Dic5—31no—
138C22⋊Dic5—32no—
139C22⋊Dic5C2×A411yescharacteristic
140#683—34no—
141#683—32no—
142#683—32no—
143#683—32no—
144#683C2×A412yescomplemented
145C4×Dic5—34no—
146C4×Dic5—32no—
147C4×Dic5—32no—
148C4×Dic5—32no—
149C4×Dic5C2×A412yescomplemented
150C24×C4—51no—
151C2×#117—151no—
152C2×#117—152no—
153C2×#117—151no—
154C2×#117—151no—
155C2×#117—152no—
156C2×#117—151no—
157C22×C4⋊C4—52no—
158C22×C22⋊C4—51no—
159C22×C22⋊C4C3×D511yescharacteristic
160C22×C42—52no—
161C22×C3×C5—42no—
162C22×C3×C5—41no—
163C22×C3×C5—82no—
164C5×A4#11711yescharacteristic, complemented
165C3×C4×C5—82no—
166C3×Dic5—42no—
167C3×Dic5—42no—
168C3×Dic5—82no—
169C22×A4—22no—
170C22×A4C4×D512yes—
171C22×A4C22×D511yescharacteristic
172C22×C3×C4—201no—
173C4×A4—22no—
174C4×A4—52no—
175C4×A4—52no—
176C4×A4—102no—
177C3×C4⋊C4—202no—
178C3×C22⋊C4—41no—
179C3×C22⋊C4—201no—
180C3×C42—202no—
181C23×C5—22no—
182C23×C5—31no—
183C23×C5—32no—
184C23×C5—31no—
185C23×C5—32no—
186C23×C5—32no—
187C23×C5—32no—
188C23×C5—32no—
189C23×C5—31no—
190C23×C5—62no—
191C23×C5—62no—
192C23×C5—62no—
193C23×C5—64no—
194C23×C5—62no—
195C23×C5—62no—
196C23×C5—64no—
197C23×C5C22×C3×C411yescharacteristic
198C23×C5C3×#4112yes—
199C23×C5C22×A411yescharacteristic
200C2×C4×C5—32no—
201C2×C4×C5—62no—
202C2×C4×C5—62no—
203C2×C4×C5—62no—
204C2×C4×C5—62no—
205C2×C4×C5C22×A412yes—
206C2×Dic5—32no—
207C2×Dic5—32no—
208C2×Dic5—38no—
209C2×Dic5—34no—
210C2×Dic5—31no—
211C2×Dic5—31no—
212C2×Dic5—32no—
213C2×Dic5—32no—
214C2×Dic5—34no—
215C2×Dic5—34no—
216C2×Dic5—32no—
217C2×Dic5—32no—
218C2×Dic5—32no—
219C2×Dic5—32no—
220C2×Dic5—32no—
221C2×Dic5—62no—
222C2×Dic5—62no—
223C2×Dic5—62no—
224C2×Dic5—62no—
225C2×Dic5C4×A414yescomplemented
226C2×Dic5C22×A412yes—
227C2×Dic5C22×A411yescharacteristic
228C2×Dic5C22×A411yescharacteristic
229C25C2×C3×D511yescharacteristic
230C23×C4—54no—
231C23×C4—51no—
232C23×C4—51no—
233C23×C4—52no—
234C23×C4—151no—
235C23×C4—151no—
236C23×C4—151no—
237C23×C4—151no—
238C23×C4—154no—
239C23×C4C2×C3×D512yes—
240#117—51no—
241#117—152no—
242#117—151no—
243#117—152no—
244#117—151no—
245#117—152no—
246#117—154no—
247#117—152no—
248#117—152no—
249#117—154no—
250#117—151no—
251C2×C4⋊C4—152no—
252C2×C4⋊C4—152no—
253C2×C4⋊C4—152no—
254C2×C4⋊C4—152no—
255C2×C22⋊C4—31no—
256C2×C22⋊C4—31no—
257C2×C22⋊C4—32no—
258C2×C22⋊C4—151no—
259C2×C22⋊C4—152no—
260C2×C22⋊C4—151no—
261C2×C42—152no—
262C2×C42—152no—
263C2×C42—152no—
264C2×C42—152no—
265C2×C3×C5—41no—
266C2×C3×C5—42no—
267C2×C3×C5—82no—
268C23×C3—41no—
269C2×A4—22no—
270C2×A4C2×C4×D511yescharacteristic
271C2×A4#71112yes—
272C2×C3×C4—42no—
273C2×C3×C4—201no—
274C2×C3×C4—201no—
275C2×C3×C4—204no—
276C2×C3×C4—202no—
277C22×C5—22no—
278C22×C5—31no—
279C22×C5—32no—
280C22×C5—32no—
281C22×C5—31no—
282C22×C5—31no—
283C22×C5—32no—
284C22×C5—32no—
285C22×C5—32no—
286C22×C5—62no—
287C22×C5—64no—
288C22×C5—62no—
289C22×C5—62no—
290C22×C5—62no—
291C22×C5—64no—
292C22×C5—62no—
293C22×C5C3×#11711yescharacteristic, complemented
294C22×C5C2×C4×A412yes—
295C22×C5C23×A411yescharacteristic
296C4×C5—22no—
297C4×C5—62no—
298Dic5—22no—
299Dic5—32no—
300Dic5—32no—
301Dic5—62no—
302Dic5C2×C4×A412yes—
303Dic5C2×C4×A412yes—
304C24—22no—
305C24—31no—
306C24—32no—
307C24—31no—
308C24—62no—
309C24C3×C4×D512yes—
310C24C22×C3×D511yescharacteristic
311C22×C4—22no—
312C22×C4—32no—
313C22×C4—32no—
314C22×C4—51no—
315C22×C4—52no—
316C22×C4—52no—
317C22×C4—62no—
318C22×C4—102no—
319C22×C4—151no—
320C22×C4—151no—
321C22×C4—151no—
322C22×C4—151no—
323C22×C4—154no—
324C22×C4—154no—
325C22×C4—152no—
326C22×C4—152no—
327C22×C4—154no—
328C22×C4—154no—
329C22×C4—154no—
330C22×C4—151no—
331C22×C4—151no—
332C22×C4—151no—
333C22×C4—152no—
334C22×C4—152no—
335C22×C4—154no—
336C22×C4—154no—
337C22×C4—154no—
338C22×C4—152no—
339C22×C4—302no—
340C4⋊C4—52no—
341C4⋊C4—152no—
342C4⋊C4—152no—
343C4⋊C4—154no—
344C4⋊C4—152no—
345C22⋊C4—32no—
346C22⋊C4—31no—
347C22⋊C4—32no—
348C22⋊C4—51no—
349C22⋊C4—152no—
350C22⋊C4—152no—
351C22⋊C4—151no—
352C22⋊C4D5×A411yescharacteristic
353C42—52no—
354C42—152no—
355C42—152no—
356C42—154no—
357C42—152no—
358C3×C5—41noHall
359C22×C3—42no—
360C22×C3—41no—
361C22×C3—82no—
362A4#402311yescharacteristic, direct product factor
363C3×C4—82no—
364C3×C4—202no—
365C3×C4—202no—
366C3×C4—402no—
367C2×C5—22no—
368C2×C5—31no—
369C2×C5—32no—
370C2×C5—31no—
371C2×C5—62no—
372C2×C5C22×C4×A411yescharacteristic
373C2×C5A4×#4112yes—
374C23—22no—
375C23—32no—
376C23—31no—
377C23—31no—
378C23—32no—
379C23—32no—
380C23—32no—
381C23—32no—
382C23—31no—
383C23—62no—
384C23—62no—
385C23—62no—
386C23—64no—
387C23—62no—
388C23—62no—
389C23—64no—
390C23C2×C3×C4×D511yescharacteristic
391C23C3×#71112yes—
392C23C2×D5×A411yescharacteristic
393C2×C4—32no—
394C2×C4—54no—
395C2×C4—51no—
396C2×C4—51no—
397C2×C4—52no—
398C2×C4—62no—
399C2×C4—62no—
400C2×C4—62no—
401C2×C4—62no—
402C2×C4—152no—
403C2×C4—152no—
404C2×C4—152no—
405C2×C4—152no—
406C2×C4—154no—
407C2×C4—154no—
408C2×C4—152no—
409C2×C4—152no—
410C2×C4—158no—
411C2×C4—154no—
412C2×C4—151no—
413C2×C4—151no—
414C2×C4—152no—
415C2×C4—152no—
416C2×C4—152no—
417C2×C4—302no—
418C2×C4—302no—
419C2×C4—302no—
420C2×C4—302no—
421C2×C4C2×D5×A412yes—
422C2×C3—41no—
423C2×C3—42no—
424C2×C3—82no—
425C5A4×#11711yescharacteristic, complemented, Sylow
426C22—22no—
427C22—31no—
428C22—32no—
429C22—32no—
430C22—31no—
431C22—31no—
432C22—32no—
433C22—32no—
434C22—32no—
435C22—62no—
436C22—62no—
437C22—64no—
438C22—62no—
439C22—62no—
440C22—64no—
441C22—62no—
442C22C3×#402311yescharacteristic, complemented
443C22C4×D5×A412yes—
444C22C22×D5×A411yescharacteristic
445C4—22no—
446C4—52no—
447C4—52no—
448C4—62no—
449C4—102no—
450C4—152no—
451C4—152no—
452C4—302no—
453C3—41noSylow
454C2—22no—
455C2—31no—
456C2—31no—
457C2—32no—
458C2—62no—
459C2C2×C4×D5×A411yescharacteristic
460C2A4×#71112yes—
subgroupG/H
center, FrattiniC22C22×D5×A4
commutatorC23×C5C22×C3×C4
FittingC22×C5×C22⋊C4C2×C3
socle, cosocleC24×C5C23×C3
prime radicalC23×C5×A4C22
hypercenterC22⋊C4D5×A4
hypocenterC22×C5C3×#117
LCSG ▷ C23×C5 ▷ C22×C5
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
4. A4×#683C2semidirect
5. C4×A4×Dic5C2semidirect
6. C22×#4023C3semidirect
9. C2×A4×Dic5C4semidirect
18. C22×#683C2×C3semidirect
19. C22×C4×Dic5C2×C3semidirect
50. C23×Dic5C3×C4semidirect
78. #4023A4direct (N × H)
144. #683C2×A4semidirect
149. C4×Dic5C2×A4semidirect
164. C5×A4#117semidirect
225. C2×Dic5C4×A4semidirect
293. C22×C5C3×#117semidirect
362. A4#4023direct (N × H)
425. C5A4×#117semidirect
442. C22C3×#4023semidirect

Maximal subgroups

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

subgroupindex
C22×A4×Dic52
C5×A4×C22⋊C42
A4×C22⋊Dic52
A4×#6832
C4×A4×Dic52
C22×#40233
C3×#40234
A4×#1175