Finite Group Databasebeta

← A4×#4075 #14514181

Subgroups — A4×#4075 #14514181

Order 1920.

916 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 99 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×C4×C5×A4C211yesmaximal, characteristic, complemented
2C2×A4×C22⋊D5C211yesmaximal, characteristic
3A4×C22⋊Dic5C211yesmaximal, characteristic, complemented
4A4×#685C212yesmaximal, complemented
5A4×#683C212yesmaximal, complemented
6C22×#4075C311yesmaximal, characteristic, complemented, Hall
7C23×C5×A4C2211yescharacteristic
8C22×D5×A4C2211yescharacteristic
9C2×C4×C5×A4—22no—
10C2×C4×C5×A4C2212yes—
11A4×C22⋊D5—22no—
12C2×A4×Dic5C2212yes—
13C2×A4×Dic5C2211yescharacteristic
14C3×#4075—41nomaximal
15A4×#123—51nomaximal, Hall
16C24×C4×C5C2×C311yescharacteristic, complemented
17C23×C22⋊D5C2×C311yescharacteristic
18C22×C22⋊Dic5C2×C311yescharacteristic, complemented
19C2×#4075—31no—
20C2×#4075—31no—
21C2×#4075—31no—
22C2×#4075—32no—
23C2×#4075—32no—
24C2×#4075—31no—
25C22×#685C2×C312yescomplemented
26C22×#683C2×C312yescomplemented
27C22×C5×A4—22no—
28C22×C5×A4D412yes—
29C22×C5×A4C2311yescharacteristic
30C2×D5×A4—21no—
31C2×D5×A4—22no—
32C22×C3×C4×C5—41no—
33C2×C3×C22⋊D5—41no—
34C4×C5×A4—22no—
35A4×Dic5—21no—
36A4×Dic5—22no—
37C3×C22⋊Dic5—41no—
38C3×#685—42no—
39C3×#683—42no—
40C2×D4×A4—51no—
41C22×C4×A4D511yescharacteristic, complemented
42A4×C4⋊C4—52no—
43A4×C22⋊C4—51no—
44A4×C22⋊C4—52no—
45C25×C5C22×C311yescharacteristic
46C24×D5C22×C311yescharacteristic
47C23×C4×C5—22no—
48C23×C4×C5—31no—
49C23×C4×C5—31no—
50C23×C4×C5—32no—
51C23×C4×C5—62no—
52C23×C4×C5C22×C312yes—
53C22×C22⋊D5—22no—
54C22×C22⋊D5—31no—
55C22×C22⋊D5—31no—
56C22×C22⋊D5—31no—
57C22×C22⋊D5—31no—
58C22×C22⋊D5—62no—
59C23×Dic5C22×C311yescharacteristic
60C23×Dic5C22×C312yes—
61C2×C22⋊Dic5—31no—
62C2×C22⋊Dic5—31no—
63C2×C22⋊Dic5—32no—
64#4075—32no—
65#4075—32no—
66#4075—34no—
67#4075—34no—
68#4075—32no—
69#4075—31no—
70#4075—31no—
71#4075—31no—
72#4075—32no—
73#4075—32no—
74#4075A411yescharacteristic, direct product factor
75C2×#685—32no—
76C2×#685—32no—
77C2×#685—32no—
78C2×#685—32no—
79C2×#683—32no—
80C2×#683—32no—
81C2×#683—32no—
82C2×#683—32no—
83C22×#123—51noSylow
84C23×C3×C5—41no—
85C22×C3×D5—41no—
86C2×C5×A4—22no—
87C2×C5×A4C2×D411yescharacteristic
88C2×C5×A4#4112yes—
89D5×A4—41no—
90C2×C3×C4×C5—42no—
91C2×C3×C4×C5—82no—
92C3×C22⋊D5—82no—
93C2×C3×Dic5—41no—
94C2×C3×Dic5—42no—
95C23×A4—51no—
96C23×A4C2×D511yescharacteristic
97D4×A4—102no—
98C2×C4×A4—22no—
99C2×C4×A4—51no—
100C2×C4×A4—52no—
101C2×C4×A4C2×D512yes—
102C3×#123—201no—
103C24×C5—22no—
104C24×C5—31no—
105C24×C5—32no—
106C24×C5—31no—
107C24×C5—62no—
108C24×C5C3×D412yes—
109C24×C5C23×C311yescharacteristic
110C23×D5—21no—
111C23×D5—22no—
112C23×D5—31no—
113C23×D5—31no—
114C23×D5—61no—
115C23×D5—62no—
116C22×C4×C5—22no—
117C22×C4×C5—32no—
118C22×C4×C5—31no—
119C22×C4×C5—32no—
120C22×C4×C5—32no—
121C22×C4×C5—32no—
122C22×C4×C5—62no—
123C22×C4×C5—62no—
124C22×C4×C5—64no—
125C22×C4×C5—64no—
126C22×C4×C5—62no—
127C22×C4×C5—62no—
128C22×C4×C5—62no—
129C22×C4×C5C2×A411yescharacteristic, complemented
130C2×C22⋊D5—32no—
131C2×C22⋊D5—31no—
132C2×C22⋊D5—31no—
133C2×C22⋊D5—31no—
134C2×C22⋊D5—62no—
135C2×C22⋊D5—62no—
136C2×C22⋊D5—62no—
137C2×C22⋊D5—62no—
138C2×C22⋊D5—62no—
139C2×C22⋊D5—62no—
140C2×C22⋊D5—62no—
141C2×C22⋊D5—62no—
142C2×C22⋊D5C2×A411yescharacteristic
143C22×Dic5—21no—
144C22×Dic5—22no—
145C22×Dic5—32no—
146C22×Dic5—31no—
147C22×Dic5—32no—
148C22×Dic5—31no—
149C22×Dic5—62no—
150C22×Dic5—61no—
151C22⋊Dic5—32no—
152C22⋊Dic5—31no—
153C22⋊Dic5—32no—
154C22⋊Dic5C2×A411yescharacteristic, complemented
155#685—34no—
156#685—32no—
157#685—32no—
158#685—32no—
159#685C2×A412yescomplemented
160#683—34no—
161#683—32no—
162#683—32no—
163#683—32no—
164#683C2×A412yescomplemented
165C23×D4—51no—
166C24×C4C3×D511yescharacteristic, complemented
167C2×#123—151no—
168C2×#123—152no—
169C2×#123—151no—
170C2×#123—152no—
171C2×#123—151no—
172C2×#123—151no—
173C22×C4⋊C4—52no—
174C22×C22⋊C4—51no—
175C22×C22⋊C4—52no—
176C22×C3×C5—42no—
177C22×C3×C5—41no—
178C22×C3×C5—82no—
179C2×C3×D5—81no—
180C2×C3×D5—82no—
181C5×A4#12311yescharacteristic, complemented
182C3×C4×C5—82no—
183C3×Dic5—82no—
184C3×Dic5—81no—
185C22×A4—22no—
186C22×A4—102no—
187C22×A4—101no—
188C22×A4C22⋊D512yes—
189C22×A4C22×D511yescharacteristic
190C2×C3×D4—201no—
191C22×C3×C4—41no—
192C4×A4—22no—
193C4×A4—102no—
194C4×A4—101no—
195C3×C4⋊C4—202no—
196C3×C22⋊C4—202no—
197C3×C22⋊C4—201no—
198C23×C5—22no—
199C23×C5—31no—
200C23×C5—32no—
201C23×C5—31no—
202C23×C5—32no—
203C23×C5—32no—
204C23×C5—32no—
205C23×C5—32no—
206C23×C5—31no—
207C23×C5—62no—
208C23×C5—62no—
209C23×C5—62no—
210C23×C5—64no—
211C23×C5—62no—
212C23×C5—62no—
213C23×C5—64no—
214C23×C5C2×C3×D411yescharacteristic
215C23×C5C3×#4112yes—
216C23×C5C22×A411yescharacteristic
217C22×D5—31no—
218C22×D5—41no—
219C22×D5—61no—
220C22×D5—61no—
221C22×D5—62no—
222C22×D5—62no—
223C22×D5—61no—
224C22×D5—61no—
225C22×D5—62no—
226C22×D5—62no—
227C22×D5—122no—
228C22×D5—121no—
229C22×D5—122no—
230C22×D5C22×A411yescharacteristic
231C2×C4×C5—22no—
232C2×C4×C5—32no—
233C2×C4×C5—62no—
234C2×C4×C5—62no—
235C2×C4×C5—64no—
236C2×C4×C5—64no—
237C2×C4×C5—62no—
238C2×C4×C5—62no—
239C2×C4×C5—62no—
240C2×C4×C5C22×A412yes—
241C22⋊D5—22no—
242C22⋊D5—62no—
243C22⋊D5—62no—
244C22⋊D5—64no—
245C22⋊D5—62no—
246C2×Dic5—31no—
247C2×Dic5—32no—
248C2×Dic5—62no—
249C2×Dic5—61no—
250C2×Dic5—62no—
251C2×Dic5—61no—
252C2×Dic5—62no—
253C2×Dic5—61no—
254C2×Dic5—62no—
255C2×Dic5—61no—
256C2×Dic5C22×A411yescharacteristic
257C2×Dic5C22×A412yes—
258C25—51no—
259C25C2×C3×D511yescharacteristic
260C22×D4—102no—
261C22×D4—151no—
262C22×D4—151no—
263C22×D4—151no—
264C22×D4—151no—
265C22×D4—302no—
266C23×C4—22no—
267C23×C4—31no—
268C23×C4—31no—
269C23×C4—32no—
270C23×C4—52no—
271C23×C4—51no—
272C23×C4—62no—
273C23×C4C2×C3×D512yes—
274#123—51no—
275#123—151no—
276#123—152no—
277#123—152no—
278#123—151no—
279#123—154no—
280#123—152no—
281#123—152no—
282#123—152no—
283#123—154no—
284#123—151no—
285C2×C4⋊C4—152no—
286C2×C4⋊C4—152no—
287C2×C4⋊C4—152no—
288C2×C4⋊C4—152no—
289C2×C22⋊C4—152no—
290C2×C22⋊C4—151no—
291C2×C22⋊C4—152no—
292C2×C22⋊C4—152no—
293C2×C22⋊C4—152no—
294C2×C22⋊C4—152no—
295C2×C22⋊C4—151no—
296C2×C3×C5—41no—
297C2×C3×C5—42no—
298C2×C3×C5—82no—
299C3×D5—161no—
300C23×C3—41no—
301C23×C3—201no—
302C2×A4—22no—
303C2×A4—201no—
304C2×A4#70912yes—
305C2×A4C2×C22⋊D511yescharacteristic
306C3×D4—402no—
307C2×C3×C4—42no—
308C2×C3×C4—82no—
309C2×C3×C4—202no—
310C2×C3×C4—201no—
311C22×C5—22no—
312C22×C5—31no—
313C22×C5—32no—
314C22×C5—32no—
315C22×C5—31no—
316C22×C5—31no—
317C22×C5—32no—
318C22×C5—32no—
319C22×C5—32no—
320C22×C5—62no—
321C22×C5—64no—
322C22×C5—62no—
323C22×C5—62no—
324C22×C5—62no—
325C22×C5—64no—
326C22×C5—62no—
327C22×C5C3×#12311yescharacteristic, complemented
328C22×C5D4×A412yes—
329C22×C5C23×A411yescharacteristic
330C2×D5—21no—
331C2×D5—22no—
332C2×D5—61no—
333C2×D5—62no—
334C2×D5—121no—
335C2×D5—121no—
336C2×D5—122no—
337C2×D5—121no—
338C2×D5—121no—
339C2×D5—122no—
340C4×C5—22no—
341C4×C5—62no—
342Dic5—21no—
343Dic5—22no—
344Dic5—61no—
345Dic5—62no—
346C24—22no—
347C24—31no—
348C24—32no—
349C24—31no—
350C24—62no—
351C24—102no—
352C24—101no—
353C24—151no—
354C24—151no—
355C24—301no—
356C24—302no—
357C24C3×C22⋊D512yes—
358C24C22×C3×D511yescharacteristic
359C2×D4—51no—
360C2×D4—151no—
361C2×D4—151no—
362C2×D4—152no—
363C2×D4—151no—
364C2×D4—302no—
365C2×D4—302no—
366C2×D4—302no—
367C2×D4—302no—
368C2×D4—302no—
369C2×D4—302no—
370C2×D4—302no—
371C2×D4—302no—
372C22×C4—22no—
373C22×C4—32no—
374C22×C4—31no—
375C22×C4—32no—
376C22×C4—32no—
377C22×C4—32no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—64no—
381C22×C4—64no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—102no—
386C22×C4—101no—
387C22×C4—152no—
388C22×C4—151no—
389C22×C4—152no—
390C22×C4—151no—
391C22×C4—301no—
392C22×C4—302no—
393C22×C4D5×A411yescharacteristic, complemented
394C4⋊C4—52no—
395C4⋊C4—152no—
396C4⋊C4—152no—
397C4⋊C4—154no—
398C4⋊C4—152no—
399C22⋊C4—51no—
400C22⋊C4—52no—
401C22⋊C4—152no—
402C22⋊C4—152no—
403C22⋊C4—154no—
404C22⋊C4—152no—
405C22⋊C4—152no—
406C22⋊C4—151no—
407C22⋊C4—152no—
408C3×C5—41noHall
409C22×C3—42no—
410C22×C3—41no—
411C22×C3—82no—
412C22×C3—401no—
413C22×C3—402no—
414A4#407511yescharacteristic, direct product factor
415C3×C4—82no—
416C3×C4—401no—
417C3×C4—402no—
418C2×C5—22no—
419C2×C5—31no—
420C2×C5—32no—
421C2×C5—31no—
422C2×C5—62no—
423C2×C5C2×D4×A411yescharacteristic
424C2×C5A4×#4112yes—
425D5—41no—
426D5—121no—
427C23—22no—
428C23—31no—
429C23—32no—
430C23—31no—
431C23—32no—
432C23—32no—
433C23—32no—
434C23—32no—
435C23—31no—
436C23—51no—
437C23—62no—
438C23—62no—
439C23—62no—
440C23—64no—
441C23—62no—
442C23—62no—
443C23—64no—
444C23—151no—
445C23—201no—
446C23—301no—
447C23—301no—
448C23—302no—
449C23—302no—
450C23—301no—
451C23—301no—
452C23—302no—
453C23—302no—
454C23—602no—
455C23—601no—
456C23—602no—
457C23C3×#70912yes—
458C23C2×C3×C22⋊D511yescharacteristic
459C23C2×D5×A411yescharacteristic
460D4—102no—
461D4—302no—
462D4—302no—
463D4—304no—
464D4—302no—
465C2×C4—22no—
466C2×C4—32no—
467C2×C4—52no—
468C2×C4—51no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—64no—
472C2×C4—64no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—152no—
477C2×C4—151no—
478C2×C4—301no—
479C2×C4—302no—
480C2×C4—301no—
481C2×C4—302no—
482C2×C4—301no—
483C2×C4—302no—
484C2×C4—301no—
485C2×C4—302no—
486C2×C4C2×D5×A412yes—
487C2×C3—42no—
488C2×C3—41no—
489C2×C3—82no—
490C2×C3—801no—
491C5A4×#12311yescharacteristic, complemented, Sylow
492C22—22no—
493C22—32no—
494C22—31no—
495C22—32no—
496C22—32no—
497C22—32no—
498C22—31no—
499C22—31no—
500C22—32no—
501C22—62no—
502C22—62no—
503C22—64no—
504C22—62no—
505C22—62no—
506C22—64no—
507C22—62no—
508C22—102no—
509C22—101no—
510C22—302no—
511C22—301no—
512C22—601no—
513C22—601no—
514C22—602no—
515C22—601no—
516C22—601no—
517C22—602no—
518C22C3×#407511yescharacteristic, complemented
519C22A4×C22⋊D512yes—
520C22C22×D5×A411yescharacteristic
521C4—22no—
522C4—62no—
523C4—102no—
524C4—101no—
525C4—302no—
526C4—301no—
527C3—41noSylow
528C2—22no—
529C2—31no—
530C2—32no—
531C2—31no—
532C2—62no—
533C2—201no—
534C2—601no—
535C2A4×#70912yes—
536C2C2×A4×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, socle, cosocleC24×C5C23×C3
FittingC24×C4×C5C2×C3
prime radicalC2×A4×C22⋊D5C2
hypercenterC22×C4D5×A4
hypocenterC22×C5C3×#123
LCSG ▷ C24×C5 ▷ C22×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×C5×A4C2semidirect
3. A4×C22⋊Dic5C2semidirect
4. A4×#685C2semidirect
5. A4×#683C2semidirect
6. C22×#4075C3semidirect
16. C24×C4×C5C2×C3semidirect
18. C22×C22⋊Dic5C2×C3semidirect
25. C22×#685C2×C3semidirect
26. C22×#683C2×C3semidirect
41. C22×C4×A4D5semidirect
74. #4075A4direct (N × H)
129. C22×C4×C5C2×A4semidirect
154. C22⋊Dic5C2×A4semidirect
159. #685C2×A4semidirect
164. #683C2×A4semidirect
166. C24×C4C3×D5semidirect
181. C5×A4#123semidirect
327. C22×C5C3×#123semidirect
393. C22×C4D5×A4semidirect
414. A4#4075direct (N × H)
491. C5A4×#123semidirect
518. C22C3×#4075semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C5×A42
C2×A4×C22⋊D52
A4×C22⋊Dic52
A4×#6852
A4×#6832
C22×#40753
C3×#40754
A4×#1235