Finite Group Databasebeta

← A4×#4076 #14514182

Subgroups — A4×#4076 #14514182

Order 1920.

1156 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 1156 conjugacy classes of subgroups — too many to draw. Only its 111 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⋊D5C212yesmaximal, complemented
3C2×A4×D20C211yesmaximal, characteristic, complemented
4A4×#685C212yesmaximal, complemented
5A4×C4⋊Dic5C211yesmaximal, characteristic, complemented
6C22×#4076C311yesmaximal, characteristic, complemented, Hall
7C23×C5×A4C2211yescharacteristic
8C22×D5×A4C2212yes—
9C2×C4×C5×A4—22no—
10C2×C4×C5×A4C2211yescharacteristic
11C2×C4×C5×A4C2211yescharacteristic
12A4×C22⋊D5—24no—
13C2×A4×Dic5C2212yes—
14A4×D20—22no—
15C3×#4076—41nomaximal
16A4×#121—51nomaximal, Hall
17C24×C4×C5C2×C311yescharacteristic, complemented
18C23×C22⋊D5C2×C312yescomplemented
19C23×D20C2×C311yescharacteristic, complemented
20C2×#4076—31no—
21C2×#4076—32no—
22C2×#4076—31no—
23C2×#4076—31no—
24C2×#4076—32no—
25C2×#4076—31no—
26C22×#685C2×C312yescomplemented
27C22×C4⋊Dic5C2×C311yescharacteristic, complemented
28C22×C5×A4—21no—
29C22×C5×A4—21no—
30C22×C5×A4D412yescomplemented
31C22×C5×A4C2311yescharacteristic
32C2×D5×A4—22no—
33C2×D5×A4—22no—
34C2×D5×A4—22no—
35C22×C3×C4×C5—41no—
36C2×C3×C22⋊D5—42no—
37C2×C3×D20—41no—
38C4×C5×A4—21no—
39C4×C5×A4D412yescomplemented
40A4×Dic5—22no—
41C3×#685—42no—
42C3×C4⋊Dic5—41no—
43C2×D4×A4—52no—
44C2×D4×A4—51no—
45C22×C4×A4D511yescharacteristic, complemented
46A4×C4⋊C4—51no—
47A4×C22⋊C4—52no—
48C25×C5C22×C311yescharacteristic
49C24×D5C22×C312yes—
50C23×C4×C5—22no—
51C23×C4×C5—31no—
52C23×C4×C5—31no—
53C23×C4×C5—31no—
54C23×C4×C5—31no—
55C23×C4×C5—62no—
56C23×C4×C5C22×C311yescharacteristic
57C23×C4×C5C22×C311yescharacteristic
58C22×C22⋊D5—24no—
59C22×C22⋊D5—32no—
60C22×C22⋊D5—32no—
61C22×C22⋊D5—32no—
62C22×C22⋊D5—32no—
63C22×C22⋊D5—64no—
64C23×Dic5C22×C312yes—
65C22×D20—22no—
66C22×D20—31no—
67C22×D20—32no—
68C22×D20—31no—
69C22×D20—62no—
70#4076—32no—
71#4076—34no—
72#4076—32no—
73#4076—32no—
74#4076—34no—
75#4076—31no—
76#4076—32no—
77#4076—31no—
78#4076—32no—
79#4076—31no—
80#4076A411yescharacteristic, direct product factor
81C2×#685—32no—
82C2×#685—32no—
83C2×#685—32no—
84C2×#685—32no—
85C2×C4⋊Dic5—31no—
86C2×C4⋊Dic5—31no—
87C2×C4⋊Dic5—32no—
88C22×#121—51noSylow
89C23×C3×C5—41no—
90C22×C3×D5—42no—
91C2×C5×A4—22no—
92C2×C5×A4C2×D411yescharacteristic
93C2×C5×A4C2×D411yescharacteristic
94C2×C5×A4#4111yescharacteristic
95D5×A4—42no—
96C2×C3×C4×C5—41no—
97C2×C3×C4×C5—41no—
98C2×C3×C4×C5—82no—
99C3×C22⋊D5—84no—
100C2×C3×Dic5—42no—
101C3×D20—82no—
102C23×A4—52no—
103C23×A4C2×D511yescharacteristic
104D4×A4—104no—
105D4×A4—102no—
106C2×C4×A4—22no—
107C2×C4×A4—52no—
108C2×C4×A4C2×D511yescharacteristic
109C2×C4×A4C2×D511yescharacteristic
110C3×#121—201no—
111C24×C5—21no—
112C24×C5—21no—
113C24×C5—31no—
114C24×C5—32no—
115C24×C5—31no—
116C24×C5—61no—
117C24×C5—61no—
118C24×C5C3×D412yescomplemented
119C24×C5C23×C311yescharacteristic
120C23×D5—22no—
121C23×D5—22no—
122C23×D5—22no—
123C23×D5—32no—
124C23×D5—32no—
125C23×D5—62no—
126C23×D5—62no—
127C23×D5—62no—
128C22×C4×C5—21no—
129C22×C4×C5—31no—
130C22×C4×C5—31no—
131C22×C4×C5—32no—
132C22×C4×C5—32no—
133C22×C4×C5—31no—
134C22×C4×C5—31no—
135C22×C4×C5—31no—
136C22×C4×C5—31no—
137C22×C4×C5—31no—
138C22×C4×C5—62no—
139C22×C4×C5—62no—
140C22×C4×C5—62no—
141C22×C4×C5—62no—
142C22×C4×C5—62no—
143C22×C4×C5—62no—
144C22×C4×C5—62no—
145C22×C4×C5—62no—
146C22×C4×C5—61no—
147C22×C4×C5C3×D412yescomplemented
148C22×C4×C5C2×A411yescharacteristic, complemented
149C2×C22⋊D5—34no—
150C2×C22⋊D5—32no—
151C2×C22⋊D5—32no—
152C2×C22⋊D5—32no—
153C2×C22⋊D5—64no—
154C2×C22⋊D5—64no—
155C2×C22⋊D5—64no—
156C2×C22⋊D5—64no—
157C2×C22⋊D5—64no—
158C2×C22⋊D5—64no—
159C2×C22⋊D5—64no—
160C2×C22⋊D5—64no—
161C2×C22⋊D5C2×A412yescomplemented
162C22×Dic5—22no—
163C22×Dic5—32no—
164C22×Dic5—32no—
165C22×Dic5—62no—
166C2×D20—32no—
167C2×D20—32no—
168C2×D20—31no—
169C2×D20—62no—
170C2×D20—62no—
171C2×D20—64no—
172C2×D20—64no—
173C2×D20—62no—
174C2×D20—62no—
175C2×D20C2×A411yescharacteristic, complemented
176#685—34no—
177#685—32no—
178#685—32no—
179#685—32no—
180#685C2×A412yescomplemented
181C4⋊Dic5—32no—
182C4⋊Dic5—31no—
183C4⋊Dic5—32no—
184C4⋊Dic5C2×A411yescharacteristic, complemented
185C23×D4—52no—
186C23×D4—51no—
187C24×C4C3×D511yescharacteristic, complemented
188C2×#121—151no—
189C2×#121—152no—
190C2×#121—151no—
191C2×#121—151no—
192C2×#121—152no—
193C2×#121—151no—
194C22×C4⋊C4—51no—
195C22×C22⋊C4—52no—
196C22×C3×C5—42no—
197C22×C3×C5—41no—
198C22×C3×C5—81no—
199C22×C3×C5—81no—
200C2×C3×D5—82no—
201C2×C3×D5—82no—
202C2×C3×D5—82no—
203C5×A4#12111yescharacteristic, complemented
204C3×C4×C5—42no—
205C3×C4×C5—81no—
206C3×Dic5—82no—
207C22×A4—21no—
208C22×A4—21no—
209C22×A4—102no—
210C22×A4—102no—
211C22×A4—102no—
212C22×A4D2012yescomplemented
213C22×A4C22×D511yescharacteristic
214C2×C3×D4—201no—
215C2×C3×D4—202no—
216C22×C3×C4—41no—
217C4×A4—21no—
218C4×A4—102no—
219C4×A4C22⋊D512yescomplemented
220C3×C4⋊C4—201no—
221C3×C22⋊C4—202no—
222C23×C5—22no—
223C23×C5—31no—
224C23×C5—31no—
225C23×C5—31no—
226C23×C5—31no—
227C23×C5—32no—
228C23×C5—32no—
229C23×C5—32no—
230C23×C5—32no—
231C23×C5—31no—
232C23×C5—61no—
233C23×C5—61no—
234C23×C5—61no—
235C23×C5—61no—
236C23×C5—62no—
237C23×C5—61no—
238C23×C5—61no—
239C23×C5—62no—
240C23×C5—62no—
241C23×C5—61no—
242C23×C5—61no—
243C23×C5—64no—
244C23×C5C2×C3×D411yescharacteristic
245C23×C5C2×C3×D411yescharacteristic
246C23×C5C3×#4111yescharacteristic
247C23×C5C22×A411yescharacteristic
248C22×D5—32no—
249C22×D5—42no—
250C22×D5—62no—
251C22×D5—62no—
252C22×D5—62no—
253C22×D5—62no—
254C22×D5—62no—
255C22×D5—62no—
256C22×D5—62no—
257C22×D5—62no—
258C22×D5—62no—
259C22×D5—62no—
260C22×D5—62no—
261C22×D5—62no—
262C22×D5—124no—
263C22×D5—122no—
264C22×D5—122no—
265C22×D5—122no—
266C22×D5C22×A412yes—
267C2×C4×C5—22no—
268C2×C4×C5—32no—
269C2×C4×C5—31no—
270C2×C4×C5—32no—
271C2×C4×C5—32no—
272C2×C4×C5—32no—
273C2×C4×C5—31no—
274C2×C4×C5—62no—
275C2×C4×C5—61no—
276C2×C4×C5—61no—
277C2×C4×C5—62no—
278C2×C4×C5—64no—
279C2×C4×C5—62no—
280C2×C4×C5—61no—
281C2×C4×C5—61no—
282C2×C4×C5C22×A411yescharacteristic
283C2×C4×C5C22×A411yescharacteristic
284C22⋊D5—24no—
285C22⋊D5—64no—
286C22⋊D5—64no—
287C22⋊D5—68no—
288C22⋊D5—64no—
289C2×Dic5—32no—
290C2×Dic5—62no—
291C2×Dic5—62no—
292C2×Dic5—62no—
293C2×Dic5—62no—
294C2×Dic5C22×A412yes—
295D20—22no—
296D20—64no—
297D20—64no—
298D20—62no—
299C25—52no—
300C25C2×C3×D511yescharacteristic
301C22×D4—104no—
302C22×D4—102no—
303C22×D4—151no—
304C22×D4—151no—
305C22×D4—152no—
306C22×D4—152no—
307C22×D4—152no—
308C22×D4—152no—
309C22×D4—152no—
310C22×D4—302no—
311C22×D4—304no—
312C23×C4—22no—
313C23×C4—31no—
314C23×C4—31no—
315C23×C4—31no—
316C23×C4—31no—
317C23×C4—52no—
318C23×C4—62no—
319C23×C4C2×C3×D511yescharacteristic
320C23×C4C2×C3×D511yescharacteristic
321#121—51no—
322#121—152no—
323#121—151no—
324#121—152no—
325#121—151no—
326#121—154no—
327#121—154no—
328#121—152no—
329#121—152no—
330#121—152no—
331#121—151no—
332C2×C4⋊C4—151no—
333C2×C4⋊C4—152no—
334C2×C4⋊C4—151no—
335C2×C22⋊C4—152no—
336C2×C22⋊C4—152no—
337C2×C22⋊C4—152no—
338C2×C22⋊C4—152no—
339C2×C3×C5—41no—
340C2×C3×C5—41no—
341C2×C3×C5—41no—
342C2×C3×C5—82no—
343C3×D5—162no—
344C23×C3—41no—
345C23×C3—202no—
346C2×A4—22no—
347C2×A4—202no—
348C2×A4C2×D2011yescharacteristic
349C2×A4#70911yescharacteristic
350C2×A4C2×C22⋊D511yescharacteristic
351C3×D4—402no—
352C3×D4—404no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—82no—
356C2×C3×C4—202no—
357C22×C5—21no—
358C22×C5—21no—
359C22×C5—31no—
360C22×C5—31no—
361C22×C5—31no—
362C22×C5—32no—
363C22×C5—31no—
364C22×C5—31no—
365C22×C5—31no—
366C22×C5—31no—
367C22×C5—31no—
368C22×C5—31no—
369C22×C5—32no—
370C22×C5—62no—
371C22×C5—62no—
372C22×C5—62no—
373C22×C5—62no—
374C22×C5—62no—
375C22×C5—62no—
376C22×C5—62no—
377C22×C5—62no—
378C22×C5—61no—
379C22×C5—61no—
380C22×C5C3×#12111yescharacteristic, complemented
381C22×C5D4×A412yescomplemented
382C22×C5C23×A411yescharacteristic
383C2×D5—22no—
384C2×D5—22no—
385C2×D5—22no—
386C2×D5—62no—
387C2×D5—62no—
388C2×D5—62no—
389C2×D5—122no—
390C2×D5—122no—
391C2×D5—122no—
392C2×D5—122no—
393C2×D5—122no—
394C2×D5—122no—
395C2×D5—122no—
396C2×D5—122no—
397C4×C5—21no—
398C4×C5—32no—
399C4×C5—61no—
400C4×C5D4×A412yescomplemented
401Dic5—22no—
402Dic5—62no—
403C24—21no—
404C24—21no—
405C24—31no—
406C24—32no—
407C24—31no—
408C24—61no—
409C24—61no—
410C24—102no—
411C24—102no—
412C24—102no—
413C24—152no—
414C24—152no—
415C24—302no—
416C24—302no—
417C24—302no—
418C24C3×D2012yescomplemented
419C24C22×C3×D511yescharacteristic
420C2×D4—52no—
421C2×D4—51no—
422C2×D4—152no—
423C2×D4—152no—
424C2×D4—152no—
425C2×D4—152no—
426C2×D4—154no—
427C2×D4—152no—
428C2×D4—151no—
429C2×D4—302no—
430C2×D4—302no—
431C2×D4—304no—
432C2×D4—302no—
433C2×D4—304no—
434C2×D4—302no—
435C2×D4—304no—
436C2×D4—304no—
437C2×D4—304no—
438C2×D4—304no—
439C2×D4—304no—
440C2×D4—304no—
441C2×D4—304no—
442C2×D4—304no—
443C22×C4—21no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—31no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—61no—
462C22×C4—102no—
463C22×C4—152no—
464C22×C4—152no—
465C22×C4—302no—
466C22×C4C3×C22⋊D512yescomplemented
467C22×C4D5×A411yescharacteristic, complemented
468C4⋊C4—51no—
469C4⋊C4—152no—
470C4⋊C4—152no—
471C4⋊C4—151no—
472C22⋊C4—52no—
473C22⋊C4—152no—
474C22⋊C4—152no—
475C22⋊C4—154no—
476C22⋊C4—152no—
477C3×C5—41noHall
478C22×C3—42no—
479C22×C3—41no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—402no—
483C22×C3—402no—
484C22×C3—402no—
485A4#407611yescharacteristic, direct product factor
486C3×C4—42no—
487C3×C4—81no—
488C3×C4—402no—
489C2×C5—22no—
490C2×C5—31no—
491C2×C5—31no—
492C2×C5—31no—
493C2×C5—31no—
494C2×C5—62no—
495C2×C5C2×D4×A411yescharacteristic
496C2×C5C2×D4×A411yescharacteristic
497C2×C5A4×#4111yescharacteristic
498D5—42no—
499D5—122no—
500C23—22no—
501C23—31no—
502C23—31no—
503C23—31no—
504C23—31no—
505C23—32no—
506C23—32no—
507C23—32no—
508C23—32no—
509C23—31no—
510C23—52no—
511C23—61no—
512C23—61no—
513C23—61no—
514C23—61no—
515C23—62no—
516C23—61no—
517C23—61no—
518C23—62no—
519C23—62no—
520C23—61no—
521C23—61no—
522C23—64no—
523C23—152no—
524C23—202no—
525C23—302no—
526C23—302no—
527C23—302no—
528C23—302no—
529C23—302no—
530C23—302no—
531C23—302no—
532C23—302no—
533C23—302no—
534C23—302no—
535C23—302no—
536C23—302no—
537C23—602no—
538C23—602no—
539C23—602no—
540C23—604no—
541C23C2×C3×D2011yescharacteristic
542C23C3×#70911yescharacteristic
543C23C2×C3×C22⋊D511yescharacteristic
544C23C2×D5×A411yescharacteristic
545D4—104no—
546D4—102no—
547D4—304no—
548D4—304no—
549D4—304no—
550D4—304no—
551D4—308no—
552D4—304no—
553D4—302no—
554C2×C4—22no—
555C2×C4—32no—
556C2×C4—31no—
557C2×C4—32no—
558C2×C4—32no—
559C2×C4—32no—
560C2×C4—31no—
561C2×C4—52no—
562C2×C4—62no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—62no—
566C2×C4—64no—
567C2×C4—62no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—152no—
571C2×C4—302no—
572C2×C4—302no—
573C2×C4—302no—
574C2×C4—302no—
575C2×C4C2×D5×A411yescharacteristic
576C2×C4C2×D5×A411yescharacteristic
577C2×C3—41no—
578C2×C3—41no—
579C2×C3—41no—
580C2×C3—82no—
581C2×C3—802no—
582C5A4×#12111yescharacteristic, complemented, Sylow
583C22—21no—
584C22—21no—
585C22—31no—
586C22—31no—
587C22—31no—
588C22—32no—
589C22—31no—
590C22—31no—
591C22—31no—
592C22—31no—
593C22—31no—
594C22—31no—
595C22—32no—
596C22—62no—
597C22—62no—
598C22—62no—
599C22—62no—
600C22—62no—
601C22—62no—
602C22—62no—
603C22—62no—
604C22—61no—
605C22—61no—
606C22—102no—
607C22—102no—
608C22—102no—
609C22—302no—
610C22—302no—
611C22—302no—
612C22—602no—
613C22—602no—
614C22—602no—
615C22—602no—
616C22—602no—
617C22—602no—
618C22—602no—
619C22—602no—
620C22C3×#407611yescharacteristic, complemented
621C22A4×D2012yescomplemented
622C22C22×D5×A411yescharacteristic
623C4—21no—
624C4—32no—
625C4—61no—
626C4—102no—
627C4—302no—
628C4A4×C22⋊D512yescomplemented
629C3—41noSylow
630C2—22no—
631C2—31no—
632C2—31no—
633C2—31no—
634C2—31no—
635C2—62no—
636C2—202no—
637C2—602no—
638C2C2×A4×D2011yescharacteristic
639C2A4×#70911yescharacteristic
640C2C2×A4×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, socle, cosocleC24×C5C23×C3
FittingC24×C4×C5C2×C3
hypercenterC22×C4D5×A4
hypocenterC22×C5C3×#121
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
2. C2×A4×C22⋊D5C2semidirect
3. C2×A4×D20C2semidirect
4. A4×#685C2semidirect
5. A4×C4⋊Dic5C2semidirect
6. C22×#4076C3semidirect
17. C24×C4×C5C2×C3semidirect
18. C23×C22⋊D5C2×C3semidirect
19. C23×D20C2×C3semidirect
26. C22×#685C2×C3semidirect
27. C22×C4⋊Dic5C2×C3semidirect
30. C22×C5×A4D4semidirect
39. C4×C5×A4D4semidirect
45. C22×C4×A4D5semidirect
80. #4076A4direct (N × H)
118. C24×C5C3×D4semidirect
147. C22×C4×C5C3×D4semidirect
148. C22×C4×C5C2×A4semidirect
161. C2×C22⋊D5C2×A4semidirect
175. C2×D20C2×A4semidirect
180. #685C2×A4semidirect
184. C4⋊Dic5C2×A4semidirect
187. C24×C4C3×D5semidirect
203. C5×A4#121semidirect
212. C22×A4D20semidirect
219. C4×A4C22⋊D5semidirect
380. C22×C5C3×#121semidirect
381. C22×C5D4×A4semidirect
400. C4×C5D4×A4semidirect
418. C24C3×D20semidirect
466. C22×C4C3×C22⋊D5semidirect
467. C22×C4D5×A4semidirect
485. A4#4076direct (N × H)
582. C5A4×#121semidirect
620. C22C3×#4076semidirect
621. C22A4×D20semidirect
628. C4A4×C22⋊D5semidirect

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
C2×A4×D202
A4×#6852
A4×C4⋊Dic52
C22×#40763
C3×#40764
A4×#1215