Finite Group Databasebeta

← A4×#4042 #14513993

Subgroups — A4×#4042 #14513993

Order 1920.

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

normal subgroups only

This group has 860 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
1C2×C4×D5×A4C211yesmaximal, characteristic
2C2×A4×Dic10C211yesmaximal, characteristic, complemented
3C5×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4A4×#685C211yesmaximal, characteristic
5A4×#685C211yesmaximal, characteristic
6A4×#683C211yesmaximal, characteristic, complemented
7A4×#683C211yesmaximal, characteristic, complemented
8C22×#4042C311yesmaximal, characteristic, complemented, Hall
9C22×D5×A4C2211yescharacteristic
10C2×C4×C5×A4C2211yescharacteristic
11C2×C4×C5×A4C2211yescharacteristic
12C2×C4×C5×A4C2211yescharacteristic
13C2×A4×Dic5C2211yescharacteristic
14C2×A4×Dic5C2211yescharacteristic
15C2×A4×Dic5C2211yescharacteristic
16C4×D5×A4—22no—
17A4×Dic10—22no—
18C3×#4042—41nomaximal
19A4×C22⋊Q8—51nomaximal, Hall
20C23×C4×D5C2×C311yescharacteristic
21C23×Dic10C2×C311yescharacteristic, complemented
22C22×C5×C4⋊C4C2×C311yescharacteristic, complemented
23C22×#685C2×C311yescharacteristic
24C22×#685C2×C311yescharacteristic
25C22×#683C2×C311yescharacteristic, complemented
26C22×#683C2×C311yescharacteristic, complemented
27C2×#4042—31no—
28C2×#4042—31no—
29C2×#4042—31no—
30C2×#4042—31no—
31C2×#4042—31no—
32C2×#4042—31no—
33C2×#4042—31no—
34C2×#4042—31no—
35C22×C5×A4C2311yescharacteristic
36C2×D5×A4—21no—
37C2×D5×A4—21no—
38C2×D5×A4Q812yescomplemented
39C2×C3×C4×D5—41no—
40C2×C3×Dic10—41no—
41C4×C5×A4—21no—
42C4×C5×A4—21no—
43C4×C5×A4—21no—
44A4×Dic5—21no—
45A4×Dic5—21no—
46A4×Dic5D412yes—
47C3×C5×C4⋊C4—41no—
48C3×#685—41no—
49C3×#685—41no—
50C3×#683—41no—
51C3×#683—41no—
52C2×Q8×A4—51no—
53C22×C4×A4—51no—
54A4×C4⋊C4—51no—
55A4×C4⋊C4—51no—
56A4×C4⋊C4D511yescharacteristic, complemented
57A4×C22⋊C4—51no—
58A4×C22⋊C4—51no—
59C24×D5C22×C311yescharacteristic
60C23×C4×C5C22×C311yescharacteristic
61C23×C4×C5C22×C311yescharacteristic
62C23×C4×C5C22×C311yescharacteristic
63C23×Dic5C22×C311yescharacteristic
64C23×Dic5C22×C311yescharacteristic
65C23×Dic5C22×C311yescharacteristic
66C22×C4×D5—22no—
67C22×C4×D5—31no—
68C22×C4×D5—31no—
69C22×C4×D5—31no—
70C22×C4×D5—31no—
71C22×C4×D5—62no—
72C22×Dic10—22no—
73C22×Dic10—31no—
74C22×Dic10—31no—
75C22×Dic10—31no—
76C22×Dic10—31no—
77C22×Dic10—62no—
78C2×C5×C4⋊C4—31no—
79C2×C5×C4⋊C4—31no—
80C2×C5×C4⋊C4—31no—
81C2×C5×C4⋊C4—31no—
82C2×#685—31no—
83C2×#685—31no—
84C2×#685—31no—
85C2×#685—31no—
86C2×#685—31no—
87C2×#685—31no—
88C2×#685—31no—
89C2×#685—31no—
90C2×#683—31no—
91C2×#683—31no—
92C2×#683—31no—
93C2×#683—31no—
94C2×#683—31no—
95C2×#683—31no—
96C2×#683—31no—
97C2×#683—31no—
98#4042—32no—
99#4042—32no—
100#4042—32no—
101#4042—32no—
102#4042—32no—
103#4042—32no—
104#4042—32no—
105#4042—31no—
106#4042—31no—
107#4042—31no—
108#4042—31no—
109#4042—31no—
110#4042—31no—
111#4042—31no—
112#4042A411yescharacteristic, direct product factor
113C22×C22⋊Q8—51noSylow
114C22×C3×D5—41no—
115C2×C5×A4C2×D411yescharacteristic
116C2×C5×A4C2×Q811yescharacteristic
117C2×C5×A4#4111yescharacteristic
118D5×A4—22no—
119C2×C3×C4×C5—41no—
120C2×C3×C4×C5—41no—
121C2×C3×C4×C5—41no—
122C2×C3×Dic5—41no—
123C2×C3×Dic5—41no—
124C2×C3×Dic5—41no—
125C3×C4×D5—82no—
126C3×Dic10—82no—
127C23×A4—51no—
128Q8×A4—102no—
129C2×C4×A4—51no—
130C2×C4×A4—51no—
131C2×C4×A4—51no—
132C2×C4×A4—102no—
133C2×C4×A4C2×D511yescharacteristic
134C2×C4×A4C2×D511yescharacteristic
135C2×C4×A4C2×D511yescharacteristic
136C3×C22⋊Q8—201no—
137C24×C5C23×C311yescharacteristic
138C23×D5—21no—
139C23×D5—21no—
140C23×D5—31no—
141C23×D5—31no—
142C23×D5—32no—
143C23×D5—61no—
144C23×D5—61no—
145C23×D5C3×Q812yescomplemented
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5—31no—
150C22×C4×C5—31no—
151C22×C4×C5—31no—
152C22×C4×C5—31no—
153C22×C4×C5—31no—
154C22×C4×C5—31no—
155C22×C4×C5—61no—
156C22×C4×C5—61no—
157C22×C4×C5—61no—
158C22×Dic5—21no—
159C22×Dic5—21no—
160C22×Dic5—31no—
161C22×Dic5—31no—
162C22×Dic5—32no—
163C22×Dic5—31no—
164C22×Dic5—31no—
165C22×Dic5—31no—
166C22×Dic5—31no—
167C22×Dic5—61no—
168C22×Dic5—61no—
169C22×Dic5C3×D412yes—
170C2×C4×D5—32no—
171C2×C4×D5—31no—
172C2×C4×D5—31no—
173C2×C4×D5—31no—
174C2×C4×D5—62no—
175C2×C4×D5—62no—
176C2×C4×D5—62no—
177C2×C4×D5—62no—
178C2×C4×D5—62no—
179C2×C4×D5—62no—
180C2×C4×D5—62no—
181C2×C4×D5—62no—
182C2×C4×D5C2×A411yescharacteristic
183C2×Dic10—32no—
184C2×Dic10—31no—
185C2×Dic10—31no—
186C2×Dic10—31no—
187C2×Dic10—62no—
188C2×Dic10—62no—
189C2×Dic10—62no—
190C2×Dic10—62no—
191C2×Dic10—62no—
192C2×Dic10—62no—
193C2×Dic10—62no—
194C2×Dic10—62no—
195C2×Dic10C2×A411yescharacteristic, complemented
196C5×C4⋊C4—32no—
197C5×C4⋊C4—31no—
198C5×C4⋊C4—31no—
199C5×C4⋊C4—31no—
200C5×C4⋊C4C2×A411yescharacteristic, complemented
201#685—32no—
202#685—32no—
203#685—31no—
204#685—31no—
205#685—31no—
206#685—31no—
207#685—31no—
208#685—31no—
209#685C2×A411yescharacteristic
210#685C2×A411yescharacteristic
211#683—32no—
212#683—32no—
213#683—31no—
214#683—31no—
215#683—31no—
216#683—31no—
217#683—31no—
218#683—31no—
219#683C2×A411yescharacteristic, complemented
220#683C2×A411yescharacteristic, complemented
221C23×Q8—51no—
222C24×C4—51no—
223C2×C22⋊Q8—151no—
224C2×C22⋊Q8—151no—
225C2×C22⋊Q8—151no—
226C2×C22⋊Q8—151no—
227C2×C22⋊Q8—151no—
228C2×C22⋊Q8—151no—
229C2×C22⋊Q8—151no—
230C2×C22⋊Q8—151no—
231C22×C4⋊C4—51no—
232C22×C4⋊C4—51no—
233C22×C4⋊C4C3×D511yescharacteristic, complemented
234C22×C22⋊C4—51no—
235C22×C22⋊C4—51no—
236C22×C3×C5—41no—
237C2×C3×D5—42no—
238C2×C3×D5—81no—
239C2×C3×D5—81no—
240C5×A4C22⋊Q811yescharacteristic, complemented
241C3×C4×C5—81no—
242C3×C4×C5—81no—
243C3×C4×C5—81no—
244C3×Dic5—42no—
245C3×Dic5—81no—
246C3×Dic5—81no—
247C22×A4—52no—
248C22×A4—101no—
249C22×A4—101no—
250C22×A4C22×D511yescharacteristic
251C2×C3×Q8—201no—
252C22×C3×C4—201no—
253C4×A4—21no—
254C4×A4—21no—
255C4×A4—21no—
256C4×A4—52no—
257C4×A4—101no—
258C4×A4—101no—
259C3×C4⋊C4—41no—
260C3×C4⋊C4—201no—
261C3×C4⋊C4—201no—
262C3×C22⋊C4—201no—
263C3×C22⋊C4—201no—
264C23×C5—31no—
265C23×C5—31no—
266C23×C5—31no—
267C23×C5—31no—
268C23×C5C2×C3×D411yescharacteristic
269C23×C5C2×C3×Q811yescharacteristic
270C23×C5C3×#4111yescharacteristic
271C22×D5—22no—
272C22×D5—32no—
273C22×D5—31no—
274C22×D5—32no—
275C22×D5—32no—
276C22×D5—32no—
277C22×D5—61no—
278C22×D5—61no—
279C22×D5—61no—
280C22×D5—61no—
281C22×D5—64no—
282C22×D5—61no—
283C22×D5—61no—
284C22×D5—62no—
285C22×D5—62no—
286C22×D5—61no—
287C22×D5—61no—
288C22×D5—62no—
289C22×D5C22×A411yescharacteristic
290C2×C4×C5—31no—
291C2×C4×C5—31no—
292C2×C4×C5—31no—
293C2×C4×C5—61no—
294C2×C4×C5—61no—
295C2×C4×C5—61no—
296C2×C4×C5—61no—
297C2×C4×C5—61no—
298C2×C4×C5—61no—
299C2×C4×C5—61no—
300C2×C4×C5—61no—
301C2×C4×C5—61no—
302C2×C4×C5—61no—
303C2×C4×C5—61no—
304C2×C4×C5—61no—
305C2×C4×C5C22×A411yescharacteristic
306C2×C4×C5C22×A411yescharacteristic
307C2×C4×C5C22×A411yescharacteristic
308C2×Dic5—32no—
309C2×Dic5—31no—
310C2×Dic5—32no—
311C2×Dic5—32no—
312C2×Dic5—32no—
313C2×Dic5—31no—
314C2×Dic5—31no—
315C2×Dic5—61no—
316C2×Dic5—61no—
317C2×Dic5—61no—
318C2×Dic5—61no—
319C2×Dic5—61no—
320C2×Dic5—61no—
321C2×Dic5—61no—
322C2×Dic5—61no—
323C2×Dic5C22×A411yescharacteristic
324C2×Dic5C22×A411yescharacteristic
325C2×Dic5C22×A411yescharacteristic
326C4×D5—22no—
327C4×D5—62no—
328C4×D5—62no—
329C4×D5—64no—
330C4×D5—62no—
331Dic10—22no—
332Dic10—62no—
333Dic10—62no—
334Dic10—64no—
335Dic10—62no—
336C25—51no—
337C22×Q8—102no—
338C22×Q8—151no—
339C22×Q8—151no—
340C22×Q8—151no—
341C22×Q8—151no—
342C22×Q8—302no—
343C23×C4—51no—
344C23×C4—51no—
345C23×C4—51no—
346C23×C4—102no—
347C23×C4—151no—
348C23×C4—151no—
349C23×C4—151no—
350C23×C4—151no—
351C23×C4—302no—
352C23×C4C2×C3×D511yescharacteristic
353C23×C4C2×C3×D511yescharacteristic
354C23×C4C2×C3×D511yescharacteristic
355C22⋊Q8—51no—
356C22⋊Q8—151no—
357C22⋊Q8—151no—
358C22⋊Q8—151no—
359C22⋊Q8—151no—
360C22⋊Q8—151no—
361C22⋊Q8—151no—
362C22⋊Q8—152no—
363C22⋊Q8—152no—
364C22⋊Q8—152no—
365C22⋊Q8—152no—
366C22⋊Q8—152no—
367C22⋊Q8—152no—
368C22⋊Q8—152no—
369C22⋊Q8—151no—
370C2×C4⋊C4—31no—
371C2×C4⋊C4—31no—
372C2×C4⋊C4—31no—
373C2×C4⋊C4—31no—
374C2×C4⋊C4—151no—
375C2×C4⋊C4—151no—
376C2×C4⋊C4—151no—
377C2×C4⋊C4—151no—
378C2×C4⋊C4—151no—
379C2×C4⋊C4—151no—
380C2×C4⋊C4—151no—
381C2×C4⋊C4—151no—
382C2×C22⋊C4—151no—
383C2×C22⋊C4—151no—
384C2×C22⋊C4—151no—
385C2×C22⋊C4—151no—
386C2×C22⋊C4—151no—
387C2×C22⋊C4—151no—
388C2×C22⋊C4—151no—
389C2×C22⋊C4—151no—
390C2×C3×C5—41no—
391C2×C3×C5—41no—
392C2×C3×C5—41no—
393C3×D5—82no—
394C23×C3—201no—
395C2×A4—102no—
396C2×A4#70911yescharacteristic
397C2×A4D4×D511yescharacteristic
398C2×A4Q8×D511yescharacteristic
399C3×Q8—402no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—201no—
404C2×C3×C4—201no—
405C2×C3×C4—201no—
406C2×C3×C4—402no—
407C22×C5—31no—
408C22×C5—31no—
409C22×C5—31no—
410C22×C5—32no—
411C22×C5—31no—
412C22×C5—31no—
413C22×C5—31no—
414C22×C5—31no—
415C22×C5—31no—
416C22×C5—31no—
417C22×C5C3×C22⋊Q811yescharacteristic, complemented
418C22×C5C23×A411yescharacteristic
419C2×D5—21no—
420C2×D5—21no—
421C2×D5—32no—
422C2×D5—62no—
423C2×D5—62no—
424C2×D5—62no—
425C2×D5—62no—
426C2×D5—62no—
427C2×D5—62no—
428C2×D5—61no—
429C2×D5—61no—
430C2×D5—62no—
431C2×D5—62no—
432C2×D5Q8×A412yescomplemented
433C4×C5—21no—
434C4×C5—21no—
435C4×C5—21no—
436C4×C5—61no—
437C4×C5—61no—
438C4×C5—61no—
439Dic5—21no—
440Dic5—21no—
441Dic5—32no—
442Dic5—61no—
443Dic5—61no—
444Dic5D4×A412yes—
445C24—52no—
446C24—101no—
447C24—101no—
448C24—151no—
449C24—151no—
450C24—152no—
451C24—301no—
452C24—301no—
453C24C22×C3×D511yescharacteristic
454C2×Q8—51no—
455C2×Q8—151no—
456C2×Q8—151no—
457C2×Q8—152no—
458C2×Q8—151no—
459C2×Q8—302no—
460C2×Q8—302no—
461C2×Q8—302no—
462C2×Q8—302no—
463C2×Q8—302no—
464C2×Q8—302no—
465C2×Q8—302no—
466C2×Q8—302no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—51no—
477C22×C4—52no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—101no—
482C22×C4—101no—
483C22×C4—151no—
484C22×C4—151no—
485C22×C4—152no—
486C22×C4—151no—
487C22×C4—151no—
488C22×C4—151no—
489C22×C4—151no—
490C22×C4—152no—
491C22×C4—151no—
492C22×C4—151no—
493C22×C4—151no—
494C22×C4—302no—
495C22×C4—302no—
496C22×C4—302no—
497C22×C4—302no—
498C22×C4—302no—
499C22×C4—302no—
500C22×C4—302no—
501C22×C4—302no—
502C22×C4—301no—
503C22×C4—301no—
504C4⋊C4—32no—
505C4⋊C4—31no—
506C4⋊C4—31no—
507C4⋊C4—31no—
508C4⋊C4—51no—
509C4⋊C4—51no—
510C4⋊C4—151no—
511C4⋊C4—151no—
512C4⋊C4—151no—
513C4⋊C4—151no—
514C4⋊C4—152no—
515C4⋊C4—152no—
516C4⋊C4—151no—
517C4⋊C4—151no—
518C4⋊C4D5×A411yescharacteristic, complemented
519C22⋊C4—51no—
520C22⋊C4—51no—
521C22⋊C4—151no—
522C22⋊C4—151no—
523C22⋊C4—151no—
524C22⋊C4—151no—
525C22⋊C4—152no—
526C22⋊C4—152no—
527C22⋊C4—151no—
528C22⋊C4—151no—
529C3×C5—41noHall
530C22×C3—41no—
531C22×C3—202no—
532C22×C3—401no—
533C22×C3—401no—
534A4#404211yescharacteristic, direct product factor
535C3×C4—81no—
536C3×C4—81no—
537C3×C4—81no—
538C3×C4—202no—
539C3×C4—401no—
540C3×C4—401no—
541C2×C5—31no—
542C2×C5—31no—
543C2×C5—31no—
544C2×C5—31no—
545C2×C5C2×D4×A411yescharacteristic
546C2×C5C2×Q8×A411yescharacteristic
547C2×C5A4×#4111yescharacteristic
548D5—22no—
549D5—62no—
550C23—31no—
551C23—31no—
552C23—31no—
553C23—31no—
554C23—51no—
555C23—102no—
556C23—152no—
557C23—152no—
558C23—152no—
559C23—152no—
560C23—151no—
561C23—301no—
562C23—301no—
563C23—301no—
564C23—301no—
565C23—302no—
566C23—301no—
567C23—301no—
568C23—302no—
569C23—302no—
570C23—301no—
571C23—301no—
572C23—304no—
573C23C3×#70911yescharacteristic
574C23C3×D4×D511yescharacteristic
575C23C3×Q8×D511yescharacteristic
576Q8—102no—
577Q8—302no—
578Q8—302no—
579Q8—302no—
580Q8—304no—
581C2×C4—31no—
582C2×C4—31no—
583C2×C4—31no—
584C2×C4—51no—
585C2×C4—51no—
586C2×C4—51no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—102no—
600C2×C4—152no—
601C2×C4—152no—
602C2×C4—152no—
603C2×C4—151no—
604C2×C4—152no—
605C2×C4—151no—
606C2×C4—151no—
607C2×C4—302no—
608C2×C4—302no—
609C2×C4—302no—
610C2×C4—304no—
611C2×C4—301no—
612C2×C4—301no—
613C2×C4—301no—
614C2×C4—301no—
615C2×C4—301no—
616C2×C4—301no—
617C2×C4—301no—
618C2×C4—301no—
619C2×C4C2×D5×A411yescharacteristic
620C2×C4C2×D5×A411yescharacteristic
621C2×C4C2×D5×A411yescharacteristic
622C2×C3—41no—
623C2×C3—41no—
624C2×C3—41no—
625C2×C3—402no—
626C5A4×C22⋊Q811yescharacteristic, complemented, Sylow
627C22—31no—
628C22—31no—
629C22—31no—
630C22—32no—
631C22—31no—
632C22—31no—
633C22—31no—
634C22—31no—
635C22—31no—
636C22—31no—
637C22—52no—
638C22—101no—
639C22—101no—
640C22—152no—
641C22—302no—
642C22—302no—
643C22—302no—
644C22—302no—
645C22—302no—
646C22—302no—
647C22—302no—
648C22—302no—
649C22—301no—
650C22—301no—
651C22C3×#404211yescharacteristic, complemented
652C22C22×D5×A411yescharacteristic
653C4—21no—
654C4—21no—
655C4—21no—
656C4—52no—
657C4—61no—
658C4—61no—
659C4—61no—
660C4—101no—
661C4—101no—
662C4—152no—
663C4—301no—
664C4—301no—
665C3—41noSylow
666C2—31no—
667C2—31no—
668C2—31no—
669C2—31no—
670C2—102no—
671C2—302no—
672C2A4×#70911yescharacteristic
673C2D4×D5×A411yescharacteristic
674C2Q8×D5×A411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, socle, cosocleC24×C5C23×C3
FittingC22×C5×C4⋊C4C2×C3
prime radicalC22×D5×A4C22
hypercenterC4⋊C4D5×A4
hypocenterC22×C5C3×C22⋊Q8
LCSG ▷ C24×C5 ▷ C22×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×A4×Dic10C2semidirect
3. C5×A4×C4⋊C4C2semidirect
6. A4×#683C2semidirect
7. A4×#683C2semidirect
8. C22×#4042C3semidirect
21. C23×Dic10C2×C3semidirect
22. C22×C5×C4⋊C4C2×C3semidirect
25. C22×#683C2×C3semidirect
26. C22×#683C2×C3semidirect
38. C2×D5×A4Q8semidirect
56. A4×C4⋊C4D5semidirect
112. #4042A4direct (N × H)
145. C23×D5C3×Q8semidirect
195. C2×Dic10C2×A4semidirect
200. C5×C4⋊C4C2×A4semidirect
219. #683C2×A4semidirect
220. #683C2×A4semidirect
233. C22×C4⋊C4C3×D5semidirect
240. C5×A4C22⋊Q8semidirect
417. C22×C5C3×C22⋊Q8semidirect
432. C2×D5Q8×A4semidirect
518. C4⋊C4D5×A4semidirect
534. A4#4042direct (N × H)
626. C5A4×C22⋊Q8semidirect
651. C22C3×#4042semidirect

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×Dic102
C5×A4×C4⋊C42
A4×#6852
A4×#6832
C22×#40423
C3×#40424
A4×C22⋊Q85