Finite Group Databasebeta

← A4×#4039 #14513999

Subgroups — A4×#4039 #14513999

Order 1920.

1156 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 1156 conjugacy classes of subgroups — too many to draw. Only its 129 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
2C2×C4×D5×A4C212yesmaximal, complemented
3C5×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4A4×#685C212yesmaximal, complemented
5C4×A4×Dic5C211yesmaximal, characteristic, complemented
6C22×#4039C311yesmaximal, characteristic, complemented, Hall
7C22×D5×A4C2212yes—
8C2×C4×C5×A4C2212yes—
9C2×C4×C5×A4C2211yescharacteristic
10C2×A4×Dic5C2211yescharacteristic
11C2×A4×Dic5C2211yescharacteristic
12A4×D20C414yescomplemented
13C4×D5×A4—24no—
14C3×#4039—41nomaximal
15C4×D4×A4—51nomaximal, Hall
16C23×D20C2×C311yescharacteristic
17C23×C4×D5C2×C312yescomplemented
18C22×C5×C4⋊C4C2×C311yescharacteristic, complemented
19C22×#685C2×C312yescomplemented
20C22×C4×Dic5C2×C311yescharacteristic, complemented
21C2×#4039—31no—
22C2×#4039—31no—
23C2×#4039—32no—
24C2×#4039—31no—
25C2×#4039—32no—
26C2×#4039—31no—
27C22×C5×A4C2311yescharacteristic
28C2×D5×A4—22no—
29C2×D5×A4—22no—
30C2×D5×A4C2×C414yescomplemented
31C2×C3×D20—41no—
32C2×C3×C4×D5—42no—
33C4×C5×A4—22no—
34C4×C5×A4C2×C412yescomplemented
35A4×Dic5—21no—
36A4×Dic5D412yescomplemented
37C3×C5×C4⋊C4—41no—
38C3×#685—42no—
39C3×C4×Dic5—41no—
40C2×D4×A4—51no—
41C22×C4×A4—52no—
42A4×C4⋊C4D511yescharacteristic, complemented
43A4×C22⋊C4—52no—
44C42×A4—51no—
45C24×D5C22×C312yes—
46C23×C4×C5C22×C312yes—
47C23×C4×C5C22×C311yescharacteristic
48C23×Dic5C22×C311yescharacteristic
49C23×Dic5C22×C311yescharacteristic
50C22×D20—31no—
51C22×D20—32no—
52C22×D20—31no—
53C22×D20—34no—
54C22×D20C3×C414yescomplemented
55C22×C4×D5—24no—
56C22×C4×D5—32no—
57C22×C4×D5—32no—
58C22×C4×D5—32no—
59C22×C4×D5—32no—
60C22×C4×D5—64no—
61C2×C5×C4⋊C4—31no—
62C2×C5×C4⋊C4—31no—
63C2×C5×C4⋊C4—32no—
64C2×#685—32no—
65C2×#685—32no—
66C2×#685—32no—
67C2×#685—32no—
68C2×C4×Dic5—31no—
69C2×C4×Dic5—31no—
70C2×C4×Dic5—31no—
71C2×C4×Dic5—31no—
72#4039—34no—
73#4039—34no—
74#4039—32no—
75#4039—32no—
76#4039—32no—
77#4039—31no—
78#4039—32no—
79#4039—31no—
80#4039—32no—
81#4039—31no—
82#4039A411yescharacteristic, direct product factor
83C22×C4×D4—51noSylow
84C22×C3×D5—42no—
85C2×C5×A4C22×C411yescharacteristic
86C2×C5×A4C2×D411yescharacteristic
87C2×C5×A4#4111yescharacteristic
88D5×A4—24no—
89C2×C3×C4×C5—41no—
90C2×C3×C4×C5—42no—
91C2×C3×Dic5—41no—
92C2×C3×Dic5—41no—
93C3×D20—44no—
94C3×C4×D5—84no—
95C23×A4—52no—
96D4×A4—54no—
97C2×C4×A4—51no—
98C2×C4×A4—51no—
99C2×C4×A4—104no—
100C2×C4×A4C2×D512yes—
101C2×C4×A4C2×D511yescharacteristic
102C3×C4×D4—201no—
103C24×C5C23×C311yescharacteristic
104C23×D5—22no—
105C23×D5—22no—
106C23×D5—32no—
107C23×D5—32no—
108C23×D5—34no—
109C23×D5—62no—
110C23×D5—62no—
111C23×D5C2×C3×C414yescomplemented
112C22×C4×C5—22no—
113C22×C4×C5—31no—
114C22×C4×C5—31no—
115C22×C4×C5—32no—
116C22×C4×C5—32no—
117C22×C4×C5—32no—
118C22×C4×C5—62no—
119C22×C4×C5C2×C3×C412yescomplemented
120C22×Dic5—21no—
121C22×Dic5—31no—
122C22×Dic5—31no—
123C22×Dic5—32no—
124C22×Dic5—31no—
125C22×Dic5—31no—
126C22×Dic5—61no—
127C22×Dic5C3×D412yescomplemented
128C2×D20—38no—
129C2×D20—34no—
130C2×D20—32no—
131C2×D20—32no—
132C2×D20—31no—
133C2×D20—34no—
134C2×D20—34no—
135C2×D20—38no—
136C2×D20—34no—
137C2×D20C2×A411yescharacteristic
138C2×C4×D5—34no—
139C2×C4×D5—32no—
140C2×C4×D5—32no—
141C2×C4×D5—32no—
142C2×C4×D5—64no—
143C2×C4×D5—64no—
144C2×C4×D5—64no—
145C2×C4×D5—64no—
146C2×C4×D5—64no—
147C2×C4×D5—64no—
148C2×C4×D5—64no—
149C2×C4×D5—64no—
150C2×C4×D5C2×A412yescomplemented
151C5×C4⋊C4—32no—
152C5×C4⋊C4—32no—
153C5×C4⋊C4—31no—
154C5×C4⋊C4C2×A411yescharacteristic, complemented
155#685—34no—
156#685—32no—
157#685—32no—
158#685—32no—
159#685C2×A412yescomplemented
160C4×Dic5—32no—
161C4×Dic5—31no—
162C4×Dic5—31no—
163C4×Dic5—31no—
164C4×Dic5C2×A411yescharacteristic, complemented
165C23×D4—51no—
166C24×C4—52no—
167C2×C4×D4—151no—
168C2×C4×D4—151no—
169C2×C4×D4—152no—
170C2×C4×D4—151no—
171C2×C4×D4—152no—
172C2×C4×D4—151no—
173C22×C4⋊C4C3×D511yescharacteristic, complemented
174C22×C22⋊C4—52no—
175C22×C42—51no—
176C22×C3×C5—41no—
177C2×C3×D5—44no—
178C2×C3×D5—82no—
179C2×C3×D5—82no—
180C5×A4C4×D411yescharacteristic, complemented
181C3×C4×C5—42no—
182C3×C4×C5—82no—
183C3×Dic5—42no—
184C3×Dic5—81no—
185C22×A4—54no—
186C22×A4—102no—
187C22×A4—102no—
188C22×A4C22×D511yescharacteristic
189C2×C3×D4—201no—
190C22×C3×C4—202no—
191C4×A4—22no—
192C4×A4—52no—
193C4×A4—101no—
194C4×A4C4×D512yescomplemented
195C3×C4⋊C4—41no—
196C3×C22⋊C4—202no—
197C3×C42—201no—
198C23×C5—31no—
199C23×C5—31no—
200C23×C5—31no—
201C23×C5—31no—
202C23×C5C22×C3×C411yescharacteristic
203C23×C5C2×C3×D411yescharacteristic
204C23×C5C3×#4111yescharacteristic
205C22×D5—24no—
206C22×D5—34no—
207C22×D5—32no—
208C22×D5—34no—
209C22×D5—34no—
210C22×D5—34no—
211C22×D5—62no—
212C22×D5—62no—
213C22×D5—62no—
214C22×D5—62no—
215C22×D5—68no—
216C22×D5—62no—
217C22×D5—62no—
218C22×D5—64no—
219C22×D5—64no—
220C22×D5—62no—
221C22×D5—62no—
222C22×D5—64no—
223C22×D5C22×A412yes—
224C2×C4×C5—32no—
225C2×C4×C5—31no—
226C2×C4×C5—32no—
227C2×C4×C5—32no—
228C2×C4×C5—32no—
229C2×C4×C5—32no—
230C2×C4×C5—62no—
231C2×C4×C5—62no—
232C2×C4×C5—62no—
233C2×C4×C5—62no—
234C2×C4×C5C22×A412yes—
235C2×C4×C5C22×A411yescharacteristic
236C2×Dic5—32no—
237C2×Dic5—31no—
238C2×Dic5—32no—
239C2×Dic5—32no—
240C2×Dic5—32no—
241C2×Dic5—31no—
242C2×Dic5—61no—
243C2×Dic5—61no—
244C2×Dic5—61no—
245C2×Dic5—61no—
246C2×Dic5C22×A411yescharacteristic
247C2×Dic5C22×A411yescharacteristic
248D20—38no—
249D20—38no—
250D20—34no—
251D20C4×A414yescomplemented
252C4×D5—24no—
253C4×D5—64no—
254C4×D5—64no—
255C4×D5—68no—
256C4×D5—64no—
257C25—52no—
258C22×D4—54no—
259C22×D4—151no—
260C22×D4—151no—
261C22×D4—152no—
262C22×D4—154no—
263C23×C4—51no—
264C23×C4—51no—
265C23×C4—104no—
266C23×C4—152no—
267C23×C4—152no—
268C23×C4—152no—
269C23×C4—152no—
270C23×C4—304no—
271C23×C4C2×C3×D512yes—
272C23×C4C2×C3×D511yescharacteristic
273C4×D4—51no—
274C4×D4—151no—
275C4×D4—152no—
276C4×D4—152no—
277C4×D4—151no—
278C4×D4—152no—
279C4×D4—152no—
280C4×D4—154no—
281C4×D4—154no—
282C4×D4—152no—
283C4×D4—151no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—32no—
287C2×C22⋊C4—152no—
288C2×C22⋊C4—152no—
289C2×C22⋊C4—152no—
290C2×C22⋊C4—152no—
291C2×C42—151no—
292C2×C42—151no—
293C2×C42—151no—
294C2×C42—151no—
295C2×C3×C5—41no—
296C2×C3×C5—41no—
297C2×C3×C5—41no—
298C3×D5—84no—
299C23×C3—202no—
300C2×A4—104no—
301C2×A4C2×C4×D511yescharacteristic
302C2×A4D4×D511yescharacteristic
303C2×A4#71311yescharacteristic
304C3×D4—204no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—201no—
308C2×C3×C4—201no—
309C2×C3×C4—404no—
310C22×C5—31no—
311C22×C5—31no—
312C22×C5—31no—
313C22×C5—32no—
314C22×C5—31no—
315C22×C5—31no—
316C22×C5—31no—
317C22×C5—31no—
318C22×C5—31no—
319C22×C5—31no—
320C22×C5C3×C4×D411yescharacteristic, complemented
321C22×C5C23×A411yescharacteristic
322C2×D5—22no—
323C2×D5—22no—
324C2×D5—34no—
325C2×D5—64no—
326C2×D5—64no—
327C2×D5—64no—
328C2×D5—64no—
329C2×D5—64no—
330C2×D5—64no—
331C2×D5—62no—
332C2×D5—62no—
333C2×D5—64no—
334C2×D5—64no—
335C2×D5C2×C4×A414yescomplemented
336C4×C5—22no—
337C4×C5—32no—
338C4×C5—62no—
339C4×C5C2×C4×A412yescomplemented
340Dic5—21no—
341Dic5—32no—
342Dic5—61no—
343Dic5D4×A412yescomplemented
344C24—54no—
345C24—102no—
346C24—102no—
347C24—152no—
348C24—152no—
349C24—154no—
350C24—302no—
351C24—302no—
352C24C22×C3×D511yescharacteristic
353C2×D4—51no—
354C2×D4—154no—
355C2×D4—154no—
356C2×D4—152no—
357C2×D4—152no—
358C2×D4—158no—
359C2×D4—154no—
360C2×D4—154no—
361C2×D4—158no—
362C2×D4—151no—
363C22×C4—22no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—32no—
367C22×C4—32no—
368C22×C4—32no—
369C22×C4—52no—
370C22×C4—52no—
371C22×C4—62no—
372C22×C4—101no—
373C22×C4—152no—
374C22×C4—152no—
375C22×C4—154no—
376C22×C4—151no—
377C22×C4—151no—
378C22×C4—151no—
379C22×C4—152no—
380C22×C4—152no—
381C22×C4—151no—
382C22×C4—304no—
383C22×C4—304no—
384C22×C4—304no—
385C22×C4—304no—
386C22×C4—304no—
387C22×C4—304no—
388C22×C4—304no—
389C22×C4—304no—
390C22×C4—301no—
391C22×C4C3×C4×D512yescomplemented
392C4⋊C4—32no—
393C4⋊C4—32no—
394C4⋊C4—31no—
395C4⋊C4D5×A411yescharacteristic, complemented
396C22⋊C4—52no—
397C22⋊C4—152no—
398C22⋊C4—152no—
399C22⋊C4—154no—
400C22⋊C4—152no—
401C42—51no—
402C42—151no—
403C42—151no—
404C42—152no—
405C42—151no—
406C3×C5—41noHall
407C22×C3—41no—
408C22×C3—204no—
409C22×C3—402no—
410C22×C3—402no—
411A4#403911yescharacteristic, direct product factor
412C3×C4—42no—
413C3×C4—82no—
414C3×C4—202no—
415C3×C4—401no—
416C2×C5—31no—
417C2×C5—31no—
418C2×C5—31no—
419C2×C5—31no—
420C2×C5C22×C4×A411yescharacteristic
421C2×C5C2×D4×A411yescharacteristic
422C2×C5A4×#4111yescharacteristic
423D5—24no—
424D5—64no—
425C23—31no—
426C23—31no—
427C23—31no—
428C23—31no—
429C23—52no—
430C23—104no—
431C23—154no—
432C23—154no—
433C23—154no—
434C23—154no—
435C23—152no—
436C23—302no—
437C23—302no—
438C23—302no—
439C23—302no—
440C23—304no—
441C23—302no—
442C23—302no—
443C23—304no—
444C23—304no—
445C23—302no—
446C23—302no—
447C23—308no—
448C23C2×C3×C4×D511yescharacteristic
449C23C3×D4×D511yescharacteristic
450C23C3×#71311yescharacteristic
451D4—54no—
452D4—158no—
453D4—158no—
454D4—154no—
455C2×C4—32no—
456C2×C4—31no—
457C2×C4—32no—
458C2×C4—32no—
459C2×C4—32no—
460C2×C4—32no—
461C2×C4—51no—
462C2×C4—51no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—104no—
468C2×C4—152no—
469C2×C4—152no—
470C2×C4—152no—
471C2×C4—151no—
472C2×C4—152no—
473C2×C4—151no—
474C2×C4—304no—
475C2×C4—304no—
476C2×C4—304no—
477C2×C4—308no—
478C2×C4—301no—
479C2×C4—301no—
480C2×C4—301no—
481C2×C4—301no—
482C2×C4C2×D5×A412yes—
483C2×C4C2×D5×A411yescharacteristic
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—41no—
487C2×C3—404no—
488C5C4×D4×A411yescharacteristic, complemented, Sylow
489C22—31no—
490C22—31no—
491C22—31no—
492C22—32no—
493C22—31no—
494C22—31no—
495C22—31no—
496C22—31no—
497C22—31no—
498C22—31no—
499C22—54no—
500C22—102no—
501C22—102no—
502C22—154no—
503C22—304no—
504C22—304no—
505C22—304no—
506C22—304no—
507C22—304no—
508C22—304no—
509C22—304no—
510C22—304no—
511C22—302no—
512C22—302no—
513C22C3×#403911yescharacteristic, complemented
514C22C22×D5×A411yescharacteristic
515C4—22no—
516C4—32no—
517C4—52no—
518C4—62no—
519C4—101no—
520C4—152no—
521C4—301no—
522C4C4×D5×A412yescomplemented
523C3—41noSylow
524C2—31no—
525C2—31no—
526C2—31no—
527C2—31no—
528C2—104no—
529C2—304no—
530C2C2×C4×D5×A411yescharacteristic
531C2D4×D5×A411yescharacteristic
532C2A4×#71311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutatorC23×C5C22×C3×C4
FittingC22×C5×C4⋊C4C2×C3
socle, cosocleC24×C5C23×C3
prime radicalC2×A4×D20C2
hypercenterC4⋊C4D5×A4
hypocenterC22×C5C3×C4×D4
LCSG ▷ C23×C5 ▷ C22×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×D5×A4C2semidirect
3. C5×A4×C4⋊C4C2semidirect
4. A4×#685C2semidirect
5. C4×A4×Dic5C2semidirect
6. C22×#4039C3semidirect
12. A4×D20C4semidirect
17. C23×C4×D5C2×C3semidirect
18. C22×C5×C4⋊C4C2×C3semidirect
19. C22×#685C2×C3semidirect
20. C22×C4×Dic5C2×C3semidirect
30. C2×D5×A4C2×C4semidirect
34. C4×C5×A4C2×C4semidirect
36. A4×Dic5D4semidirect
42. A4×C4⋊C4D5semidirect
54. C22×D20C3×C4semidirect
82. #4039A4direct (N × H)
111. C23×D5C2×C3×C4semidirect
119. C22×C4×C5C2×C3×C4semidirect
127. C22×Dic5C3×D4semidirect
150. C2×C4×D5C2×A4semidirect
154. C5×C4⋊C4C2×A4semidirect
159. #685C2×A4semidirect
164. C4×Dic5C2×A4semidirect
173. C22×C4⋊C4C3×D5semidirect
180. C5×A4C4×D4semidirect
194. C4×A4C4×D5semidirect
251. D20C4×A4semidirect
320. C22×C5C3×C4×D4semidirect
335. C2×D5C2×C4×A4semidirect
339. C4×C5C2×C4×A4semidirect
343. Dic5D4×A4semidirect
391. C22×C4C3×C4×D5semidirect
395. C4⋊C4D5×A4semidirect
411. A4#4039direct (N × H)
488. C5C4×D4×A4semidirect
513. C22C3×#4039semidirect
522. C4C4×D5×A4semidirect

Maximal subgroups

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

subgroupindex
C2×A4×D202
C2×C4×D5×A42
C5×A4×C4⋊C42
A4×#6852
C4×A4×Dic52
C22×#40393
C3×#40394
C4×D4×A45