Finite Group Databasebeta

← A4×#4070 #14514176

Subgroups — A4×#4070 #14514176

Order 1920.

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

normal subgroups only

This group has 860 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
2C2×A4×Dic10C211yesmaximal, characteristic, complemented
3A4×C22⋊Dic5C212yesmaximal
4A4×C4⋊Dic5C211yesmaximal, characteristic, complemented
5A4×#683C212yesmaximal, complemented
6C22×#4070C311yesmaximal, characteristic, complemented, Hall
7C23×C5×A4C2211yescharacteristic
8C2×C4×C5×A4—22no—
9C2×C4×C5×A4C2211yescharacteristic
10C2×C4×C5×A4C2211yescharacteristic
11C2×A4×Dic5C2212yes—
12C2×A4×Dic5C2212yes—
13A4×Dic10—22no—
14C3×#4070—41nomaximal
15A4×C22⋊Q8—51nomaximal, Hall
16C24×C4×C5C2×C311yescharacteristic
17C23×Dic10C2×C311yescharacteristic, complemented
18C22×C22⋊Dic5C2×C312yes—
19C22×C4⋊Dic5C2×C311yescharacteristic, complemented
20C2×#4070—31no—
21C2×#4070—32no—
22C2×#4070—31no—
23C2×#4070—31no—
24C2×#4070—32no—
25C2×#4070—31no—
26C22×#683C2×C312yescomplemented
27C22×C5×A4—21no—
28C22×C5×A4—21no—
29C22×C5×A4Q812yescomplemented
30C22×C5×A4C2311yescharacteristic
31C22×C3×C4×C5—41no—
32C2×C3×Dic10—41no—
33C4×C5×A4—21no—
34C4×C5×A4D412yes—
35A4×Dic5—22no—
36A4×Dic5—22no—
37C3×C22⋊Dic5—42no—
38C3×C4⋊Dic5—41no—
39C3×#683—42no—
40C2×Q8×A4—51no—
41C22×C4×A4D511yescharacteristic
42A4×C4⋊C4—52no—
43A4×C4⋊C4—51no—
44A4×C22⋊C4—52no—
45C25×C5C22×C311yescharacteristic
46C23×C4×C5—22no—
47C23×C4×C5—31no—
48C23×C4×C5—31no—
49C23×C4×C5—31no—
50C23×C4×C5—31no—
51C23×C4×C5—62no—
52C23×C4×C5C22×C311yescharacteristic
53C23×C4×C5C22×C311yescharacteristic
54C23×Dic5C22×C312yes—
55C23×Dic5C22×C312yes—
56C22×Dic10—22no—
57C22×Dic10—31no—
58C22×Dic10—31no—
59C22×Dic10—32no—
60C22×Dic10—62no—
61C2×C22⋊Dic5—32no—
62C2×C22⋊Dic5—32no—
63C2×C22⋊Dic5—32no—
64C2×C22⋊Dic5—32no—
65C2×C4⋊Dic5—31no—
66C2×C4⋊Dic5—31no—
67C2×C4⋊Dic5—32no—
68#4070—32no—
69#4070—32no—
70#4070—34no—
71#4070—34no—
72#4070—32no—
73#4070—31no—
74#4070—31no—
75#4070—32no—
76#4070—31no—
77#4070—32no—
78#4070A411yescharacteristic, direct product factor
79C2×#683—32no—
80C2×#683—32no—
81C2×#683—32no—
82C2×#683—32no—
83C22×C22⋊Q8—51noSylow
84C23×C3×C5—41no—
85C2×C5×A4—22no—
86C2×C5×A4C2×D411yescharacteristic
87C2×C5×A4C2×Q811yescharacteristic
88C2×C5×A4#4111yescharacteristic
89C2×C3×C4×C5—41no—
90C2×C3×C4×C5—41no—
91C2×C3×C4×C5—82no—
92C2×C3×Dic5—42no—
93C2×C3×Dic5—42no—
94C3×Dic10—82no—
95C23×A4C2×D511yescharacteristic
96Q8×A4—102no—
97C2×C4×A4—22no—
98C2×C4×A4—52no—
99C2×C4×A4—52no—
100C2×C4×A4C2×D511yescharacteristic
101C2×C4×A4C2×D511yescharacteristic
102C3×C22⋊Q8—201no—
103C24×C5—21no—
104C24×C5—21no—
105C24×C5—31no—
106C24×C5—32no—
107C24×C5—31no—
108C24×C5—61no—
109C24×C5—61no—
110C24×C5C3×Q812yescomplemented
111C24×C5C23×C311yescharacteristic
112C22×C4×C5—21no—
113C22×C4×C5—31no—
114C22×C4×C5—31no—
115C22×C4×C5—32no—
116C22×C4×C5—32no—
117C22×C4×C5—31no—
118C22×C4×C5—31no—
119C22×C4×C5—31no—
120C22×C4×C5—31no—
121C22×C4×C5—31no—
122C22×C4×C5—62no—
123C22×C4×C5—62no—
124C22×C4×C5—62no—
125C22×C4×C5—62no—
126C22×C4×C5—62no—
127C22×C4×C5—62no—
128C22×C4×C5—62no—
129C22×C4×C5—62no—
130C22×C4×C5—61no—
131C22×C4×C5C3×D412yes—
132C22×C4×C5C2×A411yescharacteristic
133C22×Dic5—22no—
134C22×Dic5—22no—
135C22×Dic5—32no—
136C22×Dic5—32no—
137C22×Dic5—32no—
138C22×Dic5—32no—
139C22×Dic5—62no—
140C22×Dic5—62no—
141C2×Dic10—32no—
142C2×Dic10—31no—
143C2×Dic10—32no—
144C2×Dic10—62no—
145C2×Dic10—62no—
146C2×Dic10—64no—
147C2×Dic10—64no—
148C2×Dic10—62no—
149C2×Dic10—62no—
150C2×Dic10C2×A411yescharacteristic, complemented
151C22⋊Dic5—34no—
152C22⋊Dic5—32no—
153C22⋊Dic5—32no—
154C22⋊Dic5—32no—
155C22⋊Dic5C2×A412yes—
156C4⋊Dic5—32no—
157C4⋊Dic5—31no—
158C4⋊Dic5—32no—
159C4⋊Dic5C2×A411yescharacteristic, complemented
160#683—34no—
161#683—32no—
162#683—32no—
163#683—32no—
164#683C2×A412yescomplemented
165C23×Q8—51no—
166C24×C4C3×D511yescharacteristic
167C2×C22⋊Q8—151no—
168C2×C22⋊Q8—152no—
169C2×C22⋊Q8—152no—
170C2×C22⋊Q8—151no—
171C2×C22⋊Q8—151no—
172C2×C22⋊Q8—151no—
173C22×C4⋊C4—52no—
174C22×C4⋊C4—51no—
175C22×C22⋊C4—52no—
176C22×C3×C5—42no—
177C22×C3×C5—41no—
178C22×C3×C5—81no—
179C22×C3×C5—81no—
180C5×A4C22⋊Q811yescharacteristic, complemented
181C3×C4×C5—42no—
182C3×C4×C5—81no—
183C3×Dic5—82no—
184C3×Dic5—82no—
185C22×A4—21no—
186C22×A4—21no—
187C22×A4Dic1012yescomplemented
188C22×A4C22×D511yescharacteristic
189C2×C3×Q8—201no—
190C22×C3×C4—41no—
191C4×A4—21no—
192C4×A4—102no—
193C4×A4—102no—
194C4×A4C22⋊D512yes—
195C3×C4⋊C4—202no—
196C3×C4⋊C4—201no—
197C3×C22⋊C4—202no—
198C23×C5—22no—
199C23×C5—31no—
200C23×C5—31no—
201C23×C5—31no—
202C23×C5—31no—
203C23×C5—32no—
204C23×C5—32no—
205C23×C5—32no—
206C23×C5—32no—
207C23×C5—31no—
208C23×C5—61no—
209C23×C5—61no—
210C23×C5—61no—
211C23×C5—61no—
212C23×C5—62no—
213C23×C5—61no—
214C23×C5—61no—
215C23×C5—62no—
216C23×C5—62no—
217C23×C5—61no—
218C23×C5—61no—
219C23×C5—64no—
220C23×C5C2×C3×D411yescharacteristic
221C23×C5C2×C3×Q811yescharacteristic
222C23×C5C3×#4111yescharacteristic
223C23×C5C22×A411yescharacteristic
224C2×C4×C5—22no—
225C2×C4×C5—32no—
226C2×C4×C5—31no—
227C2×C4×C5—32no—
228C2×C4×C5—32no—
229C2×C4×C5—32no—
230C2×C4×C5—31no—
231C2×C4×C5—62no—
232C2×C4×C5—61no—
233C2×C4×C5—61no—
234C2×C4×C5—62no—
235C2×C4×C5—64no—
236C2×C4×C5—62no—
237C2×C4×C5—61no—
238C2×C4×C5—61no—
239C2×C4×C5C22×A411yescharacteristic
240C2×C4×C5C22×A411yescharacteristic
241C2×Dic5—32no—
242C2×Dic5—32no—
243C2×Dic5—62no—
244C2×Dic5—62no—
245C2×Dic5—62no—
246C2×Dic5—62no—
247C2×Dic5—62no—
248C2×Dic5—62no—
249C2×Dic5—62no—
250C2×Dic5—62no—
251C2×Dic5C22×A412yes—
252C2×Dic5C22×A412yes—
253Dic10—22no—
254Dic10—64no—
255Dic10—64no—
256Dic10—62no—
257C25C2×C3×D511yescharacteristic
258C22×Q8—102no—
259C22×Q8—151no—
260C22×Q8—151no—
261C22×Q8—152no—
262C22×Q8—302no—
263C23×C4—22no—
264C23×C4—31no—
265C23×C4—31no—
266C23×C4—31no—
267C23×C4—31no—
268C23×C4—52no—
269C23×C4—52no—
270C23×C4—62no—
271C23×C4C2×C3×D511yescharacteristic
272C23×C4C2×C3×D511yescharacteristic
273C22⋊Q8—51no—
274C22⋊Q8—152no—
275C22⋊Q8—152no—
276C22⋊Q8—151no—
277C22⋊Q8—151no—
278C22⋊Q8—154no—
279C22⋊Q8—154no—
280C22⋊Q8—152no—
281C22⋊Q8—152no—
282C22⋊Q8—152no—
283C22⋊Q8—151no—
284C2×C4⋊C4—152no—
285C2×C4⋊C4—151no—
286C2×C4⋊C4—152no—
287C2×C4⋊C4—152no—
288C2×C4⋊C4—152no—
289C2×C4⋊C4—152no—
290C2×C4⋊C4—151no—
291C2×C22⋊C4—152no—
292C2×C22⋊C4—152no—
293C2×C22⋊C4—152no—
294C2×C22⋊C4—152no—
295C2×C3×C5—41no—
296C2×C3×C5—41no—
297C2×C3×C5—41no—
298C2×C3×C5—82no—
299C23×C3—41no—
300C2×A4—22no—
301C2×A4C2×Dic1011yescharacteristic
302C2×A4#70911yescharacteristic
303C2×A4C2×C22⋊D511yescharacteristic
304C3×Q8—402no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—82no—
308C2×C3×C4—202no—
309C2×C3×C4—202no—
310C22×C5—21no—
311C22×C5—21no—
312C22×C5—31no—
313C22×C5—31no—
314C22×C5—31no—
315C22×C5—32no—
316C22×C5—31no—
317C22×C5—31no—
318C22×C5—31no—
319C22×C5—31no—
320C22×C5—31no—
321C22×C5—31no—
322C22×C5—32no—
323C22×C5—62no—
324C22×C5—62no—
325C22×C5—62no—
326C22×C5—62no—
327C22×C5—62no—
328C22×C5—62no—
329C22×C5—62no—
330C22×C5—62no—
331C22×C5—61no—
332C22×C5—61no—
333C22×C5C3×C22⋊Q811yescharacteristic, complemented
334C22×C5Q8×A412yescomplemented
335C22×C5C23×A411yescharacteristic
336C4×C5—21no—
337C4×C5—32no—
338C4×C5—61no—
339C4×C5D4×A412yes—
340Dic5—22no—
341Dic5—22no—
342Dic5—62no—
343Dic5—62no—
344C24—21no—
345C24—21no—
346C24—31no—
347C24—32no—
348C24—31no—
349C24—61no—
350C24—61no—
351C24C3×Dic1012yescomplemented
352C24C22×C3×D511yescharacteristic
353C2×Q8—51no—
354C2×Q8—152no—
355C2×Q8—152no—
356C2×Q8—151no—
357C2×Q8—302no—
358C2×Q8—302no—
359C2×Q8—302no—
360C2×Q8—304no—
361C2×Q8—302no—
362C2×Q8—304no—
363C22×C4—21no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—32no—
367C22×C4—32no—
368C22×C4—31no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—31no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—61no—
382C22×C4—102no—
383C22×C4—102no—
384C22×C4—152no—
385C22×C4—152no—
386C22×C4—152no—
387C22×C4—152no—
388C22×C4—302no—
389C22×C4—302no—
390C22×C4C3×C22⋊D512yes—
391C22×C4D5×A411yescharacteristic
392C4⋊C4—52no—
393C4⋊C4—51no—
394C4⋊C4—152no—
395C4⋊C4—152no—
396C4⋊C4—152no—
397C4⋊C4—154no—
398C4⋊C4—152no—
399C4⋊C4—152no—
400C4⋊C4—151no—
401C22⋊C4—52no—
402C22⋊C4—152no—
403C22⋊C4—152no—
404C22⋊C4—154no—
405C22⋊C4—152no—
406C3×C5—41noHall
407C22×C3—42no—
408C22×C3—41no—
409C22×C3—81no—
410C22×C3—81no—
411A4#407011yescharacteristic, direct product factor
412C3×C4—42no—
413C3×C4—81no—
414C3×C4—402no—
415C3×C4—402no—
416C2×C5—22no—
417C2×C5—31no—
418C2×C5—31no—
419C2×C5—31no—
420C2×C5—31no—
421C2×C5—62no—
422C2×C5C2×D4×A411yescharacteristic
423C2×C5C2×Q8×A411yescharacteristic
424C2×C5A4×#4111yescharacteristic
425C23—22no—
426C23—31no—
427C23—31no—
428C23—31no—
429C23—31no—
430C23—32no—
431C23—32no—
432C23—32no—
433C23—32no—
434C23—31no—
435C23—61no—
436C23—61no—
437C23—61no—
438C23—61no—
439C23—62no—
440C23—61no—
441C23—61no—
442C23—62no—
443C23—62no—
444C23—61no—
445C23—61no—
446C23—64no—
447C23C2×C3×Dic1011yescharacteristic
448C23C3×#70911yescharacteristic
449C23C2×C3×C22⋊D511yescharacteristic
450C23C2×D5×A411yescharacteristic
451Q8—102no—
452Q8—304no—
453Q8—304no—
454Q8—302no—
455C2×C4—22no—
456C2×C4—32no—
457C2×C4—31no—
458C2×C4—32no—
459C2×C4—32no—
460C2×C4—32no—
461C2×C4—31no—
462C2×C4—52no—
463C2×C4—52no—
464C2×C4—62no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—62no—
468C2×C4—64no—
469C2×C4—62no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—152no—
473C2×C4—152no—
474C2×C4—302no—
475C2×C4—302no—
476C2×C4—302no—
477C2×C4—302no—
478C2×C4—302no—
479C2×C4—302no—
480C2×C4—302no—
481C2×C4—302no—
482C2×C4C2×D5×A411yescharacteristic
483C2×C4C2×D5×A411yescharacteristic
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—41no—
487C2×C3—82no—
488C5A4×C22⋊Q811yescharacteristic, complemented, Sylow
489C22—21no—
490C22—21no—
491C22—31no—
492C22—31no—
493C22—31no—
494C22—32no—
495C22—31no—
496C22—31no—
497C22—31no—
498C22—31no—
499C22—31no—
500C22—31no—
501C22—32no—
502C22—62no—
503C22—62no—
504C22—62no—
505C22—62no—
506C22—62no—
507C22—62no—
508C22—62no—
509C22—62no—
510C22—61no—
511C22—61no—
512C22C3×#407011yescharacteristic, complemented
513C22A4×Dic1012yescomplemented
514C22C22×D5×A411yescharacteristic
515C4—21no—
516C4—32no—
517C4—61no—
518C4—102no—
519C4—102no—
520C4—302no—
521C4—302no—
522C4A4×C22⋊D512yes—
523C3—41noSylow
524C2—22no—
525C2—31no—
526C2—31no—
527C2—31no—
528C2—31no—
529C2—62no—
530C2C2×A4×Dic1011yescharacteristic
531C2A4×#70911yescharacteristic
532C2C2×A4×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, socle, cosocleC24×C5C23×C3
FittingC24×C4×C5C2×C3
prime radicalC23×C5×A4C22
hypercenterC22×C4D5×A4
hypocenterC22×C5C3×C22⋊Q8
LCSG ▷ C24×C5 ▷ C22×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×A4×Dic10C2semidirect
4. A4×C4⋊Dic5C2semidirect
5. A4×#683C2semidirect
6. C22×#4070C3semidirect
17. C23×Dic10C2×C3semidirect
19. C22×C4⋊Dic5C2×C3semidirect
26. C22×#683C2×C3semidirect
29. C22×C5×A4Q8semidirect
78. #4070A4direct (N × H)
110. C24×C5C3×Q8semidirect
150. C2×Dic10C2×A4semidirect
159. C4⋊Dic5C2×A4semidirect
164. #683C2×A4semidirect
180. C5×A4C22⋊Q8semidirect
187. C22×A4Dic10semidirect
333. C22×C5C3×C22⋊Q8semidirect
334. C22×C5Q8×A4semidirect
351. C24C3×Dic10semidirect
411. A4#4070direct (N × H)
488. C5A4×C22⋊Q8semidirect
512. C22C3×#4070semidirect
513. C22A4×Dic10semidirect

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×Dic102
A4×C22⋊Dic52
A4×C4⋊Dic52
A4×#6832
C22×#40703
C3×#40704
A4×C22⋊Q85