Finite Group Databasebeta

← C2×A4×C5⋊S4 #72277319

Subgroups — C2×A4×C5⋊S4 #72277319

Order 2880.

710 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 45 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×C5×A42C211yesmaximal, characteristic, complemented
2A4×C5⋊S4C212yesmaximal, direct product factor
3C23×C5⋊S4C311yesmaximal, characteristic, complemented
4C2×A4×C22⋊D5—31nomaximal
5C2×A4×D15—41nomaximal
6C2×C3×C5⋊S4—41nomaximal
7C5×A42C2211yescharacteristic, complemented
8C2×A4×S4—51nomaximal, Hall
9C2×C5×C22⋊A4—21no—
10C23×C5×A4D311yescharacteristic, complemented
11C23×C5×A4C2×C311yescharacteristic, complemented
12C22×D5×A4—31no—
13C22×C5⋊S4—31no—
14C22×C5⋊S4—31no—
15C22×C5⋊S4—32no—
16C22×C5⋊S4C2×C312yescomplemented
17A4×C22⋊D5—32no—
18A4×C22⋊D5—32no—
19C2×A4×Dic5—31no—
20C2×C3×C5×A4—41no—
21C2×C3×C5×A4—41no—
22A4×D15—42no—
23C3×C5⋊S4—42no—
24C23×C22⋊D5—31noHall
25C2×A42D511yescharacteristic, complemented
26A4×S4—52no—
27C23×D15—41no—
28C5×C22⋊A4—21no—
29C22×C5×A4—31no—
30C22×C5×A4—31no—
31C22×C5×A4—31no—
32C22×C5×A4—31no—
33C22×C5×A4C2×D311yescharacteristic, complemented
34C22×C5×A4C22×C311yescharacteristic, complemented
35C2×D5×A4—32no—
36C2×D5×A4—61no—
37C2×D5×A4—61no—
38C2×C5⋊S4—32no—
39C2×C5⋊S4—31no—
40C2×C5⋊S4—32no—
41C2×C5⋊S4—32no—
42C2×C5⋊S4—32no—
43C2×C5⋊S4A411yescharacteristic, direct product factor
44C2×C3×C22⋊D5—121no—
45A4×Dic5—32no—
46C23×S4—51no—
47C2×D4×A4—151no—
48C2×C3×D15—161no—
49C3×C5×A4—41no—
50C3×C5×A4—41no—
51C25×C5C3×D311yescharacteristic, complemented
52C24×D5—31no—
53C22×C22⋊D5—32no—
54C22×C22⋊D5—32no—
55C22×C22⋊D5—91no—
56C22×C22⋊D5—92no—
57C22×C22⋊D5—91no—
58C22×C22⋊D5—91no—
59C22×C22⋊D5—92no—
60C22×C22⋊D5—91no—
61C23×Dic5—31no—
62C2×D3×A4—201no—
63C2×C3×S4—201no—
64A42C2×D511yescharacteristic, complemented
65C23×C3×C5—41no—
66C23×C3×C5—41no—
67C22×D15—42no—
68C22×D15—121no—
69C22×D15—122no—
70C22×D15—121no—
71C22×C3×D5—121no—
72C2×C5×A4—31no—
73C2×C5×A4—31no—
74C2×C5×A4—31no—
75C2×C5×A4—31no—
76C2×C5×A4—81no—
77C2×C5×A4—81no—
78C2×C5×A4—241no—
79C2×C5×A4S411yescharacteristic, complemented
80C2×C5×A4C2×A411yescharacteristic, complemented
81D5×A4—62no—
82C5⋊S4—32no—
83C5⋊S4C2×A412yesdirect product factor
84C3×C22⋊D5—122no—
85C3×C22⋊D5—122no—
86C2×C3×Dic5—121no—
87C2×C22⋊A4—21no—
88C23×A4—151no—
89C23×A4C3×D511yescharacteristic, complemented
90C23×A4D1511yescharacteristic, complemented
91C22×S4—52no—
92C22×S4—151no—
93C22×S4—152no—
94C22×S4—151no—
95D4×A4—152no—
96D4×A4—152no—
97C2×C4×A4—151no—
98C2×C32×C5—161no—
99C3×D15—162no—
100C24×C5—31no—
101C24×C5—31no—
102C24×C5—31no—
103C24×C5—31no—
104C24×C5—91no—
105C24×C5—91no—
106C24×C5C2×C3×D311yescharacteristic, complemented
107C23×D5—32no—
108C23×D5—61no—
109C23×D5—61no—
110C23×D5—91no—
111C23×D5—92no—
112C23×D5—91no—
113C23×D5—181no—
114C23×D5—181no—
115C2×C22⋊D5—31no—
116C2×C22⋊D5—91no—
117C2×C22⋊D5—92no—
118C2×C22⋊D5—92no—
119C2×C22⋊D5—92no—
120C2×C22⋊D5—92no—
121C2×C22⋊D5—91no—
122C2×C22⋊D5—92no—
123C2×C22⋊D5—94no—
124C2×C22⋊D5—92no—
125C2×C22⋊D5—94no—
126C2×C22⋊D5—92no—
127C2×C22⋊D5—91no—
128C2×C22⋊D5—92no—
129C2×C22⋊D5—92no—
130C2×C22⋊D5—92no—
131C2×C22⋊D5—92no—
132C2×C22⋊D5—92no—
133C2×C22⋊D5—92no—
134C22×Dic5—32no—
135C22×Dic5—91no—
136C22×Dic5—92no—
137C22×Dic5—91no—
138C2×C3×A4—41no—
139C2×C3×A4—41no—
140D3×A4—202no—
141C3×S4—202no—
142C23×D4—151noSylow
143C22×C3×C5—41no—
144C22×C3×C5—41no—
145C22×C3×C5—121no—
146C22×C3×C5—121no—
147C22×C3×C5—121no—
148C22×C3×C5—121no—
149C2×D15—41no—
150C2×D15—122no—
151C2×D15—122no—
152C2×D15—121no—
153C2×D15—122no—
154C2×D15—122no—
155C2×C3×D5—122no—
156C2×C3×D5—241no—
157C2×C3×D5—241no—
158C5×A4—81no—
159C5×A4—81no—
160C5×A4—241no—
161C5×A4C2×S411yescharacteristic, complemented
162C5×A4C22×A411yescharacteristic, complemented
163C3×Dic5—122no—
164C23×D3—201no—
165C22⋊A4—21no—
166C22×A4—31no—
167C22×A4—31no—
168C22×A4—31no—
169C22×A4—31no—
170C22×A4—152no—
171C22×A4—301no—
172C22×A4—301no—
173C22×A4C2×C3×D511yescharacteristic, complemented
174C22×A4C2×D1511yescharacteristic, complemented
175C2×S4—51no—
176C2×S4—152no—
177C2×S4—152no—
178C2×S4—152no—
179C2×S4—151no—
180C2×S4—152no—
181C2×C3×D4—601no—
182C4×A4—152no—
183C32×C5—161noHall
184C23×C5—31no—
185C23×C5—31no—
186C23×C5—31no—
187C23×C5—31no—
188C23×C5—61no—
189C23×C5—91no—
190C23×C5—91no—
191C23×C5—91no—
192C23×C5—91no—
193C23×C5—91no—
194C23×C5—91no—
195C23×C5—91no—
196C23×C5—91no—
197C23×C5—92no—
198C23×C5—91no—
199C23×C5—181no—
200C23×C5—181no—
201C23×C5C3×S411yescharacteristic, complemented
202C23×C5D3×A411yescharacteristic, complemented
203C22×D5—31no—
204C22×D5—62no—
205C22×D5—92no—
206C22×D5—92no—
207C22×D5—91no—
208C22×D5—92no—
209C22×D5—92no—
210C22×D5—181no—
211C22×D5—181no—
212C22×D5—181no—
213C22×D5—181no—
214C22×D5—184no—
215C22×D5—182no—
216C22×D5—181no—
217C22×D5—182no—
218C22×D5—181no—
219C22×D5—181no—
220C22×D5—181no—
221C22×D5—182no—
222C22⋊D5—32no—
223C22⋊D5—32no—
224C22⋊D5—92no—
225C22⋊D5—92no—
226C22⋊D5—92no—
227C22⋊D5—92no—
228C22⋊D5—92no—
229C22⋊D5—94no—
230C22⋊D5—94no—
231C22⋊D5—92no—
232C2×Dic5—31no—
233C2×Dic5—92no—
234C2×Dic5—92no—
235C2×Dic5—91no—
236C2×Dic5—92no—
237C2×Dic5—92no—
238C2×C3×D3—801no—
239C3×A4—41no—
240C3×A4—41no—
241C25—151no—
242C25C3×D1511yescharacteristic, complemented
243C22×D4—152no—
244C22×D4—152no—
245C22×D4—451no—
246C22×D4—452no—
247C22×D4—451no—
248C22×D4—451no—
249C22×D4—452no—
250C22×D4—451no—
251C23×C4—151no—
252C2×C3×C5—41no—
253C2×C3×C5—41no—
254C2×C3×C5—121no—
255C2×C3×C5—121no—
256C2×C3×C5—121no—
257C2×C3×C5—121no—
258C2×C3×C5—321no—
259D15—42no—
260D15—122no—
261C3×D5—242no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—601no—
265C22×D3—202no—
266C22×D3—601no—
267C22×D3—602no—
268C22×D3—601no—
269C2×A4—31no—
270C2×A4—31no—
271C2×A4—31no—
272C2×A4—31no—
273C2×A4—81no—
274C2×A4—81no—
275C2×A4—241no—
276C2×A4—302no—
277C2×A4C5⋊S411yescharacteristic, direct product factor
278C2×A4D5×A411yescharacteristic, complemented
279S4—52no—
280S4—152no—
281C3×D4—602no—
282C3×D4—602no—
283C2×C3×C4—601no—
284C22×C5—31no—
285C22×C5—31no—
286C22×C5—31no—
287C22×C5—31no—
288C22×C5—61no—
289C22×C5—91no—
290C22×C5—91no—
291C22×C5—91no—
292C22×C5—91no—
293C22×C5—91no—
294C22×C5—91no—
295C22×C5—91no—
296C22×C5—91no—
297C22×C5—92no—
298C22×C5—91no—
299C22×C5—181no—
300C22×C5—181no—
301C22×C5C2×C3×S411yescharacteristic, complemented
302C22×C5C2×D3×A411yescharacteristic, complemented
303C2×D5—32no—
304C2×D5—61no—
305C2×D5—61no—
306C2×D5—92no—
307C2×D5—182no—
308C2×D5—182no—
309C2×D5—182no—
310C2×D5—182no—
311C2×D5—182no—
312C2×D5—181no—
313C2×D5—182no—
314C2×D5—181no—
315C2×D5—182no—
316C2×D5—182no—
317Dic5—32no—
318Dic5—92no—
319C2×C32—161no—
320C3×D3—802no—
321C24—31no—
322C24—31no—
323C24—31no—
324C24—31no—
325C24—91no—
326C24—91no—
327C24—152no—
328C24—301no—
329C24—301no—
330C24—451no—
331C24—451no—
332C24—452no—
333C24—901no—
334C24—901no—
335C24C2×C3×D1511yescharacteristic, complemented
336C2×D4—151no—
337C2×D4—451no—
338C2×D4—452no—
339C2×D4—452no—
340C2×D4—452no—
341C2×D4—452no—
342C2×D4—452no—
343C2×D4—451no—
344C2×D4—454no—
345C2×D4—452no—
346C2×D4—452no—
347C2×D4—454no—
348C2×D4—451no—
349C2×D4—452no—
350C2×D4—452no—
351C2×D4—452no—
352C2×D4—452no—
353C2×D4—452no—
354C2×D4—452no—
355C22×C4—152no—
356C22×C4—451no—
357C22×C4—452no—
358C22×C4—451no—
359C3×C5—41no—
360C3×C5—41no—
361C3×C5—321no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—121no—
365C22×C3—121no—
366C22×C3—121no—
367C22×C3—121no—
368C22×C3—602no—
369C22×C3—1201no—
370C22×C3—1201no—
371C2×D3—201no—
372C2×D3—602no—
373C2×D3—602no—
374C2×D3—601no—
375C2×D3—602no—
376C2×D3—602no—
377A4—81no—
378A4—81no—
379A4—241no—
380A4C2×C5⋊S411yescharacteristic, direct product factor
381A4C2×D5×A411yescharacteristic, complemented
382C3×C4—602no—
383C2×C5—31no—
384C2×C5—31no—
385C2×C5—31no—
386C2×C5—31no—
387C2×C5—91no—
388C2×C5—91no—
389C2×C5A4×S411yescharacteristic, complemented
390D5—62no—
391D5—182no—
392C32—161noSylow
393C23—31no—
394C23—31no—
395C23—31no—
396C23—31no—
397C23—61no—
398C23—91no—
399C23—91no—
400C23—91no—
401C23—91no—
402C23—91no—
403C23—91no—
404C23—91no—
405C23—91no—
406C23—92no—
407C23—91no—
408C23—151no—
409C23—181no—
410C23—181no—
411C23—302no—
412C23—451no—
413C23—452no—
414C23—452no—
415C23—452no—
416C23—452no—
417C23—901no—
418C23—902no—
419C23—901no—
420C23—901no—
421C23—901no—
422C23—901no—
423C23—902no—
424C23—901no—
425C23—902no—
426C23—901no—
427C23—901no—
428C23—904no—
429C23C3×C5⋊S411yescharacteristic, complemented
430C23A4×D1511yescharacteristic, complemented
431D4—152no—
432D4—152no—
433D4—452no—
434D4—452no—
435D4—452no—
436D4—452no—
437D4—452no—
438D4—454no—
439D4—454no—
440D4—452no—
441C2×C4—151no—
442C2×C4—452no—
443C2×C4—452no—
444C2×C4—451no—
445C2×C4—452no—
446C2×C4—452no—
447C2×C3—41no—
448C2×C3—41no—
449C2×C3—121no—
450C2×C3—121no—
451C2×C3—121no—
452C2×C3—121no—
453C2×C3—321no—
454C2×C3—1202no—
455D3—202no—
456D3—602no—
457C5C2×A4×S411yescharacteristic, complemented, Sylow
458C22—31no—
459C22—31no—
460C22—31no—
461C22—31no—
462C22—61no—
463C22—91no—
464C22—91no—
465C22—91no—
466C22—91no—
467C22—91no—
468C22—91no—
469C22—91no—
470C22—91no—
471C22—92no—
472C22—91no—
473C22—152no—
474C22—181no—
475C22—181no—
476C22—301no—
477C22—301no—
478C22—452no—
479C22—902no—
480C22—902no—
481C22—902no—
482C22—902no—
483C22—901no—
484C22—902no—
485C22—902no—
486C22—902no—
487C22—902no—
488C22—901no—
489C22C2×C3×C5⋊S411yescharacteristic, complemented
490C22C2×A4×D1511yescharacteristic, complemented
491C4—152no—
492C4—452no—
493C3—41no—
494C3—41no—
495C3—321no—
496C2—31no—
497C2—31no—
498C2—31no—
499C2—31no—
500C2—91no—
501C2—91no—
502C2—302no—
503C2—902no—
504C2A4×C5⋊S411yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2A4×C5⋊S4
commutator, cosocle, hypocenterC22×C5×A4C22×C3
FrattiniC1G
Fitting, socleC25×C5C3×D3
LCSG ▷ C22×C5×A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C5×A42C2semidirect
2. A4×C5⋊S4C2direct (N × H)
3. C23×C5⋊S4C3semidirect
7. C5×A42C22semidirect
10. C23×C5×A4D3semidirect
11. C23×C5×A4C2×C3semidirect
16. C22×C5⋊S4C2×C3semidirect
25. C2×A42D5semidirect
33. C22×C5×A4C2×D3semidirect
34. C22×C5×A4C22×C3semidirect
43. C2×C5⋊S4A4direct (N × H)
51. C25×C5C3×D3semidirect
64. A42C2×D5semidirect
79. C2×C5×A4S4semidirect
80. C2×C5×A4C2×A4semidirect
83. C5⋊S4C2×A4direct (N × H)
89. C23×A4C3×D5semidirect
90. C23×A4D15semidirect
106. C24×C5C2×C3×D3semidirect
161. C5×A4C2×S4semidirect
162. C5×A4C22×A4semidirect
173. C22×A4C2×C3×D5semidirect
174. C22×A4C2×D15semidirect
201. C23×C5C3×S4semidirect
202. C23×C5D3×A4semidirect
242. C25C3×D15semidirect
277. C2×A4C5⋊S4direct (N × H)
278. C2×A4D5×A4semidirect
301. C22×C5C2×C3×S4semidirect
302. C22×C5C2×D3×A4semidirect
335. C24C2×C3×D15semidirect
380. A4C2×C5⋊S4direct (N × H)
381. A4C2×D5×A4semidirect
389. C2×C5A4×S4semidirect
429. C23C3×C5⋊S4semidirect
430. C23A4×D15semidirect
457. C5C2×A4×S4semidirect
489. C22C2×C3×C5⋊S4semidirect
490. C22C2×A4×D15semidirect
504. C2A4×C5⋊S4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×A422
A4×C5⋊S42
C23×C5⋊S43
C2×A4×C22⋊D53
C2×A4×D154
C2×C3×C5⋊S44
C2×A4×S45