Finite Group Databasebeta

← C2×C5×(C2×S4)⋊C4 #14511221

Subgroups — C2×C5×(C2×S4)⋊C4 #14511221

Order 1920.

1422 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1422 conjugacy classes of subgroups — too many to draw. Only its 152 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
1C23×C5×S4C211yesmaximal, characteristic
2C22×C4×C5×A4C211yesmaximal, characteristic, complemented
3C22×C5×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C5×(C2×S4)⋊C4C214yesmaximal, direct product factor
5C2×C5×#386—31nomaximal, Hall
6C23×C5×A4C2211yescharacteristic
7C22×C5×S4—22no—
8C22×C5×S4—22no—
9C22×C5×S4C414yescomplemented
10C22×C5×S4C2212yes—
11C2×C4×C5×A4—22no—
12C2×C4×C5×A4C2212yescomplemented
13C2×C5×SL(2,ℤ/4)—22no—
14C2×C5×SL(2,ℤ/4)C2212yescomplemented
15C2×C5×(C2×D3)⋊C4—41nomaximal
16C2×(C2×S4)⋊C4C511yesmaximal, characteristic, direct product factor, Hall
17C23×C5×D4—31no—
18C24×C4×C5D311yescharacteristic, complemented
19C22×C5×C22⋊C4—31no—
20C22×C5×C22⋊C4—31no—
21C22×C5×C22⋊C4—31no—
22C5×#386—34no—
23C5×#386—34no—
24C2×C5×#95—31no—
25C2×C5×#95—31no—
26C23×C5×D3—41no—
27C22×C5×A4C2×C412yes—
28C22×C5×A4D412yes—
29C22×C5×A4D412yes—
30C22×C5×A4C2311yescharacteristic
31C2×C5×S4—28no—
32C2×C5×S4—22no—
33C2×C5×S4—22no—
34C2×C5×S4C2×C414yescomplemented
35C22×C3×C4×C5—41no—
36C22×C5×Dic3—41no—
37C4×C5×A4—22no—
38C5×SL(2,ℤ/4)—22no—
39C5×(C2×D3)⋊C4—44no—
40C23×S4C2×C511yescharacteristic
41C22×C4×A4C2×C511yescharacteristic, complemented
42C22×SL(2,ℤ/4)C2×C511yescharacteristic, complemented
43(C2×S4)⋊C4C2×C514yesdirect product factor
44C25×C5—31no—
45C25×C5C2×D311yescharacteristic
46C22×C5×D4—32no—
47C22×C5×D4—34no—
48C22×C5×D4—34no—
49C22×C5×D4—32no—
50C22×C5×D4—62no—
51C22×C5×D4—62no—
52C22×C5×D4—62no—
53C22×C5×D4—62no—
54C23×C4×C5—22no—
55C23×C4×C5—31no—
56C23×C4×C5—31no—
57C23×C4×C5—31no—
58C23×C4×C5—32no—
59C23×C4×C5—31no—
60C23×C4×C5—31no—
61C23×C4×C5—62no—
62C23×C4×C5C2×D312yescomplemented
63C2×C5×C22⋊C4—34no—
64C2×C5×C22⋊C4—34no—
65C2×C5×C22⋊C4—32no—
66C2×C5×C22⋊C4—32no—
67C2×C5×C22⋊C4—64no—
68C2×C5×C22⋊C4—61no—
69C2×C5×C22⋊C4—62no—
70C2×C5×C22⋊C4—62no—
71C2×C5×C22⋊C4—62no—
72C2×C5×C22⋊C4—61no—
73C5×#95—34no—
74C5×#95—34no—
75C2×#386—31noSylow
76C23×C3×C5—41no—
77C22×C5×D3—42no—
78C22×C5×D3—44no—
79C22×C5×D3—82no—
80C22×C5×D3—82no—
81C2×C5×A4C22⋊C414yescomplemented
82C2×C5×A4C22×C411yescharacteristic
83C2×C5×A4C2×D411yescharacteristic
84C2×C5×A4C2×D411yescharacteristic
85C5×S4—24no—
86C2×C3×C4×C5—42no—
87C2×C3×C4×C5—82no—
88C2×C5×Dic3—42no—
89C2×C5×Dic3—82no—
90C23×A4C22×C511yescharacteristic
91C22×S4—22no—
92C22×S4—22no—
93C22×S4C4×C514yescomplemented
94C22×S4C22×C512yes—
95C2×C4×A4—22no—
96C2×C4×A4C22×C512yescomplemented
97C2×SL(2,ℤ/4)—22no—
98C2×SL(2,ℤ/4)C22×C512yescomplemented
99C2×(C2×D3)⋊C4—41no—
100C24×C5—31no—
101C24×C5—31no—
102C24×C5—32no—
103C24×C5—32no—
104C24×C5—32no—
105C24×C5—32no—
106C24×C5—34no—
107C24×C5—62no—
108C24×C5—61no—
109C24×C5—62no—
110C24×C5—62no—
111C24×C5—62no—
112C24×C5—62no—
113C24×C5—61no—
114C24×C5C4×D312yes—
115C24×C5D1212yes—
116C24×C5#6712yes—
117C24×C5C22×D311yescharacteristic
118C2×C5×D4—38no—
119C2×C5×D4—34no—
120C2×C5×D4—34no—
121C2×C5×D4—64no—
122C2×C5×D4—62no—
123C2×C5×D4—64no—
124C2×C5×D4—62no—
125C2×C5×D4—68no—
126C2×C5×D4—68no—
127C2×C5×D4—68no—
128C2×C5×D4—68no—
129C2×C5×D4—62no—
130C2×C5×D4—62no—
131C22×C4×C5—22no—
132C22×C4×C5—32no—
133C22×C4×C5—34no—
134C22×C4×C5—34no—
135C22×C4×C5—32no—
136C22×C4×C5—31no—
137C22×C4×C5—32no—
138C22×C4×C5—32no—
139C22×C4×C5—32no—
140C22×C4×C5—32no—
141C22×C4×C5—62no—
142C22×C4×C5—62no—
143C22×C4×C5—61no—
144C22×C4×C5—62no—
145C22×C4×C5—64no—
146C22×C4×C5—62no—
147C22×C4×C5—62no—
148C22×C4×C5—64no—
149C22×C4×C5—62no—
150C22×C4×C5—62no—
151C22×C4×C5—62no—
152C22×C4×C5—62no—
153C22×C4×C5—61no—
154C22×C4×C5—62no—
155C22×C4×C5—62no—
156C22×C4×C5—61no—
157C22×C4×C5S411yescharacteristic, complemented
158C5×C22⋊C4—68no—
159C5×C22⋊C4—64no—
160C5×C22⋊C4—64no—
161C5×C22⋊C4—62no—
162C5×C22⋊C4—62no—
163C5×C22⋊C4—64no—
164C23×D4—31no—
165C24×C4C5×D311yescharacteristic, complemented
166C22×C22⋊C4—31no—
167C22×C22⋊C4—31no—
168C22×C22⋊C4—31no—
169#386—34no—
170#386—34no—
171C2×#95—31no—
172C2×#95—31no—
173C22×C3×C5—42no—
174C22×C3×C5—42no—
175C22×C3×C5—42no—
176C22×C3×C5—41no—
177C2×C5×D3—44no—
178C2×C5×D3—82no—
179C2×C5×D3—88no—
180C2×C5×D3—82no—
181C5×A4C2×C22⋊C411yescharacteristic, complemented
182C3×C4×C5—82no—
183C5×Dic3—82no—
184C23×D3—41no—
185C22×A4C2×C4×C512yes—
186C22×A4C5×D412yes—
187C22×A4C5×D412yes—
188C22×A4C23×C511yescharacteristic
189C2×S4—28no—
190C2×S4—22no—
191C2×S4—22no—
192C2×S4C2×C4×C514yescomplemented
193C22×C3×C4—41no—
194C22×Dic3—41no—
195C4×A4—22no—
196SL(2,ℤ/4)—22no—
197(C2×D3)⋊C4—44no—
198C23×C5—32no—
199C23×C5—32no—
200C23×C5—32no—
201C23×C5—32no—
202C23×C5—32no—
203C23×C5—32no—
204C23×C5—31no—
205C23×C5—31no—
206C23×C5—32no—
207C23×C5—31no—
208C23×C5—32no—
209C23×C5—31no—
210C23×C5—32no—
211C23×C5—31no—
212C23×C5—34no—
213C23×C5—34no—
214C23×C5—34no—
215C23×C5—34no—
216C23×C5—61no—
217C23×C5—61no—
218C23×C5—61no—
219C23×C5—64no—
220C23×C5—62no—
221C23×C5—62no—
222C23×C5—68no—
223C23×C5—62no—
224C23×C5—62no—
225C23×C5—64no—
226C23×C5—62no—
227C23×C5—68no—
228C23×C5—64no—
229C23×C5—62no—
230C23×C5—122no—
231C23×C5—124no—
232C23×C5—124no—
233C23×C5—122no—
234C23×C5—122no—
235C23×C5—122no—
236C23×C5(C2×D3)⋊C414yescomplemented
237C23×C5C2×C4×D311yescharacteristic
238C23×C5C2×D1211yescharacteristic
239C23×C5C2×#6711yescharacteristic
240C23×C5C2×S411yescharacteristic
241C5×D4—616no—
242C5×D4—64no—
243C5×D4—64no—
244C5×D4—64no—
245C5×D4—64no—
246C2×C4×C5—22no—
247C2×C4×C5—34no—
248C2×C4×C5—34no—
249C2×C4×C5—32no—
250C2×C4×C5—62no—
251C2×C4×C5—62no—
252C2×C4×C5—64no—
253C2×C4×C5—64no—
254C2×C4×C5—62no—
255C2×C4×C5—68no—
256C2×C4×C5—62no—
257C2×C4×C5—62no—
258C2×C4×C5—62no—
259C2×C4×C5—62no—
260C2×C4×C5—62no—
261C2×C4×C5—62no—
262C2×C4×C5—62no—
263C2×C4×C5—62no—
264C2×C4×C5—62no—
265C2×C4×C5—124no—
266C2×C4×C5—122no—
267C2×C4×C5—122no—
268C2×C4×C5C2×S412yescomplemented
269C25—31no—
270C25C2×C5×D311yescharacteristic
271C22×D4—34no—
272C22×D4—32no—
273C22×D4—34no—
274C22×D4—32no—
275C22×D4—62no—
276C22×D4—62no—
277C22×D4—62no—
278C22×D4—62no—
279C23×C4—22no—
280C23×C4—31no—
281C23×C4—31no—
282C23×C4—31no—
283C23×C4—32no—
284C23×C4—31no—
285C23×C4—31no—
286C23×C4—62no—
287C23×C4C2×C5×D312yescomplemented
288C2×C22⋊C4—34no—
289C2×C22⋊C4—34no—
290C2×C22⋊C4—32no—
291C2×C22⋊C4—32no—
292C2×C22⋊C4—64no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—62no—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—62no—
297C2×C22⋊C4—61no—
298#95—34no—
299#95—34no—
300C2×C3×C5—41no—
301C2×C3×C5—41no—
302C2×C3×C5—44no—
303C2×C3×C5—41no—
304C5×D3—84no—
305C23×C3—41no—
306C22×D3—42no—
307C22×D3—44no—
308C22×D3—82no—
309C22×D3—82no—
310C2×A4C5×C22⋊C414yescomplemented
311C2×A4C22×C4×C511yescharacteristic
312C2×A4C2×C5×D411yescharacteristic
313C2×A4C2×C5×D411yescharacteristic
314S4—24no—
315C2×C3×C4—42no—
316C2×C3×C4—82no—
317C2×Dic3—42no—
318C2×Dic3—82no—
319C22×C5—34no—
320C22×C5—31no—
321C22×C5—31no—
322C22×C5—31no—
323C22×C5—31no—
324C22×C5—31no—
325C22×C5—32no—
326C22×C5—31no—
327C22×C5—31no—
328C22×C5—32no—
329C22×C5—34no—
330C22×C5—34no—
331C22×C5—34no—
332C22×C5—34no—
333C22×C5—31no—
334C22×C5—31no—
335C22×C5—32no—
336C22×C5—61no—
337C22×C5—62no—
338C22×C5—62no—
339C22×C5—62no—
340C22×C5—64no—
341C22×C5—64no—
342C22×C5—64no—
343C22×C5—122no—
344C22×C5—128no—
345C22×C5—122no—
346C22×C5—122no—
347C22×C5—128no—
348C22×C5—122no—
349C22×C5C2×(C2×D3)⋊C411yescharacteristic, complemented
350C22×C5C4×S412yes—
351C22×C5C4⋊S412yes—
352C22×C5GL(2,ℤ/4)12yes—
353C22×C5C22×S411yescharacteristic
354C4×C5—22no—
355C4×C5—64no—
356C4×C5—62no—
357C4×C5—122no—
358C4×C5—122no—
359C24—31no—
360C24—31no—
361C24—32no—
362C24—32no—
363C24—32no—
364C24—32no—
365C24—34no—
366C24—62no—
367C24—61no—
368C24—62no—
369C24—62no—
370C24—62no—
371C24—62no—
372C24—61no—
373C24C4×C5×D312yes—
374C24C5×D1212yes—
375C24C5×#6712yes—
376C24C22×C5×D311yescharacteristic
377C2×D4—38no—
378C2×D4—34no—
379C2×D4—34no—
380C2×D4—64no—
381C2×D4—62no—
382C2×D4—64no—
383C2×D4—62no—
384C2×D4—68no—
385C2×D4—68no—
386C2×D4—68no—
387C2×D4—68no—
388C2×D4—62no—
389C2×D4—62no—
390C22×C4—22no—
391C22×C4—32no—
392C22×C4—32no—
393C22×C4—34no—
394C22×C4—34no—
395C22×C4—31no—
396C22×C4—32no—
397C22×C4—32no—
398C22×C4—32no—
399C22×C4—32no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—61no—
403C22×C4—62no—
404C22×C4—64no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—64no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—61no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—61no—
416C22×C4C5×S411yescharacteristic, complemented
417C22⋊C4—68no—
418C22⋊C4—64no—
419C22⋊C4—64no—
420C22⋊C4—62no—
421C22⋊C4—62no—
422C22⋊C4—64no—
423C3×C5—41noHall
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—41no—
428C2×D3—44no—
429C2×D3—82no—
430C2×D3—82no—
431C2×D3—88no—
432A4C2×C5×C22⋊C411yescharacteristic, complemented
433C3×C4—82no—
434Dic3—82no—
435C2×C5—31no—
436C2×C5—31no—
437C2×C5—31no—
438C2×C5—34no—
439C2×C5—31no—
440C2×C5—124no—
441C2×C5(C2×S4)⋊C414yesdirect product factor
442C2×C5C2×C4×S411yescharacteristic
443C2×C5C2×C4⋊S411yescharacteristic
444C2×C5C2×GL(2,ℤ/4)11yescharacteristic
445C23—32no—
446C23—32no—
447C23—32no—
448C23—32no—
449C23—32no—
450C23—32no—
451C23—31no—
452C23—31no—
453C23—32no—
454C23—31no—
455C23—32no—
456C23—31no—
457C23—32no—
458C23—31no—
459C23—34no—
460C23—34no—
461C23—34no—
462C23—34no—
463C23—61no—
464C23—61no—
465C23—61no—
466C23—64no—
467C23—62no—
468C23—62no—
469C23—62no—
470C23—68no—
471C23—62no—
472C23—64no—
473C23—62no—
474C23—68no—
475C23—64no—
476C23—62no—
477C23—122no—
478C23—122no—
479C23—124no—
480C23—124no—
481C23—122no—
482C23—122no—
483C23C5×(C2×D3)⋊C414yescomplemented
484C23C2×C4×C5×D311yescharacteristic
485C23C2×C5×D1211yescharacteristic
486C23C2×C5×#6711yescharacteristic
487C23C2×C5×S411yescharacteristic
488D4—616no—
489D4—64no—
490D4—64no—
491D4—64no—
492D4—64no—
493C2×C4—22no—
494C2×C4—34no—
495C2×C4—34no—
496C2×C4—32no—
497C2×C4—62no—
498C2×C4—62no—
499C2×C4—64no—
500C2×C4—64no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—68no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—124no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4C2×C5×S412yescomplemented
516C2×C3—41no—
517C2×C3—41no—
518C2×C3—41no—
519C2×C3—44no—
520D3—84no—
521C5C2×(C2×S4)⋊C411yescharacteristic, direct product factor, Sylow
522C22—34no—
523C22—31no—
524C22—31no—
525C22—31no—
526C22—31no—
527C22—31no—
528C22—32no—
529C22—31no—
530C22—31no—
531C22—32no—
532C22—31no—
533C22—31no—
534C22—32no—
535C22—34no—
536C22—34no—
537C22—34no—
538C22—34no—
539C22—61no—
540C22—62no—
541C22—62no—
542C22—62no—
543C22—64no—
544C22—64no—
545C22—64no—
546C22—122no—
547C22—128no—
548C22—122no—
549C22—122no—
550C22—122no—
551C22—128no—
552C22C2×C5×(C2×D3)⋊C411yescharacteristic, complemented
553C22C4×C5×S412yes—
554C22C5×C4⋊S412yes—
555C22C5×GL(2,ℤ/4)12yes—
556C22C22×C5×S411yescharacteristic
557C4—22no—
558C4—64no—
559C4—62no—
560C4—122no—
561C4—122no—
562C3—41noSylow
563C2—31no—
564C2—31no—
565C2—31no—
566C2—34no—
567C2—31no—
568C2—124no—
569C2C5×(C2×S4)⋊C414yesdirect product factor
570C2C2×C4×C5×S411yescharacteristic
571C2C2×C5×C4⋊S411yescharacteristic
572C2C2×C5×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC23×C5C2×S4
commutatorC2×A4C22×C4×C5
FrattiniC22C22×C5×S4
FittingC24×C4×C5D3
socleC25×C5C2×D3
prime radicalC23×C5×S4C2
cosocleC22×A4C23×C5
hypercenterC22×C4×C5S4
hypocenterA4C2×C5×C22⋊C4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C23×C5 ◁ C22×C4×C5

Complements

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

NH ≅ G/Ntype
2. C22×C4×C5×A4C2semidirect
3. C22×C5×SL(2,ℤ/4)C2semidirect
4. C5×(C2×S4)⋊C4C2direct (N × H)
9. C22×C5×S4C4semidirect
12. C2×C4×C5×A4C22semidirect
14. C2×C5×SL(2,ℤ/4)C22semidirect
16. C2×(C2×S4)⋊C4C5direct (N × H)
18. C24×C4×C5D3semidirect
34. C2×C5×S4C2×C4semidirect
41. C22×C4×A4C2×C5semidirect
42. C22×SL(2,ℤ/4)C2×C5semidirect
43. (C2×S4)⋊C4C2×C5direct (N × H)
62. C23×C4×C5C2×D3semidirect
81. C2×C5×A4C22⋊C4semidirect
93. C22×S4C4×C5semidirect
96. C2×C4×A4C22×C5semidirect
98. C2×SL(2,ℤ/4)C22×C5semidirect
157. C22×C4×C5S4semidirect
165. C24×C4C5×D3semidirect
181. C5×A4C2×C22⋊C4semidirect
192. C2×S4C2×C4×C5semidirect
236. C23×C5(C2×D3)⋊C4semidirect
268. C2×C4×C5C2×S4semidirect
287. C23×C4C2×C5×D3semidirect
310. C2×A4C5×C22⋊C4semidirect
349. C22×C5C2×(C2×D3)⋊C4semidirect
416. C22×C4C5×S4semidirect
432. A4C2×C5×C22⋊C4semidirect
441. C2×C5(C2×S4)⋊C4direct (N × H)
483. C23C5×(C2×D3)⋊C4semidirect
515. C2×C4C2×C5×S4semidirect
521. C5C2×(C2×S4)⋊C4direct (N × H)
552. C22C2×C5×(C2×D3)⋊C4semidirect
569. C2C5×(C2×S4)⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C5×S42
C22×C4×C5×A42
C22×C5×SL(2,ℤ/4)2
C5×(C2×S4)⋊C42
C2×C5×#3863
C2×C5×(C2×D3)⋊C44
C2×(C2×S4)⋊C45