Finite Group Databasebeta

← D20×#239 #14483219

Subgroups — D20×#239 #14483219

Order 1920.

3296 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×#239C212yesmaximal, complemented
2#11756336C211yesmaximal, characteristic, complemented
3C4×C5×#239C211yesmaximal, characteristic, complemented
4C3×Q8×D20C211yesmaximal, characteristic, complemented
5#11755128C212yesmaximal, complemented
6#11753881C216yesmaximal, complemented
7#11753880C216yesmaximal, complemented
8D12×D20C213yesmaximal, complemented
9C4×D3×D20C213yesmaximal, complemented
10#11752550C213yesmaximal, complemented
11#11752549C213yesmaximal, complemented
12#41×D20—31nomaximal, Hall
13C2×#6821C2212yescomplemented
14C2×C5×#239C2211yescharacteristic
15C2×C3×Q8×D5C2212yescomplemented
16D5×#239—216no—
17C2×D3×D20C2213yes—
18C2×D5×D12C2216yescomplemented
19C2×C4×D3×D5C2216yescomplemented
20C2×#6769C2216yescomplemented
21C2×#6768C2216yescomplemented
22C3×C4×C5×Q8C2211yescharacteristic, complemented
23#91784C2216yescomplemented
24#91763C2213yescomplemented
25C4×D60C2213yescomplemented
26C5×Q8×Dic3C2211yescharacteristic, complemented
27C5×#893C2213yescomplemented
28C5×#892C2213yescomplemented
29C4×C5×D12C2213yescomplemented
30C42×C5×D3C2213yescomplemented
31C3×#4043C2216yescomplemented
32C3×C4×D20C2213yes—
33C3×#4015C2213yescomplemented
34#91476C22112yescomplemented
35D3×#685C2216yescomplemented
36#91460C2216yescomplemented
37#91460C2216yescomplemented
38#91450C2216yescomplemented
39C4×#1202C2216yescomplemented
40#91428C2213yescomplemented
41#91428C2216yescomplemented
42#91427C22112yescomplemented
43D3×C4⋊Dic5C2213yescomplemented
44Dic3×D20C2211yescharacteristic
45#91395C2216yescomplemented
46#91390C2216yescomplemented
47#91374C2216yescomplemented
48D4×#239—51nomaximal, Hall
49C2×D5×#41—32no—
50C4×C5×#41—31no—
51#65911—31no—
52#65910—32no—
53Q8×D20D311yescharacteristic, complemented
54#65889—33no—
55#65888—36no—
56D4×D20—33no—
57#65858—36no—
58#65811—33no—
59C2×C4×D20—33no—
60C22×D3×D5C2316yes—
61C2×C3×C5×Q8C2311yescharacteristic
62#6821—28no—
63C2×D60C2313yes—
64C2×D60C2316yes—
65C2×D60C2316yes—
66C2×C4×D15C2316yes—
67C5×#239D418yescomplemented
68C2×C5×D12C2313yes—
69C2×C4×C5×D3C2313yes—
70C2×C4×C5×D3C2313yes—
71C2×C4×C5×D3C2313yes—
72C3×Q8×D5—28no—
73C2×C3×D20C2311yescharacteristic
74C2×C3×C4×D5C2316yes—
75C2×C3×Dic10C2316yes—
76C2×#1203C23112yes—
77C2×#1202C2316yes—
78C2×#1200C2316yes—
79C2×D3×Dic5C2316yes—
80C2×D5×Dic3C2312yes—
81D3×D20—212no—
82D5×D12—224no—
83C4×D3×D5—224no—
84#6769—224no—
85#6768—224no—
86C3×C5×C4⋊C4C2313yes—
87C3×C42×C5C2313yes—
88#6714C2316yes—
89C4⋊Dic15C2311yescharacteristic
90C5×(C2×D3)⋊C4C2316yes—
91C5×C4⋊Dic3C2313yes—
92C4×C5×Dic3C2313yes—
93C4×C5×Dic3C2311yescharacteristic
94C4×C5×Dic3C2313yes—
95C3×#685C2316yes—
96C3×C4⋊Dic5C2313yes—
97C3×C4⋊Dic5C2316yes—
98Dic3⋊Dic5C2316yes—
99#6667C23112yes—
100#6667C2316yes—
101#6665C2312yes—
102C22×#239—52no—
103C3×D4×Q8—51no—
104#5817—52no—
105#5726—53no—
106#5725—53no—
107#5630—56no—
108#5627—56no—
109#5577—51no—
110C4×#239D511yescharacteristic, complemented
111D4×D12—53no—
112C4×D3×D4—53no—
113C2×C5×#41—31no—
114D5×#41—616no—
115C2×#713—32no—
116C2×Q8×D5C2×D312yescomplemented
117C2×#711—36no—
118C2×D4×D5—36no—
119C2×#709—36no—
120C22×D20—33no—
121C22×C4×D5—36no—
122C4×C5×Q8—31no—
123C4×C5×Q8C2×D311yescharacteristic, complemented
124C4×C5×D4—33no—
125C4×C5×D4—33no—
126C5×#117—33no—
127C2×C42×C5—33no—
128#4076—36no—
129#4076—36no—
130C2×#685—36no—
131C2×C4⋊Dic5—33no—
132#4043—36no—
133#4043C2×D316yescomplemented
134#4041—36no—
135#4041—36no—
136#4032—312no—
137#4028—312no—
138#4021—36no—
139#4020—33no—
140C4×D20—31no—
141C4×D20—36no—
142C4×D20—33no—
143C4×D20—36no—
144C4×D20—33no—
145C4×D20—36no—
146C4×D20C2×D313yes—
147#4015C2×D313yescomplemented
148D4×#41—151noSylow
149C22×D15C2416yes—
150C22×C5×D3C2413yes—
151C22×C3×D5C2412yes—
152C2×D3×D5—212no—
153C2×D3×D5—212no—
154C2×D3×D5—46no—
155C3×C5×Q8C2×D414yescomplemented
156C2×C3×C4×C5C2413yes—
157C2×C3×C4×C5C2413yes—
158C2×C3×C4×C5C2411yescharacteristic
159C2×Dic15C2412yes—
160D60—212no—
161D60—26no—
162D60—46no—
163C4×D15—212no—
164C2×C5×Dic3C2411yescharacteristic
165C2×C5×Dic3C2413yes—
166C2×C5×Dic3C2411yescharacteristic
167C5×D12C2×D4112yescomplemented
168C4×C5×D3—26no—
169C4×C5×D3—43no—
170C4×C5×D3C2×D4112yescomplemented
171C2×C3×Dic5C2416yes—
172C3×D20#4114yescomplemented
173C3×C4×D5—212no—
174C3×Dic10—212no—
175#1203—224no—
176#1202—212no—
177#1200—212no—
178D3×Dic5—212no—
179D5×Dic3—24no—
180C22×C3×Q8—52no—
181C2×#239—52no—
182C2×#239—56no—
183C2×#239—56no—
184C2×#239—1016no—
185C2×#239C2×D511yescharacteristic
186C2×D3×D4—53no—
187C22×D12—56no—
188C22×C4×D3—56no—
189C3×C4⋊Q8—53no—
190C3×C22⋊Q8—56no—
191C3×C4×Q8C2×D511yescharacteristic, complemented
192C3×C4×D4—53no—
193#947—56no—
194#946—56no—
195Q8×Dic3C2×D511yescharacteristic, complemented
196D4×Dic3—51no—
197#930—56no—
198C4×#67—56no—
199#895—56no—
200#895—56no—
201#895—56no—
202#894—512no—
203#893—53no—
204#893—56no—
205#893C2×D513yescomplemented
206#892C2×D513yescomplemented
207D3×C4⋊C4—53no—
208#882—512no—
209#67⋊C4—56no—
210D3×C22⋊C4—56no—
211#874—53no—
212C4×D12—53no—
213C4×D12C2×D513yescomplemented
214C42×D3C2×D513yescomplemented
215C23×D5—36no—
216C5×#41—38no—
217C2×C5×Q8C22×D311yescharacteristic
218C2×C5×D4—33no—
219C22×C4×C5—33no—
220C22×C4×C5—33no—
221C22×C4×C5—33no—
222C2×C22⋊D5—312no—
223C22×Dic5—36no—
224#713—68no—
225Q8×D5—28no—
226#711—624no—
227D4×D5—624no—
228#709—624no—
229C2×D20—36no—
230C2×D20—33no—
231C2×D20—36no—
232C2×D20—36no—
233C2×D20—612no—
234C2×D20C22×D311yescharacteristic
235C2×C4×D5—32no—
236C2×C4×D5—36no—
237C2×C4×D5—36no—
238C2×C4×D5—624no—
239C2×C4×D5C22×D316yes—
240C2×Dic10C22×D316yes—
241C5×C4⋊C4—33no—
242C5×C4⋊C4C22×D313yes—
243C5×C22⋊C4—36no—
244C42×C5—31no—
245C42×C5—33no—
246C42×C5—33no—
247C42×C5C22×D313yes—
248#685—32no—
249#685—36no—
250#685—36no—
251#685—312no—
252#685C22×D316yes—
253C4⋊Dic5—31no—
254C4⋊Dic5—36no—
255C4⋊Dic5C22×D313yes—
256C4⋊Dic5C22×D316yes—
257C22×#41—152no—
258#550—151no—
259D4×Q8—51no—
260#548—152no—
261#547—153no—
262#546—156no—
263D42—153no—
264#532—153no—
265#525—156no—
266C4×#41—31no—
267C2×C4×D4—153no—
268C22×C3×C5C2511yescharacteristic
269C2×D15—26no—
270C2×D15—26no—
271C2×D15—26no—
272C2×C5×D3—23no—
273C2×C5×D3—23no—
274C2×C5×D3C22×D416yes—
275C2×C3×D5—22no—
276C2×C3×D5—22no—
277C2×C3×D5C2×#4114yescomplemented
278D3×D5—424no—
279C3×C4×C5—23no—
280C3×C4×C5C22×D416yes—
281C3×C4×C5C2×#4112yescomplemented
282Dic15—22no—
283C3×Dic5—26no—
284C5×Dic3—23no—
285C5×Dic3—21no—
286C5×Dic3C22×D412yes—
287C23×D3—56no—
288C2×C3×Q8—56no—
289C2×C3×Q8—108no—
290C2×C3×Q8C22×D511yescharacteristic
291C2×C3×D4—51no—
292C22×C3×C4—56no—
293C2×#67—56no—
294C22×Dic3—52no—
295#239—108no—
296#239—1024no—
297#239—1024no—
298#239D2018yesdirect product factor
299D3×D4—1012no—
300C2×D12—512no—
301C2×D12—53no—
302C2×D12—56no—
303C2×D12—56no—
304C2×D12—1024no—
305C2×D12C22×D513yes—
306C2×C4×D3—56no—
307C2×C4×D3—56no—
308C2×C4×D3—56no—
309C2×C4×D3—1024no—
310C2×C4×D3C22×D513yes—
311C2×C4×D3C22×D513yes—
312C2×C4×D3C22×D513yes—
313C3×C4⋊C4—53no—
314C3×C4⋊C4—56no—
315C3×C4⋊C4C22×D513yes—
316C3×C22⋊C4—56no—
317C3×C42C22×D513yes—
318#217—52no—
319(C2×D3)⋊C4—56no—
320(C2×D3)⋊C4—56no—
321(C2×D3)⋊C4—512no—
322(C2×D3)⋊C4C22×D516yes—
323C4⋊Dic3—51no—
324C4⋊Dic3C22×D513yes—
325Dic3⋊C4—56no—
326C4×Dic3C22×D511yescharacteristic
327C4×Dic3C22×D513yes—
328C4×Dic3C22×D513yes—
329C23×C5—33no—
330C22×D5—36no—
331C22×D5—612no—
332C22×D5—612no—
333C22×D5—126no—
334C22×D5C23×D312yes—
335C5×Q8D3×D414yescomplemented
336C5×D4—312no—
337C2×C4×C5—31no—
338C2×C4×C5—312no—
339C2×C4×C5—33no—
340C2×C4×C5—31no—
341C2×C4×C5—66no—
342C2×C4×C5—123no—
343C2×C4×C5C23×D313yes—
344C2×C4×C5C23×D313yes—
345C2×C4×C5C23×D311yescharacteristic
346C22⋊D5—624no—
347C2×Dic5—32no—
348C2×Dic5—612no—
349C2×Dic5C23×D316yes—
350D20—612no—
351D20—612no—
352D20—66no—
353D20—126no—
354D20#23914yesdirect product factor
355C4×D5—212no—
356C4×D5—612no—
357C4×D5—64no—
358C4×D5—612no—
359Dic10—212no—
360C2×#41—31no—
361C2×#41—156no—
362C2×#41—156no—
363C2×#41—152no—
364C2×#41—3016no—
365C22×Q8—52no—
366C22×D4—153no—
367C22×D4—156no—
368C23×C4—156no—
369C4⋊Q8—53no—
370#127—153no—
371#124—156no—
372#123—1512no—
373C22⋊Q8—56no—
374C22⋊Q8—156no—
375#121—156no—
376#121—156no—
377#121—156no—
378#121—156no—
379#120—1512no—
380C4×Q8—31no—
381C4×Q8D3×D511yescharacteristic, complemented
382C4×D4—33no—
383C4×D4—33no—
384C4×D4—53no—
385C4×D4—156no—
386C4×D4—153no—
387C4×D4—156no—
388C4×D4—156no—
389C4×D4—151no—
390C4×D4—153no—
391#117—33no—
392C2×C4⋊C4—153no—
393C2×C22⋊C4—156no—
394C2×C42—33no—
395C2×C3×C5C23×D411yescharacteristic
396C2×C3×C5C22×#4111yescharacteristic
397C2×C3×C5#58511yescharacteristic
398D15—46no—
399C3×D5—24no—
400C5×D3—26no—
401C23×C3—52no—
402C22×D3—56no—
403C22×D3—1012no—
404C22×D3—1012no—
405C22×D3—206no—
406C22×D3C23×D513yes—
407C3×Q8—1012no—
408C3×Q8C2×D2014yescomplemented
409C3×D4—54no—
410C2×C3×C4—56no—
411C2×C3×C4—1012no—
412C2×C3×C4C23×D513yes—
413C2×C3×C4C23×D513yes—
414C2×C3×C4C23×D511yescharacteristic
415#67—1012no—
416C2×Dic3—52no—
417C2×Dic3—104no—
418C2×Dic3C23×D511yescharacteristic
419C2×Dic3C23×D513yes—
420C2×Dic3C23×D511yescharacteristic
421D12—1024no—
422D12—1012no—
423D12—106no—
424D12—206no—
425D12C2×D20112yescomplemented
426C4×D3—26no—
427C4×D3—43no—
428C4×D3—1012no—
429C4×D3—1012no—
430C4×D3—1012no—
431C4×D3C2×D20112yescomplemented
432C22×C5—36no—
433C22×C5—63no—
434C22×C5—63no—
435C22×C5C24×D311yescharacteristic
436C2×D5—22no—
437C2×D5—22no—
438C2×D5—66no—
439C2×D5—66no—
440C2×D5—66no—
441C2×D5—1224no—
442C2×D5C2×#23914yescomplemented
443C4×C5—23no—
444C4×C5—32no—
445C4×C5—63no—
446C4×C5—61no—
447C4×C5C2×D3×D416yes—
448C4×C5C2×#23912yescomplemented
449Dic5—26no—
450Dic5—62no—
451C24—156no—
452#41—38no—
453#41—3024no—
454#41—3024no—
455#41—308no—
456C2×Q8—56no—
457C2×Q8—108no—
458C2×Q8C2×D3×D511yescharacteristic
459C2×D4—33no—
460C2×D4—51no—
461C2×D4—156no—
462C2×D4—1512no—
463C2×D4—153no—
464C2×D4—156no—
465C2×D4—156no—
466C2×D4—3012no—
467C2×D4—3024no—
468C22×C4—33no—
469C22×C4—33no—
470C22×C4—33no—
471C22×C4—56no—
472C22×C4—156no—
473C22×C4—152no—
474C22×C4—156no—
475C22×C4—156no—
476C22×C4—3024no—
477C4⋊C4—33no—
478C4⋊C4—53no—
479C4⋊C4—56no—
480C4⋊C4—156no—
481C4⋊C4—151no—
482C4⋊C4C2×D3×D513yes—
483C22⋊C4—36no—
484C22⋊C4—56no—
485C22⋊C4—156no—
486C22⋊C4—1512no—
487C22⋊C4—152no—
488C22⋊C4—156no—
489C42—31no—
490C42—33no—
491C42—33no—
492C42C2×D3×D513yes—
493C3×C5D4×#4111yescharacteristic, complemented, Hall
494C22×C3—54no—
495C22×C3—102no—
496C22×C3—102no—
497C22×C3C24×D511yescharacteristic
498C2×D3—23no—
499C2×D3—23no—
500C2×D3—106no—
501C2×D3—106no—
502C2×D3—106no—
503C2×D3—2024no—
504C2×D3C22×D2016yes—
505C3×C4—23no—
506C3×C4—106no—
507C3×C4C22×D2016yes—
508C3×C4D5×#4112yescomplemented
509Dic3—23no—
510Dic3—21no—
511Dic3—102no—
512Dic3C22×D2012yes—
513C2×C5—66no—
514C2×C5C22×D3×D411yescharacteristic
515C2×C5C22×#23911yescharacteristic
516C2×C5#596811yescharacteristic
517D5—24no—
518D5—126no—
519C23—33no—
520C23—52no—
521C23—156no—
522C23—3012no—
523C23—3012no—
524C23—606no—
525Q8—1012no—
526Q8D3×D2014yescomplemented
527D4—312no—
528D4—54no—
529D4—3012no—
530D4—3024no—
531D4—3012no—
532D4—306no—
533D4—606no—
534C2×C4—31no—
535C2×C4—312no—
536C2×C4—31no—
537C2×C4—33no—
538C2×C4—56no—
539C2×C4—66no—
540C2×C4—1012no—
541C2×C4—123no—
542C2×C4—152no—
543C2×C4—3012no—
544C2×C4—304no—
545C2×C4—3012no—
546C2×C4—3012no—
547C2×C4C22×D3×D513yes—
548C2×C4C22×D3×D513yes—
549C2×C4C22×D3×D511yescharacteristic
550C2×C3—104no—
551C2×C3C23×D2011yescharacteristic
552C2×C3C2×D5×#4111yescharacteristic
553C2×C3#6628311yescharacteristic
554D3—26no—
555D3—206no—
556C5D4×#23911yescharacteristic, complemented, Sylow
557C22—36no—
558C22—54no—
559C22—63no—
560C22—63no—
561C22—102no—
562C22—102no—
563C22—306no—
564C22—306no—
565C22—306no—
566C22—6024no—
567C22C23×D3×D511yescharacteristic
568C4—23no—
569C4—32no—
570C4—61no—
571C4—63no—
572C4—106no—
573C4—302no—
574C4C2×D3×D2016yes—
575C4D5×#23912yescomplemented
576C3#41×D2011yescharacteristic, complemented, Sylow
577C2—66no—
578C2—104no—
579C2—606no—
580C2C22×D3×D2011yescharacteristic
581C2C2×D5×#23911yescharacteristic
582C2#1175706311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3×D5
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C4×C5×Q8C22
hypercenterC4×Q8D3×D5
hypocenterC3×C5D4×#41
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×D5×#239C2semidirect
2. #11756336C2semidirect
3. C4×C5×#239C2semidirect
4. C3×Q8×D20C2semidirect
5. #11755128C2semidirect
6. #11753881C2semidirect
7. #11753880C2semidirect
8. D12×D20C2semidirect
9. C4×D3×D20C2semidirect
10. #11752550C2semidirect
11. #11752549C2semidirect
13. C2×#6821C22semidirect
15. C2×C3×Q8×D5C22semidirect
18. C2×D5×D12C22semidirect
19. C2×C4×D3×D5C22semidirect
20. C2×#6769C22semidirect
21. C2×#6768C22semidirect
22. C3×C4×C5×Q8C22semidirect
23. #91784C22semidirect
24. #91763C22semidirect
25. C4×D60C22semidirect
26. C5×Q8×Dic3C22semidirect
27. C5×#893C22semidirect
28. C5×#892C22semidirect
29. C4×C5×D12C22semidirect
30. C42×C5×D3C22semidirect
31. C3×#4043C22semidirect
33. C3×#4015C22semidirect
34. #91476C22semidirect
35. D3×#685C22semidirect
36. #91460C22semidirect
37. #91460C22semidirect
38. #91450C22semidirect
39. C4×#1202C22semidirect
40. #91428C22semidirect
41. #91428C22semidirect
42. #91427C22semidirect
43. D3×C4⋊Dic5C22semidirect
45. #91395C22semidirect
46. #91390C22semidirect
47. #91374C22semidirect
53. Q8×D20D3semidirect
67. C5×#239D4semidirect
110. C4×#239D5semidirect
116. C2×Q8×D5C2×D3semidirect
123. C4×C5×Q8C2×D3semidirect
133. #4043C2×D3semidirect
147. #4015C2×D3semidirect
155. C3×C5×Q8C2×D4semidirect
167. C5×D12C2×D4semidirect
170. C4×C5×D3C2×D4semidirect
172. C3×D20#41semidirect
191. C3×C4×Q8C2×D5semidirect
195. Q8×Dic3C2×D5semidirect
205. #893C2×D5semidirect
206. #892C2×D5semidirect
213. C4×D12C2×D5semidirect
214. C42×D3C2×D5semidirect
277. C2×C3×D5C2×#41semidirect
281. C3×C4×C5C2×#41semidirect
298. #239D20direct (N × H)
335. C5×Q8D3×D4semidirect
354. D20#239direct (N × H)
381. C4×Q8D3×D5semidirect
408. C3×Q8C2×D20semidirect
425. D12C2×D20semidirect
431. C4×D3C2×D20semidirect
442. C2×D5C2×#239semidirect
448. C4×C5C2×#239semidirect
493. C3×C5D4×#41semidirect
508. C3×C4D5×#41semidirect
526. Q8D3×D20semidirect
556. C5D4×#239semidirect
575. C4D5×#239semidirect
576. C3#41×D20semidirect

Maximal subgroups

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

subgroupindex
C2×D5×#2392
#117563362
C4×C5×#2392
C3×Q8×D202
#117551282
#117538812
#117538802
D12×D202
C4×D3×D202
#117525502
#117525492
#41×D203
D4×#2395