Finite Group Databasebeta

← D5×#5577 #14482990

Subgroups — D5×#5577 #14482990

Order 1920.

3472 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×#239C212yesmaximal, complemented
2#11756336C211yesmaximal, characteristic, complemented
3C5×#5577C211yesmaximal, characteristic, complemented
4C3×C4×Q8×D5C211yesmaximal, characteristic, complemented
5#11755245C212yesmaximal, complemented
6D5×#895C216yesmaximal, complemented
7#11753692C216yesmaximal, complemented
8D5×#874C213yesmaximal, complemented
9C4×D5×D12C213yesmaximal, complemented
10#11752538C213yesmaximal, complemented
11#11752536C213yesmaximal, complemented
12D5×#550—31nomaximal, Hall
13C2×#6821C2212yescomplemented
14C2×C5×#239C2212yescomplemented
15C2×C3×Q8×D5C2211yescharacteristic
16D5×#239—216no—
17C2×D5×D12C2213yes—
18C2×D5×D12C2216yes—
19C2×D5×D12C2216yes—
20C2×C4×D3×D5C2216yes—
21C2×#6769C2216yescomplemented
22C2×#6768C2216yescomplemented
23C3×C4×C5×Q8C2211yescharacteristic, complemented
24#91784C2216yescomplemented
25#91763C2213yescomplemented
26C4×D60C2213yescomplemented
27C5×#895C2216yescomplemented
28C5×#874C2213yescomplemented
29C4×C5×D12C2213yescomplemented
30C3×Q8×Dic5C2211yescharacteristic, complemented
31C3×D5×C4⋊C4C2213yes—
32C3×#4033C2213yescomplemented
33C3×C42×D5C2213yes—
34C3×C4×Dic10C2213yescomplemented
35D5×(C2×D3)⋊C4C2216yes—
36#91466C2216yescomplemented
37#91460C2216yescomplemented
38#91451C2216yescomplemented
39#91451C2216yescomplemented
40#91451C2216yescomplemented
41C4×#1203C2216yescomplemented
42#91420C22112yescomplemented
43#91418C2213yescomplemented
44#91416C22112yescomplemented
45Dic5×D12C2213yescomplemented
46D5×C4⋊Dic3C2211yescharacteristic
47#91405C2216yescomplemented
48C2×#5577—51nomaximal, Hall
49C2×D5×#41—32no—
50C5×#550—31no—
51#66168—32no—
52#66052—33no—
53D5×#127—33no—
54D5×#121—36no—
55#65933—36no—
56#65911—31no—
57C4×Q8×D5D311yescharacteristic, complemented
58#65887—33no—
59C4×D4×D5—33no—
60C22×D3×D5C2316yes—
61C2×C3×C5×Q8C2311yescharacteristic
62#6821—28no—
63C2×D60C2313yes—
64C2×D60C2316yes—
65C2×D60C2316yes—
66C2×C4×D15C2316yes—
67C5×#239—28no—
68C2×C5×D12C2313yes—
69C2×C5×D12C2316yes—
70C2×C5×D12C2316yes—
71C2×C4×C5×D3C2316yes—
72C3×Q8×D5D418yescomplemented
73C2×C3×C4×D5C2313yes—
74C2×C3×C4×D5C2313yes—
75C2×C3×C4×D5C2311yescharacteristic
76C2×C3×Dic10C2313yes—
77C2×#1203C2316yes—
78C2×#1203C23112yes—
79C2×#1203C23112yes—
80C2×#1200C2316yes—
81C2×D3×Dic5C2316yes—
82C2×D5×Dic3C2312yes—
83D5×D12—224no—
84D5×D12—212no—
85D5×D12—412no—
86C4×D3×D5—224no—
87#6769—224no—
88#6768—224no—
89C3×C5×C4⋊C4C2313yes—
90C3×C42×C5C2313yes—
91#6714C2316yes—
92C4⋊Dic15C2311yescharacteristic
93C5×(C2×D3)⋊C4C2316yes—
94C5×C4⋊Dic3C2311yescharacteristic
95C3×C4⋊Dic5C2313yes—
96C3×#683C2316yes—
97C3×C4×Dic5C2313yes—
98C3×C4×Dic5C2313yes—
99C3×C4×Dic5C2313yes—
100#6668C2312yes—
101#6667C2316yes—
102(C2×D3)⋊Dic5C2316yes—
103C22×#239—52no—
104C2×C3×C4×Q8—51no—
105#5577—52no—
106#5577—51no—
107#5577—56no—
108#5577—53no—
109#5577—53no—
110#5577D511yescharacteristic, direct product factor
111C2×#895—56no—
112C2×#874—53no—
113C2×C4×D12—53no—
114C2×C5×#41—32no—
115D5×#41—616no—
116C2×#713—32no—
117C2×Q8×D5C2×D311yescharacteristic
118C2×#711—36no—
119C2×D4×D5—33no—
120C2×D4×D5—36no—
121C2×D4×D5—36no—
122C2×#709—36no—
123C22×C4×D5—36no—
124C5×#127—33no—
125C5×#121—36no—
126C4×C5×Q8C2×D311yescharacteristic, complemented
127C4×C5×D4—33no—
128Q8×Dic5C2×D311yescharacteristic, complemented
129#4086—36no—
130#4086—36no—
131#4086—36no—
132#4085—312no—
133#4084—36no—
134D4×Dic5—33no—
135#4076—36no—
136C4×C22⋊D5—36no—
137#4041—36no—
138#4039—33no—
139D5×C4⋊C4—31no—
140D5×C4⋊C4C2×D313yes—
141#4033C2×D313yescomplemented
142#4030—312no—
143#4027—36no—
144D5×C22⋊C4—36no—
145#4020—33no—
146C4×D20—33no—
147C42×D5C2×D313yes—
148C4×Dic10C2×D313yescomplemented
149C2×#550—151noSylow
150C22×D15C2416yes—
151C22×C5×D3C2416yes—
152C22×C3×D5C2411yescharacteristic
153C2×D3×D5—212no—
154C2×D3×D5—212no—
155C2×D3×D5—212no—
156C3×C5×Q8C2×D414yescomplemented
157C2×C3×C4×C5C2413yes—
158C2×C3×C4×C5C2413yes—
159C2×C3×C4×C5C2411yescharacteristic
160C2×Dic15C2412yes—
161D60—212no—
162D60—26no—
163D60—46no—
164C4×D15—212no—
165C2×C5×Dic3C2412yes—
166C5×D12—212no—
167C5×D12—26no—
168C5×D12—46no—
169C4×C5×D3—212no—
170C2×C3×Dic5C2413yes—
171C2×C3×Dic5C2413yes—
172C2×C3×Dic5C2411yescharacteristic
173C3×C4×D5—26no—
174C3×C4×D5C2×D4112yes—
175C3×C4×D5#4114yescomplemented
176C3×Dic10C2×D4112yescomplemented
177#1203—224no—
178#1203—212no—
179#1203—412no—
180#1200—212no—
181D3×Dic5—212no—
182D5×Dic3—24no—
183C22×C3×Q8—51no—
184C2×#239—52no—
185C2×#239—56no—
186C2×#239—56no—
187C2×#239—1016no—
188C2×#239C2×D512yescomplemented
189C22×D12—56no—
190C22×D12—53no—
191C22×D12—56no—
192C22×C4×D3—56no—
193C3×C4×Q8—51no—
194C3×C4×Q8—53no—
195C3×C4×Q8—53no—
196C3×C4×Q8C2×D511yescharacteristic, complemented
197C2×C3×C4⋊C4—53no—
198C2×C3×C42—53no—
199C2×(C2×D3)⋊C4—56no—
200C2×C4⋊Dic3—51no—
201#895—512no—
202#895—56no—
203#895—56no—
204#895—56no—
205#895—512no—
206#895C2×D516yescomplemented
207#874—56no—
208#874—56no—
209#874—56no—
210#874—53no—
211#874C2×D513yescomplemented
212C4×D12—53no—
213C4×D12—56no—
214C4×D12—53no—
215C4×D12—56no—
216C4×D12—53no—
217C4×D12C2×D513yescomplemented
218C23×D5—36no—
219C5×#41—68no—
220C2×C5×Q8C22×D311yescharacteristic
221C2×C5×D4—33no—
222C2×C5×D4—36no—
223C2×C5×D4—36no—
224C22×C4×C5—36no—
225C2×C22⋊D5—312no—
226C2×C22⋊D5—36no—
227C2×C22⋊D5—312no—
228C22×Dic5—36no—
229#713—68no—
230Q8×D5D1218yescomplemented
231#711—624no—
232D4×D5—624no—
233D4×D5—612no—
234D4×D5—1212no—
235#709—624no—
236C2×D20—36no—
237C2×D20—33no—
238C2×D20—36no—
239C2×C4×D5—32no—
240C2×C4×D5—36no—
241C2×C4×D5—36no—
242C2×C4×D5—624no—
243C2×C4×D5C22×D313yes—
244C2×C4×D5C22×D313yes—
245C2×C4×D5C22×D311yescharacteristic
246C2×Dic10C22×D313yes—
247C5×C4⋊C4—31no—
248C5×C4⋊C4C22×D313yes—
249C5×C22⋊C4—36no—
250C42×C5C22×D313yes—
251C22⋊Dic5—36no—
252#685—36no—
253#685—36no—
254C4⋊Dic5—31no—
255C4⋊Dic5C22×D313yes—
256#683—32no—
257#683C22×D316yes—
258C4×Dic5C22×D313yes—
259C4×Dic5C22×D313yes—
260C4×Dic5C22×D313yes—
261C22×#41—152no—
262#550—31no—
263#550—153no—
264#550—156no—
265#550—153no—
266#550—152no—
267#550—151no—
268C2×#127—153no—
269C2×#121—156no—
270C2×C4×Q8—51no—
271C2×C4×D4—153no—
272C22×C3×C5C2511yescharacteristic
273C2×D15—26no—
274C2×D15—26no—
275C2×D15—26no—
276C2×C5×D3—26no—
277C2×C5×D3—26no—
278C2×C5×D3—26no—
279C2×C3×D5C22×D412yes—
280C2×C3×D5C2×#4112yes—
281C2×C3×D5H(5,𝔽2)12yes—
282D3×D5—424no—
283C3×C4×C5—23no—
284C3×C4×C5C22×D416yes—
285C3×C4×C5C2×#4112yescomplemented
286Dic15—22no—
287C3×Dic5—23no—
288C3×Dic5C22×D416yes—
289C3×Dic5C2×#4112yescomplemented
290C5×Dic3—22no—
291C23×D3—56no—
292C2×C3×Q8—58no—
293C2×C3×Q8—53no—
294C2×C3×Q8C22×D511yescharacteristic
295C22×C3×C4—51no—
296C22×C3×C4—53no—
297C22×C3×C4—53no—
298C22×Dic3—52no—
299#239—28no—
300#239—1024no—
301#239—1024no—
302#239—108no—
303C2×D12—56no—
304C2×D12—512no—
305C2×D12—53no—
306C2×D12—56no—
307C2×D12—512no—
308C2×D12—56no—
309C2×D12—1012no—
310C2×D12—1024no—
311C2×D12—2012no—
312C2×D12C22×D513yes—
313C2×D12C22×D516yes—
314C2×D12C22×D516yes—
315C2×C4×D3—56no—
316C2×C4×D3—56no—
317C2×C4×D3—56no—
318C2×C4×D3—1024no—
319C2×C4×D3C22×D516yes—
320C3×C4⋊C4—56no—
321C3×C4⋊C4—53no—
322C3×C4⋊C4C22×D513yes—
323C3×C42—53no—
324C3×C42—53no—
325C3×C42—53no—
326C3×C42C22×D513yes—
327(C2×D3)⋊C4—56no—
328(C2×D3)⋊C4—56no—
329(C2×D3)⋊C4—56no—
330(C2×D3)⋊C4C22×D516yes—
331C4⋊Dic3—51no—
332C4⋊Dic3—52no—
333C4⋊Dic3C22×D511yescharacteristic
334C23×C5—36no—
335C22×D5—36no—
336C22×D5—612no—
337C22×D5—612no—
338C22×D5—612no—
339C22×D5C23×D311yescharacteristic
340C5×Q8C2×D1214yescomplemented
341C5×D4—612no—
342C5×D4—66no—
343C5×D4—126no—
344C2×C4×C5—32no—
345C2×C4×C5—612no—
346C2×C4×C5C23×D313yes—
347C2×C4×C5C23×D313yes—
348C2×C4×C5C23×D311yescharacteristic
349C22⋊D5—624no—
350C22⋊D5—612no—
351C22⋊D5—1212no—
352C2×Dic5—32no—
353C2×Dic5—612no—
354C2×Dic5C23×D313yes—
355C2×Dic5C23×D313yes—
356C2×Dic5C23×D311yescharacteristic
357D20—612no—
358D20—66no—
359D20—126no—
360C4×D5—26no—
361C4×D5—64no—
362C4×D5—612no—
363C4×D5—612no—
364C4×D5C2×D12112yes—
365C4×D5#23914yescomplemented
366Dic10C2×D12112yescomplemented
367C2×#41—32no—
368C2×#41—156no—
369C2×#41—156no—
370C2×#41—152no—
371C2×#41—3016no—
372C22×Q8—51no—
373C22×D4—153no—
374C22×D4—156no—
375C22×D4—156no—
376C23×C4—156no—
377#127—33no—
378#127—156no—
379#127—156no—
380#127—156no—
381#127—153no—
382#121—36no—
383#121—1512no—
384#121—156no—
385#121—156no—
386#121—1512no—
387#121—156no—
388C4×Q8—51no—
389C4×Q8—53no—
390C4×Q8—53no—
391C4×Q8D3×D511yescharacteristic, complemented
392C4×D4—33no—
393C4×D4—156no—
394C4×D4—153no—
395C4×D4—156no—
396C4×D4—153no—
397C4×D4—153no—
398C2×C4⋊C4—53no—
399C2×C4⋊C4—151no—
400C2×C22⋊C4—156no—
401C2×C42—53no—
402C2×C3×C5C23×D411yescharacteristic
403C2×C3×C5C22×#4111yescharacteristic
404C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
405D15—46no—
406C3×D5#55014yescomplemented
407C5×D3—46no—
408C23×C3—51no—
409C22×D3—56no—
410C22×D3—1012no—
411C22×D3—1012no—
412C22×D3—1012no—
413C22×D3C23×D516yes—
414C3×Q8—512no—
415C3×Q8D4×D514yescomplemented
416C2×C3×C4—51no—
417C2×C3×C4—53no—
418C2×C3×C4—512no—
419C2×C3×C4—54no—
420C2×C3×C4—53no—
421C2×C3×C4—106no—
422C2×C3×C4C23×D513yes—
423C2×C3×C4C23×D513yes—
424C2×C3×C4C23×D511yescharacteristic
425C2×Dic3—52no—
426C2×Dic3—104no—
427C2×Dic3C23×D512yes—
428D12—212no—
429D12—26no—
430D12—46no—
431D12—1012no—
432D12—1024no—
433D12—1012no—
434D12—106no—
435D12—206no—
436D12—2012no—
437C4×D3—212no—
438C4×D3—1012no—
439C4×D3—1012no—
440C4×D3—1012no—
441C22×C5—66no—
442C22×C5—66no—
443C22×C5—66no—
444C22×C5C24×D311yescharacteristic
445C2×D5—66no—
446C2×D5—66no—
447C2×D5—66no—
448C2×D5—1224no—
449C2×D5C22×D1212yes—
450C2×D5C2×#23912yes—
451C2×D5#100912yes—
452C4×C5—23no—
453C4×C5—62no—
454C4×C5C22×D1216yes—
455C4×C5C2×#23912yescomplemented
456Dic5—23no—
457Dic5—62no—
458Dic5C22×D1216yes—
459Dic5C2×#23912yescomplemented
460C24—156no—
461#41—68no—
462#41—3024no—
463#41—3024no—
464#41—308no—
465C2×Q8—58no—
466C2×Q8—53no—
467C2×Q8C2×D3×D511yescharacteristic
468C2×D4—33no—
469C2×D4—36no—
470C2×D4—36no—
471C2×D4—156no—
472C2×D4—1512no—
473C2×D4—1512no—
474C2×D4—153no—
475C2×D4—156no—
476C2×D4—156no—
477C2×D4—3024no—
478C2×D4—3012no—
479C2×D4—6012no—
480C22×C4—36no—
481C22×C4—51no—
482C22×C4—53no—
483C22×C4—53no—
484C22×C4—156no—
485C22×C4—152no—
486C22×C4—156no—
487C22×C4—156no—
488C22×C4—3024no—
489C4⋊C4—31no—
490C4⋊C4—56no—
491C4⋊C4—53no—
492C4⋊C4—152no—
493C4⋊C4—151no—
494C4⋊C4C2×D3×D513yes—
495C22⋊C4—36no—
496C22⋊C4—156no—
497C22⋊C4—156no—
498C22⋊C4—156no—
499C42—53no—
500C42—53no—
501C42—53no—
502C42C2×D3×D513yes—
503C3×C5C2×#55011yescharacteristic, complemented, Hall
504C22×C3—52no—
505C22×C3—52no—
506C22×C3—52no—
507C22×C3C24×D511yescharacteristic
508C2×D3—26no—
509C2×D3—26no—
510C2×D3—26no—
511C2×D3—106no—
512C2×D3—106no—
513C2×D3—106no—
514C2×D3—2024no—
515C3×C4—23no—
516C3×C4—56no—
517C3×C4—52no—
518C3×C4—103no—
519C3×C4C2×D4×D516yes—
520C3×C4D5×#4112yescomplemented
521Dic3—22no—
522Dic3—102no—
523C2×C5—126no—
524C2×C5C23×D1211yescharacteristic
525C2×C5C22×#23911yescharacteristic
526C2×C5C2×#100911yescharacteristic
527D5—126no—
528D5#557714yesdirect product factor
529C23—36no—
530C23—51no—
531C23—156no—
532C23—3012no—
533C23—3012no—
534C23—3012no—
535Q8—512no—
536Q8D5×D1214yescomplemented
537D4—612no—
538D4—66no—
539D4—126no—
540D4—3024no—
541D4—3012no—
542D4—3012no—
543D4—306no—
544D4—606no—
545D4—6012no—
546C2×C4—32no—
547C2×C4—51no—
548C2×C4—53no—
549C2×C4—512no—
550C2×C4—54no—
551C2×C4—53no—
552C2×C4—612no—
553C2×C4—106no—
554C2×C4—152no—
555C2×C4—3012no—
556C2×C4—3012no—
557C2×C4—304no—
558C2×C4—3012no—
559C2×C4C22×D3×D513yes—
560C2×C4C22×D3×D513yes—
561C2×C4C22×D3×D511yescharacteristic
562C2×C3—54no—
563C2×C3C22×D4×D511yescharacteristic
564C2×C3C2×D5×#4111yescharacteristic
565C2×C3D5×H(5,𝔽2)11yescharacteristic
566D3—46no—
567D3—206no—
568C5C2×#557711yescharacteristic, complemented, Sylow
569C22—52no—
570C22—52no—
571C22—52no—
572C22—66no—
573C22—66no—
574C22—66no—
575C22—306no—
576C22—306no—
577C22—306no—
578C22—6024no—
579C22C23×D3×D511yescharacteristic
580C4—23no—
581C4—56no—
582C4—52no—
583C4—62no—
584C4—103no—
585C4—302no—
586C4C2×D5×D1216yes—
587C4D5×#23912yescomplemented
588C3D5×#55011yescharacteristic, complemented, Sylow
589C2—54no—
590C2—126no—
591C2—606no—
592C2C22×D5×D1211yescharacteristic
593C2C2×D5×#23911yescharacteristic
594C2D5×#100911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3×D5
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C4×C5×Q8C22
hypercenterC4×Q8D3×D5
hypocenterC3×C5C2×#550
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. C5×#5577C2semidirect
4. C3×C4×Q8×D5C2semidirect
5. #11755245C2semidirect
6. D5×#895C2semidirect
7. #11753692C2semidirect
8. D5×#874C2semidirect
9. C4×D5×D12C2semidirect
10. #11752538C2semidirect
11. #11752536C2semidirect
13. C2×#6821C22semidirect
14. C2×C5×#239C22semidirect
21. C2×#6769C22semidirect
22. C2×#6768C22semidirect
23. C3×C4×C5×Q8C22semidirect
24. #91784C22semidirect
25. #91763C22semidirect
26. C4×D60C22semidirect
27. C5×#895C22semidirect
28. C5×#874C22semidirect
29. C4×C5×D12C22semidirect
30. C3×Q8×Dic5C22semidirect
32. C3×#4033C22semidirect
34. C3×C4×Dic10C22semidirect
36. #91466C22semidirect
37. #91460C22semidirect
38. #91451C22semidirect
39. #91451C22semidirect
40. #91451C22semidirect
41. C4×#1203C22semidirect
42. #91420C22semidirect
43. #91418C22semidirect
44. #91416C22semidirect
45. Dic5×D12C22semidirect
47. #91405C22semidirect
57. C4×Q8×D5D3semidirect
72. C3×Q8×D5D4semidirect
110. #5577D5direct (N × H)
126. C4×C5×Q8C2×D3semidirect
128. Q8×Dic5C2×D3semidirect
141. #4033C2×D3semidirect
148. C4×Dic10C2×D3semidirect
156. C3×C5×Q8C2×D4semidirect
175. C3×C4×D5#41semidirect
176. C3×Dic10C2×D4semidirect
188. C2×#239C2×D5semidirect
196. C3×C4×Q8C2×D5semidirect
206. #895C2×D5semidirect
211. #874C2×D5semidirect
217. C4×D12C2×D5semidirect
230. Q8×D5D12semidirect
285. C3×C4×C5C2×#41semidirect
289. C3×Dic5C2×#41semidirect
340. C5×Q8C2×D12semidirect
365. C4×D5#239semidirect
366. Dic10C2×D12semidirect
391. C4×Q8D3×D5semidirect
406. C3×D5#550semidirect
415. C3×Q8D4×D5semidirect
455. C4×C5C2×#239semidirect
459. Dic5C2×#239semidirect
503. C3×C5C2×#550semidirect
520. C3×C4D5×#41semidirect
528. D5#5577direct (N × H)
536. Q8D5×D12semidirect
568. C5C2×#5577semidirect
587. C4D5×#239semidirect
588. C3D5×#550semidirect

Maximal subgroups

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

subgroupindex
C2×D5×#2392
#117563362
C5×#55772
C3×C4×Q8×D52
#117552452
D5×#8952
#117536922
D5×#8742
C4×D5×D122
#117525382
#117525362
D5×#5503
C2×#55775