Finite Group Databasebeta

← D15×#555 #14483912

Subgroups — D15×#555 #14483912

Order 1920.

2480 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#555C211yesmaximal, characteristic, complemented
2#11756493C211yesmaximal, characteristic, complemented
3D15×C4⋊Q8C211yesmaximal, characteristic, complemented
4D15×#124C214yesmaximal, complemented
5#11756429C214yesmaximal, complemented
6#11756375C214yesmaximal, complemented
7D15×#121C214yesmaximal, complemented
8#11756322C212yesmaximal, complemented
9C4×D4×D15C212yesmaximal, complemented
10#11756291C214yesmaximal, complemented
11D15×#117C214yesmaximal, complemented
12D5×#555—31nomaximal, Hall
13C2×Q8×D15C2212yes—
14C2×D4×D15C2212yes—
15C2×D4×D15C2214yes—
16C22×C4×D15C2214yes—
17C3×C5×C4⋊Q8C2211yescharacteristic, complemented
18C3×C5×#124C2214yescomplemented
19C3×C5×#121C2214yescomplemented
20C3×C4×C5×D4C2212yescomplemented
21C3×C5×#117C2214yescomplemented
22#91836C2214yescomplemented
23#91833C2212yescomplemented
24#91828C2218yescomplemented
25#91827C2214yescomplemented
26#91825C2214yescomplemented
27D4×Dic15C2212yescomplemented
28#91819C2214yescomplemented
29C4×#1220C2214yescomplemented
30#91815C2214yescomplemented
31#91784C2214yescomplemented
32#91782C2212yescomplemented
33#91781C2214yescomplemented
34#91781C2214yescomplemented
35C4⋊C4×D15C2212yes—
36C4⋊C4×D15C2214yes—
37#91777C2214yescomplemented
38#91774C2218yescomplemented
39#91774C2218yescomplemented
40#91773C2218yescomplemented
41#91770C2214yescomplemented
42C22⋊C4×D15C2214yes—
43C22⋊C4×D15C2218yes—
44#91766C2218yescomplemented
45#91764C2214yescomplemented
46C4×D60C2212yescomplemented
47#91761C2218yescomplemented
48C42×D15C2211yescharacteristic
49C42×D15C2214yes—
50#91758C2211yescharacteristic, complemented
51D3×#555—51nomaximal, Hall
52C5×#555D311yescharacteristic, complemented
53#66068—31no—
54D5×C4⋊Q8—31no—
55D5×#124—34no—
56#66004—34no—
57#65950—34no—
58D5×#121—34no—
59#65897—32no—
60C4×D4×D5—32no—
61#65866—34no—
62D5×#117—34no—
63C23×D15C2314yes—
64C2×C3×C5×Q8C2312yes—
65C2×C3×C5×D4C2312yes—
66C2×C3×C5×D4C2314yes—
67C22×C3×C4×C5C2314yes—
68C2×#1220C2314yes—
69C2×#1220C2318yes—
70C22×Dic15C2314yes—
71Q8×D15—44no—
72D4×D15—28no—
73D4×D15—48no—
74C2×D60C2312yes—
75C2×D60C2314yes—
76C2×C4×D15—216no—
77C2×C4×D15C2314yes—
78C2×C4×D15C2311yescharacteristic
79C2×C4×D15C2314yes—
80C2×C4×D15C2314yes—
81C2×C4×D15C2314yes—
82C2×C4×D15C2312yes—
83C2×Dic30C2312yes—
84C2×Dic30C2314yes—
85C3×C5×C4⋊C4C2314yes—
86C3×C5×C4⋊C4C2312yes—
87C3×C5×C22⋊C4C2318yes—
88C3×C5×C22⋊C4C2314yes—
89C3×C42×C5C2311yescharacteristic
90C3×C42×C5C2314yes—
91#6719C2318yes—
92#6719C2314yes—
93#6714C2318yes—
94#6714C2318yes—
95#6714C2314yes—
96#6714C2314yes—
97C4⋊Dic15C2314yes—
98C4⋊Dic15C2312yes—
99#6712C2314yes—
100#6712C2318yes—
101C4×Dic15C2318yes—
102C4×Dic15C2314yes—
103C4×Dic15C2311yescharacteristic
104C4×Dic15C2312yes—
105C3×#555D511yescharacteristic, complemented
106#5734—51no—
107D3×C4⋊Q8—51no—
108D3×#124—54no—
109#5670—54no—
110#5616—54no—
111D3×#121—54no—
112#5563—52no—
113C4×D3×D4—52no—
114#5532—54no—
115D3×#117—54no—
116C2×Q8×D5—32no—
117C2×D4×D5—32no—
118C2×D4×D5—34no—
119C22×C4×D5—34no—
120C5×C4⋊Q8C2×D311yescharacteristic, complemented
121C5×#124C2×D314yescomplemented
122C5×#121C2×D314yescomplemented
123C4×C5×D4C2×D312yescomplemented
124C5×#117C2×D314yescomplemented
125#4093—34no—
126#4090—32no—
127#4085—38no—
128#4084—34no—
129#4082—34no—
130D4×Dic5—32no—
131#4076—34no—
132C4×C22⋊D5—34no—
133#4072—34no—
134#4041—34no—
135#4039—32no—
136#4038—34no—
137#4038—34no—
138D5×C4⋊C4—34no—
139D5×C4⋊C4—32no—
140#4034—34no—
141#4031—38no—
142#4031—38no—
143#4030—38no—
144#4027—34no—
145D5×C22⋊C4—38no—
146D5×C22⋊C4—34no—
147#4023—38no—
148#4021—34no—
149C4×D20—32no—
150#4018—38no—
151C42×D5—31no—
152C42×D5—34no—
153#4015—31no—
154C2×#555—151noSylow
155C23×C3×C5C2414yes—
156C22×D15—28no—
157C22×D15—28no—
158C22×D15—28no—
159C22×D15C2414yes—
160C22×D15C2411yescharacteristic
161C3×C5×Q8—42no—
162C3×C5×D4—24no—
163C3×C5×D4—44no—
164C2×C3×C4×C5—28no—
165C2×C3×C4×C5C2414yes—
166C2×C3×C4×C5C2414yes—
167C2×C3×C4×C5C2411yescharacteristic
168C2×C3×C4×C5C2412yes—
169#1220—28no—
170#1220—48no—
171C2×Dic15—28no—
172C2×Dic15C2414yes—
173C2×Dic15C2414yes—
174C2×Dic15C2411yescharacteristic
175C2×Dic15C2412yes—
176D60—24no—
177D60—44no—
178C4×D15—28no—
179C4×D15—28no—
180C4×D15—22no—
181C4×D15—28no—
182C4×D15—28no—
183C4×D15#4118yes—
184Dic30—44no—
185Dic30—42no—
186C2×D3×Q8—52no—
187C2×D3×D4—52no—
188C2×D3×D4—54no—
189C22×C4×D3—54no—
190C3×C4⋊Q8C2×D511yescharacteristic, complemented
191C3×#124C2×D514yescomplemented
192C3×#121C2×D514yescomplemented
193C3×C4×D4C2×D512yescomplemented
194C3×#117C2×D514yescomplemented
195#947—54no—
196#944—52no—
197#939—58no—
198#938—54no—
199#936—54no—
200D4×Dic3—52no—
201#930—54no—
202C4×#67—54no—
203#926—54no—
204#895—54no—
205#893—52no—
206#892—54no—
207#892—54no—
208D3×C4⋊C4—54no—
209D3×C4⋊C4—52no—
210#888—54no—
211#885—58no—
212#885—58no—
213#884—58no—
214#67⋊C4—54no—
215D3×C22⋊C4—54no—
216D3×C22⋊C4—58no—
217#877—58no—
218#875—54no—
219C4×D12—52no—
220#872—58no—
221C42×D3—54no—
222C42×D3—51no—
223#869—51no—
224C23×D5—34no—
225C2×C5×Q8C22×D312yes—
226C2×C5×D4C22×D312yes—
227C2×C5×D4C22×D314yes—
228C22×C4×C5C22×D314yes—
229C2×C22⋊D5—34no—
230C2×C22⋊D5—38no—
231C22×Dic5—34no—
232Q8×D5—124no—
233D4×D5—68no—
234D4×D5—128no—
235C2×D20—32no—
236C2×D20—34no—
237C2×C4×D5—32no—
238C2×C4×D5—34no—
239C2×C4×D5—34no—
240C2×C4×D5—34no—
241C2×C4×D5—34no—
242C2×C4×D5—31no—
243C2×C4×D5—616no—
244C2×Dic10—32no—
245C2×Dic10—34no—
246C5×C4⋊C4C22×D312yes—
247C5×C4⋊C4C22×D314yes—
248C5×C22⋊C4C22×D318yes—
249C5×C22⋊C4C22×D314yes—
250C42×C5C22×D314yes—
251C42×C5C22×D311yescharacteristic
252C22⋊Dic5—38no—
253C22⋊Dic5—34no—
254#685—38no—
255#685—38no—
256#685—34no—
257#685—34no—
258C4⋊Dic5—34no—
259C4⋊Dic5—32no—
260#683—34no—
261#683—38no—
262C4×Dic5—38no—
263C4×Dic5—34no—
264C4×Dic5—31no—
265C4×Dic5—32no—
266#555—151no—
267#555—154no—
268#555—154no—
269#555—152no—
270#555—154no—
271#555D1511yescharacteristic, direct product factor
272C2×C4⋊Q8—151no—
273C2×#124—154no—
274C2×#121—154no—
275C2×C4×D4—152no—
276C2×#117—154no—
277C22×C3×C5—24no—
278C22×C3×C5—24no—
279C22×C3×C5—24no—
280C22×C3×C5C2511yescharacteristic
281C2×D15—24no—
282C2×D15—24no—
283C2×D15—24no—
284C2×D15—416no—
285C2×D15C2×#4114yes—
286C2×D15H(5,𝔽2)12yes—
287C3×C4×C5—24no—
288C3×C4×C5—21no—
289C3×C4×C5—24no—
290C3×C4×C5C2×#4114yes—
291Dic15—24no—
292Dic15—21no—
293Dic15—24no—
294Dic15C2×#4114yes—
295C23×D3—54no—
296C2×C3×Q8C22×D512yes—
297C2×C3×D4C22×D512yes—
298C2×C3×D4C22×D514yes—
299C22×C3×C4C22×D514yes—
300C2×#67—54no—
301C2×#67—58no—
302C22×Dic3—54no—
303D3×Q8—204no—
304D3×D4—108no—
305D3×D4—208no—
306C2×D12—52no—
307C2×D12—54no—
308C2×C4×D3—52no—
309C2×C4×D3—54no—
310C2×C4×D3—51no—
311C2×C4×D3—54no—
312C2×C4×D3—54no—
313C2×C4×D3—54no—
314C2×C4×D3—1016no—
315C2×Dic6—52no—
316C2×Dic6—54no—
317C3×C4⋊C4C22×D514yes—
318C3×C4⋊C4C22×D512yes—
319C3×C22⋊C4C22×D518yes—
320C3×C22⋊C4C22×D514yes—
321C3×C42C22×D511yescharacteristic
322C3×C42C22×D514yes—
323#217—58no—
324#217—54no—
325(C2×D3)⋊C4—58no—
326(C2×D3)⋊C4—58no—
327(C2×D3)⋊C4—54no—
328(C2×D3)⋊C4—54no—
329C4⋊Dic3—54no—
330C4⋊Dic3—52no—
331Dic3⋊C4—54no—
332Dic3⋊C4—58no—
333C4×Dic3—58no—
334C4×Dic3—54no—
335C4×Dic3—51no—
336C4×Dic3—52no—
337C23×C5C23×D314yes—
338C22×D5—31no—
339C22×D5—34no—
340C22×D5—68no—
341C22×D5—68no—
342C22×D5—68no—
343C5×Q8—42no—
344C5×D4—24no—
345C5×D4—44no—
346C2×C4×C5—28no—
347C2×C4×C5C23×D314yes—
348C2×C4×C5C23×D314yes—
349C2×C4×C5C23×D311yescharacteristic
350C2×C4×C5C23×D312yes—
351C22⋊D5—68no—
352C22⋊D5—128no—
353C2×Dic5—32no—
354C2×Dic5—34no—
355C2×Dic5—34no—
356C2×Dic5—31no—
357C2×Dic5—68no—
358D20—64no—
359D20—124no—
360C4×D5—38no—
361C4×D5—68no—
362C4×D5—68no—
363C4×D5—68no—
364C4×D5—62no—
365C4×D5—68no—
366Dic10—124no—
367Dic10—122no—
368C22×Q8—152no—
369C22×D4—152no—
370C22×D4—154no—
371C23×C4—154no—
372C4⋊Q8—152no—
373C4⋊Q8—154no—
374C4⋊Q8—151no—
375C4⋊Q8C2×D1511yescharacteristic, complemented
376#124—154no—
377#124—154no—
378#124—158no—
379#124—158no—
380#124—154no—
381#124C2×D1514yescomplemented
382#121—154no—
383#121—158no—
384#121—154no—
385#121—154no—
386#121—158no—
387#121C2×D1514yescomplemented
388C4×D4—152no—
389C4×D4—154no—
390C4×D4—152no—
391C4×D4—154no—
392C4×D4—152no—
393C4×D4C2×D1512yescomplemented
394#117—154no—
395#117—154no—
396#117—154no—
397#117—158no—
398#117—158no—
399#117C2×D1514yescomplemented
400C2×C4⋊C4—152no—
401C2×C4⋊C4—154no—
402C2×C22⋊C4—158no—
403C2×C22⋊C4—154no—
404C2×C42—154no—
405C2×C42—151no—
406C2×C3×C5—44no—
407C2×C3×C5C22×#4112yes—
408C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
409D15—44no—
410D15#55514yesdirect product factor
411C23×C3C23×D514yes—
412C22×D3—51no—
413C22×D3—54no—
414C22×D3—108no—
415C22×D3—108no—
416C22×D3—108no—
417C3×Q8—42no—
418C3×D4—24no—
419C3×D4—44no—
420C2×C3×C4—28no—
421C2×C3×C4C23×D514yes—
422C2×C3×C4C23×D514yes—
423C2×C3×C4C23×D511yescharacteristic
424C2×C3×C4C23×D512yes—
425#67—108no—
426#67—208no—
427C2×Dic3—52no—
428C2×Dic3—54no—
429C2×Dic3—54no—
430C2×Dic3—51no—
431C2×Dic3—108no—
432D12—104no—
433D12—204no—
434C4×D3—58no—
435C4×D3—108no—
436C4×D3—108no—
437C4×D3—108no—
438C4×D3—102no—
439C4×D3—108no—
440Dic6—204no—
441Dic6—202no—
442C22×C5—24no—
443C22×C5—24no—
444C22×C5—24no—
445C22×C5C24×D311yescharacteristic
446C2×D5—34no—
447C2×D5—32no—
448C2×D5—64no—
449C2×D5—64no—
450C2×D5—64no—
451C2×D5—1216no—
452C4×C5—24no—
453C4×C5—21no—
454C4×C5—24no—
455C4×C5D3×#4114yes—
456Dic5—34no—
457Dic5—64no—
458Dic5—61no—
459Dic5—64no—
460C24—154no—
461C2×Q8—154no—
462C2×Q8—152no—
463C2×Q8—604no—
464C2×Q8C22×D1512yes—
465C2×D4—154no—
466C2×D4—158no—
467C2×D4—152no—
468C2×D4—154no—
469C2×D4—308no—
470C2×D4—608no—
471C2×D4C22×D1512yes—
472C2×D4C22×D1514yes—
473C22×C4—152no—
474C22×C4—154no—
475C22×C4—154no—
476C22×C4—151no—
477C22×C4—154no—
478C22×C4—154no—
479C22×C4—154no—
480C22×C4—3016no—
481C22×C4C22×D1514yes—
482C4⋊C4—154no—
483C4⋊C4—152no—
484C4⋊C4—158no—
485C4⋊C4—154no—
486C4⋊C4C22×D1512yes—
487C4⋊C4C22×D1514yes—
488C22⋊C4—158no—
489C22⋊C4—154no—
490C22⋊C4—158no—
491C22⋊C4—158no—
492C22⋊C4—154no—
493C22⋊C4—154no—
494C22⋊C4C22×D1518yes—
495C22⋊C4C22×D1514yes—
496C42—154no—
497C42—158no—
498C42—151no—
499C42—152no—
500C42C22×D1514yes—
501C42C22×D1511yescharacteristic
502C3×C5C2×#55511yescharacteristic, complemented, Hall
503C22×C3—24no—
504C22×C3—24no—
505C22×C3—24no—
506C22×C3C24×D511yescharacteristic
507C2×D3—54no—
508C2×D3—52no—
509C2×D3—104no—
510C2×D3—104no—
511C2×D3—104no—
512C2×D3—2016no—
513C3×C4—24no—
514C3×C4—21no—
515C3×C4—24no—
516C3×C4D5×#4114yes—
517Dic3—54no—
518Dic3—104no—
519Dic3—101no—
520Dic3—104no—
521C2×C5—44no—
522C2×C5C2×D3×#4112yes—
523C2×C5D3×H(5,𝔽2)11yescharacteristic
524D5—34no—
525D5—124no—
526C23—151no—
527C23—154no—
528C23—308no—
529C23—308no—
530C23—308no—
531C23C23×D1514yes—
532Q8—42no—
533Q8—604no—
534Q8—602no—
535D4—24no—
536D4—44no—
537D4—308no—
538D4—304no—
539D4—604no—
540D4—608no—
541C2×C4—28no—
542C2×C4—152no—
543C2×C4—158no—
544C2×C4—154no—
545C2×C4—151no—
546C2×C4—154no—
547C2×C4—308no—
548C2×C4—308no—
549C2×C4—308no—
550C2×C4—302no—
551C2×C4—308no—
552C2×C4—308no—
553C2×C4C23×D1514yes—
554C2×C4C23×D1511yescharacteristic
555C2×C4C23×D1514yes—
556C2×C4C23×D1512yes—
557C2×C3—44no—
558C2×C3C2×D5×#4112yes—
559C2×C3D5×H(5,𝔽2)11yescharacteristic
560D3—54no—
561D3—204no—
562C5D3×#55511yescharacteristic, complemented, Sylow
563C22—24no—
564C22—24no—
565C22—24no—
566C22—154no—
567C22—152no—
568C22—304no—
569C22—304no—
570C22—304no—
571C22—6016no—
572C22C24×D1511yescharacteristic
573C4—24no—
574C4—24no—
575C4—21no—
576C4—154no—
577C4—304no—
578C4—304no—
579C4—301no—
580C4#41×D1514yes—
581C3D5×#55511yescharacteristic, complemented, Sylow
582C2—44no—
583C2—154no—
584C2—604no—
585C2C2×#41×D1512yes—
586C2D15×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC22C24×D15
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C5×#555C2
hypercenter#555D15
hypocenterC3×C5C2×#555
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #555

Complements

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

NH ≅ G/Ntype
1. C3×C5×#555C2semidirect
2. #11756493C2semidirect
3. D15×C4⋊Q8C2semidirect
4. D15×#124C2semidirect
5. #11756429C2semidirect
6. #11756375C2semidirect
7. D15×#121C2semidirect
8. #11756322C2semidirect
9. C4×D4×D15C2semidirect
10. #11756291C2semidirect
11. D15×#117C2semidirect
17. C3×C5×C4⋊Q8C22semidirect
18. C3×C5×#124C22semidirect
19. C3×C5×#121C22semidirect
20. C3×C4×C5×D4C22semidirect
21. C3×C5×#117C22semidirect
22. #91836C22semidirect
23. #91833C22semidirect
24. #91828C22semidirect
25. #91827C22semidirect
26. #91825C22semidirect
27. D4×Dic15C22semidirect
28. #91819C22semidirect
29. C4×#1220C22semidirect
30. #91815C22semidirect
31. #91784C22semidirect
32. #91782C22semidirect
33. #91781C22semidirect
34. #91781C22semidirect
37. #91777C22semidirect
38. #91774C22semidirect
39. #91774C22semidirect
40. #91773C22semidirect
41. #91770C22semidirect
44. #91766C22semidirect
45. #91764C22semidirect
46. C4×D60C22semidirect
47. #91761C22semidirect
50. #91758C22semidirect
52. C5×#555D3semidirect
105. C3×#555D5semidirect
120. C5×C4⋊Q8C2×D3semidirect
121. C5×#124C2×D3semidirect
122. C5×#121C2×D3semidirect
123. C4×C5×D4C2×D3semidirect
124. C5×#117C2×D3semidirect
190. C3×C4⋊Q8C2×D5semidirect
191. C3×#124C2×D5semidirect
192. C3×#121C2×D5semidirect
193. C3×C4×D4C2×D5semidirect
194. C3×#117C2×D5semidirect
271. #555D15direct (N × H)
375. C4⋊Q8C2×D15semidirect
381. #124C2×D15semidirect
387. #121C2×D15semidirect
393. C4×D4C2×D15semidirect
399. #117C2×D15semidirect
410. D15#555direct (N × H)
502. C3×C5C2×#555semidirect
562. C5D3×#555semidirect
581. C3D5×#555semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5552
#117564932
D15×C4⋊Q82
D15×#1242
#117564292
#117563752
D15×#1212
#117563222
C4×D4×D152
#117562912
D15×#1172
D5×#5553
D3×#5555