Finite Group Databasebeta

← D15×#559 #14484698

Subgroups — D15×#559 #14484698

Order 1920.

2480 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#559C211yesmaximal, characteristic, complemented
2D15×#127C211yesmaximal, characteristic
3#11756472C211yesmaximal, characteristic, complemented
4#11756440C214yesmaximal, complemented
5D15×#124C214yesmaximal, complemented
6#11756423C214yesmaximal, complemented
7D15×#123C214yesmaximal, complemented
8#11756344C212yesmaximal, complemented
9C4×Q8×D15C212yesmaximal, complemented
10#11756318C214yesmaximal, complemented
11C4×D4×D15C214yesmaximal, complemented
12D5×#559—31nomaximal, Hall
13C2×Q8×D15C2212yes—
14C2×D4×D15C2214yes—
15C2×D4×D15C2212yes—
16C22×C4×D15C2214yes—
17C3×C5×#127C2211yescharacteristic
18C3×C5×#124C2214yescomplemented
19C3×C5×#123C2214yescomplemented
20C3×C4×C5×Q8C2212yescomplemented
21C3×C4×C5×D4C2214yescomplemented
22#91836C2214yescomplemented
23Q8×Dic15C2212yescomplemented
24#91829C2214yes—
25#91829C2212yes—
26#91825C2214yescomplemented
27#91824C2214yescomplemented
28D4×Dic15C2214yescomplemented
29#91818C2214yescomplemented
30C4×#1220C2218yescomplemented
31#91783C2218yescomplemented
32#91782C2214yescomplemented
33C4⋊C4×D15C2212yes—
34C4⋊C4×D15C2214yes—
35#91776C2212yescomplemented
36#91776C2214yescomplemented
37#91775C2214yescomplemented
38#91774C2218yescomplemented
39#91774C2218yescomplemented
40#91772C2218yescomplemented
41#91770C2218yescomplemented
42C22⋊C4×D15C2218yes—
43C22⋊C4×D15C2214yes—
44#91764C2214yescomplemented
45#91763C2211yescharacteristic
46C4×D60C2214yescomplemented
47C42×D15C2214yes—
48C42×D15C2211yescharacteristic
49C4×Dic30C2214yescomplemented
50C4×Dic30C2212yescomplemented
51D3×#559—51nomaximal, Hall
52C5×#559D311yescharacteristic, complemented
53D5×#127—31no—
54#66047—31no—
55#66015—34no—
56D5×#124—34no—
57#65998—34no—
58D5×#123—34no—
59#65919—32no—
60C4×Q8×D5—32no—
61#65893—34no—
62C4×D4×D5—34no—
63C23×D15C2314yes—
64C2×C3×C5×Q8C2312yes—
65C2×C3×C5×D4C2314yes—
66C2×C3×C5×D4C2312yes—
67C22×C3×C4×C5C2314yes—
68C2×#1220C2318yes—
69C2×#1220C2314yes—
70C22×Dic15C2314yes—
71Q8×D15—28no—
72D4×D15—216no—
73D4×D15—44no—
74C2×D60C2314yes—
75C2×D60C2312yes—
76C2×C4×D15—48no—
77C2×C4×D15C2314yes—
78C2×C4×D15C2314yes—
79C2×C4×D15C2311yescharacteristic
80C2×C4×D15C2318yes—
81C2×C4×D15C2312yes—
82C2×Dic30C2312yes—
83C2×Dic30C2314yes—
84C3×C5×C4⋊C4C2312yes—
85C3×C5×C4⋊C4C2314yes—
86C3×C5×C22⋊C4C2318yes—
87C3×C5×C22⋊C4C2314yes—
88C3×C42×C5C2311yescharacteristic
89C3×C42×C5C2314yes—
90#6719C2318yes—
91#6719C2314yes—
92#6714C2318yes—
93#6714C2318yes—
94#6714C2318yes—
95C4⋊Dic15C2312yes—
96C4⋊Dic15C2314yes—
97#6712C2314yes—
98#6712C2314yes—
99#6712C2314yes—
100C4×Dic15C2312yes—
101C4×Dic15C2314yes—
102C4×Dic15C2314yes—
103C4×Dic15C2314yes—
104C4×Dic15C2311yescharacteristic
105C3×#559D511yescharacteristic, complemented
106D3×#127—51no—
107#5713—51no—
108#5681—54no—
109D3×#124—54no—
110#5664—54no—
111D3×#123—54no—
112#5585—52no—
113C4×D3×Q8—52no—
114#5559—54no—
115C4×D3×D4—54no—
116C2×Q8×D5—32no—
117C2×D4×D5—34no—
118C2×D4×D5—32no—
119C22×C4×D5—34no—
120C5×#127C2×D311yescharacteristic
121C5×#124C2×D314yescomplemented
122C5×#123C2×D314yescomplemented
123C4×C5×Q8C2×D312yescomplemented
124C4×C5×D4C2×D314yescomplemented
125#4093—34no—
126Q8×Dic5—32no—
127#4086—34no—
128#4086—32no—
129#4082—34no—
130#4081—34no—
131D4×Dic5—34no—
132#4075—34no—
133C4×C22⋊D5—38no—
134#4040—38no—
135#4039—34no—
136D5×C4⋊C4—32no—
137D5×C4⋊C4—34no—
138#4033—32no—
139#4033—34no—
140#4032—34no—
141#4031—38no—
142#4031—38no—
143#4029—38no—
144#4027—38no—
145D5×C22⋊C4—38no—
146D5×C22⋊C4—34no—
147#4021—34no—
148#4020—31no—
149C4×D20—34no—
150C42×D5—31no—
151C42×D5—34no—
152C4×Dic10—34no—
153C4×Dic10—32no—
154C2×#559—151noSylow
155C23×C3×C5C2414yes—
156C22×D15—216no—
157C22×D15—28no—
158C22×D15C2414yes—
159C22×D15C2411yescharacteristic
160C3×C5×Q8—24no—
161C3×C5×D4—28no—
162C3×C5×D4—42no—
163C2×C3×C4×C5—44no—
164C2×C3×C4×C5C2411yescharacteristic
165C2×C3×C4×C5C2414yes—
166C2×C3×C4×C5C2414yes—
167C2×C3×C4×C5C2412yes—
168#1220—216no—
169#1220—44no—
170C2×Dic15—44no—
171C2×Dic15C2411yescharacteristic
172C2×Dic15C2414yes—
173C2×Dic15C2414yes—
174C2×Dic15C2412yes—
175D60—28no—
176D60—42no—
177C4×D15—28no—
178C4×D15—28no—
179C4×D15—22no—
180C4×D15—48no—
181C4×D15#4118yes—
182Dic30—24no—
183Dic30—28no—
184C2×D3×Q8—52no—
185C2×D3×D4—54no—
186C2×D3×D4—52no—
187C22×C4×D3—54no—
188C3×#127C2×D511yescharacteristic
189C3×#124C2×D514yescomplemented
190C3×#123C2×D514yescomplemented
191C3×C4×Q8C2×D512yescomplemented
192C3×C4×D4C2×D514yescomplemented
193#947—54no—
194Q8×Dic3—52no—
195#940—52no—
196#940—54no—
197#936—54no—
198#935—54no—
199D4×Dic3—54no—
200#929—54no—
201C4×#67—58no—
202#894—58no—
203#893—54no—
204D3×C4⋊C4—54no—
205D3×C4⋊C4—52no—
206#887—54no—
207#887—52no—
208#886—54no—
209#885—58no—
210#885—58no—
211#883—58no—
212#67⋊C4—58no—
213D3×C22⋊C4—54no—
214D3×C22⋊C4—58no—
215#875—54no—
216#874—51no—
217C4×D12—54no—
218C42×D3—54no—
219C42×D3—51no—
220C4×Dic6—54no—
221C4×Dic6—52no—
222C23×D5—34no—
223C2×C5×Q8C22×D312yes—
224C2×C5×D4C22×D314yes—
225C2×C5×D4C22×D312yes—
226C22×C4×C5C22×D314yes—
227C2×C22⋊D5—38no—
228C2×C22⋊D5—34no—
229C22×Dic5—34no—
230Q8×D5—68no—
231D4×D5—616no—
232D4×D5—124no—
233C2×D20—34no—
234C2×D20—32no—
235C2×C4×D5—32no—
236C2×C4×D5—34no—
237C2×C4×D5—31no—
238C2×C4×D5—34no—
239C2×C4×D5—38no—
240C2×C4×D5—128no—
241C2×Dic10—32no—
242C2×Dic10—34no—
243C5×C4⋊C4C22×D312yes—
244C5×C4⋊C4C22×D314yes—
245C5×C22⋊C4C22×D318yes—
246C5×C22⋊C4C22×D314yes—
247C42×C5C22×D314yes—
248C42×C5C22×D311yescharacteristic
249C22⋊Dic5—38no—
250C22⋊Dic5—34no—
251#685—38no—
252#685—38no—
253#685—38no—
254C4⋊Dic5—32no—
255C4⋊Dic5—34no—
256#683—34no—
257#683—34no—
258#683—34no—
259C4×Dic5—32no—
260C4×Dic5—34no—
261C4×Dic5—34no—
262C4×Dic5—34no—
263C4×Dic5—31no—
264#559—151no—
265#559—154no—
266#559—154no—
267#559—152no—
268#559—154no—
269#559D1511yescharacteristic, direct product factor
270C2×#127—151no—
271C2×#124—154no—
272C2×#123—154no—
273C2×C4×Q8—152no—
274C2×C4×D4—154no—
275C22×C3×C5—28no—
276C22×C3×C5—24no—
277C22×C3×C5C2511yescharacteristic
278C2×D15—28no—
279C2×D15—24no—
280C2×D15—416no—
281C2×D15C2×#4114yes—
282C2×D15H(5,𝔽2)12yes—
283C3×C4×C5—24no—
284C3×C4×C5—24no—
285C3×C4×C5—21no—
286C3×C4×C5C2×#4114yes—
287Dic15—24no—
288Dic15—24no—
289Dic15—21no—
290Dic15C2×#4114yes—
291C23×D3—54no—
292C2×C3×Q8C22×D512yes—
293C2×C3×D4C22×D514yes—
294C2×C3×D4C22×D512yes—
295C22×C3×C4C22×D514yes—
296C2×#67—58no—
297C2×#67—54no—
298C22×Dic3—54no—
299D3×Q8—108no—
300D3×D4—1016no—
301D3×D4—204no—
302C2×D12—54no—
303C2×D12—52no—
304C2×C4×D3—52no—
305C2×C4×D3—54no—
306C2×C4×D3—54no—
307C2×C4×D3—51no—
308C2×C4×D3—58no—
309C2×C4×D3—208no—
310C2×Dic6—52no—
311C2×Dic6—54no—
312C3×C4⋊C4C22×D512yes—
313C3×C4⋊C4C22×D514yes—
314C3×C22⋊C4C22×D518yes—
315C3×C22⋊C4C22×D514yes—
316C3×C42C22×D511yescharacteristic
317C3×C42C22×D514yes—
318#217—58no—
319#217—54no—
320(C2×D3)⋊C4—58no—
321(C2×D3)⋊C4—58no—
322(C2×D3)⋊C4—58no—
323C4⋊Dic3—52no—
324C4⋊Dic3—54no—
325Dic3⋊C4—54no—
326Dic3⋊C4—54no—
327Dic3⋊C4—54no—
328C4×Dic3—52no—
329C4×Dic3—54no—
330C4×Dic3—54no—
331C4×Dic3—54no—
332C4×Dic3—51no—
333C23×C5C23×D314yes—
334C22×D5—31no—
335C22×D5—34no—
336C22×D5—68no—
337C22×D5—616no—
338C5×Q8—24no—
339C5×D4—28no—
340C5×D4—42no—
341C2×C4×C5—44no—
342C2×C4×C5C23×D311yescharacteristic
343C2×C4×C5C23×D314yes—
344C2×C4×C5C23×D314yes—
345C2×C4×C5C23×D312yes—
346C22⋊D5—616no—
347C22⋊D5—124no—
348C2×Dic5—32no—
349C2×Dic5—31no—
350C2×Dic5—34no—
351C2×Dic5—34no—
352C2×Dic5—124no—
353D20—68no—
354D20—122no—
355C4×D5—38no—
356C4×D5—68no—
357C4×D5—62no—
358C4×D5—68no—
359C4×D5—128no—
360Dic10—68no—
361Dic10—64no—
362C22×Q8—152no—
363C22×D4—154no—
364C22×D4—152no—
365C23×C4—154no—
366#127—152no—
367#127—154no—
368#127—151no—
369#127C2×D1511yescharacteristic
370#124—154no—
371#124—154no—
372#124—158no—
373#124—158no—
374#124—154no—
375#124C2×D1514yescomplemented
376#123—154no—
377#123—154no—
378#123—158no—
379#123—158no—
380#123—154no—
381#123C2×D1514yescomplemented
382C4×Q8—152no—
383C4×Q8—154no—
384C4×Q8—152no—
385C4×Q8—152no—
386C4×Q8—154no—
387C4×Q8C2×D1512yescomplemented
388C4×D4—154no—
389C4×D4—158no—
390C4×D4—154no—
391C4×D4—158no—
392C4×D4—154no—
393C4×D4C2×D1514yescomplemented
394C2×C4⋊C4—152no—
395C2×C4⋊C4—154no—
396C2×C22⋊C4—158no—
397C2×C22⋊C4—154no—
398C2×C42—154no—
399C2×C42—151no—
400C2×C3×C5—44no—
401C2×C3×C5C22×#4112yes—
402C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
403D15—44no—
404D15#55914yesdirect product factor
405C23×C3C23×D514yes—
406C22×D3—51no—
407C22×D3—54no—
408C22×D3—108no—
409C22×D3—1016no—
410C3×Q8—24no—
411C3×D4—28no—
412C3×D4—42no—
413C2×C3×C4—44no—
414C2×C3×C4C23×D511yescharacteristic
415C2×C3×C4C23×D514yes—
416C2×C3×C4C23×D514yes—
417C2×C3×C4C23×D512yes—
418#67—1016no—
419#67—204no—
420C2×Dic3—52no—
421C2×Dic3—51no—
422C2×Dic3—54no—
423C2×Dic3—54no—
424C2×Dic3—204no—
425D12—108no—
426D12—202no—
427C4×D3—58no—
428C4×D3—108no—
429C4×D3—102no—
430C4×D3—108no—
431C4×D3—208no—
432Dic6—108no—
433Dic6—104no—
434C22×C5—28no—
435C22×C5—24no—
436C22×C5C24×D311yescharacteristic
437C2×D5—34no—
438C2×D5—32no—
439C2×D5—68no—
440C2×D5—64no—
441C2×D5—1216no—
442C4×C5—24no—
443C4×C5—24no—
444C4×C5—21no—
445C4×C5D3×#4114yes—
446Dic5—34no—
447Dic5—64no—
448Dic5—64no—
449Dic5—61no—
450C24—154no—
451C2×Q8—154no—
452C2×Q8—152no—
453C2×Q8—308no—
454C2×Q8C22×D1512yes—
455C2×D4—158no—
456C2×D4—154no—
457C2×D4—154no—
458C2×D4—152no—
459C2×D4—3016no—
460C2×D4—604no—
461C2×D4C22×D1514yes—
462C2×D4C22×D1512yes—
463C22×C4—152no—
464C22×C4—154no—
465C22×C4—154no—
466C22×C4—154no—
467C22×C4—151no—
468C22×C4—158no—
469C22×C4—608no—
470C22×C4C22×D1514yes—
471C4⋊C4—154no—
472C4⋊C4—152no—
473C4⋊C4—154no—
474C4⋊C4—154no—
475C4⋊C4—154no—
476C4⋊C4C22×D1512yes—
477C4⋊C4C22×D1514yes—
478C22⋊C4—158no—
479C22⋊C4—154no—
480C22⋊C4—158no—
481C22⋊C4—158no—
482C22⋊C4—158no—
483C22⋊C4C22×D1518yes—
484C22⋊C4C22×D1514yes—
485C42—154no—
486C42—154no—
487C42—154no—
488C42—151no—
489C42—152no—
490C42C22×D1514yes—
491C42C22×D1511yescharacteristic
492C3×C5C2×#55911yescharacteristic, complemented, Hall
493C22×C3—28no—
494C22×C3—24no—
495C22×C3C24×D511yescharacteristic
496C2×D3—54no—
497C2×D3—52no—
498C2×D3—108no—
499C2×D3—104no—
500C2×D3—2016no—
501C3×C4—24no—
502C3×C4—24no—
503C3×C4—21no—
504C3×C4D5×#4114yes—
505Dic3—54no—
506Dic3—104no—
507Dic3—104no—
508Dic3—101no—
509C2×C5—44no—
510C2×C5C2×D3×#4112yes—
511C2×C5D3×H(5,𝔽2)11yescharacteristic
512D5—34no—
513D5—124no—
514C23—151no—
515C23—154no—
516C23—308no—
517C23—3016no—
518C23C23×D1514yes—
519Q8—24no—
520Q8—308no—
521Q8—304no—
522D4—28no—
523D4—42no—
524D4—3016no—
525D4—308no—
526D4—602no—
527D4—604no—
528C2×C4—44no—
529C2×C4—152no—
530C2×C4—158no—
531C2×C4—154no—
532C2×C4—154no—
533C2×C4—151no—
534C2×C4—308no—
535C2×C4—302no—
536C2×C4—308no—
537C2×C4—608no—
538C2×C4—604no—
539C2×C4C23×D1514yes—
540C2×C4C23×D1514yes—
541C2×C4C23×D1511yescharacteristic
542C2×C4C23×D1512yes—
543C2×C3—44no—
544C2×C3C2×D5×#4112yes—
545C2×C3D5×H(5,𝔽2)11yescharacteristic
546D3—54no—
547D3—204no—
548C5D3×#55911yescharacteristic, complemented, Sylow
549C22—28no—
550C22—24no—
551C22—154no—
552C22—152no—
553C22—308no—
554C22—304no—
555C22—6016no—
556C22C24×D1511yescharacteristic
557C4—24no—
558C4—24no—
559C4—21no—
560C4—154no—
561C4—304no—
562C4—304no—
563C4—301no—
564C4#41×D1514yes—
565C3D5×#55911yescharacteristic, complemented, Sylow
566C2—44no—
567C2—154no—
568C2—604no—
569C2C2×#41×D1512yes—
570C2D15×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC22C24×D15
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C5×#559C2
prime radicalD15×#127C2
hypercenter#559D15
hypocenterC3×C5C2×#559
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #559

Complements

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

NH ≅ G/Ntype
1. C3×C5×#559C2semidirect
3. #11756472C2semidirect
4. #11756440C2semidirect
5. D15×#124C2semidirect
6. #11756423C2semidirect
7. D15×#123C2semidirect
8. #11756344C2semidirect
9. C4×Q8×D15C2semidirect
10. #11756318C2semidirect
11. C4×D4×D15C2semidirect
18. C3×C5×#124C22semidirect
19. C3×C5×#123C22semidirect
20. C3×C4×C5×Q8C22semidirect
21. C3×C4×C5×D4C22semidirect
22. #91836C22semidirect
23. Q8×Dic15C22semidirect
26. #91825C22semidirect
27. #91824C22semidirect
28. D4×Dic15C22semidirect
29. #91818C22semidirect
30. C4×#1220C22semidirect
31. #91783C22semidirect
32. #91782C22semidirect
35. #91776C22semidirect
36. #91776C22semidirect
37. #91775C22semidirect
38. #91774C22semidirect
39. #91774C22semidirect
40. #91772C22semidirect
41. #91770C22semidirect
44. #91764C22semidirect
46. C4×D60C22semidirect
49. C4×Dic30C22semidirect
50. C4×Dic30C22semidirect
52. C5×#559D3semidirect
105. C3×#559D5semidirect
121. C5×#124C2×D3semidirect
122. C5×#123C2×D3semidirect
123. C4×C5×Q8C2×D3semidirect
124. C4×C5×D4C2×D3semidirect
189. C3×#124C2×D5semidirect
190. C3×#123C2×D5semidirect
191. C3×C4×Q8C2×D5semidirect
192. C3×C4×D4C2×D5semidirect
269. #559D15direct (N × H)
375. #124C2×D15semidirect
381. #123C2×D15semidirect
387. C4×Q8C2×D15semidirect
393. C4×D4C2×D15semidirect
404. D15#559direct (N × H)
492. C3×C5C2×#559semidirect
548. C5D3×#559semidirect
565. C3D5×#559semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5592
D15×#1272
#117564722
#117564402
D15×#1242
#117564232
D15×#1232
#117563442
C4×Q8×D152
#117563182
C4×D4×D152
D5×#5593
D3×#5595