Finite Group Databasebeta

← D15×#562 #14495274

Subgroups — D15×#562 #14495274

Order 1920.

2304 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#562C211yesmaximal, characteristic, complemented
2#11756459C211yesmaximal, characteristic, complemented
3D15×#125C211yesmaximal, characteristic, complemented
4#11756443C212yesmaximal, complemented
5D15×#124C212yesmaximal, complemented
6#11756426C214yesmaximal, complemented
7D15×#123C214yesmaximal, complemented
8#11756406C214yesmaximal, complemented
9D15×C22⋊Q8C214yesmaximal, complemented
10#11756380C214yesmaximal, complemented
11D15×#121C214yesmaximal, complemented
12D5×#562—31nomaximal, Hall
13C2×Q8×D15C2212yes—
14C2×D4×D15C2214yes—
15C2×D4×D15C2212yes—
16C22×C4×D15C2214yes—
17C3×C5×#125C2211yescharacteristic, complemented
18C3×C5×#124C2212yescomplemented
19C3×C5×#123C2214yescomplemented
20C3×C5×C22⋊Q8C2214yescomplemented
21C3×C5×#121C2214yescomplemented
22#91836C2212yescomplemented
23#91835C2214yescomplemented
24#91828C2214yescomplemented
25#91828C2214yescomplemented
26#91827C2214yescomplemented
27#91825C2212yescomplemented
28#91824C2214yescomplemented
29#91819C2214yescomplemented
30#91818C2214yescomplemented
31#91813C2214yescomplemented
32#91786C2214yescomplemented
33#91785C2214yescomplemented
34#91785C2214yescomplemented
35#91784C2214yescomplemented
36#91783C2214yescomplemented
37#91783C2214yescomplemented
38C4⋊C4×D15C2214yes—
39C4⋊C4×D15C2214yes—
40C4⋊C4×D15C2212yes—
41#91779C2212yescomplemented
42#91778C2214yescomplemented
43#91775C2214yescomplemented
44#91774C2214yescomplemented
45#91774C2214yescomplemented
46#91773C2214yescomplemented
47#91773C2214yescomplemented
48#91772C2214yescomplemented
49#91772C2214yescomplemented
50C22⋊C4×D15C2214yes—
51C22⋊C4×D15C2214yes—
52C22⋊C4×D15C2214yes—
53#91767C2214yescomplemented
54#91767C2214yescomplemented
55#91764C2212yescomplemented
56C42×D15C2211yescharacteristic
57#91759C2211yescharacteristic, complemented
58D3×#562—51nomaximal, Hall
59C5×#562D311yescharacteristic, complemented
60#66034—31no—
61D5×#125—31no—
62#66018—32no—
63D5×#124—32no—
64#66001—34no—
65D5×#123—34no—
66#65981—34no—
67D5×C22⋊Q8—34no—
68#65955—34no—
69D5×#121—34no—
70C23×D15C2314yes—
71C2×C3×C5×Q8C2312yes—
72C2×C3×C5×D4C2314yes—
73C2×C3×C5×D4C2312yes—
74C22×C3×C4×C5C2314yes—
75C2×#1220C2314yes—
76C2×#1220C2314yes—
77C2×#1220C2314yes—
78C22×Dic15C2314yes—
79Q8×D15—44no—
80D4×D15—48no—
81D4×D15—44no—
82C2×D60C2312yes—
83C2×D60C2314yes—
84C2×C4×D15—48no—
85C2×C4×D15C2311yescharacteristic
86C2×C4×D15C2312yes—
87C2×C4×D15C2314yes—
88C2×C4×D15C2314yes—
89C2×C4×D15C2314yes—
90C2×C4×D15C2314yes—
91C2×Dic30C2312yes—
92C2×Dic30C2314yes—
93C3×C5×C4⋊C4C2314yes—
94C3×C5×C4⋊C4C2312yes—
95C3×C5×C4⋊C4C2314yes—
96C3×C5×C22⋊C4C2314yes—
97C3×C5×C22⋊C4C2314yes—
98C3×C5×C22⋊C4C2314yes—
99C3×C42×C5C2311yescharacteristic
100#6719C2314yes—
101#6719C2314yes—
102#6719C2314yes—
103#6714C2314yes—
104#6714C2314yes—
105#6714C2314yes—
106#6714C2314yes—
107#6714C2314yes—
108#6714C2314yes—
109C4⋊Dic15C2312yes—
110C4⋊Dic15C2314yes—
111C4⋊Dic15C2314yes—
112#6712C2314yes—
113#6712C2314yes—
114#6712C2314yes—
115#6712C2314yes—
116#6712C2314yes—
117C4×Dic15C2311yescharacteristic
118C4×Dic15C2312yes—
119C3×#562D511yescharacteristic, complemented
120#5700—51no—
121D3×#125—51no—
122#5684—52no—
123D3×#124—52no—
124#5667—54no—
125D3×#123—54no—
126#5647—54no—
127D3×C22⋊Q8—54no—
128#5621—54no—
129D3×#121—54no—
130C2×Q8×D5—32no—
131C2×D4×D5—34no—
132C2×D4×D5—32no—
133C22×C4×D5—34no—
134C5×#125C2×D311yescharacteristic, complemented
135C5×#124C2×D312yescomplemented
136C5×#123C2×D314yescomplemented
137C5×C22⋊Q8C2×D314yescomplemented
138C5×#121C2×D314yescomplemented
139#4093—32no—
140#4092—34no—
141#4085—34no—
142#4085—34no—
143#4084—34no—
144#4082—32no—
145#4081—34no—
146#4076—34no—
147#4075—34no—
148#4070—34no—
149#4043—34no—
150#4042—34no—
151#4042—34no—
152#4041—34no—
153#4040—34no—
154#4040—34no—
155D5×C4⋊C4—34no—
156D5×C4⋊C4—34no—
157D5×C4⋊C4—32no—
158#4036—32no—
159#4035—34no—
160#4032—34no—
161#4031—34no—
162#4031—34no—
163#4030—34no—
164#4030—34no—
165#4029—34no—
166#4029—34no—
167D5×C22⋊C4—34no—
168D5×C22⋊C4—34no—
169D5×C22⋊C4—34no—
170#4024—34no—
171#4024—34no—
172#4021—32no—
173C42×D5—31no—
174#4016—31no—
175C2×#562—151noSylow
176C23×C3×C5C2414yes—
177C22×D15—28no—
178C22×D15—28no—
179C22×D15—28no—
180C22×D15C2414yes—
181C22×D15C2411yescharacteristic
182C3×C5×Q8—42no—
183C3×C5×D4—42no—
184C3×C5×D4—44no—
185C2×C3×C4×C5—44no—
186C2×C3×C4×C5C2412yes—
187C2×C3×C4×C5C2411yescharacteristic
188C2×C3×C4×C5C2414yes—
189C2×C3×C4×C5C2414yes—
190#1220—44no—
191#1220—44no—
192#1220—44no—
193C2×Dic15—44no—
194C2×Dic15C2411yescharacteristic
195C2×Dic15C2412yes—
196C2×Dic15C2414yes—
197C2×Dic15C2414yes—
198D60—42no—
199D60—44no—
200C4×D15—28no—
201C4×D15—28no—
202C4×D15—24no—
203C4×D15—22no—
204C4×D15—44no—
205C4×D15—44no—
206Dic30—42no—
207Dic30—44no—
208C2×D3×Q8—52no—
209C2×D3×D4—54no—
210C2×D3×D4—52no—
211C22×C4×D3—54no—
212C3×#125C2×D511yescharacteristic, complemented
213C3×#124C2×D512yescomplemented
214C3×#123C2×D514yescomplemented
215C3×C22⋊Q8C2×D514yescomplemented
216C3×#121C2×D514yescomplemented
217#947—52no—
218#946—54no—
219#939—54no—
220#939—54no—
221#938—54no—
222#936—52no—
223#935—54no—
224#930—54no—
225#929—54no—
226#924—54no—
227#897—54no—
228#896—54no—
229#896—54no—
230#895—54no—
231#894—54no—
232#894—54no—
233D3×C4⋊C4—52no—
234D3×C4⋊C4—54no—
235D3×C4⋊C4—54no—
236#890—52no—
237#889—54no—
238#886—54no—
239#885—54no—
240#885—54no—
241#884—54no—
242#884—54no—
243#883—54no—
244#883—54no—
245D3×C22⋊C4—54no—
246D3×C22⋊C4—54no—
247D3×C22⋊C4—54no—
248#878—54no—
249#878—54no—
250#875—52no—
251C42×D3—51no—
252#870—51no—
253C23×D5—34no—
254C2×C5×Q8C22×D312yes—
255C2×C5×D4C22×D314yes—
256C2×C5×D4C22×D312yes—
257C22×C4×C5C22×D314yes—
258C2×C22⋊D5—34no—
259C2×C22⋊D5—34no—
260C2×C22⋊D5—34no—
261C22×Dic5—34no—
262Q8×D5—124no—
263D4×D5—128no—
264D4×D5—124no—
265C2×D20—32no—
266C2×D20—34no—
267C2×C4×D5—34no—
268C2×C4×D5—34no—
269C2×C4×D5—32no—
270C2×C4×D5—31no—
271C2×C4×D5—34no—
272C2×C4×D5—34no—
273C2×C4×D5—128no—
274C2×Dic10—32no—
275C2×Dic10—34no—
276C5×C4⋊C4C22×D314yes—
277C5×C4⋊C4C22×D314yes—
278C5×C4⋊C4C22×D312yes—
279C5×C22⋊C4C22×D314yes—
280C5×C22⋊C4C22×D314yes—
281C5×C22⋊C4C22×D314yes—
282C42×C5C22×D311yescharacteristic
283C22⋊Dic5—34no—
284C22⋊Dic5—34no—
285C22⋊Dic5—34no—
286#685—34no—
287#685—34no—
288#685—34no—
289#685—34no—
290#685—34no—
291#685—34no—
292C4⋊Dic5—32no—
293C4⋊Dic5—34no—
294C4⋊Dic5—34no—
295#683—34no—
296#683—34no—
297#683—34no—
298#683—34no—
299#683—34no—
300C4×Dic5—31no—
301C4×Dic5—32no—
302#562—151no—
303#562—152no—
304#562—154no—
305#562—154no—
306#562—154no—
307#562D1511yescharacteristic, direct product factor
308C2×#125—151no—
309C2×#124—152no—
310C2×#123—154no—
311C2×C22⋊Q8—154no—
312C2×#121—154no—
313C22×C3×C5—24no—
314C22×C3×C5—24no—
315C22×C3×C5—24no—
316C22×C3×C5C2511yescharacteristic
317C2×D15—24no—
318C2×D15—24no—
319C2×D15—24no—
320C2×D15—416no—
321C2×D15H(5,𝔽2)14yes—
322C2×D152−1+412yes—
323C3×C4×C5—24no—
324C3×C4×C5—22no—
325C3×C4×C5—24no—
326C3×C4×C5—21no—
327Dic15—24no—
328Dic15—22no—
329Dic15—24no—
330Dic15—21no—
331C23×D3—54no—
332C2×C3×Q8C22×D512yes—
333C2×C3×D4C22×D514yes—
334C2×C3×D4C22×D512yes—
335C22×C3×C4C22×D514yes—
336C2×#67—54no—
337C2×#67—54no—
338C2×#67—54no—
339C22×Dic3—54no—
340D3×Q8—204no—
341D3×D4—208no—
342D3×D4—204no—
343C2×D12—52no—
344C2×D12—54no—
345C2×C4×D3—51no—
346C2×C4×D3—52no—
347C2×C4×D3—54no—
348C2×C4×D3—54no—
349C2×C4×D3—54no—
350C2×C4×D3—54no—
351C2×C4×D3—208no—
352C2×Dic6—52no—
353C2×Dic6—54no—
354C3×C4⋊C4C22×D514yes—
355C3×C4⋊C4C22×D512yes—
356C3×C4⋊C4C22×D514yes—
357C3×C22⋊C4C22×D514yes—
358C3×C22⋊C4C22×D514yes—
359C3×C22⋊C4C22×D514yes—
360C3×C42C22×D511yescharacteristic
361#217—54no—
362#217—54no—
363#217—54no—
364(C2×D3)⋊C4—54no—
365(C2×D3)⋊C4—54no—
366(C2×D3)⋊C4—54no—
367(C2×D3)⋊C4—54no—
368(C2×D3)⋊C4—54no—
369(C2×D3)⋊C4—54no—
370C4⋊Dic3—52no—
371C4⋊Dic3—54no—
372C4⋊Dic3—54no—
373Dic3⋊C4—54no—
374Dic3⋊C4—54no—
375Dic3⋊C4—54no—
376Dic3⋊C4—54no—
377Dic3⋊C4—54no—
378C4×Dic3—51no—
379C4×Dic3—52no—
380C23×C5C23×D314yes—
381C22×D5—31no—
382C22×D5—34no—
383C22×D5—68no—
384C22×D5—68no—
385C22×D5—68no—
386C5×Q8—42no—
387C5×D4—42no—
388C5×D4—44no—
389C2×C4×C5—44no—
390C2×C4×C5C23×D312yes—
391C2×C4×C5C23×D311yescharacteristic
392C2×C4×C5C23×D314yes—
393C2×C4×C5C23×D314yes—
394C22⋊D5—124no—
395C22⋊D5—124no—
396C22⋊D5—124no—
397C2×Dic5—31no—
398C2×Dic5—32no—
399C2×Dic5—34no—
400C2×Dic5—34no—
401C2×Dic5—124no—
402D20—122no—
403D20—124no—
404C4×D5—62no—
405C4×D5—68no—
406C4×D5—68no—
407C4×D5—64no—
408C4×D5—124no—
409C4×D5—124no—
410Dic10—122no—
411Dic10—124no—
412C22×Q8—152no—
413C22×D4—154no—
414C22×D4—152no—
415C23×C4—154no—
416#125—154no—
417#125—152no—
418#125—151no—
419#125C2×D1511yescharacteristic, complemented
420#124—152no—
421#124—152no—
422#124—154no—
423#124—154no—
424#124—152no—
425#124C2×D1512yescomplemented
426#123—154no—
427#123—154no—
428#123—154no—
429#123—154no—
430#123—154no—
431#123—154no—
432#123—154no—
433#123C2×D1514yescomplemented
434C22⋊Q8—154no—
435C22⋊Q8—154no—
436C22⋊Q8—154no—
437C22⋊Q8—154no—
438C22⋊Q8—154no—
439C22⋊Q8—154no—
440C22⋊Q8—154no—
441C22⋊Q8C2×D1514yescomplemented
442#121—154no—
443#121—154no—
444#121—154no—
445#121—154no—
446#121—154no—
447#121—154no—
448#121—154no—
449#121C2×D1514yescomplemented
450C2×C4⋊C4—154no—
451C2×C4⋊C4—154no—
452C2×C4⋊C4—152no—
453C2×C22⋊C4—154no—
454C2×C22⋊C4—154no—
455C2×C22⋊C4—154no—
456C2×C42—151no—
457C2×C3×C5—44no—
458C2×C3×C5C2×H(5,𝔽2)12yes—
459C2×C3×C5C2×2−1+411yescharacteristic
460D15—44no—
461D15#56214yesdirect product factor
462C23×C3C23×D514yes—
463C22×D3—51no—
464C22×D3—54no—
465C22×D3—108no—
466C22×D3—108no—
467C22×D3—108no—
468C3×Q8—42no—
469C3×D4—42no—
470C3×D4—44no—
471C2×C3×C4—44no—
472C2×C3×C4C23×D512yes—
473C2×C3×C4C23×D511yescharacteristic
474C2×C3×C4C23×D514yes—
475C2×C3×C4C23×D514yes—
476#67—204no—
477#67—204no—
478#67—204no—
479C2×Dic3—51no—
480C2×Dic3—52no—
481C2×Dic3—54no—
482C2×Dic3—54no—
483C2×Dic3—204no—
484D12—202no—
485D12—204no—
486C4×D3—102no—
487C4×D3—108no—
488C4×D3—108no—
489C4×D3—104no—
490C4×D3—204no—
491C4×D3—204no—
492Dic6—202no—
493Dic6—204no—
494C22×C5—24no—
495C22×C5—24no—
496C22×C5—24no—
497C22×C5C24×D311yescharacteristic
498C2×D5—34no—
499C2×D5—32no—
500C2×D5—64no—
501C2×D5—64no—
502C2×D5—64no—
503C2×D5—1216no—
504C4×C5—24no—
505C4×C5—22no—
506C4×C5—24no—
507C4×C5—21no—
508Dic5—64no—
509Dic5—62no—
510Dic5—64no—
511Dic5—61no—
512C24—154no—
513C2×Q8—154no—
514C2×Q8—152no—
515C2×Q8—604no—
516C2×Q8C22×D1512yes—
517C2×D4—154no—
518C2×D4—154no—
519C2×D4—154no—
520C2×D4—154no—
521C2×D4—152no—
522C2×D4—608no—
523C2×D4—604no—
524C2×D4C22×D1514yes—
525C2×D4C22×D1512yes—
526C22×C4—154no—
527C22×C4—151no—
528C22×C4—152no—
529C22×C4—154no—
530C22×C4—154no—
531C22×C4—154no—
532C22×C4—154no—
533C22×C4—608no—
534C22×C4C22×D1514yes—
535C4⋊C4—154no—
536C4⋊C4—154no—
537C4⋊C4—154no—
538C4⋊C4—154no—
539C4⋊C4—154no—
540C4⋊C4—154no—
541C4⋊C4—154no—
542C4⋊C4—152no—
543C4⋊C4C22×D1514yes—
544C4⋊C4C22×D1514yes—
545C4⋊C4C22×D1512yes—
546C22⋊C4—154no—
547C22⋊C4—154no—
548C22⋊C4—154no—
549C22⋊C4—154no—
550C22⋊C4—154no—
551C22⋊C4—154no—
552C22⋊C4—154no—
553C22⋊C4—154no—
554C22⋊C4—154no—
555C22⋊C4C22×D1514yes—
556C22⋊C4C22×D1514yes—
557C22⋊C4C22×D1514yes—
558C42—152no—
559C42—151no—
560C42C22×D1511yescharacteristic
561C3×C5C2×#56211yescharacteristic, complemented, Hall
562C22×C3—24no—
563C22×C3—24no—
564C22×C3—24no—
565C22×C3C24×D511yescharacteristic
566C2×D3—54no—
567C2×D3—52no—
568C2×D3—104no—
569C2×D3—104no—
570C2×D3—104no—
571C2×D3—2016no—
572C3×C4—24no—
573C3×C4—22no—
574C3×C4—24no—
575C3×C4—21no—
576Dic3—104no—
577Dic3—102no—
578Dic3—104no—
579Dic3—101no—
580C2×C5—44no—
581C2×C5D3×H(5,𝔽2)12yes—
582C2×C5D3×2−1+411yescharacteristic
583D5—34no—
584D5—124no—
585C23—151no—
586C23—154no—
587C23—308no—
588C23—308no—
589C23—308no—
590C23C23×D1514yes—
591Q8—42no—
592Q8—604no—
593Q8—602no—
594D4—44no—
595D4—42no—
596D4—604no—
597D4—602no—
598D4—604no—
599D4—604no—
600D4—604no—
601C2×C4—44no—
602C2×C4—151no—
603C2×C4—152no—
604C2×C4—154no—
605C2×C4—154no—
606C2×C4—308no—
607C2×C4—308no—
608C2×C4—304no—
609C2×C4—302no—
610C2×C4—604no—
611C2×C4—604no—
612C2×C4—604no—
613C2×C4C23×D1511yescharacteristic
614C2×C4C23×D1512yes—
615C2×C4C23×D1514yes—
616C2×C4C23×D1514yes—
617C2×C3—44no—
618C2×C3D5×H(5,𝔽2)12yes—
619C2×C3D5×2−1+411yescharacteristic
620D3—54no—
621D3—204no—
622C5D3×#56211yescharacteristic, complemented, Sylow
623C22—24no—
624C22—24no—
625C22—24no—
626C22—154no—
627C22—152no—
628C22—304no—
629C22—304no—
630C22—304no—
631C22—6016no—
632C22C24×D1511yescharacteristic
633C4—24no—
634C4—24no—
635C4—22no—
636C4—21no—
637C4—304no—
638C4—304no—
639C4—302no—
640C4—301no—
641C3D5×#56211yescharacteristic, complemented, Sylow
642C2—44no—
643C2—154no—
644C2—604no—
645C2D15×H(5,𝔽2)12yes—
646C2D15×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D15
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C5×#562C2
hypercenter#562D15
hypocenterC3×C5C2×#562
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #562

Complements

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

NH ≅ G/Ntype
1. C3×C5×#562C2semidirect
2. #11756459C2semidirect
3. D15×#125C2semidirect
4. #11756443C2semidirect
5. D15×#124C2semidirect
6. #11756426C2semidirect
7. D15×#123C2semidirect
8. #11756406C2semidirect
9. D15×C22⋊Q8C2semidirect
10. #11756380C2semidirect
11. D15×#121C2semidirect
17. C3×C5×#125C22semidirect
18. C3×C5×#124C22semidirect
19. C3×C5×#123C22semidirect
20. C3×C5×C22⋊Q8C22semidirect
21. C3×C5×#121C22semidirect
22. #91836C22semidirect
23. #91835C22semidirect
24. #91828C22semidirect
25. #91828C22semidirect
26. #91827C22semidirect
27. #91825C22semidirect
28. #91824C22semidirect
29. #91819C22semidirect
30. #91818C22semidirect
31. #91813C22semidirect
32. #91786C22semidirect
33. #91785C22semidirect
34. #91785C22semidirect
35. #91784C22semidirect
36. #91783C22semidirect
37. #91783C22semidirect
41. #91779C22semidirect
42. #91778C22semidirect
43. #91775C22semidirect
44. #91774C22semidirect
45. #91774C22semidirect
46. #91773C22semidirect
47. #91773C22semidirect
48. #91772C22semidirect
49. #91772C22semidirect
53. #91767C22semidirect
54. #91767C22semidirect
55. #91764C22semidirect
57. #91759C22semidirect
59. C5×#562D3semidirect
119. C3×#562D5semidirect
134. C5×#125C2×D3semidirect
135. C5×#124C2×D3semidirect
136. C5×#123C2×D3semidirect
137. C5×C22⋊Q8C2×D3semidirect
138. C5×#121C2×D3semidirect
212. C3×#125C2×D5semidirect
213. C3×#124C2×D5semidirect
214. C3×#123C2×D5semidirect
215. C3×C22⋊Q8C2×D5semidirect
216. C3×#121C2×D5semidirect
307. #562D15direct (N × H)
419. #125C2×D15semidirect
425. #124C2×D15semidirect
433. #123C2×D15semidirect
441. C22⋊Q8C2×D15semidirect
449. #121C2×D15semidirect
461. D15#562direct (N × H)
561. C3×C5C2×#562semidirect
622. C5D3×#562semidirect
641. C3D5×#562semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5622
#117564592
D15×#1252
#117564432
D15×#1242
#117564262
D15×#1232
#117564062
D15×C22⋊Q82
#117563802
D15×#1212
D5×#5623
D3×#5625