Finite Group Databasebeta

← D15×#533 #14480110

Subgroups — D15×#533 #14480110

Order 1920.

2336 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#533C211yesmaximal, characteristic, complemented
2#11756487C212yesmaximal, complemented
3D15×C4⋊Q8C212yesmaximal, complemented
4#11756447C212yesmaximal, complemented
5D15×#125C212yesmaximal, complemented
6D15×C22⋊Q8C214yesmaximal
7#11756381C214yesmaximal, complemented
8#11756337C214yesmaximal, complemented
9C4×Q8×D15C214yesmaximal, complemented
10D15×#117C212yesmaximal
11#11756276C212yesmaximal, complemented
12C2×C42×D15C211yesmaximal, characteristic
13#11756229C211yesmaximal, characteristic, complemented
14D5×#533—31nomaximal, Hall
15C2×Q8×D15C2214yes—
16C22×C4×D15C2212yes—
17C22×C4×D15C2211yescharacteristic
18C3×C5×C4⋊Q8C2212yescomplemented
19C3×C5×#125C2212yescomplemented
20C3×C5×C22⋊Q8C2214yes—
21C3×C4×C5×Q8C2214yescomplemented
22C3×C5×#117C2212yes—
23C2×C3×C42×C5C2211yescharacteristic
24#91835C2214yes—
25Q8×Dic15C2214yescomplemented
26#91833C2214yescomplemented
27#91815C2212yes—
28#91813C2214yes—
29C2×C4×Dic15C2212yes—
30C2×C4×Dic15C2211yescharacteristic
31#91786C2214yes—
32#91785C2218yes—
33#91781C2214yes—
34C4⋊C4×D15C2218yes—
35C4⋊C4×D15C2214yes—
36C4⋊C4×D15C2214yes—
37#91779C2214yescomplemented
38#91778C2218yescomplemented
39#91777C2214yescomplemented
40#91777C2214yescomplemented
41#91776C2218yescomplemented
42#91776C2214yescomplemented
43C22⋊C4×D15C2214yes—
44#91767C2218yes—
45#91766C2214yes—
46#91761C2214yes—
47C42×D15C2212yes—
48C42×D15C2214yes—
49C42×D15C2212yes—
50C42×D15C2211yescharacteristic
51C42×D15C2211yescharacteristic
52C42×D15C2211yescharacteristic
53C42×D15C2211yescharacteristic
54#91759C2212yescomplemented
55#91758C2212yescomplemented
56C4×Dic30C2214yescomplemented
57C4×Dic30C2218yescomplemented
58D3×#533—51nomaximal, Hall
59C5×#533D311yescharacteristic, complemented
60#66062—32no—
61D5×C4⋊Q8—32no—
62#66022—32no—
63D5×#125—32no—
64D5×C22⋊Q8—34no—
65#65956—34no—
66#65912—34no—
67C4×Q8×D5—34no—
68D5×#117—32no—
69#65851—32no—
70C2×C42×D5—31no—
71#65804—31no—
72C23×D15C2311yescharacteristic
73C2×C3×C5×Q8C2314yes—
74C22×C3×C4×C5C2312yes—
75C22×C3×C4×C5C2311yescharacteristic
76C22×Dic15C2312yes—
77C22×Dic15C2311yescharacteristic
78Q8×D15—216no—
79C2×C4×D15—28no—
80C2×C4×D15Q818yescomplemented
81C2×C4×D15C2312yes—
82C2×C4×D15C2312yes—
83C2×C4×D15C2312yes—
84C2×C4×D15C2318yes—
85C2×C4×D15C2311yescharacteristic
86C2×C4×D15C2311yescharacteristic
87C2×C4×D15C2311yescharacteristic
88C2×C4×D15C2311yescharacteristic
89C2×C4×D15C2312yes—
90C2×Dic30C2314yes—
91C2×Dic30C2318yes—
92C3×C5×C4⋊C4C2314yes—
93C3×C5×C4⋊C4C2318yes—
94C3×C5×C4⋊C4C2314yes—
95C3×C5×C22⋊C4C2314yes—
96C3×C42×C5C2312yes—
97C3×C42×C5C2314yes—
98C3×C42×C5C2311yescharacteristic
99C3×C42×C5C2311yescharacteristic
100#6719C2314yes—
101#6714C2318yes—
102C4⋊Dic15C2314yes—
103C4⋊Dic15C2318yes—
104C4⋊Dic15C2314yes—
105#6712C2318yes—
106#6712C2318yes—
107#6712C2318yes—
108#6712C2318yes—
109C4×Dic15C2312yes—
110C4×Dic15C2314yes—
111C4×Dic15C2312yes—
112C4×Dic15C2312yes—
113C4×Dic15C2311yescharacteristic
114C4×Dic15C2312yes—
115C4×Dic15C2312yes—
116C4×Dic15C2311yescharacteristic
117C4×Dic15C2318yes—
118C3×#533D511yescharacteristic, complemented
119#5728—52no—
120D3×C4⋊Q8—52no—
121#5688—52no—
122D3×#125—52no—
123D3×C22⋊Q8—54no—
124#5622—54no—
125#5578—54no—
126C4×D3×Q8—54no—
127D3×#117—52no—
128#5517—52no—
129C2×C42×D3—51no—
130#5470—51no—
131C2×Q8×D5—34no—
132C22×C4×D5—31no—
133C22×C4×D5—32no—
134C5×C4⋊Q8C2×D312yescomplemented
135C5×#125C2×D312yescomplemented
136C5×C22⋊Q8C2×D314yes—
137C4×C5×Q8C2×D314yescomplemented
138C5×#117C2×D312yes—
139C2×C42×C5C2×D311yescharacteristic
140#4092—34no—
141Q8×Dic5—34no—
142#4090—34no—
143#4072—32no—
144#4070—34no—
145C2×C4×Dic5—31no—
146C2×C4×Dic5—32no—
147#4043—34no—
148#4042—38no—
149#4038—34no—
150D5×C4⋊C4—34no—
151D5×C4⋊C4—34no—
152D5×C4⋊C4—38no—
153#4036—34no—
154#4035—38no—
155#4034—34no—
156#4034—34no—
157#4033—34no—
158#4033—38no—
159D5×C22⋊C4—34no—
160#4024—38no—
161#4023—34no—
162#4018—34no—
163C42×D5—31no—
164C42×D5—31no—
165C42×D5—32no—
166C42×D5—32no—
167C42×D5—31no—
168C42×D5—31no—
169C42×D5—34no—
170#4016—32no—
171#4015—32no—
172C4×Dic10—34no—
173C4×Dic10—38no—
174C2×#533—151noSylow
175C23×C3×C5C2411yescharacteristic
176C22×D15—24no—
177C22×D15C2×Q814yes—
178C22×D15C2411yescharacteristic
179C22×D15C2411yescharacteristic
180C3×C5×Q8—28no—
181C2×C3×C4×C5—24no—
182C2×C3×C4×C5C2×Q814yescomplemented
183C2×C3×C4×C5C2412yes—
184C2×C3×C4×C5C2418yes—
185C2×C3×C4×C5C2411yescharacteristic
186C2×C3×C4×C5C2411yescharacteristic
187C2×C3×C4×C5C2412yes—
188C2×Dic15—24no—
189C2×Dic15C2×Q814yescomplemented
190C2×Dic15C2412yes—
191C2×Dic15C2418yes—
192C2×Dic15C2411yescharacteristic
193C2×Dic15C2411yescharacteristic
194C2×Dic15C2412yes—
195C4×D15—24no—
196C4×D15—216no—
197C4×D15—24no—
198C4×D15—24no—
199C4×D15C2×Q814yes—
200C4×D15C2×Q814yes—
201C4×D15C2×Q814yescomplemented
202C4×D15C2×Q814yescomplemented
203C4×D15#4118yes—
204Dic30—28no—
205Dic30—216no—
206C2×D3×Q8—54no—
207C22×C4×D3—51no—
208C22×C4×D3—52no—
209C3×C4⋊Q8C2×D512yescomplemented
210C3×#125C2×D512yescomplemented
211C3×C22⋊Q8C2×D514yes—
212C3×C4×Q8C2×D514yescomplemented
213C3×#117C2×D512yes—
214C2×C3×C42C2×D511yescharacteristic
215#946—54no—
216Q8×Dic3—54no—
217#944—54no—
218#926—52no—
219#924—54no—
220C2×C4×Dic3—51no—
221C2×C4×Dic3—52no—
222#897—54no—
223#896—58no—
224#892—54no—
225D3×C4⋊C4—54no—
226D3×C4⋊C4—58no—
227D3×C4⋊C4—54no—
228#890—54no—
229#889—58no—
230#888—54no—
231#888—54no—
232#887—58no—
233#887—54no—
234D3×C22⋊C4—54no—
235#878—58no—
236#877—54no—
237#872—54no—
238C42×D3—51no—
239C42×D3—51no—
240C42×D3—51no—
241C42×D3—51no—
242C42×D3—52no—
243C42×D3—52no—
244C42×D3—54no—
245#870—52no—
246#869—52no—
247C4×Dic6—54no—
248C4×Dic6—58no—
249C23×D5—31no—
250C2×C5×Q8C22×D314yes—
251C22×C4×C5C22×D312yes—
252C22×C4×C5C22×D311yescharacteristic
253C22×Dic5—31no—
254C22×Dic5—32no—
255Q8×D5—616no—
256C2×C4×D5—38no—
257C2×C4×D5—32no—
258C2×C4×D5—31no—
259C2×C4×D5—31no—
260C2×C4×D5—31no—
261C2×C4×D5—31no—
262C2×C4×D5—32no—
263C2×C4×D5—32no—
264C2×C4×D5—32no—
265C2×C4×D5—38no—
266C2×C4×D5—68no—
267C2×Dic10—34no—
268C2×Dic10—38no—
269C5×C4⋊C4C22×D314yes—
270C5×C4⋊C4C22×D318yes—
271C5×C4⋊C4C22×D314yes—
272C5×C22⋊C4C22×D314yes—
273C42×C5C22×D312yes—
274C42×C5C22×D311yescharacteristic
275C42×C5C22×D311yescharacteristic
276C42×C5C22×D314yes—
277C22⋊Dic5—34no—
278#685—38no—
279C4⋊Dic5—34no—
280C4⋊Dic5—38no—
281C4⋊Dic5—34no—
282#683—38no—
283#683—38no—
284#683—38no—
285#683—38no—
286C4×Dic5—32no—
287C4×Dic5—34no—
288C4×Dic5—32no—
289C4×Dic5—32no—
290C4×Dic5—31no—
291C4×Dic5—32no—
292C4×Dic5—32no—
293C4×Dic5—31no—
294C4×Dic5—38no—
295#533—152no—
296#533—152no—
297#533—154no—
298#533—154no—
299#533—152no—
300#533—151no—
301#533D1511yescharacteristic, direct product factor
302C2×C4⋊Q8—152no—
303C2×#125—152no—
304C2×C22⋊Q8—154no—
305C2×C4×Q8—154no—
306C2×#117—152no—
307C22×C42—151no—
308C22×C3×C5—22no—
309C22×C3×C5C22×Q812yes—
310C22×C3×C5C2511yescharacteristic
311C2×D15—22no—
312C2×D15—28no—
313C2×D15C22×Q812yes—
314C2×D15C22×Q812yes—
315C2×D15C2×#4114yes—
316C3×C4×C5—28no—
317C3×C4×C5—22no—
318C3×C4×C5C22×Q812yes—
319C3×C4×C5C22×Q812yes—
320C3×C4×C5C2×#4114yes—
321Dic15—28no—
322Dic15—22no—
323Dic15C22×Q812yes—
324Dic15C22×Q812yes—
325Dic15C2×#4114yes—
326C23×D3—51no—
327C2×C3×Q8C22×D514yes—
328C22×C3×C4C22×D512yes—
329C22×C3×C4C22×D511yescharacteristic
330C22×Dic3—51no—
331C22×Dic3—52no—
332D3×Q8—1016no—
333C2×C4×D3—58no—
334C2×C4×D3—51no—
335C2×C4×D3—51no—
336C2×C4×D3—51no—
337C2×C4×D3—52no—
338C2×C4×D3—51no—
339C2×C4×D3—52no—
340C2×C4×D3—52no—
341C2×C4×D3—52no—
342C2×C4×D3—58no—
343C2×C4×D3—108no—
344C2×Dic6—54no—
345C2×Dic6—58no—
346C3×C4⋊C4C22×D514yes—
347C3×C4⋊C4C22×D518yes—
348C3×C4⋊C4C22×D514yes—
349C3×C22⋊C4C22×D514yes—
350C3×C42C22×D512yes—
351C3×C42C22×D514yes—
352C3×C42C22×D511yescharacteristic
353C3×C42C22×D511yescharacteristic
354#217—54no—
355(C2×D3)⋊C4—58no—
356C4⋊Dic3—54no—
357C4⋊Dic3—58no—
358C4⋊Dic3—54no—
359Dic3⋊C4—58no—
360Dic3⋊C4—58no—
361Dic3⋊C4—58no—
362Dic3⋊C4—58no—
363C4×Dic3—52no—
364C4×Dic3—54no—
365C4×Dic3—52no—
366C4×Dic3—52no—
367C4×Dic3—51no—
368C4×Dic3—52no—
369C4×Dic3—52no—
370C4×Dic3—51no—
371C4×Dic3—58no—
372C23×C5C23×D311yescharacteristic
373C22×D5—31no—
374C22×D5—34no—
375C22×D5—31no—
376C22×D5—64no—
377C5×Q8—28no—
378C2×C4×C5—24no—
379C2×C4×C5D3×Q814yescomplemented
380C2×C4×C5C23×D312yes—
381C2×C4×C5C23×D318yes—
382C2×C4×C5C23×D311yescharacteristic
383C2×C4×C5C23×D312yes—
384C2×C4×C5C23×D311yescharacteristic
385C2×Dic5—34no—
386C2×Dic5—31no—
387C2×Dic5—31no—
388C2×Dic5—32no—
389C2×Dic5—32no—
390C2×Dic5—38no—
391C2×Dic5—64no—
392C4×D5—34no—
393C4×D5—34no—
394C4×D5—34no—
395C4×D5—34no—
396C4×D5—38no—
397C4×D5—64no—
398C4×D5—64no—
399C4×D5—616no—
400C4×D5—64no—
401Dic10—616no—
402Dic10—68no—
403C22×Q8—154no—
404C23×C4—151no—
405C23×C4—152no—
406C4⋊Q8—154no—
407C4⋊Q8—154no—
408C4⋊Q8—154no—
409C4⋊Q8—152no—
410C4⋊Q8C2×D1512yescomplemented
411#125—158no—
412#125—154no—
413#125—152no—
414#125C2×D1512yescomplemented
415C22⋊Q8—154no—
416C22⋊Q8—154no—
417C22⋊Q8—158no—
418C22⋊Q8—154no—
419C22⋊Q8—158no—
420C22⋊Q8C2×D1514yes—
421C4×Q8—154no—
422C4×Q8—154no—
423C4×Q8—158no—
424C4×Q8—158no—
425C4×Q8—154no—
426C4×Q8C2×D1514yescomplemented
427#117—152no—
428#117—154no—
429#117—154no—
430#117—154no—
431#117C2×D1512yes—
432C2×C4⋊C4—154no—
433C2×C4⋊C4—158no—
434C2×C4⋊C4—154no—
435C2×C22⋊C4—154no—
436C2×C42—152no—
437C2×C42—151no—
438C2×C42—154no—
439C2×C42—152no—
440C2×C42—151no—
441C2×C42—151no—
442C2×C42—151no—
443C2×C42—151no—
444C2×C42—152no—
445C2×C42C2×D1511yescharacteristic
446C2×C3×C5—22no—
447C2×C3×C5C23×Q811yescharacteristic
448C2×C3×C5C22×#4112yes—
449D15—22no—
450D15#53314yesdirect product factor
451C23×C3C23×D511yescharacteristic
452C22×D3—51no—
453C22×D3—54no—
454C22×D3—51no—
455C22×D3—104no—
456C3×Q8—28no—
457C2×C3×C4—24no—
458C2×C3×C4Q8×D514yescomplemented
459C2×C3×C4C23×D512yes—
460C2×C3×C4C23×D518yes—
461C2×C3×C4C23×D511yescharacteristic
462C2×C3×C4C23×D512yes—
463C2×C3×C4C23×D511yescharacteristic
464C2×Dic3—54no—
465C2×Dic3—51no—
466C2×Dic3—51no—
467C2×Dic3—52no—
468C2×Dic3—52no—
469C2×Dic3—58no—
470C2×Dic3—104no—
471C4×D3—54no—
472C4×D3—54no—
473C4×D3—54no—
474C4×D3—54no—
475C4×D3—58no—
476C4×D3—104no—
477C4×D3—104no—
478C4×D3—1016no—
479C4×D3—104no—
480Dic6—1016no—
481Dic6—108no—
482C22×C5—22no—
483C22×C5C2×D3×Q812yes—
484C22×C5C24×D311yescharacteristic
485C2×D5—32no—
486C2×D5—34no—
487C2×D5—32no—
488C2×D5—68no—
489C2×D5—62no—
490C4×C5—28no—
491C4×C5—22no—
492C4×C5C2×D3×Q812yes—
493C4×C5C2×D3×Q812yes—
494C4×C5D3×#4114yes—
495Dic5—32no—
496Dic5—32no—
497Dic5—34no—
498Dic5—68no—
499Dic5—62no—
500C24—151no—
501C2×Q8—158no—
502C2×Q8—154no—
503C2×Q8—3016no—
504C2×Q8C22×D1514yes—
505C22×C4—151no—
506C22×C4—158no—
507C22×C4—151no—
508C22×C4—152no—
509C22×C4—151no—
510C22×C4—151no—
511C22×C4—151no—
512C22×C4—152no—
513C22×C4—152no—
514C22×C4—152no—
515C22×C4—152no—
516C22×C4—158no—
517C22×C4—308no—
518C22×C4C22×D1512yes—
519C22×C4C22×D1511yescharacteristic
520C4⋊C4—158no—
521C4⋊C4—154no—
522C4⋊C4—158no—
523C4⋊C4—158no—
524C4⋊C4—158no—
525C4⋊C4—158no—
526C4⋊C4—154no—
527C4⋊C4C22×D1514yes—
528C4⋊C4C22×D1518yes—
529C4⋊C4C22×D1514yes—
530C22⋊C4—154no—
531C22⋊C4—158no—
532C22⋊C4C22×D1514yes—
533C42—152no—
534C42—152no—
535C42—152no—
536C42—151no—
537C42—152no—
538C42—151no—
539C42—158no—
540C42—152no—
541C42—154no—
542C42C22×D1511yescharacteristic
543C42C22×D1511yescharacteristic
544C42C22×D1512yes—
545C42C22×D1514yes—
546C3×C5C2×#53311yescharacteristic, complemented, Hall
547C22×C3—22no—
548C22×C3C2×Q8×D512yes—
549C22×C3C24×D511yescharacteristic
550C2×D3—52no—
551C2×D3—54no—
552C2×D3—52no—
553C2×D3—108no—
554C2×D3—102no—
555C3×C4—28no—
556C3×C4—22no—
557C3×C4C2×Q8×D512yes—
558C3×C4C2×Q8×D512yes—
559C3×C4D5×#4114yes—
560Dic3—52no—
561Dic3—52no—
562Dic3—54no—
563Dic3—108no—
564Dic3—102no—
565C2×C5—22no—
566C2×C5C22×D3×Q811yescharacteristic
567C2×C5C2×D3×#4112yes—
568D5—34no—
569D5—62no—
570C23—151no—
571C23—154no—
572C23—151no—
573C23—304no—
574C23C23×D1511yescharacteristic
575Q8—28no—
576Q8—3016no—
577Q8—308no—
578C2×C4—24no—
579C2×C4—154no—
580C2×C4—151no—
581C2×C4—151no—
582C2×C4—152no—
583C2×C4—154no—
584C2×C4—154no—
585C2×C4—154no—
586C2×C4—154no—
587C2×C4—158no—
588C2×C4—152no—
589C2×C4—158no—
590C2×C4—304no—
591C2×C4—304no—
592C2×C4—3016no—
593C2×C4—304no—
594C2×C4—304no—
595C2×C4Q8×D1514yescomplemented
596C2×C4C23×D1512yes—
597C2×C4C23×D1518yes—
598C2×C4C23×D1511yescharacteristic
599C2×C4C23×D1512yes—
600C2×C4C23×D1511yescharacteristic
601C2×C3—22no—
602C2×C3C22×Q8×D511yescharacteristic
603C2×C3C2×D5×#4112yes—
604D3—54no—
605D3—102no—
606C5D3×#53311yescharacteristic, complemented, Sylow
607C22—22no—
608C22—152no—
609C22—154no—
610C22—152no—
611C22—308no—
612C22—302no—
613C22C2×Q8×D1512yes—
614C22C24×D1511yescharacteristic
615C4—22no—
616C4—28no—
617C4—152no—
618C4—152no—
619C4—154no—
620C4—302no—
621C4—308no—
622C4C2×Q8×D1512yes—
623C4C2×Q8×D1512yes—
624C4#41×D1514yes—
625C3D5×#53311yescharacteristic, complemented, Sylow
626C2—22no—
627C2—154no—
628C2—302no—
629C2C22×Q8×D1511yescharacteristic
630C2C2×#41×D1512yes—
subgroupG/H
centerC2×C4C23×D15
commutator, socle, cosocleC22×C3×C5C25
FrattiniC22C24×D15
FittingC3×C5×#533C2
prime radicalC23×D15C23
hypercenter#533D15
hypocenterC3×C5C2×#533
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C3×C5×#533C2semidirect
2. #11756487C2semidirect
3. D15×C4⋊Q8C2semidirect
4. #11756447C2semidirect
5. D15×#125C2semidirect
7. #11756381C2semidirect
8. #11756337C2semidirect
9. C4×Q8×D15C2semidirect
11. #11756276C2semidirect
13. #11756229C2semidirect
18. C3×C5×C4⋊Q8C22semidirect
19. C3×C5×#125C22semidirect
21. C3×C4×C5×Q8C22semidirect
25. Q8×Dic15C22semidirect
26. #91833C22semidirect
37. #91779C22semidirect
38. #91778C22semidirect
39. #91777C22semidirect
40. #91777C22semidirect
41. #91776C22semidirect
42. #91776C22semidirect
54. #91759C22semidirect
55. #91758C22semidirect
56. C4×Dic30C22semidirect
57. C4×Dic30C22semidirect
59. C5×#533D3semidirect
80. C2×C4×D15Q8semidirect
118. C3×#533D5semidirect
134. C5×C4⋊Q8C2×D3semidirect
135. C5×#125C2×D3semidirect
137. C4×C5×Q8C2×D3semidirect
182. C2×C3×C4×C5C2×Q8semidirect
189. C2×Dic15C2×Q8semidirect
201. C4×D15C2×Q8semidirect
202. C4×D15C2×Q8semidirect
209. C3×C4⋊Q8C2×D5semidirect
210. C3×#125C2×D5semidirect
212. C3×C4×Q8C2×D5semidirect
301. #533D15direct (N × H)
379. C2×C4×C5D3×Q8semidirect
410. C4⋊Q8C2×D15semidirect
414. #125C2×D15semidirect
426. C4×Q8C2×D15semidirect
450. D15#533direct (N × H)
458. C2×C3×C4Q8×D5semidirect
546. C3×C5C2×#533semidirect
595. C2×C4Q8×D15semidirect
606. C5D3×#533semidirect
625. C3D5×#533semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5332
#117564872
D15×C4⋊Q82
#117564472
D15×#1252
D15×C22⋊Q82
#117563812
#117563372
C4×Q8×D152
D15×#1172
#117562762
C2×C42×D152
#117562292
D5×#5333
D3×#5335