Finite Group Databasebeta

← D15×#544 #14481728

Subgroups — D15×#544 #14481728

Order 1920.

2160 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#544C211yesmaximal, characteristic, complemented
2#11756488C214yesmaximal, complemented
3D15×C4⋊Q8C214yesmaximal, complemented
4#11756448C214yesmaximal, complemented
5D15×#125C214yesmaximal, complemented
6D15×C22⋊Q8C214yesmaximal
7#11756382C214yesmaximal, complemented
8D15×#117C212yesmaximal
9#11756278C212yesmaximal, complemented
10C2×C4⋊C4×D15C211yesmaximal, characteristic
11#11756259C211yesmaximal, characteristic, complemented
12D5×#544—31nomaximal, Hall
13C2×Q8×D15C2214yes—
14C22×C4×D15C2212yes—
15C22×C4×D15C2211yescharacteristic
16C3×C5×C4⋊Q8C2214yescomplemented
17C3×C5×#125C2214yescomplemented
18C3×C5×C22⋊Q8C2214yes—
19C3×C5×#117C2212yes—
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21#91835C2214yes—
22#91833C2218yescomplemented
23#91815C2212yes—
24C2×C4⋊Dic15C2211yescharacteristic
25#91813C2214yes—
26C2×#6712C2212yes—
27#91786C2214yes—
28#91785C2218yes—
29#91781C2212yes—
30#91781C2212yes—
31C4⋊C4×D15C2214yes—
32C4⋊C4×D15C2218yes—
33C4⋊C4×D15C2212yes—
34C4⋊C4×D15C2214yes—
35C4⋊C4×D15C2214yes—
36C4⋊C4×D15C2212yes—
37#91779C2214yescomplemented
38#91779C2214yescomplemented
39#91778C2218yescomplemented
40#91778C2218yescomplemented
41#91777C2218yescomplemented
42#91777C2214yescomplemented
43#91777C2214yescomplemented
44C22⋊C4×D15C2214yes—
45#91767C2218yes—
46#91766C2214yes—
47#91761C2214yes—
48C42×D15C2214yes—
49#91759C2214yescomplemented
50#91758C2214yescomplemented
51D3×#544—51nomaximal, Hall
52C5×#544D311yescharacteristic, complemented
53#66063—34no—
54D5×C4⋊Q8—34no—
55#66023—34no—
56D5×#125—34no—
57D5×C22⋊Q8—34no—
58#65957—34no—
59D5×#117—32no—
60#65853—32no—
61C2×D5×C4⋊C4—31no—
62#65834—31no—
63C23×D15C2311yescharacteristic
64C2×C3×C5×Q8C2314yes—
65C22×C3×C4×C5C2312yes—
66C22×C3×C4×C5C2311yescharacteristic
67C22×Dic15C2312yes—
68C22×Dic15C2311yescharacteristic
69Q8×D15—48no—
70C2×C4×D15—44no—
71C2×C4×D15Q818yescomplemented
72C2×C4×D15C2314yes—
73C2×C4×D15C2314yes—
74C2×C4×D15C2314yes—
75C2×C4×D15C2314yes—
76C2×C4×D15C2312yes—
77C2×C4×D15C2312yes—
78C2×Dic30C2318yes—
79C2×Dic30C2314yes—
80C3×C5×C4⋊C4C2312yes—
81C3×C5×C4⋊C4C2318yes—
82C3×C5×C4⋊C4C2314yes—
83C3×C5×C4⋊C4C2314yes—
84C3×C5×C4⋊C4C2312yes—
85C3×C5×C22⋊C4C2314yes—
86C3×C42×C5C2314yes—
87#6719C2314yes—
88#6714C2314yes—
89#6714C2314yes—
90C4⋊Dic15C2312yes—
91C4⋊Dic15C2314yes—
92C4⋊Dic15C2318yes—
93C4⋊Dic15C2312yes—
94C4⋊Dic15C2314yes—
95#6712C2318yes—
96#6712C2318yes—
97#6712C2314yes—
98#6712C2314yes—
99#6712C2318yes—
100#6712C2318yes—
101C4×Dic15C2314yes—
102C4×Dic15C2314yes—
103C4×Dic15C2314yes—
104C3×#544D511yescharacteristic, complemented
105#5729—54no—
106D3×C4⋊Q8—54no—
107#5689—54no—
108D3×#125—54no—
109D3×C22⋊Q8—54no—
110#5623—54no—
111D3×#117—52no—
112#5519—52no—
113C2×D3×C4⋊C4—51no—
114#5500—51no—
115C2×Q8×D5—34no—
116C22×C4×D5—31no—
117C22×C4×D5—32no—
118C5×C4⋊Q8C2×D314yescomplemented
119C5×#125C2×D314yescomplemented
120C5×C22⋊Q8C2×D314yes—
121C5×#117C2×D312yes—
122C2×C5×C4⋊C4C2×D311yescharacteristic
123#4092—34no—
124#4090—38no—
125#4072—32no—
126C2×C4⋊Dic5—31no—
127#4070—34no—
128C2×#683—32no—
129#4043—34no—
130#4042—38no—
131#4038—32no—
132#4038—32no—
133D5×C4⋊C4—32no—
134D5×C4⋊C4—34no—
135D5×C4⋊C4—38no—
136D5×C4⋊C4—34no—
137D5×C4⋊C4—32no—
138D5×C4⋊C4—34no—
139#4036—34no—
140#4036—34no—
141#4035—38no—
142#4035—38no—
143#4034—34no—
144#4034—34no—
145#4034—38no—
146D5×C22⋊C4—34no—
147#4024—38no—
148#4023—34no—
149#4018—34no—
150C42×D5—34no—
151#4016—34no—
152#4015—34no—
153C2×#544—151noSylow
154C23×C3×C5C2411yescharacteristic
155C22×D15—22no—
156C22×D15—22no—
157C22×D15C2×Q814yes—
158C22×D15C2411yescharacteristic
159C22×D15C2411yescharacteristic
160C3×C5×Q8—44no—
161C2×C3×C4×C5—42no—
162C2×C3×C4×C5C2×Q814yescomplemented
163C2×C3×C4×C5C2414yes—
164C2×C3×C4×C5C2414yes—
165C2×C3×C4×C5C2414yes—
166C2×C3×C4×C5C2412yes—
167C2×Dic15—42no—
168C2×Dic15C2×Q814yescomplemented
169C2×Dic15C2414yes—
170C2×Dic15C2414yes—
171C2×Dic15C2414yes—
172C2×Dic15C2412yes—
173C4×D15—28no—
174C4×D15—28no—
175C4×D15—28no—
176C4×D15—44no—
177C4×D15C2×Q818yes—
178C4×D15C2×Q818yescomplemented
179Dic30—44no—
180Dic30—48no—
181C2×D3×Q8—54no—
182C22×C4×D3—51no—
183C22×C4×D3—52no—
184C3×C4⋊Q8C2×D514yescomplemented
185C3×#125C2×D514yescomplemented
186C3×C22⋊Q8C2×D514yes—
187C3×#117C2×D512yes—
188C2×C3×C4⋊C4C2×D511yescharacteristic
189#946—54no—
190#944—58no—
191#926—52no—
192C2×C4⋊Dic3—51no—
193#924—54no—
194C2×Dic3⋊C4—52no—
195#897—54no—
196#896—58no—
197#892—52no—
198#892—52no—
199D3×C4⋊C4—52no—
200D3×C4⋊C4—54no—
201D3×C4⋊C4—54no—
202D3×C4⋊C4—58no—
203D3×C4⋊C4—54no—
204D3×C4⋊C4—52no—
205#890—54no—
206#890—54no—
207#889—58no—
208#889—58no—
209#888—54no—
210#888—58no—
211#888—54no—
212D3×C22⋊C4—54no—
213#878—58no—
214#877—54no—
215#872—54no—
216C42×D3—54no—
217#870—54no—
218#869—54no—
219C23×D5—31no—
220C2×C5×Q8C22×D314yes—
221C22×C4×C5C22×D312yes—
222C22×C4×C5C22×D311yescharacteristic
223C22×Dic5—31no—
224C22×Dic5—32no—
225Q8×D5—128no—
226C2×C4×D5—38no—
227C2×C4×D5—32no—
228C2×C4×D5—32no—
229C2×C4×D5—34no—
230C2×C4×D5—34no—
231C2×C4×D5—34no—
232C2×C4×D5—34no—
233C2×C4×D5—124no—
234C2×Dic10—38no—
235C2×Dic10—34no—
236C5×C4⋊C4C22×D314yes—
237C5×C4⋊C4C22×D312yes—
238C5×C4⋊C4C22×D312yes—
239C5×C4⋊C4C22×D314yes—
240C5×C4⋊C4C22×D318yes—
241C5×C22⋊C4C22×D314yes—
242C42×C5C22×D314yes—
243C22⋊Dic5—34no—
244#685—34no—
245#685—34no—
246C4⋊Dic5—32no—
247C4⋊Dic5—34no—
248C4⋊Dic5—38no—
249C4⋊Dic5—32no—
250C4⋊Dic5—34no—
251#683—38no—
252#683—38no—
253#683—34no—
254#683—34no—
255#683—38no—
256#683—38no—
257C4×Dic5—34no—
258C4×Dic5—34no—
259C4×Dic5—34no—
260#544—154no—
261#544—154no—
262#544—154no—
263#544—152no—
264#544—151no—
265#544D1511yescharacteristic, direct product factor
266C2×C4⋊Q8—154no—
267C2×#125—154no—
268C2×C22⋊Q8—154no—
269C2×#117—152no—
270C22×C4⋊C4—151no—
271C22×C3×C5—21no—
272C22×C3×C5—21no—
273C22×C3×C5C22×Q812yes—
274C22×C3×C5C2511yescharacteristic
275C2×D15—21no—
276C2×D15—21no—
277C2×D15—28no—
278C2×D15C22×Q812yes—
279C2×D15C22×Q812yes—
280C2×D15H(5,𝔽2)12yes—
281C2×D152−1+412yes—
282C3×C4×C5—24no—
283C3×C4×C5—24no—
284C3×C4×C5—24no—
285C3×C4×C5C22×Q814yes—
286Dic15—24no—
287Dic15—24no—
288Dic15—24no—
289Dic15C22×Q814yes—
290C23×D3—51no—
291C2×C3×Q8C22×D514yes—
292C22×C3×C4C22×D512yes—
293C22×C3×C4C22×D511yescharacteristic
294C22×Dic3—51no—
295C22×Dic3—52no—
296D3×Q8—208no—
297C2×C4×D3—58no—
298C2×C4×D3—52no—
299C2×C4×D3—52no—
300C2×C4×D3—54no—
301C2×C4×D3—54no—
302C2×C4×D3—54no—
303C2×C4×D3—54no—
304C2×C4×D3—204no—
305C2×Dic6—58no—
306C2×Dic6—54no—
307C3×C4⋊C4C22×D512yes—
308C3×C4⋊C4C22×D518yes—
309C3×C4⋊C4C22×D514yes—
310C3×C4⋊C4C22×D514yes—
311C3×C4⋊C4C22×D512yes—
312C3×C22⋊C4C22×D514yes—
313C3×C42C22×D514yes—
314#217—54no—
315(C2×D3)⋊C4—54no—
316(C2×D3)⋊C4—54no—
317C4⋊Dic3—52no—
318C4⋊Dic3—54no—
319C4⋊Dic3—58no—
320C4⋊Dic3—52no—
321C4⋊Dic3—54no—
322Dic3⋊C4—58no—
323Dic3⋊C4—58no—
324Dic3⋊C4—54no—
325Dic3⋊C4—54no—
326Dic3⋊C4—58no—
327Dic3⋊C4—58no—
328C4×Dic3—54no—
329C4×Dic3—54no—
330C4×Dic3—54no—
331C23×C5C23×D311yescharacteristic
332C22×D5—31no—
333C22×D5—34no—
334C22×D5—31no—
335C22×D5—62no—
336C22×D5—62no—
337C5×Q8—44no—
338C2×C4×C5—42no—
339C2×C4×C5D3×Q814yescomplemented
340C2×C4×C5C23×D314yes—
341C2×C4×C5C23×D314yes—
342C2×C4×C5C23×D314yes—
343C2×C4×C5C23×D312yes—
344C2×Dic5—34no—
345C2×Dic5—32no—
346C2×Dic5—34no—
347C2×Dic5—34no—
348C2×Dic5—34no—
349C2×Dic5—122no—
350C4×D5—38no—
351C4×D5—38no—
352C4×D5—68no—
353C4×D5—68no—
354C4×D5—68no—
355C4×D5—124no—
356Dic10—124no—
357Dic10—128no—
358C22×Q8—154no—
359C23×C4—151no—
360C23×C4—152no—
361C4⋊Q8—158no—
362C4⋊Q8—158no—
363C4⋊Q8—154no—
364C4⋊Q8—154no—
365C4⋊Q8—154no—
366C4⋊Q8C2×D1514yescomplemented
367#125—158no—
368#125—158no—
369#125—154no—
370#125—154no—
371#125—154no—
372#125C2×D1514yescomplemented
373C22⋊Q8—154no—
374C22⋊Q8—154no—
375C22⋊Q8—158no—
376C22⋊Q8—154no—
377C22⋊Q8—158no—
378C22⋊Q8C2×D1514yes—
379#117—152no—
380#117—152no—
381#117—152no—
382#117—154no—
383#117—154no—
384#117C2×D1512yes—
385C2×C4⋊C4—152no—
386C2×C4⋊C4—151no—
387C2×C4⋊C4—154no—
388C2×C4⋊C4—154no—
389C2×C4⋊C4—152no—
390C2×C4⋊C4—152no—
391C2×C4⋊C4—154no—
392C2×C4⋊C4—158no—
393C2×C4⋊C4C2×D1511yescharacteristic
394C2×C22⋊C4—154no—
395C2×C42—154no—
396C2×C3×C5—22no—
397C2×C3×C5C23×Q811yescharacteristic
398C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
399C2×C3×C5C2×2−1+411yescharacteristic
400D15—22no—
401D15#54414yesdirect product factor
402C23×C3C23×D511yescharacteristic
403C22×D3—51no—
404C22×D3—54no—
405C22×D3—51no—
406C22×D3—102no—
407C22×D3—102no—
408C3×Q8—44no—
409C2×C3×C4—42no—
410C2×C3×C4Q8×D514yescomplemented
411C2×C3×C4C23×D514yes—
412C2×C3×C4C23×D514yes—
413C2×C3×C4C23×D514yes—
414C2×C3×C4C23×D512yes—
415C2×Dic3—54no—
416C2×Dic3—52no—
417C2×Dic3—54no—
418C2×Dic3—54no—
419C2×Dic3—54no—
420C2×Dic3—202no—
421C4×D3—58no—
422C4×D3—58no—
423C4×D3—108no—
424C4×D3—108no—
425C4×D3—108no—
426C4×D3—204no—
427Dic6—204no—
428Dic6—208no—
429C22×C5—21no—
430C22×C5—21no—
431C22×C5C2×D3×Q812yes—
432C22×C5C24×D311yescharacteristic
433C2×D5—32no—
434C2×D5—32no—
435C2×D5—32no—
436C2×D5—32no—
437C2×D5—68no—
438C2×D5—61no—
439C2×D5—61no—
440C4×C5—24no—
441C4×C5—24no—
442C4×C5—24no—
443C4×C5C2×D3×Q814yes—
444Dic5—34no—
445Dic5—64no—
446Dic5—64no—
447Dic5—64no—
448C24—151no—
449C2×Q8—154no—
450C2×Q8—158no—
451C2×Q8—608no—
452C2×Q8C22×D1514yes—
453C22×C4—151no—
454C22×C4—158no—
455C22×C4—152no—
456C22×C4—152no—
457C22×C4—152no—
458C22×C4—154no—
459C22×C4—154no—
460C22×C4—154no—
461C22×C4—154no—
462C22×C4—604no—
463C22×C4C22×D1512yes—
464C22×C4C22×D1511yescharacteristic
465C4⋊C4—158no—
466C4⋊C4—154no—
467C4⋊C4—154no—
468C4⋊C4—152no—
469C4⋊C4—154no—
470C4⋊C4—152no—
471C4⋊C4—158no—
472C4⋊C4—154no—
473C4⋊C4—158no—
474C4⋊C4—158no—
475C4⋊C4—158no—
476C4⋊C4C22×D1514yes—
477C4⋊C4C22×D1512yes—
478C4⋊C4C22×D1512yes—
479C4⋊C4C22×D1514yes—
480C4⋊C4C22×D1518yes—
481C22⋊C4—154no—
482C22⋊C4—154no—
483C22⋊C4—154no—
484C22⋊C4C22×D1514yes—
485C42—154no—
486C42—154no—
487C42—154no—
488C42C22×D1514yes—
489C3×C5C2×#54411yescharacteristic, complemented, Hall
490C22×C3—21no—
491C22×C3—21no—
492C22×C3C2×Q8×D512yes—
493C22×C3C24×D511yescharacteristic
494C2×D3—52no—
495C2×D3—52no—
496C2×D3—52no—
497C2×D3—52no—
498C2×D3—108no—
499C2×D3—101no—
500C2×D3—101no—
501C3×C4—24no—
502C3×C4—24no—
503C3×C4—24no—
504C3×C4C2×Q8×D514yes—
505Dic3—54no—
506Dic3—104no—
507Dic3—104no—
508Dic3—104no—
509C2×C5—22no—
510C2×C5C22×D3×Q811yescharacteristic
511C2×C5D3×H(5,𝔽2)11yescharacteristic
512C2×C5D3×2−1+411yescharacteristic
513D5—34no—
514D5—62no—
515C23—151no—
516C23—154no—
517C23—151no—
518C23—302no—
519C23—302no—
520C23C23×D1511yescharacteristic
521Q8—44no—
522Q8—604no—
523Q8—608no—
524C2×C4—42no—
525C2×C4—154no—
526C2×C4—152no—
527C2×C4—158no—
528C2×C4—158no—
529C2×C4—154no—
530C2×C4—154no—
531C2×C4—154no—
532C2×C4—308no—
533C2×C4—308no—
534C2×C4—308no—
535C2×C4—604no—
536C2×C4—602no—
537C2×C4Q8×D1514yescomplemented
538C2×C4C23×D1514yes—
539C2×C4C23×D1514yes—
540C2×C4C23×D1514yes—
541C2×C4C23×D1512yes—
542C2×C3—22no—
543C2×C3C22×Q8×D511yescharacteristic
544C2×C3D5×H(5,𝔽2)11yescharacteristic
545C2×C3D5×2−1+411yescharacteristic
546D3—54no—
547D3—102no—
548C5D3×#54411yescharacteristic, complemented, Sylow
549C22—21no—
550C22—21no—
551C22—152no—
552C22—152no—
553C22—152no—
554C22—152no—
555C22—308no—
556C22—301no—
557C22—301no—
558C22C2×Q8×D1512yes—
559C22C24×D1511yescharacteristic
560C4—24no—
561C4—24no—
562C4—24no—
563C4—154no—
564C4—304no—
565C4—304no—
566C4—304no—
567C4C2×Q8×D1514yes—
568C3D5×#54411yescharacteristic, complemented, Sylow
569C2—22no—
570C2—154no—
571C2—302no—
572C2C22×Q8×D1511yescharacteristic
573C2D15×H(5,𝔽2)11yescharacteristic
574C2D15×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D15
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C5×#544C2
prime radicalC23×D15C23
hypercenter#544D15
hypocenterC3×C5C2×#544
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #544

Complements

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

NH ≅ G/Ntype
1. C3×C5×#544C2semidirect
2. #11756488C2semidirect
3. D15×C4⋊Q8C2semidirect
4. #11756448C2semidirect
5. D15×#125C2semidirect
7. #11756382C2semidirect
9. #11756278C2semidirect
11. #11756259C2semidirect
16. C3×C5×C4⋊Q8C22semidirect
17. C3×C5×#125C22semidirect
22. #91833C22semidirect
37. #91779C22semidirect
38. #91779C22semidirect
39. #91778C22semidirect
40. #91778C22semidirect
41. #91777C22semidirect
42. #91777C22semidirect
43. #91777C22semidirect
49. #91759C22semidirect
50. #91758C22semidirect
52. C5×#544D3semidirect
71. C2×C4×D15Q8semidirect
104. C3×#544D5semidirect
118. C5×C4⋊Q8C2×D3semidirect
119. C5×#125C2×D3semidirect
162. C2×C3×C4×C5C2×Q8semidirect
168. C2×Dic15C2×Q8semidirect
178. C4×D15C2×Q8semidirect
184. C3×C4⋊Q8C2×D5semidirect
185. C3×#125C2×D5semidirect
265. #544D15direct (N × H)
339. C2×C4×C5D3×Q8semidirect
366. C4⋊Q8C2×D15semidirect
372. #125C2×D15semidirect
401. D15#544direct (N × H)
410. C2×C3×C4Q8×D5semidirect
489. C3×C5C2×#544semidirect
537. C2×C4Q8×D15semidirect
548. C5D3×#544semidirect
568. C3D5×#544semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5442
#117564882
D15×C4⋊Q82
#117564482
D15×#1252
D15×C22⋊Q82
#117563822
D15×#1172
#117562782
C2×C4⋊C4×D152
#117562592
D5×#5443
D3×#5445