Finite Group Databasebeta

← D15×#476 #14429861

Subgroups — D15×#476 #14429861

Order 1920.

960 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D15×C4⋊Q8C211yesmaximal, characteristic, complemented
2C4×D4×D15C211yesmaximal, characteristic
3C3×C5×#476C211yesmaximal, characteristic, complemented
4#11751185C211yesmaximal, characteristic, complemented
5#11751109C211yesmaximal, characteristic, complemented
6#11750971C212yesmaximal, complemented
7D15×#106C212yesmaximal, complemented
8#11750943C211yesmaximal, characteristic, complemented
9D15×C4⋊C8C211yesmaximal, characteristic, complemented
10D15×#102C212yesmaximal
11#11750862C212yesmaximal, complemented
12D5×#476—31nomaximal, Hall
13C2×Q8×D15—21no—
14C2×D4×D15C2211yescharacteristic
15C22×C4×D15—21no—
16C3×C5×C4⋊Q8C2211yescharacteristic, complemented
17C3×C4×C5×D4C2211yescharacteristic
18#91833—21no—
19D4×Dic15C2211yescharacteristic
20C4×#1220—21no—
21C2×C8×D15C2212yes—
22#91782C2211yescharacteristic
23C4⋊C4×D15—21no—
24C4⋊C4×D15C2211yescharacteristic
25C4⋊C4×D15C2212yes—
26#91777—21no—
27#91777C2212yescomplemented
28#91770—21no—
29C22⋊C4×D15—21no—
30C4×D60C2211yescharacteristic
31C42×D15C2211yescharacteristic
32#91758C2211yescharacteristic, complemented
33C3×C5×#106C2212yescomplemented
34C3×C5×C4⋊C8C2211yescharacteristic, complemented
35C3×C5×#102C2212yes—
36#91116C2212yes—
37#91105C2212yes—
38#91101C2212yescomplemented
39#91098C2212yescomplemented
40#91093C2212yes—
41#91092C2212yescomplemented
42#91092C2212yescomplemented
43C4⋊(C3⋊#168)C2211yescharacteristic, complemented
44D3×#476—51nomaximal, Hall
45D5×C4⋊Q8—31no—
46C4×D4×D5—31no—
47C5×#476D311yescharacteristic, complemented
48#65373—31no—
49#65297—31no—
50#65159—32no—
51D5×#106—32no—
52#65131—31no—
53D5×C4⋊C8—31no—
54D5×#102—32no—
55#65050—32no—
56C23×D15—21no—
57C2×C3×C5×Q8—21no—
58C2×C3×C5×D4C2311yescharacteristic
59C22×C3×C4×C5—21no—
60C2×#1220—21no—
61C22×Dic15—21no—
62Q8×D15—44no—
63D4×D15—22no—
64D4×D15Q814yescomplemented
65C2×D60C2311yescharacteristic
66C2×C4×D15—21no—
67C2×C4×D15—22no—
68C2×C4×D15—21no—
69C2×C4×D15—21no—
70C2×C4×D15—44no—
71C2×C4×D15D411yescharacteristic
72C2×C4×D15D411yescharacteristic
73C2×C4×D15C2311yescharacteristic
74C2×Dic30—21no—
75C2×Dic30—22no—
76C2×C3×C5×C8C2312yes—
77C3×C5×C4⋊C4—21no—
78C3×C5×C4⋊C4C2312yes—
79C3×C5×C4⋊C4C2311yescharacteristic
80C3×C5×C22⋊C4—21no—
81C3×C42×C5C2311yescharacteristic
82#6719—21no—
83#6714—21no—
84#6714—21no—
85C4⋊Dic15—21no—
86C4⋊Dic15C2312yes—
87C4⋊Dic15C2311yescharacteristic
88#6712—21no—
89#6712—22no—
90#6712—22no—
91C4×Dic15D411yescharacteristic
92C4×Dic15D411yescharacteristic
93C4×Dic15C2311yescharacteristic
94C2×C3⋊#168C2312yes—
95C8×D15—24no—
96D3×C4⋊Q8—51no—
97C4×D3×D4—51no—
98C3×#476D511yescharacteristic, complemented
99#5084—51no—
100#5008—51no—
101#4870—52no—
102D3×#106—52no—
103#4842—51no—
104D3×C4⋊C8—51no—
105D3×#102—52no—
106#4761—52no—
107C2×Q8×D5—61no—
108C2×D4×D5—31no—
109C22×C4×D5—61no—
110C5×C4⋊Q8C2×D311yescharacteristic, complemented
111C4×C5×D4C2×D311yescharacteristic
112#4090—61no—
113D4×Dic5—31no—
114C4×C22⋊D5—61no—
115C2×C8×D5—32no—
116#4039—31no—
117D5×C4⋊C4—32no—
118D5×C4⋊C4—31no—
119D5×C4⋊C4—61no—
120#4034—32no—
121#4034—61no—
122#4027—61no—
123D5×C22⋊C4—61no—
124C4×D20—31no—
125C42×D5—31no—
126#4015—31no—
127C5×#106C2×D312yescomplemented
128C5×C4⋊C8C2×D311yescharacteristic, complemented
129C5×#102C2×D312yes—
130#3964—32no—
131#3953—32no—
132#3949—32no—
133#3946—32no—
134#3941—32no—
135#3940—32no—
136#3940—32no—
137C4⋊#168—31no—
138C2×#476—151noSylow
139C23×C3×C5—21no—
140C22×D15—21no—
141C22×D15—24no—
142C22×D15—42no—
143C22×D15—42no—
144C22×D15C2×D411yescharacteristic
145C3×C5×Q8—42no—
146C3×C5×D4—21no—
147C3×C5×D4C2×Q812yescomplemented
148C2×C3×C4×C5—22no—
149C2×C3×C4×C5—21no—
150C2×C3×C4×C5—42no—
151C2×C3×C4×C5C2×D411yescharacteristic
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5C2411yescharacteristic
154#1220—24no—
155C2×Dic15—22no—
156C2×Dic15—21no—
157C2×Dic15—42no—
158C2×Dic15C2×D411yescharacteristic
159C2×Dic15C2×D411yescharacteristic
160C2×Dic15C2×D411yescharacteristic
161D60—21no—
162D60C2×Q812yescomplemented
163C4×D15—22no—
164C4×D15—42no—
165C4×D15—44no—
166C4×D15—42no—
167C4×D15—42no—
168C4×D15SD1614yes—
169C4×D15C2×Q812yes—
170C4×D15#4112yes—
171Dic30—44no—
172Dic30—42no—
173C3×C5×C8—22no—
174C3⋊#168—22no—
175C2×D3×Q8—101no—
176C2×D3×D4—51no—
177C22×C4×D3—101no—
178C3×C4⋊Q8C2×D511yescharacteristic, complemented
179C3×C4×D4C2×D511yescharacteristic
180#944—101no—
181D4×Dic3—51no—
182C4×#67—101no—
183C2×D3×C8—52no—
184#893—51no—
185D3×C4⋊C4—52no—
186D3×C4⋊C4—51no—
187D3×C4⋊C4—101no—
188#888—52no—
189#888—101no—
190#67⋊C4—101no—
191D3×C22⋊C4—101no—
192C4×D12—51no—
193C42×D3—51no—
194#869—51no—
195C3×#106C2×D512yescomplemented
196C3×C4⋊C8C2×D511yescharacteristic, complemented
197C3×#102C2×D512yes—
198#832—52no—
199#821—52no—
200#817—52no—
201Dic3⋊C8—52no—
202#809—52no—
203#808—52no—
204#808—52no—
205C4⋊(C3⋊C8)—51no—
206C23×D5—61no—
207C2×C5×Q8—21no—
208C2×C5×D4C22×D311yescharacteristic
209C22×C4×C5—21no—
210C2×C22⋊D5—61no—
211C22×Dic5—61no—
212Q8×D5—124no—
213D4×D5—34no—
214D4×D5—62no—
215C2×D20—31no—
216C2×C4×D5—31no—
217C2×C4×D5—31no—
218C2×C4×D5—31no—
219C2×C4×D5—61no—
220C2×C4×D5—61no—
221C2×C4×D5—62no—
222C2×C4×D5—61no—
223C2×C4×D5—124no—
224C2×Dic10—62no—
225C2×Dic10—61no—
226C2×C5×C8C22×D312yes—
227C5×C4⋊C4—21no—
228C5×C4⋊C4C22×D311yescharacteristic
229C5×C4⋊C4C22×D312yes—
230C5×C22⋊C4—21no—
231C42×C5C22×D311yescharacteristic
232C22⋊Dic5—61no—
233#685—61no—
234#685—61no—
235C4⋊Dic5—31no—
236C4⋊Dic5—32no—
237C4⋊Dic5—61no—
238#683—61no—
239#683—62no—
240#683—62no—
241C4×Dic5—31no—
242C4×Dic5—31no—
243C4×Dic5—31no—
244C2×#168—32no—
245C8×D5—64no—
246C2×C4⋊Q8—151no—
247C2×C4×D4—151no—
248#476—151no—
249#476—151no—
250#476—152no—
251#476—151no—
252#476—152no—
253#476D1511yescharacteristic, direct product factor
254C2×#106—152no—
255C2×C4⋊C8—151no—
256C2×#102—152no—
257C22×C3×C5—22no—
258C22×C3×C5—41no—
259C22×C3×C5—41no—
260C22×C3×C5C22×D411yescharacteristic
261C2×D15—22no—
262C2×D15—41no—
263C2×D15—41no—
264C2×D15—48no—
265C2×D15C22⋊Q812yes—
266C2×D15C2×SD1612yes—
267C2×D15AGL(1,ℤ/8)12yes—
268C3×C4×C5—21no—
269C3×C4×C5—41no—
270C3×C4×C5—42no—
271C3×C4×C5C2×SD1612yes—
272C3×C4×C5C22×Q811yescharacteristic
273C3×C4×C5C2×#4111yescharacteristic
274Dic15—21no—
275Dic15—41no—
276Dic15—42no—
277Dic15C22⋊Q812yes—
278Dic15C2×SD1612yes—
279C23×D3—101no—
280C2×C3×Q8—21no—
281C2×C3×D4C22×D511yescharacteristic
282C22×C3×C4—21no—
283C2×#67—101no—
284C22×Dic3—101no—
285D3×Q8—204no—
286D3×D4—54no—
287D3×D4—102no—
288C2×D12—51no—
289C2×C4×D3—51no—
290C2×C4×D3—51no—
291C2×C4×D3—51no—
292C2×C4×D3—101no—
293C2×C4×D3—101no—
294C2×C4×D3—102no—
295C2×C4×D3—101no—
296C2×C4×D3—204no—
297C2×Dic6—102no—
298C2×Dic6—101no—
299C2×C3×C8C22×D512yes—
300C3×C4⋊C4—21no—
301C3×C4⋊C4C22×D511yescharacteristic
302C3×C4⋊C4C22×D512yes—
303C3×C22⋊C4—21no—
304C3×C42C22×D511yescharacteristic
305#217—101no—
306(C2×D3)⋊C4—101no—
307(C2×D3)⋊C4—101no—
308C4⋊Dic3—51no—
309C4⋊Dic3—52no—
310C4⋊Dic3—101no—
311Dic3⋊C4—101no—
312Dic3⋊C4—102no—
313Dic3⋊C4—102no—
314C4×Dic3—51no—
315C4×Dic3—51no—
316C4×Dic3—51no—
317C2×C3⋊C8—52no—
318D3×C8—104no—
319C23×C5—21no—
320C22×D5—31no—
321C22×D5—64no—
322C22×D5—61no—
323C22×D5—122no—
324C22×D5—122no—
325C5×Q8—42no—
326C5×D4—21no—
327C5×D4D3×Q812yescomplemented
328C2×C4×C5—22no—
329C2×C4×C5—21no—
330C2×C4×C5—42no—
331C2×C4×C5D3×D411yescharacteristic
332C2×C4×C5D3×D411yescharacteristic
333C2×C4×C5C23×D311yescharacteristic
334C22⋊D5—64no—
335C2×Dic5—31no—
336C2×Dic5—31no—
337C2×Dic5—31no—
338C2×Dic5—62no—
339C2×Dic5—61no—
340C2×Dic5—122no—
341D20—32no—
342D20—61no—
343C4×D5—34no—
344C4×D5—32no—
345C4×D5—32no—
346C4×D5—62no—
347C4×D5—122no—
348C4×D5—122no—
349C4×D5—124no—
350C4×D5—122no—
351Dic10—124no—
352Dic10—122no—
353C5×C8—22no—
354#168—62no—
355C22×Q8—301no—
356C22×D4—151no—
357C23×C4—301no—
358C22×C8—152no—
359C4⋊Q8—152no—
360C4⋊Q8—151no—
361C4⋊Q8—301no—
362C4⋊Q8—301no—
363C4⋊Q8C2×D1511yescharacteristic, complemented
364C4×D4—151no—
365C4×D4—151no—
366C4×D4—151no—
367C4×D4—301no—
368C4×D4—301no—
369C4×D4C2×D1511yescharacteristic
370C2×C4⋊C4—151no—
371C2×C4⋊C4—152no—
372C2×C4⋊C4—301no—
373C2×C22⋊C4—301no—
374C2×C42—151no—
375#106—152no—
376#106—152no—
377#106—152no—
378#106C2×D1512yescomplemented
379C4⋊C8—152no—
380C4⋊C8—151no—
381C4⋊C8C2×D1511yescharacteristic, complemented
382#102—152no—
383#102—152no—
384#102—152no—
385#102C2×D1512yes—
386C2×C3×C5—42no—
387C2×C3×C5C2×C22⋊Q811yescharacteristic
388C2×C3×C5C22×SD1611yescharacteristic
389C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
390D15—42no—
391D15#47614yesdirect product factor
392C23×C3—21no—
393C22×D3—51no—
394C22×D3—104no—
395C22×D3—101no—
396C22×D3—202no—
397C22×D3—202no—
398C3×Q8—42no—
399C3×D4—21no—
400C3×D4Q8×D512yescomplemented
401C2×C3×C4—22no—
402C2×C3×C4—21no—
403C2×C3×C4—42no—
404C2×C3×C4D4×D511yescharacteristic
405C2×C3×C4D4×D511yescharacteristic
406C2×C3×C4C23×D511yescharacteristic
407#67—104no—
408C2×Dic3—51no—
409C2×Dic3—51no—
410C2×Dic3—51no—
411C2×Dic3—102no—
412C2×Dic3—101no—
413C2×Dic3—202no—
414D12—52no—
415D12—101no—
416C4×D3—52no—
417C4×D3—52no—
418C4×D3—54no—
419C4×D3—102no—
420C4×D3—202no—
421C4×D3—202no—
422C4×D3—204no—
423C4×D3—202no—
424Dic6—204no—
425Dic6—202no—
426C3×C8—22no—
427C3⋊C8—102no—
428C22×C5—22no—
429C22×C5—41no—
430C22×C5—41no—
431C22×C5C2×D3×D411yescharacteristic
432C2×D5—32no—
433C2×D5—32no—
434C2×D5—32no—
435C2×D5—62no—
436C2×D5—128no—
437C2×D5—121no—
438C2×D5—121no—
439C4×C5—21no—
440C4×C5—41no—
441C4×C5—42no—
442C4×C5D3×SD1612yes—
443C4×C5C2×D3×Q811yescharacteristic
444C4×C5D3×#4111yescharacteristic
445Dic5—32no—
446Dic5—32no—
447Dic5—61no—
448Dic5—121no—
449Dic5—122no—
450C24—301no—
451C2×Q8—21no—
452C2×Q8—302no—
453C2×Q8—301no—
454C2×Q8—604no—
455C2×D4—154no—
456C2×D4—151no—
457C2×D4—302no—
458C2×D4—301no—
459C2×D4C22×D1511yescharacteristic
460C22×C4—21no—
461C22×C4—151no—
462C22×C4—151no—
463C22×C4—151no—
464C22×C4—301no—
465C22×C4—301no—
466C22×C4—302no—
467C22×C4—301no—
468C22×C4—301no—
469C22×C4—604no—
470C2×C8—152no—
471C2×C8—304no—
472C2×C8C22×D1512yes—
473C4⋊C4—21no—
474C4⋊C4—152no—
475C4⋊C4—151no—
476C4⋊C4—301no—
477C4⋊C4—302no—
478C4⋊C4—302no—
479C4⋊C4—301no—
480C4⋊C4C22×D1512yes—
481C4⋊C4C22×D1511yescharacteristic
482C22⋊C4—21no—
483C22⋊C4—301no—
484C22⋊C4—301no—
485C22⋊C4—301no—
486C42—151no—
487C42—151no—
488C42—151no—
489C42C22×D1511yescharacteristic
490C3×C5C2×#47611yescharacteristic, complemented, Hall
491C22×C3—22no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3C2×D4×D511yescharacteristic
495C2×D3—52no—
496C2×D3—52no—
497C2×D3—52no—
498C2×D3—102no—
499C2×D3—208no—
500C2×D3—201no—
501C2×D3—201no—
502C3×C4—21no—
503C3×C4—41no—
504C3×C4—42no—
505C3×C4D5×SD1612yes—
506C3×C4C2×Q8×D511yescharacteristic
507C3×C4D5×#4111yescharacteristic
508Dic3—52no—
509Dic3—52no—
510Dic3—101no—
511Dic3—201no—
512Dic3—202no—
513C2×C5—42no—
514C2×C5D3×C22⋊Q811yescharacteristic
515C2×C5C2×D3×SD1611yescharacteristic
516C2×C5D3×AGL(1,ℤ/8)11yescharacteristic
517D5—34no—
518D5—122no—
519C23—21no—
520C23—151no—
521C23—304no—
522C23—301no—
523C23—602no—
524C23—602no—
525Q8—42no—
526Q8—602no—
527Q8—604no—
528D4—21no—
529D4—152no—
530D4—304no—
531D4—301no—
532D4Q8×D1512yescomplemented
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—42no—
536C2×C4—151no—
537C2×C4—151no—
538C2×C4—154no—
539C2×C4—152no—
540C2×C4—152no—
541C2×C4—151no—
542C2×C4—302no—
543C2×C4—301no—
544C2×C4—302no—
545C2×C4—602no—
546C2×C4—602no—
547C2×C4—604no—
548C2×C4—602no—
549C2×C4—602no—
550C2×C4D4×D1511yescharacteristic
551C2×C4D4×D1511yescharacteristic
552C2×C4C23×D1511yescharacteristic
553C8—22no—
554C8—302no—
555C2×C3—42no—
556C2×C3D5×C22⋊Q811yescharacteristic
557C2×C3C2×D5×SD1611yescharacteristic
558C2×C3D5×AGL(1,ℤ/8)11yescharacteristic
559D3—54no—
560D3—202no—
561C5D3×#47611yescharacteristic, complemented, Sylow
562C22—22no—
563C22—41no—
564C22—41no—
565C22—152no—
566C22—152no—
567C22—152no—
568C22—302no—
569C22—608no—
570C22—601no—
571C22—601no—
572C22C2×D4×D1511yescharacteristic
573C4—21no—
574C4—41no—
575C4—42no—
576C4—152no—
577C4—152no—
578C4—301no—
579C4—601no—
580C4—602no—
581C4SD16×D1512yes—
582C4C2×Q8×D1511yescharacteristic
583C4#41×D1511yescharacteristic
584C3D5×#47611yescharacteristic, complemented, Sylow
585C2—42no—
586C2—154no—
587C2—602no—
588C2D15×C22⋊Q811yescharacteristic
589C2C2×SD16×D1511yescharacteristic
590C2D15×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#476C2
socleC22×C3×C5C22×D4
prime radicalC2×D4×D15C22
hypercenter#476D15
hypocenterC3×C5C2×#476
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. D15×C4⋊Q8C2semidirect
3. C3×C5×#476C2semidirect
4. #11751185C2semidirect
5. #11751109C2semidirect
6. #11750971C2semidirect
7. D15×#106C2semidirect
8. #11750943C2semidirect
9. D15×C4⋊C8C2semidirect
11. #11750862C2semidirect
16. C3×C5×C4⋊Q8C22semidirect
27. #91777C22semidirect
32. #91758C22semidirect
33. C3×C5×#106C22semidirect
34. C3×C5×C4⋊C8C22semidirect
38. #91101C22semidirect
39. #91098C22semidirect
41. #91092C22semidirect
42. #91092C22semidirect
43. C4⋊(C3⋊#168)C22semidirect
47. C5×#476D3semidirect
64. D4×D15Q8semidirect
98. C3×#476D5semidirect
110. C5×C4⋊Q8C2×D3semidirect
127. C5×#106C2×D3semidirect
128. C5×C4⋊C8C2×D3semidirect
147. C3×C5×D4C2×Q8semidirect
162. D60C2×Q8semidirect
178. C3×C4⋊Q8C2×D5semidirect
195. C3×#106C2×D5semidirect
196. C3×C4⋊C8C2×D5semidirect
253. #476D15direct (N × H)
327. C5×D4D3×Q8semidirect
363. C4⋊Q8C2×D15semidirect
378. #106C2×D15semidirect
381. C4⋊C8C2×D15semidirect
391. D15#476direct (N × H)
400. C3×D4Q8×D5semidirect
490. C3×C5C2×#476semidirect
532. D4Q8×D15semidirect
561. C5D3×#476semidirect
584. C3D5×#476semidirect

Maximal subgroups

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

subgroupindex
D15×C4⋊Q82
C4×D4×D152
C3×C5×#4762
#117511852
#117511092
#117509712
D15×#1062
#117509432
D15×C4⋊C82
D15×#1022
#117508622
D5×#4763
D3×#4765