Finite Group Databasebeta

← D15×#419 #14412421

Subgroups — D15×#419 #14412421

Order 1920.

1504 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×D15C211yesmaximal, characteristic
2C2×M16×D15C211yesmaximal, characteristic
3D15×#117C211yesmaximal, characteristic
4C3×C5×#419C211yesmaximal, characteristic, complemented
5#11751326C211yesmaximal, characteristic, complemented
6#11751221C211yesmaximal, characteristic, complemented
7#11751105C211yesmaximal, characteristic, complemented
8#11750903C214yesmaximal, complemented
9D15×#103C214yesmaximal, complemented
10D5×#419—31nomaximal, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C2×Q8×D15—24no—
13C2×Q8×D15—21no—
14C2×Q8×D15—21no—
15C2×Q8×D15C418yescomplemented
16C2×Q8×D15C2212yes—
17C2×Q8×D15C2212yes—
18C22×C4×D15—21no—
19C22×C4×D15C2211yescharacteristic
20C22×Dic30—21no—
21C22×Dic30C2211yescharacteristic
22C2×C3×C5×M16C2211yescharacteristic
23C3×C5×#117C2211yescharacteristic
24#91815C2211yescharacteristic
25C2×#6710C2211yescharacteristic
26M16×D15—24no—
27C2×#6705C2212yes—
28C2×C8×D15C2212yes—
29#91781C2212yes—
30C4⋊C4×D15C2212yes—
31C22⋊C4×D15—21no—
32#91766—21no—
33#91761—21no—
34C42×D15—21no—
35C3×C5×#103C2214yescomplemented
36#91119C2214yescomplemented
37#91100C2214yescomplemented
38#91094C2214yescomplemented
39D3×#419—51nomaximal, Hall
40C22×Q8×D5—31no—
41C2×D5×M16—31no—
42D5×#117—31no—
43C5×#419D311yescharacteristic, complemented
44#65514—31no—
45#65409—31no—
46#65293—31no—
47#65091—34no—
48D5×#103—34no—
49C23×D15D411yescharacteristic
50C2×C3×C5×Q8—22no—
51C2×C3×C5×Q8—21no—
52C2×C3×C5×Q8C2×C414yescomplemented
53C2×C3×C5×Q8C2312yes—
54C22×C3×C4×C5—21no—
55C22×C3×C4×C5C2311yescharacteristic
56C22×Dic15—21no—
57C22×Dic15D411yescharacteristic
58Q8×D15—28no—
59Q8×D15—28no—
60Q8×D15—24no—
61Q8×D15—24no—
62Q8×D15C2×C418yes—
63Q8×D15C2×C418yescomplemented
64C2×C4×D15—28no—
65C2×C4×D15—22no—
66C2×C4×D15—22no—
67C2×C4×D15—22no—
68C2×C4×D15C2×C414yes—
69C2×C4×D15D411yescharacteristic
70C2×C4×D15D411yescharacteristic
71C2×C4×D15D414yes—
72C2×C4×D15C2311yescharacteristic
73C2×C4×D15C2311yescharacteristic
74C2×Dic30—21no—
75C2×Dic30—22no—
76C2×Dic30—28no—
77C2×Dic30—22no—
78C2×Dic30—22no—
79C2×Dic30C2×C414yescomplemented
80C2×Dic30C2312yes—
81C3×C5×M16—22no—
82C2×C3×C5×C8C2312yes—
83C3×C5×C4⋊C4C2312yes—
84C3×C5×C22⋊C4—21no—
85C3×C42×C5—21no—
86#6719—21no—
87#6714—22no—
88C4⋊Dic15C2312yes—
89#6712—22no—
90C4×Dic15—21no—
91C4×Dic15—22no—
92#6710—22no—
93C2×C3⋊#168C2312yes—
94#6705—24no—
95C8×D15—24no—
96C22×D3×Q8—51no—
97C2×D3×M16—51no—
98D3×#117—51no—
99C3×#419D511yescharacteristic, complemented
100#5225—51no—
101#5120—51no—
102#5004—51no—
103#4802—54no—
104D3×#103—54no—
105C22×C5×Q8C2×D311yescharacteristic
106C2×Q8×D5—38no—
107C2×Q8×D5—32no—
108C2×Q8×D5—32no—
109C2×Q8×D5—61no—
110C2×Q8×D5—61no—
111C2×Q8×D5—64no—
112C22×C4×D5—31no—
113C22×C4×D5—61no—
114C22×Dic10—31no—
115C22×Dic10—61no—
116C2×C5×M16C2×D311yescharacteristic
117C5×#117C2×D311yescharacteristic
118#4072—31no—
119C2×#681—31no—
120D5×M16—64no—
121C2×#676—32no—
122C2×C8×D5—32no—
123#4038—32no—
124D5×C4⋊C4—32no—
125D5×C22⋊C4—61no—
126#4023—61no—
127#4018—61no—
128C42×D5—61no—
129C5×#103C2×D314yescomplemented
130#3967—34no—
131#3948—34no—
132#3942—34no—
133C2×#419—151noSylow
134C23×C3×C5C2×D411yescharacteristic
135C22×D15—24no—
136C22×D15C22⋊C414yes—
137C22×D15C2×D411yescharacteristic
138C22×D15C2×D411yescharacteristic
139C3×C5×Q8—24no—
140C3×C5×Q8—22no—
141C3×C5×Q8C22×C414yes—
142C2×C3×C4×C5—24no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5C22×C412yes—
146C2×C3×C4×C5C2×D411yescharacteristic
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2411yescharacteristic
149C2×Dic15—24no—
150C2×Dic15—22no—
151C2×Dic15—22no—
152C2×Dic15C22⋊C414yes—
153C2×Dic15C2×D411yescharacteristic
154C2×Dic15C2×D411yescharacteristic
155C4×D15—28no—
156C4×D15—28no—
157C4×D15—44no—
158C4×D15C22⋊C414yes—
159C4×D15C22⋊C414yes—
160C4×D15C22×C412yes—
161C4×D15C22×C412yes—
162C4×D15C2×D412yes—
163C4×D15C2×D412yes—
164Dic30—24no—
165Dic30—22no—
166Dic30—28no—
167Dic30—28no—
168Dic30C22×C414yes—
169C3×C5×C8—22no—
170C3⋊#168—22no—
171C22×C3×Q8C2×D511yescharacteristic
172C2×D3×Q8—58no—
173C2×D3×Q8—52no—
174C2×D3×Q8—52no—
175C2×D3×Q8—101no—
176C2×D3×Q8—101no—
177C2×D3×Q8—104no—
178C22×C4×D3—51no—
179C22×C4×D3—101no—
180C22×Dic6—51no—
181C22×Dic6—101no—
182C2×C3×M16C2×D511yescharacteristic
183C3×#117C2×D511yescharacteristic
184#926—51no—
185C2×#208—51no—
186D3×M16—104no—
187C2×ΓL(1,𝔽25)—52no—
188C2×D3×C8—52no—
189#892—52no—
190D3×C4⋊C4—52no—
191D3×C22⋊C4—101no—
192#877—101no—
193#872—101no—
194C42×D3—101no—
195C3×#103C2×D514yescomplemented
196#835—54no—
197#816—54no—
198#810—54no—
199C23×D5—31no—
200C2×C5×Q8—22no—
201C2×C5×Q8—21no—
202C2×C5×Q8C4×D314yescomplemented
203C2×C5×Q8C22×D312yes—
204C22×C4×C5—21no—
205C22×C4×C5C22×D311yescharacteristic
206C22×Dic5—31no—
207C22×Dic5—61no—
208Q8×D5—38no—
209Q8×D5—38no—
210Q8×D5—68no—
211Q8×D5—68no—
212Q8×D5—64no—
213Q8×D5—64no—
214C2×C4×D5—34no—
215C2×C4×D5—31no—
216C2×C4×D5—31no—
217C2×C4×D5—34no—
218C2×C4×D5—31no—
219C2×C4×D5—31no—
220C2×C4×D5—68no—
221C2×C4×D5—62no—
222C2×C4×D5—62no—
223C2×C4×D5—62no—
224C2×Dic10—34no—
225C2×Dic10—32no—
226C2×Dic10—68no—
227C2×Dic10—62no—
228C2×Dic10—62no—
229C2×Dic10—61no—
230C2×Dic10—62no—
231C5×M16—22no—
232C2×C5×C8C22×D312yes—
233C5×C4⋊C4C22×D312yes—
234C5×C22⋊C4—21no—
235C42×C5—21no—
236C22⋊Dic5—61no—
237#685—62no—
238C4⋊Dic5—32no—
239#683—62no—
240C4×Dic5—62no—
241C4×Dic5—61no—
242#681—62no—
243C2×#168—32no—
244#676—64no—
245C8×D5—64no—
246C23×Q8—151no—
247C22×M16—151no—
248C2×#117—151no—
249#419—151no—
250#419—151no—
251#419—151no—
252#419—154no—
253#419D1511yescharacteristic, direct product factor
254C2×#103—154no—
255C22×C3×C5—22no—
256C22×C3×C5C2×C22⋊C412yes—
257C22×C3×C5C22×D411yescharacteristic
258C2×D15—22no—
259C2×D15—28no—
260C2×D15C2×C22⋊C412yes—
261C2×D15C2×C22⋊C412yes—
262C2×D15#13714yes—
263C3×C4×C5—24no—
264C3×C4×C5—42no—
265C3×C4×C5C2×C22⋊C412yes—
266C3×C4×C5C23×C411yescharacteristic
267C3×C4×C5C22×D411yescharacteristic
268Dic15—24no—
269Dic15—42no—
270Dic15C2×C22⋊C412yes—
271Dic15C2×C22⋊C412yes—
272C23×D3—51no—
273C2×C3×Q8—21no—
274C2×C3×Q8—22no—
275C2×C3×Q8C4×D514yescomplemented
276C2×C3×Q8C22×D512yes—
277C22×C3×C4—21no—
278C22×C3×C4C22×D511yescharacteristic
279C22×Dic3—51no—
280C22×Dic3—101no—
281D3×Q8—58no—
282D3×Q8—58no—
283D3×Q8—104no—
284D3×Q8—104no—
285D3×Q8—108no—
286D3×Q8—108no—
287C2×C4×D3—54no—
288C2×C4×D3—51no—
289C2×C4×D3—51no—
290C2×C4×D3—54no—
291C2×C4×D3—51no—
292C2×C4×D3—51no—
293C2×C4×D3—102no—
294C2×C4×D3—102no—
295C2×C4×D3—108no—
296C2×C4×D3—102no—
297C2×Dic6—54no—
298C2×Dic6—52no—
299C2×Dic6—102no—
300C2×Dic6—102no—
301C2×Dic6—108no—
302C2×Dic6—102no—
303C2×Dic6—101no—
304C3×M16—22no—
305C2×C3×C8C22×D512yes—
306C3×C4⋊C4C22×D512yes—
307C3×C22⋊C4—21no—
308C3×C42—21no—
309#217—101no—
310(C2×D3)⋊C4—102no—
311C4⋊Dic3—52no—
312Dic3⋊C4—102no—
313C4×Dic3—102no—
314C4×Dic3—101no—
315#208—102no—
316C2×C3⋊C8—52no—
317ΓL(1,𝔽25)—104no—
318D3×C8—104no—
319C23×C5D3×D411yescharacteristic
320C22×D5—31no—
321C22×D5—34no—
322C22×D5—31no—
323C22×D5—64no—
324C5×Q8—24no—
325C5×Q8—22no—
326C5×Q8C2×C4×D314yes—
327C2×C4×C5—24no—
328C2×C4×C5—22no—
329C2×C4×C5—22no—
330C2×C4×C5C2×C4×D312yes—
331C2×C4×C5D3×D411yescharacteristic
332C2×C4×C5D3×D412yes—
333C2×C4×C5C23×D311yescharacteristic
334C2×Dic5—34no—
335C2×Dic5—31no—
336C2×Dic5—31no—
337C2×Dic5—64no—
338C2×Dic5—62no—
339C2×Dic5—62no—
340C4×D5—34no—
341C4×D5—34no—
342C4×D5—32no—
343C4×D5—32no—
344C4×D5—32no—
345C4×D5—32no—
346C4×D5—68no—
347C4×D5—68no—
348C4×D5—124no—
349Dic10—34no—
350Dic10—68no—
351Dic10—68no—
352Dic10—64no—
353Dic10—62no—
354C5×C8—22no—
355#168—62no—
356C22×Q8—158no—
357C22×Q8—152no—
358C22×Q8—152no—
359C22×Q8—151no—
360C22×Q8—301no—
361C22×Q8—301no—
362C22×Q8—301no—
363C22×Q8—304no—
364C22×Q8C2×D1511yescharacteristic
365C23×C4—151no—
366C23×C4—301no—
367C2×M16—151no—
368C2×M16—152no—
369C2×M16—304no—
370C2×M16C2×D1511yescharacteristic
371C22×C8—152no—
372#117—151no—
373#117—152no—
374#117—301no—
375#117—301no—
376#117C2×D1511yescharacteristic
377C2×C4⋊C4—152no—
378C2×C22⋊C4—301no—
379C2×C42—301no—
380#103—154no—
381#103—154no—
382#103—154no—
383#103C2×D1514yescomplemented
384C2×C3×C5—22no—
385C2×C3×C5C22×C22⋊C411yescharacteristic
386C2×C3×C5C2×#13712yes—
387D15—22no—
388D15#41914yesdirect product factor
389C23×C3D4×D511yescharacteristic
390C22×D3—51no—
391C22×D3—51no—
392C22×D3—54no—
393C22×D3—104no—
394C3×Q8—22no—
395C3×Q8—24no—
396C3×Q8C2×C4×D514yes—
397C2×C3×C4—22no—
398C2×C3×C4—24no—
399C2×C3×C4—22no—
400C2×C3×C4C2×C4×D512yes—
401C2×C3×C4D4×D512yes—
402C2×C3×C4D4×D511yescharacteristic
403C2×C3×C4C23×D511yescharacteristic
404C2×Dic3—51no—
405C2×Dic3—51no—
406C2×Dic3—54no—
407C2×Dic3—102no—
408C2×Dic3—104no—
409C2×Dic3—102no—
410C4×D3—52no—
411C4×D3—52no—
412C4×D3—52no—
413C4×D3—52no—
414C4×D3—54no—
415C4×D3—54no—
416C4×D3—108no—
417C4×D3—108no—
418C4×D3—204no—
419Dic6—54no—
420Dic6—108no—
421Dic6—108no—
422Dic6—102no—
423Dic6—104no—
424C3×C8—22no—
425C3⋊C8—102no—
426C22×C5—22no—
427C22×C5D3×C22⋊C412yes—
428C22×C5C2×D3×D411yescharacteristic
429C2×D5—32no—
430C2×D5—34no—
431C2×D5—32no—
432C2×D5—68no—
433C2×D5—62no—
434C4×C5—24no—
435C4×C5—42no—
436C4×C5D3×C22⋊C412yes—
437C4×C5C22×C4×D311yescharacteristic
438C4×C5C2×D3×D411yescharacteristic
439Dic5—32no—
440Dic5—32no—
441Dic5—64no—
442Dic5—122no—
443C24—151no—
444C2×Q8—22no—
445C2×Q8—21no—
446C2×Q8—158no—
447C2×Q8—158no—
448C2×Q8—154no—
449C2×Q8—152no—
450C2×Q8—308no—
451C2×Q8—302no—
452C2×Q8—302no—
453C2×Q8—308no—
454C2×Q8—308no—
455C2×Q8—304no—
456C2×Q8—304no—
457C2×Q8—302no—
458C2×Q8—301no—
459C2×Q8C4×D1514yescomplemented
460C2×Q8C22×D1512yes—
461C22×C4—21no—
462C22×C4—151no—
463C22×C4—154no—
464C22×C4—151no—
465C22×C4—151no—
466C22×C4—154no—
467C22×C4—151no—
468C22×C4—151no—
469C22×C4—308no—
470C22×C4—302no—
471C22×C4—302no—
472C22×C4—302no—
473C22×C4—301no—
474C22×C4C22×D1511yescharacteristic
475M16—22no—
476M16—304no—
477M16—302no—
478C2×C8—152no—
479C2×C8—304no—
480C2×C8C22×D1512yes—
481C4⋊C4—152no—
482C4⋊C4—302no—
483C4⋊C4C22×D1512yes—
484C22⋊C4—21no—
485C22⋊C4—302no—
486C22⋊C4—301no—
487C42—21no—
488C42—302no—
489C42—301no—
490C3×C5C2×#41911yescharacteristic, complemented, Hall
491C22×C3—22no—
492C22×C3D5×C22⋊C412yes—
493C22×C3C2×D4×D511yescharacteristic
494C2×D3—52no—
495C2×D3—54no—
496C2×D3—52no—
497C2×D3—108no—
498C2×D3—102no—
499C3×C4—24no—
500C3×C4—42no—
501C3×C4D5×C22⋊C412yes—
502C3×C4C22×C4×D511yescharacteristic
503C3×C4C2×D4×D511yescharacteristic
504Dic3—52no—
505Dic3—52no—
506Dic3—104no—
507Dic3—202no—
508C2×C5—22no—
509C2×C5C2×D3×C22⋊C411yescharacteristic
510C2×C5D3×#13712yes—
511D5—34no—
512D5—62no—
513C23—151no—
514C23—154no—
515C23—151no—
516C23—304no—
517C23D4×D1511yescharacteristic
518Q8—24no—
519Q8—22no—
520Q8—154no—
521Q8—308no—
522Q8—308no—
523Q8—304no—
524Q8—302no—
525Q8C2×C4×D1514yes—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—154no—
530C2×C4—151no—
531C2×C4—151no—
532C2×C4—154no—
533C2×C4—154no—
534C2×C4—152no—
535C2×C4—152no—
536C2×C4—152no—
537C2×C4—152no—
538C2×C4—308no—
539C2×C4—308no—
540C2×C4—304no—
541C2×C4—302no—
542C2×C4—302no—
543C2×C4—604no—
544C2×C4C2×C4×D1512yes—
545C2×C4D4×D1511yescharacteristic
546C2×C4D4×D1512yes—
547C2×C4C23×D1511yescharacteristic
548C8—22no—
549C8—302no—
550C2×C3—22no—
551C2×C3C2×D5×C22⋊C411yescharacteristic
552C2×C3D5×#13712yes—
553D3—54no—
554D3—102no—
555C5D3×#41911yescharacteristic, complemented, Sylow
556C22—22no—
557C22—152no—
558C22—154no—
559C22—152no—
560C22—308no—
561C22—302no—
562C22C22⋊C4×D1512yes—
563C22C2×D4×D1511yescharacteristic
564C4—24no—
565C4—42no—
566C4—152no—
567C4—152no—
568C4—304no—
569C4—602no—
570C4C22⋊C4×D1512yes—
571C4C22×C4×D1511yescharacteristic
572C4C2×D4×D1511yescharacteristic
573C3D5×#41911yescharacteristic, complemented, Sylow
574C2—22no—
575C2—154no—
576C2—302no—
577C2C2×C22⋊C4×D1511yescharacteristic
578C2D15×#13712yes—
subgroupG/H
centerC22C2×D4×D15
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C23×D15
FittingC3×C5×#419C2
socleC22×C3×C5C22×D4
prime radicalC23×D15D4
cosocleC2×C3×C4×C5C24
hypercenter#419D15
hypocenterC3×C5C2×#419
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
4. C3×C5×#419C2semidirect
5. #11751326C2semidirect
6. #11751221C2semidirect
7. #11751105C2semidirect
8. #11750903C2semidirect
9. D15×#103C2semidirect
15. C2×Q8×D15C4semidirect
35. C3×C5×#103C22semidirect
36. #91119C22semidirect
37. #91100C22semidirect
38. #91094C22semidirect
43. C5×#419D3semidirect
52. C2×C3×C5×Q8C2×C4semidirect
63. Q8×D15C2×C4semidirect
79. C2×Dic30C2×C4semidirect
99. C3×#419D5semidirect
129. C5×#103C2×D3semidirect
195. C3×#103C2×D5semidirect
202. C2×C5×Q8C4×D3semidirect
253. #419D15direct (N × H)
275. C2×C3×Q8C4×D5semidirect
383. #103C2×D15semidirect
388. D15#419direct (N × H)
459. C2×Q8C4×D15semidirect
490. C3×C5C2×#419semidirect
555. C5D3×#419semidirect
573. C3D5×#419semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D152
C2×M16×D152
D15×#1172
C3×C5×#4192
#117513262
#117512212
#117511052
#117509032
D15×#1032
D5×#4193
D3×#4195