Finite Group Databasebeta

← D15×#390 #14320765

Subgroups — D15×#390 #14320765

Order 1920.

2720 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D15C211yesmaximal, characteristic
2C2×C4⋊C4×D15C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D15C214yesmaximal, complemented
4C2×C42×D15C211yesmaximal, characteristic, complemented
5C3×C5×#390C211yesmaximal, characteristic, complemented
6#11751322C211yesmaximal, characteristic, complemented
7#11751089C211yesmaximal, characteristic, complemented
8#11751058C214yesmaximal, complemented
9#11751040C211yesmaximal, characteristic, complemented
10D5×#390—31nomaximal, Hall
11C24×D15C2212yes—
12C22×C3×C5×D4C2211yescharacteristic
13C22×#1220C2212yes—
14C2×D4×D15—24no—
15C2×D4×D15—24no—
16C2×D4×D15C418yescomplemented
17C22×D60C2211yescharacteristic
18C22×C4×D15C2212yes—
19C22×C4×D15C2212yes—
20C22×C4×D15C2211yescharacteristic
21C2×C3×C5×C4⋊C4C2211yescharacteristic, complemented
22C2×C3×C5×C22⋊C4C2214yescomplemented
23C2×C3×C42×C5C2211yescharacteristic, complemented
24C2×#6719C2214yescomplemented
25C2×#6714C2214yescomplemented
26C2×#6714C2214yescomplemented
27C2×C4⋊Dic15C2211yescharacteristic, complemented
28C2×#6712C2212yescomplemented
29C2×C4×Dic15C2212yescomplemented
30C2×C4×Dic15C2211yescharacteristic, complemented
31C4⋊C4×D15—24no—
32C22⋊C4×D15—216no—
33C42×D15—24no—
34D3×#390—51nomaximal, Hall
35C22×D4×D5—31no—
36C2×D5×C4⋊C4—31no—
37C2×D5×C22⋊C4—34no—
38C2×C42×D5—31no—
39C5×#390D311yescharacteristic, complemented
40#65510—31no—
41#65277—31no—
42#65246—34no—
43#65228—31no—
44C24×C3×C5C2312yes—
45C23×D15—28no—
46C23×D15—24no—
47C23×D15—24no—
48C23×D15—28no—
49C23×D15C2×C418yes—
50C23×D15C2312yes—
51C23×D15C2311yescharacteristic
52C2×C3×C5×D4—22no—
53C2×C3×C5×D4—22no—
54C2×C3×C5×D4C2×C414yescomplemented
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2311yescharacteristic
58C2×#1220—24no—
59C2×#1220—24no—
60C2×#1220C2×C418yescomplemented
61C22×Dic15C2312yes—
62C22×Dic15C2312yes—
63C22×Dic15C2311yescharacteristic
64D4×D15—232no—
65C2×D60—22no—
66C2×D60—22no—
67C2×D60C2×C414yescomplemented
68C2×C4×D15—24no—
69C2×C4×D15—24no—
70C2×C4×D15—28no—
71C2×C4×D15—24no—
72C2×C4×D15C2×C414yes—
73C2×C4×D15D414yes—
74C2×C4×D15D414yes—
75C2×C4×D15D418yescomplemented
76C3×C5×C4⋊C4—22no—
77C3×C5×C22⋊C4—28no—
78C3×C42×C5—22no—
79#6719—28no—
80#6714—28no—
81#6714—28no—
82C4⋊Dic15—22no—
83#6712—24no—
84C4×Dic15—22no—
85C4×Dic15—24no—
86C22×D3×D4—51no—
87C2×D3×C4⋊C4—51no—
88C2×D3×C22⋊C4—54no—
89C2×C42×D3—51no—
90C3×#390D511yescharacteristic, complemented
91#5221—51no—
92#4988—51no—
93#4957—54no—
94#4939—51no—
95C24×D5—32no—
96C22×C5×D4C2×D311yescharacteristic
97C22×C22⋊D5—32no—
98C2×D4×D5—38no—
99C2×D4×D5—64no—
100C2×D4×D5—64no—
101C22×D20—31no—
102C22×C4×D5—31no—
103C22×C4×D5—32no—
104C22×C4×D5—32no—
105C2×C5×C4⋊C4C2×D311yescharacteristic, complemented
106C2×C5×C22⋊C4C2×D314yescomplemented
107C2×C42×C5C2×D311yescharacteristic, complemented
108C2×C22⋊Dic5—34no—
109C2×#685—34no—
110C2×#685—34no—
111C2×C4⋊Dic5—31no—
112C2×#683—32no—
113C2×C4×Dic5—32no—
114C2×C4×Dic5—31no—
115D5×C4⋊C4—64no—
116D5×C22⋊C4—616no—
117C42×D5—64no—
118C2×#390—151noSylow
119C23×C3×C5—24no—
120C23×C3×C5—22no—
121C23×C3×C5—22no—
122C23×C3×C5—24no—
123C23×C3×C5C22×C414yes—
124C23×C3×C5C2411yescharacteristic
125C22×D15—24no—
126C22×D15—22no—
127C22×D15—22no—
128C22×D15—24no—
129C22×D15—216no—
130C22×D15—232no—
131C22×D15—416no—
132C22×D15—48no—
133C22×D15—48no—
134C22×D15C22×C414yes—
135C22×D15C22×C412yes—
136C22×D15C2×D412yes—
137C22×D15C2×D412yes—
138C22×D15C2×D414yes—
139C22×D15#4114yes—
140C3×C5×D4—28no—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5—24no—
145C2×C3×C4×C5C22×C412yes—
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2×D414yescomplemented
149#1220—216no—
150C2×Dic15—22no—
151C2×Dic15—22no—
152C2×Dic15—22no—
153C2×Dic15—24no—
154C2×Dic15C22×C412yes—
155C2×Dic15C2×D412yes—
156C2×Dic15C2×D412yes—
157C2×Dic15C2×D414yescomplemented
158D60—28no—
159C4×D15—216no—
160C4×D15—48no—
161C4×D15C22⋊C4116yescomplemented
162C24×D3—52no—
163C22×C3×D4C2×D511yescharacteristic
164C22×#67—52no—
165C2×D3×D4—58no—
166C2×D3×D4—104no—
167C2×D3×D4—104no—
168C22×D12—51no—
169C22×C4×D3—51no—
170C22×C4×D3—52no—
171C22×C4×D3—52no—
172C2×C3×C4⋊C4C2×D511yescharacteristic, complemented
173C2×C3×C22⋊C4C2×D514yescomplemented
174C2×C3×C42C2×D511yescharacteristic, complemented
175C2×#217—54no—
176C2×(C2×D3)⋊C4—54no—
177C2×(C2×D3)⋊C4—54no—
178C2×C4⋊Dic3—51no—
179C2×Dic3⋊C4—52no—
180C2×C4×Dic3—52no—
181C2×C4×Dic3—51no—
182D3×C4⋊C4—104no—
183D3×C22⋊C4—1016no—
184C42×D3—104no—
185C24×C5C22×D312yes—
186C23×D5—31no—
187C23×D5—38no—
188C23×D5—32no—
189C23×D5—64no—
190C23×D5—64no—
191C23×D5—68no—
192C23×D5—68no—
193C2×C5×D4—22no—
194C2×C5×D4—22no—
195C2×C5×D4C4×D314yescomplemented
196C22×C4×C5C22×D312yes—
197C22×C4×C5C22×D312yes—
198C22×C4×C5C22×D311yescharacteristic
199C2×C22⋊D5—38no—
200C2×C22⋊D5—64no—
201C2×C22⋊D5—64no—
202C22×Dic5—31no—
203C22×Dic5—32no—
204C22×Dic5—32no—
205D4×D5—632no—
206C2×D20—34no—
207C2×D20—62no—
208C2×D20—62no—
209C2×C4×D5—34no—
210C2×C4×D5—34no—
211C2×C4×D5—34no—
212C2×C4×D5—38no—
213C2×C4×D5—64no—
214C2×C4×D5—64no—
215C2×C4×D5—68no—
216C2×C4×D5—64no—
217C5×C4⋊C4—22no—
218C5×C22⋊C4—28no—
219C42×C5—22no—
220C22⋊Dic5—68no—
221#685—68no—
222#685—68no—
223C4⋊Dic5—62no—
224#683—64no—
225C4×Dic5—64no—
226C4×Dic5—62no—
227C23×D4—151no—
228C22×C4⋊C4—151no—
229C22×C22⋊C4—154no—
230C22×C42—151no—
231#390—151no—
232#390—151no—
233#390—154no—
234#390—151no—
235#390D1511yescharacteristic, direct product factor
236C22×C3×C5—24no—
237C22×C3×C5—28no—
238C22×C3×C5—44no—
239C22×C3×C5—42no—
240C22×C3×C5—42no—
241C22×C3×C5C23×C411yescharacteristic
242C22×C3×C5C22×D411yescharacteristic
243C22×C3×C5C22×D411yescharacteristic
244C22×C3×C5C22×D412yes—
245C22×C3×C5C2×#4112yes—
246C2×D15—24no—
247C2×D15—28no—
248C2×D15—44no—
249C2×D15—42no—
250C2×D15—42no—
251C2×D15—432no—
252C2×D15C2×C22⋊C414yes—
253C2×D15C4×D418yes—
254C2×D15#12118yes—
255C2×D15#12414yes—
256C2×D15#12714yes—
257C3×C4×C5—24no—
258C3×C4×C5—42no—
259C3×C4×C5C2×C22⋊C414yescomplemented
260Dic15—24no—
261Dic15—42no—
262Dic15C2×C22⋊C414yescomplemented
263C24×C3C22×D512yes—
264C23×D3—51no—
265C23×D3—58no—
266C23×D3—52no—
267C23×D3—108no—
268C23×D3—104no—
269C23×D3—104no—
270C23×D3—108no—
271C2×C3×D4—22no—
272C2×C3×D4—22no—
273C2×C3×D4C4×D514yescomplemented
274C22×C3×C4C22×D512yes—
275C22×C3×C4C22×D512yes—
276C22×C3×C4C22×D511yescharacteristic
277C2×#67—58no—
278C2×#67—104no—
279C2×#67—104no—
280C22×Dic3—51no—
281C22×Dic3—52no—
282C22×Dic3—52no—
283D3×D4—1032no—
284C2×D12—54no—
285C2×D12—102no—
286C2×D12—102no—
287C2×C4×D3—54no—
288C2×C4×D3—54no—
289C2×C4×D3—54no—
290C2×C4×D3—58no—
291C2×C4×D3—104no—
292C2×C4×D3—104no—
293C2×C4×D3—108no—
294C2×C4×D3—104no—
295C3×C4⋊C4—22no—
296C3×C22⋊C4—28no—
297C3×C42—22no—
298#217—108no—
299(C2×D3)⋊C4—108no—
300(C2×D3)⋊C4—108no—
301C4⋊Dic3—102no—
302Dic3⋊C4—104no—
303C4×Dic3—104no—
304C4×Dic3—102no—
305C23×C5—24no—
306C23×C5—22no—
307C23×C5—22no—
308C23×C5—24no—
309C23×C5C2×C4×D314yes—
310C23×C5C23×D311yescharacteristic
311C22×D5—32no—
312C22×D5—32no—
313C22×D5—34no—
314C22×D5—32no—
315C22×D5—34no—
316C22×D5—34no—
317C22×D5—616no—
318C22×D5—632no—
319C22×D5—64no—
320C22×D5—62no—
321C22×D5—62no—
322C22×D5—64no—
323C22×D5—128no—
324C22×D5—128no—
325C22×D5—1216no—
326C5×D4—28no—
327C2×C4×C5—22no—
328C2×C4×C5—22no—
329C2×C4×C5—22no—
330C2×C4×C5—24no—
331C2×C4×C5C2×C4×D312yes—
332C2×C4×C5D3×D412yes—
333C2×C4×C5D3×D412yes—
334C2×C4×C5D3×D414yescomplemented
335C22⋊D5—616no—
336C2×Dic5—32no—
337C2×Dic5—32no—
338C2×Dic5—32no—
339C2×Dic5—34no—
340C2×Dic5—62no—
341C2×Dic5—62no—
342C2×Dic5—62no—
343C2×Dic5—64no—
344D20—68no—
345C4×D5—316no—
346C4×D5—616no—
347C4×D5—128no—
348C25—152no—
349C22×D4—152no—
350C22×D4—158no—
351C22×D4—151no—
352C22×D4—304no—
353C22×D4—304no—
354C22×D4C2×D1511yescharacteristic
355C23×C4—151no—
356C23×C4—152no—
357C23×C4—152no—
358C2×C4⋊C4—152no—
359C2×C4⋊C4—151no—
360C2×C4⋊C4—304no—
361C2×C4⋊C4C2×D1511yescharacteristic, complemented
362C2×C22⋊C4—154no—
363C2×C22⋊C4—154no—
364C2×C22⋊C4—154no—
365C2×C22⋊C4—3016no—
366C2×C22⋊C4C2×D1514yescomplemented
367C2×C42—152no—
368C2×C42—151no—
369C2×C42—304no—
370C2×C42C2×D1511yescharacteristic, complemented
371C2×C3×C5—44no—
372C2×C3×C5C22×C22⋊C411yescharacteristic
373C2×C3×C5C2×C4×D412yes—
374C2×C3×C5C2×#12112yes—
375C2×C3×C5C2×#12411yescharacteristic
376C2×C3×C5C2×#12711yescharacteristic
377D15—44no—
378D15#39018yesdirect product factor
379C23×C3—24no—
380C23×C3—22no—
381C23×C3—22no—
382C23×C3—24no—
383C23×C3C2×C4×D514yes—
384C23×C3C23×D511yescharacteristic
385C22×D3—52no—
386C22×D3—52no—
387C22×D3—54no—
388C22×D3—52no—
389C22×D3—54no—
390C22×D3—54no—
391C22×D3—1016no—
392C22×D3—1032no—
393C22×D3—104no—
394C22×D3—102no—
395C22×D3—102no—
396C22×D3—104no—
397C22×D3—208no—
398C22×D3—2016no—
399C22×D3—208no—
400C3×D4—28no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—24no—
405C2×C3×C4C2×C4×D512yes—
406C2×C3×C4D4×D512yes—
407C2×C3×C4D4×D512yes—
408C2×C3×C4D4×D514yescomplemented
409#67—1016no—
410C2×Dic3—52no—
411C2×Dic3—52no—
412C2×Dic3—52no—
413C2×Dic3—54no—
414C2×Dic3—102no—
415C2×Dic3—102no—
416C2×Dic3—102no—
417C2×Dic3—104no—
418D12—108no—
419C4×D3—516no—
420C4×D3—1016no—
421C4×D3—208no—
422C22×C5—24no—
423C22×C5—28no—
424C22×C5—42no—
425C22×C5—42no—
426C22×C5—44no—
427C22×C5C22×C4×D311yescharacteristic
428C22×C5C2×D3×D411yescharacteristic
429C22×C5C2×D3×D412yes—
430C22×C5C2×D3×D411yescharacteristic
431C22×C5D3×#4112yes—
432C2×D5—34no—
433C2×D5—34no—
434C2×D5—34no—
435C2×D5—38no—
436C2×D5—38no—
437C2×D5—64no—
438C2×D5—68no—
439C2×D5—1232no—
440C2×D5—122no—
441C2×D5—122no—
442C2×D5—124no—
443C4×C5—24no—
444C4×C5—42no—
445C4×C5D3×C22⋊C414yescomplemented
446Dic5—34no—
447Dic5—64no—
448Dic5—122no—
449C24—151no—
450C24—158no—
451C24—152no—
452C24—308no—
453C24—304no—
454C24—304no—
455C24—308no—
456C24C22×D1512yes—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—158no—
460C2×D4—154no—
461C2×D4—3032no—
462C2×D4—304no—
463C2×D4—304no—
464C2×D4—302no—
465C2×D4—302no—
466C2×D4C4×D1514yescomplemented
467C22×C4—151no—
468C22×C4—154no—
469C22×C4—154no—
470C22×C4—154no—
471C22×C4—152no—
472C22×C4—158no—
473C22×C4—152no—
474C22×C4—304no—
475C22×C4—304no—
476C22×C4—308no—
477C22×C4—304no—
478C22×C4C22×D1512yes—
479C22×C4C22×D1512yes—
480C22×C4C22×D1511yescharacteristic
481C4⋊C4—22no—
482C4⋊C4—304no—
483C4⋊C4—302no—
484C22⋊C4—28no—
485C22⋊C4—308no—
486C22⋊C4—308no—
487C22⋊C4—308no—
488C42—22no—
489C42—304no—
490C42—302no—
491C3×C5C2×#39011yescharacteristic, complemented, Hall
492C22×C3—24no—
493C22×C3—28no—
494C22×C3—42no—
495C22×C3—44no—
496C22×C3—42no—
497C22×C3C22×C4×D511yescharacteristic
498C22×C3C2×D4×D511yescharacteristic
499C22×C3C2×D4×D512yes—
500C22×C3C2×D4×D511yescharacteristic
501C22×C3D5×#4112yes—
502C2×D3—54no—
503C2×D3—54no—
504C2×D3—58no—
505C2×D3—58no—
506C2×D3—54no—
507C2×D3—104no—
508C2×D3—108no—
509C2×D3—2032no—
510C2×D3—202no—
511C2×D3—204no—
512C2×D3—202no—
513C3×C4—24no—
514C3×C4—42no—
515C3×C4D5×C22⋊C414yescomplemented
516Dic3—54no—
517Dic3—104no—
518Dic3—202no—
519C2×C5—44no—
520C2×C5C2×D3×C22⋊C411yescharacteristic
521C2×C5C4×D3×D412yes—
522C2×C5D3×#12112yes—
523C2×C5D3×#12411yescharacteristic
524C2×C5D3×#12711yescharacteristic
525D5—38no—
526D5—124no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—152no—
532C23—154no—
533C23—152no—
534C23—152no—
535C23—154no—
536C23—154no—
537C23—3016no—
538C23—3032no—
539C23—304no—
540C23—302no—
541C23—302no—
542C23—304no—
543C23—6016no—
544C23—608no—
545C23—608no—
546C23C2×C4×D1514yes—
547C23C23×D1511yescharacteristic
548D4—28no—
549D4—3016no—
550D4—308no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—152no—
556C2×C4—152no—
557C2×C4—152no—
558C2×C4—1516no—
559C2×C4—154no—
560C2×C4—3016no—
561C2×C4—302no—
562C2×C4—302no—
563C2×C4—304no—
564C2×C4—302no—
565C2×C4—608no—
566C2×C4C2×C4×D1512yes—
567C2×C4D4×D1512yes—
568C2×C4D4×D1512yes—
569C2×C4D4×D1514yescomplemented
570C2×C3—44no—
571C2×C3C2×D5×C22⋊C411yescharacteristic
572C2×C3C4×D4×D512yes—
573C2×C3D5×#12112yes—
574C2×C3D5×#12411yescharacteristic
575C2×C3D5×#12711yescharacteristic
576D3—58no—
577D3—204no—
578C5D3×#39011yescharacteristic, complemented, Sylow
579C22—28no—
580C22—24no—
581C22—44no—
582C22—42no—
583C22—42no—
584C22—158no—
585C22—158no—
586C22—154no—
587C22—154no—
588C22—154no—
589C22—308no—
590C22—304no—
591C22—6032no—
592C22—604no—
593C22—602no—
594C22—602no—
595C22C22×C4×D1511yescharacteristic
596C22C2×D4×D1512yes—
597C22C2×D4×D1511yescharacteristic
598C22C2×D4×D1511yescharacteristic
599C22#41×D1512yes—
600C4—24no—
601C4—42no—
602C4—154no—
603C4—304no—
604C4—602no—
605C4C22⋊C4×D1514yescomplemented
606C3D5×#39011yescharacteristic, complemented, Sylow
607C2—44no—
608C2—158no—
609C2—604no—
610C2C2×C22⋊C4×D1511yescharacteristic
611C2C4×D4×D1512yes—
612C2D15×#12112yes—
613C2D15×#12411yescharacteristic
614C2D15×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#390C2
socle, cosocleC23×C3×C5C24
prime radicalC22×D4×D15C2
hypercenter#390D15
hypocenterC3×C5C2×#390
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D15C2semidirect
3. C2×C22⋊C4×D15C2semidirect
4. C2×C42×D15C2semidirect
5. C3×C5×#390C2semidirect
6. #11751322C2semidirect
7. #11751089C2semidirect
8. #11751058C2semidirect
9. #11751040C2semidirect
16. C2×D4×D15C4semidirect
21. C2×C3×C5×C4⋊C4C22semidirect
22. C2×C3×C5×C22⋊C4C22semidirect
23. C2×C3×C42×C5C22semidirect
24. C2×#6719C22semidirect
25. C2×#6714C22semidirect
26. C2×#6714C22semidirect
27. C2×C4⋊Dic15C22semidirect
28. C2×#6712C22semidirect
29. C2×C4×Dic15C22semidirect
30. C2×C4×Dic15C22semidirect
39. C5×#390D3semidirect
54. C2×C3×C5×D4C2×C4semidirect
60. C2×#1220C2×C4semidirect
67. C2×D60C2×C4semidirect
75. C2×C4×D15D4semidirect
90. C3×#390D5semidirect
105. C2×C5×C4⋊C4C2×D3semidirect
106. C2×C5×C22⋊C4C2×D3semidirect
107. C2×C42×C5C2×D3semidirect
148. C2×C3×C4×C5C2×D4semidirect
157. C2×Dic15C2×D4semidirect
161. C4×D15C22⋊C4semidirect
172. C2×C3×C4⋊C4C2×D5semidirect
173. C2×C3×C22⋊C4C2×D5semidirect
174. C2×C3×C42C2×D5semidirect
195. C2×C5×D4C4×D3semidirect
235. #390D15direct (N × H)
259. C3×C4×C5C2×C22⋊C4semidirect
262. Dic15C2×C22⋊C4semidirect
273. C2×C3×D4C4×D5semidirect
334. C2×C4×C5D3×D4semidirect
361. C2×C4⋊C4C2×D15semidirect
366. C2×C22⋊C4C2×D15semidirect
370. C2×C42C2×D15semidirect
378. D15#390direct (N × H)
408. C2×C3×C4D4×D5semidirect
445. C4×C5D3×C22⋊C4semidirect
466. C2×D4C4×D15semidirect
491. C3×C5C2×#390semidirect
515. C3×C4D5×C22⋊C4semidirect
569. C2×C4D4×D15semidirect
578. C5D3×#390semidirect
605. C4C22⋊C4×D15semidirect
606. C3D5×#390semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D152
C2×C4⋊C4×D152
C2×C22⋊C4×D152
C2×C42×D152
C3×C5×#3902
#117513222
#117510892
#117510582
#117510402
D5×#3903
D3×#3905