Finite Group Databasebeta

← D15×#461 #14425862

Subgroups — D15×#461 #14425862

Order 1920.

1088 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D15C212yesmaximal
2D15×C4⋊Q8C211yesmaximal, characteristic, complemented
3C4×D4×D15C211yesmaximal, characteristic
4C3×C5×#461C211yesmaximal, characteristic, complemented
5#11751263C212yesmaximal, complemented
6#11751184C211yesmaximal, characteristic, complemented
7#11751120C211yesmaximal, characteristic, complemented
8#11750942C211yesmaximal, characteristic, complemented
9D15×C4⋊C8C211yesmaximal, characteristic, complemented
10#11750919C212yesmaximal, complemented
11D15×#103C212yesmaximal, complemented
12D5×#461—31nomaximal, Hall
13C2×Q8×D15C2212yes—
14C2×D4×D15C2211yescharacteristic
15C22×C4×D15—21no—
16C2×C3×C5×SD16C2212yes—
17C3×C5×C4⋊Q8C2211yescharacteristic, complemented
18C3×C4×C5×D4C2211yescharacteristic
19#91833C2212yescomplemented
20C2×#6717C2212yes—
21D4×Dic15C2211yescharacteristic
22C2×#6716C2212yes—
23C4×#1220—21no—
24SD16×D15—28no—
25C2×#6706C2212yes—
26C2×C8×D15C2212yes—
27#91782C2211yescharacteristic
28C4⋊C4×D15—21no—
29C4⋊C4×D15—21no—
30C4⋊C4×D15C2211yescharacteristic
31#91777—21no—
32#91777—21no—
33#91770—21no—
34C22⋊C4×D15—21no—
35C4×D60C2211yescharacteristic
36C42×D15C2211yescharacteristic
37#91758C2211yescharacteristic, complemented
38C3×C5×C4⋊C8C2211yescharacteristic, complemented
39C3×C5×#103C2212yescomplemented
40#91119C2212yescomplemented
41#91100C2212yescomplemented
42#91098C2212yescomplemented
43#91094C2212yescomplemented
44C4⋊(C3⋊#168)C2211yescharacteristic, complemented
45D3×#461—51nomaximal, Hall
46C2×D5×SD16—32no—
47D5×C4⋊Q8—31no—
48C4×D4×D5—31no—
49C5×#461D311yescharacteristic, complemented
50#65451—32no—
51#65372—31no—
52#65308—31no—
53#65130—31no—
54D5×C4⋊C8—31no—
55#65107—32no—
56D5×#103—32no—
57C23×D15—21no—
58C2×C3×C5×Q8C2312yes—
59C2×C3×C5×D4C2311yescharacteristic
60C22×C3×C4×C5—21no—
61C2×#1220—21no—
62C22×Dic15—21no—
63Q8×D15—24no—
64Q8×D15—24no—
65D4×D15—22no—
66D4×D15D414yes—
67C2×D60C2311yescharacteristic
68C2×C4×D15—21no—
69C2×C4×D15—22no—
70C2×C4×D15—21no—
71C2×C4×D15—21no—
72C2×C4×D15—44no—
73C2×C4×D15D411yescharacteristic
74C2×C4×D15D411yescharacteristic
75C2×C4×D15C2311yescharacteristic
76C2×Dic30—22no—
77C2×Dic30C2312yes—
78C3×C5×SD16—24no—
79C2×C3×C5×C8C2312yes—
80C3×C5×C4⋊C4—21no—
81C3×C5×C4⋊C4—21no—
82C3×C5×C4⋊C4C2311yescharacteristic
83C3×C5×C22⋊C4—21no—
84C3×C42×C5C2311yescharacteristic
85#6719—21no—
86#6717—24no—
87#6716—24no—
88#6714—21no—
89#6714—21no—
90C4⋊Dic15—21no—
91C4⋊Dic15—21no—
92C4⋊Dic15C2311yescharacteristic
93#6712—21no—
94#6712—22no—
95#6712—22no—
96C4×Dic15D411yescharacteristic
97C4×Dic15D411yescharacteristic
98C4×Dic15C2311yescharacteristic
99C2×C3⋊#168C2312yes—
100#6706—24no—
101C8×D15—24no—
102C2×D3×SD16—52no—
103D3×C4⋊Q8—51no—
104C4×D3×D4—51no—
105C3×#461D511yescharacteristic, complemented
106#5162—52no—
107#5083—51no—
108#5019—51no—
109#4841—51no—
110D3×C4⋊C8—51no—
111#4818—52no—
112D3×#103—52no—
113C2×Q8×D5—32no—
114C2×D4×D5—31no—
115C22×C4×D5—61no—
116C2×C5×SD16C2×D312yes—
117C5×C4⋊Q8C2×D311yescharacteristic, complemented
118C4×C5×D4C2×D311yescharacteristic
119#4090—32no—
120C2×#688—32no—
121D4×Dic5—31no—
122C2×#687—32no—
123C4×C22⋊D5—61no—
124D5×SD16—68no—
125C2×#677—32no—
126C2×C8×D5—32no—
127#4039—31no—
128D5×C4⋊C4—31no—
129D5×C4⋊C4—61no—
130D5×C4⋊C4—61no—
131#4034—61no—
132#4034—61no—
133#4027—61no—
134D5×C22⋊C4—61no—
135C4×D20—31no—
136C42×D5—31no—
137#4015—31no—
138C5×C4⋊C8C2×D311yescharacteristic, complemented
139C5×#103C2×D312yescomplemented
140#3967—32no—
141#3948—32no—
142#3946—32no—
143#3942—32no—
144C4⋊#168—31no—
145C2×#461—151noSylow
146C23×C3×C5—21no—
147C22×D15—21no—
148C22×D15—24no—
149C22×D15—42no—
150C22×D15—42no—
151C22×D15C2×D411yescharacteristic
152C3×C5×Q8—22no—
153C3×C5×Q8—22no—
154C3×C5×D4—21no—
155C3×C5×D4C2×D412yes—
156C2×C3×C4×C5—22no—
157C2×C3×C4×C5—21no—
158C2×C3×C4×C5—42no—
159C2×C3×C4×C5C2×D411yescharacteristic
160C2×C3×C4×C5C2×D411yescharacteristic
161C2×C3×C4×C5C2411yescharacteristic
162#1220—24no—
163C2×Dic15—22no—
164C2×Dic15—21no—
165C2×Dic15—42no—
166C2×Dic15C2×D411yescharacteristic
167C2×Dic15C2×D411yescharacteristic
168C2×Dic15C2×D411yescharacteristic
169D60—21no—
170D60C2×D412yes—
171C4×D15—22no—
172C4×D15—44no—
173C4×D15—42no—
174C4×D15—42no—
175C4×D15—42no—
176C4×D15SD1614yescomplemented
177C4×D15C2×D412yes—
178C4×D15#4112yes—
179Dic30—22no—
180Dic30—22no—
181Dic30—44no—
182C3×C5×C8—22no—
183C3⋊#168—22no—
184C2×D3×Q8—52no—
185C2×D3×D4—51no—
186C22×C4×D3—101no—
187C2×C3×SD16C2×D512yes—
188C3×C4⋊Q8C2×D511yescharacteristic, complemented
189C3×C4×D4C2×D511yescharacteristic
190#944—52no—
191C2×#215—52no—
192D4×Dic3—51no—
193C2×#214—52no—
194C4×#67—101no—
195D3×SD16—108no—
196C2×#204—52no—
197C2×D3×C8—52no—
198#893—51no—
199D3×C4⋊C4—51no—
200D3×C4⋊C4—101no—
201D3×C4⋊C4—101no—
202#888—101no—
203#888—101no—
204#67⋊C4—101no—
205D3×C22⋊C4—101no—
206C4×D12—51no—
207C42×D3—51no—
208#869—51no—
209C3×C4⋊C8C2×D511yescharacteristic, complemented
210C3×#103C2×D512yescomplemented
211#835—52no—
212#816—52no—
213Dic3⋊C8—52no—
214#810—52no—
215C4⋊(C3⋊C8)—51no—
216C23×D5—61no—
217C2×C5×Q8C22×D312yes—
218C2×C5×D4C22×D311yescharacteristic
219C22×C4×C5—21no—
220C2×C22⋊D5—61no—
221C22×Dic5—61no—
222Q8×D5—64no—
223Q8×D5—64no—
224D4×D5—34no—
225D4×D5—62no—
226C2×D20—31no—
227C2×C4×D5—31no—
228C2×C4×D5—31no—
229C2×C4×D5—31no—
230C2×C4×D5—61no—
231C2×C4×D5—61no—
232C2×C4×D5—62no—
233C2×C4×D5—61no—
234C2×C4×D5—124no—
235C2×Dic10—32no—
236C2×Dic10—62no—
237C5×SD16—24no—
238C2×C5×C8C22×D312yes—
239C5×C4⋊C4—21no—
240C5×C4⋊C4—21no—
241C5×C4⋊C4C22×D311yescharacteristic
242C5×C22⋊C4—21no—
243C42×C5C22×D311yescharacteristic
244C22⋊Dic5—61no—
245#688—64no—
246#687—64no—
247#685—61no—
248#685—61no—
249C4⋊Dic5—31no—
250C4⋊Dic5—61no—
251C4⋊Dic5—61no—
252#683—61no—
253#683—62no—
254#683—62no—
255C4×Dic5—31no—
256C4×Dic5—31no—
257C4×Dic5—31no—
258C2×#168—32no—
259#677—64no—
260C8×D5—64no—
261C22×SD16—152no—
262C2×C4⋊Q8—151no—
263C2×C4×D4—151no—
264#461—152no—
265#461—151no—
266#461—151no—
267#461—151no—
268#461—152no—
269#461D1511yescharacteristic, direct product factor
270C2×C4⋊C8—151no—
271C2×#103—152no—
272C22×C3×C5—22no—
273C22×C3×C5—41no—
274C22×C3×C5—41no—
275C22×C3×C5C22×D411yescharacteristic
276C2×D15—22no—
277C2×D15—41no—
278C2×D15—41no—
279C2×D15—48no—
280C2×D15#12112yes—
281C2×D15C2×SD1612yes—
282C2×D15#13712yes—
283C3×C4×C5—21no—
284C3×C4×C5—42no—
285C3×C4×C5—41no—
286C3×C4×C5C2×SD1612yescomplemented
287C3×C4×C5C22×D411yescharacteristic
288C3×C4×C5C2×#4111yescharacteristic
289Dic15—21no—
290Dic15—42no—
291Dic15—41no—
292Dic15#12112yes—
293Dic15C2×SD1612yescomplemented
294C23×D3—101no—
295C2×C3×Q8C22×D512yes—
296C2×C3×D4C22×D511yescharacteristic
297C22×C3×C4—21no—
298C2×#67—101no—
299C22×Dic3—101no—
300D3×Q8—104no—
301D3×Q8—104no—
302D3×D4—54no—
303D3×D4—102no—
304C2×D12—51no—
305C2×C4×D3—51no—
306C2×C4×D3—51no—
307C2×C4×D3—51no—
308C2×C4×D3—101no—
309C2×C4×D3—101no—
310C2×C4×D3—102no—
311C2×C4×D3—101no—
312C2×C4×D3—204no—
313C2×Dic6—52no—
314C2×Dic6—102no—
315C3×SD16—24no—
316C2×C3×C8C22×D512yes—
317C3×C4⋊C4—21no—
318C3×C4⋊C4—21no—
319C3×C4⋊C4C22×D511yescharacteristic
320C3×C22⋊C4—21no—
321C3×C42C22×D511yescharacteristic
322#217—101no—
323#215—104no—
324#214—104no—
325(C2×D3)⋊C4—101no—
326(C2×D3)⋊C4—101no—
327C4⋊Dic3—51no—
328C4⋊Dic3—101no—
329C4⋊Dic3—101no—
330Dic3⋊C4—101no—
331Dic3⋊C4—102no—
332Dic3⋊C4—102no—
333C4×Dic3—51no—
334C4×Dic3—51no—
335C4×Dic3—51no—
336C2×C3⋊C8—52no—
337#204—104no—
338D3×C8—104no—
339C23×C5—21no—
340C22×D5—31no—
341C22×D5—64no—
342C22×D5—61no—
343C22×D5—122no—
344C22×D5—122no—
345C5×Q8—22no—
346C5×Q8—22no—
347C5×D4—21no—
348C5×D4D3×D412yes—
349C2×C4×C5—22no—
350C2×C4×C5—21no—
351C2×C4×C5—42no—
352C2×C4×C5D3×D411yescharacteristic
353C2×C4×C5D3×D411yescharacteristic
354C2×C4×C5C23×D311yescharacteristic
355C22⋊D5—64no—
356C2×Dic5—31no—
357C2×Dic5—31no—
358C2×Dic5—31no—
359C2×Dic5—62no—
360C2×Dic5—61no—
361C2×Dic5—122no—
362D20—32no—
363D20—61no—
364C4×D5—34no—
365C4×D5—32no—
366C4×D5—32no—
367C4×D5—62no—
368C4×D5—122no—
369C4×D5—122no—
370C4×D5—124no—
371C4×D5—122no—
372Dic10—62no—
373Dic10—62no—
374Dic10—124no—
375C5×C8—22no—
376#168—62no—
377C22×Q8—152no—
378C22×D4—151no—
379C23×C4—301no—
380C2×SD16—152no—
381C2×SD16—152no—
382C2×SD16—152no—
383C2×SD16—308no—
384C2×SD16C2×D1512yes—
385C22×C8—152no—
386C4⋊Q8—152no—
387C4⋊Q8—151no—
388C4⋊Q8—301no—
389C4⋊Q8—301no—
390C4⋊Q8C2×D1511yescharacteristic, complemented
391C4×D4—151no—
392C4×D4—151no—
393C4×D4—151no—
394C4×D4—301no—
395C4×D4—301no—
396C4×D4C2×D1511yescharacteristic
397C2×C4⋊C4—151no—
398C2×C4⋊C4—301no—
399C2×C4⋊C4—301no—
400C2×C22⋊C4—301no—
401C2×C42—151no—
402C4⋊C8—152no—
403C4⋊C8—151no—
404C4⋊C8C2×D1511yescharacteristic, complemented
405#103—152no—
406#103—152no—
407#103—152no—
408#103C2×D1512yescomplemented
409C2×C3×C5—42no—
410C2×C3×C5C2×#12111yescharacteristic
411C2×C3×C5C22×SD1611yescharacteristic
412C2×C3×C5C2×#13711yescharacteristic
413D15—42no—
414D15#46114yesdirect product factor
415C23×C3—21no—
416C22×D3—51no—
417C22×D3—104no—
418C22×D3—101no—
419C22×D3—202no—
420C22×D3—202no—
421C3×Q8—22no—
422C3×Q8—22no—
423C3×D4—21no—
424C3×D4D4×D512yes—
425C2×C3×C4—22no—
426C2×C3×C4—21no—
427C2×C3×C4—42no—
428C2×C3×C4D4×D511yescharacteristic
429C2×C3×C4D4×D511yescharacteristic
430C2×C3×C4C23×D511yescharacteristic
431#67—104no—
432C2×Dic3—51no—
433C2×Dic3—51no—
434C2×Dic3—51no—
435C2×Dic3—102no—
436C2×Dic3—101no—
437C2×Dic3—202no—
438D12—52no—
439D12—101no—
440C4×D3—52no—
441C4×D3—52no—
442C4×D3—54no—
443C4×D3—102no—
444C4×D3—202no—
445C4×D3—202no—
446C4×D3—204no—
447C4×D3—202no—
448Dic6—102no—
449Dic6—102no—
450Dic6—204no—
451C3×C8—22no—
452C3⋊C8—102no—
453C22×C5—22no—
454C22×C5—41no—
455C22×C5—41no—
456C22×C5C2×D3×D411yescharacteristic
457C2×D5—32no—
458C2×D5—32no—
459C2×D5—32no—
460C2×D5—62no—
461C2×D5—128no—
462C2×D5—121no—
463C2×D5—121no—
464C4×C5—21no—
465C4×C5—41no—
466C4×C5—42no—
467C4×C5D3×SD1612yescomplemented
468C4×C5C2×D3×D411yescharacteristic
469C4×C5D3×#4111yescharacteristic
470Dic5—32no—
471Dic5—32no—
472Dic5—61no—
473Dic5—121no—
474Dic5—122no—
475C24—301no—
476C2×Q8—152no—
477C2×Q8—302no—
478C2×Q8—304no—
479C2×Q8—304no—
480C2×Q8C22×D1512yes—
481C2×D4—154no—
482C2×D4—151no—
483C2×D4—302no—
484C2×D4—301no—
485C2×D4C22×D1511yescharacteristic
486C22×C4—21no—
487C22×C4—151no—
488C22×C4—151no—
489C22×C4—151no—
490C22×C4—301no—
491C22×C4—301no—
492C22×C4—302no—
493C22×C4—301no—
494C22×C4—301no—
495C22×C4—604no—
496SD16—24no—
497SD16—304no—
498SD16—304no—
499SD16—304no—
500C2×C8—152no—
501C2×C8—304no—
502C2×C8C22×D1512yes—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—151no—
506C4⋊C4—301no—
507C4⋊C4—302no—
508C4⋊C4—302no—
509C4⋊C4—301no—
510C4⋊C4—301no—
511C4⋊C4C22×D1511yescharacteristic
512C22⋊C4—21no—
513C22⋊C4—301no—
514C22⋊C4—301no—
515C22⋊C4—301no—
516C42—151no—
517C42—151no—
518C42—151no—
519C42C22×D1511yescharacteristic
520C3×C5C2×#46111yescharacteristic, complemented, Hall
521C22×C3—22no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3C2×D4×D511yescharacteristic
525C2×D3—52no—
526C2×D3—52no—
527C2×D3—52no—
528C2×D3—102no—
529C2×D3—208no—
530C2×D3—201no—
531C2×D3—201no—
532C3×C4—21no—
533C3×C4—41no—
534C3×C4—42no—
535C3×C4D5×SD1612yescomplemented
536C3×C4C2×D4×D511yescharacteristic
537C3×C4D5×#4111yescharacteristic
538Dic3—52no—
539Dic3—52no—
540Dic3—101no—
541Dic3—201no—
542Dic3—202no—
543C2×C5—42no—
544C2×C5D3×#12111yescharacteristic
545C2×C5C2×D3×SD1611yescharacteristic
546C2×C5D3×#13711yescharacteristic
547D5—34no—
548D5—122no—
549C23—21no—
550C23—151no—
551C23—304no—
552C23—301no—
553C23—602no—
554C23—602no—
555Q8—22no—
556Q8—22no—
557Q8—302no—
558Q8—302no—
559Q8—604no—
560D4—21no—
561D4—152no—
562D4—304no—
563D4—301no—
564D4D4×D1512yes—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—151no—
569C2×C4—151no—
570C2×C4—154no—
571C2×C4—152no—
572C2×C4—152no—
573C2×C4—151no—
574C2×C4—302no—
575C2×C4—301no—
576C2×C4—302no—
577C2×C4—602no—
578C2×C4—602no—
579C2×C4—604no—
580C2×C4—602no—
581C2×C4—602no—
582C2×C4D4×D1511yescharacteristic
583C2×C4D4×D1511yescharacteristic
584C2×C4C23×D1511yescharacteristic
585C8—22no—
586C8—302no—
587C2×C3—42no—
588C2×C3D5×#12111yescharacteristic
589C2×C3C2×D5×SD1611yescharacteristic
590C2×C3D5×#13711yescharacteristic
591D3—54no—
592D3—202no—
593C5D3×#46111yescharacteristic, complemented, Sylow
594C22—22no—
595C22—41no—
596C22—41no—
597C22—152no—
598C22—152no—
599C22—152no—
600C22—302no—
601C22—608no—
602C22—601no—
603C22—601no—
604C22C2×D4×D1511yescharacteristic
605C4—21no—
606C4—42no—
607C4—41no—
608C4—152no—
609C4—152no—
610C4—301no—
611C4—602no—
612C4—601no—
613C4SD16×D1512yescomplemented
614C4C2×D4×D1511yescharacteristic
615C4#41×D1511yescharacteristic
616C3D5×#46111yescharacteristic, complemented, Sylow
617C2—42no—
618C2—154no—
619C2—602no—
620C2D15×#12111yescharacteristic
621C2C2×SD16×D1511yescharacteristic
622C2D15×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#461C2
socleC22×C3×C5C22×D4
prime radicalC2×D4×D15C22
hypercenter#461D15
hypocenterC3×C5C2×#461
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. D15×C4⋊Q8C2semidirect
4. C3×C5×#461C2semidirect
5. #11751263C2semidirect
6. #11751184C2semidirect
7. #11751120C2semidirect
8. #11750942C2semidirect
9. D15×C4⋊C8C2semidirect
10. #11750919C2semidirect
11. D15×#103C2semidirect
17. C3×C5×C4⋊Q8C22semidirect
19. #91833C22semidirect
37. #91758C22semidirect
38. C3×C5×C4⋊C8C22semidirect
39. C3×C5×#103C22semidirect
40. #91119C22semidirect
41. #91100C22semidirect
42. #91098C22semidirect
43. #91094C22semidirect
44. C4⋊(C3⋊#168)C22semidirect
49. C5×#461D3semidirect
105. C3×#461D5semidirect
117. C5×C4⋊Q8C2×D3semidirect
138. C5×C4⋊C8C2×D3semidirect
139. C5×#103C2×D3semidirect
176. C4×D15SD16semidirect
188. C3×C4⋊Q8C2×D5semidirect
209. C3×C4⋊C8C2×D5semidirect
210. C3×#103C2×D5semidirect
269. #461D15direct (N × H)
286. C3×C4×C5C2×SD16semidirect
293. Dic15C2×SD16semidirect
390. C4⋊Q8C2×D15semidirect
404. C4⋊C8C2×D15semidirect
408. #103C2×D15semidirect
414. D15#461direct (N × H)
467. C4×C5D3×SD16semidirect
520. C3×C5C2×#461semidirect
535. C3×C4D5×SD16semidirect
593. C5D3×#461semidirect
613. C4SD16×D15semidirect
616. C3D5×#461semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D152
D15×C4⋊Q82
C4×D4×D152
C3×C5×#4612
#117512632
#117511842
#117511202
#117509422
D15×C4⋊C82
#117509192
D15×#1032
D5×#4613
D3×#4615