Finite Group Databasebeta

← D5×#4856 #14439912

Subgroups — D5×#4856 #14439912

Order 1920.

768 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#890C211yesmaximal, characteristic
2D5×#888C211yesmaximal, characteristic
3#11750957C211yesmaximal, characteristic, complemented
4C5×#4856C211yesmaximal, characteristic, complemented
5C3×D5×#106C211yesmaximal, characteristic
6#11748038C211yesmaximal, characteristic, complemented
7#11748030C211yesmaximal, characteristic, complemented
8D5×#818C211yesmaximal, characteristic
9D5×#815C211yesmaximal, characteristic
10D5×#808C211yesmaximal, characteristic
11D5×#807C211yesmaximal, characteristic
12#11747425C211yesmaximal, characteristic, complemented
13#11747398C211yesmaximal, characteristic, complemented
14#11747163C211yesmaximal, characteristic, complemented
15#11747162C211yesmaximal, characteristic, complemented
16D5×#501—31nomaximal, Hall
17C2×D5×Dic6—21no—
18#91779C2211yescharacteristic
19#91777C2211yescharacteristic
20C5×#890C2211yescharacteristic
21C5×#888C2211yescharacteristic
22C2×C3×C8×D5C2211yescharacteristic
23C3×D5×C4⋊C4C2211yescharacteristic
24C3×D5×C4⋊C4C2211yescharacteristic
25#91470C2211yescharacteristic
26#91469C2211yescharacteristic
27D5×C4⋊Dic3—21no—
28D5×C4⋊Dic3C2211yescharacteristic
29D5×C4⋊Dic3C2211yescharacteristic
30D5×Dic3⋊C4—21no—
31D5×Dic3⋊C4—21no—
32C4×D5×Dic3C2211yescharacteristic
33#91384C2211yescharacteristic
34#91381C2211yescharacteristic
35#91335—21no—
36#91334—21no—
37#91329—21no—
38#91328—21no—
39C2×D5×C3⋊C8C2211yescharacteristic
40C3×C5×#106C2211yescharacteristic
41#91102C2211yescharacteristic
42#91099C2211yescharacteristic
43#91092C2211yescharacteristic
44#91091C2211yescharacteristic
45C5×#818C2211yescharacteristic
46C5×#815C2211yescharacteristic
47C5×#808C2211yescharacteristic
48C5×#807C2211yescharacteristic
49C3×#3949C2211yescharacteristic
50C3×#3940C2211yescharacteristic
51C3×#3940C2211yescharacteristic
52#90990C2211yescharacteristic
53#90989C2211yescharacteristic
54#90988C2211yescharacteristic
55#90988C2211yescharacteristic
56#90987C2211yescharacteristic
57#90986C2211yescharacteristic
58#90954C2211yescharacteristic
59#90953C2211yescharacteristic
60C2×#4856—51nomaximal, Hall
61D5×C4⋊Q8—31no—
62D5×#125—31no—
63C5×#501—31no—
64#65369—31no—
65#65354—31no—
66D5×#107—31no—
67D5×#107—31no—
68#65165—31no—
69#65165—31no—
70D5×#106—31no—
71D5×#106D311yescharacteristic
72#65145—31no—
73#65145—31no—
74D5×#97—31no—
75#64991—31no—
76C2×Dic30—21no—
77C2×C5×Dic6—21no—
78C2×C3×C4×D5—21no—
79C2×C3×C4×D5—21no—
80C2×C3×C4×D5C2311yescharacteristic
81C2×#1204—21no—
82C2×#1204—21no—
83C2×D5×Dic3—21no—
84C2×D5×Dic3—21no—
85C2×D5×Dic3D411yescharacteristic
86C2×D5×Dic3D411yescharacteristic
87D5×Dic6—44no—
88C2×C3×C5×C8C2311yescharacteristic
89C3×C5×C4⋊C4C2311yescharacteristic
90C3×C5×C4⋊C4C2311yescharacteristic
91C4⋊Dic15—21no—
92C4⋊Dic15C2311yescharacteristic
93C4⋊Dic15C2311yescharacteristic
94#6712—21no—
95#6712—21no—
96C4×Dic15C2311yescharacteristic
97C2×C3⋊#168C2311yescharacteristic
98C5×C4⋊Dic3—21no—
99C5×C4⋊Dic3C2311yescharacteristic
100C5×C4⋊Dic3C2311yescharacteristic
101C5×Dic3⋊C4—21no—
102C5×Dic3⋊C4—21no—
103C4×C5×Dic3C2311yescharacteristic
104C2×C5×C3⋊C8C2311yescharacteristic
105C3×C4⋊Dic5C2311yescharacteristic
106C3×C4⋊Dic5C2311yescharacteristic
107C3×#683—21no—
108C3×#683—21no—
109C2×C3×#168C2311yescharacteristic
110C3×C8×D5Q814yescomplemented
111Dic3⋊Dic5—21no—
112Dic3⋊Dic5—21no—
113#6669—21no—
114#6669—21no—
115#6668—21no—
116#6668—21no—
117#6668—21no—
118#6668—21no—
119Dic3×Dic5D411yescharacteristic
120Dic3×Dic5D411yescharacteristic
121D5×C3⋊C8Q814yescomplemented
122C2×#890—51no—
123C2×#888—51no—
124C2×C3×#106—51no—
125C2×#818—51no—
126C2×#815—51no—
127C2×#808—51no—
128C2×#807—51no—
129#4856—51no—
130#4856—51no—
131#4856—51no—
132#4856—51no—
133#4856—51no—
134#4856—51no—
135#4856—51no—
136#4856D511yescharacteristic, direct product factor
137C2×Q8×D5—61no—
138C5×C4⋊Q8—31no—
139C5×#125—31no—
140#4090—61no—
141C2×C8×D5—31no—
142C2×C8×D5C2×D311yescharacteristic
143D5×C4⋊C4—31no—
144D5×C4⋊C4—31no—
145D5×C4⋊C4—61no—
146D5×C4⋊C4—61no—
147D5×C4⋊C4—61no—
148D5×C4⋊C4C2×D311yescharacteristic
149D5×C4⋊C4C2×D311yescharacteristic
150#4036—31no—
151#4036—31no—
152#4035—61no—
153#4035—61no—
154#4034—31no—
155#4034—31no—
156#4034—61no—
157C42×D5—31no—
158#4016—31no—
159#4015—31no—
160C5×#107—31no—
161C5×#107—31no—
162C5×#106—31no—
163C5×#106C2×D311yescharacteristic
164C5×#97—31no—
165#3950—31no—
166#3950—31no—
167#3949—31no—
168#3949C2×D311yescharacteristic
169#3947—31no—
170#3947—31no—
171#3940—31no—
172#3940—31no—
173#3940C2×D311yescharacteristic
174#3940C2×D311yescharacteristic
175#3939—31no—
176#3939—31no—
177#3939—31no—
178#3939—31no—
179#3935—31no—
180C2×#501—151noSylow
181C22×C3×D5C2×D411yescharacteristic
182C2×C3×C4×C5—21no—
183C2×C3×C4×C5—21no—
184C2×C3×C4×C5C2411yescharacteristic
185C2×Dic15—21no—
186C2×Dic15—21no—
187C2×Dic15C2×D411yescharacteristic
188C2×Dic15C2×D411yescharacteristic
189Dic30—42no—
190C2×C5×Dic3—21no—
191C2×C5×Dic3—21no—
192C2×C5×Dic3C2×D411yescharacteristic
193C2×C5×Dic3C2×D411yescharacteristic
194C5×Dic6—42no—
195C2×C3×Dic5—21no—
196C2×C3×Dic5—21no—
197C2×C3×Dic5C2×D411yescharacteristic
198C3×C4×D5—42no—
199C3×C4×D5—42no—
200C3×C4×D5C2×Q812yes—
201C3×C4×D5C2×Q812yes—
202#1204—42no—
203#1204—42no—
204D5×Dic3—22no—
205D5×Dic3—22no—
206D5×Dic3—42no—
207D5×Dic3—42no—
208C3×C5×C8C2×Q812yescomplemented
209C3⋊#168C2×Q812yescomplemented
210C3×#168C2×Q812yescomplemented
211C5×C3⋊C8C2×Q812yescomplemented
212C22×Dic6—101no—
213C22×C3×C8—51no—
214C2×C3×C4⋊C4—51no—
215C2×C3×C4⋊C4—51no—
216C2×C4⋊Dic3—51no—
217C2×C4⋊Dic3—51no—
218C2×C4⋊Dic3—101no—
219C2×Dic3⋊C4—101no—
220C2×Dic3⋊C4—101no—
221C2×C4×Dic3—51no—
222C22×C3⋊C8—51no—
223#890—51no—
224#890—51no—
225#890—51no—
226#890—101no—
227#890—101no—
228#890C2×D511yescharacteristic
229#888—51no—
230#888—51no—
231#888—51no—
232#888—101no—
233#888—101no—
234#888C2×D511yescharacteristic
235C3×#106—51no—
236C3×#106—51no—
237C3×#106—51no—
238C3×#106C2×D511yescharacteristic
239#818—51no—
240#818—51no—
241#818—51no—
242#818C2×D511yescharacteristic
243#815—51no—
244#815—51no—
245#815—51no—
246#815C2×D511yescharacteristic
247#808—51no—
248#808—51no—
249#808—51no—
250#808C2×D511yescharacteristic
251#807—51no—
252#807—51no—
253#807—51no—
254#807C2×D511yescharacteristic
255C2×C5×Q8—61no—
256Q8×D5—124no—
257C2×C4×D5—21no—
258C2×C4×D5—21no—
259C2×C4×D5—31no—
260C2×C4×D5—31no—
261C2×C4×D5—61no—
262C2×C4×D5—61no—
263C2×C4×D5C22×D311yescharacteristic
264C2×Dic10—61no—
265C2×Dic10—61no—
266C2×Dic10—61no—
267C2×C5×C8—31no—
268C2×C5×C8C22×D311yescharacteristic
269C5×C4⋊C4—31no—
270C5×C4⋊C4—31no—
271C5×C4⋊C4—61no—
272C5×C4⋊C4—61no—
273C5×C4⋊C4—61no—
274C5×C4⋊C4C22×D311yescharacteristic
275C5×C4⋊C4C22×D311yescharacteristic
276C42×C5—31no—
277C4⋊Dic5—31no—
278C4⋊Dic5—31no—
279C4⋊Dic5—61no—
280C4⋊Dic5—61no—
281C4⋊Dic5—61no—
282C4⋊Dic5C22×D311yescharacteristic
283C4⋊Dic5C22×D311yescharacteristic
284#683—21no—
285#683—21no—
286#683—61no—
287#683—61no—
288#683—61no—
289#683—61no—
290#683—61no—
291#683—61no—
292#683—61no—
293#683—61no—
294C4×Dic5—31no—
295C4×Dic5—31no—
296C4×Dic5—31no—
297C2×#168—31no—
298C2×#168C22×D311yescharacteristic
299C8×D5—34no—
300C8×D5Dic614yescomplemented
301C2×C4⋊Q8—151no—
302C2×#125—151no—
303#501—31no—
304#501—151no—
305#501—151no—
306#501—151no—
307#501—151no—
308#501—151no—
309#501—151no—
310#501—151no—
311C2×#107—151no—
312C2×#107—151no—
313C2×#106—51no—
314C2×#106—151no—
315C2×#97—151no—
316C22×C3×C5C22×D411yescharacteristic
317C2×C3×D5C4⋊Q812yes—
318C2×C3×D5AGL(1,ℤ/8)12yes—
319C2×C3×D5#13712yes—
320C3×C4×C5—41no—
321C3×C4×C5—41no—
322C3×C4×C5C22×Q811yescharacteristic
323C3×C4×C5C22×Q811yescharacteristic
324Dic15—21no—
325Dic15—21no—
326Dic15—41no—
327Dic15—41no—
328C3×Dic5—41no—
329C3×Dic5—41no—
330C3×Dic5C4⋊Q812yes—
331C5×Dic3—21no—
332C5×Dic3—21no—
333C5×Dic3—41no—
334C5×Dic3—41no—
335C22×C3×C4—51no—
336C22×C3×C4—101no—
337C22×C3×C4—101no—
338C22×Dic3—51no—
339C22×Dic3—51no—
340C22×Dic3—101no—
341C22×Dic3—101no—
342C2×Dic6—21no—
343C2×Dic6—101no—
344C2×Dic6—101no—
345C2×Dic6—101no—
346C2×Dic6—204no—
347C2×C3×C8—51no—
348C2×C3×C8—54no—
349C2×C3×C8C22×D511yescharacteristic
350C3×C4⋊C4—51no—
351C3×C4⋊C4—51no—
352C3×C4⋊C4—101no—
353C3×C4⋊C4—101no—
354C3×C4⋊C4C22×D511yescharacteristic
355C3×C4⋊C4C22×D511yescharacteristic
356C4⋊Dic3—21no—
357C4⋊Dic3—51no—
358C4⋊Dic3—51no—
359C4⋊Dic3—101no—
360C4⋊Dic3—101no—
361C4⋊Dic3—101no—
362C4⋊Dic3—101no—
363C4⋊Dic3—101no—
364C4⋊Dic3C22×D511yescharacteristic
365C4⋊Dic3C22×D511yescharacteristic
366Dic3⋊C4—21no—
367Dic3⋊C4—21no—
368Dic3⋊C4—101no—
369Dic3⋊C4—101no—
370Dic3⋊C4—101no—
371Dic3⋊C4—101no—
372Dic3⋊C4—101no—
373Dic3⋊C4—101no—
374C4×Dic3—51no—
375C4×Dic3—51no—
376C4×Dic3—51no—
377C4×Dic3C22×D511yescharacteristic
378C2×C3⋊C8—51no—
379C2×C3⋊C8—54no—
380C2×C3⋊C8C22×D511yescharacteristic
381C22×D5D3×D411yescharacteristic
382C5×Q8—122no—
383C2×C4×C5—21no—
384C2×C4×C5—21no—
385C2×C4×C5—31no—
386C2×C4×C5—31no—
387C2×C4×C5—61no—
388C2×C4×C5—61no—
389C2×C4×C5C23×D311yescharacteristic
390C2×Dic5—21no—
391C2×Dic5—21no—
392C2×Dic5—31no—
393C2×Dic5—31no—
394C2×Dic5—61no—
395C2×Dic5—61no—
396C2×Dic5D3×D411yescharacteristic
397C4×D5—42no—
398C4×D5—42no—
399C4×D5—62no—
400C4×D5—62no—
401C4×D5—122no—
402C4×D5—122no—
403C4×D5C2×Dic612yes—
404C4×D5D3×Q812yes—
405Dic10—122no—
406Dic10—122no—
407Dic10—122no—
408C5×C8—32no—
409C5×C8C2×Dic612yescomplemented
410#168—32no—
411#168C2×Dic612yescomplemented
412C22×Q8—301no—
413C22×C8—51no—
414C22×C8—151no—
415C4⋊Q8—31no—
416C4⋊Q8—151no—
417C4⋊Q8—151no—
418C4⋊Q8—151no—
419C4⋊Q8—301no—
420C4⋊Q8—301no—
421#125—31no—
422#125—151no—
423#125—151no—
424#125—151no—
425#125—301no—
426#125—301no—
427C2×C4⋊C4—51no—
428C2×C4⋊C4—51no—
429C2×C4⋊C4—151no—
430C2×C4⋊C4—151no—
431C2×C4⋊C4—301no—
432C2×C4⋊C4—301no—
433C2×C4⋊C4—301no—
434C2×C42—151no—
435#107—31no—
436#107—31no—
437#107—151no—
438#107—151no—
439#107—151no—
440#107—151no—
441#107—151no—
442#107—151no—
443#106—31no—
444#106—51no—
445#106—51no—
446#106—51no—
447#106—151no—
448#106—151no—
449#106—151no—
450#106D3×D511yescharacteristic
451#97—31no—
452#97—151no—
453#97—151no—
454#97—151no—
455C2×C3×C5C2×C4⋊Q811yescharacteristic
456C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
457C2×C3×C5C2×#13711yescharacteristic
458C3×D5#50114yescomplemented
459C23×C3—51no—
460C2×C3×C4—21no—
461C2×C3×C4—21no—
462C2×C3×C4—51no—
463C2×C3×C4—52no—
464C2×C3×C4—52no—
465C2×C3×C4—101no—
466C2×C3×C4—101no—
467C2×C3×C4—202no—
468C2×C3×C4—202no—
469C2×C3×C4C23×D511yescharacteristic
470C2×Dic3—21no—
471C2×Dic3—21no—
472C2×Dic3—51no—
473C2×Dic3—51no—
474C2×Dic3—102no—
475C2×Dic3—102no—
476C2×Dic3—101no—
477C2×Dic3—101no—
478C2×Dic3—202no—
479C2×Dic3—202no—
480C2×Dic3D4×D511yescharacteristic
481C2×Dic3D4×D511yescharacteristic
482Dic6—42no—
483Dic6—202no—
484Dic6—202no—
485Dic6—202no—
486C3×C8—52no—
487C3×C8Q8×D512yescomplemented
488C3⋊C8—52no—
489C3⋊C8Q8×D512yescomplemented
490C22×C5C2×D3×D411yescharacteristic
491C2×D5#88812yes—
492C2×D5#91412yes—
493C2×D5#91512yes—
494C4×C5—41no—
495C4×C5—41no—
496C4×C5—61no—
497C4×C5—61no—
498C4×C5—121no—
499C4×C5—121no—
500C4×C5C22×Dic611yescharacteristic
501C4×C5C2×D3×Q811yescharacteristic
502Dic5—41no—
503Dic5—41no—
504Dic5—61no—
505Dic5—61no—
506Dic5—121no—
507Dic5—121no—
508Dic5#88812yes—
509C2×Q8—61no—
510C2×Q8—301no—
511C2×Q8—301no—
512C2×Q8—301no—
513C2×Q8—604no—
514C22×C4—51no—
515C22×C4—101no—
516C22×C4—101no—
517C22×C4—151no—
518C22×C4—151no—
519C22×C4—301no—
520C22×C4—301no—
521C2×C8—31no—
522C2×C8—51no—
523C2×C8—54no—
524C2×C8—154no—
525C2×C8—151no—
526C2×C8C2×D3×D511yescharacteristic
527C4⋊C4—31no—
528C4⋊C4—31no—
529C4⋊C4—51no—
530C4⋊C4—51no—
531C4⋊C4—61no—
532C4⋊C4—61no—
533C4⋊C4—61no—
534C4⋊C4—101no—
535C4⋊C4—101no—
536C4⋊C4—151no—
537C4⋊C4—151no—
538C4⋊C4—301no—
539C4⋊C4—301no—
540C4⋊C4—301no—
541C4⋊C4—301no—
542C4⋊C4—301no—
543C4⋊C4—301no—
544C4⋊C4—301no—
545C4⋊C4—301no—
546C4⋊C4—301no—
547C4⋊C4—301no—
548C4⋊C4—301no—
549C4⋊C4C2×D3×D511yescharacteristic
550C4⋊C4C2×D3×D511yescharacteristic
551C42—31no—
552C42—151no—
553C42—151no—
554C42—151no—
555C3×C5C2×#50111yescharacteristic, complemented, Hall
556C22×C3—52no—
557C22×C3—52no—
558C22×C3—52no—
559C22×C3C2×D4×D511yescharacteristic
560C3×C4—41no—
561C3×C4—41no—
562C3×C4—52no—
563C3×C4—201no—
564C3×C4—201no—
565C3×C4C2×Q8×D511yescharacteristic
566C3×C4C2×Q8×D511yescharacteristic
567Dic3—21no—
568Dic3—21no—
569Dic3—41no—
570Dic3—41no—
571Dic3—101no—
572Dic3—101no—
573Dic3—201no—
574Dic3—201no—
575C2×C5C2×#88811yescharacteristic
576C2×C5C2×#91411yescharacteristic
577C2×C5C2×#91511yescharacteristic
578D5#485614yesdirect product factor
579C23—51no—
580Q8—122no—
581Q8—602no—
582Q8—602no—
583Q8—602no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—31no—
587C2×C4—31no—
588C2×C4—51no—
589C2×C4—52no—
590C2×C4—52no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—101no—
594C2×C4—101no—
595C2×C4—151no—
596C2×C4—151no—
597C2×C4—202no—
598C2×C4—202no—
599C2×C4—302no—
600C2×C4—302no—
601C2×C4—301no—
602C2×C4—301no—
603C2×C4—602no—
604C2×C4—602no—
605C2×C4C22×D3×D511yescharacteristic
606C8—32no—
607C8—52no—
608C8—152no—
609C8D5×Dic612yescomplemented
610C2×C3—54no—
611C2×C3D5×C4⋊Q811yescharacteristic
612C2×C3D5×AGL(1,ℤ/8)11yescharacteristic
613C2×C3D5×#13711yescharacteristic
614C5C2×#485611yescharacteristic, complemented, Sylow
615C22—52no—
616C22—52no—
617C22—52no—
618C22D3×D4×D511yescharacteristic
619C4—41no—
620C4—41no—
621C4—52no—
622C4—61no—
623C4—61no—
624C4—121no—
625C4—121no—
626C4—201no—
627C4—201no—
628C4—301no—
629C4—301no—
630C4—601no—
631C4—601no—
632C4C2×D5×Dic611yescharacteristic
633C4D3×Q8×D511yescharacteristic
634C3D5×#50111yescharacteristic, complemented, Sylow
635C2—54no—
636C2D5×#88811yescharacteristic
637C2D5×#91411yescharacteristic
638C2D5×#91511yescharacteristic
subgroupG/H
centerC22D3×D4×D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#106C22
socleC22×C3×C5C22×D4
prime radicalC22×C3×D5C2×D4
hypercenter#106D3×D5
hypocenterC3×C5C2×#501
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
3. #11750957C2semidirect
4. C5×#4856C2semidirect
6. #11748038C2semidirect
7. #11748030C2semidirect
12. #11747425C2semidirect
13. #11747398C2semidirect
14. #11747163C2semidirect
15. #11747162C2semidirect
110. C3×C8×D5Q8semidirect
121. D5×C3⋊C8Q8semidirect
136. #4856D5direct (N × H)
208. C3×C5×C8C2×Q8semidirect
209. C3⋊#168C2×Q8semidirect
210. C3×#168C2×Q8semidirect
211. C5×C3⋊C8C2×Q8semidirect
300. C8×D5Dic6semidirect
409. C5×C8C2×Dic6semidirect
411. #168C2×Dic6semidirect
458. C3×D5#501semidirect
487. C3×C8Q8×D5semidirect
489. C3⋊C8Q8×D5semidirect
555. C3×C5C2×#501semidirect
578. D5#4856direct (N × H)
609. C8D5×Dic6semidirect
614. C5C2×#4856semidirect
634. C3D5×#501semidirect

Maximal subgroups

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

subgroupindex
D5×#8902
D5×#8882
#117509572
C5×#48562
C3×D5×#1062
#117480382
#117480302
D5×#8182
D5×#8152
D5×#8082
D5×#8072
#117474252
#117473982
#117471632
#117471622
D5×#5013
C2×#48565