Finite Group Databasebeta

← C5×#24086 #11946806

Subgroups — C5×#24086 #11946806

Order 1280.

1014 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3495C211yesmaximal, characteristic
2C5×#3092C211yesmaximal, characteristic
3C2×C5×#428C211yesmaximal, characteristic, complemented
4C5×#2502C211yesmaximal, characteristic, complemented
5C5×#2060C212yesmaximal, complemented
6C5×#2038C212yesmaximal, complemented
7C5×#1901C211yesmaximal, characteristic, complemented
8C5×#1871C211yesmaximal, characteristic
9C5×#1859C212yesmaximal, complemented
10C5×#1857C212yesmaximal, complemented
11C5×#1826C211yesmaximal, characteristic, complemented
12C22×C5×M16C2211yescharacteristic
13C5×#556—21no—
14C5×#554—21no—
15C5×#552—21no—
16C5×#552—22no—
17C5×#552—21no—
18C5×#551—22no—
19C5×#544C2211yescharacteristic
20C5×#542—21no—
21C5×#541—21no—
22C5×#536C2211yescharacteristic
23C5×#533—21no—
24C5×#529—21no—
25C5×#525C2211yescharacteristic
26C2×C5×C22⋊Q8C2211yescharacteristic
27C5×#520C2211yescharacteristic
28C5×#519C2211yescharacteristic
29C2×C5×#117C2211yescharacteristic
30C22×C5×C4⋊C4C2211yescharacteristic
31C5×#484—21no—
32C5×#483—21no—
33C5×#481—21no—
34C5×#480—21no—
35C5×#428—21no—
36C5×#428—21no—
37C5×#428C2212yes—
38C5×#428C2212yes—
39C5×#428C2212yes—
40C2×C5×#107—21no—
41C2×C5×#106—21no—
42C2×C5×#104C2212yes—
43C5×#417C2212yes—
44C2×C5×#101C2212yes—
45C5×#407C2211yescharacteristic
46C5×#398—22no—
47C5×#391C2212yes—
48C5×#385—22no—
49C5×#384C2212yes—
50C4×C5×C22⋊C4C2211yescharacteristic
51C5×#343C2212yes—
52C5×#339C2212yes—
53C5×#339C2212yes—
54C5×#339C2212yes—
55#24086C511yesmaximal, characteristic, direct product factor, Sylow
56C2×C5×#41C2311yescharacteristic
57C22×C5×Q8C2311yescharacteristic
58C23×C4×C5—21no—
59C23×C4×C5C2311yescharacteristic
60C2×C5×M16—21no—
61C2×C5×M16—24no—
62C2×C5×M16C2312yes—
63C2×C5×M16C2312yes—
64C22×C5×C8—21no—
65C5×C4⋊Q8—21no—
66C5×C4⋊Q8—21no—
67C5×C4⋊Q8—21no—
68C5×#126—22no—
69C5×#126—22no—
70C5×#126—22no—
71C5×#126—22no—
72C5×#125—22no—
73C5×#125—22no—
74C5×#125—22no—
75C5×#125—21no—
76C5×#125—21no—
77C5×#124—22no—
78C5×#124—21no—
79C5×#123—22no—
80C5×#123—22no—
81C5×#123—21no—
82C5×#123—22no—
83C5×C22⋊Q8—22no—
84C5×C22⋊Q8—22no—
85C5×C22⋊Q8—21no—
86C5×C22⋊Q8—21no—
87C5×C22⋊Q8—21no—
88C5×C22⋊Q8—21no—
89C5×C22⋊Q8—22no—
90C5×C22⋊Q8—21no—
91C5×C22⋊Q8—21no—
92C5×C22⋊Q8D411yescharacteristic
93C5×C22⋊Q8D411yescharacteristic
94C5×#121D411yescharacteristic
95C5×#121D411yescharacteristic
96C5×#120—21no—
97C4×C5×Q8—22no—
98C4×C5×Q8—21no—
99C4×C5×Q8—21no—
100C4×C5×Q8—21no—
101C4×C5×Q8—21no—
102C4×C5×Q8—21no—
103C4×C5×D4—22no—
104C4×C5×D4—21no—
105C4×C5×D4—22no—
106C4×C5×D4—21no—
107C4×C5×D4—21no—
108C5×#117—22no—
109C5×#117—22no—
110C5×#117—21no—
111C5×#117—21no—
112C5×#117—22no—
113C5×#117—21no—
114C5×#117D411yescharacteristic
115C5×#117D411yescharacteristic
116C5×#117C2311yescharacteristic
117C5×#117C2311yescharacteristic
118C5×#117C2311yescharacteristic
119C2×C5×C4⋊C4—21no—
120C2×C5×C4⋊C4—21no—
121C2×C5×C4⋊C4—21no—
122C2×C5×C4⋊C4—21no—
123C2×C5×C4⋊C4—21no—
124C2×C5×C4⋊C4—21no—
125C2×C5×C4⋊C4—24no—
126C2×C5×C4⋊C4D412yes—
127C2×C5×C4⋊C4D411yescharacteristic
128C2×C5×C4⋊C4D411yescharacteristic
129C2×C5×C4⋊C4C2312yes—
130C2×C5×C4⋊C4C2311yescharacteristic
131C2×C5×C22⋊C4—21no—
132C2×C5×C22⋊C4—21no—
133C2×C5×C22⋊C4D412yes—
134C2×C42×C5—21no—
135C2×C42×C5C2312yes—
136C5×#107—22no—
137C5×#107—22no—
138C5×#107—22no—
139C5×#106—22no—
140C5×#106—22no—
141C5×#106—22no—
142C5×#104—24no—
143C5×#103—22no—
144C5×#102—22no—
145C5×#101—24no—
146C5×C22⋊C8—21no—
147C5×C22⋊C8—21no—
148C5×#95—21no—
149C5×#95—24no—
150#3495C2×C511yescharacteristic
151#3092C2×C511yescharacteristic
152C2×#428C2×C511yescharacteristic, complemented
153#2502C2×C511yescharacteristic, complemented
154#2060C2×C512yescomplemented
155#2038C2×C512yescomplemented
156#1901C2×C511yescharacteristic, complemented
157#1871C2×C511yescharacteristic
158#1859C2×C512yescomplemented
159#1857C2×C512yescomplemented
160#1826C2×C511yescharacteristic, complemented
161C24×C5C2×D411yescharacteristic
162C5×#41—22no—
163C5×#41—22no—
164C2×C5×Q8—22no—
165C2×C5×Q8—21no—
166C2×C5×Q8—21no—
167C2×C5×Q8—22no—
168C2×C5×Q8—22no—
169C2×C5×Q8C2×D411yescharacteristic
170C2×C5×D4—21no—
171C2×C5×D4—21no—
172C2×C5×D4—21no—
173C2×C5×D4C2×D411yescharacteristic
174C22×C4×C5—22no—
175C22×C4×C5—22no—
176C22×C4×C5—21no—
177C22×C4×C5—22no—
178C22×C4×C5—22no—
179C22×C4×C5—21no—
180C22×C4×C5—21no—
181C22×C4×C5—22no—
182C22×C4×C5—41no—
183C22×C4×C5—41no—
184C22×C4×C5—82no—
185C22×C4×C5C2×D412yes—
186C22×C4×C5C2×D411yescharacteristic
187C22×C4×C5C2×D411yescharacteristic
188C22×C4×C5C2×D411yescharacteristic
189C22×C4×C5C2×D411yescharacteristic
190C22×C4×C5C2×D411yescharacteristic
191C22×C4×C5C2×D412yes—
192C22×C4×C5C2411yescharacteristic
193C5×M16—24no—
194C5×M16—22no—
195C5×M16—22no—
196C5×M16—22no—
197C5×M16#4114yes—
198C2×C5×C8—24no—
199C2×C5×C8—22no—
200C2×C5×C8—22no—
201C5×C4⋊C4—22no—
202C5×C4⋊C4—22no—
203C5×C4⋊C4—22no—
204C5×C4⋊C4—22no—
205C5×C4⋊C4—22no—
206C5×C4⋊C4—22no—
207C5×C4⋊C4—22no—
208C5×C4⋊C4—22no—
209C5×C4⋊C4—21no—
210C5×C4⋊C4—21no—
211C5×C4⋊C4—22no—
212C5×C4⋊C4—22no—
213C5×C4⋊C4—21no—
214C5×C4⋊C4—21no—
215C5×C4⋊C4—21no—
216C5×C4⋊C4—21no—
217C5×C4⋊C4—22no—
218C5×C4⋊C4—22no—
219C5×C4⋊C4—41no—
220C5×C4⋊C4—41no—
221C5×C4⋊C4—84no—
222C5×C4⋊C4C2×D411yescharacteristic
223C5×C4⋊C4C2×D411yescharacteristic
224C5×C4⋊C4C2×D411yescharacteristic
225C5×C4⋊C4C2×D411yescharacteristic
226C5×C4⋊C4C2×D411yescharacteristic
227C5×C4⋊C4C2×D411yescharacteristic
228C5×C22⋊C4—24no—
229C5×C22⋊C4—22no—
230C5×C22⋊C4—22no—
231C5×C22⋊C4—22no—
232C5×C22⋊C4—21no—
233C5×C22⋊C4—21no—
234C5×C22⋊C4—22no—
235C5×C22⋊C4—22no—
236C5×C22⋊C4—21no—
237C5×C22⋊C4—21no—
238C5×C22⋊C4—21no—
239C5×C22⋊C4—21no—
240C5×C22⋊C4—21no—
241C5×C22⋊C4—21no—
242C5×C22⋊C4—21no—
243C42×C5—22no—
244C42×C5—22no—
245C42×C5—22no—
246C42×C5—21no—
247C42×C5—21no—
248C42×C5—21no—
249C42×C5—21no—
250C42×C5—21no—
251C42×C5—22no—
252C42×C5—22no—
253C22×M16C22×C511yescharacteristic
254#556—21no—
255#554—21no—
256#552—21no—
257#552—22no—
258#552—21no—
259#551—22no—
260#544C22×C511yescharacteristic
261#542—21no—
262#541—21no—
263#536C22×C511yescharacteristic
264#533—21no—
265#529—21no—
266#525C22×C511yescharacteristic
267C2×C22⋊Q8C22×C511yescharacteristic
268#520C22×C511yescharacteristic
269#519C22×C511yescharacteristic
270C2×#117C22×C511yescharacteristic
271C22×C4⋊C4C22×C511yescharacteristic
272#484—21no—
273#483—21no—
274#481—21no—
275#480—21no—
276#428—21no—
277#428—21no—
278#428C22×C512yes—
279#428C22×C512yes—
280#428C22×C512yes—
281C2×#107—21no—
282C2×#106—21no—
283C2×#104C22×C512yes—
284#417C22×C512yes—
285C2×#101C22×C512yes—
286#407C22×C511yescharacteristic
287#398—22no—
288#391C22×C512yes—
289#385—22no—
290#384C22×C512yes—
291C4×C22⋊C4C22×C511yescharacteristic
292#343C22×C512yes—
293#339C22×C512yes—
294#339C22×C512yes—
295#339C22×C512yes—
296C23×C5—22no—
297C23×C5—22no—
298C23×C5—21no—
299C23×C5—41no—
300C23×C5#12011yescharacteristic
301C23×C5#12011yescharacteristic
302C23×C5C22×D411yescharacteristic
303C5×Q8—22no—
304C5×Q8—41no—
305C5×Q8—41no—
306C5×Q8—42no—
307C5×Q8—82no—
308C5×D4—22no—
309C5×D4—42no—
310C5×D4—42no—
311C5×D4—81no—
312C2×C4×C5—22no—
313C2×C4×C5—21no—
314C2×C4×C5—21no—
315C2×C4×C5—21no—
316C2×C4×C5—21no—
317C2×C4×C5—22no—
318C2×C4×C5—22no—
319C2×C4×C5—24no—
320C2×C4×C5—21no—
321C2×C4×C5—21no—
322C2×C4×C5—21no—
323C2×C4×C5—21no—
324C2×C4×C5—21no—
325C2×C4×C5—22no—
326C2×C4×C5—21no—
327C2×C4×C5—21no—
328C2×C4×C5—22no—
329C2×C4×C5—24no—
330C2×C4×C5—42no—
331C2×C4×C5—42no—
332C2×C4×C5—42no—
333C2×C4×C5—41no—
334C2×C4×C5—41no—
335C2×C4×C5—42no—
336C2×C4×C5—44no—
337C2×C4×C5—42no—
338C2×C4×C5—81no—
339C2×C4×C5—82no—
340C2×C4×C5—82no—
341C2×C4×C5—82no—
342C2×C4×C5#12011yescharacteristic
343C2×C4×C5#12011yescharacteristic
344C2×C4×C5C22×D411yescharacteristic
345C2×C4×C5C22×D411yescharacteristic
346C2×C4×C5C2×#4112yes—
347C2×C4×C52−1+412yes—
348C5×C8—24no—
349C5×C8—42no—
350C2×#41C23×C511yescharacteristic
351C22×Q8C23×C511yescharacteristic
352C23×C4—21no—
353C23×C4C23×C511yescharacteristic
354C2×M16—21no—
355C2×M16—24no—
356C2×M16C23×C512yes—
357C2×M16C23×C512yes—
358C22×C8—21no—
359C4⋊Q8—21no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362#126—22no—
363#126—22no—
364#126—22no—
365#126—22no—
366#125—22no—
367#125—22no—
368#125—22no—
369#125—21no—
370#125—21no—
371#124—22no—
372#124—21no—
373#123—22no—
374#123—22no—
375#123—21no—
376#123—22no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—22no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386C22⋊Q8C5×D411yescharacteristic
387C22⋊Q8C5×D411yescharacteristic
388#121C5×D411yescharacteristic
389#121C5×D411yescharacteristic
390#120—21no—
391C4×Q8—22no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×D4—22no—
398C4×D4—21no—
399C4×D4—22no—
400C4×D4—21no—
401C4×D4—21no—
402#117—22no—
403#117—22no—
404#117—21no—
405#117—21no—
406#117—22no—
407#117—21no—
408#117C5×D411yescharacteristic
409#117C5×D411yescharacteristic
410#117C23×C511yescharacteristic
411#117C23×C511yescharacteristic
412#117C23×C511yescharacteristic
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—24no—
420C2×C4⋊C4C5×D412yes—
421C2×C4⋊C4C5×D411yescharacteristic
422C2×C4⋊C4C5×D411yescharacteristic
423C2×C4⋊C4C23×C512yes—
424C2×C4⋊C4C23×C511yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4C5×D412yes—
428C2×C42—21no—
429C2×C42C23×C512yes—
430#107—22no—
431#107—22no—
432#107—22no—
433#106—22no—
434#106—22no—
435#106—22no—
436#104—24no—
437#103—22no—
438#102—22no—
439#101—24no—
440C22⋊C8—21no—
441C22⋊C8—21no—
442#95—21no—
443#95—24no—
444C22×C5—21no—
445C22×C5—21no—
446C22×C5—21no—
447C22×C5—21no—
448C22×C5—21no—
449C22×C5—21no—
450C22×C5—21no—
451C22×C5—44no—
452C22×C5—41no—
453C22×C5—41no—
454C22×C5—41no—
455C22×C5#45612yes—
456C22×C5C2×#12011yescharacteristic
457C22×C5#54712yes—
458C4×C5—21no—
459C4×C5—21no—
460C4×C5—44no—
461C4×C5—42no—
462C4×C5—41no—
463C4×C5—41no—
464C4×C5—41no—
465C4×C5—41no—
466C4×C5—42no—
467C4×C5—41no—
468C4×C5—41no—
469C4×C5—41no—
470C4×C5—82no—
471C4×C5#52511yescharacteristic
472C4×C5#53611yescharacteristic
473C4×C5#54712yes—
474C24C2×C5×D411yescharacteristic
475#41—22no—
476#41—22no—
477C2×Q8—22no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—22no—
481C2×Q8—22no—
482C2×Q8C2×C5×D411yescharacteristic
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×C5×D411yescharacteristic
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—82no—
498C22×C4C2×C5×D412yes—
499C22×C4C2×C5×D411yescharacteristic
500C22×C4C2×C5×D411yescharacteristic
501C22×C4C2×C5×D411yescharacteristic
502C22×C4C2×C5×D411yescharacteristic
503C22×C4C2×C5×D411yescharacteristic
504C22×C4C2×C5×D412yes—
505C22×C4C24×C511yescharacteristic
506M16—24no—
507M16—22no—
508M16—22no—
509M16—22no—
510M16C5×#4114yes—
511C2×C8—24no—
512C2×C8—22no—
513C2×C8—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—84no—
535C4⋊C4C2×C5×D411yescharacteristic
536C4⋊C4C2×C5×D411yescharacteristic
537C4⋊C4C2×C5×D411yescharacteristic
538C4⋊C4C2×C5×D411yescharacteristic
539C4⋊C4C2×C5×D411yescharacteristic
540C4⋊C4C2×C5×D411yescharacteristic
541C22⋊C4—24no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—21no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—22no—
565C42—22no—
566C2×C5—22no—
567C2×C5—22no—
568C2×C5—41no—
569C2×C5—81no—
570C2×C5#240611yescharacteristic
571C2×C5C2×#45611yescharacteristic
572C2×C5#302111yescharacteristic
573C23—22no—
574C23—22no—
575C23—21no—
576C23—41no—
577C23C5×#12011yescharacteristic
578C23C5×#12011yescharacteristic
579C23C22×C5×D411yescharacteristic
580Q8—22no—
581Q8—41no—
582Q8—41no—
583Q8—42no—
584Q8—82no—
585D4—22no—
586D4—42no—
587D4—42no—
588D4—81no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—81no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4C5×#12011yescharacteristic
620C2×C4C5×#12011yescharacteristic
621C2×C4C22×C5×D411yescharacteristic
622C2×C4C22×C5×D411yescharacteristic
623C2×C4C2×C5×#4112yes—
624C2×C4C5×2−1+412yes—
625C8—24no—
626C8—42no—
627C5#2408611yescharacteristic, direct product factor, Sylow
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—44no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22C5×#45612yes—
640C22C2×C5×#12011yescharacteristic
641C22C5×#54712yes—
642C4—21no—
643C4—21no—
644C4—44no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—42no—
653C4—41no—
654C4—82no—
655C4C5×#52511yescharacteristic
656C4C5×#53611yescharacteristic
657C4C5×#54712yes—
658C2—22no—
659C2—22no—
660C2—41no—
661C2—81no—
662C2C5×#240611yescharacteristic
663C2C2×C5×#45611yescharacteristic
664C2C5×#302111yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC5×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C5×#428C2semidirect
4. C5×#2502C2semidirect
5. C5×#2060C2semidirect
6. C5×#2038C2semidirect
7. C5×#1901C2semidirect
9. C5×#1859C2semidirect
10. C5×#1857C2semidirect
11. C5×#1826C2semidirect
55. #24086C5direct (N × H)
152. C2×#428C2×C5semidirect
153. #2502C2×C5semidirect
154. #2060C2×C5semidirect
155. #2038C2×C5semidirect
156. #1901C2×C5semidirect
158. #1859C2×C5semidirect
159. #1857C2×C5semidirect
160. #1826C2×C5semidirect
627. C5#24086direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#34952
C5×#30922
C2×C5×#4282
C5×#25022
C5×#20602
C5×#20382
C5×#19012
C5×#18712
C5×#18592
C5×#18572
C5×#18262
#240865