Finite Group Databasebeta

← C5×#24085 #11946805

Subgroups — C5×#24085 #11946805

Order 1280.

998 conjugacy classes of subgroups.

The subgroup lattice has 664 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×#2503C211yesmaximal, characteristic, complemented
5C5×#2061C212yesmaximal, complemented
6C5×#2038C212yesmaximal, complemented
7C5×#1901C211yesmaximal, characteristic, complemented
8C5×#1871C211yesmaximal, characteristic
9C5×#1860C212yesmaximal, 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×#400—22no—
47C5×#391C2212yes—
48C5×#382C2212yes—
49C5×#381—22no—
50C4×C5×C22⋊C4C2211yescharacteristic
51C5×#343C2212yes—
52C5×#339C2212yes—
53C5×#339C2212yes—
54C5×#339C2212yes—
55#24085C511yesmaximal, 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⋊C4D412yes—
126C2×C5×C4⋊C4D411yescharacteristic
127C2×C5×C4⋊C4D411yescharacteristic
128C2×C5×C4⋊C4C2312yes—
129C2×C5×C4⋊C4C2311yescharacteristic
130C2×C5×C22⋊C4—21no—
131C2×C5×C22⋊C4—21no—
132C2×C5×C22⋊C4D412yes—
133C2×C42×C5—21no—
134C2×C42×C5C2312yes—
135C5×#107—22no—
136C5×#107—22no—
137C5×#107—22no—
138C5×#106—22no—
139C5×#106—22no—
140C5×#106—22no—
141C5×#104—24no—
142C5×#103—22no—
143C5×#102—22no—
144C5×#101—24no—
145C5×C22⋊C8—21no—
146C5×C22⋊C8—21no—
147C5×#95—21no—
148C5×#95—24no—
149C5×#95—24no—
150#3495C2×C511yescharacteristic
151#3092C2×C511yescharacteristic
152C2×#428C2×C511yescharacteristic, complemented
153#2503C2×C511yescharacteristic, complemented
154#2061C2×C512yescomplemented
155#2038C2×C512yescomplemented
156#1901C2×C511yescharacteristic, complemented
157#1871C2×C511yescharacteristic
158#1860C2×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—21no—
179C22×C4×C5—21no—
180C22×C4×C5—22no—
181C22×C4×C5—22no—
182C22×C4×C5—44no—
183C22×C4×C5—41no—
184C22×C4×C5—41no—
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⋊C4C2×D411yescharacteristic
222C5×C4⋊C4C2×D411yescharacteristic
223C5×C4⋊C4C2×D411yescharacteristic
224C5×C4⋊C4C2×D411yescharacteristic
225C5×C4⋊C4C2×D411yescharacteristic
226C5×C4⋊C4C2×D411yescharacteristic
227C5×C22⋊C4—22no—
228C5×C22⋊C4—22no—
229C5×C22⋊C4—22no—
230C5×C22⋊C4—21no—
231C5×C22⋊C4—21no—
232C5×C22⋊C4—22no—
233C5×C22⋊C4—22no—
234C5×C22⋊C4—21no—
235C5×C22⋊C4—21no—
236C5×C22⋊C4—21no—
237C5×C22⋊C4—21no—
238C5×C22⋊C4—21no—
239C5×C22⋊C4—21no—
240C5×C22⋊C4—21no—
241C5×C22⋊C4—42no—
242C42×C5—22no—
243C42×C5—22no—
244C42×C5—22no—
245C42×C5—21no—
246C42×C5—21no—
247C42×C5—21no—
248C42×C5—21no—
249C42×C5—21no—
250C42×C5—22no—
251C42×C5—22no—
252C22×M16C22×C511yescharacteristic
253#556—21no—
254#554—21no—
255#552—21no—
256#552—22no—
257#552—21no—
258#551—22no—
259#544C22×C511yescharacteristic
260#542—21no—
261#541—21no—
262#536C22×C511yescharacteristic
263#533—21no—
264#529—21no—
265#525C22×C511yescharacteristic
266C2×C22⋊Q8C22×C511yescharacteristic
267#520C22×C511yescharacteristic
268#519C22×C511yescharacteristic
269C2×#117C22×C511yescharacteristic
270C22×C4⋊C4C22×C511yescharacteristic
271#484—21no—
272#483—21no—
273#481—21no—
274#480—21no—
275#428—21no—
276#428—21no—
277#428C22×C512yes—
278#428C22×C512yes—
279#428C22×C512yes—
280C2×#107—21no—
281C2×#106—21no—
282C2×#104C22×C512yes—
283#417C22×C512yes—
284C2×#101C22×C512yes—
285#407C22×C511yescharacteristic
286#400—22no—
287#391C22×C512yes—
288#382C22×C512yes—
289#381—22no—
290C4×C22⋊C4C22×C511yescharacteristic
291#343C22×C512yes—
292#339C22×C512yes—
293#339C22×C512yes—
294#339C22×C512yes—
295C23×C5—22no—
296C23×C5—22no—
297C23×C5—21no—
298C23×C5—41no—
299C23×C5#12011yescharacteristic
300C23×C5#12011yescharacteristic
301C23×C5C22×D411yescharacteristic
302C5×Q8—22no—
303C5×Q8—41no—
304C5×Q8—41no—
305C5×Q8—42no—
306C5×Q8—82no—
307C5×D4—22no—
308C5×D4—42no—
309C5×D4—42no—
310C5×D4—81no—
311C2×C4×C5—22no—
312C2×C4×C5—21no—
313C2×C4×C5—21no—
314C2×C4×C5—21no—
315C2×C4×C5—21no—
316C2×C4×C5—22no—
317C2×C4×C5—22no—
318C2×C4×C5—21no—
319C2×C4×C5—21no—
320C2×C4×C5—21no—
321C2×C4×C5—21no—
322C2×C4×C5—21no—
323C2×C4×C5—22no—
324C2×C4×C5—21no—
325C2×C4×C5—21no—
326C2×C4×C5—22no—
327C2×C4×C5—24no—
328C2×C4×C5—42no—
329C2×C4×C5—41no—
330C2×C4×C5—41no—
331C2×C4×C5—42no—
332C2×C4×C5—42no—
333C2×C4×C5—42no—
334C2×C4×C5—44no—
335C2×C4×C5—42no—
336C2×C4×C5—42no—
337C2×C4×C5—81no—
338C2×C4×C5—82no—
339C2×C4×C5—82no—
340C2×C4×C5—82no—
341C2×C4×C5#12011yescharacteristic
342C2×C4×C5#12011yescharacteristic
343C2×C4×C5C22×D411yescharacteristic
344C2×C4×C5C22×D411yescharacteristic
345C2×C4×C5C2×#4112yes—
346C2×C4×C52−1+412yes—
347C5×C8—24no—
348C5×C8—42no—
349C2×#41C23×C511yescharacteristic
350C22×Q8C23×C511yescharacteristic
351C23×C4—21no—
352C23×C4C23×C511yescharacteristic
353C2×M16—21no—
354C2×M16—24no—
355C2×M16C23×C512yes—
356C2×M16C23×C512yes—
357C22×C8—21no—
358C4⋊Q8—21no—
359C4⋊Q8—21no—
360C4⋊Q8—21no—
361#126—22no—
362#126—22no—
363#126—22no—
364#126—22no—
365#125—22no—
366#125—22no—
367#125—22no—
368#125—21no—
369#125—21no—
370#124—22no—
371#124—21no—
372#123—22no—
373#123—22no—
374#123—21no—
375#123—22no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—22no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8C5×D411yescharacteristic
386C22⋊Q8C5×D411yescharacteristic
387#121C5×D411yescharacteristic
388#121C5×D411yescharacteristic
389#120—21no—
390C4×Q8—22no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×D4—22no—
397C4×D4—21no—
398C4×D4—22no—
399C4×D4—21no—
400C4×D4—21no—
401#117—22no—
402#117—22no—
403#117—21no—
404#117—21no—
405#117—22no—
406#117—21no—
407#117C5×D411yescharacteristic
408#117C5×D411yescharacteristic
409#117C23×C511yescharacteristic
410#117C23×C511yescharacteristic
411#117C23×C511yescharacteristic
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4C5×D412yes—
419C2×C4⋊C4C5×D411yescharacteristic
420C2×C4⋊C4C5×D411yescharacteristic
421C2×C4⋊C4C23×C512yes—
422C2×C4⋊C4C23×C511yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4C5×D412yes—
426C2×C42—21no—
427C2×C42C23×C512yes—
428#107—22no—
429#107—22no—
430#107—22no—
431#106—22no—
432#106—22no—
433#106—22no—
434#104—24no—
435#103—22no—
436#102—22no—
437#101—24no—
438C22⋊C8—21no—
439C22⋊C8—21no—
440#95—21no—
441#95—24no—
442#95—24no—
443C22×C5—21no—
444C22×C5—21no—
445C22×C5—21no—
446C22×C5—21no—
447C22×C5—21no—
448C22×C5—21no—
449C22×C5—21no—
450C22×C5—44no—
451C22×C5—41no—
452C22×C5—41no—
453C22×C5—41no—
454C22×C5#45412yes—
455C22×C5C2×#12011yescharacteristic
456C22×C5#54712yes—
457C4×C5—21no—
458C4×C5—21no—
459C4×C5—42no—
460C4×C5—41no—
461C4×C5—41no—
462C4×C5—41no—
463C4×C5—41no—
464C4×C5—42no—
465C4×C5—41no—
466C4×C5—41no—
467C4×C5—41no—
468C4×C5—82no—
469C4×C5—82no—
470C4×C5#52511yescharacteristic
471C4×C5#53611yescharacteristic
472C4×C5#54712yes—
473C24C2×C5×D411yescharacteristic
474#41—22no—
475#41—22no—
476C2×Q8—22no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—22no—
480C2×Q8—22no—
481C2×Q8C2×C5×D411yescharacteristic
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4C2×C5×D411yescharacteristic
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—44no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4C2×C5×D412yes—
498C22×C4C2×C5×D411yescharacteristic
499C22×C4C2×C5×D411yescharacteristic
500C22×C4C2×C5×D411yescharacteristic
501C22×C4C2×C5×D411yescharacteristic
502C22×C4C2×C5×D411yescharacteristic
503C22×C4C2×C5×D412yes—
504C22×C4C24×C511yescharacteristic
505M16—24no—
506M16—22no—
507M16—22no—
508M16—22no—
509M16C5×#4114yes—
510C2×C8—24no—
511C2×C8—22no—
512C2×C8—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4C2×C5×D411yescharacteristic
534C4⋊C4C2×C5×D411yescharacteristic
535C4⋊C4C2×C5×D411yescharacteristic
536C4⋊C4C2×C5×D411yescharacteristic
537C4⋊C4C2×C5×D411yescharacteristic
538C4⋊C4C2×C5×D411yescharacteristic
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—42no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—22no—
563C42—22no—
564C2×C5—22no—
565C2×C5—22no—
566C2×C5—41no—
567C2×C5—81no—
568C2×C5#240611yescharacteristic
569C2×C5C2×#45411yescharacteristic
570C2×C5#302211yescharacteristic
571C23—22no—
572C23—22no—
573C23—21no—
574C23—41no—
575C23C5×#12011yescharacteristic
576C23C5×#12011yescharacteristic
577C23C22×C5×D411yescharacteristic
578Q8—22no—
579Q8—41no—
580Q8—41no—
581Q8—42no—
582Q8—82no—
583D4—22no—
584D4—42no—
585D4—42no—
586D4—81no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—81no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4C5×#12011yescharacteristic
618C2×C4C5×#12011yescharacteristic
619C2×C4C22×C5×D411yescharacteristic
620C2×C4C22×C5×D411yescharacteristic
621C2×C4C2×C5×#4112yes—
622C2×C4C5×2−1+412yes—
623C8—24no—
624C8—42no—
625C5#2408511yescharacteristic, direct product factor, Sylow
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—44no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C5×#45412yes—
638C22C2×C5×#12011yescharacteristic
639C22C5×#54712yes—
640C4—21no—
641C4—21no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—82no—
652C4—82no—
653C4C5×#52511yescharacteristic
654C4C5×#53611yescharacteristic
655C4C5×#54712yes—
656C2—22no—
657C2—22no—
658C2—41no—
659C2—81no—
660C2C5×#240611yescharacteristic
661C2C2×C5×#45411yescharacteristic
662C2C5×#302211yescharacteristic
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×#2503C2semidirect
5. C5×#2061C2semidirect
6. C5×#2038C2semidirect
7. C5×#1901C2semidirect
9. C5×#1860C2semidirect
10. C5×#1857C2semidirect
11. C5×#1826C2semidirect
55. #24085C5direct (N × H)
152. C2×#428C2×C5semidirect
153. #2503C2×C5semidirect
154. #2061C2×C5semidirect
155. #2038C2×C5semidirect
156. #1901C2×C5semidirect
158. #1860C2×C5semidirect
159. #1857C2×C5semidirect
160. #1826C2×C5semidirect
625. C5#24085direct (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×#25032
C5×#20612
C5×#20382
C5×#19012
C5×#18712
C5×#18602
C5×#18572
C5×#18262
#240855