Finite Group Databasebeta

← C5×#18586 #11941306

Subgroups — C5×#18586 #11941306

Order 1280.

964 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3520C211yesmaximal, characteristic
2C5×#3239C211yesmaximal, characteristic, complemented
3C5×#3165C211yesmaximal, characteristic, complemented
4C2×C5×#461C211yesmaximal, characteristic, complemented
5C5×#1654C212yesmaximal, complemented
6C5×#1651C212yesmaximal, complemented
7C5×#1650C212yesmaximal, complemented
8C5×#1647C212yesmaximal, complemented
9C5×#1552C211yesmaximal, characteristic, complemented
10C5×#1490C211yesmaximal, characteristic, complemented
11C5×#1483C211yesmaximal, characteristic, complemented
12C22×C5×SD16—21no—
13C5×#556C2211yescharacteristic
14C5×#554—21no—
15C5×#552—21no—
16C5×#551—22no—
17C5×D4×Q8C2211yescharacteristic
18C5×#548—21no—
19C5×#547C2211yescharacteristic
20C5×#547C2211yescharacteristic
21C5×#546—21no—
22C5×#542—21no—
23C5×#539—22no—
24C5×#533C2211yescharacteristic
25C2×C5×C4⋊Q8C2211yescharacteristic
26C5×#529C2211yescharacteristic
27C5×#525—21no—
28C2×C5×C22⋊Q8—21no—
29C5×#520—21no—
30C2×C4×C5×D4C2211yescharacteristic
31C5×#501—21no—
32C5×#501—22no—
33C5×#500C2212yescomplemented
34C5×#498C2212yescomplemented
35C5×#489—21no—
36C5×#484—21no—
37C5×#481—21no—
38C5×#475C2212yescomplemented
39C5×#474C2212yescomplemented
40C5×#461—21no—
41C5×#461—21no—
42C5×#461C2212yes—
43C5×#461C2212yes—
44C5×#432C2211yescharacteristic
45C5×#428—22no—
46C2×C5×C4⋊C8C2211yescharacteristic
47C2×C5×#103—21no—
48C4×C5×M16C2211yescharacteristic
49C5×#332C2212yescomplemented
50C5×#332C2212yescomplemented
51C5×Q8⋊C8C2212yescomplemented
52C5×D4⋊C8C2212yes—
53C5×D4⋊C8C2212yes—
54C5×D4⋊C8C2212yescomplemented
55#18586C511yesmaximal, characteristic, direct product factor, Sylow
56C2×C5×#41—21no—
57C2×C5×#41—21no—
58C22×C5×Q8—21no—
59C22×C5×Q8—21no—
60C22×C5×D4D411yescharacteristic
61C23×C4×C5—21no—
62C2×C5×SD16—22no—
63C2×C5×SD16—22no—
64C2×C5×SD16—44no—
65C2×C5×M16—21no—
66C22×C5×C8—21no—
67C5×C4⋊Q8—21no—
68C5×C4⋊Q8—21no—
69C5×C4⋊Q8—21no—
70C5×C4⋊Q8—21no—
71C5×C4⋊Q8C2312yes—
72C5×C4⋊Q8C2311yescharacteristic
73C5×C4⋊Q8C2311yescharacteristic
74C5×#126—22no—
75C5×#126—22no—
76C5×#126—21no—
77C5×#125—22no—
78C5×#125D411yescharacteristic
79C5×#125D412yes—
80C5×#124—22no—
81C5×#124—21no—
82C5×#124D411yescharacteristic
83C5×#123—22no—
84C5×#123—22no—
85C5×#123—21no—
86C5×#123—22no—
87C5×#123—22no—
88C5×C22⋊Q8—22no—
89C5×C22⋊Q8—22no—
90C5×C22⋊Q8—22no—
91C5×C22⋊Q8—21no—
92C5×C22⋊Q8—21no—
93C5×C22⋊Q8—21no—
94C5×C22⋊Q8—21no—
95C5×C22⋊Q8—21no—
96C5×C22⋊Q8—22no—
97C5×C22⋊Q8—21no—
98C5×C22⋊Q8D411yescharacteristic
99C5×#121—22no—
100C5×#121—21no—
101C5×#121—21no—
102C5×#121D411yescharacteristic
103C5×#120—22no—
104C4×C5×Q8—21no—
105C4×C5×Q8—21no—
106C4×C5×Q8—21no—
107C4×C5×Q8C2311yescharacteristic
108C4×C5×D4—22no—
109C4×C5×D4—22no—
110C4×C5×D4—22no—
111C4×C5×D4—21no—
112C4×C5×D4—21no—
113C4×C5×D4—21no—
114C4×C5×D4—21no—
115C4×C5×D4D411yescharacteristic
116C4×C5×D4D411yescharacteristic
117C4×C5×D4C2311yescharacteristic
118C4×C5×D4C2311yescharacteristic
119C4×C5×D4C2311yescharacteristic
120C5×#117—22no—
121C5×#117—21no—
122C5×#117—21no—
123C5×#117D412yes—
124C2×C5×C4⋊C4—22no—
125C2×C5×C4⋊C4—21no—
126C2×C5×C4⋊C4—21no—
127C2×C5×C4⋊C4—21no—
128C2×C5×C4⋊C4—21no—
129C2×C5×C4⋊C4D411yescharacteristic
130C2×C5×C22⋊C4—22no—
131C2×C5×C22⋊C4—21no—
132C2×C42×C5C2311yescharacteristic
133C5×#107—24no—
134C5×#107—22no—
135C5×#106—24no—
136C5×#106—22no—
137C5×C4⋊C8—21no—
138C5×C4⋊C8C2312yes—
139C5×C4⋊C8C2312yes—
140C5×#103—22no—
141C5×#103—22no—
142C5×#103—22no—
143C5×#102—22no—
144C5×C22⋊C8—21no—
145C5×#97—21no—
146C5×#97—21no—
147C4×C5×C8C2312yes—
148#3520C2×C511yescharacteristic
149#3239C2×C511yescharacteristic, complemented
150#3165C2×C511yescharacteristic, complemented
151C2×#461C2×C511yescharacteristic, complemented
152#1654C2×C512yescomplemented
153#1651C2×C512yescomplemented
154#1650C2×C512yescomplemented
155#1647C2×C512yescomplemented
156#1552C2×C511yescharacteristic, complemented
157#1490C2×C511yescharacteristic, complemented
158#1483C2×C511yescharacteristic, complemented
159C24×C5—21no—
160C5×#41—42no—
161C5×#41—42no—
162C5×#41—82no—
163C2×C5×Q8—22no—
164C2×C5×Q8—21no—
165C2×C5×Q8—21no—
166C2×C5×Q8—22no—
167C2×C5×Q8—42no—
168C2×C5×Q8—42no—
169C2×C5×Q8—41no—
170C2×C5×Q8—82no—
171C2×C5×Q8C2×D411yescharacteristic
172C2×C5×D4—22no—
173C2×C5×D4—22no—
174C2×C5×D4—21no—
175C2×C5×D4—22no—
176C2×C5×D4—21no—
177C2×C5×D4—41no—
178C2×C5×D4C2×D411yescharacteristic
179C2×C5×D4C2×D411yescharacteristic
180C2×C5×D4C2×D411yescharacteristic
181C22×C4×C5—22no—
182C22×C4×C5—22no—
183C22×C4×C5—22no—
184C22×C4×C5—21no—
185C22×C4×C5—21no—
186C22×C4×C5—21no—
187C22×C4×C5—21no—
188C22×C4×C5—21no—
189C22×C4×C5—21no—
190C22×C4×C5—41no—
191C22×C4×C5—82no—
192C22×C4×C5C2×D411yescharacteristic
193C22×C4×C5C2×D411yescharacteristic
194C22×C4×C5C2×D411yescharacteristic
195C5×SD16—44no—
196C5×SD16—44no—
197C5×M16—24no—
198C2×C5×C8—22no—
199C2×C5×C8—22no—
200C2×C5×C8—22no—
201C2×C5×C8—42no—
202C5×C4⋊C4—22no—
203C5×C4⋊C4—22no—
204C5×C4⋊C4—22no—
205C5×C4⋊C4—22no—
206C5×C4⋊C4—21no—
207C5×C4⋊C4—22no—
208C5×C4⋊C4—21no—
209C5×C4⋊C4—21no—
210C5×C4⋊C4—21no—
211C5×C4⋊C4—22no—
212C5×C4⋊C4—21no—
213C5×C4⋊C4—21no—
214C5×C4⋊C4—21no—
215C5×C4⋊C4—22no—
216C5×C4⋊C4—22no—
217C5×C4⋊C4—41no—
218C5×C4⋊C4—41no—
219C5×C4⋊C4C2×D411yescharacteristic
220C5×C4⋊C4C2×D411yescharacteristic
221C5×C4⋊C4C2×D411yescharacteristic
222C5×C4⋊C4C2×D411yescharacteristic
223C5×C4⋊C4C2×D412yes—
224C5×C4⋊C4C2×D412yes—
225C5×C22⋊C4—22no—
226C5×C22⋊C4—22no—
227C5×C22⋊C4—22no—
228C5×C22⋊C4—22no—
229C5×C22⋊C4—22no—
230C5×C22⋊C4—21no—
231C5×C22⋊C4—22no—
232C5×C22⋊C4—21no—
233C5×C22⋊C4—21no—
234C5×C22⋊C4—21no—
235C5×C22⋊C4—21no—
236C5×C22⋊C4—21no—
237C5×C22⋊C4—21no—
238C5×C22⋊C4—21no—
239C5×C22⋊C4—21no—
240C42×C5—22no—
241C42×C5—22no—
242C42×C5—21no—
243C42×C5C2×D411yescharacteristic
244C42×C5C2×D411yescharacteristic
245C42×C5C2×D411yescharacteristic
246C42×C5C2411yescharacteristic
247C22×SD16—21no—
248#556C22×C511yescharacteristic
249#554—21no—
250#552—21no—
251#551—22no—
252D4×Q8C22×C511yescharacteristic
253#548—21no—
254#547C22×C511yescharacteristic
255#547C22×C511yescharacteristic
256#546—21no—
257#542—21no—
258#539—22no—
259#533C22×C511yescharacteristic
260C2×C4⋊Q8C22×C511yescharacteristic
261#529C22×C511yescharacteristic
262#525—21no—
263C2×C22⋊Q8—21no—
264#520—21no—
265C2×C4×D4C22×C511yescharacteristic
266#501—22no—
267#501—21no—
268#500C22×C512yescomplemented
269#498C22×C512yescomplemented
270#489—21no—
271#484—21no—
272#481—21no—
273#475C22×C512yescomplemented
274#474C22×C512yescomplemented
275#461—21no—
276#461—21no—
277#461C22×C512yes—
278#461C22×C512yes—
279#432C22×C511yescharacteristic
280#428—22no—
281C2×C4⋊C8C22×C511yescharacteristic
282C2×#103—21no—
283C4×M16C22×C511yescharacteristic
284#332C22×C512yescomplemented
285#332C22×C512yescomplemented
286Q8⋊C8C22×C512yescomplemented
287D4⋊C8C22×C512yes—
288D4⋊C8C22×C512yes—
289D4⋊C8C22×C512yescomplemented
290C23×C5—21no—
291C23×C5—21no—
292C23×C5—21no—
293C23×C5—42no—
294C23×C5—42no—
295C23×C5—81no—
296C23×C5#12011yescharacteristic
297C5×Q8—22no—
298C5×Q8—41no—
299C5×Q8—41no—
300C5×Q8—42no—
301C5×Q8—42no—
302C5×Q8—42no—
303C5×Q8—82no—
304C5×Q8—82no—
305C5×D4—21no—
306C5×D4—21no—
307C5×D4—21no—
308C5×D4—22no—
309C5×D4—42no—
310C5×D4—44no—
311C5×D4—42no—
312C5×D4—82no—
313C5×D4AGL(1,ℤ/8)12yes—
314C2×C4×C5—22no—
315C2×C4×C5—21no—
316C2×C4×C5—21no—
317C2×C4×C5—22no—
318C2×C4×C5—21no—
319C2×C4×C5—22no—
320C2×C4×C5—21no—
321C2×C4×C5—21no—
322C2×C4×C5—22no—
323C2×C4×C5—21no—
324C2×C4×C5—21no—
325C2×C4×C5—44no—
326C2×C4×C5—41no—
327C2×C4×C5—41no—
328C2×C4×C5—42no—
329C2×C4×C5—42no—
330C2×C4×C5—42no—
331C2×C4×C5—81no—
332C2×C4×C5—81no—
333C2×C4×C5—81no—
334C2×C4×C5—82no—
335C2×C4×C5—82no—
336C2×C4×C5—82no—
337C2×C4×C5#12011yescharacteristic
338C2×C4×C5#12011yescharacteristic
339C2×C4×C5#12011yescharacteristic
340C2×C4×C5C22×D411yescharacteristic
341C2×C4×C5C22×D411yescharacteristic
342C2×C4×C5C22×D411yescharacteristic
343C5×C8—24no—
344C5×C8—42no—
345C5×C8—42no—
346C2×#41—21no—
347C2×#41—21no—
348C22×Q8—21no—
349C22×Q8—21no—
350C22×D4C5×D411yescharacteristic
351C23×C4—21no—
352C2×SD16—22no—
353C2×SD16—22no—
354C2×SD16—44no—
355C2×M16—21no—
356C22×C8—21no—
357C4⋊Q8—21no—
358C4⋊Q8—21no—
359C4⋊Q8—21no—
360C4⋊Q8—21no—
361C4⋊Q8C23×C512yes—
362C4⋊Q8C23×C511yescharacteristic
363C4⋊Q8C23×C511yescharacteristic
364#126—22no—
365#126—22no—
366#126—21no—
367#125—22no—
368#125C5×D411yescharacteristic
369#125C5×D412yes—
370#124—22no—
371#124—21no—
372#124C5×D411yescharacteristic
373#123—22no—
374#123—22no—
375#123—21no—
376#123—22no—
377#123—22no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386C22⋊Q8—22no—
387C22⋊Q8—21no—
388C22⋊Q8C5×D411yescharacteristic
389#121—22no—
390#121—21no—
391#121—21no—
392#121C5×D411yescharacteristic
393#120—22no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×Q8C23×C511yescharacteristic
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4C5×D411yescharacteristic
406C4×D4C5×D411yescharacteristic
407C4×D4C23×C511yescharacteristic
408C4×D4C23×C511yescharacteristic
409C4×D4C23×C511yescharacteristic
410#117—22no—
411#117—21no—
412#117—21no—
413#117C5×D412yes—
414C2×C4⋊C4—22no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4C5×D411yescharacteristic
420C2×C22⋊C4—22no—
421C2×C22⋊C4—21no—
422C2×C42C23×C511yescharacteristic
423#107—24no—
424#107—22no—
425#106—24no—
426#106—22no—
427C4⋊C8—21no—
428C4⋊C8C23×C512yes—
429C4⋊C8C23×C512yes—
430#103—22no—
431#103—22no—
432#103—22no—
433#102—22no—
434C22⋊C8—21no—
435#97—21no—
436#97—21no—
437C4×C8C23×C512yes—
438C22×C5—21no—
439C22×C5—21no—
440C22×C5—22no—
441C22×C5—41no—
442C22×C5—41no—
443C22×C5—41no—
444C22×C5—41no—
445C22×C5—41no—
446C22×C5—41no—
447C22×C5—41no—
448C22×C5—41no—
449C22×C5—84no—
450C22×C5#45612yes—
451C22×C5C2×#12011yescharacteristic
452C4×C5—21no—
453C4×C5—21no—
454C4×C5—21no—
455C4×C5—21no—
456C4×C5—41no—
457C4×C5—42no—
458C4×C5—41no—
459C4×C5—41no—
460C4×C5—42no—
461C4×C5—41no—
462C4×C5—41no—
463C4×C5—82no—
464C4×C5C2×AGL(1,ℤ/8)11yescharacteristic
465C4×C5#57511yescharacteristic
466C4×C5#57712yes—
467C24—21no—
468#41—42no—
469#41—42no—
470#41—82no—
471C2×Q8—22no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—22no—
475C2×Q8—42no—
476C2×Q8—42no—
477C2×Q8—41no—
478C2×Q8—82no—
479C2×Q8C2×C5×D411yescharacteristic
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—22no—
484C2×D4—21no—
485C2×D4—41no—
486C2×D4C2×C5×D411yescharacteristic
487C2×D4C2×C5×D411yescharacteristic
488C2×D4C2×C5×D411yescharacteristic
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—41no—
499C22×C4—82no—
500C22×C4C2×C5×D411yescharacteristic
501C22×C4C2×C5×D411yescharacteristic
502C22×C4C2×C5×D411yescharacteristic
503SD16—44no—
504SD16—44no—
505M16—24no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—42no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4C2×C5×D411yescharacteristic
528C4⋊C4C2×C5×D411yescharacteristic
529C4⋊C4C2×C5×D411yescharacteristic
530C4⋊C4C2×C5×D412yes—
531C4⋊C4C2×C5×D411yescharacteristic
532C4⋊C4C2×C5×D412yes—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C42—22no—
549C42—22no—
550C42—21no—
551C42C2×C5×D411yescharacteristic
552C42C2×C5×D411yescharacteristic
553C42C2×C5×D411yescharacteristic
554C42C24×C511yescharacteristic
555C2×C5—22no—
556C2×C5—42no—
557C2×C5—81no—
558C2×C5—81no—
559C2×C5#300611yescharacteristic
560C2×C5#300911yescharacteristic
561C2×C5C2×#45611yescharacteristic
562C23—21no—
563C23—21no—
564C23—21no—
565C23—42no—
566C23—42no—
567C23—81no—
568C23C5×#12011yescharacteristic
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—82no—
576Q8—82no—
577D4—21no—
578D4—21no—
579D4—21no—
580D4—22no—
581D4—42no—
582D4—44no—
583D4—42no—
584D4—82no—
585D4C5×AGL(1,ℤ/8)12yes—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4C5×#12011yescharacteristic
610C2×C4C5×#12011yescharacteristic
611C2×C4C5×#12011yescharacteristic
612C2×C4C22×C5×D411yescharacteristic
613C2×C4C22×C5×D411yescharacteristic
614C2×C4C22×C5×D411yescharacteristic
615C8—24no—
616C8—42no—
617C8—42no—
618C5#1858611yescharacteristic, direct product factor, Sylow
619C22—21no—
620C22—21no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—84no—
631C22C5×#45612yes—
632C22C2×C5×#12011yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—82no—
645C4C2×C5×AGL(1,ℤ/8)11yescharacteristic
646C4C5×#57511yescharacteristic
647C4C5×#57712yes—
648C2—22no—
649C2—42no—
650C2—81no—
651C2—81no—
652C2C5×#300611yescharacteristic
653C2C5×#300911yescharacteristic
654C2C2×C5×#45611yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC42C24×C5
prime radicalC5×#3520C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#3239C2semidirect
3. C5×#3165C2semidirect
4. C2×C5×#461C2semidirect
5. C5×#1654C2semidirect
6. C5×#1651C2semidirect
7. C5×#1650C2semidirect
8. C5×#1647C2semidirect
9. C5×#1552C2semidirect
10. C5×#1490C2semidirect
11. C5×#1483C2semidirect
33. C5×#500C22semidirect
34. C5×#498C22semidirect
38. C5×#475C22semidirect
39. C5×#474C22semidirect
49. C5×#332C22semidirect
50. C5×#332C22semidirect
51. C5×Q8⋊C8C22semidirect
54. C5×D4⋊C8C22semidirect
55. #18586C5direct (N × H)
149. #3239C2×C5semidirect
150. #3165C2×C5semidirect
151. C2×#461C2×C5semidirect
152. #1654C2×C5semidirect
153. #1651C2×C5semidirect
154. #1650C2×C5semidirect
155. #1647C2×C5semidirect
156. #1552C2×C5semidirect
157. #1490C2×C5semidirect
158. #1483C2×C5semidirect
268. #500C22×C5semidirect
269. #498C22×C5semidirect
273. #475C22×C5semidirect
274. #474C22×C5semidirect
284. #332C22×C5semidirect
285. #332C22×C5semidirect
286. Q8⋊C8C22×C5semidirect
289. D4⋊C8C22×C5semidirect
618. C5#18586direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35202
C5×#32392
C5×#31652
C2×C5×#4612
C5×#16542
C5×#16512
C5×#16502
C5×#16472
C5×#15522
C5×#14902
C5×#14832
#185865