Finite Group Databasebeta

← C5×#18689 #11941409

Subgroups — C5×#18689 #11941409

Order 1280.

1006 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×#3543C211yesmaximal, characteristic
2C5×#3239C211yesmaximal, characteristic, complemented
3C5×#3218C211yesmaximal, characteristic, complemented
4C2×C5×#486C211yesmaximal, characteristic, complemented
5C5×#1642C212yesmaximal, complemented
6C5×#1641C212yesmaximal, complemented
7C5×#1640C212yesmaximal, complemented
8C5×#1639C212yesmaximal, complemented
9C5×#1555C211yesmaximal, characteristic, complemented
10C5×#1494C211yesmaximal, characteristic, complemented
11C5×#1494C211yesmaximal, characteristic, complemented
12C5×#560—22no—
13C5×#559C2211yescharacteristic
14C5×#550C2211yescharacteristic
15C5×#550C2211yescharacteristic
16C5×#546—21no—
17C5×#546—21no—
18C5×D42—21no—
19C5×D42—21no—
20C5×D42C2211yescharacteristic
21C5×#540—21no—
22C5×#540—21no—
23C5×#535—21no—
24C5×#535—21no—
25C5×#534—22no—
26C2×C5×C4⋊Q8C2211yescharacteristic, complemented
27C2×C5×#127C2211yescharacteristic
28C5×#529C2211yescharacteristic
29C5×#529C2211yescharacteristic
30C5×#501—21no—
31C5×#497—21no—
32C5×#491—21no—
33C5×#489—21no—
34C5×#486—22no—
35C5×#486C2212yes—
36C5×#486C2212yescomplemented
37C5×#484—21no—
38C5×#481—21no—
39C5×#475C2212yescomplemented
40C5×#474C2212yescomplemented
41C5×#462C2212yescomplemented
42C5×#461C2212yescomplemented
43C5×#451—21no—
44C5×#448—21no—
45C5×#432C2211yescharacteristic, complemented
46C5×#432C2211yescharacteristic, complemented
47C2×C5×#102—22no—
48C2×C4×C5×C8C2211yescharacteristic, complemented
49C5×#331C2212yes—
50C5×#331C2212yescomplemented
51C5×Q8⋊C8C2212yescomplemented
52C5×Q8⋊C8C2212yescomplemented
53C5×D4⋊C8C2212yescomplemented
54C5×D4⋊C8C2212yescomplemented
55#18689C511yesmaximal, characteristic, direct product factor, Sylow
56C2×C5×#41—21no—
57C2×C5×#41—21no—
58C22×C5×Q8—21no—
59C22×C5×D4—22no—
60C22×C5×D4—21no—
61C22×C5×D4—21no—
62C22×C5×D4—21no—
63C22×C5×D4—21no—
64C22×C5×D4D412yes—
65C2×C5×Q16—22no—
66C2×C5×SD16—22no—
67C22×C5×C8—21no—
68C5×C4⋊Q8—21no—
69C5×C4⋊Q8—21no—
70C5×C4⋊Q8—21no—
71C5×C4⋊Q8C2312yes—
72C5×#127—21no—
73C5×#127—21no—
74C5×#127—21no—
75C5×#127—21no—
76C5×#127D412yes—
77C5×#127C2311yescharacteristic
78C5×#127C2311yescharacteristic
79C5×#126—21no—
80C5×#126—21no—
81C5×#125D411yescharacteristic
82C5×#125D411yescharacteristic
83C5×#124—21no—
84C5×#124—21no—
85C5×#124D411yescharacteristic
86C5×#124D411yescharacteristic
87C5×#123—22no—
88C5×#123—22no—
89C5×#123—22no—
90C5×#123—22no—
91C5×#123—21no—
92C5×#123—21no—
93C5×C22⋊Q8D411yescharacteristic
94C5×C22⋊Q8D411yescharacteristic
95C5×#121—22no—
96C5×#121—22no—
97C5×#121—22no—
98C5×#121—22no—
99C5×#121—22no—
100C5×#121—22no—
101C5×#121—22no—
102C5×#121—22no—
103C5×#121—21no—
104C5×#121—21no—
105C5×#121—21no—
106C5×#121—21no—
107C5×#121D411yescharacteristic
108C5×#121D411yescharacteristic
109C5×#120—22no—
110C5×#120—22no—
111C5×#120—22no—
112C5×#120—22no—
113C5×#120—22no—
114C5×#120—22no—
115C4×C5×Q8C2311yescharacteristic
116C4×C5×Q8C2311yescharacteristic
117C4×C5×D4—21no—
118C4×C5×D4—21no—
119C4×C5×D4—21no—
120C4×C5×D4—21no—
121C4×C5×D4—21no—
122C4×C5×D4—21no—
123C4×C5×D4C2311yescharacteristic
124C4×C5×D4C2311yescharacteristic
125C5×#117—21no—
126C5×#117—21no—
127C2×C5×C4⋊C4—21no—
128C2×C5×C4⋊C4—21no—
129C2×C5×C22⋊C4—22no—
130C2×C42×C5C2311yescharacteristic
131C5×#107—22no—
132C5×#106—22no—
133C5×C4⋊C8C2312yes—
134C5×C4⋊C8C2312yes—
135C5×#103—22no—
136C5×#103—22no—
137C5×#103—22no—
138C5×#102—24no—
139C5×#102—24no—
140C5×#102—22no—
141C5×C22⋊C8—21no—
142C5×C22⋊C8—21no—
143C5×#97—21no—
144C5×#97—21no—
145C4×C5×C8—21no—
146C4×C5×C8C2312yes—
147#3543C2×C511yescharacteristic
148#3239C2×C511yescharacteristic, complemented
149#3218C2×C511yescharacteristic, complemented
150C2×#486C2×C511yescharacteristic, complemented
151#1642C2×C512yescomplemented
152#1641C2×C512yescomplemented
153#1640C2×C512yescomplemented
154#1639C2×C512yescomplemented
155#1555C2×C511yescharacteristic, complemented
156#1494C2×C511yescharacteristic, complemented
157#1494C2×C511yescharacteristic, complemented
158C24×C5—22no—
159C24×C5—22no—
160C5×#41—82no—
161C5×#41—82no—
162C2×C5×Q8—22no—
163C2×C5×Q8—41no—
164C2×C5×Q8—82no—
165C2×C5×Q8C2×D411yescharacteristic
166C2×C5×Q8C2×D411yescharacteristic
167C2×C5×D4—21no—
168C2×C5×D4—21no—
169C2×C5×D4—22no—
170C2×C5×D4—22no—
171C2×C5×D4—22no—
172C2×C5×D4—22no—
173C2×C5×D4—22no—
174C2×C5×D4—22no—
175C2×C5×D4—22no—
176C2×C5×D4—22no—
177C2×C5×D4—22no—
178C2×C5×D4—21no—
179C2×C5×D4—21no—
180C2×C5×D4—21no—
181C2×C5×D4—21no—
182C2×C5×D4—44no—
183C2×C5×D4—82no—
184C2×C5×D4—82no—
185C2×C5×D4—82no—
186C2×C5×D4—82no—
187C2×C5×D4—84no—
188C2×C5×D4C2×D411yescharacteristic
189C2×C5×D4C2×D411yescharacteristic
190C2×C5×D4C2×D412yes—
191C2×C5×D4C2×D412yes—
192C22×C4×C5—22no—
193C22×C4×C5—22no—
194C22×C4×C5—22no—
195C22×C4×C5—21no—
196C22×C4×C5—21no—
197C22×C4×C5—41no—
198C22×C4×C5C2×D411yescharacteristic
199C22×C4×C5C2×D411yescharacteristic
200C22×C4×C5C2×D411yescharacteristic
201C5×Q16—82no—
202C5×SD16—82no—
203C2×C5×C8—22no—
204C2×C5×C8—22no—
205C2×C5×C8—22no—
206C2×C5×C8—42no—
207C5×C4⋊C4—21no—
208C5×C4⋊C4—21no—
209C5×C4⋊C4—21no—
210C5×C4⋊C4—21no—
211C5×C4⋊C4—21no—
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×C22⋊C4—22no—
224C5×C22⋊C4—22no—
225C5×C22⋊C4—22no—
226C5×C22⋊C4—22no—
227C5×C22⋊C4—22no—
228C5×C22⋊C4—22no—
229C5×C22⋊C4—21no—
230C5×C22⋊C4—21no—
231C5×C22⋊C4—21no—
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—
239C42×C5—21no—
240C42×C5—21no—
241C42×C5C2×D411yescharacteristic
242C42×C5C2×D411yescharacteristic
243C42×C5C2×D411yescharacteristic
244C42×C5C2411yescharacteristic
245#560—22no—
246#559C22×C511yescharacteristic
247#550C22×C511yescharacteristic
248#550C22×C511yescharacteristic
249#546—21no—
250#546—21no—
251D42—21no—
252D42—21no—
253D42C22×C511yescharacteristic
254#540—21no—
255#540—21no—
256#535—21no—
257#535—21no—
258#534—22no—
259C2×C4⋊Q8C22×C511yescharacteristic, complemented
260C2×#127C22×C511yescharacteristic
261#529C22×C511yescharacteristic
262#529C22×C511yescharacteristic
263#501—21no—
264#497—21no—
265#491—21no—
266#489—21no—
267#486—22no—
268#486C22×C512yes—
269#486C22×C512yescomplemented
270#484—21no—
271#481—21no—
272#475C22×C512yescomplemented
273#474C22×C512yescomplemented
274#462C22×C512yescomplemented
275#461C22×C512yescomplemented
276#451—21no—
277#448—21no—
278#432C22×C511yescharacteristic, complemented
279#432C22×C511yescharacteristic, complemented
280C2×#102—22no—
281C2×C4×C8C22×C511yescharacteristic, complemented
282#331C22×C512yes—
283#331C22×C512yescomplemented
284Q8⋊C8C22×C512yescomplemented
285Q8⋊C8C22×C512yescomplemented
286D4⋊C8C22×C512yescomplemented
287D4⋊C8C22×C512yescomplemented
288C23×C5—21no—
289C23×C5—21no—
290C23×C5—22no—
291C23×C5—22no—
292C23×C5—44no—
293C23×C5—82no—
294C23×C5—82no—
295C23×C5—82no—
296C23×C5—82no—
297C23×C5—82no—
298C23×C5#12011yescharacteristic
299C5×Q8—22no—
300C5×Q8—22no—
301C5×Q8—42no—
302C5×Q8—42no—
303C5×Q8—82no—
304C5×D4—22no—
305C5×D4—22no—
306C5×D4—42no—
307C5×D4—42no—
308C5×D4—42no—
309C5×D4—42no—
310C5×D4—42no—
311C5×D4—42no—
312C5×D4—42no—
313C5×D4—42no—
314C5×D4—82no—
315C5×D4—82no—
316C5×D4—82no—
317C5×D4—82no—
318C5×D4—82no—
319C5×D4—82no—
320C5×D4—82no—
321C5×D4—82no—
322C5×D4—82no—
323C2×C4×C5—22no—
324C2×C4×C5—22no—
325C2×C4×C5—22no—
326C2×C4×C5—21no—
327C2×C4×C5—21no—
328C2×C4×C5—21no—
329C2×C4×C5—21no—
330C2×C4×C5—21no—
331C2×C4×C5—21no—
332C2×C4×C5—42no—
333C2×C4×C5—42no—
334C2×C4×C5—42no—
335C2×C4×C5—82no—
336C2×C4×C5—82no—
337C2×C4×C5—82no—
338C2×C4×C5—82no—
339C2×C4×C5#12011yescharacteristic
340C2×C4×C5#12011yescharacteristic
341C2×C4×C5#12011yescharacteristic
342C2×C4×C5C22×D411yescharacteristic
343C2×C4×C5C22×D411yescharacteristic
344C2×C4×C5C22×D411yescharacteristic
345C5×C8—42no—
346C5×C8—42no—
347C5×C8—42no—
348C2×#41—21no—
349C2×#41—21no—
350C22×Q8—21no—
351C22×D4—22no—
352C22×D4—21no—
353C22×D4—21no—
354C22×D4—21no—
355C22×D4—21no—
356C22×D4C5×D412yes—
357C2×Q16—22no—
358C2×SD16—22no—
359C22×C8—21no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—21no—
363C4⋊Q8C23×C512yes—
364#127—21no—
365#127—21no—
366#127—21no—
367#127—21no—
368#127C5×D412yes—
369#127C23×C511yescharacteristic
370#127C23×C511yescharacteristic
371#126—21no—
372#126—21no—
373#125C5×D411yescharacteristic
374#125C5×D411yescharacteristic
375#124—21no—
376#124—21no—
377#124C5×D411yescharacteristic
378#124C5×D411yescharacteristic
379#123—22no—
380#123—22no—
381#123—22no—
382#123—22no—
383#123—21no—
384#123—21no—
385C22⋊Q8C5×D411yescharacteristic
386C22⋊Q8C5×D411yescharacteristic
387#121—22no—
388#121—22no—
389#121—22no—
390#121—22no—
391#121—22no—
392#121—22no—
393#121—22no—
394#121—22no—
395#121—21no—
396#121—21no—
397#121—21no—
398#121—21no—
399#121C5×D411yescharacteristic
400#121C5×D411yescharacteristic
401#120—22no—
402#120—22no—
403#120—22no—
404#120—22no—
405#120—22no—
406#120—22no—
407C4×Q8C23×C511yescharacteristic
408C4×Q8C23×C511yescharacteristic
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—21no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4C23×C511yescharacteristic
416C4×D4C23×C511yescharacteristic
417#117—21no—
418#117—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C22⋊C4—22no—
422C2×C42C23×C511yescharacteristic
423#107—22no—
424#106—22no—
425C4⋊C8C23×C512yes—
426C4⋊C8C23×C512yes—
427#103—22no—
428#103—22no—
429#103—22no—
430#102—24no—
431#102—24no—
432#102—22no—
433C22⋊C8—21no—
434C22⋊C8—21no—
435#97—21no—
436#97—21no—
437C4×C8—21no—
438C4×C8C23×C512yes—
439C22×C5—21no—
440C22×C5—21no—
441C22×C5—41no—
442C22×C5—42no—
443C22×C5—41no—
444C22×C5—42no—
445C22×C5—41no—
446C22×C5—42no—
447C22×C5—41no—
448C22×C5—42no—
449C22×C5—42no—
450C22×C5—41no—
451C22×C5—41no—
452C22×C5—42no—
453C22×C5—164no—
454C22×C5—164no—
455C22×C5#45512yes—
456C22×C5C2×#12011yescharacteristic
457C4×C5—21no—
458C4×C5—21no—
459C4×C5—21no—
460C4×C5—21no—
461C4×C5—41no—
462C4×C5—41no—
463C4×C5—41no—
464C4×C5—41no—
465C4×C5—41no—
466C4×C5—41no—
467C4×C5—82no—
468C4×C5#57612yes—
469C4×C5#57712yes—
470C24—22no—
471C24—22no—
472#41—82no—
473#41—82no—
474C2×Q8—22no—
475C2×Q8—41no—
476C2×Q8—82no—
477C2×Q8C2×C5×D411yescharacteristic
478C2×Q8C2×C5×D411yescharacteristic
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—44no—
495C2×D4—82no—
496C2×D4—82no—
497C2×D4—82no—
498C2×D4—82no—
499C2×D4—84no—
500C2×D4C2×C5×D411yescharacteristic
501C2×D4C2×C5×D411yescharacteristic
502C2×D4C2×C5×D412yes—
503C2×D4C2×C5×D412yes—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—41no—
510C22×C4C2×C5×D411yescharacteristic
511C22×C4C2×C5×D411yescharacteristic
512C22×C4C2×C5×D411yescharacteristic
513Q16—82no—
514SD16—82no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—42no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4C2×C5×D411yescharacteristic
532C4⋊C4C2×C5×D411yescharacteristic
533C4⋊C4C2×C5×D411yescharacteristic
534C4⋊C4C2×C5×D411yescharacteristic
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C42—21no—
552C42—21no—
553C42C2×C5×D411yescharacteristic
554C42C2×C5×D411yescharacteristic
555C42C2×C5×D411yescharacteristic
556C42C24×C511yescharacteristic
557C2×C5—22no—
558C2×C5—82no—
559C2×C5—81no—
560C2×C5—81no—
561C2×C5—82no—
562C2×C5#300811yescharacteristic
563C2×C5#300911yescharacteristic
564C2×C5C2×#45511yescharacteristic
565C23—21no—
566C23—21no—
567C23—22no—
568C23—22no—
569C23—44no—
570C23—82no—
571C23—82no—
572C23—82no—
573C23—82no—
574C23—82no—
575C23C5×#12011yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—42no—
579Q8—42no—
580Q8—82no—
581D4—22no—
582D4—22no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—82no—
592D4—82no—
593D4—82no—
594D4—82no—
595D4—82no—
596D4—82no—
597D4—82no—
598D4—82no—
599D4—82no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4C5×#12011yescharacteristic
617C2×C4C5×#12011yescharacteristic
618C2×C4C5×#12011yescharacteristic
619C2×C4C22×C5×D411yescharacteristic
620C2×C4C22×C5×D411yescharacteristic
621C2×C4C22×C5×D411yescharacteristic
622C8—42no—
623C8—42no—
624C8—42no—
625C5#1868911yescharacteristic, direct product factor, Sylow
626C22—21no—
627C22—21no—
628C22—41no—
629C22—42no—
630C22—41no—
631C22—42no—
632C22—41no—
633C22—42no—
634C22—41no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22—41no—
639C22—42no—
640C22—164no—
641C22—164no—
642C22C5×#45512yes—
643C22C2×C5×#12011yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—82no—
655C4C5×#57612yes—
656C4C5×#57712yes—
657C2—22no—
658C2—81no—
659C2—81no—
660C2—82no—
661C2—82no—
662C2C5×#300811yescharacteristic
663C2C5×#300911yescharacteristic
664C2C2×C5×#45511yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC42C24×C5
prime radicalC5×#3543C2
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×#3218C2semidirect
4. C2×C5×#486C2semidirect
5. C5×#1642C2semidirect
6. C5×#1641C2semidirect
7. C5×#1640C2semidirect
8. C5×#1639C2semidirect
9. C5×#1555C2semidirect
10. C5×#1494C2semidirect
11. C5×#1494C2semidirect
26. C2×C5×C4⋊Q8C22semidirect
36. C5×#486C22semidirect
39. C5×#475C22semidirect
40. C5×#474C22semidirect
41. C5×#462C22semidirect
42. C5×#461C22semidirect
45. C5×#432C22semidirect
46. C5×#432C22semidirect
48. C2×C4×C5×C8C22semidirect
50. C5×#331C22semidirect
51. C5×Q8⋊C8C22semidirect
52. C5×Q8⋊C8C22semidirect
53. C5×D4⋊C8C22semidirect
54. C5×D4⋊C8C22semidirect
55. #18689C5direct (N × H)
148. #3239C2×C5semidirect
149. #3218C2×C5semidirect
150. C2×#486C2×C5semidirect
151. #1642C2×C5semidirect
152. #1641C2×C5semidirect
153. #1640C2×C5semidirect
154. #1639C2×C5semidirect
155. #1555C2×C5semidirect
156. #1494C2×C5semidirect
157. #1494C2×C5semidirect
259. C2×C4⋊Q8C22×C5semidirect
269. #486C22×C5semidirect
272. #475C22×C5semidirect
273. #474C22×C5semidirect
274. #462C22×C5semidirect
275. #461C22×C5semidirect
278. #432C22×C5semidirect
279. #432C22×C5semidirect
281. C2×C4×C8C22×C5semidirect
283. #331C22×C5semidirect
284. Q8⋊C8C22×C5semidirect
285. Q8⋊C8C22×C5semidirect
286. D4⋊C8C22×C5semidirect
287. D4⋊C8C22×C5semidirect
625. C5#18689direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35432
C5×#32392
C5×#32182
C2×C5×#4862
C5×#16422
C5×#16412
C5×#16402
C5×#16392
C5×#15552
C5×#14942
#186895