Finite Group Databasebeta

← C5×#18565 #11941285

Subgroups — C5×#18565 #11941285

Order 1280.

986 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3522C211yesmaximal, characteristic
2C5×#3238C211yesmaximal, characteristic, complemented
3C5×#3152C211yesmaximal, characteristic, complemented
4C2×C5×#477C211yesmaximal, characteristic, complemented
5C5×#1633C212yesmaximal, complemented
6C5×#1632C212yesmaximal, complemented
7C5×#1631C212yesmaximal, complemented
8C5×#1630C212yesmaximal, complemented
9C5×#1551C211yesmaximal, characteristic, complemented
10C5×#1491C211yesmaximal, characteristic, complemented
11C5×#1483C211yesmaximal, characteristic, complemented
12C2×C5×#137—22no—
13C5×#559—21no—
14C5×#559C2211yescharacteristic
15C5×Q82C2211yescharacteristic
16C5×#554—21no—
17C5×#554C2211yescharacteristic
18C5×#550C2211yescharacteristic
19C5×#548—21no—
20C5×#547—22no—
21C5×#547—21no—
22C5×#542—22no—
23C5×#542—21no—
24C5×#532—21no—
25C5×#532C2211yescharacteristic
26C2×C5×C4⋊Q8C2211yescharacteristic
27C5×#529—21no—
28C5×#529C2211yescharacteristic
29C5×#520—21no—
30C2×C4×C5×Q8C2211yescharacteristic
31C5×#501—21no—
32C5×#497—22no—
33C5×#494C2212yes—
34C5×#492C2212yes—
35C5×#489—21no—
36C5×#483—21no—
37C5×#480—21no—
38C5×#477—21no—
39C5×#477—21no—
40C5×#477C2212yes—
41C5×#477C2212yes—
42C5×#477C2212yes—
43C5×#476C2212yes—
44C5×#432C2211yescharacteristic
45C2×C5×#107—21no—
46C2×C5×C4⋊C8C2211yescharacteristic
47C2×C5×#103—21no—
48C4×C5×M16C2211yescharacteristic
49C5×#333C2212yes—
50C5×#331C2212yes—
51C5×Q8⋊C8C2212yes—
52C5×Q8⋊C8C2212yes—
53C5×Q8⋊C8C2212yes—
54C5×D4⋊C8C2212yes—
55#18565C511yesmaximal, characteristic, direct product factor, Sylow
56C2×C5×#41—22no—
57C2×C5×#41—21no—
58C2×C5×#41D412yes—
59C22×C5×Q8—21no—
60C22×C5×Q8D411yescharacteristic
61C5×#137—48no—
62C2×C5×Q16—24no—
63C2×C5×SD16—24no—
64C2×C5×M16—21no—
65C22×C5×C8—21no—
66C5×C4⋊Q8—22no—
67C5×C4⋊Q8—21no—
68C5×C4⋊Q8—21no—
69C5×C4⋊Q8—21no—
70C5×C4⋊Q8—21no—
71C5×C4⋊Q8—21no—
72C5×C4⋊Q8C2312yes—
73C5×C4⋊Q8C2311yescharacteristic
74C5×#127—21no—
75C5×#127C2311yescharacteristic
76C5×#126—22no—
77C5×#126—21no—
78C5×#125—21no—
79C5×#125D411yescharacteristic
80C5×#124—22no—
81C5×#124—22no—
82C5×#124—21no—
83C5×#124—21no—
84C5×#124D411yescharacteristic
85C5×#124D412yes—
86C5×#123—22no—
87C5×#123—22no—
88C5×#123—22no—
89C5×#123—22no—
90C5×#123—21no—
91C5×C22⋊Q8—22no—
92C5×C22⋊Q8—22no—
93C5×C22⋊Q8—22no—
94C5×C22⋊Q8—22no—
95C5×C22⋊Q8—21no—
96C5×C22⋊Q8D411yescharacteristic
97C5×#121—22no—
98C5×#121—22no—
99C5×#121—21no—
100C5×#121—21no—
101C5×#121—21no—
102C5×#121D411yescharacteristic
103C4×C5×Q8—21no—
104C4×C5×Q8—21no—
105C4×C5×Q8—21no—
106C4×C5×Q8—21no—
107C4×C5×Q8D411yescharacteristic
108C4×C5×Q8D411yescharacteristic
109C4×C5×Q8C2311yescharacteristic
110C4×C5×Q8C2311yescharacteristic
111C4×C5×Q8C2311yescharacteristic
112C4×C5×D4—22no—
113C4×C5×D4—22no—
114C4×C5×D4—22no—
115C4×C5×D4—21no—
116C4×C5×D4—21no—
117C4×C5×D4—21no—
118C4×C5×D4—21no—
119C4×C5×D4—21no—
120C4×C5×D4—21no—
121C4×C5×D4—21no—
122C4×C5×D4C2311yescharacteristic
123C5×#117—22no—
124C5×#117—21no—
125C2×C5×C4⋊C4—22no—
126C2×C5×C4⋊C4—21no—
127C2×C5×C4⋊C4—21no—
128C2×C5×C4⋊C4—21no—
129C2×C5×C4⋊C4—21no—
130C2×C5×C4⋊C4D411yescharacteristic
131C2×C42×C5—21no—
132C2×C42×C5C2311yescharacteristic
133C5×#107—22no—
134C5×#107—22no—
135C5×#107—22no—
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#3522C2×C511yescharacteristic
149#3238C2×C511yescharacteristic, complemented
150#3152C2×C511yescharacteristic, complemented
151C2×#477C2×C511yescharacteristic, complemented
152#1633C2×C512yescomplemented
153#1632C2×C512yescomplemented
154#1631C2×C512yescomplemented
155#1630C2×C512yescomplemented
156#1551C2×C511yescharacteristic, complemented
157#1491C2×C511yescharacteristic, complemented
158#1483C2×C511yescharacteristic, complemented
159C5×#41—44no—
160C5×#41—42no—
161C5×#41—42no—
162C5×#41—84no—
163C2×C5×Q8—22no—
164C2×C5×Q8—21no—
165C2×C5×Q8—22no—
166C2×C5×Q8—22no—
167C2×C5×Q8—41no—
168C2×C5×Q8—44no—
169C2×C5×Q8—41no—
170C2×C5×Q8C2×D411yescharacteristic
171C2×C5×Q8C2×D411yescharacteristic
172C2×C5×Q8C2×D411yescharacteristic
173C2×C5×Q8C2×D412yes—
174C2×C5×D4—22no—
175C2×C5×D4—22no—
176C2×C5×D4—21no—
177C2×C5×D4—22no—
178C2×C5×D4—21no—
179C2×C5×D4—21no—
180C2×C5×D4C2×D411yescharacteristic
181C2×C5×D4C2×D412yes—
182C22×C4×C5—22no—
183C22×C4×C5—21no—
184C22×C4×C5—21no—
185C22×C4×C5—22no—
186C22×C4×C5—22no—
187C22×C4×C5—21no—
188C22×C4×C5—22no—
189C22×C4×C5—41no—
190C22×C4×C5C2×D411yescharacteristic
191C22×C4×C5C2×D411yescharacteristic
192C22×C4×C5C2×D411yescharacteristic
193C5×Q16—48no—
194C5×SD16—48no—
195C5×M16—24no—
196C2×C5×C8—22no—
197C2×C5×C8—22no—
198C2×C5×C8—22no—
199C2×C5×C8—42no—
200C5×C4⋊C4—22no—
201C5×C4⋊C4—22no—
202C5×C4⋊C4—22no—
203C5×C4⋊C4—22no—
204C5×C4⋊C4—21no—
205C5×C4⋊C4—21no—
206C5×C4⋊C4—21no—
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—21no—
229C5×C22⋊C4—22no—
230C5×C22⋊C4—21no—
231C5×C22⋊C4—21no—
232C5×C22⋊C4—22no—
233C5×C22⋊C4—21no—
234C5×C22⋊C4—21no—
235C42×C5—22no—
236C42×C5—21no—
237C42×C5—21no—
238C42×C5—21no—
239C42×C5—21no—
240C42×C5—21no—
241C42×C5C2×D411yescharacteristic
242C42×C5C2×D411yescharacteristic
243C42×C5C2×D411yescharacteristic
244C42×C5C2411yescharacteristic
245C2×#137—22no—
246#559—21no—
247#559C22×C511yescharacteristic
248Q82C22×C511yescharacteristic
249#554—21no—
250#554C22×C511yescharacteristic
251#550C22×C511yescharacteristic
252#548—21no—
253#547—22no—
254#547—21no—
255#542—22no—
256#542—21no—
257#532—21no—
258#532C22×C511yescharacteristic
259C2×C4⋊Q8C22×C511yescharacteristic
260#529—21no—
261#529C22×C511yescharacteristic
262#520—21no—
263C2×C4×Q8C22×C511yescharacteristic
264#501—21no—
265#497—22no—
266#494C22×C512yes—
267#492C22×C512yes—
268#489—21no—
269#483—21no—
270#480—21no—
271#477—21no—
272#477—21no—
273#477C22×C512yes—
274#477C22×C512yes—
275#477C22×C512yes—
276#476C22×C512yes—
277#432C22×C511yescharacteristic
278C2×#107—21no—
279C2×C4⋊C8C22×C511yescharacteristic
280C2×#103—21no—
281C4×M16C22×C511yescharacteristic
282#333C22×C512yes—
283#331C22×C512yes—
284Q8⋊C8C22×C512yes—
285Q8⋊C8C22×C512yes—
286Q8⋊C8C22×C512yes—
287D4⋊C8C22×C512yes—
288C23×C5—21no—
289C23×C5—22no—
290C23×C5#12011yescharacteristic
291C5×Q8—21no—
292C5×Q8—21no—
293C5×Q8—21no—
294C5×Q8—22no—
295C5×Q8—42no—
296C5×Q8—42no—
297C5×Q8—44no—
298C5×Q8—82no—
299C5×Q8—82no—
300C5×Q8#13712yescomplemented
301C5×D4—22no—
302C5×D4—42no—
303C5×D4—44no—
304C5×D4—44no—
305C5×D4—42no—
306C5×D4—41no—
307C5×D4—41no—
308C5×D4—42no—
309C5×D4—82no—
310C2×C4×C5—21no—
311C2×C4×C5—22no—
312C2×C4×C5—21no—
313C2×C4×C5—21no—
314C2×C4×C5—22no—
315C2×C4×C5—21no—
316C2×C4×C5—21no—
317C2×C4×C5—21no—
318C2×C4×C5—21no—
319C2×C4×C5—22no—
320C2×C4×C5—21no—
321C2×C4×C5—21no—
322C2×C4×C5—42no—
323C2×C4×C5—44no—
324C2×C4×C5—42no—
325C2×C4×C5—41no—
326C2×C4×C5—41no—
327C2×C4×C5—42no—
328C2×C4×C5—82no—
329C2×C4×C5—82no—
330C2×C4×C5—82no—
331C2×C4×C5—82no—
332C2×C4×C5#12011yescharacteristic
333C2×C4×C5#12011yescharacteristic
334C2×C4×C5#12011yescharacteristic
335C2×C4×C5C22×D411yescharacteristic
336C2×C4×C5C22×D411yescharacteristic
337C2×C4×C5C22×D411yescharacteristic
338C5×C8—24no—
339C5×C8—42no—
340C5×C8—42no—
341C2×#41—22no—
342C2×#41—21no—
343C2×#41C5×D412yes—
344C22×Q8—21no—
345C22×Q8C5×D411yescharacteristic
346#137—48no—
347C2×Q16—24no—
348C2×SD16—24no—
349C2×M16—21no—
350C22×C8—21no—
351C4⋊Q8—22no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—21no—
356C4⋊Q8—21no—
357C4⋊Q8C23×C512yes—
358C4⋊Q8C23×C511yescharacteristic
359#127—21no—
360#127C23×C511yescharacteristic
361#126—22no—
362#126—21no—
363#125—21no—
364#125C5×D411yescharacteristic
365#124—22no—
366#124—22no—
367#124—21no—
368#124—21no—
369#124C5×D411yescharacteristic
370#124C5×D412yes—
371#123—22no—
372#123—22no—
373#123—22no—
374#123—22no—
375#123—21no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—21no—
381C22⋊Q8C5×D411yescharacteristic
382#121—22no—
383#121—22no—
384#121—21no—
385#121—21no—
386#121—21no—
387#121C5×D411yescharacteristic
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8—21no—
391C4×Q8—21no—
392C4×Q8C5×D411yescharacteristic
393C4×Q8C5×D411yescharacteristic
394C4×Q8C23×C511yescharacteristic
395C4×Q8C23×C511yescharacteristic
396C4×Q8C23×C511yescharacteristic
397C4×D4—22no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—21no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4C23×C511yescharacteristic
408#117—22no—
409#117—21no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4C5×D411yescharacteristic
416C2×C42—21no—
417C2×C42C23×C511yescharacteristic
418#107—22no—
419#107—22no—
420#107—22no—
421#106—22no—
422C4⋊C8—21no—
423C4⋊C8C23×C512yes—
424C4⋊C8C23×C512yes—
425#103—22no—
426#103—22no—
427#103—22no—
428#102—22no—
429C22⋊C8—21no—
430#97—21no—
431#97—21no—
432C4×C8C23×C512yes—
433C22×C5—21no—
434C22×C5—21no—
435C22×C5—42no—
436C22×C5—41no—
437C22×C5—41no—
438C22×C5—42no—
439C22×C5—42no—
440C22×C5—41no—
441C22×C5#45312yes—
442C22×C5C2×#12011yescharacteristic
443C4×C5—21no—
444C4×C5—22no—
445C4×C5—21no—
446C4×C5—21no—
447C4×C5—21no—
448C4×C5—41no—
449C4×C5—41no—
450C4×C5—41no—
451C4×C5—41no—
452C4×C5—41no—
453C4×C5—42no—
454C4×C5—41no—
455C4×C5—82no—
456C4×C5C2×#13711yescharacteristic
457C4×C5#57511yescharacteristic
458C4×C5#57812yes—
459#41—44no—
460#41—42no—
461#41—42no—
462#41—84no—
463C2×Q8—22no—
464C2×Q8—21no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—41no—
468C2×Q8—44no—
469C2×Q8—41no—
470C2×Q8C2×C5×D411yescharacteristic
471C2×Q8C2×C5×D411yescharacteristic
472C2×Q8C2×C5×D411yescharacteristic
473C2×Q8C2×C5×D412yes—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—21no—
477C2×D4—22no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4C2×C5×D411yescharacteristic
481C2×D4C2×C5×D412yes—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—41no—
490C22×C4C2×C5×D411yescharacteristic
491C22×C4C2×C5×D411yescharacteristic
492C22×C4C2×C5×D411yescharacteristic
493Q16—48no—
494SD16—48no—
495M16—24no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—42no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4C2×C5×D411yescharacteristic
520C4⋊C4C2×C5×D411yescharacteristic
521C4⋊C4C2×C5×D411yescharacteristic
522C4⋊C4C2×C5×D411yescharacteristic
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—21no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C42—22no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42C2×C5×D411yescharacteristic
542C42C2×C5×D411yescharacteristic
543C42C2×C5×D411yescharacteristic
544C42C24×C511yescharacteristic
545C2×C5—22no—
546C2×C5—81no—
547C2×C5—82no—
548C2×C5#300711yescharacteristic
549C2×C5#301111yescharacteristic
550C2×C5C2×#45311yescharacteristic
551C23—21no—
552C23—22no—
553C23C5×#12011yescharacteristic
554Q8—21no—
555Q8—21no—
556Q8—21no—
557Q8—22no—
558Q8—42no—
559Q8—42no—
560Q8—44no—
561Q8—82no—
562Q8—82no—
563Q8C5×#13712yescomplemented
564D4—22no—
565D4—42no—
566D4—44no—
567D4—44no—
568D4—42no—
569D4—41no—
570D4—41no—
571D4—42no—
572D4—82no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4C5×#12011yescharacteristic
596C2×C4C5×#12011yescharacteristic
597C2×C4C5×#12011yescharacteristic
598C2×C4C22×C5×D411yescharacteristic
599C2×C4C22×C5×D411yescharacteristic
600C2×C4C22×C5×D411yescharacteristic
601C8—24no—
602C8—42no—
603C8—42no—
604C5#1856511yescharacteristic, direct product factor, Sylow
605C22—21no—
606C22—21no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22C5×#45312yes—
614C22C2×C5×#12011yescharacteristic
615C4—21no—
616C4—22no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—82no—
628C4C2×C5×#13711yescharacteristic
629C4C5×#57511yescharacteristic
630C4C5×#57812yes—
631C2—22no—
632C2—81no—
633C2—82no—
634C2C5×#300711yescharacteristic
635C2C5×#301111yescharacteristic
636C2C2×C5×#45311yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC42C24×C5
prime radicalC5×#3522C2
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×#3238C2semidirect
3. C5×#3152C2semidirect
4. C2×C5×#477C2semidirect
5. C5×#1633C2semidirect
6. C5×#1632C2semidirect
7. C5×#1631C2semidirect
8. C5×#1630C2semidirect
9. C5×#1551C2semidirect
10. C5×#1491C2semidirect
11. C5×#1483C2semidirect
55. #18565C5direct (N × H)
149. #3238C2×C5semidirect
150. #3152C2×C5semidirect
151. C2×#477C2×C5semidirect
152. #1633C2×C5semidirect
153. #1632C2×C5semidirect
154. #1631C2×C5semidirect
155. #1630C2×C5semidirect
156. #1551C2×C5semidirect
157. #1491C2×C5semidirect
158. #1483C2×C5semidirect
300. C5×Q8#137semidirect
563. Q8C5×#137semidirect
604. C5#18565direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35222
C5×#32382
C5×#31522
C2×C5×#4772
C5×#16332
C5×#16322
C5×#16312
C5×#16302
C5×#15512
C5×#14912
C5×#14832
#185655