Finite Group Databasebeta

← C5×#18580 #11941300

Subgroups — C5×#18580 #11941300

Order 1280.

962 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3523C211yesmaximal, characteristic, complemented
2C5×#3217C211yesmaximal, characteristic
3C5×#3138C211yesmaximal, characteristic, complemented
4C2×C5×#460C211yesmaximal, characteristic, complemented
5C5×#1628C212yesmaximal, complemented
6C5×#1627C212yesmaximal, complemented
7C5×#1626C212yesmaximal, complemented
8C5×#1626C212yesmaximal, complemented
9C5×#1552C211yesmaximal, characteristic, complemented
10C5×#1491C211yesmaximal, characteristic, complemented
11C5×#1483C211yesmaximal, characteristic, complemented
12C22×C5×SD16—21no—
13C5×#557—21no—
14C5×#557C2211yescharacteristic, complemented
15C5×#557C2211yescharacteristic, complemented
16C5×#556—21no—
17C5×#556C2211yescharacteristic, complemented
18C5×#552—22no—
19C5×#552—21no—
20C5×D4×Q8—21no—
21C5×D4×Q8C2211yescharacteristic, complemented
22C5×#541—22no—
23C5×#541—21no—
24C5×#533—21no—
25C5×#533C2211yescharacteristic
26C2×C5×#127C2211yescharacteristic
27C5×#529—21no—
28C5×#529C2211yescharacteristic
29C5×#519—21no—
30C2×C4×C5×Q8C2211yescharacteristic
31C5×#496—21no—
32C5×#490—22no—
33C5×#490—21no—
34C5×#486C2212yes—
35C5×#486C2212yes—
36C5×#460—21no—
37C5×#460—21no—
38C5×#460C2212yes—
39C5×#460C2212yes—
40C5×#460C2212yes—
41C5×#459C2212yes—
42C5×#450—21no—
43C5×#447—21no—
44C5×#432C2211yescharacteristic
45C2×C5×C4⋊C8C2211yescharacteristic
46C5×#418—22no—
47C2×C5×#102—21no—
48C4×C5×M16C2211yescharacteristic
49C5×#332C2212yescomplemented
50C5×#332C2212yescomplemented
51C5×Q8⋊C8C2212yescomplemented
52C5×Q8⋊C8C2212yescomplemented
53C5×Q8⋊C8C2212yescomplemented
54C5×D4⋊C8C2212yescomplemented
55#18580C511yesmaximal, characteristic, direct product factor, Sylow
56C22×C5×Q8—21no—
57C22×C5×Q8D411yescharacteristic
58C22×C5×D4—21no—
59C22×C5×D4—21no—
60C22×C5×D4—21no—
61C2×C5×SD16—22no—
62C2×C5×SD16—22no—
63C2×C5×SD16—22no—
64C2×C5×SD16—44no—
65C2×C5×D8—22no—
66C2×C5×M16—21no—
67C22×C5×C8—21no—
68C5×C4⋊Q8—22no—
69C5×C4⋊Q8—21no—
70C5×C4⋊Q8—21no—
71C5×C4⋊Q8—21no—
72C5×C4⋊Q8C2311yescharacteristic
73C5×C4⋊Q8C2311yescharacteristic
74C5×#127—21no—
75C5×#127—21no—
76C5×#127—21no—
77C5×#127C2312yes—
78C5×#126—22no—
79C5×#126—22no—
80C5×#126—22no—
81C5×#126—22no—
82C5×#126—21no—
83C5×#125—22no—
84C5×#125—22no—
85C5×#125—22no—
86C5×#125—21no—
87C5×#125—22no—
88C5×#125—22no—
89C5×#125—21no—
90C5×#125D411yescharacteristic
91C5×#125D412yes—
92C5×#124—21no—
93C5×#124D411yescharacteristic
94C5×#123—22no—
95C5×#123—21no—
96C5×C22⋊Q8—22no—
97C5×C22⋊Q8—22no—
98C5×C22⋊Q8—21no—
99C5×C22⋊Q8—21no—
100C5×C22⋊Q8—21no—
101C5×C22⋊Q8—21no—
102C5×C22⋊Q8—22no—
103C5×C22⋊Q8D411yescharacteristic
104C5×#121D411yescharacteristic
105C4×C5×Q8—22no—
106C4×C5×Q8—22no—
107C4×C5×Q8—21no—
108C4×C5×Q8—21no—
109C4×C5×Q8—21no—
110C4×C5×Q8—21no—
111C4×C5×Q8—21no—
112C4×C5×Q8—21no—
113C4×C5×Q8—21no—
114C4×C5×Q8—21no—
115C4×C5×Q8—21no—
116C4×C5×Q8D411yescharacteristic
117C4×C5×Q8D411yescharacteristic
118C4×C5×Q8C2311yescharacteristic
119C4×C5×Q8C2311yescharacteristic
120C4×C5×Q8C2311yescharacteristic
121C4×C5×D4—22no—
122C4×C5×D4—21no—
123C4×C5×D4—21no—
124C4×C5×D4C2311yescharacteristic
125C5×#117—22no—
126C5×#117—22no—
127C5×#117—22no—
128C5×#117—21no—
129C5×#117—21no—
130C5×#117D412yes—
131C2×C5×C4⋊C4—21no—
132C2×C5×C4⋊C4D411yescharacteristic
133C2×C42×C5—21no—
134C2×C42×C5C2311yescharacteristic
135C5×C4⋊C8—21no—
136C5×C4⋊C8C2312yes—
137C5×C4⋊C8C2312yes—
138C5×#102—22no—
139C5×#102—22no—
140C5×#102—24no—
141C5×#102—24no—
142C5×#102—22no—
143C5×#102—22no—
144C5×C22⋊C8—21no—
145C5×#97—21no—
146C5×#97—21no—
147C4×C5×C8C2312yes—
148#3523C2×C511yescharacteristic, complemented
149#3217C2×C511yescharacteristic
150#3138C2×C511yescharacteristic, complemented
151C2×#460C2×C511yescharacteristic, complemented
152#1628C2×C512yescomplemented
153#1627C2×C512yescomplemented
154#1626C2×C512yescomplemented
155#1626C2×C512yescomplemented
156#1552C2×C511yescharacteristic, complemented
157#1491C2×C511yescharacteristic, complemented
158#1483C2×C511yescharacteristic, complemented
159C24×C5—41no—
160C2×C5×Q8—22no—
161C2×C5×Q8—21no—
162C2×C5×Q8—21no—
163C2×C5×Q8—21no—
164C2×C5×Q8—22no—
165C2×C5×Q8—41no—
166C2×C5×Q8—42no—
167C2×C5×Q8—42no—
168C2×C5×Q8C2×D411yescharacteristic
169C2×C5×Q8C2×D411yescharacteristic
170C2×C5×Q8C2×D411yescharacteristic
171C2×C5×D4—21no—
172C2×C5×D4—22no—
173C2×C5×D4—22no—
174C2×C5×D4—22no—
175C2×C5×D4—42no—
176C2×C5×D4—44no—
177C2×C5×D4—42no—
178C2×C5×D4—41no—
179C2×C5×D4—41no—
180C2×C5×D4—41no—
181C2×C5×D4—82no—
182C2×C5×D4C2×D411yescharacteristic
183C22×C4×C5—22no—
184C22×C4×C5—21no—
185C22×C4×C5—21no—
186C22×C4×C5—21no—
187C22×C4×C5C2×D411yescharacteristic
188C22×C4×C5C2×D411yescharacteristic
189C22×C4×C5C2×D411yescharacteristic
190C5×SD16—44no—
191C5×SD16—44no—
192C5×SD16—82no—
193C5×D8—82no—
194C5×M16—42no—
195C2×C5×C8—22no—
196C2×C5×C8—22no—
197C2×C5×C8—22no—
198C2×C5×C8—42no—
199C5×C4⋊C4—22no—
200C5×C4⋊C4—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—21no—
208C5×C4⋊C4—21no—
209C5×C4⋊C4—21no—
210C5×C4⋊C4—22no—
211C5×C4⋊C4—21no—
212C5×C4⋊C4—21no—
213C5×C4⋊C4—21no—
214C5×C4⋊C4—22no—
215C5×C4⋊C4—21no—
216C5×C4⋊C4—21no—
217C5×C4⋊C4—21no—
218C5×C4⋊C4—21no—
219C5×C4⋊C4—21no—
220C5×C4⋊C4C2×D411yescharacteristic
221C5×C4⋊C4C2×D411yescharacteristic
222C5×C4⋊C4C2×D412yes—
223C5×C4⋊C4C2×D412yes—
224C5×C4⋊C4C2×D411yescharacteristic
225C5×C4⋊C4C2×D411yescharacteristic
226C5×C22⋊C4—22no—
227C5×C22⋊C4—22no—
228C5×C22⋊C4—22no—
229C5×C22⋊C4—21no—
230C5×C22⋊C4—22no—
231C5×C22⋊C4—21no—
232C5×C22⋊C4—21no—
233C5×C22⋊C4—21no—
234C5×C22⋊C4—21no—
235C42×C5—22no—
236C42×C5—22no—
237C42×C5—22no—
238C42×C5—21no—
239C42×C5—21no—
240C42×C5—22no—
241C42×C5—21no—
242C42×C5—21no—
243C42×C5—21no—
244C42×C5C2×D411yescharacteristic
245C42×C5C2×D411yescharacteristic
246C42×C5C2×D411yescharacteristic
247C42×C5C2411yescharacteristic
248C22×SD16—21no—
249#557—21no—
250#557C22×C511yescharacteristic, complemented
251#557C22×C511yescharacteristic, complemented
252#556—21no—
253#556C22×C511yescharacteristic, complemented
254#552—22no—
255#552—21no—
256D4×Q8—21no—
257D4×Q8C22×C511yescharacteristic, complemented
258#541—22no—
259#541—21no—
260#533—21no—
261#533C22×C511yescharacteristic
262C2×#127C22×C511yescharacteristic
263#529—21no—
264#529C22×C511yescharacteristic
265#519—21no—
266C2×C4×Q8C22×C511yescharacteristic
267#496—21no—
268#490—22no—
269#490—21no—
270#486C22×C512yes—
271#486C22×C512yes—
272#460—21no—
273#460—21no—
274#460C22×C512yes—
275#460C22×C512yes—
276#460C22×C512yes—
277#459C22×C512yes—
278#450—21no—
279#447—21no—
280#432C22×C511yescharacteristic
281C2×C4⋊C8C22×C511yescharacteristic
282#418—22no—
283C2×#102—21no—
284C4×M16C22×C511yescharacteristic
285#332C22×C512yescomplemented
286#332C22×C512yescomplemented
287Q8⋊C8C22×C512yescomplemented
288Q8⋊C8C22×C512yescomplemented
289Q8⋊C8C22×C512yescomplemented
290D4⋊C8C22×C512yescomplemented
291C23×C5—21no—
292C23×C5—42no—
293C23×C5—82no—
294C23×C5—82no—
295C23×C5—82no—
296C23×C5#12011yescharacteristic
297C5×Q8—21no—
298C5×Q8—21no—
299C5×Q8—21no—
300C5×Q8—22no—
301C5×Q8—42no—
302C5×Q8—44no—
303C5×Q8—41no—
304C5×Q8—41no—
305C5×Q8—42no—
306C5×Q8AGL(1,ℤ/8)12yes—
307C5×D4—22no—
308C5×D4—44no—
309C5×D4—42no—
310C5×D4—42no—
311C5×D4—42no—
312C5×D4—42no—
313C5×D4—42no—
314C5×D4—82no—
315C5×D4—82no—
316C5×D4—82no—
317C2×C4×C5—21no—
318C2×C4×C5—22no—
319C2×C4×C5—21no—
320C2×C4×C5—21no—
321C2×C4×C5—22no—
322C2×C4×C5—21no—
323C2×C4×C5—21no—
324C2×C4×C5—22no—
325C2×C4×C5—21no—
326C2×C4×C5—21no—
327C2×C4×C5—21no—
328C2×C4×C5—21no—
329C2×C4×C5—22no—
330C2×C4×C5—42no—
331C2×C4×C5—44no—
332C2×C4×C5—42no—
333C2×C4×C5—41no—
334C2×C4×C5—41no—
335C2×C4×C5#12011yescharacteristic
336C2×C4×C5#12011yescharacteristic
337C2×C4×C5#12011yescharacteristic
338C2×C4×C5C22×D411yescharacteristic
339C2×C4×C5C22×D411yescharacteristic
340C2×C4×C5C22×D411yescharacteristic
341C5×C8—42no—
342C5×C8—42no—
343C5×C8—42no—
344C22×Q8—21no—
345C22×Q8C5×D411yescharacteristic
346C22×D4—21no—
347C22×D4—21no—
348C22×D4—21no—
349C2×SD16—22no—
350C2×SD16—22no—
351C2×SD16—22no—
352C2×SD16—44no—
353C2×D8—22no—
354C2×M16—21no—
355C22×C8—21no—
356C4⋊Q8—22no—
357C4⋊Q8—21no—
358C4⋊Q8—21no—
359C4⋊Q8—21no—
360C4⋊Q8C23×C511yescharacteristic
361C4⋊Q8C23×C511yescharacteristic
362#127—21no—
363#127—21no—
364#127—21no—
365#127C23×C512yes—
366#126—22no—
367#126—22no—
368#126—22no—
369#126—22no—
370#126—21no—
371#125—22no—
372#125—22no—
373#125—22no—
374#125—21no—
375#125—22no—
376#125—22no—
377#125—21no—
378#125C5×D411yescharacteristic
379#125C5×D412yes—
380#124—21no—
381#124C5×D411yescharacteristic
382#123—22no—
383#123—21no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—21no—
387C22⋊Q8—21no—
388C22⋊Q8—21no—
389C22⋊Q8—21no—
390C22⋊Q8—22no—
391C22⋊Q8C5×D411yescharacteristic
392#121C5×D411yescharacteristic
393C4×Q8—22no—
394C4×Q8—22no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8—21no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×Q8—21no—
403C4×Q8—21no—
404C4×Q8C5×D411yescharacteristic
405C4×Q8C5×D411yescharacteristic
406C4×Q8C23×C511yescharacteristic
407C4×Q8C23×C511yescharacteristic
408C4×Q8C23×C511yescharacteristic
409C4×D4—22no—
410C4×D4—21no—
411C4×D4—21no—
412C4×D4C23×C511yescharacteristic
413#117—22no—
414#117—22no—
415#117—22no—
416#117—21no—
417#117—21no—
418#117C5×D412yes—
419C2×C4⋊C4—21no—
420C2×C4⋊C4C5×D411yescharacteristic
421C2×C42—21no—
422C2×C42C23×C511yescharacteristic
423C4⋊C8—21no—
424C4⋊C8C23×C512yes—
425C4⋊C8C23×C512yes—
426#102—22no—
427#102—22no—
428#102—24no—
429#102—24no—
430#102—22no—
431#102—22no—
432C22⋊C8—21no—
433#97—21no—
434#97—21no—
435C4×C8C23×C512yes—
436C22×C5—21no—
437C22×C5—21no—
438C22×C5—41no—
439C22×C5—41no—
440C22×C5—41no—
441C22×C5—82no—
442C22×C5—82no—
443C22×C5—82no—
444C22×C5—164no—
445C22×C5#45412yes—
446C22×C5C2×#12011yescharacteristic
447C4×C5—21no—
448C4×C5—22no—
449C4×C5—21no—
450C4×C5—21no—
451C4×C5—21no—
452C4×C5—41no—
453C4×C5—42no—
454C4×C5—41no—
455C4×C5—41no—
456C4×C5—42no—
457C4×C5—41no—
458C4×C5—41no—
459C4×C5—41no—
460C4×C5C2×AGL(1,ℤ/8)11yescharacteristic
461C4×C5#57511yescharacteristic
462C4×C5#57712yes—
463C24—41no—
464C2×Q8—22no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—22no—
469C2×Q8—41no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8C2×C5×D411yescharacteristic
473C2×Q8C2×C5×D411yescharacteristic
474C2×Q8C2×C5×D411yescharacteristic
475C2×D4—21no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—42no—
480C2×D4—44no—
481C2×D4—42no—
482C2×D4—41no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—82no—
486C2×D4C2×C5×D411yescharacteristic
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4C2×C5×D411yescharacteristic
492C22×C4C2×C5×D411yescharacteristic
493C22×C4C2×C5×D411yescharacteristic
494SD16—44no—
495SD16—44no—
496SD16—82no—
497D8—82no—
498M16—42no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—42no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4C2×C5×D411yescharacteristic
525C4⋊C4C2×C5×D411yescharacteristic
526C4⋊C4C2×C5×D412yes—
527C4⋊C4C2×C5×D412yes—
528C4⋊C4C2×C5×D411yescharacteristic
529C4⋊C4C2×C5×D411yescharacteristic
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—22no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—21no—
543C42—21no—
544C42—22no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42C2×C5×D411yescharacteristic
549C42C2×C5×D411yescharacteristic
550C42C2×C5×D411yescharacteristic
551C42C24×C511yescharacteristic
552C2×C5—22no—
553C2×C5—81no—
554C2×C5—162no—
555C2×C5#300611yescharacteristic
556C2×C5#301011yescharacteristic
557C2×C5C2×#45411yescharacteristic
558C23—21no—
559C23—42no—
560C23—82no—
561C23—82no—
562C23—82no—
563C23C5×#12011yescharacteristic
564Q8—21no—
565Q8—21no—
566Q8—21no—
567Q8—22no—
568Q8—42no—
569Q8—44no—
570Q8—41no—
571Q8—41no—
572Q8—42no—
573Q8C5×AGL(1,ℤ/8)12yes—
574D4—22no—
575D4—44no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—82no—
582D4—82no—
583D4—82no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4C5×#12011yescharacteristic
603C2×C4C5×#12011yescharacteristic
604C2×C4C5×#12011yescharacteristic
605C2×C4C22×C5×D411yescharacteristic
606C2×C4C22×C5×D411yescharacteristic
607C2×C4C22×C5×D411yescharacteristic
608C8—42no—
609C8—42no—
610C8—42no—
611C5#1858011yescharacteristic, direct product factor, Sylow
612C22—21no—
613C22—21no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—164no—
621C22C5×#45412yes—
622C22C2×C5×#12011yescharacteristic
623C4—21no—
624C4—22no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4C2×C5×AGL(1,ℤ/8)11yescharacteristic
637C4C5×#57511yescharacteristic
638C4C5×#57712yes—
639C2—22no—
640C2—81no—
641C2—162no—
642C2C5×#300611yescharacteristic
643C2C5×#301011yescharacteristic
644C2C2×C5×#45411yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC42C24×C5
prime radicalC5×#3217C2
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
1. C5×#3523C2semidirect
3. C5×#3138C2semidirect
4. C2×C5×#460C2semidirect
5. C5×#1628C2semidirect
6. C5×#1627C2semidirect
7. C5×#1626C2semidirect
8. C5×#1626C2semidirect
9. C5×#1552C2semidirect
10. C5×#1491C2semidirect
11. C5×#1483C2semidirect
14. C5×#557C22semidirect
15. C5×#557C22semidirect
17. C5×#556C22semidirect
21. C5×D4×Q8C22semidirect
49. C5×#332C22semidirect
50. C5×#332C22semidirect
51. C5×Q8⋊C8C22semidirect
52. C5×Q8⋊C8C22semidirect
53. C5×Q8⋊C8C22semidirect
54. C5×D4⋊C8C22semidirect
55. #18580C5direct (N × H)
148. #3523C2×C5semidirect
150. #3138C2×C5semidirect
151. C2×#460C2×C5semidirect
152. #1628C2×C5semidirect
153. #1627C2×C5semidirect
154. #1626C2×C5semidirect
155. #1626C2×C5semidirect
156. #1552C2×C5semidirect
157. #1491C2×C5semidirect
158. #1483C2×C5semidirect
250. #557C22×C5semidirect
251. #557C22×C5semidirect
253. #556C22×C5semidirect
257. D4×Q8C22×C5semidirect
285. #332C22×C5semidirect
286. #332C22×C5semidirect
287. Q8⋊C8C22×C5semidirect
288. Q8⋊C8C22×C5semidirect
289. Q8⋊C8C22×C5semidirect
290. D4⋊C8C22×C5semidirect
611. C5#18580direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35232
C5×#32172
C5×#31382
C2×C5×#4602
C5×#16282
C5×#16272
C5×#16262
C5×#15522
C5×#14912
C5×#14832
#185805