Finite Group Databasebeta

← C5×#18255 #11940975

Subgroups — C5×#18255 #11940975

Order 1280.

1260 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3537C211yesmaximal, characteristic
2C5×#3004C211yesmaximal, characteristic, complemented
3C2×C5×#448C211yesmaximal, characteristic, complemented
4C5×#2436C211yesmaximal, characteristic, complemented
5C5×#1606C214yesmaximal, complemented
6C5×#1601C214yesmaximal, complemented
7C5×#1507C211yesmaximal, characteristic, complemented
8C5×#1501C211yesmaximal, characteristic, complemented
9C5×#1471C211yesmaximal, characteristic, complemented
10C23×C5×Q8C2211yescharacteristic, complemented
11C2×C5×#137—21no—
12C22×C5×SD16—21no—
13C5×#560—22no—
14C5×#553—21no—
15C5×#553—21no—
16C5×D42—22no—
17C5×D42—21no—
18C5×D42—21no—
19C5×#540—21no—
20C5×#538—21no—
21C5×#538C2212yes—
22C5×#535—21no—
23C5×#534—21no—
24C5×#534C2212yes—
25C2×C5×#124—21no—
26C5×#525C2211yescharacteristic
27C2×C5×#121C2211yescharacteristic
28C2×C5×#121C2211yescharacteristic
29C5×#451—21no—
30C5×#451C2212yescomplemented
31C5×#448—22no—
32C5×#448—21no—
33C5×#448C2214yescomplemented
34C5×#448C2212yescomplemented
35C5×#419—21no—
36C2×C5×#103—21no—
37C5×#407C2211yescharacteristic, complemented
38C2×C5×C22⋊C8C2211yescharacteristic, complemented
39C5×#393C2214yescomplemented
40C5×#391—21no—
41C5×#386—22no—
42C4×C5×C22⋊C4C2211yescharacteristic, complemented
43C5×#328C2212yescomplemented
44C5×#327C2212yescomplemented
45C5×#327C2214yescomplemented
46C5×#323C2214yescomplemented
47#18255C511yesmaximal, characteristic, direct product factor, Sylow
48C2×C5×#41D411yescharacteristic
49C22×C5×Q8—21no—
50C22×C5×Q8—21no—
51C22×C5×Q8—21no—
52C22×C5×Q8—21no—
53C22×C5×Q8—21no—
54C22×C5×Q8—44no—
55C22×C5×Q8C2312yes—
56C22×C5×D4—22no—
57C22×C5×D4—21no—
58C22×C5×D4—21no—
59C22×C5×D4—22no—
60C22×C5×D4D412yes—
61C22×C5×D4D411yescharacteristic
62C22×C5×D4D411yescharacteristic
63C23×C4×C5—21no—
64C23×C4×C5C2311yescharacteristic
65C5×#137—44no—
66C2×C5×Q16—22no—
67C2×C5×SD16—24no—
68C2×C5×SD16—22no—
69C2×C5×SD16—44no—
70C2×C5×M16—21no—
71C22×C5×C8—21no—
72C5×#127—22no—
73C5×#127—21no—
74C5×#126—22no—
75C5×#126—22no—
76C5×#125—21no—
77C5×#124—22no—
78C5×#124—44no—
79C5×#123—22no—
80C5×#123—21no—
81C5×#123—22no—
82C5×#123—22no—
83C5×#123—22no—
84C5×C22⋊Q8D411yescharacteristic
85C5×C22⋊Q8C2311yescharacteristic
86C5×#121—22no—
87C5×#121—21no—
88C5×#121—21no—
89C5×#121—22no—
90C5×#121—22no—
91C5×#121—22no—
92C5×#121—22no—
93C5×#121—21no—
94C5×#121—21no—
95C5×#121—21no—
96C5×#121—21no—
97C5×#121—21no—
98C5×#121—21no—
99C5×#121—22no—
100C5×#121D412yes—
101C5×#121D411yescharacteristic
102C5×#121C2312yes—
103C5×#121C2311yescharacteristic
104C5×#120—22no—
105C5×#120—22no—
106C5×#120—22no—
107C5×#120—21no—
108C5×#120—22no—
109C5×#120—22no—
110C5×#120—22no—
111C4×C5×D4—22no—
112C4×C5×D4—21no—
113C4×C5×D4—21no—
114C4×C5×D4—21no—
115C4×C5×D4—21no—
116C4×C5×D4—22no—
117C5×#117—21no—
118C5×#117D411yescharacteristic
119C2×C5×C4⋊C4D411yescharacteristic
120C2×C5×C4⋊C4D411yescharacteristic
121C2×C5×C22⋊C4—21no—
122C2×C5×C22⋊C4—21no—
123C2×C5×C22⋊C4—24no—
124C2×C5×C22⋊C4C2312yes—
125C2×C42×C5—21no—
126C5×#103—24no—
127C5×#103—22no—
128C5×#103—22no—
129C5×C22⋊C8—21no—
130C5×C22⋊C8—21no—
131C5×C22⋊C8C2312yes—
132C5×C22⋊C8C2312yes—
133C5×#95—24no—
134C5×#95C2312yes—
135#3537C2×C511yescharacteristic
136#3004C2×C511yescharacteristic, complemented
137C2×#448C2×C511yescharacteristic, complemented
138#2436C2×C511yescharacteristic, complemented
139#1606C2×C514yescomplemented
140#1601C2×C514yescomplemented
141#1507C2×C511yescharacteristic, complemented
142#1501C2×C511yescharacteristic, complemented
143#1471C2×C511yescharacteristic, complemented
144C24×C5—22no—
145C24×C5—21no—
146C24×C5C2×D411yescharacteristic
147C24×C5C2×D412yes—
148C5×#41—42no—
149C2×C5×Q8—24no—
150C2×C5×Q8—24no—
151C2×C5×Q8—24no—
152C2×C5×Q8—22no—
153C2×C5×Q8—22no—
154C2×C5×Q8—44no—
155C2×C5×Q8—42no—
156C2×C5×Q8—42no—
157C2×C5×Q8—42no—
158C2×C5×Q8—44no—
159C2×C5×Q8—44no—
160C2×C5×Q8—41no—
161C2×C5×Q8—41no—
162C2×C5×Q8C2×D411yescharacteristic
163C2×C5×D4—22no—
164C2×C5×D4—21no—
165C2×C5×D4—21no—
166C2×C5×D4—22no—
167C2×C5×D4—22no—
168C2×C5×D4—22no—
169C2×C5×D4—22no—
170C2×C5×D4—22no—
171C2×C5×D4—22no—
172C2×C5×D4—21no—
173C2×C5×D4—22no—
174C2×C5×D4—22no—
175C2×C5×D4—21no—
176C2×C5×D4—21no—
177C2×C5×D4—21no—
178C2×C5×D4—42no—
179C2×C5×D4—42no—
180C2×C5×D4—44no—
181C2×C5×D4—82no—
182C2×C5×D4—82no—
183C2×C5×D4—84no—
184C2×C5×D4—84no—
185C2×C5×D4C2×D411yescharacteristic
186C2×C5×D4C2×D412yes—
187C22×C4×C5—22no—
188C22×C4×C5—22no—
189C22×C4×C5—22no—
190C22×C4×C5—21no—
191C22×C4×C5—21no—
192C22×C4×C5—22no—
193C22×C4×C5—42no—
194C22×C4×C5—44no—
195C22×C4×C5—41no—
196C22×C4×C5—41no—
197C22×C4×C5C2×D411yescharacteristic
198C22×C4×C5C2×D411yescharacteristic
199C22×C4×C5C2×D411yescharacteristic
200C22×C4×C5C2×D412yes—
201C22×C4×C5C2×D411yescharacteristic
202C22×C4×C5C2×D411yescharacteristic
203C22×C4×C5C2411yescharacteristic
204C5×Q16—44no—
205C5×SD16—48no—
206C5×SD16—44no—
207C5×M16—42no—
208C2×C5×C8—22no—
209C2×C5×C8—22no—
210C2×C5×C8—42no—
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—21no—
218C5×C4⋊C4C2×D412yes—
219C5×C4⋊C4C2×D411yescharacteristic
220C5×C4⋊C4C2×D411yescharacteristic
221C5×C22⋊C4—22no—
222C5×C22⋊C4—22no—
223C5×C22⋊C4—22no—
224C5×C22⋊C4—22no—
225C5×C22⋊C4—22no—
226C5×C22⋊C4—21no—
227C5×C22⋊C4—21no—
228C5×C22⋊C4—21no—
229C5×C22⋊C4—22no—
230C5×C22⋊C4—21no—
231C5×C22⋊C4—22no—
232C5×C22⋊C4—22no—
233C5×C22⋊C4—22no—
234C5×C22⋊C4—21no—
235C5×C22⋊C4—48no—
236C5×C22⋊C4—42no—
237C42×C5—22no—
238C42×C5—21no—
239C42×C5—42no—
240C23×Q8C22×C511yescharacteristic, complemented
241C2×#137—21no—
242C22×SD16—21no—
243#560—22no—
244#553—21no—
245#553—21no—
246D42—22no—
247D42—21no—
248D42—21no—
249#540—21no—
250#538—21no—
251#538C22×C512yes—
252#535—21no—
253#534—21no—
254#534C22×C512yes—
255C2×#124—21no—
256#525C22×C511yescharacteristic
257C2×#121C22×C511yescharacteristic
258C2×#121C22×C511yescharacteristic
259#451—21no—
260#451C22×C512yescomplemented
261#448—22no—
262#448—21no—
263#448C22×C512yescomplemented
264#448C22×C514yescomplemented
265#419—21no—
266C2×#103—21no—
267#407C22×C511yescharacteristic, complemented
268C2×C22⋊C8C22×C511yescharacteristic, complemented
269#393C22×C514yescomplemented
270#391—21no—
271#386—22no—
272C4×C22⋊C4C22×C511yescharacteristic, complemented
273#328C22×C512yescomplemented
274#327C22×C512yescomplemented
275#327C22×C514yescomplemented
276#323C22×C514yescomplemented
277C23×C5—22no—
278C23×C5—22no—
279C23×C5—22no—
280C23×C5—22no—
281C23×C5—21no—
282C23×C5—24no—
283C23×C5—41no—
284C23×C5—42no—
285C23×C5—42no—
286C23×C5—42no—
287C23×C5—81no—
288C23×C5—81no—
289C23×C5—82no—
290C23×C5—82no—
291C23×C5—82no—
292C23×C5#12011yescharacteristic
293C23×C5#12011yescharacteristic
294C23×C5C22×D411yescharacteristic
295C5×Q8—24no—
296C5×Q8—24no—
297C5×Q8—41no—
298C5×Q8—42no—
299C5×Q8—42no—
300C5×Q8—44no—
301C5×Q8—44no—
302C5×D4—42no—
303C5×D4—41no—
304C5×D4—44no—
305C5×D4—42no—
306C5×D4—44no—
307C5×D4—42no—
308C5×D4—42no—
309C5×D4—42no—
310C5×D4—81no—
311C5×D4—82no—
312C5×D4—82no—
313C5×D4—81no—
314C5×D4—82no—
315C5×D4—82no—
316C5×D4—82no—
317C5×D4—82no—
318C5×D4—82no—
319C2×C4×C5—22no—
320C2×C4×C5—22no—
321C2×C4×C5—22no—
322C2×C4×C5—24no—
323C2×C4×C5—21no—
324C2×C4×C5—22no—
325C2×C4×C5—22no—
326C2×C4×C5—21no—
327C2×C4×C5—21no—
328C2×C4×C5—41no—
329C2×C4×C5—41no—
330C2×C4×C5—41no—
331C2×C4×C5—42no—
332C2×C4×C5—44no—
333C2×C4×C5—44no—
334C2×C4×C5—42no—
335C2×C4×C5—42no—
336C2×C4×C5—42no—
337C2×C4×C5—42no—
338C2×C4×C5—42no—
339C2×C4×C5—42no—
340C2×C4×C5—81no—
341C2×C4×C5—82no—
342C2×C4×C5—82no—
343C2×C4×C5#12011yescharacteristic
344C2×C4×C5#12011yescharacteristic
345C2×C4×C5C22×D411yescharacteristic
346C2×C4×C5C22×D411yescharacteristic
347C5×C8—42no—
348C5×C8—42no—
349C2×#41C5×D411yescharacteristic
350C22×Q8—21no—
351C22×Q8—21no—
352C22×Q8—21no—
353C22×Q8—21no—
354C22×Q8—21no—
355C22×Q8—44no—
356C22×Q8C23×C512yes—
357C22×D4—22no—
358C22×D4—21no—
359C22×D4—21no—
360C22×D4—22no—
361C22×D4C5×D411yescharacteristic
362C22×D4C5×D412yes—
363C22×D4C5×D411yescharacteristic
364C23×C4—21no—
365C23×C4C23×C511yescharacteristic
366#137—44no—
367C2×Q16—22no—
368C2×SD16—24no—
369C2×SD16—22no—
370C2×SD16—44no—
371C2×M16—21no—
372C22×C8—21no—
373#127—22no—
374#127—21no—
375#126—22no—
376#126—22no—
377#125—21no—
378#124—22no—
379#124—44no—
380#123—22no—
381#123—21no—
382#123—22no—
383#123—22no—
384#123—22no—
385C22⋊Q8C5×D411yescharacteristic
386C22⋊Q8C23×C511yescharacteristic
387#121—22no—
388#121—21no—
389#121—21no—
390#121—22no—
391#121—22no—
392#121—22no—
393#121—22no—
394#121—21no—
395#121—21no—
396#121—21no—
397#121—21no—
398#121—21no—
399#121—21no—
400#121—22no—
401#121C5×D412yes—
402#121C5×D411yescharacteristic
403#121C23×C512yes—
404#121C23×C511yescharacteristic
405#120—22no—
406#120—22no—
407#120—22no—
408#120—21no—
409#120—22no—
410#120—22no—
411#120—22no—
412C4×D4—22no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—22no—
418#117—21no—
419#117C5×D411yescharacteristic
420C2×C4⋊C4C5×D411yescharacteristic
421C2×C4⋊C4C5×D411yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—24no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4C23×C512yes—
426C2×C42—21no—
427#103—24no—
428#103—22no—
429#103—22no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8C23×C512yes—
433C22⋊C8C23×C512yes—
434#95—24no—
435#95C23×C512yes—
436C22×C5—21no—
437C22×C5—21no—
438C22×C5—21no—
439C22×C5—21no—
440C22×C5—21no—
441C22×C5—21no—
442C22×C5—21no—
443C22×C5—44no—
444C22×C5—42no—
445C22×C5—42no—
446C22×C5—41no—
447C22×C5—42no—
448C22×C5—42no—
449C22×C5—41no—
450C22×C5—41no—
451C22×C5—42no—
452C22×C5—42no—
453C22×C5—84no—
454C22×C5—162no—
455C22×C5—164no—
456C22×C5#45512yes—
457C22×C5C2×#12011yescharacteristic
458C22×C5#57712yes—
459C4×C5—21no—
460C4×C5—21no—
461C4×C5—21no—
462C4×C5—42no—
463C4×C5—41no—
464C4×C5—42no—
465C4×C5—41no—
466C4×C5—44no—
467C4×C5—41no—
468C4×C5—82no—
469C4×C5B(4,𝔽2)12yes—
470C24—22no—
471C24—21no—
472C24C2×C5×D411yescharacteristic
473C24C2×C5×D412yes—
474#41—42no—
475C2×Q8—24no—
476C2×Q8—24no—
477C2×Q8—24no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—44no—
481C2×Q8—42no—
482C2×Q8—42no—
483C2×Q8—42no—
484C2×Q8—44no—
485C2×Q8—44no—
486C2×Q8—41no—
487C2×Q8—41no—
488C2×Q8C2×C5×D411yescharacteristic
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—21no—
499C2×D4—22no—
500C2×D4—22no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4—42no—
505C2×D4—42no—
506C2×D4—44no—
507C2×D4—82no—
508C2×D4—82no—
509C2×D4—84no—
510C2×D4—84no—
511C2×D4C2×C5×D411yescharacteristic
512C2×D4C2×C5×D412yes—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—22no—
519C22×C4—42no—
520C22×C4—44no—
521C22×C4—41no—
522C22×C4—41no—
523C22×C4C2×C5×D411yescharacteristic
524C22×C4C2×C5×D411yescharacteristic
525C22×C4C2×C5×D411yescharacteristic
526C22×C4C2×C5×D412yes—
527C22×C4C2×C5×D411yescharacteristic
528C22×C4C2×C5×D411yescharacteristic
529C22×C4C24×C511yescharacteristic
530Q16—44no—
531SD16—48no—
532SD16—44no—
533M16—42no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—42no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4C2×C5×D412yes—
545C4⋊C4C2×C5×D411yescharacteristic
546C4⋊C4C2×C5×D411yescharacteristic
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—22no—
556C22⋊C4—21no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—21no—
561C22⋊C4—48no—
562C22⋊C4—42no—
563C42—22no—
564C42—21no—
565C42—42no—
566C2×C5—22no—
567C2×C5—22no—
568C2×C5—41no—
569C2×C5—82no—
570C2×C5—82no—
571C2×C5—81no—
572C2×C5#300911yescharacteristic
573C2×C5C2×#45511yescharacteristic
574C2×C5C2×B(4,𝔽2)11yescharacteristic
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—21no—
580C23—24no—
581C23—41no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—81no—
586C23—81no—
587C23—82no—
588C23—82no—
589C23—82no—
590C23C5×#12011yescharacteristic
591C23C5×#12011yescharacteristic
592C23C22×C5×D411yescharacteristic
593Q8—24no—
594Q8—24no—
595Q8—41no—
596Q8—42no—
597Q8—42no—
598Q8—44no—
599Q8—44no—
600D4—42no—
601D4—41no—
602D4—44no—
603D4—42no—
604D4—44no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—81no—
609D4—82no—
610D4—82no—
611D4—81no—
612D4—82no—
613D4—82no—
614D4—82no—
615D4—82no—
616D4—82no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—81no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4C5×#12011yescharacteristic
642C2×C4C5×#12011yescharacteristic
643C2×C4C22×C5×D411yescharacteristic
644C2×C4C22×C5×D411yescharacteristic
645C8—42no—
646C8—42no—
647C5#1825511yescharacteristic, direct product factor, Sylow
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—44no—
656C22—42no—
657C22—42no—
658C22—41no—
659C22—42no—
660C22—42no—
661C22—41no—
662C22—41no—
663C22—42no—
664C22—42no—
665C22—84no—
666C22—162no—
667C22—164no—
668C22C5×#45512yes—
669C22C2×C5×#12011yescharacteristic
670C22C5×#57712yes—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—42no—
675C4—41no—
676C4—42no—
677C4—41no—
678C4—44no—
679C4—41no—
680C4—82no—
681C4C5×B(4,𝔽2)12yes—
682C2—22no—
683C2—22no—
684C2—41no—
685C2—82no—
686C2—82no—
687C2—81no—
688C2C5×#300911yescharacteristic
689C2C2×C5×#45511yescharacteristic
690C2C2×C5×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC5×#3537C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#3004C2semidirect
3. C2×C5×#448C2semidirect
4. C5×#2436C2semidirect
5. C5×#1606C2semidirect
6. C5×#1601C2semidirect
7. C5×#1507C2semidirect
8. C5×#1501C2semidirect
9. C5×#1471C2semidirect
10. C23×C5×Q8C22semidirect
30. C5×#451C22semidirect
33. C5×#448C22semidirect
34. C5×#448C22semidirect
37. C5×#407C22semidirect
38. C2×C5×C22⋊C8C22semidirect
39. C5×#393C22semidirect
42. C4×C5×C22⋊C4C22semidirect
43. C5×#328C22semidirect
44. C5×#327C22semidirect
45. C5×#327C22semidirect
46. C5×#323C22semidirect
47. #18255C5direct (N × H)
136. #3004C2×C5semidirect
137. C2×#448C2×C5semidirect
138. #2436C2×C5semidirect
139. #1606C2×C5semidirect
140. #1601C2×C5semidirect
141. #1507C2×C5semidirect
142. #1501C2×C5semidirect
143. #1471C2×C5semidirect
240. C23×Q8C22×C5semidirect
260. #451C22×C5semidirect
263. #448C22×C5semidirect
264. #448C22×C5semidirect
267. #407C22×C5semidirect
268. C2×C22⋊C8C22×C5semidirect
269. #393C22×C5semidirect
272. C4×C22⋊C4C22×C5semidirect
273. #328C22×C5semidirect
274. #327C22×C5semidirect
275. #327C22×C5semidirect
276. #323C22×C5semidirect
647. C5#18255direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35372
C5×#30042
C2×C5×#4482
C5×#24362
C5×#16062
C5×#16012
C5×#15072
C5×#15012
C5×#14712
#182555