Finite Group Databasebeta

← C5×#18466 #11941186

Subgroups — C5×#18466 #11941186

Order 1280.

954 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3558C211yesmaximal, characteristic
2C5×#3234C211yesmaximal, characteristic, complemented
3C5×#3232C211yesmaximal, characteristic, complemented
4C5×#2820C211yesmaximal, characteristic, complemented
5C5×#1618C212yesmaximal, complemented
6C5×#1613C212yesmaximal, complemented
7C5×#1612C212yesmaximal, complemented
8C5×#1607C212yesmaximal, complemented
9C5×#1517C211yesmaximal, characteristic, complemented
10C5×#1512C211yesmaximal, characteristic, complemented
11C5×#1469C211yesmaximal, characteristic, complemented
12C5×#562—21no—
13C5×H(3,𝔽4)—21no—
14C5×#559—21no—
15C5×#548—21no—
16C5×#546—21no—
17C5×#546—22no—
18C5×D42—21no—
19C5×#542—21no—
20C5×#538—22no—
21C5×#538C2212yes—
22C5×#535—21no—
23C5×#534C2212yes—
24C5×#532C2211yescharacteristic
25C5×#529C2211yescharacteristic
26C2×C5×#125C2211yescharacteristic, complemented
27C2×C5×#124C2211yescharacteristic
28C5×#501—21no—
29C5×#499—21no—
30C5×#490—21no—
31C5×#488—21no—
32C5×#484C2211yescharacteristic, complemented
33C5×#483C2211yescharacteristic, complemented
34C5×#481C2211yescharacteristic
35C5×#481C2212yescomplemented
36C5×#480C2211yescharacteristic
37C5×#480C2212yescomplemented
38C5×#479—21no—
39C5×#478—21no—
40C5×#478—22no—
41C5×#432C2211yescharacteristic, complemented
42C5×#431C2211yescharacteristic, complemented
43C5×#399C2212yescomplemented
44C5×#397C2212yescomplemented
45C5×#396—22no—
46C5×#395—21no—
47C5×#394—21no—
48C5×#382C2211yescharacteristic, complemented
49C5×#328C2212yescomplemented
50C5×#327C2212yes—
51C5×#327C2212yescomplemented
52C5×#327C2212yescomplemented
53C5×#323C2212yescomplemented
54C5×#323C2212yescomplemented
55#18466C511yesmaximal, characteristic, direct product factor, Sylow
56C2×C5×#41—21no—
57C2×C5×#41—21no—
58C22×C5×Q8D411yescharacteristic
59C22×C5×D4—21no—
60C22×C5×D4—22no—
61C22×C5×D4D411yescharacteristic
62C5×C4⋊Q8D411yescharacteristic
63C5×#127—21no—
64C5×#127D411yescharacteristic
65C5×#126—22no—
66C5×#126—21no—
67C5×#125—21no—
68C5×#125—21no—
69C5×#125—21no—
70C5×#125—21no—
71C5×#125D411yescharacteristic
72C5×#124—21no—
73C5×#124—22no—
74C5×#124—22no—
75C5×#124—22no—
76C5×#124—21no—
77C5×#124—21no—
78C5×#124—21no—
79C5×#124—21no—
80C5×#124—21no—
81C5×#124D411yescharacteristic
82C5×#123—22no—
83C5×#123—21no—
84C5×#123—22no—
85C5×#123—22no—
86C5×#123—22no—
87C5×C22⋊Q8—22no—
88C5×C22⋊Q8—21no—
89C5×C22⋊Q8—22no—
90C5×C22⋊Q8C2311yescharacteristic
91C5×#121—22no—
92C5×#121—22no—
93C5×#121—22no—
94C5×#121—21no—
95C5×#121—22no—
96C5×#121—21no—
97C5×#121—22no—
98C5×#121C2311yescharacteristic
99C5×#121C2312yes—
100C5×#120—22no—
101C5×#120—22no—
102C5×#120—22no—
103C5×#120—22no—
104C5×#120—22no—
105C4×C5×Q8—21no—
106C4×C5×Q8D411yescharacteristic
107C4×C5×D4—22no—
108C4×C5×D4—21no—
109C4×C5×D4—21no—
110C4×C5×D4—22no—
111C4×C5×D4—21no—
112C4×C5×D4D411yescharacteristic
113C4×C5×D4D412yes—
114C5×#117—21no—
115C2×C5×C4⋊C4—21no—
116C2×C5×C4⋊C4—21no—
117C2×C5×C4⋊C4—21no—
118C2×C5×C4⋊C4—22no—
119C2×C5×C4⋊C4C2311yescharacteristic
120C2×C5×C4⋊C4C2311yescharacteristic
121C2×C5×C22⋊C4—22no—
122C2×C5×C22⋊C4—22no—
123C2×C5×C22⋊C4—22no—
124C2×C5×C22⋊C4D412yes—
125C2×C5×C22⋊C4C2312yes—
126C2×C42×C5C2311yescharacteristic
127C5×#107—22no—
128C5×#107—21no—
129C5×#107—21no—
130C5×#106—22no—
131C5×#106—21no—
132C5×#106—21no—
133C5×C4⋊C8—21no—
134C5×C4⋊C8—21no—
135C5×#103—21no—
136C5×#103—21no—
137C5×#102—21no—
138C5×#102—22no—
139C5×#102—21no—
140C5×#102—22no—
141C5×C22⋊C8C2312yes—
142C5×C22⋊C8C2311yescharacteristic
143C5×C22⋊C8C2311yescharacteristic
144C5×#97—21no—
145C4×C5×C8—21no—
146C5×#95—21no—
147C5×#95—22no—
148C5×#95C2312yes—
149#3558C2×C511yescharacteristic
150#3234C2×C511yescharacteristic, complemented
151#3232C2×C511yescharacteristic, complemented
152#2820C2×C511yescharacteristic, complemented
153#1618C2×C512yescomplemented
154#1613C2×C512yescomplemented
155#1612C2×C512yescomplemented
156#1607C2×C512yescomplemented
157#1517C2×C511yescharacteristic, complemented
158#1512C2×C511yescharacteristic, complemented
159#1469C2×C511yescharacteristic, complemented
160C24×C5—22no—
161C24×C5C2×D412yes—
162C5×#41—81no—
163C5×#41—81no—
164C5×#41—82no—
165C2×C5×Q8—21no—
166C2×C5×Q8—21no—
167C2×C5×Q8—21no—
168C2×C5×Q8—81no—
169C2×C5×Q8C2×D411yescharacteristic
170C2×C5×D4—22no—
171C2×C5×D4—21no—
172C2×C5×D4—22no—
173C2×C5×D4—22no—
174C2×C5×D4—22no—
175C2×C5×D4—22no—
176C2×C5×D4—22no—
177C2×C5×D4—21no—
178C2×C5×D4—21no—
179C2×C5×D4—21no—
180C2×C5×D4—21no—
181C2×C5×D4—82no—
182C2×C5×D4—81no—
183C2×C5×D4—82no—
184C2×C5×D4—82no—
185C2×C5×D4C2×D411yescharacteristic
186C2×C5×D4C2×D412yes—
187C22×C4×C5—22no—
188C22×C4×C5—22no—
189C22×C4×C5—21no—
190C22×C4×C5—22no—
191C22×C4×C5—21no—
192C22×C4×C5—21no—
193C22×C4×C5—22no—
194C22×C4×C5C2×D411yescharacteristic
195C22×C4×C5C2×D412yes—
196C22×C4×C5C2×D411yescharacteristic
197C22×C4×C5C2411yescharacteristic
198C2×C5×C8—21no—
199C2×C5×C8—21no—
200C2×C5×C8—22no—
201C5×C4⋊C4—22no—
202C5×C4⋊C4—22no—
203C5×C4⋊C4—21no—
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—41no—
213C5×C4⋊C4—41no—
214C5×C4⋊C4—41no—
215C5×C4⋊C4—41no—
216C5×C4⋊C4—81no—
217C5×C4⋊C4—82no—
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—22no—
229C5×C22⋊C4—22no—
230C5×C22⋊C4—21no—
231C5×C22⋊C4—21no—
232C5×C22⋊C4—22no—
233C5×C22⋊C4—22no—
234C5×C22⋊C4—22no—
235C5×C22⋊C4—21no—
236C5×C22⋊C4—22no—
237C5×C22⋊C4—82no—
238C5×C22⋊C4—82no—
239C42×C5—22no—
240C42×C5—21no—
241C42×C5C2×D411yescharacteristic
242C42×C5C2×D411yescharacteristic
243C42×C5C2×D411yescharacteristic
244C42×C5C2×D411yescharacteristic
245#562—21no—
246H(3,𝔽4)—21no—
247#559—21no—
248#548—21no—
249#546—21no—
250#546—22no—
251D42—21no—
252#542—21no—
253#538—22no—
254#538C22×C512yes—
255#535—21no—
256#534C22×C512yes—
257#532C22×C511yescharacteristic
258#529C22×C511yescharacteristic
259C2×#125C22×C511yescharacteristic, complemented
260C2×#124C22×C511yescharacteristic
261#501—21no—
262#499—21no—
263#490—21no—
264#488—21no—
265#484C22×C511yescharacteristic, complemented
266#483C22×C511yescharacteristic, complemented
267#481C22×C511yescharacteristic
268#481C22×C512yescomplemented
269#480C22×C511yescharacteristic
270#480C22×C512yescomplemented
271#479—21no—
272#478—21no—
273#478—22no—
274#432C22×C511yescharacteristic, complemented
275#431C22×C511yescharacteristic, complemented
276#399C22×C512yescomplemented
277#397C22×C512yescomplemented
278#396—22no—
279#395—21no—
280#394—21no—
281#382C22×C511yescharacteristic, complemented
282#328C22×C512yescomplemented
283#327C22×C512yes—
284#327C22×C512yescomplemented
285#327C22×C512yescomplemented
286#323C22×C512yescomplemented
287#323C22×C512yescomplemented
288C23×C5—22no—
289C23×C5—21no—
290C23×C5—22no—
291C23×C5—22no—
292C23×C5—22no—
293C23×C5—42no—
294C23×C5—42no—
295C23×C5—82no—
296C23×C5—82no—
297C23×C5—82no—
298C23×C5C22×D411yescharacteristic
299C5×Q8—41no—
300C5×Q8—42no—
301C5×Q8—81no—
302C5×Q8—81no—
303C5×D4—42no—
304C5×D4—41no—
305C5×D4—42no—
306C5×D4—42no—
307C5×D4—41no—
308C5×D4—42no—
309C5×D4—41no—
310C5×D4—42no—
311C5×D4—81no—
312C5×D4—81no—
313C5×D4—82no—
314C5×D4—82no—
315C5×D4—82no—
316C5×D4—82no—
317C5×D4—82no—
318C2×C4×C5—21no—
319C2×C4×C5—21no—
320C2×C4×C5—21no—
321C2×C4×C5—22no—
322C2×C4×C5—22no—
323C2×C4×C5—21no—
324C2×C4×C5—21no—
325C2×C4×C5—41no—
326C2×C4×C5—41no—
327C2×C4×C5—42no—
328C2×C4×C5—42no—
329C2×C4×C5—42no—
330C2×C4×C5—42no—
331C2×C4×C5—42no—
332C2×C4×C5—41no—
333C2×C4×C5—41no—
334C2×C4×C5—41no—
335C2×C4×C5—42no—
336C2×C4×C5—41no—
337C2×C4×C5—82no—
338C2×C4×C5—82no—
339C2×C4×C5—82no—
340C2×C4×C5#12011yescharacteristic
341C2×C4×C5#12011yescharacteristic
342C2×C4×C5#12011yescharacteristic
343C2×C4×C5#12011yescharacteristic
344C2×C4×C5C22×D411yescharacteristic
345C2×C4×C5C22×D411yescharacteristic
346C5×C8—42no—
347C5×C8—41no—
348C5×C8—41no—
349C2×#41—21no—
350C2×#41—21no—
351C22×Q8C5×D411yescharacteristic
352C22×D4—21no—
353C22×D4—22no—
354C22×D4C5×D411yescharacteristic
355C4⋊Q8C5×D411yescharacteristic
356#127—21no—
357#127C5×D411yescharacteristic
358#126—22no—
359#126—21no—
360#125—21no—
361#125—21no—
362#125—21no—
363#125—21no—
364#125C5×D411yescharacteristic
365#124—21no—
366#124—22no—
367#124—22no—
368#124—22no—
369#124—21no—
370#124—21no—
371#124—21no—
372#124—21no—
373#124—21no—
374#124C5×D411yescharacteristic
375#123—22no—
376#123—21no—
377#123—22no—
378#123—22no—
379#123—22no—
380C22⋊Q8—22no—
381C22⋊Q8—21no—
382C22⋊Q8—22no—
383C22⋊Q8C23×C511yescharacteristic
384#121—22no—
385#121—22no—
386#121—22no—
387#121—21no—
388#121—22no—
389#121—21no—
390#121—22no—
391#121C23×C511yescharacteristic
392#121C23×C512yes—
393#120—22no—
394#120—22no—
395#120—22no—
396#120—22no—
397#120—22no—
398C4×Q8—21no—
399C4×Q8C5×D411yescharacteristic
400C4×D4—22no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—22no—
404C4×D4—21no—
405C4×D4C5×D411yescharacteristic
406C4×D4C5×D412yes—
407#117—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4C23×C511yescharacteristic
413C2×C4⋊C4C23×C511yescharacteristic
414C2×C22⋊C4—22no—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—22no—
417C2×C22⋊C4C5×D412yes—
418C2×C22⋊C4C23×C512yes—
419C2×C42C23×C511yescharacteristic
420#107—22no—
421#107—21no—
422#107—21no—
423#106—22no—
424#106—21no—
425#106—21no—
426C4⋊C8—21no—
427C4⋊C8—21no—
428#103—21no—
429#103—21no—
430#102—21no—
431#102—22no—
432#102—21no—
433#102—22no—
434C22⋊C8C23×C512yes—
435C22⋊C8C23×C511yescharacteristic
436C22⋊C8C23×C511yescharacteristic
437#97—21no—
438C4×C8—21no—
439#95—21no—
440#95—22no—
441#95C23×C512yes—
442C22×C5—21no—
443C22×C5—21no—
444C22×C5—42no—
445C22×C5—42no—
446C22×C5—41no—
447C22×C5—41no—
448C22×C5—42no—
449C22×C5—42no—
450C22×C5—42no—
451C22×C5—41no—
452C22×C5—42no—
453C22×C5—82no—
454C22×C5—82no—
455C22×C5—164no—
456C22×C5C2×#12011yescharacteristic
457C22×C5#57611yescharacteristic
458C22×C5#57711yescharacteristic
459C4×C5—21no—
460C4×C5—21no—
461C4×C5—21no—
462C4×C5—21no—
463C4×C5—21no—
464C4×C5—21no—
465C4×C5—41no—
466C4×C5—42no—
467C4×C5—41no—
468C4×C5—42no—
469C4×C5—41no—
470C4×C5—82no—
471C4×C5—81no—
472C4×C5—81no—
473C24—22no—
474C24C2×C5×D412yes—
475#41—81no—
476#41—81no—
477#41—82no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—81no—
482C2×Q8C2×C5×D411yescharacteristic
483C2×D4—22no—
484C2×D4—21no—
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—82no—
495C2×D4—81no—
496C2×D4—82no—
497C2×D4—82no—
498C2×D4C2×C5×D411yescharacteristic
499C2×D4C2×C5×D412yes—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4C2×C5×D411yescharacteristic
508C22×C4C2×C5×D412yes—
509C22×C4C2×C5×D411yescharacteristic
510C22×C4C24×C511yescharacteristic
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—81no—
530C4⋊C4—82no—
531C4⋊C4C2×C5×D412yes—
532C4⋊C4C2×C5×D411yescharacteristic
533C4⋊C4C2×C5×D411yescharacteristic
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—21no—
549C22⋊C4—22no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C42—22no—
553C42—21no—
554C42C2×C5×D411yescharacteristic
555C42C2×C5×D411yescharacteristic
556C42C2×C5×D411yescharacteristic
557C42C2×C5×D411yescharacteristic
558C2×C5—21no—
559C2×C5—21no—
560C2×C5—82no—
561C2×C5—82no—
562C2×C5—81no—
563C2×C5#301211yescharacteristic
564C2×C5#301911yescharacteristic
565C2×C5#302811yescharacteristic
566C23—22no—
567C23—21no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—42no—
572C23—42no—
573C23—82no—
574C23—82no—
575C23—82no—
576C23C22×C5×D411yescharacteristic
577Q8—41no—
578Q8—42no—
579Q8—81no—
580Q8—81no—
581D4—42no—
582D4—41no—
583D4—42no—
584D4—42no—
585D4—41no—
586D4—42no—
587D4—41no—
588D4—42no—
589D4—81no—
590D4—81no—
591D4—82no—
592D4—82no—
593D4—82no—
594D4—82no—
595D4—82no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4C5×#12011yescharacteristic
619C2×C4C5×#12011yescharacteristic
620C2×C4C5×#12011yescharacteristic
621C2×C4C5×#12011yescharacteristic
622C2×C4C22×C5×D411yescharacteristic
623C2×C4C22×C5×D411yescharacteristic
624C8—42no—
625C8—41no—
626C8—41no—
627C5#1846611yescharacteristic, direct product factor, Sylow
628C22—21no—
629C22—21no—
630C22—42no—
631C22—42no—
632C22—41no—
633C22—41no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22—42no—
639C22—82no—
640C22—82no—
641C22—164no—
642C22C2×C5×#12011yescharacteristic
643C22C5×#57611yescharacteristic
644C22C5×#57711yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—41no—
652C4—42no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—82no—
657C4—81no—
658C4—81no—
659C2—21no—
660C2—21no—
661C2—82no—
662C2—81no—
663C2—82no—
664C2C5×#301211yescharacteristic
665C2C5×#301911yescharacteristic
666C2C5×#302811yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC5×#3558C2
LCSG ▷ C22×C4 ▷ 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×#3234C2semidirect
3. C5×#3232C2semidirect
4. C5×#2820C2semidirect
5. C5×#1618C2semidirect
6. C5×#1613C2semidirect
7. C5×#1612C2semidirect
8. C5×#1607C2semidirect
9. C5×#1517C2semidirect
10. C5×#1512C2semidirect
11. C5×#1469C2semidirect
26. C2×C5×#125C22semidirect
32. C5×#484C22semidirect
33. C5×#483C22semidirect
35. C5×#481C22semidirect
37. C5×#480C22semidirect
41. C5×#432C22semidirect
42. C5×#431C22semidirect
43. C5×#399C22semidirect
44. C5×#397C22semidirect
48. C5×#382C22semidirect
49. C5×#328C22semidirect
51. C5×#327C22semidirect
52. C5×#327C22semidirect
53. C5×#323C22semidirect
54. C5×#323C22semidirect
55. #18466C5direct (N × H)
150. #3234C2×C5semidirect
151. #3232C2×C5semidirect
152. #2820C2×C5semidirect
153. #1618C2×C5semidirect
154. #1613C2×C5semidirect
155. #1612C2×C5semidirect
156. #1607C2×C5semidirect
157. #1517C2×C5semidirect
158. #1512C2×C5semidirect
159. #1469C2×C5semidirect
259. C2×#125C22×C5semidirect
265. #484C22×C5semidirect
266. #483C22×C5semidirect
268. #481C22×C5semidirect
270. #480C22×C5semidirect
274. #432C22×C5semidirect
275. #431C22×C5semidirect
276. #399C22×C5semidirect
277. #397C22×C5semidirect
281. #382C22×C5semidirect
282. #328C22×C5semidirect
284. #327C22×C5semidirect
285. #327C22×C5semidirect
286. #323C22×C5semidirect
287. #323C22×C5semidirect
627. C5#18466direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35582
C5×#32342
C5×#32322
C5×#28202
C5×#16182
C5×#16132
C5×#16122
C5×#16072
C5×#15172
C5×#15122
C5×#14692
#184665