Finite Group Databasebeta

← C5×#18275 #11940995

Subgroups — C5×#18275 #11940995

Order 1280.

938 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3561C211yesmaximal, characteristic
2C5×#3113C211yesmaximal, characteristic, complemented
3C5×#3108C211yesmaximal, characteristic, complemented
4C5×#2592C211yesmaximal, characteristic, complemented
5C5×#1612C212yesmaximal, complemented
6C5×#1608C212yesmaximal, complemented
7C5×#1606C212yesmaximal, complemented
8C5×#1602C212yesmaximal, complemented
9C5×#1509C211yesmaximal, characteristic, complemented
10C5×#1503C211yesmaximal, characteristic, complemented
11C5×#1469C211yesmaximal, characteristic, complemented
12C5×#563—21no—
13C5×#562—21no—
14C5×#556—21no—
15C5×#551—22no—
16C5×D4×Q8—21no—
17C5×#548—21no—
18C5×#546—21no—
19C5×#543C2212yes—
20C5×#542—21no—
21C5×#542—22no—
22C5×#538C2212yes—
23C5×#536—21no—
24C5×#533C2211yescharacteristic
25C5×#529C2211yescharacteristic
26C2×C5×#124C2211yescharacteristic
27C2×C4×C5×Q8C2211yescharacteristic
28C5×#483C2211yescharacteristic
29C5×#483C2212yes—
30C5×#480C2211yescharacteristic
31C5×#479—21no—
32C5×#475—21no—
33C5×#464—21no—
34C5×#462—21no—
35C5×#451C2211yescharacteristic
36C5×#451C2212yes—
37C5×#448C2211yescharacteristic
38C5×#441—22no—
39C4×C5×Q16—21no—
40C4×C5×SD16—21no—
41C5×#432C2211yescharacteristic
42C5×#431C2211yescharacteristic
43C5×#397C2212yes—
44C5×#393C2212yes—
45C5×#391—21no—
46C5×#390—21no—
47C5×#388—22no—
48C4×C5×C4⋊C4C2211yescharacteristic
49C5×#328C2212yes—
50C5×#328C2212yes—
51C5×#328C2212yes—
52C5×#327C2212yes—
53C5×#323C2212yes—
54C5×#323C2212yes—
55#18275C511yesmaximal, characteristic, direct product factor, Sylow
56C2×C5×#41—21no—
57C22×C5×Q8—21no—
58C22×C5×Q8D411yescharacteristic
59C22×C5×Q8C2311yescharacteristic
60C22×C5×D4D411yescharacteristic
61C2×C5×Q16—21no—
62C2×C5×Q16—22no—
63C2×C5×SD16—21no—
64C5×C4⋊Q8—21no—
65C5×C4⋊Q8—22no—
66C5×C4⋊Q8D411yescharacteristic
67C5×C4⋊Q8D411yescharacteristic
68C5×#126—22no—
69C5×#126—22no—
70C5×#126—22no—
71C5×#126—21no—
72C5×#125—21no—
73C5×#125D411yescharacteristic
74C5×#124—21no—
75C5×#124—22no—
76C5×#124—22no—
77C5×#124—21no—
78C5×#124—21no—
79C5×#124—21no—
80C5×#124—21no—
81C5×#124D411yescharacteristic
82C5×#123—22no—
83C5×#123—22no—
84C5×#123—22no—
85C5×#123—21no—
86C5×#123—22no—
87C5×C22⋊Q8—22no—
88C5×C22⋊Q8—21no—
89C5×C22⋊Q8—22no—
90C5×C22⋊Q8—22no—
91C5×C22⋊Q8—22no—
92C5×C22⋊Q8—21no—
93C5×C22⋊Q8—22no—
94C5×C22⋊Q8—22no—
95C5×C22⋊Q8—22no—
96C5×C22⋊Q8C2311yescharacteristic
97C5×C22⋊Q8C2312yes—
98C5×#121—21no—
99C5×#121—22no—
100C5×#121—22no—
101C5×#121C2311yescharacteristic
102C5×#120—22no—
103C4×C5×Q8—21no—
104C4×C5×Q8—21no—
105C4×C5×Q8—21no—
106C4×C5×Q8—21no—
107C4×C5×Q8—21no—
108C4×C5×Q8—21no—
109C4×C5×Q8D411yescharacteristic
110C4×C5×Q8D412yes—
111C4×C5×D4—21no—
112C4×C5×D4—22no—
113C4×C5×D4—21no—
114C4×C5×D4—22no—
115C4×C5×D4D411yescharacteristic
116C5×#117—22no—
117C5×#117—21no—
118C5×#117—21no—
119C2×C5×C4⋊C4—21no—
120C2×C5×C4⋊C4—21no—
121C2×C5×C4⋊C4C2311yescharacteristic
122C2×C5×C22⋊C4—22no—
123C2×C5×C22⋊C4—22no—
124C2×C5×C22⋊C4—22no—
125C2×C5×C22⋊C4D412yes—
126C2×C5×C22⋊C4C2312yes—
127C2×C42×C5—21no—
128C2×C42×C5—21no—
129C2×C42×C5C2311yescharacteristic
130C5×#107—21no—
131C5×#106—21no—
132C5×#106—22no—
133C5×C4⋊C8—21no—
134C5×C4⋊C8—21no—
135C5×#103—21no—
136C5×#103—22no—
137C5×#103—21no—
138C5×#103—21no—
139C5×#103—22no—
140C5×#102—21no—
141C5×C22⋊C8C2312yes—
142C5×C22⋊C8C2311yescharacteristic
143C5×C22⋊C8C2311yescharacteristic
144C5×#97—21no—
145C4×C5×C8—21no—
146C5×#95—22no—
147C5×#95—22no—
148C5×#95C2312yes—
149#3561C2×C511yescharacteristic
150#3113C2×C511yescharacteristic, complemented
151#3108C2×C511yescharacteristic, complemented
152#2592C2×C511yescharacteristic, complemented
153#1612C2×C512yescomplemented
154#1608C2×C512yescomplemented
155#1606C2×C512yescomplemented
156#1602C2×C512yescomplemented
157#1509C2×C511yescharacteristic, complemented
158#1503C2×C511yescharacteristic, complemented
159#1469C2×C511yescharacteristic, complemented
160C24×C5C2×D412yes—
161C5×#41—82no—
162C2×C5×Q8—22no—
163C2×C5×Q8—21no—
164C2×C5×Q8—21no—
165C2×C5×Q8—21no—
166C2×C5×Q8—21no—
167C2×C5×Q8—21no—
168C2×C5×Q8—42no—
169C2×C5×Q8—42no—
170C2×C5×Q8—82no—
171C2×C5×Q8C2×D411yescharacteristic
172C2×C5×Q8C2×D412yes—
173C2×C5×D4—21no—
174C2×C5×D4—22no—
175C2×C5×D4—22no—
176C2×C5×D4—21no—
177C2×C5×D4—21no—
178C2×C5×D4—42no—
179C2×C5×D4C2×D411yescharacteristic
180C22×C4×C5—21no—
181C22×C4×C5—22no—
182C22×C4×C5—22no—
183C22×C4×C5—22no—
184C22×C4×C5—21no—
185C22×C4×C5—21no—
186C22×C4×C5—22no—
187C22×C4×C5C2×D411yescharacteristic
188C22×C4×C5C2×D411yescharacteristic
189C22×C4×C5C2×D412yes—
190C22×C4×C5C2411yescharacteristic
191C5×Q16—42no—
192C5×Q16—44no—
193C5×SD16—42no—
194C2×C5×C8—21no—
195C2×C5×C8—21no—
196C2×C5×C8—22no—
197C5×C4⋊C4—22no—
198C5×C4⋊C4—22no—
199C5×C4⋊C4—22no—
200C5×C4⋊C4—22no—
201C5×C4⋊C4—21no—
202C5×C4⋊C4—21no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4—21no—
205C5×C4⋊C4—21no—
206C5×C4⋊C4—21no—
207C5×C4⋊C4—22no—
208C5×C4⋊C4—21no—
209C5×C4⋊C4—22no—
210C5×C4⋊C4—21no—
211C5×C4⋊C4—21no—
212C5×C4⋊C4—42no—
213C5×C4⋊C4—41no—
214C5×C4⋊C4—41no—
215C5×C4⋊C4C2×D411yescharacteristic
216C5×C4⋊C4C2×D411yescharacteristic
217C5×C4⋊C4C2×D412yes—
218C5×C22⋊C4—22no—
219C5×C22⋊C4—22no—
220C5×C22⋊C4—22no—
221C5×C22⋊C4—22no—
222C5×C22⋊C4—22no—
223C5×C22⋊C4—21no—
224C5×C22⋊C4—22no—
225C5×C22⋊C4—22no—
226C5×C22⋊C4—22no—
227C5×C22⋊C4—21no—
228C5×C22⋊C4—21no—
229C5×C22⋊C4—21no—
230C5×C22⋊C4—22no—
231C5×C22⋊C4—22no—
232C5×C22⋊C4—21no—
233C5×C22⋊C4—44no—
234C5×C22⋊C4—82no—
235C42×C5—22no—
236C42×C5—22no—
237C42×C5—21no—
238C42×C5—41no—
239C42×C5—41no—
240C42×C5—81no—
241C42×C5C2×D411yescharacteristic
242C42×C5C2×D411yescharacteristic
243C42×C5C2×D411yescharacteristic
244C42×C5C2×D411yescharacteristic
245#563—21no—
246#562—21no—
247#556—21no—
248#551—22no—
249D4×Q8—21no—
250#548—21no—
251#546—21no—
252#543C22×C512yes—
253#542—21no—
254#542—22no—
255#538C22×C512yes—
256#536—21no—
257#533C22×C511yescharacteristic
258#529C22×C511yescharacteristic
259C2×#124C22×C511yescharacteristic
260C2×C4×Q8C22×C511yescharacteristic
261#483C22×C511yescharacteristic
262#483C22×C512yes—
263#480C22×C511yescharacteristic
264#479—21no—
265#475—21no—
266#464—21no—
267#462—21no—
268#451C22×C511yescharacteristic
269#451C22×C512yes—
270#448C22×C511yescharacteristic
271#441—22no—
272C4×Q16—21no—
273C4×SD16—21no—
274#432C22×C511yescharacteristic
275#431C22×C511yescharacteristic
276#397C22×C512yes—
277#393C22×C512yes—
278#391—21no—
279#390—21no—
280#388—22no—
281C4×C4⋊C4C22×C511yescharacteristic
282#328C22×C512yes—
283#328C22×C512yes—
284#328C22×C512yes—
285#327C22×C512yes—
286#323C22×C512yes—
287#323C22×C512yes—
288C23×C5—22no—
289C23×C5—21no—
290C23×C5—22no—
291C23×C5—22no—
292C23×C5—42no—
293C23×C5—42no—
294C23×C5C22×D411yescharacteristic
295C5×Q8—22no—
296C5×Q8—22no—
297C5×Q8—41no—
298C5×Q8—42no—
299C5×Q8—41no—
300C5×Q8—41no—
301C5×Q8—42no—
302C5×Q8—42no—
303C5×Q8—81no—
304C5×Q8—82no—
305C5×D4—41no—
306C5×D4—41no—
307C5×D4—41no—
308C5×D4—42no—
309C5×D4—81no—
310C5×D4—82no—
311C5×D4—82no—
312C2×C4×C5—21no—
313C2×C4×C5—21no—
314C2×C4×C5—22no—
315C2×C4×C5—22no—
316C2×C4×C5—21no—
317C2×C4×C5—22no—
318C2×C4×C5—21no—
319C2×C4×C5—22no—
320C2×C4×C5—22no—
321C2×C4×C5—21no—
322C2×C4×C5—41no—
323C2×C4×C5—41no—
324C2×C4×C5—41no—
325C2×C4×C5—42no—
326C2×C4×C5—42no—
327C2×C4×C5—42no—
328C2×C4×C5—42no—
329C2×C4×C5—41no—
330C2×C4×C5—42no—
331C2×C4×C5—41no—
332C2×C4×C5—41no—
333C2×C4×C5—82no—
334C2×C4×C5—82no—
335C2×C4×C5—82no—
336C2×C4×C5#12011yescharacteristic
337C2×C4×C5#12011yescharacteristic
338C2×C4×C5#12011yescharacteristic
339C2×C4×C5#12011yescharacteristic
340C2×C4×C5C22×D411yescharacteristic
341C2×C4×C5C22×D411yescharacteristic
342C5×C8—41no—
343C5×C8—41no—
344C5×C8—42no—
345C2×#41—21no—
346C22×Q8—21no—
347C22×Q8C5×D411yescharacteristic
348C22×Q8C23×C511yescharacteristic
349C22×D4C5×D411yescharacteristic
350C2×Q16—21no—
351C2×Q16—22no—
352C2×SD16—21no—
353C4⋊Q8—21no—
354C4⋊Q8—22no—
355C4⋊Q8C5×D411yescharacteristic
356C4⋊Q8C5×D411yescharacteristic
357#126—22no—
358#126—22no—
359#126—22no—
360#126—21no—
361#125—21no—
362#125C5×D411yescharacteristic
363#124—21no—
364#124—22no—
365#124—22no—
366#124—21no—
367#124—21no—
368#124—21no—
369#124—21no—
370#124C5×D411yescharacteristic
371#123—22no—
372#123—22no—
373#123—22no—
374#123—21no—
375#123—22no—
376C22⋊Q8—22no—
377C22⋊Q8—21no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C22⋊Q8—21no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8C23×C511yescharacteristic
386C22⋊Q8C23×C512yes—
387#121—21no—
388#121—22no—
389#121—22no—
390#121C23×C511yescharacteristic
391#120—22no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8C5×D411yescharacteristic
399C4×Q8C5×D412yes—
400C4×D4—21no—
401C4×D4—22no—
402C4×D4—21no—
403C4×D4—22no—
404C4×D4C5×D411yescharacteristic
405#117—22no—
406#117—21no—
407#117—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4C23×C511yescharacteristic
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4C5×D412yes—
415C2×C22⋊C4C23×C512yes—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42C23×C511yescharacteristic
419#107—21no—
420#106—21no—
421#106—22no—
422C4⋊C8—21no—
423C4⋊C8—21no—
424#103—21no—
425#103—22no—
426#103—21no—
427#103—21no—
428#103—22no—
429#102—21no—
430C22⋊C8C23×C512yes—
431C22⋊C8C23×C511yescharacteristic
432C22⋊C8C23×C511yescharacteristic
433#97—21no—
434C4×C8—21no—
435#95—22no—
436#95—22no—
437#95C23×C512yes—
438C22×C5—21no—
439C22×C5—21no—
440C22×C5—41no—
441C22×C5—42no—
442C22×C5—41no—
443C22×C5—42no—
444C22×C5—42no—
445C22×C5—41no—
446C22×C5—82no—
447C22×C5—82no—
448C22×C5C2×#12011yescharacteristic
449C22×C5#57711yescharacteristic
450C22×C5#57811yescharacteristic
451C4×C5—21no—
452C4×C5—21no—
453C4×C5—21no—
454C4×C5—21no—
455C4×C5—21no—
456C4×C5—41no—
457C4×C5—41no—
458C4×C5—42no—
459C4×C5—42no—
460C4×C5—42no—
461C4×C5—41no—
462C4×C5—42no—
463C4×C5—82no—
464C4×C5—81no—
465C4×C5#45512yes—
466C24C2×C5×D412yes—
467#41—82no—
468C2×Q8—22no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—42no—
475C2×Q8—42no—
476C2×Q8—82no—
477C2×Q8C2×C5×D411yescharacteristic
478C2×Q8C2×C5×D412yes—
479C2×D4—21no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—42no—
485C2×D4C2×C5×D411yescharacteristic
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4C2×C5×D411yescharacteristic
494C22×C4C2×C5×D411yescharacteristic
495C22×C4C2×C5×D412yes—
496C22×C4C24×C511yescharacteristic
497Q16—42no—
498Q16—44no—
499SD16—42no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—42no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4C2×C5×D411yescharacteristic
522C4⋊C4C2×C5×D412yes—
523C4⋊C4C2×C5×D411yescharacteristic
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—21no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—44no—
540C22⋊C4—82no—
541C42—22no—
542C42—22no—
543C42—21no—
544C42—41no—
545C42—41no—
546C42—81no—
547C42C2×C5×D411yescharacteristic
548C42C2×C5×D411yescharacteristic
549C42C2×C5×D411yescharacteristic
550C42C2×C5×D411yescharacteristic
551C2×C5—21no—
552C2×C5—21no—
553C2×C5—81no—
554C2×C5—82no—
555C2×C5#301311yescharacteristic
556C2×C5C2×#45511yescharacteristic
557C2×C5#302511yescharacteristic
558C23—22no—
559C23—21no—
560C23—22no—
561C23—22no—
562C23—42no—
563C23—42no—
564C23C22×C5×D411yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—42no—
569Q8—41no—
570Q8—41no—
571Q8—42no—
572Q8—42no—
573Q8—81no—
574Q8—82no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—42no—
579D4—81no—
580D4—82no—
581D4—82no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4C5×#12011yescharacteristic
607C2×C4C5×#12011yescharacteristic
608C2×C4C5×#12011yescharacteristic
609C2×C4C5×#12011yescharacteristic
610C2×C4C22×C5×D411yescharacteristic
611C2×C4C22×C5×D411yescharacteristic
612C8—41no—
613C8—41no—
614C8—42no—
615C5#1827511yescharacteristic, direct product factor, Sylow
616C22—21no—
617C22—21no—
618C22—41no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—41no—
624C22—82no—
625C22—82no—
626C22C2×C5×#12011yescharacteristic
627C22C5×#57711yescharacteristic
628C22C5×#57811yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—41no—
640C4—42no—
641C4—82no—
642C4—81no—
643C4C5×#45512yes—
644C2—21no—
645C2—21no—
646C2—82no—
647C2—81no—
648C2C5×#301311yescharacteristic
649C2C2×C5×#45511yescharacteristic
650C2C5×#302511yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC5×#3561C2
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×#3113C2semidirect
3. C5×#3108C2semidirect
4. C5×#2592C2semidirect
5. C5×#1612C2semidirect
6. C5×#1608C2semidirect
7. C5×#1606C2semidirect
8. C5×#1602C2semidirect
9. C5×#1509C2semidirect
10. C5×#1503C2semidirect
11. C5×#1469C2semidirect
55. #18275C5direct (N × H)
150. #3113C2×C5semidirect
151. #3108C2×C5semidirect
152. #2592C2×C5semidirect
153. #1612C2×C5semidirect
154. #1608C2×C5semidirect
155. #1606C2×C5semidirect
156. #1602C2×C5semidirect
157. #1509C2×C5semidirect
158. #1503C2×C5semidirect
159. #1469C2×C5semidirect
615. C5#18275direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35612
C5×#31132
C5×#31082
C5×#25922
C5×#16122
C5×#16082
C5×#16062
C5×#16022
C5×#15092
C5×#15032
C5×#14692
#182755