Finite Group Databasebeta

← C5×#18572 #11941292

Subgroups — C5×#18572 #11941292

Order 1280.

954 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3522C211yesmaximal, characteristic
2C5×#3239C211yesmaximal, characteristic, complemented
3C5×#3141C211yesmaximal, characteristic, complemented
4C2×C5×#462C211yesmaximal, characteristic, complemented
5C5×#1645C212yesmaximal, complemented
6C5×#1644C212yesmaximal, complemented
7C5×#1641C212yesmaximal, complemented
8C5×#1640C212yesmaximal, complemented
9C5×#1551C211yesmaximal, characteristic, complemented
10C5×#1491C211yesmaximal, characteristic, complemented
11C5×#1483C211yesmaximal, characteristic, complemented
12C22×C5×Q16—21no—
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×#489—21no—
33C5×#489—22no—
34C5×#487C2212yes—
35C5×#486C2212yes—
36C5×#484—21no—
37C5×#481—21no—
38C5×#475C2212yes—
39C5×#474C2212yes—
40C5×#462—21no—
41C5×#462—21no—
42C5×#462C2212yes—
43C5×#462C2212yes—
44C5×#432C2211yescharacteristic
45C2×C5×C4⋊C8C2211yescharacteristic
46C5×#417—22no—
47C2×C5×#103—21no—
48C4×C5×M16C2211yescharacteristic
49C5×#333C2212yes—
50C5×#331C2212yes—
51C5×Q8⋊C8C2212yes—
52C5×Q8⋊C8C2212yes—
53C5×Q8⋊C8C2212yes—
54C5×D4⋊C8C2212yes—
55#18572C511yesmaximal, characteristic, direct product factor, Sylow
56C2×C5×#41—22no—
57C2×C5×#41—21no—
58C2×C5×#41D412yes—
59C22×C5×Q8—21no—
60C22×C5×Q8D411yescharacteristic
61C2×C5×Q16—22no—
62C2×C5×Q16—22no—
63C2×C5×Q16—44no—
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×#106—22no—
135C5×C4⋊C8—21no—
136C5×C4⋊C8C2312yes—
137C5×C4⋊C8C2312yes—
138C5×#103—24no—
139C5×#103—22no—
140C5×#103—22no—
141C5×#103—22no—
142C5×#102—24no—
143C5×#102—22no—
144C5×C22⋊C8—21no—
145C5×#97—21no—
146C5×#97—21no—
147C4×C5×C8C2312yes—
148#3522C2×C511yescharacteristic
149#3239C2×C511yescharacteristic, complemented
150#3141C2×C511yescharacteristic, complemented
151C2×#462C2×C511yescharacteristic, complemented
152#1645C2×C512yescomplemented
153#1644C2×C512yescomplemented
154#1641C2×C512yescomplemented
155#1640C2×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—42no—
169C2×C5×Q8—42no—
170C2×C5×Q8—41no—
171C2×C5×Q8C2×D411yescharacteristic
172C2×C5×Q8C2×D411yescharacteristic
173C2×C5×Q8C2×D411yescharacteristic
174C2×C5×Q8C2×D412yes—
175C2×C5×D4—22no—
176C2×C5×D4—22no—
177C2×C5×D4—21no—
178C2×C5×D4—22no—
179C2×C5×D4—21no—
180C2×C5×D4—21no—
181C2×C5×D4C2×D411yescharacteristic
182C2×C5×D4C2×D412yes—
183C22×C4×C5—22no—
184C22×C4×C5—21no—
185C22×C4×C5—21no—
186C22×C4×C5—22no—
187C22×C4×C5—22no—
188C22×C4×C5—21no—
189C22×C4×C5—22no—
190C22×C4×C5—41no—
191C22×C4×C5C2×D411yescharacteristic
192C22×C4×C5C2×D411yescharacteristic
193C22×C4×C5C2×D411yescharacteristic
194C5×Q16—44no—
195C5×Q16—44no—
196C5×M16—42no—
197C2×C5×C8—22no—
198C2×C5×C8—22no—
199C2×C5×C8—22no—
200C2×C5×C8—42no—
201C5×C4⋊C4—22no—
202C5×C4⋊C4—22no—
203C5×C4⋊C4—22no—
204C5×C4⋊C4—22no—
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—21no—
216C5×C4⋊C4—22no—
217C5×C4⋊C4—22no—
218C5×C4⋊C4—41no—
219C5×C4⋊C4—41no—
220C5×C4⋊C4C2×D411yescharacteristic
221C5×C4⋊C4C2×D411yescharacteristic
222C5×C4⋊C4C2×D411yescharacteristic
223C5×C4⋊C4C2×D411yescharacteristic
224C5×C22⋊C4—22no—
225C5×C22⋊C4—22no—
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—22no—
234C5×C22⋊C4—21no—
235C5×C22⋊C4—21no—
236C42×C5—22no—
237C42×C5—21no—
238C42×C5—21no—
239C42×C5—21no—
240C42×C5—21no—
241C42×C5—21no—
242C42×C5C2×D411yescharacteristic
243C42×C5C2×D411yescharacteristic
244C42×C5C2×D411yescharacteristic
245C42×C5C2411yescharacteristic
246C22×Q16—21no—
247#559—21no—
248#559C22×C511yescharacteristic
249Q82C22×C511yescharacteristic
250#554—21no—
251#554C22×C511yescharacteristic
252#550C22×C511yescharacteristic
253#548—21no—
254#547—22no—
255#547—21no—
256#542—22no—
257#542—21no—
258#532—21no—
259#532C22×C511yescharacteristic
260C2×C4⋊Q8C22×C511yescharacteristic
261#529—21no—
262#529C22×C511yescharacteristic
263#520—21no—
264C2×C4×Q8C22×C511yescharacteristic
265#501—21no—
266#489—22no—
267#489—21no—
268#487C22×C512yes—
269#486C22×C512yes—
270#484—21no—
271#481—21no—
272#475C22×C512yes—
273#474C22×C512yes—
274#462—21no—
275#462—21no—
276#462C22×C512yes—
277#462C22×C512yes—
278#432C22×C511yescharacteristic
279C2×C4⋊C8C22×C511yescharacteristic
280#417—22no—
281C2×#103—21no—
282C4×M16C22×C511yescharacteristic
283#333C22×C512yes—
284#331C22×C512yes—
285Q8⋊C8C22×C512yes—
286Q8⋊C8C22×C512yes—
287Q8⋊C8C22×C512yes—
288D4⋊C8C22×C512yes—
289C23×C5—21no—
290C23×C5—22no—
291C23×C5#12011yescharacteristic
292C5×Q8—21no—
293C5×Q8—21no—
294C5×Q8—21no—
295C5×Q8—22no—
296C5×Q8—42no—
297C5×Q8—42no—
298C5×Q8—42no—
299C5×Q8—42no—
300C5×Q8—82no—
301C5×Q8—82no—
302C5×Q8AGL(1,ℤ/8)12yes—
303C5×D4—22no—
304C5×D4—42no—
305C5×D4—44no—
306C5×D4—44no—
307C5×D4—42no—
308C5×D4—41no—
309C5×D4—41no—
310C5×D4—42no—
311C5×D4—82no—
312C2×C4×C5—21no—
313C2×C4×C5—22no—
314C2×C4×C5—21no—
315C2×C4×C5—21no—
316C2×C4×C5—22no—
317C2×C4×C5—21no—
318C2×C4×C5—21no—
319C2×C4×C5—21no—
320C2×C4×C5—21no—
321C2×C4×C5—22no—
322C2×C4×C5—21no—
323C2×C4×C5—21no—
324C2×C4×C5—42no—
325C2×C4×C5—44no—
326C2×C4×C5—42no—
327C2×C4×C5—41no—
328C2×C4×C5—41no—
329C2×C4×C5—42no—
330C2×C4×C5—82no—
331C2×C4×C5—82no—
332C2×C4×C5—82no—
333C2×C4×C5—82no—
334C2×C4×C5#12011yescharacteristic
335C2×C4×C5#12011yescharacteristic
336C2×C4×C5#12011yescharacteristic
337C2×C4×C5C22×D411yescharacteristic
338C2×C4×C5C22×D411yescharacteristic
339C2×C4×C5C22×D411yescharacteristic
340C5×C8—42no—
341C5×C8—42no—
342C5×C8—42no—
343C2×#41—22no—
344C2×#41—21no—
345C2×#41C5×D412yes—
346C22×Q8—21no—
347C22×Q8C5×D411yescharacteristic
348C2×Q16—22no—
349C2×Q16—22no—
350C2×Q16—44no—
351C2×M16—21no—
352C22×C8—21no—
353C4⋊Q8—22no—
354C4⋊Q8—21no—
355C4⋊Q8—21no—
356C4⋊Q8—21no—
357C4⋊Q8—21no—
358C4⋊Q8—21no—
359C4⋊Q8C23×C512yes—
360C4⋊Q8C23×C511yescharacteristic
361#127—21no—
362#127C23×C511yescharacteristic
363#126—22no—
364#126—21no—
365#125—21no—
366#125C5×D411yescharacteristic
367#124—22no—
368#124—22no—
369#124—21no—
370#124—21no—
371#124C5×D411yescharacteristic
372#124C5×D412yes—
373#123—22no—
374#123—22no—
375#123—22no—
376#123—22no—
377#123—21no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382C22⋊Q8—21no—
383C22⋊Q8C5×D411yescharacteristic
384#121—22no—
385#121—22no—
386#121—21no—
387#121—21no—
388#121—21no—
389#121C5×D411yescharacteristic
390C4×Q8—21no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8C5×D411yescharacteristic
395C4×Q8C5×D411yescharacteristic
396C4×Q8C23×C511yescharacteristic
397C4×Q8C23×C511yescharacteristic
398C4×Q8C23×C511yescharacteristic
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—22no—
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4C23×C511yescharacteristic
410#117—22no—
411#117—21no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4C5×D411yescharacteristic
418C2×C42—21no—
419C2×C42C23×C511yescharacteristic
420#107—22no—
421#106—22no—
422C4⋊C8—21no—
423C4⋊C8C23×C512yes—
424C4⋊C8C23×C512yes—
425#103—24no—
426#103—22no—
427#103—22no—
428#103—22no—
429#102—24no—
430#102—22no—
431C22⋊C8—21no—
432#97—21no—
433#97—21no—
434C4×C8C23×C512yes—
435C22×C5—21no—
436C22×C5—21no—
437C22×C5—42no—
438C22×C5—41no—
439C22×C5—41no—
440C22×C5—42no—
441C22×C5—42no—
442C22×C5—41no—
443C22×C5#45512yes—
444C22×C5C2×#12011yescharacteristic
445C4×C5—21no—
446C4×C5—22no—
447C4×C5—21no—
448C4×C5—21no—
449C4×C5—21no—
450C4×C5—41no—
451C4×C5—41no—
452C4×C5—41no—
453C4×C5—41no—
454C4×C5—41no—
455C4×C5—42no—
456C4×C5—41no—
457C4×C5—82no—
458C4×C5C2×AGL(1,ℤ/8)11yescharacteristic
459C4×C5#57511yescharacteristic
460C4×C5#57812yes—
461#41—44no—
462#41—42no—
463#41—42no—
464#41—84no—
465C2×Q8—22no—
466C2×Q8—21no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—41no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—41no—
473C2×Q8C2×C5×D411yescharacteristic
474C2×Q8C2×C5×D411yescharacteristic
475C2×Q8C2×C5×D411yescharacteristic
476C2×Q8C2×C5×D412yes—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—21no—
480C2×D4—22no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4C2×C5×D411yescharacteristic
484C2×D4C2×C5×D412yes—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—41no—
493C22×C4C2×C5×D411yescharacteristic
494C22×C4C2×C5×D411yescharacteristic
495C22×C4C2×C5×D411yescharacteristic
496Q16—44no—
497Q16—44no—
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—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—41no—
521C4⋊C4—41no—
522C4⋊C4C2×C5×D411yescharacteristic
523C4⋊C4C2×C5×D411yescharacteristic
524C4⋊C4C2×C5×D411yescharacteristic
525C4⋊C4C2×C5×D411yescharacteristic
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—21no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C42—22no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42C2×C5×D411yescharacteristic
545C42C2×C5×D411yescharacteristic
546C42C2×C5×D411yescharacteristic
547C42C24×C511yescharacteristic
548C2×C5—22no—
549C2×C5—81no—
550C2×C5—82no—
551C2×C5#300611yescharacteristic
552C2×C5#301111yescharacteristic
553C2×C5C2×#45511yescharacteristic
554C23—21no—
555C23—22no—
556C23C5×#12011yescharacteristic
557Q8—21no—
558Q8—21no—
559Q8—21no—
560Q8—22no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—82no—
566Q8—82no—
567Q8C5×AGL(1,ℤ/8)12yes—
568D4—22no—
569D4—42no—
570D4—44no—
571D4—44no—
572D4—42no—
573D4—41no—
574D4—41no—
575D4—42no—
576D4—82no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4C5×#12011yescharacteristic
600C2×C4C5×#12011yescharacteristic
601C2×C4C5×#12011yescharacteristic
602C2×C4C22×C5×D411yescharacteristic
603C2×C4C22×C5×D411yescharacteristic
604C2×C4C22×C5×D411yescharacteristic
605C8—42no—
606C8—42no—
607C8—42no—
608C5#1857211yescharacteristic, direct product factor, Sylow
609C22—21no—
610C22—21no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22C5×#45512yes—
618C22C2×C5×#12011yescharacteristic
619C4—21no—
620C4—22no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—82no—
632C4C2×C5×AGL(1,ℤ/8)11yescharacteristic
633C4C5×#57511yescharacteristic
634C4C5×#57812yes—
635C2—22no—
636C2—81no—
637C2—82no—
638C2C5×#300611yescharacteristic
639C2C5×#301111yescharacteristic
640C2C2×C5×#45511yescharacteristic
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×#3239C2semidirect
3. C5×#3141C2semidirect
4. C2×C5×#462C2semidirect
5. C5×#1645C2semidirect
6. C5×#1644C2semidirect
7. C5×#1641C2semidirect
8. C5×#1640C2semidirect
9. C5×#1551C2semidirect
10. C5×#1491C2semidirect
11. C5×#1483C2semidirect
55. #18572C5direct (N × H)
149. #3239C2×C5semidirect
150. #3141C2×C5semidirect
151. C2×#462C2×C5semidirect
152. #1645C2×C5semidirect
153. #1644C2×C5semidirect
154. #1641C2×C5semidirect
155. #1640C2×C5semidirect
156. #1551C2×C5semidirect
157. #1491C2×C5semidirect
158. #1483C2×C5semidirect
608. C5#18572direct (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×#32392
C5×#31412
C2×C5×#4622
C5×#16452
C5×#16442
C5×#16412
C5×#16402
C5×#15512
C5×#14912
C5×#14832
#185725