Finite Group Databasebeta

← C5×#18587 #11941307

Subgroups — C5×#18587 #11941307

Order 1280.

944 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×#3520C211yesmaximal, characteristic
2C5×#3218C211yesmaximal, characteristic, complemented
3C5×#3165C211yesmaximal, characteristic, complemented
4C2×C5×#474C211yesmaximal, characteristic, complemented
5C5×#1654C212yesmaximal, complemented
6C5×#1652C212yesmaximal, complemented
7C5×#1649C212yesmaximal, complemented
8C5×#1647C212yesmaximal, complemented
9C5×#1552C211yesmaximal, characteristic, complemented
10C5×#1490C211yesmaximal, characteristic, complemented
11C5×#1483C211yesmaximal, characteristic, complemented
12C5×#556C2211yescharacteristic
13C5×#554—21no—
14C5×#552—21no—
15C5×#551—22no—
16C5×D4×Q8C2211yescharacteristic
17C5×#548—21no—
18C5×#547C2211yescharacteristic
19C5×#547C2211yescharacteristic
20C5×#546—21no—
21C5×#542—21no—
22C5×#539—22no—
23C5×#533C2211yescharacteristic
24C2×C5×C4⋊Q8C2211yescharacteristic
25C5×#529C2211yescharacteristic
26C5×#525—21no—
27C2×C5×C22⋊Q8—21no—
28C5×#520—21no—
29C2×C4×C5×D4C2211yescharacteristic
30C5×#501—22no—
31C5×#500C2212yescomplemented
32C5×#498C2212yescomplemented
33C5×#497—21no—
34C5×#491—21no—
35C5×#474—21no—
36C5×#474—21no—
37C5×#474C2212yes—
38C5×#474C2212yes—
39C5×#462C2212yescomplemented
40C5×#461C2212yescomplemented
41C5×#451—21no—
42C5×#448—21no—
43C5×#432C2211yescharacteristic
44C5×#428—22no—
45C2×C5×#107—21no—
46C2×C5×C4⋊C8C2211yescharacteristic
47C2×C5×#102—21no—
48C4×C5×M16C2211yescharacteristic
49C5×#332C2212yescomplemented
50C5×#332C2212yescomplemented
51C5×Q8⋊C8C2212yescomplemented
52C5×D4⋊C8C2212yes—
53C5×D4⋊C8C2212yes—
54C5×D4⋊C8C2212yescomplemented
55#18587C511yesmaximal, characteristic, direct product factor, Sylow
56C2×C5×#41—21no—
57C2×C5×#41—21no—
58C22×C5×Q8—21no—
59C22×C5×Q8—21no—
60C22×C5×D4D411yescharacteristic
61C23×C4×C5—21no—
62C2×C5×Q16—22no—
63C2×C5×SD16—22no—
64C2×C5×M16—21no—
65C22×C5×C8—21no—
66C5×C4⋊Q8—21no—
67C5×C4⋊Q8—21no—
68C5×C4⋊Q8—21no—
69C5×C4⋊Q8—21no—
70C5×C4⋊Q8C2312yes—
71C5×C4⋊Q8C2311yescharacteristic
72C5×C4⋊Q8C2311yescharacteristic
73C5×#126—22no—
74C5×#126—22no—
75C5×#126—21no—
76C5×#125—22no—
77C5×#125D411yescharacteristic
78C5×#125D412yes—
79C5×#124—22no—
80C5×#124—21no—
81C5×#124D411yescharacteristic
82C5×#123—22no—
83C5×#123—22no—
84C5×#123—21no—
85C5×#123—22no—
86C5×#123—22no—
87C5×C22⋊Q8—22no—
88C5×C22⋊Q8—22no—
89C5×C22⋊Q8—22no—
90C5×C22⋊Q8—21no—
91C5×C22⋊Q8—21no—
92C5×C22⋊Q8—21no—
93C5×C22⋊Q8—21no—
94C5×C22⋊Q8—21no—
95C5×C22⋊Q8—22no—
96C5×C22⋊Q8—21no—
97C5×C22⋊Q8D411yescharacteristic
98C5×#121—22no—
99C5×#121—21no—
100C5×#121—21no—
101C5×#121D411yescharacteristic
102C5×#120—22no—
103C4×C5×Q8—21no—
104C4×C5×Q8—21no—
105C4×C5×Q8—21no—
106C4×C5×Q8C2311yescharacteristic
107C4×C5×D4—22no—
108C4×C5×D4—22no—
109C4×C5×D4—22no—
110C4×C5×D4—21no—
111C4×C5×D4—21no—
112C4×C5×D4—21no—
113C4×C5×D4—21no—
114C4×C5×D4D411yescharacteristic
115C4×C5×D4D411yescharacteristic
116C4×C5×D4C2311yescharacteristic
117C4×C5×D4C2311yescharacteristic
118C4×C5×D4C2311yescharacteristic
119C5×#117—22no—
120C5×#117—21no—
121C5×#117—21no—
122C5×#117D412yes—
123C2×C5×C4⋊C4—22no—
124C2×C5×C4⋊C4—21no—
125C2×C5×C4⋊C4—21no—
126C2×C5×C4⋊C4—21no—
127C2×C5×C4⋊C4—21no—
128C2×C5×C4⋊C4D411yescharacteristic
129C2×C5×C22⋊C4—22no—
130C2×C5×C22⋊C4—21no—
131C2×C42×C5C2311yescharacteristic
132C5×#107—22no—
133C5×#107—22no—
134C5×#107—24no—
135C5×#106—24no—
136C5×C4⋊C8—21no—
137C5×C4⋊C8C2312yes—
138C5×C4⋊C8C2312yes—
139C5×#103—22no—
140C5×#103—22no—
141C5×#102—22no—
142C5×#102—22no—
143C5×C22⋊C8—21no—
144C5×#97—21no—
145C5×#97—21no—
146C4×C5×C8C2312yes—
147#3520C2×C511yescharacteristic
148#3218C2×C511yescharacteristic, complemented
149#3165C2×C511yescharacteristic, complemented
150C2×#474C2×C511yescharacteristic, complemented
151#1654C2×C512yescomplemented
152#1652C2×C512yescomplemented
153#1649C2×C512yescomplemented
154#1647C2×C512yescomplemented
155#1552C2×C511yescharacteristic, complemented
156#1490C2×C511yescharacteristic, complemented
157#1483C2×C511yescharacteristic, complemented
158C24×C5—21no—
159C5×#41—42no—
160C5×#41—42no—
161C5×#41—82no—
162C2×C5×Q8—22no—
163C2×C5×Q8—21no—
164C2×C5×Q8—21no—
165C2×C5×Q8—22no—
166C2×C5×Q8—41no—
167C2×C5×Q8—82no—
168C2×C5×Q8—82no—
169C2×C5×Q8C2×D411yescharacteristic
170C2×C5×D4—22no—
171C2×C5×D4—22no—
172C2×C5×D4—21no—
173C2×C5×D4—22no—
174C2×C5×D4—21no—
175C2×C5×D4—41no—
176C2×C5×D4C2×D411yescharacteristic
177C2×C5×D4C2×D411yescharacteristic
178C2×C5×D4C2×D411yescharacteristic
179C22×C4×C5—22no—
180C22×C4×C5—22no—
181C22×C4×C5—22no—
182C22×C4×C5—21no—
183C22×C4×C5—21no—
184C22×C4×C5—21no—
185C22×C4×C5—21no—
186C22×C4×C5—21no—
187C22×C4×C5—21no—
188C22×C4×C5—41no—
189C22×C4×C5—82no—
190C22×C4×C5C2×D411yescharacteristic
191C22×C4×C5C2×D411yescharacteristic
192C22×C4×C5C2×D411yescharacteristic
193C5×Q16—82no—
194C5×SD16—82no—
195C5×M16—24no—
196C2×C5×C8—22no—
197C2×C5×C8—22no—
198C2×C5×C8—22no—
199C2×C5×C8—42no—
200C5×C4⋊C4—22no—
201C5×C4⋊C4—22no—
202C5×C4⋊C4—22no—
203C5×C4⋊C4—22no—
204C5×C4⋊C4—21no—
205C5×C4⋊C4—22no—
206C5×C4⋊C4—21no—
207C5×C4⋊C4—21no—
208C5×C4⋊C4—21no—
209C5×C4⋊C4—22no—
210C5×C4⋊C4—21no—
211C5×C4⋊C4—21no—
212C5×C4⋊C4—21no—
213C5×C4⋊C4—22no—
214C5×C4⋊C4—22no—
215C5×C4⋊C4—41no—
216C5×C4⋊C4—41no—
217C5×C4⋊C4C2×D411yescharacteristic
218C5×C4⋊C4C2×D411yescharacteristic
219C5×C4⋊C4C2×D411yescharacteristic
220C5×C4⋊C4C2×D411yescharacteristic
221C5×C4⋊C4C2×D412yes—
222C5×C4⋊C4C2×D412yes—
223C5×C22⋊C4—22no—
224C5×C22⋊C4—22no—
225C5×C22⋊C4—22no—
226C5×C22⋊C4—22no—
227C5×C22⋊C4—22no—
228C5×C22⋊C4—21no—
229C5×C22⋊C4—22no—
230C5×C22⋊C4—21no—
231C5×C22⋊C4—21no—
232C5×C22⋊C4—21no—
233C5×C22⋊C4—21no—
234C5×C22⋊C4—21no—
235C5×C22⋊C4—21no—
236C5×C22⋊C4—21no—
237C5×C22⋊C4—21no—
238C42×C5—22no—
239C42×C5—22no—
240C42×C5—21no—
241C42×C5C2×D411yescharacteristic
242C42×C5C2×D411yescharacteristic
243C42×C5C2×D411yescharacteristic
244C42×C5C2411yescharacteristic
245#556C22×C511yescharacteristic
246#554—21no—
247#552—21no—
248#551—22no—
249D4×Q8C22×C511yescharacteristic
250#548—21no—
251#547C22×C511yescharacteristic
252#547C22×C511yescharacteristic
253#546—21no—
254#542—21no—
255#539—22no—
256#533C22×C511yescharacteristic
257C2×C4⋊Q8C22×C511yescharacteristic
258#529C22×C511yescharacteristic
259#525—21no—
260C2×C22⋊Q8—21no—
261#520—21no—
262C2×C4×D4C22×C511yescharacteristic
263#501—22no—
264#500C22×C512yescomplemented
265#498C22×C512yescomplemented
266#497—21no—
267#491—21no—
268#474—21no—
269#474—21no—
270#474C22×C512yes—
271#474C22×C512yes—
272#462C22×C512yescomplemented
273#461C22×C512yescomplemented
274#451—21no—
275#448—21no—
276#432C22×C511yescharacteristic
277#428—22no—
278C2×#107—21no—
279C2×C4⋊C8C22×C511yescharacteristic
280C2×#102—21no—
281C4×M16C22×C511yescharacteristic
282#332C22×C512yescomplemented
283#332C22×C512yescomplemented
284Q8⋊C8C22×C512yescomplemented
285D4⋊C8C22×C512yes—
286D4⋊C8C22×C512yes—
287D4⋊C8C22×C512yescomplemented
288C23×C5—21no—
289C23×C5—21no—
290C23×C5—21no—
291C23×C5—42no—
292C23×C5—42no—
293C23×C5—81no—
294C23×C5#12011yescharacteristic
295C5×Q8—22no—
296C5×Q8—41no—
297C5×Q8—41no—
298C5×Q8—42no—
299C5×Q8—42no—
300C5×Q8—42no—
301C5×Q8—82no—
302C5×Q8—82no—
303C5×D4—21no—
304C5×D4—21no—
305C5×D4—21no—
306C5×D4—22no—
307C5×D4—42no—
308C5×D4—44no—
309C5×D4—42no—
310C5×D4—82no—
311C5×D4#13712yes—
312C2×C4×C5—22no—
313C2×C4×C5—21no—
314C2×C4×C5—21no—
315C2×C4×C5—22no—
316C2×C4×C5—21no—
317C2×C4×C5—22no—
318C2×C4×C5—21no—
319C2×C4×C5—21no—
320C2×C4×C5—22no—
321C2×C4×C5—21no—
322C2×C4×C5—21no—
323C2×C4×C5—44no—
324C2×C4×C5—41no—
325C2×C4×C5—41no—
326C2×C4×C5—42no—
327C2×C4×C5—42no—
328C2×C4×C5—42no—
329C2×C4×C5—81no—
330C2×C4×C5—81no—
331C2×C4×C5—81no—
332C2×C4×C5—82no—
333C2×C4×C5—82no—
334C2×C4×C5—82no—
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—24no—
342C5×C8—42no—
343C5×C8—42no—
344C2×#41—21no—
345C2×#41—21no—
346C22×Q8—21no—
347C22×Q8—21no—
348C22×D4C5×D411yescharacteristic
349C23×C4—21no—
350C2×Q16—22no—
351C2×SD16—22no—
352C2×M16—21no—
353C22×C8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—21no—
356C4⋊Q8—21no—
357C4⋊Q8—21no—
358C4⋊Q8C23×C512yes—
359C4⋊Q8C23×C511yescharacteristic
360C4⋊Q8C23×C511yescharacteristic
361#126—22no—
362#126—22no—
363#126—21no—
364#125—22no—
365#125C5×D411yescharacteristic
366#125C5×D412yes—
367#124—22no—
368#124—21no—
369#124C5×D411yescharacteristic
370#123—22no—
371#123—22no—
372#123—21no—
373#123—22no—
374#123—22no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—22no—
384C22⋊Q8—21no—
385C22⋊Q8C5×D411yescharacteristic
386#121—22no—
387#121—21no—
388#121—21no—
389#121C5×D411yescharacteristic
390#120—22no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8C23×C511yescharacteristic
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—22no—
398C4×D4—21no—
399C4×D4—21no—
400C4×D4—21no—
401C4×D4—21no—
402C4×D4C5×D411yescharacteristic
403C4×D4C5×D411yescharacteristic
404C4×D4C23×C511yescharacteristic
405C4×D4C23×C511yescharacteristic
406C4×D4C23×C511yescharacteristic
407#117—22no—
408#117—21no—
409#117—21no—
410#117C5×D412yes—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4C5×D411yescharacteristic
417C2×C22⋊C4—22no—
418C2×C22⋊C4—21no—
419C2×C42C23×C511yescharacteristic
420#107—22no—
421#107—22no—
422#107—24no—
423#106—24no—
424C4⋊C8—21no—
425C4⋊C8C23×C512yes—
426C4⋊C8C23×C512yes—
427#103—22no—
428#103—22no—
429#102—22no—
430#102—22no—
431C22⋊C8—21no—
432#97—21no—
433#97—21no—
434C4×C8C23×C512yes—
435C22×C5—21no—
436C22×C5—21no—
437C22×C5—22no—
438C22×C5—41no—
439C22×C5—41no—
440C22×C5—41no—
441C22×C5—41no—
442C22×C5—41no—
443C22×C5—41no—
444C22×C5—41no—
445C22×C5—41no—
446C22×C5—84no—
447C22×C5#45612yes—
448C22×C5C2×#12011yescharacteristic
449C4×C5—21no—
450C4×C5—21no—
451C4×C5—21no—
452C4×C5—21no—
453C4×C5—41no—
454C4×C5—42no—
455C4×C5—41no—
456C4×C5—41no—
457C4×C5—42no—
458C4×C5—41no—
459C4×C5—41no—
460C4×C5—82no—
461C4×C5C2×#13711yescharacteristic
462C4×C5#57511yescharacteristic
463C4×C5#57612yes—
464C24—21no—
465#41—42no—
466#41—42no—
467#41—82no—
468C2×Q8—22no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—22no—
472C2×Q8—41no—
473C2×Q8—82no—
474C2×Q8—82no—
475C2×Q8C2×C5×D411yescharacteristic
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—21no—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4—41no—
482C2×D4C2×C5×D411yescharacteristic
483C2×D4C2×C5×D411yescharacteristic
484C2×D4C2×C5×D411yescharacteristic
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—41no—
495C22×C4—82no—
496C22×C4C2×C5×D411yescharacteristic
497C22×C4C2×C5×D411yescharacteristic
498C22×C4C2×C5×D411yescharacteristic
499Q16—82no—
500SD16—82no—
501M16—24no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—42no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4C2×C5×D411yescharacteristic
524C4⋊C4C2×C5×D411yescharacteristic
525C4⋊C4C2×C5×D412yes—
526C4⋊C4C2×C5×D411yescharacteristic
527C4⋊C4C2×C5×D411yescharacteristic
528C4⋊C4C2×C5×D412yes—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—21no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C42—22no—
545C42—22no—
546C42—21no—
547C42C2×C5×D411yescharacteristic
548C42C2×C5×D411yescharacteristic
549C42C2×C5×D411yescharacteristic
550C42C24×C511yescharacteristic
551C2×C5—22no—
552C2×C5—42no—
553C2×C5—81no—
554C2×C5—81no—
555C2×C5#300711yescharacteristic
556C2×C5#300811yescharacteristic
557C2×C5C2×#45611yescharacteristic
558C23—21no—
559C23—21no—
560C23—21no—
561C23—42no—
562C23—42no—
563C23—81no—
564C23C5×#12011yescharacteristic
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—82no—
572Q8—82no—
573D4—21no—
574D4—21no—
575D4—21no—
576D4—22no—
577D4—42no—
578D4—44no—
579D4—42no—
580D4—82no—
581D4C5×#13712yes—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4C5×#12011yescharacteristic
606C2×C4C5×#12011yescharacteristic
607C2×C4C5×#12011yescharacteristic
608C2×C4C22×C5×D411yescharacteristic
609C2×C4C22×C5×D411yescharacteristic
610C2×C4C22×C5×D411yescharacteristic
611C8—24no—
612C8—42no—
613C8—42no—
614C5#1858711yescharacteristic, direct product factor, Sylow
615C22—21no—
616C22—21no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—84no—
627C22C5×#45612yes—
628C22C2×C5×#12011yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—42no—
638C4—41no—
639C4—41no—
640C4—82no—
641C4C2×C5×#13711yescharacteristic
642C4C5×#57511yescharacteristic
643C4C5×#57612yes—
644C2—22no—
645C2—42no—
646C2—81no—
647C2—81no—
648C2C5×#300711yescharacteristic
649C2C5×#300811yescharacteristic
650C2C2×C5×#45611yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC42C24×C5
prime radicalC5×#3520C2
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×#3218C2semidirect
3. C5×#3165C2semidirect
4. C2×C5×#474C2semidirect
5. C5×#1654C2semidirect
6. C5×#1652C2semidirect
7. C5×#1649C2semidirect
8. C5×#1647C2semidirect
9. C5×#1552C2semidirect
10. C5×#1490C2semidirect
11. C5×#1483C2semidirect
31. C5×#500C22semidirect
32. C5×#498C22semidirect
39. C5×#462C22semidirect
40. C5×#461C22semidirect
49. C5×#332C22semidirect
50. C5×#332C22semidirect
51. C5×Q8⋊C8C22semidirect
54. C5×D4⋊C8C22semidirect
55. #18587C5direct (N × H)
148. #3218C2×C5semidirect
149. #3165C2×C5semidirect
150. C2×#474C2×C5semidirect
151. #1654C2×C5semidirect
152. #1652C2×C5semidirect
153. #1649C2×C5semidirect
154. #1647C2×C5semidirect
155. #1552C2×C5semidirect
156. #1490C2×C5semidirect
157. #1483C2×C5semidirect
264. #500C22×C5semidirect
265. #498C22×C5semidirect
272. #462C22×C5semidirect
273. #461C22×C5semidirect
282. #332C22×C5semidirect
283. #332C22×C5semidirect
284. Q8⋊C8C22×C5semidirect
287. D4⋊C8C22×C5semidirect
614. C5#18587direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35202
C5×#32182
C5×#31652
C2×C5×#4742
C5×#16542
C5×#16522
C5×#16492
C5×#16472
C5×#15522
C5×#14902
C5×#14832
#185875