Finite Group Databasebeta

← C5×#18264 #11940984

Subgroups — C5×#18264 #11940984

Order 1280.

1068 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3537C211yesmaximal, characteristic
2C5×#3092C211yesmaximal, characteristic, complemented
3C2×C5×#480C211yesmaximal, characteristic, complemented
4C5×#2504C211yesmaximal, characteristic, complemented
5C5×#1612C214yesmaximal, complemented
6C5×#1607C214yesmaximal, complemented
7C5×#1507C211yesmaximal, characteristic, complemented
8C5×#1501C211yesmaximal, characteristic, complemented
9C5×#1471C211yesmaximal, characteristic, complemented
10C5×#560—22no—
11C5×#553—21no—
12C5×#553—21no—
13C5×D42—22no—
14C5×D42—21no—
15C5×D42—21no—
16C5×#540—21no—
17C5×#538—21no—
18C5×#538C2212yes—
19C5×#535—21no—
20C5×#534—21no—
21C5×#534C2212yes—
22C5×#525C2211yescharacteristic
23C2×C5×#121C2211yescharacteristic
24C2×C5×#121C2211yescharacteristic
25C22×C5×C4⋊C4C2211yescharacteristic, complemented
26C5×#484—21no—
27C5×#483C2212yescomplemented
28C5×#481—21no—
29C5×#480—22no—
30C5×#480C2214yescomplemented
31C5×#480C2212yescomplemented
32C5×#428—21no—
33C2×C5×#107—21no—
34C5×#417—21no—
35C2×C5×#102—21no—
36C5×#407C2211yescharacteristic, complemented
37C2×C5×C22⋊C8C2211yescharacteristic, complemented
38C5×#397C2214yescomplemented
39C5×#390—21no—
40C5×#386—22no—
41C5×#382—21no—
42C4×C5×C22⋊C4C2211yescharacteristic, complemented
43C5×#328C2212yescomplemented
44C5×#327C2212yescomplemented
45C5×#327C2214yescomplemented
46C5×#323C2214yescomplemented
47#18264C511yesmaximal, characteristic, direct product factor, Sylow
48C2×C5×#41D411yescharacteristic
49C22×C5×D4—22no—
50C22×C5×D4—21no—
51C22×C5×D4—21no—
52C22×C5×D4—22no—
53C22×C5×D4D412yes—
54C22×C5×D4D411yescharacteristic
55C22×C5×D4D411yescharacteristic
56C23×C4×C5—21no—
57C23×C4×C5C2311yescharacteristic
58C2×C5×M16—21no—
59C22×C5×C8—21no—
60C5×#127—22no—
61C5×#127—21no—
62C5×#126—22no—
63C5×#126—22no—
64C5×#125—21no—
65C5×#124—22no—
66C5×#123—22no—
67C5×#123—21no—
68C5×#123—22no—
69C5×#123—22no—
70C5×#123—22no—
71C5×C22⋊Q8D411yescharacteristic
72C5×C22⋊Q8C2311yescharacteristic
73C5×#121—22no—
74C5×#121—21no—
75C5×#121—21no—
76C5×#121—22no—
77C5×#121—22no—
78C5×#121—22no—
79C5×#121—22no—
80C5×#121—21no—
81C5×#121—21no—
82C5×#121—21no—
83C5×#121—21no—
84C5×#121—21no—
85C5×#121—21no—
86C5×#121—22no—
87C5×#121D412yes—
88C5×#121D411yescharacteristic
89C5×#121C2312yes—
90C5×#121C2311yescharacteristic
91C5×#120—22no—
92C5×#120—22no—
93C5×#120—22no—
94C5×#120—21no—
95C5×#120—22no—
96C5×#120—22no—
97C5×#120—22no—
98C4×C5×D4—22no—
99C4×C5×D4—21no—
100C4×C5×D4—21no—
101C4×C5×D4—21no—
102C4×C5×D4—21no—
103C4×C5×D4—22no—
104C5×#117—21no—
105C5×#117D411yescharacteristic
106C2×C5×C4⋊C4—21no—
107C2×C5×C4⋊C4—21no—
108C2×C5×C4⋊C4—21no—
109C2×C5×C4⋊C4—21no—
110C2×C5×C4⋊C4—21no—
111C2×C5×C4⋊C4D411yescharacteristic
112C2×C5×C4⋊C4D411yescharacteristic
113C2×C5×C4⋊C4C2312yes—
114C2×C5×C22⋊C4—21no—
115C2×C5×C22⋊C4—21no—
116C2×C5×C22⋊C4—24no—
117C2×C5×C22⋊C4C2312yes—
118C2×C42×C5—21no—
119C5×#107—24no—
120C5×#107—22no—
121C5×#106—22no—
122C5×#103—22no—
123C5×#102—24no—
124C5×#102—22no—
125C5×C22⋊C8—21no—
126C5×C22⋊C8—21no—
127C5×C22⋊C8C2312yes—
128C5×C22⋊C8C2312yes—
129C5×#95—24no—
130C5×#95C2312yes—
131#3537C2×C511yescharacteristic
132#3092C2×C511yescharacteristic, complemented
133C2×#480C2×C511yescharacteristic, complemented
134#2504C2×C511yescharacteristic, complemented
135#1612C2×C514yescomplemented
136#1607C2×C514yescomplemented
137#1507C2×C511yescharacteristic, complemented
138#1501C2×C511yescharacteristic, complemented
139#1471C2×C511yescharacteristic, complemented
140C24×C5—22no—
141C24×C5—21no—
142C24×C5C2×D411yescharacteristic
143C24×C5C2×D412yes—
144C5×#41—42no—
145C2×C5×Q8C2×D411yescharacteristic
146C2×C5×D4—22no—
147C2×C5×D4—21no—
148C2×C5×D4—21no—
149C2×C5×D4—22no—
150C2×C5×D4—22no—
151C2×C5×D4—22no—
152C2×C5×D4—22no—
153C2×C5×D4—22no—
154C2×C5×D4—22no—
155C2×C5×D4—21no—
156C2×C5×D4—22no—
157C2×C5×D4—22no—
158C2×C5×D4—21no—
159C2×C5×D4—21no—
160C2×C5×D4—21no—
161C2×C5×D4—42no—
162C2×C5×D4—42no—
163C2×C5×D4—44no—
164C2×C5×D4—82no—
165C2×C5×D4—82no—
166C2×C5×D4—84no—
167C2×C5×D4—84no—
168C2×C5×D4C2×D411yescharacteristic
169C2×C5×D4C2×D412yes—
170C22×C4×C5—22no—
171C22×C4×C5—22no—
172C22×C4×C5—21no—
173C22×C4×C5—21no—
174C22×C4×C5—22no—
175C22×C4×C5—22no—
176C22×C4×C5—42no—
177C22×C4×C5—41no—
178C22×C4×C5—41no—
179C22×C4×C5—82no—
180C22×C4×C5C2×D411yescharacteristic
181C22×C4×C5C2×D411yescharacteristic
182C22×C4×C5C2×D411yescharacteristic
183C22×C4×C5C2×D412yes—
184C22×C4×C5C2×D411yescharacteristic
185C22×C4×C5C2×D411yescharacteristic
186C22×C4×C5C2411yescharacteristic
187C5×M16—42no—
188C2×C5×C8—22no—
189C2×C5×C8—22no—
190C2×C5×C8—42no—
191C5×C4⋊C4—22no—
192C5×C4⋊C4—22no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—21no—
195C5×C4⋊C4—21no—
196C5×C4⋊C4—21no—
197C5×C4⋊C4—21no—
198C5×C4⋊C4—22no—
199C5×C4⋊C4—22no—
200C5×C4⋊C4—41no—
201C5×C4⋊C4—41no—
202C5×C4⋊C4C2×D412yes—
203C5×C4⋊C4C2×D411yescharacteristic
204C5×C4⋊C4C2×D411yescharacteristic
205C5×C22⋊C4—22no—
206C5×C22⋊C4—22no—
207C5×C22⋊C4—22no—
208C5×C22⋊C4—22no—
209C5×C22⋊C4—22no—
210C5×C22⋊C4—21no—
211C5×C22⋊C4—21no—
212C5×C22⋊C4—21no—
213C5×C22⋊C4—22no—
214C5×C22⋊C4—21no—
215C5×C22⋊C4—22no—
216C5×C22⋊C4—22no—
217C5×C22⋊C4—22no—
218C5×C22⋊C4—21no—
219C5×C22⋊C4—42no—
220C5×C22⋊C4—84no—
221C42×C5—22no—
222C42×C5—21no—
223C42×C5—42no—
224#560—22no—
225#553—21no—
226#553—21no—
227D42—22no—
228D42—21no—
229D42—21no—
230#540—21no—
231#538—21no—
232#538C22×C512yes—
233#535—21no—
234#534—21no—
235#534C22×C512yes—
236#525C22×C511yescharacteristic
237C2×#121C22×C511yescharacteristic
238C2×#121C22×C511yescharacteristic
239C22×C4⋊C4C22×C511yescharacteristic, complemented
240#484—21no—
241#483C22×C512yescomplemented
242#481—21no—
243#480—22no—
244#480C22×C514yescomplemented
245#480C22×C512yescomplemented
246#428—21no—
247C2×#107—21no—
248#417—21no—
249C2×#102—21no—
250#407C22×C511yescharacteristic, complemented
251C2×C22⋊C8C22×C511yescharacteristic, complemented
252#397C22×C514yescomplemented
253#390—21no—
254#386—22no—
255#382—21no—
256C4×C22⋊C4C22×C511yescharacteristic, complemented
257#328C22×C512yescomplemented
258#327C22×C512yescomplemented
259#327C22×C514yescomplemented
260#323C22×C514yescomplemented
261C23×C5—22no—
262C23×C5—22no—
263C23×C5—22no—
264C23×C5—22no—
265C23×C5—21no—
266C23×C5—24no—
267C23×C5—41no—
268C23×C5—42no—
269C23×C5—42no—
270C23×C5—42no—
271C23×C5—81no—
272C23×C5—81no—
273C23×C5—82no—
274C23×C5—82no—
275C23×C5—82no—
276C23×C5#12011yescharacteristic
277C23×C5#12011yescharacteristic
278C23×C5C22×D411yescharacteristic
279C5×Q8—41no—
280C5×D4—42no—
281C5×D4—41no—
282C5×D4—44no—
283C5×D4—42no—
284C5×D4—44no—
285C5×D4—42no—
286C5×D4—42no—
287C5×D4—42no—
288C5×D4—81no—
289C5×D4—82no—
290C5×D4—82no—
291C5×D4—81no—
292C5×D4—82no—
293C5×D4—82no—
294C5×D4—82no—
295C5×D4—82no—
296C5×D4—82no—
297C2×C4×C5—22no—
298C2×C4×C5—22no—
299C2×C4×C5—22no—
300C2×C4×C5—21no—
301C2×C4×C5—22no—
302C2×C4×C5—22no—
303C2×C4×C5—21no—
304C2×C4×C5—21no—
305C2×C4×C5—41no—
306C2×C4×C5—41no—
307C2×C4×C5—41no—
308C2×C4×C5—42no—
309C2×C4×C5—42no—
310C2×C4×C5—42no—
311C2×C4×C5—42no—
312C2×C4×C5—42no—
313C2×C4×C5—42no—
314C2×C4×C5—44no—
315C2×C4×C5—81no—
316C2×C4×C5—82no—
317C2×C4×C5—82no—
318C2×C4×C5—82no—
319C2×C4×C5—82no—
320C2×C4×C5#12011yescharacteristic
321C2×C4×C5#12011yescharacteristic
322C2×C4×C5C22×D411yescharacteristic
323C2×C4×C5C22×D411yescharacteristic
324C5×C8—42no—
325C5×C8—42no—
326C2×#41C5×D411yescharacteristic
327C22×D4—22no—
328C22×D4—21no—
329C22×D4—21no—
330C22×D4—22no—
331C22×D4C5×D412yes—
332C22×D4C5×D411yescharacteristic
333C22×D4C5×D411yescharacteristic
334C23×C4—21no—
335C23×C4C23×C511yescharacteristic
336C2×M16—21no—
337C22×C8—21no—
338#127—22no—
339#127—21no—
340#126—22no—
341#126—22no—
342#125—21no—
343#124—22no—
344#123—22no—
345#123—21no—
346#123—22no—
347#123—22no—
348#123—22no—
349C22⋊Q8C5×D411yescharacteristic
350C22⋊Q8C23×C511yescharacteristic
351#121—22no—
352#121—21no—
353#121—21no—
354#121—22no—
355#121—22no—
356#121—22no—
357#121—22no—
358#121—21no—
359#121—21no—
360#121—21no—
361#121—21no—
362#121—21no—
363#121—21no—
364#121—22no—
365#121C5×D412yes—
366#121C5×D411yescharacteristic
367#121C23×C512yes—
368#121C23×C511yescharacteristic
369#120—22no—
370#120—22no—
371#120—22no—
372#120—21no—
373#120—22no—
374#120—22no—
375#120—22no—
376C4×D4—22no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—22no—
382#117—21no—
383#117C5×D411yescharacteristic
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4C5×D411yescharacteristic
390C2×C4⋊C4C5×D411yescharacteristic
391C2×C4⋊C4C23×C512yes—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4C23×C512yes—
396C2×C42—21no—
397#107—24no—
398#107—22no—
399#106—22no—
400#103—22no—
401#102—24no—
402#102—22no—
403C22⋊C8—21no—
404C22⋊C8—21no—
405C22⋊C8C23×C512yes—
406C22⋊C8C23×C512yes—
407#95—24no—
408#95C23×C512yes—
409C22×C5—21no—
410C22×C5—21no—
411C22×C5—21no—
412C22×C5—21no—
413C22×C5—21no—
414C22×C5—21no—
415C22×C5—21no—
416C22×C5—44no—
417C22×C5—42no—
418C22×C5—42no—
419C22×C5—41no—
420C22×C5—42no—
421C22×C5—42no—
422C22×C5—41no—
423C22×C5—41no—
424C22×C5—42no—
425C22×C5—42no—
426C22×C5—84no—
427C22×C5—162no—
428C22×C5—164no—
429C22×C5#45512yes—
430C22×C5C2×#12011yescharacteristic
431C22×C5#57612yes—
432C4×C5—21no—
433C4×C5—21no—
434C4×C5—21no—
435C4×C5—42no—
436C4×C5—41no—
437C4×C5—42no—
438C4×C5—41no—
439C4×C5—41no—
440C4×C5—82no—
441C4×C5—82no—
442C4×C5#45812yes—
443C24—22no—
444C24—21no—
445C24C2×C5×D411yescharacteristic
446C24C2×C5×D412yes—
447#41—42no—
448C2×Q8C2×C5×D411yescharacteristic
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—44no—
467C2×D4—82no—
468C2×D4—82no—
469C2×D4—84no—
470C2×D4—84no—
471C2×D4C2×C5×D411yescharacteristic
472C2×D4C2×C5×D412yes—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—42no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—82no—
483C22×C4C2×C5×D411yescharacteristic
484C22×C4C2×C5×D411yescharacteristic
485C22×C4C2×C5×D411yescharacteristic
486C22×C4C2×C5×D412yes—
487C22×C4C2×C5×D411yescharacteristic
488C22×C4C2×C5×D411yescharacteristic
489C22×C4C24×C511yescharacteristic
490M16—42no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—42no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4C2×C5×D411yescharacteristic
506C4⋊C4C2×C5×D412yes—
507C4⋊C4C2×C5×D411yescharacteristic
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—22no—
517C22⋊C4—21no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—42no—
523C22⋊C4—84no—
524C42—22no—
525C42—21no—
526C42—42no—
527C2×C5—22no—
528C2×C5—22no—
529C2×C5—41no—
530C2×C5—82no—
531C2×C5—82no—
532C2×C5—81no—
533C2×C5#300811yescharacteristic
534C2×C5C2×#45511yescharacteristic
535C2×C5C2×#45811yescharacteristic
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—21no—
541C23—24no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—81no—
547C23—81no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23C5×#12011yescharacteristic
552C23C5×#12011yescharacteristic
553C23C22×C5×D411yescharacteristic
554Q8—41no—
555D4—42no—
556D4—41no—
557D4—44no—
558D4—42no—
559D4—44no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—81no—
564D4—82no—
565D4—82no—
566D4—81no—
567D4—82no—
568D4—82no—
569D4—82no—
570D4—82no—
571D4—82no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—81no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4C5×#12011yescharacteristic
596C2×C4C5×#12011yescharacteristic
597C2×C4C22×C5×D411yescharacteristic
598C2×C4C22×C5×D411yescharacteristic
599C8—42no—
600C8—42no—
601C5#1826411yescharacteristic, direct product factor, Sylow
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—84no—
620C22—162no—
621C22—164no—
622C22C5×#45512yes—
623C22C2×C5×#12011yescharacteristic
624C22C5×#57612yes—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—82no—
634C4—82no—
635C4C5×#45812yes—
636C2—22no—
637C2—22no—
638C2—41no—
639C2—82no—
640C2—82no—
641C2—81no—
642C2C5×#300811yescharacteristic
643C2C2×C5×#45511yescharacteristic
644C2C2×C5×#45811yescharacteristic
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×#3092C2semidirect
3. C2×C5×#480C2semidirect
4. C5×#2504C2semidirect
5. C5×#1612C2semidirect
6. C5×#1607C2semidirect
7. C5×#1507C2semidirect
8. C5×#1501C2semidirect
9. C5×#1471C2semidirect
25. C22×C5×C4⋊C4C22semidirect
27. C5×#483C22semidirect
30. C5×#480C22semidirect
31. C5×#480C22semidirect
36. C5×#407C22semidirect
37. C2×C5×C22⋊C8C22semidirect
38. C5×#397C22semidirect
42. C4×C5×C22⋊C4C22semidirect
43. C5×#328C22semidirect
44. C5×#327C22semidirect
45. C5×#327C22semidirect
46. C5×#323C22semidirect
47. #18264C5direct (N × H)
132. #3092C2×C5semidirect
133. C2×#480C2×C5semidirect
134. #2504C2×C5semidirect
135. #1612C2×C5semidirect
136. #1607C2×C5semidirect
137. #1507C2×C5semidirect
138. #1501C2×C5semidirect
139. #1471C2×C5semidirect
239. C22×C4⋊C4C22×C5semidirect
241. #483C22×C5semidirect
244. #480C22×C5semidirect
245. #480C22×C5semidirect
250. #407C22×C5semidirect
251. C2×C22⋊C8C22×C5semidirect
252. #397C22×C5semidirect
256. C4×C22⋊C4C22×C5semidirect
257. #328C22×C5semidirect
258. #327C22×C5semidirect
259. #327C22×C5semidirect
260. #323C22×C5semidirect
601. C5#18264direct (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×#30922
C2×C5×#4802
C5×#25042
C5×#16122
C5×#16072
C5×#15072
C5×#15012
C5×#14712
#182645