Finite Group Databasebeta

← C5×#25282 #11948002

Subgroups — C5×#25282 #11948002

Order 1280.

784 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
1C2×C5×#488C211yesmaximal, characteristic
2C2×C5×#479C211yesmaximal, characteristic, complemented
3C5×#2545C211yesmaximal, characteristic
4C5×#2370C211yesmaximal, characteristic, complemented
5C5×#2067C211yesmaximal, characteristic
6C5×#2066C211yesmaximal, characteristic, complemented
7C5×#2043C211yesmaximal, characteristic, complemented
8C5×#2042C211yesmaximal, characteristic
9C5×#1983C211yesmaximal, characteristic
10C5×#1983C211yesmaximal, characteristic
11C5×#1981C211yesmaximal, characteristic, complemented
12C5×#1939C211yesmaximal, characteristic, complemented
13C5×#1928C211yesmaximal, characteristic, complemented
14C5×#1926C211yesmaximal, characteristic
15C4×C5×#103C211yesmaximal, characteristic, complemented
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C5×#125C2211yescharacteristic
18C2×C5×#124C2211yescharacteristic
19C2×C5×#124C2211yescharacteristic
20C2×C4×C5×Q8C2211yescharacteristic
21C5×#488—24no—
22C5×#479—24no—
23C2×C5×#107C2211yescharacteristic
24C2×C5×#106C2211yescharacteristic
25C2×C5×C4⋊C8C2211yescharacteristic
26C2×C5×C4⋊C8C2211yescharacteristic
27C2×C5×C4⋊C8C2211yescharacteristic
28C2×C5×#103C2211yescharacteristic
29C2×C5×#103C2211yescharacteristic
30C2×C5×#103C2211yescharacteristic
31C2×C5×#103C2211yescharacteristic
32C2×C5×#103C2211yescharacteristic
33C2×C5×#103C2211yescharacteristic
34C2×C5×#103C2211yescharacteristic
35C2×C5×#103C2211yescharacteristic
36C2×C5×#102C2211yescharacteristic
37C2×C5×#102C2211yescharacteristic
38C2×C5×#102C2211yescharacteristic
39C2×C5×#102C2211yescharacteristic
40C2×C4×C5×C8C2211yescharacteristic
41C5×#396—21no—
42C5×#396—21no—
43C5×#391C2211yescharacteristic
44C5×#390C2211yescharacteristic
45C5×#389—21no—
46C5×#389C2211yescharacteristic
47C5×#389C2211yescharacteristic
48C5×#388—21no—
49C5×#388—21no—
50C5×#387—21no—
51C5×#387—21no—
52C4×C5×C4⋊C4—21no—
53C4×C5×C4⋊C4C2211yescharacteristic
54C4×C5×C4⋊C4C2211yescharacteristic
55C4×C5×C4⋊C4C2211yescharacteristic
56C4×C5×C4⋊C4C2211yescharacteristic
57C5×#340C2211yescharacteristic
58C5×#340C2211yescharacteristic
59C5×#336C2211yescharacteristic
60C5×#336C2211yescharacteristic
61#25282C511yesmaximal, characteristic, direct product factor, Sylow
62C22×C5×Q8C2311yescharacteristic
63C22×C5×Q8C2311yescharacteristic
64C22×C5×D4C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C22×C5×C8C2311yescharacteristic
67C22×C5×C8C2311yescharacteristic
68C22×C5×C8C2311yescharacteristic
69C5×C4⋊Q8—22no—
70C5×C4⋊Q8—41no—
71C5×#125—22no—
72C5×#125D414yes—
73C5×#124—22no—
74C5×#124—22no—
75C5×#124—22no—
76C5×#124—41no—
77C4×C5×Q8—22no—
78C4×C5×Q8—22no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4—21no—
82C2×C5×C4⋊C4—21no—
83C2×C5×C4⋊C4—21no—
84C2×C5×C4⋊C4—21no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4D411yescharacteristic
87C2×C5×C4⋊C4D411yescharacteristic
88C2×C5×C4⋊C4C2311yescharacteristic
89C2×C5×C4⋊C4C2311yescharacteristic
90C2×C5×C4⋊C4C2311yescharacteristic
91C2×C5×C4⋊C4C2311yescharacteristic
92C2×C5×C4⋊C4C2311yescharacteristic
93C2×C5×C22⋊C4—21no—
94C2×C5×C22⋊C4—21no—
95C2×C5×C22⋊C4—21no—
96C2×C5×C22⋊C4—21no—
97C2×C5×C22⋊C4—21no—
98C2×C5×C22⋊C4—21no—
99C2×C42×C5—21no—
100C2×C42×C5—21no—
101C2×C42×C5—21no—
102C2×C42×C5—21no—
103C2×C42×C5C2311yescharacteristic
104C2×C42×C5C2311yescharacteristic
105C2×C42×C5C2311yescharacteristic
106C5×#107—22no—
107C5×#106—22no—
108C5×C4⋊C8—22no—
109C5×C4⋊C8—41no—
110C5×C4⋊C8—41no—
111C5×#103—22no—
112C5×#103—22no—
113C5×#103—22no—
114C5×#103—22no—
115C5×#103—22no—
116C5×#103—22no—
117C5×#103—22no—
118C5×#103—41no—
119C5×#102—22no—
120C5×#102—22no—
121C5×#102—22no—
122C5×#102—41no—
123C4×C5×C8—22no—
124C5×#95—21no—
125C5×#95—21no—
126C5×#95—21no—
127C5×#95—21no—
128C5×#95—21no—
129C5×#95—21no—
130C5×#95—21no—
131C5×#95D411yescharacteristic
132C5×#95D411yescharacteristic
133C2×#488C2×C511yescharacteristic
134C2×#479C2×C511yescharacteristic, complemented
135#2545C2×C511yescharacteristic
136#2370C2×C511yescharacteristic, complemented
137#2067C2×C511yescharacteristic
138#2066C2×C511yescharacteristic, complemented
139#2043C2×C511yescharacteristic, complemented
140#2042C2×C511yescharacteristic
141#1983C2×C511yescharacteristic
142#1983C2×C511yescharacteristic
143#1981C2×C511yescharacteristic, complemented
144#1939C2×C511yescharacteristic, complemented
145#1928C2×C511yescharacteristic, complemented
146#1926C2×C511yescharacteristic
147C4×#103C2×C511yescharacteristic, complemented
148C24×C5—21no—
149C2×C5×Q8—21no—
150C2×C5×Q8—21no—
151C2×C5×Q8—21no—
152C2×C5×Q8—21no—
153C2×C5×Q8—21no—
154C2×C5×Q8—21no—
155C2×C5×Q8—21no—
156C2×C5×Q8—21no—
157C2×C5×Q8—21no—
158C2×C5×Q8—21no—
159C2×C5×Q8—21no—
160C2×C5×Q8#4112yes—
161C2×C5×D4—21no—
162C2×C5×D4—21no—
163C2×C5×D4—21no—
164C2×C5×D4—21no—
165C2×C5×D4—21no—
166C2×C5×D4—21no—
167C22×C4×C5—21no—
168C22×C4×C5—21no—
169C22×C4×C5—21no—
170C22×C4×C5—21no—
171C22×C4×C5—21no—
172C22×C4×C5—21no—
173C22×C4×C5—21no—
174C22×C4×C5C2×D411yescharacteristic
175C22×C4×C5C2×D411yescharacteristic
176C22×C4×C5C2×D411yescharacteristic
177C22×C4×C5C2×D411yescharacteristic
178C22×C4×C5C2×D411yescharacteristic
179C22×C4×C5C2×D411yescharacteristic
180C22×C4×C5C2411yescharacteristic
181C2×C5×C8—21no—
182C2×C5×C8—21no—
183C2×C5×C8—21no—
184C2×C5×C8—21no—
185C2×C5×C8—21no—
186C2×C5×C8—21no—
187C2×C5×C8—21no—
188C2×C5×C8—21no—
189C2×C5×C8—21no—
190C2×C5×C8—21no—
191C2×C5×C8—21no—
192C2×C5×C8—21no—
193C5×C4⋊C4—24no—
194C5×C4⋊C4—24no—
195C5×C4⋊C4—22no—
196C5×C4⋊C4—21no—
197C5×C4⋊C4—21no—
198C5×C4⋊C4—21no—
199C5×C4⋊C4—21no—
200C5×C4⋊C4—21no—
201C5×C4⋊C4—21no—
202C5×C4⋊C4—21no—
203C5×C4⋊C4—41no—
204C5×C4⋊C4—42no—
205C5×C4⋊C4—42no—
206C5×C4⋊C4—42no—
207C5×C4⋊C4—42no—
208C5×C4⋊C4—81no—
209C5×C4⋊C4C2×D412yes—
210C5×C4⋊C4C2×D412yes—
211C5×C4⋊C4#4112yes—
212C5×C22⋊C4—42no—
213C5×C22⋊C4—42no—
214C5×C22⋊C4—42no—
215C5×C22⋊C4—42no—
216C5×C22⋊C4—42no—
217C5×C22⋊C4—42no—
218C42×C5—21no—
219C42×C5—21no—
220C42×C5—21no—
221C42×C5—21no—
222C42×C5—42no—
223C42×C5—42no—
224C42×C5—81no—
225C42×C5—81no—
226C42×C5C2×D412yes—
227C42×C5#4112yes—
228C2×C4⋊Q8C22×C511yescharacteristic
229C2×#125C22×C511yescharacteristic
230C2×#124C22×C511yescharacteristic
231C2×#124C22×C511yescharacteristic
232C2×C4×Q8C22×C511yescharacteristic
233#488—24no—
234#479—24no—
235C2×#107C22×C511yescharacteristic
236C2×#106C22×C511yescharacteristic
237C2×C4⋊C8C22×C511yescharacteristic
238C2×C4⋊C8C22×C511yescharacteristic
239C2×C4⋊C8C22×C511yescharacteristic
240C2×#103C22×C511yescharacteristic
241C2×#103C22×C511yescharacteristic
242C2×#103C22×C511yescharacteristic
243C2×#103C22×C511yescharacteristic
244C2×#103C22×C511yescharacteristic
245C2×#103C22×C511yescharacteristic
246C2×#103C22×C511yescharacteristic
247C2×#103C22×C511yescharacteristic
248C2×#102C22×C511yescharacteristic
249C2×#102C22×C511yescharacteristic
250C2×#102C22×C511yescharacteristic
251C2×#102C22×C511yescharacteristic
252C2×C4×C8C22×C511yescharacteristic
253#396—21no—
254#396—21no—
255#391C22×C511yescharacteristic
256#390C22×C511yescharacteristic
257#389—21no—
258#389C22×C511yescharacteristic
259#389C22×C511yescharacteristic
260#388—21no—
261#388—21no—
262#387—21no—
263#387—21no—
264C4×C4⋊C4—21no—
265C4×C4⋊C4C22×C511yescharacteristic
266C4×C4⋊C4C22×C511yescharacteristic
267C4×C4⋊C4C22×C511yescharacteristic
268C4×C4⋊C4C22×C511yescharacteristic
269#340C22×C511yescharacteristic
270#340C22×C511yescharacteristic
271#336C22×C511yescharacteristic
272#336C22×C511yescharacteristic
273C23×C5—41no—
274C23×C5—41no—
275C23×C5—41no—
276C23×C5—41no—
277C23×C5—41no—
278C23×C5—41no—
279C23×C5—41no—
280C23×C5C22×D411yescharacteristic
281C5×Q8—22no—
282C5×Q8—22no—
283C5×Q8—41no—
284C5×Q8—41no—
285C5×Q8—41no—
286C5×Q8—41no—
287C5×Q8—41no—
288C5×Q8—41no—
289C5×D4—41no—
290C5×D4—41no—
291C5×D4—41no—
292C5×D4—41no—
293C2×C4×C5—21no—
294C2×C4×C5—21no—
295C2×C4×C5—21no—
296C2×C4×C5—21no—
297C2×C4×C5—21no—
298C2×C4×C5—21no—
299C2×C4×C5—22no—
300C2×C4×C5—22no—
301C2×C4×C5—21no—
302C2×C4×C5—21no—
303C2×C4×C5—21no—
304C2×C4×C5—22no—
305C2×C4×C5—22no—
306C2×C4×C5—21no—
307C2×C4×C5—21no—
308C2×C4×C5—21no—
309C2×C4×C5—21no—
310C2×C4×C5—21no—
311C2×C4×C5—21no—
312C2×C4×C5—41no—
313C2×C4×C5—41no—
314C2×C4×C5—41no—
315C2×C4×C5—41no—
316C2×C4×C5—41no—
317C2×C4×C5—41no—
318C2×C4×C5—41no—
319C2×C4×C5—41no—
320C2×C4×C5—41no—
321C2×C4×C5—41no—
322C2×C4×C5—41no—
323C2×C4×C5—41no—
324C2×C4×C5—41no—
325C2×C4×C5—41no—
326C2×C4×C5—41no—
327C2×C4×C5—41no—
328C2×C4×C5—41no—
329C2×C4×C5#12112yes—
330C2×C4×C5#12112yes—
331C2×C4×C5#13512yes—
332C2×C4×C5C22×D411yescharacteristic
333C2×C4×C5C2×#4111yescharacteristic
334C2×C4×C5C2×#4111yescharacteristic
335C2×C4×C5C2×#4111yescharacteristic
336C2×C4×C5H(5,𝔽2)11yescharacteristic
337C2×C4×C52−1+411yescharacteristic
338C5×C8—41no—
339C5×C8—41no—
340C5×C8—41no—
341C5×C8—41no—
342C22×Q8C23×C511yescharacteristic
343C22×Q8C23×C511yescharacteristic
344C22×D4C23×C511yescharacteristic
345C22×C8C23×C511yescharacteristic
346C22×C8C23×C511yescharacteristic
347C22×C8C23×C511yescharacteristic
348C22×C8C23×C511yescharacteristic
349C4⋊Q8—22no—
350C4⋊Q8—41no—
351#125—22no—
352#125C5×D414yes—
353#124—22no—
354#124—22no—
355#124—22no—
356#124—41no—
357C4×Q8—22no—
358C4×Q8—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4C5×D411yescharacteristic
367C2×C4⋊C4C5×D411yescharacteristic
368C2×C4⋊C4C23×C511yescharacteristic
369C2×C4⋊C4C23×C511yescharacteristic
370C2×C4⋊C4C23×C511yescharacteristic
371C2×C4⋊C4C23×C511yescharacteristic
372C2×C4⋊C4C23×C511yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42C23×C511yescharacteristic
384C2×C42C23×C511yescharacteristic
385C2×C42C23×C511yescharacteristic
386#107—22no—
387#106—22no—
388C4⋊C8—22no—
389C4⋊C8—41no—
390C4⋊C8—41no—
391#103—22no—
392#103—22no—
393#103—22no—
394#103—22no—
395#103—22no—
396#103—22no—
397#103—22no—
398#103—41no—
399#102—22no—
400#102—22no—
401#102—22no—
402#102—41no—
403C4×C8—22no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95C5×D411yescharacteristic
412#95C5×D411yescharacteristic
413C22×C5—81no—
414C22×C5—81no—
415C22×C5—81no—
416C22×C5—81no—
417C22×C5—81no—
418C22×C5—81no—
419C22×C5—81no—
420C22×C5C2×#12111yescharacteristic
421C22×C5#54611yescharacteristic
422C22×C5#54811yescharacteristic
423C22×C5C2×#13511yescharacteristic
424C22×C5#57511yescharacteristic
425C22×C5#57711yescharacteristic
426C22×C5#57811yescharacteristic
427C4×C5—22no—
428C4×C5—22no—
429C4×C5—22no—
430C4×C5—42no—
431C4×C5—42no—
432C4×C5—42no—
433C4×C5—41no—
434C4×C5—42no—
435C4×C5—41no—
436C4×C5—41no—
437C4×C5—81no—
438C4×C5—81no—
439C4×C5—81no—
440C4×C5#52911yescharacteristic
441C4×C5#54211yescharacteristic
442C4×C5#54811yescharacteristic
443C4×C5#55011yescharacteristic
444C24—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8C5×#4112yes—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4C2×C5×D411yescharacteristic
471C22×C4C2×C5×D411yescharacteristic
472C22×C4C2×C5×D411yescharacteristic
473C22×C4C2×C5×D411yescharacteristic
474C22×C4C2×C5×D411yescharacteristic
475C22×C4C2×C5×D411yescharacteristic
476C22×C4C24×C511yescharacteristic
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—41no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—81no—
505C4⋊C4C2×C5×D412yes—
506C4⋊C4C2×C5×D412yes—
507C4⋊C4C5×#4112yes—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—42no—
519C42—42no—
520C42—81no—
521C42—81no—
522C42C2×C5×D412yes—
523C42C5×#4112yes—
524C2×C5—161no—
525C2×C5#242611yescharacteristic
526C2×C5#303511yescharacteristic
527C2×C5#304611yescharacteristic
528C2×C5#330111yescharacteristic
529C2×C5#332711yescharacteristic
530C2×C5#336411yescharacteristic
531C2×C5#337611yescharacteristic
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23C22×C5×D411yescharacteristic
540Q8—22no—
541Q8—22no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—41no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4C5×#12112yes—
589C2×C4C5×#12112yes—
590C2×C4C5×#13512yes—
591C2×C4C22×C5×D411yescharacteristic
592C2×C4C2×C5×#4111yescharacteristic
593C2×C4C2×C5×#4111yescharacteristic
594C2×C4C2×C5×#4111yescharacteristic
595C2×C4C5×H(5,𝔽2)11yescharacteristic
596C2×C4C5×2−1+411yescharacteristic
597C8—41no—
598C8—41no—
599C8—41no—
600C8—41no—
601C5#2528211yescharacteristic, direct product factor, Sylow
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22C2×C5×#12111yescharacteristic
610C22C5×#54611yescharacteristic
611C22C5×#54811yescharacteristic
612C22C2×C5×#13511yescharacteristic
613C22C5×#57511yescharacteristic
614C22C5×#57711yescharacteristic
615C22C5×#57811yescharacteristic
616C4—22no—
617C4—22no—
618C4—22no—
619C4—42no—
620C4—41no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4C5×#52911yescharacteristic
630C4C5×#54211yescharacteristic
631C4C5×#54811yescharacteristic
632C4C5×#55011yescharacteristic
633C2—161no—
634C2C5×#242611yescharacteristic
635C2C5×#303511yescharacteristic
636C2C5×#304611yescharacteristic
637C2C5×#330111yescharacteristic
638C2C5×#332711yescharacteristic
639C2C5×#336411yescharacteristic
640C2C5×#337611yescharacteristic
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC22×C5×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C4×C5×C4⋊C4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C5×#479C2semidirect
4. C5×#2370C2semidirect
6. C5×#2066C2semidirect
7. C5×#2043C2semidirect
11. C5×#1981C2semidirect
12. C5×#1939C2semidirect
13. C5×#1928C2semidirect
15. C4×C5×#103C2semidirect
61. #25282C5direct (N × H)
134. C2×#479C2×C5semidirect
136. #2370C2×C5semidirect
138. #2066C2×C5semidirect
139. #2043C2×C5semidirect
143. #1981C2×C5semidirect
144. #1939C2×C5semidirect
145. #1928C2×C5semidirect
147. C4×#103C2×C5semidirect
601. C5#25282direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#4882
C2×C5×#4792
C5×#25452
C5×#23702
C5×#20672
C5×#20662
C5×#20432
C5×#20422
C5×#19832
C5×#19812
C5×#19392
C5×#19282
C5×#19262
C4×C5×#1032
#252825