Finite Group Databasebeta

← C5×#29665 #11952385

Subgroups — C5×#29665 #11952385

Order 1280.

1164 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#529C211yesmaximal, characteristic
2C2×C5×#462C211yesmaximal, characteristic, complemented
3C2×C5×#461C211yesmaximal, characteristic, complemented
4C5×#2857C211yesmaximal, characteristic, complemented
5C5×#2026C212yesmaximal, complemented
6C5×#2015C212yesmaximal, complemented
7C5×#2001C212yesmaximal, complemented
8C5×#2000C212yesmaximal
9C5×#1803C211yesmaximal, characteristic, complemented
10C5×#1802C211yesmaximal, characteristic, complemented
11C5×#1797C211yesmaximal, characteristic, complemented
12C22×C5×Q16C2212yes—
13C22×C5×SD16C2212yes—
14C2×C5×C4⋊Q8C2211yescharacteristic
15C5×#529—24no—
16C5×#529—24no—
17C2×C5×#126—21no—
18C2×C5×#125C2211yescharacteristic
19C2×C5×#124C2211yescharacteristic
20C2×C5×#123—21no—
21C2×C5×C22⋊Q8C2211yescharacteristic
22C2×C5×C22⋊Q8C2212yes—
23C2×C5×#121C2211yescharacteristic
24C2×C4×C5×Q8C2211yescharacteristic
25C2×C4×C5×D4—21no—
26C2×C4×C5×D4C2211yescharacteristic
27C2×C5×#117—21no—
28C22×C42×C5C2211yescharacteristic
29C5×#462—24no—
30C5×#461—24no—
31C2×C5×C4⋊C8C2211yescharacteristic
32C2×C5×C4⋊C8C2211yescharacteristic
33C2×C5×#103C2212yes—
34C2×C5×#103C2212yes—
35C2×C5×#103C2212yes—
36C2×C5×#102C2212yes—
37C2×C5×C22⋊C8C2212yes—
38C5×#391C2212yes—
39C5×#389—21no—
40C5×#389—21no—
41C5×#384C2211yescharacteristic
42C5×#380—21no—
43C5×#380—21no—
44C5×#340C2212yes—
45C5×#340C2212yes—
46C5×#336C2212yes—
47#29665C511yesmaximal, characteristic, direct product factor, Sylow
48C22×C5×Q8C2312yes—
49C22×C5×Q8C2311yescharacteristic
50C22×C5×D4—21no—
51C22×C5×D4C2311yescharacteristic
52C23×C4×C5—21no—
53C23×C4×C5D411yescharacteristic
54C23×C4×C5D411yescharacteristic
55C23×C4×C5C2311yescharacteristic
56C2×C5×Q16—24no—
57C2×C5×Q16—42no—
58C2×C5×Q16—42no—
59C2×C5×SD16—24no—
60C2×C5×SD16—42no—
61C2×C5×SD16—42no—
62C22×C5×C8C2312yes—
63C22×C5×C8C2312yes—
64C5×C4⋊Q8—22no—
65C5×C4⋊Q8—22no—
66C5×#126—28no—
67C5×#125—22no—
68C5×#125—22no—
69C5×#124—22no—
70C5×#124—22no—
71C5×#123—28no—
72C5×C22⋊Q8—22no—
73C5×C22⋊Q8—22no—
74C5×C22⋊Q8—42no—
75C5×C22⋊Q8—42no—
76C5×#121—22no—
77C5×#121—22no—
78C4×C5×Q8—22no—
79C4×C5×Q8D414yes—
80C4×C5×D4—28no—
81C4×C5×D4—22no—
82C4×C5×D4D414yes—
83C5×#117—28no—
84C2×C5×C4⋊C4—21no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4—21no—
87C2×C5×C4⋊C4—21no—
88C2×C5×C4⋊C4—21no—
89C2×C5×C4⋊C4—21no—
90C2×C5×C4⋊C4—21no—
91C2×C5×C4⋊C4—21no—
92C2×C5×C4⋊C4—22no—
93C2×C5×C4⋊C4C2312yes—
94C2×C5×C4⋊C4C2311yescharacteristic
95C2×C5×C4⋊C4C2311yescharacteristic
96C2×C5×C22⋊C4—21no—
97C2×C5×C22⋊C4—21no—
98C2×C5×C22⋊C4—21no—
99C2×C5×C22⋊C4—21no—
100C2×C5×C22⋊C4—21no—
101C2×C5×C22⋊C4—22no—
102C2×C42×C5—21no—
103C2×C42×C5—22no—
104C2×C42×C5—22no—
105C2×C42×C5D411yescharacteristic
106C2×C42×C5D411yescharacteristic
107C2×C42×C5C2311yescharacteristic
108C2×C42×C5C2311yescharacteristic
109C5×C4⋊C8—22no—
110C5×C4⋊C8—22no—
111C5×#103—24no—
112C5×#103—24no—
113C5×#103—42no—
114C5×#102—42no—
115C5×C22⋊C8—42no—
116C5×#95—22no—
117C5×#95—22no—
118C2×#529C2×C511yescharacteristic
119C2×#462C2×C511yescharacteristic, complemented
120C2×#461C2×C511yescharacteristic, complemented
121#2857C2×C511yescharacteristic, complemented
122#2026C2×C512yescomplemented
123#2015C2×C512yescomplemented
124#2001C2×C512yescomplemented
125#2000C2×C512yes—
126#1803C2×C511yescharacteristic, complemented
127#1802C2×C511yescharacteristic, complemented
128#1797C2×C511yescharacteristic, complemented
129C24×C5—21no—
130C24×C5C2×D411yescharacteristic
131C2×C5×Q8—21no—
132C2×C5×Q8—22no—
133C2×C5×Q8—22no—
134C2×C5×Q8—21no—
135C2×C5×Q8—21no—
136C2×C5×Q8—21no—
137C2×C5×Q8—22no—
138C2×C5×Q8—22no—
139C2×C5×Q8—22no—
140C2×C5×Q8—21no—
141C2×C5×Q8—22no—
142C2×C5×Q8C2×D412yes—
143C2×C5×D4—24no—
144C2×C5×D4—21no—
145C2×C5×D4—21no—
146C2×C5×D4—21no—
147C2×C5×D4—21no—
148C2×C5×D4—21no—
149C2×C5×D4—42no—
150C2×C5×D4—42no—
151C2×C5×D4C2×D412yes—
152C22×C4×C5—22no—
153C22×C4×C5—21no—
154C22×C4×C5—21no—
155C22×C4×C5—21no—
156C22×C4×C5—22no—
157C22×C4×C5—22no—
158C22×C4×C5—24no—
159C22×C4×C5—22no—
160C22×C4×C5—22no—
161C22×C4×C5—21no—
162C22×C4×C5—21no—
163C22×C4×C5—21no—
164C22×C4×C5—21no—
165C22×C4×C5—21no—
166C22×C4×C5—21no—
167C22×C4×C5—22no—
168C22×C4×C5—22no—
169C22×C4×C5—41no—
170C22×C4×C5—42no—
171C22×C4×C5—42no—
172C22×C4×C5C2×D411yescharacteristic
173C22×C4×C5C2×D411yescharacteristic
174C22×C4×C5C2×D411yescharacteristic
175C22×C4×C5C2×D411yescharacteristic
176C22×C4×C5C2×D411yescharacteristic
177C22×C4×C5C2411yescharacteristic
178C5×Q16—48no—
179C5×SD16—48no—
180C2×C5×C8—22no—
181C2×C5×C8—22no—
182C2×C5×C8—22no—
183C2×C5×C8—22no—
184C2×C5×C8—22no—
185C2×C5×C8—22no—
186C5×C4⋊C4—24no—
187C5×C4⋊C4—24no—
188C5×C4⋊C4—24no—
189C5×C4⋊C4—24no—
190C5×C4⋊C4—21no—
191C5×C4⋊C4—21no—
192C5×C4⋊C4—22no—
193C5×C4⋊C4—22no—
194C5×C4⋊C4—42no—
195C5×C4⋊C4—44no—
196C5×C4⋊C4—42no—
197C5×C4⋊C4—42no—
198C5×C4⋊C4—42no—
199C5×C4⋊C4C2×D412yes—
200C5×C4⋊C4C2×D412yes—
201C5×C22⋊C4—24no—
202C5×C22⋊C4—24no—
203C5×C22⋊C4—24no—
204C5×C22⋊C4—24no—
205C5×C22⋊C4—24no—
206C5×C22⋊C4—44no—
207C42×C5—22no—
208C42×C5—22no—
209C42×C5—24no—
210C42×C5—21no—
211C42×C5—21no—
212C42×C5C2×D412yes—
213C42×C5C2×D412yes—
214C22×Q16C22×C512yes—
215C22×SD16C22×C512yes—
216C2×C4⋊Q8C22×C511yescharacteristic
217#529—24no—
218#529—24no—
219C2×#126—21no—
220C2×#125C22×C511yescharacteristic
221C2×#124C22×C511yescharacteristic
222C2×#123—21no—
223C2×C22⋊Q8C22×C511yescharacteristic
224C2×C22⋊Q8C22×C512yes—
225C2×#121C22×C511yescharacteristic
226C2×C4×Q8C22×C511yescharacteristic
227C2×C4×D4—21no—
228C2×C4×D4C22×C511yescharacteristic
229C2×#117—21no—
230C22×C42C22×C511yescharacteristic
231#462—24no—
232#461—24no—
233C2×C4⋊C8C22×C511yescharacteristic
234C2×C4⋊C8C22×C511yescharacteristic
235C2×#103C22×C512yes—
236C2×#103C22×C512yes—
237C2×#103C22×C512yes—
238C2×#102C22×C512yes—
239C2×C22⋊C8C22×C512yes—
240#391C22×C512yes—
241#389—21no—
242#389—21no—
243#384C22×C511yescharacteristic
244#380—21no—
245#380—21no—
246#340C22×C512yes—
247#340C22×C512yes—
248#336C22×C512yes—
249C23×C5—21no—
250C23×C5—21no—
251C23×C5—21no—
252C23×C5—21no—
253C23×C5—21no—
254C23×C5—21no—
255C23×C5—22no—
256C23×C5—41no—
257C23×C5—41no—
258C23×C5—41no—
259C23×C5—41no—
260C23×C5—41no—
261C23×C5—41no—
262C23×C5#13712yes—
263C23×C5C22×D411yescharacteristic
264C5×Q8—22no—
265C5×Q8—22no—
266C5×Q8—41no—
267C5×Q8—41no—
268C5×Q8—42no—
269C5×Q8—42no—
270C5×Q8—42no—
271C5×Q8—42no—
272C5×D4—22no—
273C5×D4—22no—
274C5×D4—48no—
275C5×D4—41no—
276C5×D4—41no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
282C2×C4×C5—22no—
283C2×C4×C5—22no—
284C2×C4×C5—22no—
285C2×C4×C5—21no—
286C2×C4×C5—21no—
287C2×C4×C5—21no—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
291C2×C4×C5—44no—
292C2×C4×C5—41no—
293C2×C4×C5—41no—
294C2×C4×C5—41no—
295C2×C4×C5—41no—
296C2×C4×C5—41no—
297C2×C4×C5—41no—
298C2×C4×C5—41no—
299C2×C4×C5—41no—
300C2×C4×C5—41no—
301C2×C4×C5—41no—
302C2×C4×C5—44no—
303C2×C4×C5—48no—
304C2×C4×C5—42no—
305C2×C4×C5—42no—
306C2×C4×C5—42no—
307C2×C4×C5—42no—
308C2×C4×C5—42no—
309C2×C4×C5—42no—
310C2×C4×C5#12012yes—
311C2×C4×C5#12012yes—
312C2×C4×C5#13512yes—
313C2×C4×C5#13712yes—
314C2×C4×C5C22×D411yescharacteristic
315C2×C4×C5C22×D411yescharacteristic
316C2×C4×C5H(5,𝔽2)11yescharacteristic
317C2×C4×C5H(5,𝔽2)11yescharacteristic
318C2×C4×C5H(5,𝔽2)11yescharacteristic
319C2×C4×C52−1+411yescharacteristic
320C5×C8—42no—
321C5×C8—42no—
322C22×Q8C23×C512yes—
323C22×Q8C23×C511yescharacteristic
324C22×D4—21no—
325C22×D4C23×C511yescharacteristic
326C23×C4—21no—
327C23×C4C5×D411yescharacteristic
328C23×C4C5×D411yescharacteristic
329C23×C4C23×C511yescharacteristic
330C2×Q16—24no—
331C2×Q16—42no—
332C2×Q16—42no—
333C2×SD16—24no—
334C2×SD16—42no—
335C2×SD16—42no—
336C22×C8C23×C512yes—
337C22×C8C23×C512yes—
338C4⋊Q8—22no—
339C4⋊Q8—22no—
340#126—28no—
341#125—22no—
342#125—22no—
343#124—22no—
344#124—22no—
345#123—28no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348C22⋊Q8—42no—
349C22⋊Q8—42no—
350#121—22no—
351#121—22no—
352C4×Q8—22no—
353C4×Q8C5×D414yes—
354C4×D4—28no—
355C4×D4—22no—
356C4×D4C5×D414yes—
357#117—28no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4C23×C512yes—
368C2×C4⋊C4C23×C511yescharacteristic
369C2×C4⋊C4C23×C511yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42C5×D411yescharacteristic
380C2×C42C5×D411yescharacteristic
381C2×C42C23×C511yescharacteristic
382C2×C42C23×C511yescharacteristic
383C4⋊C8—22no—
384C4⋊C8—22no—
385#103—24no—
386#103—24no—
387#103—42no—
388#102—42no—
389C22⋊C8—42no—
390#95—22no—
391#95—22no—
392C22×C5—22no—
393C22×C5—22no—
394C22×C5—22no—
395C22×C5—41no—
396C22×C5—41no—
397C22×C5—41no—
398C22×C5—41no—
399C22×C5—42no—
400C22×C5—42no—
401C22×C5—42no—
402C22×C5—81no—
403C22×C5—81no—
404C22×C5—81no—
405C22×C5—81no—
406C22×C5C2×#12011yescharacteristic
407C22×C5#53511yescharacteristic
408C22×C5#53711yescharacteristic
409C22×C5C2×#13511yescharacteristic
410C22×C5C2×#13711yescharacteristic
411C22×C5C2×#13711yescharacteristic
412C22×C5#57511yescharacteristic
413C4×C5—22no—
414C4×C5—41no—
415C4×C5—42no—
416C4×C5—41no—
417C4×C5—42no—
418C4×C5—41no—
419C4×C5—42no—
420C4×C5—82no—
421C4×C5—82no—
422C4×C5#45214yes—
423C4×C5#53511yescharacteristic
424C4×C5#53511yescharacteristic
425C4×C5#53711yescharacteristic
426C4×C5#53711yescharacteristic
427C24—21no—
428C24C2×C5×D411yescharacteristic
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8C2×C5×D412yes—
441C2×D4—24no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4C2×C5×D412yes—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—41no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4C2×C5×D411yescharacteristic
471C22×C4C2×C5×D411yescharacteristic
472C22×C4C2×C5×D411yescharacteristic
473C22×C4C2×C5×D411yescharacteristic
474C22×C4C2×C5×D411yescharacteristic
475C22×C4C24×C511yescharacteristic
476Q16—48no—
477SD16—48no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4C2×C5×D412yes—
498C4⋊C4C2×C5×D412yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—44no—
505C42—22no—
506C42—22no—
507C42—24no—
508C42—21no—
509C42—21no—
510C42C2×C5×D412yes—
511C42C2×C5×D412yes—
512C2×C5—42no—
513C2×C5—82no—
514C2×C5#261511yescharacteristic
515C2×C5C2×#45211yescharacteristic
516C2×C5#300711yescharacteristic
517C2×C5#321211yescharacteristic
518C2×C5#321811yescharacteristic
519C2×C5#324811yescharacteristic
520C2×C5#325211yescharacteristic
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23C5×#13712yes—
535C23C22×C5×D411yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540Q8—42no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544D4—22no—
545D4—22no—
546D4—48no—
547D4—41no—
548D4—41no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4C5×#12012yes—
583C2×C4C5×#12012yes—
584C2×C4C5×#13512yes—
585C2×C4C5×#13712yes—
586C2×C4C22×C5×D411yescharacteristic
587C2×C4C22×C5×D411yescharacteristic
588C2×C4C5×H(5,𝔽2)11yescharacteristic
589C2×C4C5×H(5,𝔽2)11yescharacteristic
590C2×C4C5×H(5,𝔽2)11yescharacteristic
591C2×C4C5×2−1+411yescharacteristic
592C8—42no—
593C8—42no—
594C5#2966511yescharacteristic, direct product factor, Sylow
595C22—22no—
596C22—22no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22C2×C5×#12011yescharacteristic
610C22C5×#53511yescharacteristic
611C22C5×#53711yescharacteristic
612C22C2×C5×#13511yescharacteristic
613C22C2×C5×#13711yescharacteristic
614C22C2×C5×#13711yescharacteristic
615C22C5×#57511yescharacteristic
616C4—22no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—42no—
623C4—82no—
624C4—82no—
625C4C5×#45214yes—
626C4C5×#53511yescharacteristic
627C4C5×#53511yescharacteristic
628C4C5×#53711yescharacteristic
629C4C5×#53711yescharacteristic
630C2—42no—
631C2—82no—
632C2C5×#261511yescharacteristic
633C2C2×C5×#45211yescharacteristic
634C2C5×#300711yescharacteristic
635C2C5×#321211yescharacteristic
636C2C5×#321811yescharacteristic
637C2C5×#324811yescharacteristic
638C2C5×#325211yescharacteristic
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC2×C5×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#462C2semidirect
3. C2×C5×#461C2semidirect
4. C5×#2857C2semidirect
5. C5×#2026C2semidirect
6. C5×#2015C2semidirect
7. C5×#2001C2semidirect
9. C5×#1803C2semidirect
10. C5×#1802C2semidirect
11. C5×#1797C2semidirect
47. #29665C5direct (N × H)
119. C2×#462C2×C5semidirect
120. C2×#461C2×C5semidirect
121. #2857C2×C5semidirect
122. #2026C2×C5semidirect
123. #2015C2×C5semidirect
124. #2001C2×C5semidirect
126. #1803C2×C5semidirect
127. #1802C2×C5semidirect
128. #1797C2×C5semidirect
594. C5#29665direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5292
C2×C5×#4622
C2×C5×#4612
C5×#28572
C5×#20262
C5×#20152
C5×#20012
C5×#20002
C5×#18032
C5×#18022
C5×#17972
#296655