Finite Group Databasebeta

← C5×#29689 #11952409

Subgroups — C5×#29689 #11952409

Order 1280.

1100 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#529C211yesmaximal, characteristic
2C2×C5×#475C211yesmaximal, characteristic, complemented
3C2×C5×#474C211yesmaximal, characteristic, complemented
4C5×#2857C211yesmaximal, characteristic, complemented
5C5×#2059C212yesmaximal, complemented
6C5×#2042C212yesmaximal, complemented
7C5×#2036C212yesmaximal, complemented
8C5×#2034C212yesmaximal
9C5×#1803C211yesmaximal, characteristic, complemented
10C5×#1802C211yesmaximal, characteristic, complemented
11C5×#1797C211yesmaximal, characteristic, complemented
12C2×C5×C4⋊Q8C2211yescharacteristic
13C5×#529—24no—
14C5×#529—24no—
15C2×C5×#126—21no—
16C2×C5×#125C2211yescharacteristic
17C2×C5×#124C2211yescharacteristic
18C2×C5×#123—21no—
19C2×C5×C22⋊Q8—21no—
20C2×C5×C22⋊Q8C2211yescharacteristic
21C2×C5×#121C2211yescharacteristic
22C2×C4×C5×Q8C2211yescharacteristic
23C2×C4×C5×D4—21no—
24C2×C4×C5×D4C2211yescharacteristic
25C2×C5×#117—21no—
26C22×C42×C5C2211yescharacteristic
27C5×#475—24no—
28C5×#474—24no—
29C2×C5×#107C2212yes—
30C2×C5×#106C2212yes—
31C2×C5×C4⋊C8C2211yescharacteristic
32C2×C5×C4⋊C8C2211yescharacteristic
33C2×C5×#103C2212yes—
34C2×C5×#103C2212yes—
35C2×C5×#102C2212yes—
36C2×C5×#102C2212yes—
37C2×C5×C22⋊C8C2212yes—
38C5×#391—21no—
39C5×#389—21no—
40C5×#389C2212yes—
41C5×#384C2211yescharacteristic
42C5×#380—21no—
43C5×#380C2212yes—
44C5×#340C2212yes—
45C5×#340C2212yes—
46C5×#336C2212yes—
47#29689C511yesmaximal, characteristic, direct product factor, Sylow
48C22×C5×Q8—21no—
49C22×C5×Q8C2311yescharacteristic
50C22×C5×D4—21no—
51C22×C5×D4C2311yescharacteristic
52C23×C4×C5—21no—
53C23×C4×C5D411yescharacteristic
54C23×C4×C5D411yescharacteristic
55C23×C4×C5C2311yescharacteristic
56C22×C5×C8C2312yes—
57C22×C5×C8C2312yes—
58C5×C4⋊Q8—22no—
59C5×C4⋊Q8—22no—
60C5×#126—28no—
61C5×#125—22no—
62C5×#125—22no—
63C5×#124—22no—
64C5×#124—22no—
65C5×#123—28no—
66C5×C22⋊Q8—22no—
67C5×C22⋊Q8—22no—
68C5×C22⋊Q8—44no—
69C5×#121—22no—
70C5×#121—22no—
71C4×C5×Q8—22no—
72C4×C5×Q8D414yes—
73C4×C5×D4—28no—
74C4×C5×D4—22no—
75C4×C5×D4D414yes—
76C5×#117—28no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4—21no—
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—22no—
85C2×C5×C4⋊C4C2312yes—
86C2×C5×C4⋊C4C2312yes—
87C2×C5×C4⋊C4C2311yescharacteristic
88C2×C5×C4⋊C4C2311yescharacteristic
89C2×C5×C22⋊C4—21no—
90C2×C5×C22⋊C4—21no—
91C2×C5×C22⋊C4—21no—
92C2×C5×C22⋊C4—21no—
93C2×C5×C22⋊C4—21no—
94C2×C5×C22⋊C4—22no—
95C2×C42×C5—21no—
96C2×C42×C5—22no—
97C2×C42×C5—22no—
98C2×C42×C5D411yescharacteristic
99C2×C42×C5D411yescharacteristic
100C2×C42×C5C2311yescharacteristic
101C2×C42×C5C2311yescharacteristic
102C5×#107—24no—
103C5×#106—24no—
104C5×C4⋊C8—22no—
105C5×C4⋊C8—22no—
106C5×#103—24no—
107C5×#103—42no—
108C5×#102—24no—
109C5×#102—42no—
110C5×C22⋊C8—42no—
111C5×#95—22no—
112C5×#95—22no—
113C2×#529C2×C511yescharacteristic
114C2×#475C2×C511yescharacteristic, complemented
115C2×#474C2×C511yescharacteristic, complemented
116#2857C2×C511yescharacteristic, complemented
117#2059C2×C512yescomplemented
118#2042C2×C512yescomplemented
119#2036C2×C512yescomplemented
120#2034C2×C512yes—
121#1803C2×C511yescharacteristic, complemented
122#1802C2×C511yescharacteristic, complemented
123#1797C2×C511yescharacteristic, complemented
124C24×C5—21no—
125C24×C5C2×D411yescharacteristic
126C2×C5×Q8—21no—
127C2×C5×Q8—21no—
128C2×C5×Q8—21no—
129C2×C5×Q8—21no—
130C2×C5×Q8—21no—
131C2×C5×Q8—42no—
132C2×C5×Q8—42no—
133C2×C5×Q8—42no—
134C2×C5×Q8C2×D412yes—
135C2×C5×D4—24no—
136C2×C5×D4—21no—
137C2×C5×D4—21no—
138C2×C5×D4—21no—
139C2×C5×D4—21no—
140C2×C5×D4—21no—
141C2×C5×D4—42no—
142C2×C5×D4—42no—
143C2×C5×D4C2×D412yes—
144C22×C4×C5—22no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—22no—
149C22×C4×C5—22no—
150C22×C4×C5—24no—
151C22×C4×C5—22no—
152C22×C4×C5—22no—
153C22×C4×C5—21no—
154C22×C4×C5—21no—
155C22×C4×C5—21no—
156C22×C4×C5—21no—
157C22×C4×C5—21no—
158C22×C4×C5—21no—
159C22×C4×C5—22no—
160C22×C4×C5—22no—
161C22×C4×C5—41no—
162C22×C4×C5—42no—
163C22×C4×C5—42no—
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2411yescharacteristic
170C2×C5×C8—22no—
171C2×C5×C8—22no—
172C2×C5×C8—22no—
173C2×C5×C8—22no—
174C2×C5×C8—22no—
175C2×C5×C8—22no—
176C5×C4⋊C4—24no—
177C5×C4⋊C4—24no—
178C5×C4⋊C4—24no—
179C5×C4⋊C4—24no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4—22no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—22no—
185C5×C4⋊C4—22no—
186C5×C4⋊C4—42no—
187C5×C4⋊C4—44no—
188C5×C4⋊C4—42no—
189C5×C4⋊C4—42no—
190C5×C4⋊C4C2×D412yes—
191C5×C4⋊C4C2×D412yes—
192C5×C22⋊C4—24no—
193C5×C22⋊C4—24no—
194C5×C22⋊C4—24no—
195C5×C22⋊C4—24no—
196C5×C22⋊C4—24no—
197C5×C22⋊C4—44no—
198C42×C5—22no—
199C42×C5—22no—
200C42×C5—24no—
201C42×C5—21no—
202C42×C5—21no—
203C42×C5C2×D412yes—
204C42×C5C2×D412yes—
205C2×C4⋊Q8C22×C511yescharacteristic
206#529—24no—
207#529—24no—
208C2×#126—21no—
209C2×#125C22×C511yescharacteristic
210C2×#124C22×C511yescharacteristic
211C2×#123—21no—
212C2×C22⋊Q8—21no—
213C2×C22⋊Q8C22×C511yescharacteristic
214C2×#121C22×C511yescharacteristic
215C2×C4×Q8C22×C511yescharacteristic
216C2×C4×D4—21no—
217C2×C4×D4C22×C511yescharacteristic
218C2×#117—21no—
219C22×C42C22×C511yescharacteristic
220#475—24no—
221#474—24no—
222C2×#107C22×C512yes—
223C2×#106C22×C512yes—
224C2×C4⋊C8C22×C511yescharacteristic
225C2×C4⋊C8C22×C511yescharacteristic
226C2×#103C22×C512yes—
227C2×#103C22×C512yes—
228C2×#102C22×C512yes—
229C2×#102C22×C512yes—
230C2×C22⋊C8C22×C512yes—
231#391—21no—
232#389—21no—
233#389C22×C512yes—
234#384C22×C511yescharacteristic
235#380—21no—
236#380C22×C512yes—
237#340C22×C512yes—
238#340C22×C512yes—
239#336C22×C512yes—
240C23×C5—21no—
241C23×C5—21no—
242C23×C5—21no—
243C23×C5—21no—
244C23×C5—21no—
245C23×C5—21no—
246C23×C5—22no—
247C23×C5—41no—
248C23×C5—41no—
249C23×C5—41no—
250C23×C5—41no—
251C23×C5—41no—
252C23×C5—41no—
253C23×C5#13712yes—
254C23×C5C22×D411yescharacteristic
255C5×Q8—22no—
256C5×Q8—22no—
257C5×Q8—41no—
258C5×Q8—41no—
259C5×Q8—84no—
260C5×D4—22no—
261C5×D4—22no—
262C5×D4—48no—
263C5×D4—41no—
264C5×D4—41no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—21no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—44no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—41no—
290C2×C4×C5—44no—
291C2×C4×C5—48no—
292C2×C4×C5—42no—
293C2×C4×C5—42no—
294C2×C4×C5—42no—
295C2×C4×C5—42no—
296C2×C4×C5—42no—
297C2×C4×C5—42no—
298C2×C4×C5#12012yes—
299C2×C4×C5#12012yes—
300C2×C4×C5#13512yes—
301C2×C4×C5AGL(1,ℤ/8)12yes—
302C2×C4×C5C22×D411yescharacteristic
303C2×C4×C5C22×D411yescharacteristic
304C2×C4×C5H(5,𝔽2)11yescharacteristic
305C2×C4×C52−1+411yescharacteristic
306C2×C4×C52−1+411yescharacteristic
307C2×C4×C52−1+411yescharacteristic
308C5×C8—42no—
309C5×C8—42no—
310C22×Q8—21no—
311C22×Q8C23×C511yescharacteristic
312C22×D4—21no—
313C22×D4C23×C511yescharacteristic
314C23×C4—21no—
315C23×C4C5×D411yescharacteristic
316C23×C4C5×D411yescharacteristic
317C23×C4C23×C511yescharacteristic
318C22×C8C23×C512yes—
319C22×C8C23×C512yes—
320C4⋊Q8—22no—
321C4⋊Q8—22no—
322#126—28no—
323#125—22no—
324#125—22no—
325#124—22no—
326#124—22no—
327#123—28no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—44no—
331#121—22no—
332#121—22no—
333C4×Q8—22no—
334C4×Q8C5×D414yes—
335C4×D4—28no—
336C4×D4—22no—
337C4×D4C5×D414yes—
338#117—28no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4C23×C512yes—
348C2×C4⋊C4C23×C512yes—
349C2×C4⋊C4C23×C511yescharacteristic
350C2×C4⋊C4C23×C511yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—22no—
357C2×C42—21no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42C5×D411yescharacteristic
361C2×C42C5×D411yescharacteristic
362C2×C42C23×C511yescharacteristic
363C2×C42C23×C511yescharacteristic
364#107—24no—
365#106—24no—
366C4⋊C8—22no—
367C4⋊C8—22no—
368#103—24no—
369#103—42no—
370#102—24no—
371#102—42no—
372C22⋊C8—42no—
373#95—22no—
374#95—22no—
375C22×C5—22no—
376C22×C5—22no—
377C22×C5—22no—
378C22×C5—41no—
379C22×C5—41no—
380C22×C5—41no—
381C22×C5—41no—
382C22×C5—42no—
383C22×C5—42no—
384C22×C5—42no—
385C22×C5—81no—
386C22×C5—81no—
387C22×C5—81no—
388C22×C5—81no—
389C22×C5C2×#12011yescharacteristic
390C22×C5#53611yescharacteristic
391C22×C5#53711yescharacteristic
392C22×C5C2×#13511yescharacteristic
393C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
394C22×C5C2×#13711yescharacteristic
395C22×C5#57511yescharacteristic
396C4×C5—22no—
397C4×C5—41no—
398C4×C5—42no—
399C4×C5—41no—
400C4×C5—42no—
401C4×C5—41no—
402C4×C5—42no—
403C4×C5—82no—
404C4×C5—82no—
405C4×C5#44914yes—
406C4×C5#53611yescharacteristic
407C4×C5#53611yescharacteristic
408C4×C5#53711yescharacteristic
409C4×C5#53711yescharacteristic
410C24—21no—
411C24C2×C5×D411yescharacteristic
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8C2×C5×D412yes—
421C2×D4—24no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4C2×C5×D412yes—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4C2×C5×D411yescharacteristic
451C22×C4C2×C5×D411yescharacteristic
452C22×C4C2×C5×D411yescharacteristic
453C22×C4C2×C5×D411yescharacteristic
454C22×C4C2×C5×D411yescharacteristic
455C22×C4C24×C511yescharacteristic
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4C2×C5×D412yes—
477C4⋊C4C2×C5×D412yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—44no—
484C42—22no—
485C42—22no—
486C42—24no—
487C42—21no—
488C42—21no—
489C42C2×C5×D412yes—
490C42C2×C5×D412yes—
491C2×C5—42no—
492C2×C5—82no—
493C2×C5#261611yescharacteristic
494C2×C5C2×#44911yescharacteristic
495C2×C5#300711yescharacteristic
496C2×C5#323311yescharacteristic
497C2×C5#323911yescharacteristic
498C2×C5#324811yescharacteristic
499C2×C5#325111yescharacteristic
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C5×#13712yes—
514C23C22×C5×D411yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—41no—
518Q8—41no—
519Q8—84no—
520D4—22no—
521D4—22no—
522D4—48no—
523D4—41no—
524D4—41no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—48no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4C5×#12012yes—
559C2×C4C5×#12012yes—
560C2×C4C5×#13512yes—
561C2×C4C5×AGL(1,ℤ/8)12yes—
562C2×C4C22×C5×D411yescharacteristic
563C2×C4C22×C5×D411yescharacteristic
564C2×C4C5×H(5,𝔽2)11yescharacteristic
565C2×C4C5×2−1+411yescharacteristic
566C2×C4C5×2−1+411yescharacteristic
567C2×C4C5×2−1+411yescharacteristic
568C8—42no—
569C8—42no—
570C5#2968911yescharacteristic, direct product factor, Sylow
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×C5×#12011yescharacteristic
586C22C5×#53611yescharacteristic
587C22C5×#53711yescharacteristic
588C22C2×C5×#13511yescharacteristic
589C22C2×C5×AGL(1,ℤ/8)11yescharacteristic
590C22C2×C5×#13711yescharacteristic
591C22C5×#57511yescharacteristic
592C4—22no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—42no—
597C4—41no—
598C4—42no—
599C4—82no—
600C4—82no—
601C4C5×#44914yes—
602C4C5×#53611yescharacteristic
603C4C5×#53611yescharacteristic
604C4C5×#53711yescharacteristic
605C4C5×#53711yescharacteristic
606C2—42no—
607C2—82no—
608C2C5×#261611yescharacteristic
609C2C2×C5×#44911yescharacteristic
610C2C5×#300711yescharacteristic
611C2C5×#323311yescharacteristic
612C2C5×#323911yescharacteristic
613C2C5×#324811yescharacteristic
614C2C5×#325111yescharacteristic
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×#475C2semidirect
3. C2×C5×#474C2semidirect
4. C5×#2857C2semidirect
5. C5×#2059C2semidirect
6. C5×#2042C2semidirect
7. C5×#2036C2semidirect
9. C5×#1803C2semidirect
10. C5×#1802C2semidirect
11. C5×#1797C2semidirect
47. #29689C5direct (N × H)
114. C2×#475C2×C5semidirect
115. C2×#474C2×C5semidirect
116. #2857C2×C5semidirect
117. #2059C2×C5semidirect
118. #2042C2×C5semidirect
119. #2036C2×C5semidirect
121. #1803C2×C5semidirect
122. #1802C2×C5semidirect
123. #1797C2×C5semidirect
570. C5#29689direct (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×#4752
C2×C5×#4742
C5×#28572
C5×#20592
C5×#20422
C5×#20362
C5×#20342
C5×#18032
C5×#18022
C5×#17972
#296895