Finite Group Databasebeta

← C5×#24263 #11946983

Subgroups — C5×#24263 #11946983

Order 1280.

1502 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3496C211yesmaximal, characteristic
2C2×C5×#458C212yesmaximal, complemented
3C5×#2618C212yesmaximal, complemented
4C5×#2322C211yesmaximal, characteristic, complemented
5C5×#2044C212yesmaximal, complemented
6C5×#2022C212yesmaximal, complemented
7C5×#1964C212yesmaximal, complemented
8C5×#1799C211yesmaximal, characteristic, complemented
9C5×#1790C212yesmaximal, complemented
10C2×C5×H(5,𝔽2)C2212yes—
11C5×#547—22no—
12C5×#546—24no—
13C5×D42—22no—
14C5×#537—21no—
15C5×#534—21no—
16C5×#534C2212yes—
17C5×#532—21no—
18C2×C5×#123C2211yescharacteristic
19C2×C5×#123C2214yescomplemented
20C2×C5×#121—21no—
21C2×C5×#121C2212yes—
22C2×C4×C5×D4C2211yescharacteristic
23C22×C5×C4⋊C4C2211yescharacteristic, complemented
24C5×#458—28no—
25C2×C5×#99C2212yes—
26C2×C5×#99C2214yescomplemented
27C2×C5×#99C2214yescomplemented
28C5×#398—22no—
29C5×#396—24no—
30C5×#394—22no—
31C5×#389—21no—
32C5×#388—21no—
33C5×#387—21no—
34C5×#386C2211yescharacteristic
35C5×#386C2214yescomplemented
36C5×#385—21no—
37C5×#385C2211yescharacteristic, complemented
38C5×#381C2211yescharacteristic, complemented
39C5×#380C2212yescomplemented
40C4×C5×C22⋊C4C2211yescharacteristic, complemented
41C5×#342C2212yescomplemented
42#24263C511yesmaximal, characteristic, direct product factor, Sylow
43C5×H(5,𝔽2)—24no—
44C5×H(5,𝔽2)—24no—
45C5×H(5,𝔽2)—44no—
46C2×C5×#41—22no—
47C2×C5×#41—24no—
48C22×C5×D4—22no—
49C22×C5×D4—24no—
50C22×C5×D4—21no—
51C22×C5×D4D412yes—
52C22×C5×D4C2314yes—
53C22×C5×D4C2311yescharacteristic
54C23×C4×C5D411yescharacteristic
55C23×C4×C5C2311yescharacteristic
56C23×C4×C5C2311yescharacteristic
57C23×C4×C5C2311yescharacteristic
58C5×C4⋊Q8—21no—
59C5×#127—21no—
60C5×#124—22no—
61C5×#123—24no—
62C5×#123—22no—
63C5×#123—24no—
64C5×#123—28no—
65C5×#123—44no—
66C5×#123D414yescomplemented
67C5×C22⋊Q8—24no—
68C5×#121—22no—
69C5×#121—22no—
70C5×#121—24no—
71C5×#121—22no—
72C5×#121—22no—
73C5×#121—22no—
74C5×#121—22no—
75C5×#121—22no—
76C5×#121—22no—
77C5×#120—24no—
78C5×#120—24no—
79C4×C5×D4—22no—
80C4×C5×D4—22no—
81C4×C5×D4—22no—
82C4×C5×D4—22no—
83C2×C5×C4⋊C4—21no—
84C2×C5×C4⋊C4—21no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4—21no—
87C2×C5×C4⋊C4—22no—
88C2×C5×C4⋊C4—22no—
89C2×C5×C4⋊C4—24no—
90C2×C5×C4⋊C4—22no—
91C2×C5×C4⋊C4—21no—
92C2×C5×C4⋊C4D411yescharacteristic
93C2×C5×C4⋊C4D412yes—
94C2×C5×C22⋊C4—22no—
95C2×C5×C22⋊C4—21no—
96C2×C5×C22⋊C4—24no—
97C2×C5×C22⋊C4—21no—
98C2×C5×C22⋊C4—24no—
99C2×C5×C22⋊C4D411yescharacteristic
100C2×C5×C22⋊C4D414yes—
101C2×C5×C22⋊C4C2311yescharacteristic
102C2×C5×C22⋊C4C2314yes—
103C2×C5×C22⋊C4C2311yescharacteristic
104C2×C5×C22⋊C4C2311yescharacteristic
105C2×C42×C5—21no—
106C2×C42×C5D411yescharacteristic
107C5×#99—24no—
108C5×#99—28no—
109C5×#99—44no—
110C5×#95—21no—
111C5×#95—22no—
112C5×#95—21no—
113C5×#95—21no—
114C5×#95—21no—
115C5×#95—21no—
116C5×#95—24no—
117#3496C2×C511yescharacteristic
118C2×#458C2×C512yescomplemented
119#2618C2×C512yescomplemented
120#2322C2×C511yescharacteristic, complemented
121#2044C2×C512yescomplemented
122#2022C2×C512yescomplemented
123#1964C2×C512yescomplemented
124#1799C2×C511yescharacteristic, complemented
125#1790C2×C512yescomplemented
126C24×C5—22no—
127C24×C5C2×D414yes—
128C24×C5C2×D411yescharacteristic
129C24×C5C2411yescharacteristic
130C5×#41—48no—
131C5×#41—48no—
132C5×#41—44no—
133C5×#41—44no—
134C2×C5×Q8—22no—
135C2×C5×D4—22no—
136C2×C5×D4—22no—
137C2×C5×D4—24no—
138C2×C5×D4—22no—
139C2×C5×D4—24no—
140C2×C5×D4—24no—
141C2×C5×D4—24no—
142C2×C5×D4—24no—
143C2×C5×D4—24no—
144C2×C5×D4—24no—
145C2×C5×D4—22no—
146C2×C5×D4—22no—
147C2×C5×D4—22no—
148C2×C5×D4—22no—
149C2×C5×D4—48no—
150C2×C5×D4—48no—
151C2×C5×D4—44no—
152C2×C5×D4—44no—
153C2×C5×D4—82no—
154C2×C5×D4—82no—
155C2×C5×D4C2×D412yes—
156C2×C5×D4#4112yes—
157C22×C4×C5—22no—
158C22×C4×C5—24no—
159C22×C4×C5—21no—
160C22×C4×C5—22no—
161C22×C4×C5—21no—
162C22×C4×C5—21no—
163C22×C4×C5—22no—
164C22×C4×C5—21no—
165C22×C4×C5—21no—
166C22×C4×C5—21no—
167C22×C4×C5—24no—
168C22×C4×C5—22no—
169C22×C4×C5—21no—
170C22×C4×C5—81no—
171C22×C4×C5—81no—
172C22×C4×C5—81no—
173C22×C4×C5C2×D414yes—
174C22×C4×C5C2×D411yescharacteristic
175C22×C4×C5C2×D411yescharacteristic
176C22×C4×C5C2×D411yescharacteristic
177C22×C4×C5C2×D411yescharacteristic
178C22×C4×C5C2×D411yescharacteristic
179C22×C4×C5C2×D412yes—
180C22×C4×C5C2×D412yes—
181C22×C4×C5C2×D412yescomplemented
182C5×C4⋊C4—22no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—22no—
185C5×C4⋊C4—22no—
186C5×C4⋊C4—22no—
187C5×C4⋊C4—24no—
188C5×C4⋊C4—48no—
189C5×C4⋊C4—41no—
190C5×C4⋊C4—41no—
191C5×C4⋊C4—42no—
192C5×C4⋊C4—82no—
193C5×C4⋊C4—81no—
194C5×C22⋊C4—24no—
195C5×C22⋊C4—24no—
196C5×C22⋊C4—24no—
197C5×C22⋊C4—22no—
198C5×C22⋊C4—21no—
199C5×C22⋊C4—24no—
200C5×C22⋊C4—22no—
201C5×C22⋊C4—24no—
202C5×C22⋊C4—24no—
203C5×C22⋊C4—24no—
204C5×C22⋊C4—22no—
205C5×C22⋊C4—48no—
206C5×C22⋊C4—81no—
207C5×C22⋊C4—84no—
208C5×C22⋊C4C2×D412yescomplemented
209C42×C5—21no—
210C42×C5—21no—
211C42×C5—81no—
212C2×H(5,𝔽2)C22×C512yes—
213#547—22no—
214#546—24no—
215D42—22no—
216#537—21no—
217#534—21no—
218#534C22×C512yes—
219#532—21no—
220C2×#123C22×C511yescharacteristic
221C2×#123C22×C514yescomplemented
222C2×#121—21no—
223C2×#121C22×C512yes—
224C2×C4×D4C22×C511yescharacteristic
225C22×C4⋊C4C22×C511yescharacteristic, complemented
226#458—28no—
227C2×#99C22×C512yes—
228C2×#99C22×C514yescomplemented
229C2×#99C22×C514yescomplemented
230#398—22no—
231#396—24no—
232#394—22no—
233#389—21no—
234#388—21no—
235#387—21no—
236#386C22×C511yescharacteristic
237#386C22×C514yescomplemented
238#385—21no—
239#385C22×C511yescharacteristic, complemented
240#381C22×C511yescharacteristic, complemented
241#380C22×C512yescomplemented
242C4×C22⋊C4C22×C511yescharacteristic, complemented
243#342C22×C512yescomplemented
244C23×C5—21no—
245C23×C5—21no—
246C23×C5—24no—
247C23×C5—24no—
248C23×C5—24no—
249C23×C5—41no—
250C23×C5—41no—
251C23×C5—41no—
252C23×C5—41no—
253C23×C5—44no—
254C23×C5—44no—
255C23×C5—44no—
256C23×C5—82no—
257C23×C5—82no—
258C23×C5#12112yes—
259C23×C5C22×D411yescharacteristic
260C23×C5C22×D412yes—
261C23×C5C22×D411yescharacteristic
262C23×C5C2×#4111yescharacteristic
263C23×C5H(5,𝔽2)12yes—
264C5×Q8—44no—
265C5×D4—24no—
266C5×D4—24no—
267C5×D4—44no—
268C5×D4—44no—
269C5×D4—48no—
270C5×D4—44no—
271C5×D4—44no—
272C5×D4—44no—
273C5×D4—44no—
274C5×D4—82no—
275C5×D4—82no—
276C5×D4—82no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—24no—
282C2×C4×C5—22no—
283C2×C4×C5—22no—
284C2×C4×C5—24no—
285C2×C4×C5—24no—
286C2×C4×C5—24no—
287C2×C4×C5—41no—
288C2×C4×C5—42no—
289C2×C4×C5—41no—
290C2×C4×C5—42no—
291C2×C4×C5—42no—
292C2×C4×C5—48no—
293C2×C4×C5—41no—
294C2×C4×C5—41no—
295C2×C4×C5—44no—
296C2×C4×C5—41no—
297C2×C4×C5—41no—
298C2×C4×C5—42no—
299C2×C4×C5—41no—
300C2×C4×C5—41no—
301C2×C4×C5—41no—
302C2×C4×C5—44no—
303C2×C4×C5—41no—
304C2×C4×C5—44no—
305C2×C4×C5—41no—
306C2×C4×C5—41no—
307C2×C4×C5—44no—
308C2×C4×C5—41no—
309C2×C4×C5—41no—
310C2×C4×C5—44no—
311C2×C4×C5—81no—
312C2×C4×C5#12011yescharacteristic
313C2×C4×C5#12011yescharacteristic
314C2×C4×C5#12012yes—
315C2×C4×C5#12112yes—
316H(5,𝔽2)—24no—
317H(5,𝔽2)—24no—
318H(5,𝔽2)—44no—
319C2×#41—22no—
320C2×#41—24no—
321C22×D4—22no—
322C22×D4—24no—
323C22×D4—21no—
324C22×D4C5×D412yes—
325C22×D4C23×C514yes—
326C22×D4C23×C511yescharacteristic
327C23×C4C5×D411yescharacteristic
328C23×C4C23×C511yescharacteristic
329C23×C4C23×C511yescharacteristic
330C23×C4C23×C511yescharacteristic
331C4⋊Q8—21no—
332#127—21no—
333#124—22no—
334#123—24no—
335#123—22no—
336#123—24no—
337#123—28no—
338#123—44no—
339#123C5×D414yescomplemented
340C22⋊Q8—24no—
341#121—22no—
342#121—22no—
343#121—24no—
344#121—22no—
345#121—22no—
346#121—22no—
347#121—22no—
348#121—22no—
349#121—22no—
350#120—24no—
351#120—24no—
352C4×D4—22no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4C5×D411yescharacteristic
366C2×C4⋊C4C5×D412yes—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4C5×D411yescharacteristic
373C2×C22⋊C4C5×D414yes—
374C2×C22⋊C4C23×C514yes—
375C2×C22⋊C4C23×C511yescharacteristic
376C2×C22⋊C4C23×C511yescharacteristic
377C2×C22⋊C4C23×C511yescharacteristic
378C2×C42—21no—
379C2×C42C5×D411yescharacteristic
380#99—24no—
381#99—28no—
382#99—44no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—24no—
390C22×C5—21no—
391C22×C5—22no—
392C22×C5—21no—
393C22×C5—22no—
394C22×C5—22no—
395C22×C5—41no—
396C22×C5—41no—
397C22×C5—42no—
398C22×C5—42no—
399C22×C5—41no—
400C22×C5—41no—
401C22×C5—44no—
402C22×C5—44no—
403C22×C5—44no—
404C22×C5—81no—
405C22×C5—84no—
406C22×C5—84no—
407C22×C5—164no—
408C22×C5C2×#12011yescharacteristic
409C22×C5C2×#12111yescharacteristic
410C22×C5#53511yescharacteristic
411C22×C5D4212yes—
412C22×C5#54612yes—
413C4×C5—22no—
414C4×C5—21no—
415C4×C5—21no—
416C4×C5—22no—
417C4×C5—42no—
418C4×C5—42no—
419C4×C5—44no—
420C4×C5—41no—
421C4×C5—41no—
422C4×C5—84no—
423C4×C5—81no—
424C4×C5—81no—
425C24—22no—
426C24C2×C5×D414yes—
427C24C2×C5×D411yescharacteristic
428C24C24×C511yescharacteristic
429#41—48no—
430#41—48no—
431#41—44no—
432#41—44no—
433C2×Q8—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—48no—
449C2×D4—48no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—82no—
453C2×D4—82no—
454C2×D4C2×C5×D412yes—
455C2×D4C5×#4112yes—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4C2×C5×D414yes—
473C22×C4C2×C5×D411yescharacteristic
474C22×C4C2×C5×D411yescharacteristic
475C22×C4C2×C5×D411yescharacteristic
476C22×C4C2×C5×D411yescharacteristic
477C22×C4C2×C5×D412yes—
478C22×C4C2×C5×D411yescharacteristic
479C22×C4C2×C5×D412yes—
480C22×C4C2×C5×D412yescomplemented
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—48no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—42no—
491C4⋊C4—82no—
492C4⋊C4—81no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—22no—
503C22⋊C4—24no—
504C22⋊C4—48no—
505C22⋊C4—81no—
506C22⋊C4—84no—
507C22⋊C4C2×C5×D412yescomplemented
508C42—21no—
509C42—21no—
510C42—81no—
511C2×C5—21no—
512C2×C5—22no—
513C2×C5—21no—
514C2×C5—22no—
515C2×C5—42no—
516C2×C5—84no—
517C2×C5#240811yescharacteristic
518C2×C5#302611yescharacteristic
519C2×C5#302811yescharacteristic
520C23—21no—
521C23—21no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—82no—
533C23—82no—
534C23C5×#12112yes—
535C23C22×C5×D411yescharacteristic
536C23C22×C5×D412yes—
537C23C22×C5×D411yescharacteristic
538C23C2×C5×#4111yescharacteristic
539C23C5×H(5,𝔽2)12yes—
540Q8—44no—
541D4—24no—
542D4—24no—
543D4—44no—
544D4—44no—
545D4—48no—
546D4—44no—
547D4—44no—
548D4—44no—
549D4—44no—
550D4—82no—
551D4—82no—
552D4—82no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—48no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—81no—
588C2×C4C5×#12011yescharacteristic
589C2×C4C5×#12011yescharacteristic
590C2×C4C5×#12012yes—
591C2×C4C5×#12112yes—
592C5#2426311yescharacteristic, direct product factor, Sylow
593C22—21no—
594C22—22no—
595C22—21no—
596C22—22no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22—42no—
601C22—42no—
602C22—41no—
603C22—41no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—81no—
608C22—84no—
609C22—84no—
610C22—164no—
611C22C2×C5×#12011yescharacteristic
612C22C2×C5×#12111yescharacteristic
613C22C5×#53511yescharacteristic
614C22C5×D4212yes—
615C22C5×#54612yes—
616C4—21no—
617C4—21no—
618C4—22no—
619C4—22no—
620C4—42no—
621C4—41no—
622C4—41no—
623C4—44no—
624C4—42no—
625C4—84no—
626C4—81no—
627C4—81no—
628C2—21no—
629C2—22no—
630C2—21no—
631C2—22no—
632C2—42no—
633C2—84no—
634C2C5×#240811yescharacteristic
635C2C5×#302611yescharacteristic
636C2C5×#302811yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC24C24×C5
prime radicalC5×#3496C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#458C2semidirect
3. C5×#2618C2semidirect
4. C5×#2322C2semidirect
5. C5×#2044C2semidirect
6. C5×#2022C2semidirect
7. C5×#1964C2semidirect
8. C5×#1799C2semidirect
9. C5×#1790C2semidirect
19. C2×C5×#123C22semidirect
23. C22×C5×C4⋊C4C22semidirect
26. C2×C5×#99C22semidirect
27. C2×C5×#99C22semidirect
35. C5×#386C22semidirect
37. C5×#385C22semidirect
38. C5×#381C22semidirect
39. C5×#380C22semidirect
40. C4×C5×C22⋊C4C22semidirect
41. C5×#342C22semidirect
42. #24263C5direct (N × H)
66. C5×#123D4semidirect
118. C2×#458C2×C5semidirect
119. #2618C2×C5semidirect
120. #2322C2×C5semidirect
121. #2044C2×C5semidirect
122. #2022C2×C5semidirect
123. #1964C2×C5semidirect
124. #1799C2×C5semidirect
125. #1790C2×C5semidirect
181. C22×C4×C5C2×D4semidirect
208. C5×C22⋊C4C2×D4semidirect
221. C2×#123C22×C5semidirect
225. C22×C4⋊C4C22×C5semidirect
228. C2×#99C22×C5semidirect
229. C2×#99C22×C5semidirect
237. #386C22×C5semidirect
239. #385C22×C5semidirect
240. #381C22×C5semidirect
241. #380C22×C5semidirect
242. C4×C22⋊C4C22×C5semidirect
243. #342C22×C5semidirect
339. #123C5×D4semidirect
480. C22×C4C2×C5×D4semidirect
507. C22⋊C4C2×C5×D4semidirect
592. C5#24263direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#34962
C2×C5×#4582
C5×#26182
C5×#23222
C5×#20442
C5×#20222
C5×#19642
C5×#17992
C5×#17902
#242635