Finite Group Databasebeta

← C5×#18297 #11941017

Subgroups — C5×#18297 #11941017

Order 1280.

1084 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#538C211yesmaximal, characteristic
2C5×#3189C211yesmaximal, characteristic, complemented
3C5×#3187C211yesmaximal, characteristic, complemented
4C5×#2626C211yesmaximal, characteristic, complemented
5C5×#1618C214yesmaximal, complemented
6C5×#1606C214yesmaximal, complemented
7C5×#1520C211yesmaximal, characteristic, complemented
8C5×#1519C211yesmaximal, characteristic, complemented
9C2×C5×#323C211yesmaximal, characteristic, complemented
10C5×#538—22no—
11C5×#538—22no—
12C5×#538—22no—
13C5×#538C2214yes—
14C2×C5×#126—21no—
15C2×C5×#124—21no—
16C2×C5×#123—21no—
17C2×C5×C22⋊Q8C2211yescharacteristic
18C2×C5×C22⋊Q8C2211yescharacteristic
19C2×C5×#121—21no—
20C2×C5×#121C2211yescharacteristic
21C2×C5×#120—21no—
22C2×C4×C5×D4—21no—
23C22×C5×C22⋊C4C2211yescharacteristic
24C5×#484—21no—
25C5×#484C2212yescomplemented
26C5×#483—21no—
27C5×#481C2212yescomplemented
28C5×#467—22no—
29C5×#465—22no—
30C5×#451—21no—
31C5×#451C2212yescomplemented
32C5×#450—21no—
33C5×#448C2212yescomplemented
34C2×C5×C22⋊C8C2211yescharacteristic
35C2×C5×C22⋊C8C2211yescharacteristic
36C5×#399C2212yescomplemented
37C5×#396—22no—
38C5×#393C2212yescomplemented
39C5×#385—21no—
40C5×#380C2211yescharacteristic
41C5×#379—21no—
42C5×#328C2214yescomplemented
43C5×#327C2214yescomplemented
44C5×#323C2214yescomplemented
45#18297C511yesmaximal, characteristic, direct product factor, Sylow
46C25×C5D411yescharacteristic
47C22×C5×Q8D411yescharacteristic
48C22×C5×Q8C2311yescharacteristic
49C22×C5×D4—21no—
50C22×C5×D4—21no—
51C22×C5×D4—21no—
52C22×C5×D4D411yescharacteristic
53C23×C4×C5—21no—
54C23×C4×C5D411yescharacteristic
55C23×C4×C5C2311yescharacteristic
56C2×C5×Q16—22no—
57C2×C5×SD16—22no—
58C22×C5×C8—21no—
59C22×C5×C8—21no—
60C5×#126—24no—
61C5×#126—24no—
62C5×#124—24no—
63C5×#124—24no—
64C5×#123—24no—
65C5×#123—24no—
66C5×C22⋊Q8—21no—
67C5×C22⋊Q8—21no—
68C5×C22⋊Q8—21no—
69C5×C22⋊Q8—21no—
70C5×C22⋊Q8—21no—
71C5×C22⋊Q8—21no—
72C5×C22⋊Q8D412yes—
73C5×C22⋊Q8C2312yes—
74C5×#121—24no—
75C5×#121—24no—
76C5×#121—21no—
77C5×#121—21no—
78C5×#121D412yes—
79C5×#121C2312yes—
80C5×#120—24no—
81C5×#120—24no—
82C4×C5×D4—24no—
83C4×C5×D4—24no—
84C2×C5×C4⋊C4—21no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4—21no—
87C2×C5×C4⋊C4—21no—
88C2×C5×C4⋊C4—22no—
89C2×C5×C4⋊C4D411yescharacteristic
90C2×C5×C4⋊C4D411yescharacteristic
91C2×C5×C4⋊C4C2311yescharacteristic
92C2×C5×C22⋊C4—21no—
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×C5×C22⋊C4—21no—
100C2×C5×C22⋊C4—21no—
101C2×C5×C22⋊C4—21no—
102C2×C5×C22⋊C4—21no—
103C2×C5×C22⋊C4—22no—
104C2×C5×C22⋊C4—22no—
105C2×C5×C22⋊C4D412yes—
106C2×C5×C22⋊C4C2312yes—
107C2×C42×C5—21no—
108C5×#107—22no—
109C5×#106—22no—
110C5×#103—22no—
111C5×#103—22no—
112C5×#103—22no—
113C5×#102—22no—
114C5×C22⋊C8—21no—
115C5×C22⋊C8—21no—
116C5×C22⋊C8C2312yes—
117C5×C22⋊C8C2312yes—
118C5×#95—22no—
119C5×#95C2312yes—
120C2×#538C2×C511yescharacteristic
121#3189C2×C511yescharacteristic, complemented
122#3187C2×C511yescharacteristic, complemented
123#2626C2×C511yescharacteristic, complemented
124#1618C2×C514yescomplemented
125#1606C2×C514yescomplemented
126#1520C2×C511yescharacteristic, complemented
127#1519C2×C511yescharacteristic, complemented
128C2×#323C2×C511yescharacteristic, complemented
129C24×C5—21no—
130C24×C5—21no—
131C24×C5—21no—
132C24×C5—22no—
133C24×C5—22no—
134C24×C5—81no—
135C24×C5—81no—
136C24×C5C2×D412yes—
137C24×C5C2×D411yescharacteristic
138C2×C5×Q8—21no—
139C2×C5×Q8—21no—
140C2×C5×Q8—21no—
141C2×C5×Q8—81no—
142C2×C5×Q8—81no—
143C2×C5×Q8C2×D412yes—
144C2×C5×D4—22no—
145C2×C5×D4—22no—
146C2×C5×D4—24no—
147C2×C5×D4—24no—
148C2×C5×D4—21no—
149C2×C5×D4—42no—
150C2×C5×D4—42no—
151C2×C5×D4—81no—
152C2×C5×D4—82no—
153C2×C5×D4—82no—
154C2×C5×D4C2×D412yes—
155C22×C4×C5—21no—
156C22×C4×C5—21no—
157C22×C4×C5—21no—
158C22×C4×C5—21no—
159C22×C4×C5—24no—
160C22×C4×C5—21no—
161C22×C4×C5—21no—
162C22×C4×C5—22no—
163C22×C4×C5—21no—
164C22×C4×C5—81no—
165C22×C4×C5—81no—
166C22×C4×C5—82no—
167C22×C4×C5C2×D412yes—
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5C2×D411yescharacteristic
171C22×C4×C5C2×D411yescharacteristic
172C22×C4×C5C2×D411yescharacteristic
173C22×C4×C5C2411yescharacteristic
174C5×Q16—82no—
175C5×SD16—82no—
176C2×C5×C8—22no—
177C2×C5×C8—22no—
178C2×C5×C8—81no—
179C2×C5×C8—81no—
180C5×C4⋊C4—24no—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—21no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—22no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—41no—
188C5×C4⋊C4—41no—
189C5×C4⋊C4—81no—
190C5×C4⋊C4—82no—
191C5×C4⋊C4C2×D412yes—
192C5×C4⋊C4C2×D412yes—
193C5×C22⋊C4—24no—
194C5×C22⋊C4—24no—
195C5×C22⋊C4—24no—
196C5×C22⋊C4—22no—
197C5×C22⋊C4—22no—
198C5×C22⋊C4—22no—
199C5×C22⋊C4—22no—
200C5×C22⋊C4—22no—
201C5×C22⋊C4—22no—
202C5×C22⋊C4—22no—
203C5×C22⋊C4—22no—
204C5×C22⋊C4—41no—
205C5×C22⋊C4—41no—
206C5×C22⋊C4—42no—
207C5×C22⋊C4—82no—
208C5×C22⋊C4—82no—
209C42×C5—24no—
210#538—22no—
211#538—22no—
212#538—22no—
213#538C22×C514yes—
214C2×#126—21no—
215C2×#124—21no—
216C2×#123—21no—
217C2×C22⋊Q8C22×C511yescharacteristic
218C2×C22⋊Q8C22×C511yescharacteristic
219C2×#121—21no—
220C2×#121C22×C511yescharacteristic
221C2×#120—21no—
222C2×C4×D4—21no—
223C22×C22⋊C4C22×C511yescharacteristic
224#484—21no—
225#484C22×C512yescomplemented
226#483—21no—
227#481C22×C512yescomplemented
228#467—22no—
229#465—22no—
230#451—21no—
231#451C22×C512yescomplemented
232#450—21no—
233#448C22×C512yescomplemented
234C2×C22⋊C8C22×C511yescharacteristic
235C2×C22⋊C8C22×C511yescharacteristic
236#399C22×C512yescomplemented
237#396—22no—
238#393C22×C512yescomplemented
239#385—21no—
240#380C22×C511yescharacteristic
241#379—21no—
242#328C22×C514yescomplemented
243#327C22×C514yescomplemented
244#323C22×C514yescomplemented
245C23×C5—22no—
246C23×C5—22no—
247C23×C5—22no—
248C23×C5—22no—
249C23×C5—41no—
250C23×C5—41no—
251C23×C5—42no—
252C23×C5—42no—
253C23×C5—42no—
254C23×C5—42no—
255C23×C5—42no—
256C23×C5—42no—
257C23×C5—42no—
258C23×C5—42no—
259C23×C5—42no—
260C23×C5—81no—
261C23×C5—81no—
262C23×C5—84no—
263C23×C5—84no—
264C23×C5#12011yescharacteristic
265C23×C5#12011yescharacteristic
266C23×C5#13512yes—
267C23×C5#13712yes—
268C23×C5C22×D411yescharacteristic
269C5×Q8—42no—
270C5×Q8—42no—
271C5×Q8—42no—
272C5×Q8—81no—
273C5×D4—42no—
274C5×D4—42no—
275C5×D4—42no—
276C5×D4—44no—
277C5×D4—81no—
278C5×D4—84no—
279C5×D4—84no—
280C2×C4×C5—22no—
281C2×C4×C5—22no—
282C2×C4×C5—22no—
283C2×C4×C5—22no—
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—42no—
291C2×C4×C5—42no—
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—42no—
299C2×C4×C5—41no—
300C2×C4×C5—41no—
301C2×C4×C5—42no—
302C2×C4×C5—81no—
303C2×C4×C5—84no—
304C2×C4×C5#12011yescharacteristic
305C2×C4×C5#12011yescharacteristic
306C2×C4×C5C22×D411yescharacteristic
307C2×C4×C5C22×D411yescharacteristic
308C5×C8—42no—
309C5×C8—42no—
310C25C5×D411yescharacteristic
311C22×Q8C5×D411yescharacteristic
312C22×Q8C23×C511yescharacteristic
313C22×D4—21no—
314C22×D4—21no—
315C22×D4—21no—
316C22×D4C5×D411yescharacteristic
317C23×C4—21no—
318C23×C4C5×D411yescharacteristic
319C23×C4C23×C511yescharacteristic
320C2×Q16—22no—
321C2×SD16—22no—
322C22×C8—21no—
323C22×C8—21no—
324#126—24no—
325#126—24no—
326#124—24no—
327#124—24no—
328#123—24no—
329#123—24no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8C5×D412yes—
337C22⋊Q8C23×C512yes—
338#121—24no—
339#121—24no—
340#121—21no—
341#121—21no—
342#121C5×D412yes—
343#121C23×C512yes—
344#120—24no—
345#120—24no—
346C4×D4—24no—
347C4×D4—24no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4C5×D411yescharacteristic
354C2×C4⋊C4C5×D411yescharacteristic
355C2×C4⋊C4C23×C511yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4C5×D412yes—
370C2×C22⋊C4C23×C512yes—
371C2×C42—21no—
372#107—22no—
373#106—22no—
374#103—22no—
375#103—22no—
376#103—22no—
377#102—22no—
378C22⋊C8—21no—
379C22⋊C8—21no—
380C22⋊C8C23×C512yes—
381C22⋊C8C23×C512yes—
382#95—22no—
383#95C23×C512yes—
384C22×C5—21no—
385C22×C5—21no—
386C22×C5—21no—
387C22×C5—21no—
388C22×C5—21no—
389C22×C5—21no—
390C22×C5—42no—
391C22×C5—42no—
392C22×C5—44no—
393C22×C5—42no—
394C22×C5—42no—
395C22×C5—42no—
396C22×C5—42no—
397C22×C5—42no—
398C22×C5—42no—
399C22×C5—82no—
400C22×C5—84no—
401C22×C5—84no—
402C22×C5—84no—
403C22×C5#45212yes—
404C22×C5#45212yes—
405C22×C5C2×#12011yescharacteristic
406C22×C5C2×#13511yescharacteristic
407C22×C5C2×#13711yescharacteristic
408C4×C5—21no—
409C4×C5—21no—
410C4×C5—21no—
411C4×C5—21no—
412C4×C5—42no—
413C4×C5—42no—
414C4×C5—42no—
415C4×C5—42no—
416C4×C5—82no—
417C4×C5—81no—
418C4×C5—81no—
419C24—21no—
420C24—21no—
421C24—21no—
422C24—22no—
423C24—22no—
424C24—81no—
425C24—81no—
426C24C2×C5×D412yes—
427C24C2×C5×D411yescharacteristic
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—81no—
432C2×Q8—81no—
433C2×Q8C2×C5×D412yes—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—24no—
438C2×D4—21no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—81no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4C2×C5×D412yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—82no—
457C22×C4C2×C5×D412yes—
458C22×C4C2×C5×D411yescharacteristic
459C22×C4C2×C5×D411yescharacteristic
460C22×C4C2×C5×D411yescharacteristic
461C22×C4C2×C5×D411yescharacteristic
462C22×C4C2×C5×D411yescharacteristic
463C22×C4C24×C511yescharacteristic
464Q16—82no—
465SD16—82no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—81no—
469C2×C8—81no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—81no—
480C4⋊C4—82no—
481C4⋊C4C2×C5×D412yes—
482C4⋊C4C2×C5×D412yes—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—42no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C42—24no—
500C2×C5—21no—
501C2×C5—21no—
502C2×C5—22no—
503C2×C5—22no—
504C2×C5—82no—
505C2×C5—82no—
506C2×C5C2×#45211yescharacteristic
507C2×C5#301911yescharacteristic
508C2×C5#302611yescharacteristic
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—81no—
525C23—81no—
526C23—84no—
527C23—84no—
528C23C5×#12011yescharacteristic
529C23C5×#12011yescharacteristic
530C23C5×#13512yes—
531C23C5×#13712yes—
532C23C22×C5×D411yescharacteristic
533Q8—42no—
534Q8—42no—
535Q8—42no—
536Q8—81no—
537D4—42no—
538D4—44no—
539D4—42no—
540D4—42no—
541D4—81no—
542D4—84no—
543D4—84no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—81no—
567C2×C4—84no—
568C2×C4C5×#12011yescharacteristic
569C2×C4C5×#12011yescharacteristic
570C2×C4C22×C5×D411yescharacteristic
571C2×C4C22×C5×D411yescharacteristic
572C8—42no—
573C8—42no—
574C5#1829711yescharacteristic, direct product factor, Sylow
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—42no—
582C22—42no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—82no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22C5×#45212yes—
595C22C5×#45212yes—
596C22C2×C5×#12011yescharacteristic
597C22C2×C5×#13511yescharacteristic
598C22C2×C5×#13711yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—82no—
608C4—81no—
609C4—81no—
610C2—21no—
611C2—21no—
612C2—22no—
613C2—22no—
614C2—82no—
615C2—82no—
616C2C2×C5×#45211yescharacteristic
617C2C5×#301911yescharacteristic
618C2C5×#302611yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC2×C5×#538C2
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×#3189C2semidirect
3. C5×#3187C2semidirect
4. C5×#2626C2semidirect
5. C5×#1618C2semidirect
6. C5×#1606C2semidirect
7. C5×#1520C2semidirect
8. C5×#1519C2semidirect
9. C2×C5×#323C2semidirect
25. C5×#484C22semidirect
27. C5×#481C22semidirect
31. C5×#451C22semidirect
33. C5×#448C22semidirect
36. C5×#399C22semidirect
38. C5×#393C22semidirect
42. C5×#328C22semidirect
43. C5×#327C22semidirect
44. C5×#323C22semidirect
45. #18297C5direct (N × H)
121. #3189C2×C5semidirect
122. #3187C2×C5semidirect
123. #2626C2×C5semidirect
124. #1618C2×C5semidirect
125. #1606C2×C5semidirect
126. #1520C2×C5semidirect
127. #1519C2×C5semidirect
128. C2×#323C2×C5semidirect
225. #484C22×C5semidirect
227. #481C22×C5semidirect
231. #451C22×C5semidirect
233. #448C22×C5semidirect
236. #399C22×C5semidirect
238. #393C22×C5semidirect
242. #328C22×C5semidirect
243. #327C22×C5semidirect
244. #323C22×C5semidirect
574. C5#18297direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5382
C5×#31892
C5×#31872
C5×#26262
C5×#16182
C5×#16062
C5×#15202
C5×#15192
C2×C5×#3232
#182975