Finite Group Databasebeta

← C5×#18259 #11940979

Subgroups — C5×#18259 #11940979

Order 1280.

1132 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3538C211yesmaximal, characteristic
2C5×#3004C211yesmaximal, characteristic
3C2×C5×#451C211yesmaximal, characteristic, complemented
4C5×#2434C211yesmaximal, characteristic, complemented
5C5×#1605C214yesmaximal, complemented
6C5×#1603C214yesmaximal, complemented
7C5×#1508C211yesmaximal, characteristic, complemented
8C5×#1501C211yesmaximal, characteristic
9C5×#1472C211yesmaximal, characteristic, complemented
10C23×C5×Q8C2211yescharacteristic
11C2×C5×#137—21no—
12C22×C5×Q16—21no—
13C5×#563—22no—
14C5×#552—21no—
15C5×#552—21no—
16C5×#551—22no—
17C5×D4×Q8—21no—
18C5×D4×Q8—21no—
19C5×#544—21no—
20C5×#544C2212yes—
21C5×#541—21no—
22C5×#539—21no—
23C5×#539C2212yes—
24C5×#536—21no—
25C2×C5×C4⋊Q8—21no—
26C5×#525C2211yescharacteristic
27C2×C5×C22⋊Q8C2211yescharacteristic
28C2×C5×C22⋊Q8C2211yescharacteristic
29C5×#451—22no—
30C5×#451—21no—
31C5×#451C2214yes—
32C5×#451C2212yes—
33C5×#448—21no—
34C5×#448C2212yes—
35C5×#419—21no—
36C2×C5×#103—21no—
37C5×#407C2211yescharacteristic
38C2×C5×C22⋊C8C2211yescharacteristic
39C5×#395C2214yes—
40C5×#391—21no—
41C5×#385—22no—
42C4×C5×C22⋊C4C2211yescharacteristic
43C5×#328C2212yes—
44C5×#328C2214yes—
45C5×#327C2212yes—
46C5×#324C2214yes—
47#18259C511yesmaximal, characteristic, direct product factor, Sylow
48C2×C5×#41D411yescharacteristic
49C22×C5×Q8—21no—
50C22×C5×Q8—21no—
51C22×C5×Q8—21no—
52C22×C5×Q8—21no—
53C22×C5×Q8—21no—
54C22×C5×Q8—21no—
55C22×C5×Q8—44no—
56C22×C5×Q8D411yescharacteristic
57C22×C5×Q8D411yescharacteristic
58C22×C5×Q8C2312yes—
59C23×C4×C5—21no—
60C23×C4×C5C2311yescharacteristic
61C5×#137—44no—
62C2×C5×Q16—24no—
63C2×C5×Q16—22no—
64C2×C5×Q16—44no—
65C2×C5×SD16—22no—
66C2×C5×M16—21no—
67C22×C5×C8—21no—
68C5×C4⋊Q8—21no—
69C5×C4⋊Q8—22no—
70C5×C4⋊Q8—22no—
71C5×C4⋊Q8—22no—
72C5×C4⋊Q8—44no—
73C5×#126—22no—
74C5×#126—22no—
75C5×#126—22no—
76C5×#126—22no—
77C5×#125—21no—
78C5×#125—22no—
79C5×#125—22no—
80C5×#125—22no—
81C5×#125—22no—
82C5×#124—22no—
83C5×#123—22no—
84C5×#123—22no—
85C5×#123—22no—
86C5×#123—21no—
87C5×#123—22no—
88C5×C22⋊Q8—21no—
89C5×C22⋊Q8—21no—
90C5×C22⋊Q8—22no—
91C5×C22⋊Q8—22no—
92C5×C22⋊Q8—22no—
93C5×C22⋊Q8—21no—
94C5×C22⋊Q8—21no—
95C5×C22⋊Q8—21no—
96C5×C22⋊Q8—21no—
97C5×C22⋊Q8—21no—
98C5×C22⋊Q8—21no—
99C5×C22⋊Q8—22no—
100C5×C22⋊Q8D412yes—
101C5×C22⋊Q8D411yescharacteristic
102C5×C22⋊Q8C2312yes—
103C5×C22⋊Q8C2311yescharacteristic
104C5×#121D411yescharacteristic
105C5×#121C2311yescharacteristic
106C5×#120—21no—
107C4×C5×Q8—21no—
108C4×C5×Q8—21no—
109C4×C5×D4—22no—
110C4×C5×D4—21no—
111C4×C5×D4—21no—
112C4×C5×D4—22no—
113C5×#117—21no—
114C5×#117—22no—
115C5×#117—22no—
116C5×#117D411yescharacteristic
117C2×C5×C4⋊C4—21no—
118C2×C5×C4⋊C4—24no—
119C2×C5×C4⋊C4D411yescharacteristic
120C2×C5×C4⋊C4D411yescharacteristic
121C2×C5×C4⋊C4C2312yes—
122C2×C5×C22⋊C4—21no—
123C2×C5×C22⋊C4—21no—
124C2×C5×C22⋊C4D412yes—
125C2×C42×C5—21no—
126C5×#103—24no—
127C5×#103—22no—
128C5×#103—22no—
129C5×C22⋊C8—21no—
130C5×C22⋊C8—21no—
131C5×C22⋊C8C2312yes—
132C5×C22⋊C8C2312yes—
133C5×#95—24no—
134C5×#95C2312yes—
135#3538C2×C511yescharacteristic
136#3004C2×C511yescharacteristic
137C2×#451C2×C511yescharacteristic, complemented
138#2434C2×C511yescharacteristic, complemented
139#1605C2×C514yescomplemented
140#1603C2×C514yescomplemented
141#1508C2×C511yescharacteristic, complemented
142#1501C2×C511yescharacteristic
143#1472C2×C511yescharacteristic, complemented
144C24×C5C2×D411yescharacteristic
145C5×#41—42no—
146C2×C5×Q8—24no—
147C2×C5×Q8—22no—
148C2×C5×Q8—21no—
149C2×C5×Q8—21no—
150C2×C5×Q8—21no—
151C2×C5×Q8—24no—
152C2×C5×Q8—24no—
153C2×C5×Q8—22no—
154C2×C5×Q8—22no—
155C2×C5×Q8—44no—
156C2×C5×Q8—42no—
157C2×C5×Q8—42no—
158C2×C5×Q8—42no—
159C2×C5×Q8—42no—
160C2×C5×Q8—42no—
161C2×C5×Q8—44no—
162C2×C5×Q8—44no—
163C2×C5×Q8—41no—
164C2×C5×Q8—41no—
165C2×C5×Q8—82no—
166C2×C5×Q8C2×D412yes—
167C2×C5×Q8C2×D411yescharacteristic
168C2×C5×D4—21no—
169C2×C5×D4—21no—
170C2×C5×D4—21no—
171C2×C5×D4C2×D411yescharacteristic
172C22×C4×C5—21no—
173C22×C4×C5—22no—
174C22×C4×C5—22no—
175C22×C4×C5—22no—
176C22×C4×C5—21no—
177C22×C4×C5—21no—
178C22×C4×C5—42no—
179C22×C4×C5—44no—
180C22×C4×C5—41no—
181C22×C4×C5—41no—
182C22×C4×C5C2×D412yes—
183C22×C4×C5C2×D411yescharacteristic
184C22×C4×C5C2×D411yescharacteristic
185C22×C4×C5C2×D411yescharacteristic
186C22×C4×C5C2×D412yes—
187C22×C4×C5C2×D411yescharacteristic
188C22×C4×C5C2×D411yescharacteristic
189C22×C4×C5C2411yescharacteristic
190C5×Q16—48no—
191C5×Q16—44no—
192C5×SD16—44no—
193C5×M16—42no—
194C2×C5×C8—22no—
195C2×C5×C8—22no—
196C2×C5×C8—42no—
197C5×C4⋊C4—22no—
198C5×C4⋊C4—22no—
199C5×C4⋊C4—22no—
200C5×C4⋊C4—22no—
201C5×C4⋊C4—22no—
202C5×C4⋊C4—22no—
203C5×C4⋊C4—22no—
204C5×C4⋊C4—22no—
205C5×C4⋊C4—22no—
206C5×C4⋊C4—21no—
207C5×C4⋊C4—21no—
208C5×C4⋊C4—21no—
209C5×C4⋊C4—21no—
210C5×C4⋊C4—22no—
211C5×C4⋊C4—22no—
212C5×C4⋊C4—22no—
213C5×C4⋊C4—21no—
214C5×C4⋊C4—48no—
215C5×C4⋊C4C2×D412yes—
216C5×C4⋊C4C2×D411yescharacteristic
217C5×C4⋊C4C2×D411yescharacteristic
218C5×C22⋊C4—24no—
219C5×C22⋊C4—22no—
220C5×C22⋊C4—22no—
221C5×C22⋊C4—22no—
222C5×C22⋊C4—22no—
223C5×C22⋊C4—22no—
224C5×C22⋊C4—22no—
225C5×C22⋊C4—21no—
226C5×C22⋊C4—21no—
227C5×C22⋊C4—22no—
228C5×C22⋊C4—21no—
229C5×C22⋊C4—21no—
230C5×C22⋊C4—21no—
231C42×C5—22no—
232C42×C5—22no—
233C42×C5—22no—
234C42×C5—21no—
235C42×C5—42no—
236C23×Q8C22×C511yescharacteristic
237C2×#137—21no—
238C22×Q16—21no—
239#563—22no—
240#552—21no—
241#552—21no—
242#551—22no—
243D4×Q8—21no—
244D4×Q8—21no—
245#544—21no—
246#544C22×C512yes—
247#541—21no—
248#539—21no—
249#539C22×C512yes—
250#536—21no—
251C2×C4⋊Q8—21no—
252#525C22×C511yescharacteristic
253C2×C22⋊Q8C22×C511yescharacteristic
254C2×C22⋊Q8C22×C511yescharacteristic
255#451—22no—
256#451—21no—
257#451C22×C514yes—
258#451C22×C512yes—
259#448—21no—
260#448C22×C512yes—
261#419—21no—
262C2×#103—21no—
263#407C22×C511yescharacteristic
264C2×C22⋊C8C22×C511yescharacteristic
265#395C22×C514yes—
266#391—21no—
267#385—22no—
268C4×C22⋊C4C22×C511yescharacteristic
269#328C22×C514yes—
270#328C22×C512yes—
271#327C22×C512yes—
272#324C22×C514yes—
273C23×C5—22no—
274C23×C5—22no—
275C23×C5—21no—
276C23×C5—41no—
277C23×C5#12011yescharacteristic
278C23×C5#12011yescharacteristic
279C23×C5C22×D411yescharacteristic
280C5×Q8—24no—
281C5×Q8—24no—
282C5×Q8—42no—
283C5×Q8—41no—
284C5×Q8—42no—
285C5×Q8—42no—
286C5×Q8—44no—
287C5×Q8—44no—
288C5×Q8—42no—
289C5×Q8—42no—
290C5×Q8—81no—
291C5×Q8—82no—
292C5×D4—41no—
293C5×D4—42no—
294C5×D4—42no—
295C5×D4—81no—
296C2×C4×C5—22no—
297C2×C4×C5—22no—
298C2×C4×C5—22no—
299C2×C4×C5—24no—
300C2×C4×C5—24no—
301C2×C4×C5—21no—
302C2×C4×C5—21no—
303C2×C4×C5—22no—
304C2×C4×C5—22no—
305C2×C4×C5—21no—
306C2×C4×C5—22no—
307C2×C4×C5—22no—
308C2×C4×C5—24no—
309C2×C4×C5—41no—
310C2×C4×C5—41no—
311C2×C4×C5—41no—
312C2×C4×C5—42no—
313C2×C4×C5—42no—
314C2×C4×C5—44no—
315C2×C4×C5—44no—
316C2×C4×C5—42no—
317C2×C4×C5—42no—
318C2×C4×C5—42no—
319C2×C4×C5—42no—
320C2×C4×C5—42no—
321C2×C4×C5—81no—
322C2×C4×C5—82no—
323C2×C4×C5#12011yescharacteristic
324C2×C4×C5#12011yescharacteristic
325C2×C4×C5C22×D411yescharacteristic
326C2×C4×C5C22×D411yescharacteristic
327C5×C8—42no—
328C5×C8—42no—
329C2×#41C5×D411yescharacteristic
330C22×Q8—21no—
331C22×Q8—21no—
332C22×Q8—21no—
333C22×Q8—21no—
334C22×Q8—21no—
335C22×Q8—21no—
336C22×Q8—44no—
337C22×Q8C5×D411yescharacteristic
338C22×Q8C5×D411yescharacteristic
339C22×Q8C23×C512yes—
340C23×C4—21no—
341C23×C4C23×C511yescharacteristic
342#137—44no—
343C2×Q16—24no—
344C2×Q16—22no—
345C2×Q16—44no—
346C2×SD16—22no—
347C2×M16—21no—
348C22×C8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—22no—
351C4⋊Q8—22no—
352C4⋊Q8—22no—
353C4⋊Q8—44no—
354#126—22no—
355#126—22no—
356#126—22no—
357#126—22no—
358#125—21no—
359#125—22no—
360#125—22no—
361#125—22no—
362#125—22no—
363#124—22no—
364#123—22no—
365#123—22no—
366#123—22no—
367#123—21no—
368#123—22no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8—22no—
381C22⋊Q8C5×D412yes—
382C22⋊Q8C5×D411yescharacteristic
383C22⋊Q8C23×C512yes—
384C22⋊Q8C23×C511yescharacteristic
385#121C5×D411yescharacteristic
386#121C23×C511yescharacteristic
387#120—21no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×D4—22no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—22no—
394#117—21no—
395#117—22no—
396#117—22no—
397#117C5×D411yescharacteristic
398C2×C4⋊C4—21no—
399C2×C4⋊C4—24no—
400C2×C4⋊C4C5×D411yescharacteristic
401C2×C4⋊C4C5×D411yescharacteristic
402C2×C4⋊C4C23×C512yes—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4C5×D412yes—
406C2×C42—21no—
407#103—24no—
408#103—22no—
409#103—22no—
410C22⋊C8—21no—
411C22⋊C8—21no—
412C22⋊C8C23×C512yes—
413C22⋊C8C23×C512yes—
414#95—24no—
415#95C23×C512yes—
416C22×C5—21no—
417C22×C5—21no—
418C22×C5—21no—
419C22×C5—21no—
420C22×C5—21no—
421C22×C5—21no—
422C22×C5—21no—
423C22×C5—44no—
424C22×C5—41no—
425C22×C5—41no—
426C22×C5—41no—
427C22×C5#45612yes—
428C22×C5C2×#12011yescharacteristic
429C22×C5#57812yes—
430C4×C5—21no—
431C4×C5—21no—
432C4×C5—21no—
433C4×C5—44no—
434C4×C5—42no—
435C4×C5—41no—
436C4×C5—42no—
437C4×C5—41no—
438C4×C5—44no—
439C4×C5—41no—
440C4×C5—42no—
441C4×C5—42no—
442C4×C5B(4,𝔽2)12yes—
443C24C2×C5×D411yescharacteristic
444#41—42no—
445C2×Q8—24no—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—24no—
451C2×Q8—24no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—44no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—44no—
461C2×Q8—44no—
462C2×Q8—41no—
463C2×Q8—41no—
464C2×Q8—82no—
465C2×Q8C2×C5×D412yes—
466C2×Q8C2×C5×D411yescharacteristic
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4C2×C5×D411yescharacteristic
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4C2×C5×D412yes—
482C22×C4C2×C5×D411yescharacteristic
483C22×C4C2×C5×D411yescharacteristic
484C22×C4C2×C5×D411yescharacteristic
485C22×C4C2×C5×D412yes—
486C22×C4C2×C5×D411yescharacteristic
487C22×C4C2×C5×D411yescharacteristic
488C22×C4C24×C511yescharacteristic
489Q16—48no—
490Q16—44no—
491SD16—44no—
492M16—42no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—42no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—48no—
514C4⋊C4C2×C5×D412yes—
515C4⋊C4C2×C5×D411yescharacteristic
516C4⋊C4C2×C5×D411yescharacteristic
517C22⋊C4—24no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—22no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—42no—
535C2×C5—22no—
536C2×C5—22no—
537C2×C5—41no—
538C2×C5—81no—
539C2×C5#301111yescharacteristic
540C2×C5C2×#45611yescharacteristic
541C2×C5C2×B(4,𝔽2)11yescharacteristic
542C23—22no—
543C23—22no—
544C23—21no—
545C23—41no—
546C23C5×#12011yescharacteristic
547C23C5×#12011yescharacteristic
548C23C22×C5×D411yescharacteristic
549Q8—24no—
550Q8—24no—
551Q8—42no—
552Q8—41no—
553Q8—42no—
554Q8—42no—
555Q8—44no—
556Q8—44no—
557Q8—42no—
558Q8—42no—
559Q8—81no—
560Q8—82no—
561D4—41no—
562D4—42no—
563D4—42no—
564D4—81no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—81no—
591C2×C4—82no—
592C2×C4C5×#12011yescharacteristic
593C2×C4C5×#12011yescharacteristic
594C2×C4C22×C5×D411yescharacteristic
595C2×C4C22×C5×D411yescharacteristic
596C8—42no—
597C8—42no—
598C5#1825911yescharacteristic, direct product factor, Sylow
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—44no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22C5×#45612yes—
611C22C2×C5×#12011yescharacteristic
612C22C5×#57812yes—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—44no—
617C4—42no—
618C4—41no—
619C4—42no—
620C4—41no—
621C4—44no—
622C4—41no—
623C4—42no—
624C4—42no—
625C4C5×B(4,𝔽2)12yes—
626C2—22no—
627C2—22no—
628C2—41no—
629C2—81no—
630C2C5×#301111yescharacteristic
631C2C2×C5×#45611yescharacteristic
632C2C2×C5×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC5×#525C22
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
3. C2×C5×#451C2semidirect
4. C5×#2434C2semidirect
5. C5×#1605C2semidirect
6. C5×#1603C2semidirect
7. C5×#1508C2semidirect
9. C5×#1472C2semidirect
47. #18259C5direct (N × H)
137. C2×#451C2×C5semidirect
138. #2434C2×C5semidirect
139. #1605C2×C5semidirect
140. #1603C2×C5semidirect
141. #1508C2×C5semidirect
143. #1472C2×C5semidirect
598. C5#18259direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35382
C5×#30042
C2×C5×#4512
C5×#24342
C5×#16052
C5×#16032
C5×#15082
C5×#15012
C5×#14722
#182595