Finite Group Databasebeta

← C5×#18589 #11941309

Subgroups — C5×#18589 #11941309

Order 1280.

878 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3523C211yesmaximal, characteristic
2C5×#3218C211yesmaximal, characteristic
3C5×#3165C211yesmaximal, characteristic, complemented
4C2×C5×#475C211yesmaximal, characteristic, complemented
5C5×#1653C212yesmaximal, complemented
6C5×#1651C212yesmaximal, complemented
7C5×#1650C212yesmaximal, complemented
8C5×#1648C212yesmaximal, complemented
9C5×#1552C211yesmaximal, characteristic, complemented
10C5×#1491C211yesmaximal, characteristic, complemented
11C5×#1483C211yesmaximal, characteristic
12C5×#557—21no—
13C5×#557C2211yescharacteristic
14C5×#557C2211yescharacteristic
15C5×#556—21no—
16C5×#556C2211yescharacteristic
17C5×#552—22no—
18C5×#552—21no—
19C5×D4×Q8—21no—
20C5×D4×Q8C2211yescharacteristic
21C5×#541—22no—
22C5×#541—21no—
23C5×#533—21no—
24C5×#533C2211yescharacteristic
25C2×C5×C4⋊Q8C2211yescharacteristic
26C5×#529—21no—
27C5×#529C2211yescharacteristic
28C5×#519—21no—
29C2×C4×C5×Q8C2211yescharacteristic
30C5×#501—22no—
31C5×#500C2212yes—
32C5×#498C2212yes—
33C5×#497—21no—
34C5×#491—21no—
35C5×#475—21no—
36C5×#475—21no—
37C5×#475C2212yes—
38C5×#475C2212yes—
39C5×#462C2212yes—
40C5×#461C2212yes—
41C5×#451—21no—
42C5×#448—21no—
43C5×#432C2211yescharacteristic
44C5×#428—22no—
45C2×C5×#106—21no—
46C2×C5×C4⋊C8C2211yescharacteristic
47C2×C5×#103—21no—
48C4×C5×M16C2211yescharacteristic
49C5×#332C2212yes—
50C5×#332C2212yes—
51C5×Q8⋊C8C2212yes—
52C5×Q8⋊C8C2212yes—
53C5×Q8⋊C8C2212yes—
54C5×D4⋊C8C2212yes—
55#18589C511yesmaximal, characteristic, direct product factor, Sylow
56C22×C5×Q8—21no—
57C22×C5×Q8—21no—
58C22×C5×Q8D411yescharacteristic
59C2×C5×Q16—22no—
60C2×C5×SD16—22no—
61C2×C5×M16—21no—
62C22×C5×C8—21no—
63C5×C4⋊Q8—22no—
64C5×C4⋊Q8—21no—
65C5×C4⋊Q8—21no—
66C5×C4⋊Q8—21no—
67C5×C4⋊Q8—21no—
68C5×C4⋊Q8—21no—
69C5×C4⋊Q8—21no—
70C5×C4⋊Q8C2312yes—
71C5×C4⋊Q8C2311yescharacteristic
72C5×C4⋊Q8C2311yescharacteristic
73C5×#126—22no—
74C5×#126—22no—
75C5×#126—22no—
76C5×#126—22no—
77C5×#126—21no—
78C5×#125—22no—
79C5×#125—22no—
80C5×#125—22no—
81C5×#125—21no—
82C5×#125—22no—
83C5×#125—22no—
84C5×#125—21no—
85C5×#125D411yescharacteristic
86C5×#125D412yes—
87C5×#124—21no—
88C5×#124D411yescharacteristic
89C5×#123—22no—
90C5×#123—21no—
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—22no—
98C5×C22⋊Q8D411yescharacteristic
99C5×#121D411yescharacteristic
100C4×C5×Q8—22no—
101C4×C5×Q8—22no—
102C4×C5×Q8—21no—
103C4×C5×Q8—21no—
104C4×C5×Q8—21no—
105C4×C5×Q8—21no—
106C4×C5×Q8—21no—
107C4×C5×Q8—21no—
108C4×C5×Q8—21no—
109C4×C5×Q8—21no—
110C4×C5×Q8—21no—
111C4×C5×Q8D411yescharacteristic
112C4×C5×Q8D411yescharacteristic
113C4×C5×Q8C2311yescharacteristic
114C4×C5×Q8C2311yescharacteristic
115C4×C5×Q8C2311yescharacteristic
116C4×C5×D4—22no—
117C4×C5×D4—21no—
118C4×C5×D4—21no—
119C4×C5×D4C2311yescharacteristic
120C5×#117—22no—
121C5×#117—22no—
122C5×#117—22no—
123C5×#117—21no—
124C5×#117—21no—
125C5×#117D412yes—
126C2×C5×C4⋊C4—21no—
127C2×C5×C4⋊C4—21no—
128C2×C5×C4⋊C4—21no—
129C2×C5×C4⋊C4D411yescharacteristic
130C2×C42×C5—21no—
131C2×C42×C5C2311yescharacteristic
132C5×#107—24no—
133C5×#106—22no—
134C5×#106—22no—
135C5×#106—24no—
136C5×C4⋊C8—21no—
137C5×C4⋊C8C2312yes—
138C5×C4⋊C8C2312yes—
139C5×#103—22no—
140C5×#103—22no—
141C5×#103—22no—
142C5×#103—22no—
143C5×C22⋊C8—21no—
144C5×#97—21no—
145C5×#97—21no—
146C4×C5×C8C2312yes—
147#3523C2×C511yescharacteristic
148#3218C2×C511yescharacteristic
149#3165C2×C511yescharacteristic, complemented
150C2×#475C2×C511yescharacteristic, complemented
151#1653C2×C512yescomplemented
152#1651C2×C512yescomplemented
153#1650C2×C512yescomplemented
154#1648C2×C512yescomplemented
155#1552C2×C511yescharacteristic, complemented
156#1491C2×C511yescharacteristic, complemented
157#1483C2×C511yescharacteristic
158C2×C5×Q8—22no—
159C2×C5×Q8—21no—
160C2×C5×Q8—21no—
161C2×C5×Q8—21no—
162C2×C5×Q8—22no—
163C2×C5×Q8—22no—
164C2×C5×Q8—41no—
165C2×C5×Q8—42no—
166C2×C5×Q8—42no—
167C2×C5×Q8—41no—
168C2×C5×Q8—82no—
169C2×C5×Q8C2×D411yescharacteristic
170C2×C5×Q8C2×D411yescharacteristic
171C2×C5×Q8C2×D411yescharacteristic
172C2×C5×D4—21no—
173C2×C5×D4C2×D411yescharacteristic
174C22×C4×C5—22no—
175C22×C4×C5—21no—
176C22×C4×C5—21no—
177C22×C4×C5—21no—
178C22×C4×C5—41no—
179C22×C4×C5C2×D411yescharacteristic
180C22×C4×C5C2×D411yescharacteristic
181C22×C4×C5C2×D411yescharacteristic
182C5×Q16—82no—
183C5×SD16—82no—
184C5×M16—24no—
185C2×C5×C8—22no—
186C2×C5×C8—22no—
187C2×C5×C8—22no—
188C2×C5×C8—42no—
189C5×C4⋊C4—22no—
190C5×C4⋊C4—22no—
191C5×C4⋊C4—22no—
192C5×C4⋊C4—22no—
193C5×C4⋊C4—22no—
194C5×C4⋊C4—22no—
195C5×C4⋊C4—22no—
196C5×C4⋊C4—22no—
197C5×C4⋊C4—21no—
198C5×C4⋊C4—21no—
199C5×C4⋊C4—21no—
200C5×C4⋊C4—22no—
201C5×C4⋊C4—21no—
202C5×C4⋊C4—21no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4—22no—
205C5×C4⋊C4—21no—
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—41no—
213C5×C4⋊C4—41no—
214C5×C4⋊C4C2×D411yescharacteristic
215C5×C4⋊C4C2×D411yescharacteristic
216C5×C4⋊C4C2×D412yes—
217C5×C4⋊C4C2×D412yes—
218C5×C4⋊C4C2×D411yescharacteristic
219C5×C4⋊C4C2×D411yescharacteristic
220C5×C22⋊C4—22no—
221C5×C22⋊C4—22no—
222C5×C22⋊C4—22no—
223C5×C22⋊C4—21no—
224C5×C22⋊C4—22no—
225C5×C22⋊C4—21no—
226C5×C22⋊C4—21no—
227C5×C22⋊C4—21no—
228C5×C22⋊C4—21no—
229C42×C5—22no—
230C42×C5—22no—
231C42×C5—22no—
232C42×C5—21no—
233C42×C5—21no—
234C42×C5—22no—
235C42×C5—21no—
236C42×C5—21no—
237C42×C5—21no—
238C42×C5C2×D411yescharacteristic
239C42×C5C2×D411yescharacteristic
240C42×C5C2×D411yescharacteristic
241C42×C5C2411yescharacteristic
242#557—21no—
243#557C22×C511yescharacteristic
244#557C22×C511yescharacteristic
245#556—21no—
246#556C22×C511yescharacteristic
247#552—22no—
248#552—21no—
249D4×Q8—21no—
250D4×Q8C22×C511yescharacteristic
251#541—22no—
252#541—21no—
253#533—21no—
254#533C22×C511yescharacteristic
255C2×C4⋊Q8C22×C511yescharacteristic
256#529—21no—
257#529C22×C511yescharacteristic
258#519—21no—
259C2×C4×Q8C22×C511yescharacteristic
260#501—22no—
261#500C22×C512yes—
262#498C22×C512yes—
263#497—21no—
264#491—21no—
265#475—21no—
266#475—21no—
267#475C22×C512yes—
268#475C22×C512yes—
269#462C22×C512yes—
270#461C22×C512yes—
271#451—21no—
272#448—21no—
273#432C22×C511yescharacteristic
274#428—22no—
275C2×#106—21no—
276C2×C4⋊C8C22×C511yescharacteristic
277C2×#103—21no—
278C4×M16C22×C511yescharacteristic
279#332C22×C512yes—
280#332C22×C512yes—
281Q8⋊C8C22×C512yes—
282Q8⋊C8C22×C512yes—
283Q8⋊C8C22×C512yes—
284D4⋊C8C22×C512yes—
285C23×C5—21no—
286C23×C5#12011yescharacteristic
287C5×Q8—21no—
288C5×Q8—21no—
289C5×Q8—21no—
290C5×Q8—22no—
291C5×Q8—42no—
292C5×Q8—44no—
293C5×Q8—41no—
294C5×Q8—41no—
295C5×Q8—42no—
296C5×Q8—42no—
297C5×Q8—42no—
298C5×Q8—82no—
299C5×Q8#13712yes—
300C5×D4—22no—
301C5×D4—42no—
302C2×C4×C5—21no—
303C2×C4×C5—22no—
304C2×C4×C5—21no—
305C2×C4×C5—21no—
306C2×C4×C5—22no—
307C2×C4×C5—21no—
308C2×C4×C5—21no—
309C2×C4×C5—22no—
310C2×C4×C5—21no—
311C2×C4×C5—21no—
312C2×C4×C5—21no—
313C2×C4×C5—21no—
314C2×C4×C5—22no—
315C2×C4×C5—42no—
316C2×C4×C5—44no—
317C2×C4×C5—42no—
318C2×C4×C5—41no—
319C2×C4×C5—41no—
320C2×C4×C5—42no—
321C2×C4×C5—82no—
322C2×C4×C5#12011yescharacteristic
323C2×C4×C5#12011yescharacteristic
324C2×C4×C5#12011yescharacteristic
325C2×C4×C5C22×D411yescharacteristic
326C2×C4×C5C22×D411yescharacteristic
327C2×C4×C5C22×D411yescharacteristic
328C5×C8—24no—
329C5×C8—42no—
330C5×C8—42no—
331C22×Q8—21no—
332C22×Q8—21no—
333C22×Q8C5×D411yescharacteristic
334C2×Q16—22no—
335C2×SD16—22no—
336C2×M16—21no—
337C22×C8—21no—
338C4⋊Q8—22no—
339C4⋊Q8—21no—
340C4⋊Q8—21no—
341C4⋊Q8—21no—
342C4⋊Q8—21no—
343C4⋊Q8—21no—
344C4⋊Q8—21no—
345C4⋊Q8C23×C512yes—
346C4⋊Q8C23×C511yescharacteristic
347C4⋊Q8C23×C511yescharacteristic
348#126—22no—
349#126—22no—
350#126—22no—
351#126—22no—
352#126—21no—
353#125—22no—
354#125—22no—
355#125—22no—
356#125—21no—
357#125—22no—
358#125—22no—
359#125—21no—
360#125C5×D411yescharacteristic
361#125C5×D412yes—
362#124—21no—
363#124C5×D411yescharacteristic
364#123—22no—
365#123—21no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8—22no—
373C22⋊Q8C5×D411yescharacteristic
374#121C5×D411yescharacteristic
375C4×Q8—22no—
376C4×Q8—22no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8—21no—
386C4×Q8C5×D411yescharacteristic
387C4×Q8C5×D411yescharacteristic
388C4×Q8C23×C511yescharacteristic
389C4×Q8C23×C511yescharacteristic
390C4×Q8C23×C511yescharacteristic
391C4×D4—22no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4C23×C511yescharacteristic
395#117—22no—
396#117—22no—
397#117—22no—
398#117—21no—
399#117—21no—
400#117C5×D412yes—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4C5×D411yescharacteristic
405C2×C42—21no—
406C2×C42C23×C511yescharacteristic
407#107—24no—
408#106—22no—
409#106—22no—
410#106—24no—
411C4⋊C8—21no—
412C4⋊C8C23×C512yes—
413C4⋊C8C23×C512yes—
414#103—22no—
415#103—22no—
416#103—22no—
417#103—22no—
418C22⋊C8—21no—
419#97—21no—
420#97—21no—
421C4×C8C23×C512yes—
422C22×C5—21no—
423C22×C5—21no—
424C22×C5—41no—
425C22×C5—41no—
426C22×C5—41no—
427C22×C5#45612yes—
428C22×C5C2×#12011yescharacteristic
429C4×C5—21no—
430C4×C5—22no—
431C4×C5—21no—
432C4×C5—21no—
433C4×C5—21no—
434C4×C5—41no—
435C4×C5—42no—
436C4×C5—41no—
437C4×C5—41no—
438C4×C5—42no—
439C4×C5—41no—
440C4×C5—41no—
441C4×C5—41no—
442C4×C5—82no—
443C4×C5C2×#13711yescharacteristic
444C4×C5#57511yescharacteristic
445C4×C5#57712yes—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—41no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—41no—
456C2×Q8—82no—
457C2×Q8C2×C5×D411yescharacteristic
458C2×Q8C2×C5×D411yescharacteristic
459C2×Q8C2×C5×D411yescharacteristic
460C2×D4—21no—
461C2×D4C2×C5×D411yescharacteristic
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—41no—
467C22×C4C2×C5×D411yescharacteristic
468C22×C4C2×C5×D411yescharacteristic
469C22×C4C2×C5×D411yescharacteristic
470Q16—82no—
471SD16—82no—
472M16—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—42no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4C2×C5×D411yescharacteristic
503C4⋊C4C2×C5×D411yescharacteristic
504C4⋊C4C2×C5×D411yescharacteristic
505C4⋊C4C2×C5×D411yescharacteristic
506C4⋊C4C2×C5×D412yes—
507C4⋊C4C2×C5×D412yes—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—21no—
512C22⋊C4—22no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—21no—
521C42—21no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42C2×C5×D411yescharacteristic
527C42C2×C5×D411yescharacteristic
528C42C2×C5×D411yescharacteristic
529C42C24×C511yescharacteristic
530C2×C5—22no—
531C2×C5—81no—
532C2×C5#300711yescharacteristic
533C2×C5#301011yescharacteristic
534C2×C5C2×#45611yescharacteristic
535C23—21no—
536C23C5×#12011yescharacteristic
537Q8—21no—
538Q8—21no—
539Q8—21no—
540Q8—22no—
541Q8—42no—
542Q8—44no—
543Q8—41no—
544Q8—41no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—82no—
549Q8C5×#13712yes—
550D4—22no—
551D4—42no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—82no—
572C2×C4C5×#12011yescharacteristic
573C2×C4C5×#12011yescharacteristic
574C2×C4C5×#12011yescharacteristic
575C2×C4C22×C5×D411yescharacteristic
576C2×C4C22×C5×D411yescharacteristic
577C2×C4C22×C5×D411yescharacteristic
578C8—24no—
579C8—42no—
580C8—42no—
581C5#1858911yescharacteristic, direct product factor, Sylow
582C22—21no—
583C22—21no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C5×#45612yes—
588C22C2×C5×#12011yescharacteristic
589C4—21no—
590C4—22no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—41no—
595C4—42no—
596C4—41no—
597C4—41no—
598C4—42no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—82no—
603C4C2×C5×#13711yescharacteristic
604C4C5×#57511yescharacteristic
605C4C5×#57712yes—
606C2—22no—
607C2—81no—
608C2C5×#300711yescharacteristic
609C2C5×#301011yescharacteristic
610C2C2×C5×#45611yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC42C24×C5
prime radicalC5×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
3. C5×#3165C2semidirect
4. C2×C5×#475C2semidirect
5. C5×#1653C2semidirect
6. C5×#1651C2semidirect
7. C5×#1650C2semidirect
8. C5×#1648C2semidirect
9. C5×#1552C2semidirect
10. C5×#1491C2semidirect
55. #18589C5direct (N × H)
149. #3165C2×C5semidirect
150. C2×#475C2×C5semidirect
151. #1653C2×C5semidirect
152. #1651C2×C5semidirect
153. #1650C2×C5semidirect
154. #1648C2×C5semidirect
155. #1552C2×C5semidirect
156. #1491C2×C5semidirect
581. C5#18589direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35232
C5×#32182
C5×#31652
C2×C5×#4752
C5×#16532
C5×#16512
C5×#16502
C5×#16482
C5×#15522
C5×#14912
C5×#14832
#185895