Finite Group Databasebeta

← C5×#19048 #11941768

Subgroups — C5×#19048 #11941768

Order 1280.

908 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3561C211yesmaximal, characteristic, complemented
2C5×#3213C211yesmaximal, characteristic, complemented
3C5×#3180C211yesmaximal, characteristic, complemented
4C5×#3135C211yesmaximal, characteristic, complemented
5C5×#1664C212yesmaximal, complemented
6C5×#1662C212yesmaximal, complemented
7C5×#1657C212yesmaximal, complemented
8C5×#1656C212yesmaximal, complemented
9C5×#1532C211yesmaximal, characteristic, complemented
10C5×#1531C211yesmaximal, characteristic, complemented
11C2×C5×#329C211yesmaximal, characteristic, complemented
12C5×#563C2212yes—
13C5×#562C2212yes—
14C5×#556—21no—
15C5×#551—22no—
16C5×D4×Q8—21no—
17C5×#548—21no—
18C5×#546—21no—
19C5×#543—21no—
20C5×#542—21no—
21C5×#542—22no—
22C5×#538—21no—
23C5×#536—21no—
24C5×#533C2211yescharacteristic
25C5×#532C2211yescharacteristic
26C5×#529C2211yescharacteristic
27C2×C5×#124C2211yescharacteristic
28C5×#501C2211yescharacteristic, complemented
29C5×#496C2212yes—
30C5×#496C2211yescharacteristic, complemented
31C5×#491C2212yescomplemented
32C5×#490C2212yescomplemented
33C5×#490C2211yescharacteristic, complemented
34C5×#489—21no—
35C5×#489—21no—
36C5×#489C2212yes—
37C5×#489C2211yescharacteristic, complemented
38C5×#466—21no—
39C5×#465—21no—
40C5×#463—22no—
41C5×#448—21no—
42C5×#447—21no—
43C5×#442—21no—
44C5×#440—21no—
45C5×#432C2211yescharacteristic
46C5×#416—22no—
47C2×C5×#97C2211yescharacteristic
48C2×C5×#97C2211yescharacteristic
49C5×#330C2212yescomplemented
50C5×#330C2212yescomplemented
51C5×#330C2212yescomplemented
52C5×#329C2212yes—
53C5×#329C2212yes—
54C5×#329C2212yescomplemented
55#19048C511yesmaximal, characteristic, direct product factor, Sylow
56C2×C5×#41—21no—
57C2×C5×#41—21no—
58C22×C5×Q8—21no—
59C22×C5×Q8D411yescharacteristic
60C22×C5×D4D411yescharacteristic
61C2×C5×SD16—21no—
62C2×C5×SD16—22no—
63C2×C5×D8—21no—
64C2×C5×D8—22no—
65C22×C5×C8—21no—
66C22×C5×C8—21no—
67C5×C4⋊Q8—21no—
68C5×C4⋊Q8—22no—
69C5×C4⋊Q8D411yescharacteristic
70C5×C4⋊Q8D411yescharacteristic
71C5×C4⋊Q8C2311yescharacteristic
72C5×#127C2311yescharacteristic
73C5×#126—22no—
74C5×#126—22no—
75C5×#126—22no—
76C5×#126—21no—
77C5×#125C2312yes—
78C5×#125C2311yescharacteristic
79C5×#124—21no—
80C5×#124—22no—
81C5×#124—22no—
82C5×#124—21no—
83C5×#124—21no—
84C5×#124—21no—
85C5×#124D412yes—
86C5×#124C2312yes—
87C5×#124C2311yescharacteristic
88C5×#123—22no—
89C5×#123—22no—
90C5×#123—21no—
91C5×#123—22no—
92C5×#123—22no—
93C5×C22⋊Q8—22no—
94C5×C22⋊Q8—22no—
95C5×C22⋊Q8—21no—
96C5×C22⋊Q8—22no—
97C5×C22⋊Q8—22no—
98C5×C22⋊Q8—22no—
99C5×C22⋊Q8—21no—
100C5×C22⋊Q8—21no—
101C5×C22⋊Q8—22no—
102C5×C22⋊Q8—22no—
103C5×C22⋊Q8D411yescharacteristic
104C5×#121—21no—
105C5×#121—22no—
106C5×#121—22no—
107C5×#121—21no—
108C5×#121—21no—
109C5×#121D411yescharacteristic
110C5×#120—22no—
111C4×C5×Q8—21no—
112C4×C5×Q8—21no—
113C4×C5×Q8—21no—
114C4×C5×Q8D411yescharacteristic
115C4×C5×D4—22no—
116C4×C5×D4—21no—
117C4×C5×D4—22no—
118C4×C5×D4—21no—
119C4×C5×D4—21no—
120C4×C5×D4—21no—
121C4×C5×D4D411yescharacteristic
122C5×#117—22no—
123C5×#117—21no—
124C5×#117D412yes—
125C2×C5×C22⋊C4—22no—
126C2×C5×C22⋊C4—21no—
127C2×C5×C22⋊C4—21no—
128C2×C42×C5C2311yescharacteristic
129C5×#107—21no—
130C5×#106—21no—
131C5×C4⋊C8—21no—
132C5×#103—21no—
133C5×#103—22no—
134C5×#103—22no—
135C5×#103—22no—
136C5×#102—21no—
137C5×#102—22no—
138C5×#102—22no—
139C5×#102—22no—
140C5×#102—21no—
141C5×#102—21no—
142C5×C22⋊C8—21no—
143C5×#97—21no—
144C5×#97—21no—
145C5×#97C2312yes—
146C5×#97C2311yescharacteristic
147C5×#97C2311yescharacteristic
148C5×#97C2312yes—
149#3561C2×C511yescharacteristic, complemented
150#3213C2×C511yescharacteristic, complemented
151#3180C2×C511yescharacteristic, complemented
152#3135C2×C511yescharacteristic, complemented
153#1664C2×C512yescomplemented
154#1662C2×C512yescomplemented
155#1657C2×C512yescomplemented
156#1656C2×C512yescomplemented
157#1532C2×C511yescharacteristic, complemented
158#1531C2×C511yescharacteristic, complemented
159C2×#329C2×C511yescharacteristic, complemented
160C24×C5—21no—
161C5×#41—44no—
162C5×#41—82no—
163C2×C5×Q8—22no—
164C2×C5×Q8—21no—
165C2×C5×Q8—21no—
166C2×C5×Q8—21no—
167C2×C5×Q8—21no—
168C2×C5×Q8—82no—
169C2×C5×Q8—81no—
170C2×C5×Q8C2×D411yescharacteristic
171C2×C5×Q8C2×D412yes—
172C2×C5×D4—21no—
173C2×C5×D4—22no—
174C2×C5×D4—22no—
175C2×C5×D4—21no—
176C2×C5×D4—21no—
177C2×C5×D4—21no—
178C2×C5×D4—21no—
179C2×C5×D4—41no—
180C2×C5×D4—81no—
181C2×C5×D4C2×D411yescharacteristic
182C2×C5×D4C2×D412yes—
183C22×C4×C5—22no—
184C22×C4×C5—22no—
185C22×C4×C5—22no—
186C22×C4×C5—21no—
187C22×C4×C5—21no—
188C22×C4×C5—41no—
189C22×C4×C5C2×D411yescharacteristic
190C22×C4×C5C2×D411yescharacteristic
191C22×C4×C5C2×D411yescharacteristic
192C5×SD16—42no—
193C5×SD16—82no—
194C5×D8—42no—
195C5×D8—82no—
196C2×C5×C8—21no—
197C2×C5×C8—21no—
198C2×C5×C8—22no—
199C2×C5×C8—22no—
200C2×C5×C8—42no—
201C2×C5×C8—42no—
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—21no—
211C5×C4⋊C4—21no—
212C5×C4⋊C4—22no—
213C5×C4⋊C4—22no—
214C5×C4⋊C4—21no—
215C5×C4⋊C4—21no—
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—22no—
224C5×C22⋊C4—22no—
225C5×C22⋊C4—21no—
226C5×C22⋊C4—22no—
227C5×C22⋊C4—22no—
228C5×C22⋊C4—22no—
229C5×C22⋊C4—21no—
230C5×C22⋊C4—21no—
231C5×C22⋊C4—22no—
232C5×C22⋊C4—22no—
233C5×C22⋊C4—21no—
234C5×C22⋊C4—21no—
235C5×C22⋊C4—41no—
236C5×C22⋊C4—41no—
237C42×C5—22no—
238C42×C5—22no—
239C42×C5—21no—
240C42×C5C2×D411yescharacteristic
241C42×C5C2×D411yescharacteristic
242C42×C5C2×D411yescharacteristic
243C42×C5C2411yescharacteristic
244#563C22×C512yes—
245#562C22×C512yes—
246#556—21no—
247#551—22no—
248D4×Q8—21no—
249#548—21no—
250#546—21no—
251#543—21no—
252#542—21no—
253#542—22no—
254#538—21no—
255#536—21no—
256#533C22×C511yescharacteristic
257#532C22×C511yescharacteristic
258#529C22×C511yescharacteristic
259C2×#124C22×C511yescharacteristic
260#501C22×C511yescharacteristic, complemented
261#496C22×C512yes—
262#496C22×C511yescharacteristic, complemented
263#491C22×C512yescomplemented
264#490C22×C512yescomplemented
265#490C22×C511yescharacteristic, complemented
266#489—21no—
267#489—21no—
268#489C22×C512yes—
269#489C22×C511yescharacteristic, complemented
270#466—21no—
271#465—21no—
272#463—22no—
273#448—21no—
274#447—21no—
275#442—21no—
276#440—21no—
277#432C22×C511yescharacteristic
278#416—22no—
279C2×#97C22×C511yescharacteristic
280C2×#97C22×C511yescharacteristic
281#330C22×C512yescomplemented
282#330C22×C512yescomplemented
283#330C22×C512yescomplemented
284#329C22×C512yes—
285#329C22×C512yes—
286#329C22×C512yescomplemented
287C23×C5—22no—
288C23×C5—21no—
289C23×C5—42no—
290C23×C5—41no—
291C23×C5—81no—
292C23×C5—81no—
293C23×C5#12011yescharacteristic
294C5×Q8—41no—
295C5×Q8—42no—
296C5×Q8—42no—
297C5×Q8—42no—
298C5×Q8—42no—
299C5×Q8—81no—
300C5×Q8—82no—
301C5×D4—41no—
302C5×D4—42no—
303C5×D4—42no—
304C5×D4—41no—
305C5×D4—42no—
306C5×D4—41no—
307C5×D4—42no—
308C5×D4—42no—
309C5×D4—82no—
310C5×D4—82no—
311C5×D4—82no—
312C2×C4×C5—22no—
313C2×C4×C5—21no—
314C2×C4×C5—22no—
315C2×C4×C5—21no—
316C2×C4×C5—22no—
317C2×C4×C5—22no—
318C2×C4×C5—21no—
319C2×C4×C5—41no—
320C2×C4×C5—41no—
321C2×C4×C5—42no—
322C2×C4×C5—41no—
323C2×C4×C5—82no—
324C2×C4×C5—81no—
325C2×C4×C5—82no—
326C2×C4×C5—82no—
327C2×C4×C5—82no—
328C2×C4×C5#12011yescharacteristic
329C2×C4×C5#12011yescharacteristic
330C2×C4×C5#12011yescharacteristic
331C2×C4×C5C22×D411yescharacteristic
332C2×C4×C5C22×D411yescharacteristic
333C2×C4×C5C22×D411yescharacteristic
334C5×C8—22no—
335C5×C8—42no—
336C5×C8—41no—
337C5×C8—42no—
338C2×#41—21no—
339C2×#41—21no—
340C22×Q8—21no—
341C22×Q8C5×D411yescharacteristic
342C22×D4C5×D411yescharacteristic
343C2×SD16—21no—
344C2×SD16—22no—
345C2×D8—21no—
346C2×D8—22no—
347C22×C8—21no—
348C22×C8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—22no—
351C4⋊Q8C5×D411yescharacteristic
352C4⋊Q8C5×D411yescharacteristic
353C4⋊Q8C23×C511yescharacteristic
354#127C23×C511yescharacteristic
355#126—22no—
356#126—22no—
357#126—22no—
358#126—21no—
359#125C23×C511yescharacteristic
360#125C23×C512yes—
361#124—21no—
362#124—22no—
363#124—22no—
364#124—21no—
365#124—21no—
366#124—21no—
367#124C5×D412yes—
368#124C23×C512yes—
369#124C23×C511yescharacteristic
370#123—22no—
371#123—22no—
372#123—21no—
373#123—22no—
374#123—22no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8—21no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8C5×D411yescharacteristic
386#121—21no—
387#121—22no—
388#121—22no—
389#121—21no—
390#121—21no—
391#121C5×D411yescharacteristic
392#120—22no—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8C5×D411yescharacteristic
397C4×D4—22no—
398C4×D4—21no—
399C4×D4—22no—
400C4×D4—21no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4C5×D411yescharacteristic
404#117—22no—
405#117—21no—
406#117C5×D412yes—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C42C23×C511yescharacteristic
411#107—21no—
412#106—21no—
413C4⋊C8—21no—
414#103—21no—
415#103—22no—
416#103—22no—
417#103—22no—
418#102—21no—
419#102—22no—
420#102—22no—
421#102—22no—
422#102—21no—
423#102—21no—
424C22⋊C8—21no—
425#97—21no—
426#97—21no—
427#97C23×C512yes—
428#97C23×C511yescharacteristic
429#97C23×C511yescharacteristic
430#97C23×C512yes—
431C22×C5—21no—
432C22×C5—21no—
433C22×C5—41no—
434C22×C5—42no—
435C22×C5—42no—
436C22×C5—41no—
437C22×C5—41no—
438C22×C5—42no—
439C22×C5—81no—
440C22×C5—81no—
441C22×C5—81no—
442C22×C5—162no—
443C22×C5#45412yes—
444C22×C5C2×#12011yescharacteristic
445C4×C5—21no—
446C4×C5—21no—
447C4×C5—21no—
448C4×C5—21no—
449C4×C5—21no—
450C4×C5—21no—
451C4×C5—41no—
452C4×C5—41no—
453C4×C5—42no—
454C4×C5—41no—
455C4×C5—42no—
456C4×C5—42no—
457C4×C5—81no—
458C24—21no—
459#41—44no—
460#41—82no—
461C2×Q8—22no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—82no—
467C2×Q8—81no—
468C2×Q8C2×C5×D411yescharacteristic
469C2×Q8C2×C5×D412yes—
470C2×D4—21no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—41no—
478C2×D4—81no—
479C2×D4C2×C5×D411yescharacteristic
480C2×D4C2×C5×D412yes—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—41no—
487C22×C4C2×C5×D411yescharacteristic
488C22×C4C2×C5×D411yescharacteristic
489C22×C4C2×C5×D411yescharacteristic
490SD16—42no—
491SD16—82no—
492D8—42no—
493D8—82no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—42no—
499C2×C8—42no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4C2×C5×D412yes—
515C4⋊C4C2×C5×D412yes—
516C4⋊C4C2×C5×D411yescharacteristic
517C4⋊C4C2×C5×D411yescharacteristic
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—21no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C42—22no—
536C42—22no—
537C42—21no—
538C42C2×C5×D411yescharacteristic
539C42C2×C5×D411yescharacteristic
540C42C2×C5×D411yescharacteristic
541C42C24×C511yescharacteristic
542C2×C5—22no—
543C2×C5—81no—
544C2×C5—82no—
545C2×C5—161no—
546C2×C5C2×#45411yescharacteristic
547C2×C5#301811yescharacteristic
548C2×C5#301911yescharacteristic
549C23—22no—
550C23—21no—
551C23—42no—
552C23—41no—
553C23—81no—
554C23—81no—
555C23C5×#12011yescharacteristic
556Q8—41no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—81no—
562Q8—82no—
563D4—41no—
564D4—42no—
565D4—42no—
566D4—41no—
567D4—42no—
568D4—41no—
569D4—42no—
570D4—42no—
571D4—82no—
572D4—82no—
573D4—82no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—82no—
586C2×C4—81no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4C5×#12011yescharacteristic
591C2×C4C5×#12011yescharacteristic
592C2×C4C5×#12011yescharacteristic
593C2×C4C22×C5×D411yescharacteristic
594C2×C4C22×C5×D411yescharacteristic
595C2×C4C22×C5×D411yescharacteristic
596C8—22no—
597C8—42no—
598C8—41no—
599C8—42no—
600C5#1904811yescharacteristic, direct product factor, Sylow
601C22—21no—
602C22—21no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—41no—
607C22—41no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—162no—
613C22C5×#45412yes—
614C22C2×C5×#12011yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—41no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—42no—
626C4—42no—
627C4—81no—
628C2—22no—
629C2—82no—
630C2—81no—
631C2—161no—
632C2C2×C5×#45411yescharacteristic
633C2C5×#301811yescharacteristic
634C2C5×#301911yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC42C24×C5
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
1. C5×#3561C2semidirect
2. C5×#3213C2semidirect
3. C5×#3180C2semidirect
4. C5×#3135C2semidirect
5. C5×#1664C2semidirect
6. C5×#1662C2semidirect
7. C5×#1657C2semidirect
8. C5×#1656C2semidirect
9. C5×#1532C2semidirect
10. C5×#1531C2semidirect
11. C2×C5×#329C2semidirect
28. C5×#501C22semidirect
30. C5×#496C22semidirect
31. C5×#491C22semidirect
32. C5×#490C22semidirect
33. C5×#490C22semidirect
37. C5×#489C22semidirect
49. C5×#330C22semidirect
50. C5×#330C22semidirect
51. C5×#330C22semidirect
54. C5×#329C22semidirect
55. #19048C5direct (N × H)
149. #3561C2×C5semidirect
150. #3213C2×C5semidirect
151. #3180C2×C5semidirect
152. #3135C2×C5semidirect
153. #1664C2×C5semidirect
154. #1662C2×C5semidirect
155. #1657C2×C5semidirect
156. #1656C2×C5semidirect
157. #1532C2×C5semidirect
158. #1531C2×C5semidirect
159. C2×#329C2×C5semidirect
260. #501C22×C5semidirect
262. #496C22×C5semidirect
263. #491C22×C5semidirect
264. #490C22×C5semidirect
265. #490C22×C5semidirect
269. #489C22×C5semidirect
281. #330C22×C5semidirect
282. #330C22×C5semidirect
283. #330C22×C5semidirect
286. #329C22×C5semidirect
600. C5#19048direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35612
C5×#32132
C5×#31802
C5×#31352
C5×#16642
C5×#16622
C5×#16572
C5×#16562
C5×#15322
C5×#15312
C2×C5×#3292
#190485