Finite Group Databasebeta

← C5×#19049 #11941769

Subgroups — C5×#19049 #11941769

Order 1280.

886 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
1C5×#3561C211yesmaximal, characteristic
2C5×#3215C211yesmaximal, characteristic, complemented
3C5×#3181C211yesmaximal, characteristic, complemented
4C5×#3162C211yesmaximal, characteristic, complemented
5C5×#1664C212yesmaximal, complemented
6C5×#1661C212yesmaximal, complemented
7C5×#1661C212yesmaximal, complemented
8C5×#1660C212yesmaximal, 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×#533C2211yescharacteristic
26C5×#529C2211yescharacteristic
27C2×C5×#124C2211yescharacteristic
28C5×#501—21no—
29C5×#501—21no—
30C5×#501C2211yescharacteristic, complemented
31C5×#501C2212yescomplemented
32C5×#501C2212yescomplemented
33C5×#497C2212yes—
34C5×#497C2211yescharacteristic, complemented
35C5×#491C2211yescharacteristic, complemented
36C5×#489C2212yes—
37C5×#489C2211yescharacteristic, complemented
38C5×#467—21no—
39C5×#465—21no—
40C5×#464—22no—
41C5×#451—21no—
42C5×#450—21no—
43C5×#441—21no—
44C5×#440—21no—
45C5×#432C2211yescharacteristic
46C5×#427—22no—
47C2×C5×#97C2211yescharacteristic
48C2×C5×#97C2211yescharacteristic
49C5×#330C2212yescomplemented
50C5×#330C2212yescomplemented
51C5×#330C2212yescomplemented
52C5×#329C2212yes—
53C5×#329C2212yes—
54C5×#329C2212yescomplemented
55#19049C511yesmaximal, characteristic, direct product factor, Sylow
56C2×C5×#41—21no—
57C22×C5×Q8—21no—
58C22×C5×Q8D411yescharacteristic
59C22×C5×D4D411yescharacteristic
60C2×C5×Q16—21no—
61C2×C5×Q16—22no—
62C2×C5×SD16—21no—
63C2×C5×SD16—22no—
64C22×C5×C8—21no—
65C22×C5×C8—21no—
66C5×C4⋊Q8—21no—
67C5×C4⋊Q8—22no—
68C5×C4⋊Q8D411yescharacteristic
69C5×C4⋊Q8D411yescharacteristic
70C5×C4⋊Q8C2311yescharacteristic
71C5×C4⋊Q8C2311yescharacteristic
72C5×#126—22no—
73C5×#126—22no—
74C5×#126—22no—
75C5×#126—21no—
76C5×#125—21no—
77C5×#125C2311yescharacteristic
78C5×#125C2312yes—
79C5×#124—21no—
80C5×#124—22no—
81C5×#124—22no—
82C5×#124—21no—
83C5×#124—21no—
84C5×#124D412yes—
85C5×#124C2312yes—
86C5×#124C2311yescharacteristic
87C5×#123—22no—
88C5×#123—22no—
89C5×#123—21no—
90C5×#123—22no—
91C5×#123—22no—
92C5×C22⋊Q8—22no—
93C5×C22⋊Q8—22no—
94C5×C22⋊Q8—21no—
95C5×C22⋊Q8—22no—
96C5×C22⋊Q8—22no—
97C5×C22⋊Q8—22no—
98C5×C22⋊Q8—21no—
99C5×C22⋊Q8—21no—
100C5×C22⋊Q8—22no—
101C5×C22⋊Q8—22no—
102C5×C22⋊Q8—21no—
103C5×C22⋊Q8—21no—
104C5×C22⋊Q8D411yescharacteristic
105C5×#121—21no—
106C5×#121—22no—
107C5×#121—22no—
108C5×#121D411yescharacteristic
109C5×#120—22no—
110C4×C5×Q8—21no—
111C4×C5×Q8—21no—
112C4×C5×Q8—21no—
113C4×C5×Q8—21no—
114C4×C5×Q8—21no—
115C4×C5×Q8D411yescharacteristic
116C4×C5×D4—22no—
117C4×C5×D4—21no—
118C4×C5×D4—22no—
119C4×C5×D4—21no—
120C4×C5×D4D411yescharacteristic
121C5×#117—22no—
122C5×#117—21no—
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—22no—
130C5×#107—22no—
131C5×#107—21no—
132C5×#106—22no—
133C5×#106—22no—
134C5×#106—21no—
135C5×C4⋊C8—21no—
136C5×#103—21no—
137C5×#103—22no—
138C5×#103—21no—
139C5×#103—21no—
140C5×#102—21no—
141C5×#102—22no—
142C5×C22⋊C8—21no—
143C5×#97—21no—
144C5×#97—21no—
145C5×#97C2312yes—
146C5×#97C2311yescharacteristic
147C5×#97C2311yescharacteristic
148C5×#97C2312yes—
149#3561C2×C511yescharacteristic
150#3215C2×C511yescharacteristic, complemented
151#3181C2×C511yescharacteristic, complemented
152#3162C2×C511yescharacteristic, complemented
153#1664C2×C512yescomplemented
154#1661C2×C512yescomplemented
155#1661C2×C512yescomplemented
156#1660C2×C512yescomplemented
157#1532C2×C511yescharacteristic, complemented
158#1531C2×C511yescharacteristic, complemented
159C2×#329C2×C511yescharacteristic, complemented
160C24×C5—21no—
161C5×#41—82no—
162C2×C5×Q8—22no—
163C2×C5×Q8—21no—
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—81no—
177C2×C5×D4C2×D411yescharacteristic
178C2×C5×D4C2×D412yes—
179C22×C4×C5—22no—
180C22×C4×C5—22no—
181C22×C4×C5—22no—
182C22×C4×C5—21no—
183C22×C4×C5—21no—
184C22×C4×C5C2×D411yescharacteristic
185C22×C4×C5C2×D411yescharacteristic
186C22×C4×C5C2×D411yescharacteristic
187C5×Q16—42no—
188C5×Q16—82no—
189C5×SD16—42no—
190C5×SD16—82no—
191C2×C5×C8—24no—
192C2×C5×C8—21no—
193C2×C5×C8—22no—
194C2×C5×C8—21no—
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—21no—
202C5×C4⋊C4—21no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4—21no—
205C5×C4⋊C4—21no—
206C5×C4⋊C4—21no—
207C5×C4⋊C4—22no—
208C5×C4⋊C4—22no—
209C5×C4⋊C4—21no—
210C5×C4⋊C4—21no—
211C5×C4⋊C4—21no—
212C5×C4⋊C4—21no—
213C5×C4⋊C4—41no—
214C5×C4⋊C4—41no—
215C5×C4⋊C4C2×D412yes—
216C5×C4⋊C4C2×D412yes—
217C5×C4⋊C4C2×D411yescharacteristic
218C5×C4⋊C4C2×D411yescharacteristic
219C5×C22⋊C4—22no—
220C5×C22⋊C4—22no—
221C5×C22⋊C4—22no—
222C5×C22⋊C4—22no—
223C5×C22⋊C4—22no—
224C5×C22⋊C4—21no—
225C5×C22⋊C4—22no—
226C5×C22⋊C4—22no—
227C5×C22⋊C4—22no—
228C5×C22⋊C4—21no—
229C5×C22⋊C4—21no—
230C5×C22⋊C4—22no—
231C5×C22⋊C4—22no—
232C5×C22⋊C4—21no—
233C5×C22⋊C4—21no—
234C5×C22⋊C4—41no—
235C42×C5—22no—
236C42×C5—22no—
237C42×C5—21no—
238C42×C5—41no—
239C42×C5C2×D411yescharacteristic
240C42×C5C2×D411yescharacteristic
241C42×C5C2×D411yescharacteristic
242C42×C5C2411yescharacteristic
243#563C22×C512yes—
244#562C22×C512yes—
245#556—21no—
246#551—22no—
247D4×Q8—21no—
248#548—21no—
249#546—21no—
250#543—21no—
251#542—21no—
252#542—22no—
253#538—21no—
254#536—21no—
255#533C22×C511yescharacteristic
256#533C22×C511yescharacteristic
257#529C22×C511yescharacteristic
258C2×#124C22×C511yescharacteristic
259#501—21no—
260#501—21no—
261#501C22×C512yescomplemented
262#501C22×C512yescomplemented
263#501C22×C511yescharacteristic, complemented
264#497C22×C512yes—
265#497C22×C511yescharacteristic, complemented
266#491C22×C511yescharacteristic, complemented
267#489C22×C512yes—
268#489C22×C511yescharacteristic, complemented
269#467—21no—
270#465—21no—
271#464—22no—
272#451—21no—
273#450—21no—
274#441—21no—
275#440—21no—
276#432C22×C511yescharacteristic
277#427—22no—
278C2×#97C22×C511yescharacteristic
279C2×#97C22×C511yescharacteristic
280#330C22×C512yescomplemented
281#330C22×C512yescomplemented
282#330C22×C512yescomplemented
283#329C22×C512yes—
284#329C22×C512yes—
285#329C22×C512yescomplemented
286C23×C5—22no—
287C23×C5—21no—
288C23×C5—42no—
289C23×C5—81no—
290C23×C5—81no—
291C23×C5#12011yescharacteristic
292C5×Q8—41no—
293C5×Q8—42no—
294C5×Q8—42no—
295C5×Q8—42no—
296C5×Q8—41no—
297C5×Q8—42no—
298C5×Q8—41no—
299C5×Q8—81no—
300C5×Q8—82no—
301C5×D4—41no—
302C5×D4—42no—
303C5×D4—42no—
304C5×D4—42no—
305C5×D4—82no—
306C5×D4—82no—
307C2×C4×C5—22no—
308C2×C4×C5—21no—
309C2×C4×C5—22no—
310C2×C4×C5—21no—
311C2×C4×C5—22no—
312C2×C4×C5—22no—
313C2×C4×C5—21no—
314C2×C4×C5—41no—
315C2×C4×C5—41no—
316C2×C4×C5—42no—
317C2×C4×C5—41no—
318C2×C4×C5—41no—
319C2×C4×C5—81no—
320C2×C4×C5—82no—
321C2×C4×C5—82no—
322C2×C4×C5—82no—
323C2×C4×C5#12011yescharacteristic
324C2×C4×C5#12011yescharacteristic
325C2×C4×C5#12011yescharacteristic
326C2×C4×C5C22×D411yescharacteristic
327C2×C4×C5C22×D411yescharacteristic
328C2×C4×C5C22×D411yescharacteristic
329C5×C8—24no—
330C5×C8—22no—
331C5×C8—41no—
332C5×C8—42no—
333C2×#41—21no—
334C22×Q8—21no—
335C22×Q8C5×D411yescharacteristic
336C22×D4C5×D411yescharacteristic
337C2×Q16—21no—
338C2×Q16—22no—
339C2×SD16—21no—
340C2×SD16—22no—
341C22×C8—21no—
342C22×C8—21no—
343C4⋊Q8—21no—
344C4⋊Q8—22no—
345C4⋊Q8C5×D411yescharacteristic
346C4⋊Q8C5×D411yescharacteristic
347C4⋊Q8C23×C511yescharacteristic
348C4⋊Q8C23×C511yescharacteristic
349#126—22no—
350#126—22no—
351#126—22no—
352#126—21no—
353#125—21no—
354#125C23×C511yescharacteristic
355#125C23×C512yes—
356#124—21no—
357#124—22no—
358#124—22no—
359#124—21no—
360#124—21no—
361#124C5×D412yes—
362#124C23×C512yes—
363#124C23×C511yescharacteristic
364#123—22no—
365#123—22no—
366#123—21no—
367#123—22no—
368#123—22no—
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371C22⋊Q8—21no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8C5×D411yescharacteristic
382#121—21no—
383#121—22no—
384#121—22no—
385#121C5×D411yescharacteristic
386#120—22no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8—21no—
391C4×Q8—21no—
392C4×Q8C5×D411yescharacteristic
393C4×D4—22no—
394C4×D4—21no—
395C4×D4—22no—
396C4×D4—21no—
397C4×D4C5×D411yescharacteristic
398#117—22no—
399#117—21no—
400#117—21no—
401#117C5×D412yes—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C42C23×C511yescharacteristic
406#107—22no—
407#107—22no—
408#107—21no—
409#106—22no—
410#106—22no—
411#106—21no—
412C4⋊C8—21no—
413#103—21no—
414#103—22no—
415#103—21no—
416#103—21no—
417#102—21no—
418#102—22no—
419C22⋊C8—21no—
420#97—21no—
421#97—21no—
422#97C23×C512yes—
423#97C23×C511yescharacteristic
424#97C23×C511yescharacteristic
425#97C23×C512yes—
426C22×C5—21no—
427C22×C5—21no—
428C22×C5—41no—
429C22×C5—42no—
430C22×C5—42no—
431C22×C5—41no—
432C22×C5—41no—
433C22×C5—42no—
434C22×C5—162no—
435C22×C5#45612yes—
436C22×C5C2×#12011yescharacteristic
437C4×C5—21no—
438C4×C5—21no—
439C4×C5—21no—
440C4×C5—21no—
441C4×C5—21no—
442C4×C5—21no—
443C4×C5—41no—
444C4×C5—41no—
445C4×C5—42no—
446C4×C5—41no—
447C4×C5—42no—
448C4×C5—42no—
449C4×C5—81no—
450C4×C5—81no—
451C24—21no—
452#41—82no—
453C2×Q8—22no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—82no—
460C2×Q8—81no—
461C2×Q8C2×C5×D411yescharacteristic
462C2×Q8C2×C5×D412yes—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—81no—
468C2×D4C2×C5×D411yescharacteristic
469C2×D4C2×C5×D412yes—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4C2×C5×D411yescharacteristic
476C22×C4C2×C5×D411yescharacteristic
477C22×C4C2×C5×D411yescharacteristic
478Q16—42no—
479Q16—82no—
480SD16—42no—
481SD16—82no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—22no—
487C2×C8—42no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4C2×C5×D411yescharacteristic
507C4⋊C4C2×C5×D411yescharacteristic
508C4⋊C4C2×C5×D412yes—
509C4⋊C4C2×C5×D412yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—21no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—41no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—41no—
530C42C2×C5×D411yescharacteristic
531C42C2×C5×D411yescharacteristic
532C42C2×C5×D411yescharacteristic
533C42C24×C511yescharacteristic
534C2×C5—22no—
535C2×C5—81no—
536C2×C5—82no—
537C2×C5C2×#45611yescharacteristic
538C2×C5#301811yescharacteristic
539C2×C5#301911yescharacteristic
540C23—22no—
541C23—21no—
542C23—42no—
543C23—81no—
544C23—81no—
545C23C5×#12011yescharacteristic
546Q8—41no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—41no—
551Q8—42no—
552Q8—41no—
553Q8—81no—
554Q8—82no—
555D4—41no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—82no—
560D4—82no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4C5×#12011yescharacteristic
578C2×C4C5×#12011yescharacteristic
579C2×C4C5×#12011yescharacteristic
580C2×C4C22×C5×D411yescharacteristic
581C2×C4C22×C5×D411yescharacteristic
582C2×C4C22×C5×D411yescharacteristic
583C8—24no—
584C8—22no—
585C8—41no—
586C8—42no—
587C5#1904911yescharacteristic, direct product factor, Sylow
588C22—21no—
589C22—21no—
590C22—41no—
591C22—42no—
592C22—42no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—162no—
597C22C5×#45612yes—
598C22C2×C5×#12011yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—42no—
611C4—81no—
612C4—81no—
613C2—22no—
614C2—82no—
615C2—81no—
616C2C2×C5×#45611yescharacteristic
617C2C5×#301811yescharacteristic
618C2C5×#301911yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC42C24×C5
prime radicalC5×#3561C2
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
2. C5×#3215C2semidirect
3. C5×#3181C2semidirect
4. C5×#3162C2semidirect
5. C5×#1664C2semidirect
6. C5×#1661C2semidirect
7. C5×#1661C2semidirect
8. C5×#1660C2semidirect
9. C5×#1532C2semidirect
10. C5×#1531C2semidirect
11. C2×C5×#329C2semidirect
30. C5×#501C22semidirect
31. C5×#501C22semidirect
32. C5×#501C22semidirect
34. C5×#497C22semidirect
35. C5×#491C22semidirect
37. C5×#489C22semidirect
49. C5×#330C22semidirect
50. C5×#330C22semidirect
51. C5×#330C22semidirect
54. C5×#329C22semidirect
55. #19049C5direct (N × H)
150. #3215C2×C5semidirect
151. #3181C2×C5semidirect
152. #3162C2×C5semidirect
153. #1664C2×C5semidirect
154. #1661C2×C5semidirect
155. #1661C2×C5semidirect
156. #1660C2×C5semidirect
157. #1532C2×C5semidirect
158. #1531C2×C5semidirect
159. C2×#329C2×C5semidirect
261. #501C22×C5semidirect
262. #501C22×C5semidirect
263. #501C22×C5semidirect
265. #497C22×C5semidirect
266. #491C22×C5semidirect
268. #489C22×C5semidirect
280. #330C22×C5semidirect
281. #330C22×C5semidirect
282. #330C22×C5semidirect
285. #329C22×C5semidirect
587. C5#19049direct (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×#32152
C5×#31812
C5×#31622
C5×#16642
C5×#16612
C5×#16602
C5×#15322
C5×#15312
C2×C5×#3292
#190495