Finite Group Databasebeta

← C5×#19072 #11941792

Subgroups — C5×#19072 #11941792

Order 1280.

992 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3545C211yesmaximal, characteristic, complemented
2C5×#3180C212yesmaximal, complemented
3C5×#3153C211yesmaximal, characteristic, complemented
4C5×#1661C212yesmaximal, complemented
5C5×#1660C211yesmaximal, characteristic, complemented
6C5×#1659C211yesmaximal, characteristic, complemented
7C5×#1657C211yesmaximal, characteristic, complemented
8C5×#1656C212yesmaximal, complemented
9C5×#1655C211yesmaximal, characteristic, complemented
10C5×#1524C212yesmaximal, complemented
11C2×C5×#329C211yesmaximal, characteristic, complemented
12C2×C5×#137—21no—
13C2×C5×AGL(1,ℤ/8)—21no—
14C5×#563C2211yescharacteristic
15C5×#562C2212yes—
16C5×H(3,𝔽4)C2211yescharacteristic
17C5×#555—21no—
18C5×#551—21no—
19C5×#551—22no—
20C5×#546—22no—
21C5×#542—22no—
22C5×#538—22no—
23C5×#536—21no—
24C5×#534—21no—
25C5×#532C2211yescharacteristic, complemented
26C5×#529C2212yes—
27C2×C5×#124C2211yescharacteristic
28C5×#501C2212yescomplemented
29C5×#497C2211yescharacteristic
30C5×#497C2211yescharacteristic, complemented
31C5×#496C2211yescharacteristic
32C5×#496C2212yescomplemented
33C5×#496C2211yescharacteristic, complemented
34C5×#495—21no—
35C5×#495—21no—
36C5×#490C2212yescomplemented
37C5×#489C2212yescomplemented
38C5×#466—22no—
39C5×#465—22no—
40C5×#442—22no—
41C5×#440—22no—
42C4×C5×M16C2211yescharacteristic, complemented
43C2×C5×#97C2212yescomplemented
44C5×#330C2212yescomplemented
45C5×#330C2212yescomplemented
46C5×#329C2211yescharacteristic
47C5×#329C2211yescharacteristic
48C5×#329C2212yescomplemented
49C5×#329C2212yescomplemented
50C5×#329C2212yescomplemented
51#19072C511yesmaximal, characteristic, direct product factor, Sylow
52C2×C5×#41—21no—
53C2×C5×#41—21no—
54C22×C5×Q8D411yescharacteristic
55C22×C5×D4—21no—
56C22×C5×D4D411yescharacteristic
57C5×#137—44no—
58C5×AGL(1,ℤ/8)—44no—
59C2×C5×Q16—21no—
60C2×C5×Q16—21no—
61C2×C5×SD16—22no—
62C2×C5×SD16—21no—
63C2×C5×SD16—21no—
64C2×C5×SD16—21no—
65C2×C5×SD16—21no—
66C2×C5×D8—22no—
67C2×C5×D8—21no—
68C2×C5×D8—21no—
69C2×C5×M16—21no—
70C22×C5×C8—22no—
71C5×C4⋊Q8—21no—
72C5×C4⋊Q8—21no—
73C5×C4⋊Q8C2311yescharacteristic
74C5×#127C2311yescharacteristic
75C5×#126—22no—
76C5×#126—22no—
77C5×#126—22no—
78C5×#125C2312yes—
79C5×#124—21no—
80C5×#124—22no—
81C5×#124—21no—
82C5×#124—22no—
83C5×#124—22no—
84C5×#124—21no—
85C5×#124D411yescharacteristic
86C5×#124D411yescharacteristic
87C5×#124C2311yescharacteristic
88C5×#124C2312yes—
89C5×#124C2311yescharacteristic
90C5×#123—21no—
91C5×#123—22no—
92C5×#123—22no—
93C5×#123—21no—
94C5×#123—22no—
95C5×#123—22no—
96C5×#123—22no—
97C5×C22⋊Q8—22no—
98C5×C22⋊Q8—22no—
99C5×C22⋊Q8—22no—
100C5×C22⋊Q8—22no—
101C5×C22⋊Q8D412yes—
102C5×#121—22no—
103C5×#121—22no—
104C5×#121—22no—
105C5×#121—22no—
106C5×#121—22no—
107C5×#121D412yes—
108C5×#120—21no—
109C5×#120—22no—
110C5×#120—21no—
111C5×#120—22no—
112C4×C5×Q8D412yes—
113C4×C5×D4—22no—
114C4×C5×D4—22no—
115C4×C5×D4—22no—
116C4×C5×D4—22no—
117C4×C5×D4D412yes—
118C5×#117—21no—
119C5×#117—22no—
120C5×#117—22no—
121C2×C5×C22⋊C4—21no—
122C2×C5×C22⋊C4—22no—
123C2×C5×C22⋊C4—22no—
124C2×C42×C5C2311yescharacteristic
125C5×#107—22no—
126C5×#106—22no—
127C5×#103—22no—
128C5×#102—22no—
129C5×#102—22no—
130C5×#102—22no—
131C5×#97—22no—
132C5×#97C2312yes—
133C5×#97C2312yes—
134C5×#97C2311yescharacteristic
135C5×#97C2311yescharacteristic
136C4×C5×C8—21no—
137#3545C2×C511yescharacteristic, complemented
138#3180C2×C512yescomplemented
139#3153C2×C511yescharacteristic, complemented
140#1661C2×C512yescomplemented
141#1660C2×C511yescharacteristic, complemented
142#1659C2×C511yescharacteristic, complemented
143#1657C2×C511yescharacteristic, complemented
144#1656C2×C512yescomplemented
145#1655C2×C511yescharacteristic, complemented
146#1524C2×C512yescomplemented
147C2×#329C2×C511yescharacteristic, complemented
148C24×C5—21no—
149C24×C5—21no—
150C5×#41—42no—
151C5×#41—42no—
152C5×#41—82no—
153C2×C5×Q8—21no—
154C2×C5×Q8—21no—
155C2×C5×Q8—21no—
156C2×C5×Q8—42no—
157C2×C5×Q8C2×D412yes—
158C2×C5×Q8C2×D411yescharacteristic
159C2×C5×Q8C2×D411yescharacteristic
160C2×C5×D4—22no—
161C2×C5×D4—22no—
162C2×C5×D4—22no—
163C2×C5×D4—21no—
164C2×C5×D4—22no—
165C2×C5×D4—21no—
166C2×C5×D4—22no—
167C2×C5×D4—21no—
168C2×C5×D4—42no—
169C2×C5×D4—41no—
170C2×C5×D4—82no—
171C2×C5×D4C2×D412yes—
172C2×C5×D4C2×D411yescharacteristic
173C2×C5×D4C2×D411yescharacteristic
174C22×C4×C5—21no—
175C22×C4×C5—22no—
176C22×C4×C5—22no—
177C22×C4×C5—21no—
178C22×C4×C5—22no—
179C22×C4×C5—41no—
180C22×C4×C5C2×D411yescharacteristic
181C22×C4×C5C2×D412yes—
182C5×Q16—42no—
183C5×Q16—42no—
184C5×SD16—44no—
185C5×SD16—42no—
186C5×SD16—42no—
187C5×SD16—42no—
188C5×SD16—42no—
189C5×D8—44no—
190C5×D8—42no—
191C5×D8—42no—
192C5×M16—24no—
193C2×C5×C8—22no—
194C2×C5×C8—21no—
195C2×C5×C8—21no—
196C2×C5×C8—22no—
197C2×C5×C8—44no—
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⋊C4C2×D412yes—
207C5×C4⋊C4C2×D412yes—
208C5×C22⋊C4—22no—
209C5×C22⋊C4—21no—
210C5×C22⋊C4—21no—
211C5×C22⋊C4—22no—
212C5×C22⋊C4—22no—
213C5×C22⋊C4—22no—
214C5×C22⋊C4—22no—
215C5×C22⋊C4—22no—
216C5×C22⋊C4—22no—
217C5×C22⋊C4—22no—
218C5×C22⋊C4—22no—
219C5×C22⋊C4—22no—
220C5×C22⋊C4—22no—
221C5×C22⋊C4—22no—
222C5×C22⋊C4—22no—
223C5×C22⋊C4—42no—
224C42×C5—21no—
225C42×C5—21no—
226C42×C5—22no—
227C42×C5C2×D411yescharacteristic
228C42×C5C2×D412yes—
229C42×C5C2411yescharacteristic
230C2×#137—21no—
231C2×AGL(1,ℤ/8)—21no—
232#563C22×C511yescharacteristic
233#562C22×C512yes—
234H(3,𝔽4)C22×C511yescharacteristic
235#555—21no—
236#551—21no—
237#551—22no—
238#546—22no—
239#542—22no—
240#538—22no—
241#536—21no—
242#534—21no—
243#532C22×C511yescharacteristic, complemented
244#529C22×C512yes—
245C2×#124C22×C511yescharacteristic
246#501C22×C512yescomplemented
247#497C22×C511yescharacteristic
248#497C22×C511yescharacteristic, complemented
249#496C22×C511yescharacteristic
250#496C22×C511yescharacteristic, complemented
251#496C22×C512yescomplemented
252#495—21no—
253#495—21no—
254#490C22×C512yescomplemented
255#489C22×C512yescomplemented
256#466—22no—
257#465—22no—
258#442—22no—
259#440—22no—
260C4×M16C22×C511yescharacteristic, complemented
261C2×#97C22×C512yescomplemented
262#330C22×C512yescomplemented
263#330C22×C512yescomplemented
264#329C22×C511yescharacteristic
265#329C22×C511yescharacteristic
266#329C22×C512yescomplemented
267#329C22×C512yescomplemented
268#329C22×C512yescomplemented
269C23×C5—21no—
270C23×C5—22no—
271C23×C5—21no—
272C23×C5—42no—
273C23×C5—41no—
274C23×C5—82no—
275C23×C5—81no—
276C23×C5—82no—
277C23×C5#12011yescharacteristic
278C5×Q8—41no—
279C5×Q8—41no—
280C5×Q8—42no—
281C5×Q8—41no—
282C5×Q8—41no—
283C5×Q8—81no—
284C5×Q8—81no—
285C5×D4—41no—
286C5×D4—41no—
287C5×D4—42no—
288C5×D4—41no—
289C5×D4—42no—
290C5×D4—41no—
291C5×D4—42no—
292C5×D4—44no—
293C5×D4—81no—
294C5×D4—82no—
295C5×D4—82no—
296C5×D4—82no—
297C5×D4—81no—
298C5×D4—82no—
299C2×C4×C5—22no—
300C2×C4×C5—22no—
301C2×C4×C5—21no—
302C2×C4×C5—22no—
303C2×C4×C5—22no—
304C2×C4×C5—21no—
305C2×C4×C5—42no—
306C2×C4×C5—42no—
307C2×C4×C5—41no—
308C2×C4×C5—82no—
309C2×C4×C5—82no—
310C2×C4×C5—82no—
311C2×C4×C5—81no—
312C2×C4×C5—82no—
313C2×C4×C5#12012yes—
314C2×C4×C5#12011yescharacteristic
315C2×C4×C5C22×D412yes—
316C2×C4×C5C22×D411yescharacteristic
317C5×C8—24no—
318C5×C8—22no—
319C5×C8—22no—
320C5×C8—42no—
321C2×#41—21no—
322C2×#41—21no—
323C22×Q8C5×D411yescharacteristic
324C22×D4—21no—
325C22×D4C5×D411yescharacteristic
326#137—44no—
327AGL(1,ℤ/8)—44no—
328C2×Q16—21no—
329C2×Q16—21no—
330C2×SD16—21no—
331C2×SD16—21no—
332C2×SD16—22no—
333C2×SD16—21no—
334C2×SD16—21no—
335C2×D8—21no—
336C2×D8—21no—
337C2×D8—22no—
338C2×M16—21no—
339C22×C8—22no—
340C4⋊Q8—21no—
341C4⋊Q8—21no—
342C4⋊Q8C23×C511yescharacteristic
343#127C23×C511yescharacteristic
344#126—22no—
345#126—22no—
346#126—22no—
347#125C23×C512yes—
348#124—21no—
349#124—22no—
350#124—21no—
351#124—22no—
352#124—22no—
353#124—21no—
354#124C5×D411yescharacteristic
355#124C5×D411yescharacteristic
356#124C23×C511yescharacteristic
357#124C23×C511yescharacteristic
358#124C23×C512yes—
359#123—21no—
360#123—22no—
361#123—22no—
362#123—21no—
363#123—22no—
364#123—22no—
365#123—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C22⋊Q8C5×D412yes—
371#121—22no—
372#121—22no—
373#121—22no—
374#121—22no—
375#121—22no—
376#121C5×D412yes—
377#120—21no—
378#120—22no—
379#120—21no—
380#120—22no—
381C4×Q8C5×D412yes—
382C4×D4—22no—
383C4×D4—22no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4C5×D412yes—
387#117—21no—
388#117—22no—
389#117—22no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C42C23×C511yescharacteristic
394#107—22no—
395#106—22no—
396#103—22no—
397#102—22no—
398#102—22no—
399#102—22no—
400#97—22no—
401#97C23×C512yes—
402#97C23×C512yes—
403#97C23×C511yescharacteristic
404#97C23×C511yescharacteristic
405C4×C8—21no—
406C22×C5—22no—
407C22×C5—42no—
408C22×C5—41no—
409C22×C5—41no—
410C22×C5—42no—
411C22×C5—42no—
412C22×C5—42no—
413C22×C5—42no—
414C22×C5—81no—
415C22×C5—82no—
416C22×C5—162no—
417C22×C5—162no—
418C22×C5#45312yes—
419C22×C5C2×#12011yescharacteristic
420C4×C5—21no—
421C4×C5—21no—
422C4×C5—22no—
423C4×C5—22no—
424C4×C5—41no—
425C4×C5—41no—
426C4×C5—42no—
427C4×C5—42no—
428C4×C5—42no—
429C4×C5—81no—
430C24—21no—
431C24—21no—
432#41—42no—
433#41—42no—
434#41—82no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—42no—
439C2×Q8C2×C5×D412yes—
440C2×Q8C2×C5×D411yescharacteristic
441C2×Q8C2×C5×D411yescharacteristic
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—42no—
451C2×D4—41no—
452C2×D4—82no—
453C2×D4C2×C5×D412yes—
454C2×D4C2×C5×D411yescharacteristic
455C2×D4C2×C5×D411yescharacteristic
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—41no—
462C22×C4C2×C5×D411yescharacteristic
463C22×C4C2×C5×D412yes—
464Q16—42no—
465Q16—42no—
466SD16—42no—
467SD16—42no—
468SD16—42no—
469SD16—42no—
470SD16—44no—
471D8—42no—
472D8—42no—
473D8—44no—
474M16—24no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—44no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4C2×C5×D412yes—
489C4⋊C4C2×C5×D412yes—
490C22⋊C4—22no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—42no—
506C42—21no—
507C42—21no—
508C42—22no—
509C42C2×C5×D411yescharacteristic
510C42C2×C5×D412yes—
511C42C24×C511yescharacteristic
512C2×C5—22no—
513C2×C5—82no—
514C2×C5—81no—
515C2×C5—81no—
516C2×C5—161no—
517C2×C5C2×#45311yescharacteristic
518C2×C5#301812yes—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—42no—
523C23—41no—
524C23—82no—
525C23—81no—
526C23—82no—
527C23C5×#12011yescharacteristic
528Q8—41no—
529Q8—41no—
530Q8—42no—
531Q8—41no—
532Q8—41no—
533Q8—81no—
534Q8—81no—
535D4—41no—
536D4—41no—
537D4—42no—
538D4—41no—
539D4—42no—
540D4—41no—
541D4—42no—
542D4—44no—
543D4—81no—
544D4—82no—
545D4—81no—
546D4—82no—
547D4—82no—
548D4—82no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—81no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4C5×#12012yes—
564C2×C4C5×#12011yescharacteristic
565C2×C4C22×C5×D412yes—
566C2×C4C22×C5×D411yescharacteristic
567C8—24no—
568C8—22no—
569C8—22no—
570C8—42no—
571C5#1907211yescharacteristic, direct product factor, Sylow
572C22—22no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—81no—
581C22—82no—
582C22—162no—
583C22—162no—
584C22C5×#45312yes—
585C22C2×C5×#12011yescharacteristic
586C4—21no—
587C4—21no—
588C4—22no—
589C4—22no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—81no—
596C2—22no—
597C2—81no—
598C2—82no—
599C2—81no—
600C2—161no—
601C2C2×C5×#45311yescharacteristic
602C2C5×#301812yes—
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×#3545C2semidirect
2. C5×#3180C2semidirect
3. C5×#3153C2semidirect
4. C5×#1661C2semidirect
5. C5×#1660C2semidirect
6. C5×#1659C2semidirect
7. C5×#1657C2semidirect
8. C5×#1656C2semidirect
9. C5×#1655C2semidirect
10. C5×#1524C2semidirect
11. C2×C5×#329C2semidirect
25. C5×#532C22semidirect
28. C5×#501C22semidirect
30. C5×#497C22semidirect
32. C5×#496C22semidirect
33. C5×#496C22semidirect
36. C5×#490C22semidirect
37. C5×#489C22semidirect
42. C4×C5×M16C22semidirect
43. C2×C5×#97C22semidirect
44. C5×#330C22semidirect
45. C5×#330C22semidirect
48. C5×#329C22semidirect
49. C5×#329C22semidirect
50. C5×#329C22semidirect
51. #19072C5direct (N × H)
137. #3545C2×C5semidirect
138. #3180C2×C5semidirect
139. #3153C2×C5semidirect
140. #1661C2×C5semidirect
141. #1660C2×C5semidirect
142. #1659C2×C5semidirect
143. #1657C2×C5semidirect
144. #1656C2×C5semidirect
145. #1655C2×C5semidirect
146. #1524C2×C5semidirect
147. C2×#329C2×C5semidirect
243. #532C22×C5semidirect
246. #501C22×C5semidirect
248. #497C22×C5semidirect
250. #496C22×C5semidirect
251. #496C22×C5semidirect
254. #490C22×C5semidirect
255. #489C22×C5semidirect
260. C4×M16C22×C5semidirect
261. C2×#97C22×C5semidirect
262. #330C22×C5semidirect
263. #330C22×C5semidirect
266. #329C22×C5semidirect
267. #329C22×C5semidirect
268. #329C22×C5semidirect
571. C5#19072direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35452
C5×#31802
C5×#31532
C5×#16612
C5×#16602
C5×#16592
C5×#16572
C5×#16562
C5×#16552
C5×#15242
C2×C5×#3292
#190725