Finite Group Databasebeta

← C5×#18427 #11941147

Subgroups — C5×#18427 #11941147

Order 1280.

996 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3558C211yesmaximal, characteristic
2C5×#3218C211yesmaximal, characteristic, complemented
3C5×#3206C211yesmaximal, characteristic, complemented
4C5×#2640C211yesmaximal, characteristic, complemented
5C5×#1606C214yesmaximal, complemented
6C5×#1601C214yesmaximal, complemented
7C5×#1505C211yesmaximal, characteristic, complemented
8C5×#1500C211yesmaximal, characteristic, complemented
9C5×#1469C211yesmaximal, characteristic, complemented
10C5×#562—21no—
11C5×H(3,𝔽4)—21no—
12C5×#559—21no—
13C5×#548—21no—
14C5×#546—21no—
15C5×#546—22no—
16C5×D42—21no—
17C5×#542—21no—
18C5×#538—22no—
19C5×#538C2212yes—
20C5×#535—21no—
21C5×#534C2212yes—
22C5×#532C2211yescharacteristic
23C2×C5×C4⋊Q8C2211yescharacteristic, complemented
24C5×#529C2211yescharacteristic
25C2×C5×#124C2211yescharacteristic
26C5×#497—21no—
27C5×#492—21no—
28C5×#491—21no—
29C5×#487—21no—
30C5×#462—21no—
31C5×#461—22no—
32C5×#461—21no—
33C5×#451C2212yescomplemented
34C5×#448C2212yes—
35C5×#448C2214yescomplemented
36C5×#432C2211yescharacteristic, complemented
37C5×#431C2211yescharacteristic, complemented
38C5×#397—22no—
39C5×#393C2214yescomplemented
40C5×#391—21no—
41C5×#390—21no—
42C5×#376C2211yescharacteristic, complemented
43C5×#328C2212yescomplemented
44C5×#327C2212yes—
45C5×#327C2214yescomplemented
46C5×#323C2214yescomplemented
47#18427C511yesmaximal, characteristic, direct product factor, Sylow
48C2×C5×#41—21no—
49C2×C5×#41—21no—
50C22×C5×Q8D411yescharacteristic
51C22×C5×Q8C2312yes—
52C22×C5×D4—21no—
53C22×C5×D4—22no—
54C22×C5×D4D411yescharacteristic
55C2×C5×Q16—22no—
56C2×C5×SD16—22no—
57C2×C5×SD16—24no—
58C5×C4⋊Q8—21no—
59C5×C4⋊Q8—21no—
60C5×C4⋊Q8—21no—
61C5×C4⋊Q8—21no—
62C5×C4⋊Q8D411yescharacteristic
63C5×#127—21no—
64C5×#127D411yescharacteristic
65C5×#126—22no—
66C5×#126—21no—
67C5×#125D411yescharacteristic
68C5×#124—21no—
69C5×#124—22no—
70C5×#124—22no—
71C5×#124—22no—
72C5×#124—21no—
73C5×#124—21no—
74C5×#124—21no—
75C5×#124—21no—
76C5×#124—21no—
77C5×#124D411yescharacteristic
78C5×#123—22no—
79C5×#123—21no—
80C5×#123—22no—
81C5×#123—22no—
82C5×#123—22no—
83C5×C22⋊Q8—22no—
84C5×C22⋊Q8—21no—
85C5×C22⋊Q8—22no—
86C5×C22⋊Q8C2311yescharacteristic
87C5×#121—22no—
88C5×#121—22no—
89C5×#121—22no—
90C5×#121—21no—
91C5×#121—22no—
92C5×#121—21no—
93C5×#121—22no—
94C5×#121C2311yescharacteristic
95C5×#121C2312yes—
96C5×#120—22no—
97C5×#120—22no—
98C5×#120—22no—
99C5×#120—22no—
100C5×#120—22no—
101C4×C5×Q8—21no—
102C4×C5×Q8D411yescharacteristic
103C4×C5×D4—22no—
104C4×C5×D4—21no—
105C4×C5×D4—21no—
106C4×C5×D4—22no—
107C4×C5×D4—21no—
108C4×C5×D4D411yescharacteristic
109C4×C5×D4D412yes—
110C5×#117—21no—
111C2×C5×C4⋊C4—21no—
112C2×C5×C4⋊C4—21no—
113C2×C5×C22⋊C4—22no—
114C2×C5×C22⋊C4—24no—
115C2×C5×C22⋊C4D412yes—
116C2×C5×C22⋊C4C2312yes—
117C2×C42×C5—21no—
118C2×C42×C5C2311yescharacteristic
119C5×C4⋊C8—21no—
120C5×C4⋊C8—21no—
121C5×#103—22no—
122C5×#103—24no—
123C5×#103—22no—
124C5×C22⋊C8C2312yes—
125C5×C22⋊C8C2312yes—
126C5×#97—21no—
127C4×C5×C8—21no—
128C5×#95—21no—
129C5×#95—24no—
130C5×#95C2312yes—
131#3558C2×C511yescharacteristic
132#3218C2×C511yescharacteristic, complemented
133#3206C2×C511yescharacteristic, complemented
134#2640C2×C511yescharacteristic, complemented
135#1606C2×C514yescomplemented
136#1601C2×C514yescomplemented
137#1505C2×C511yescharacteristic, complemented
138#1500C2×C511yescharacteristic, complemented
139#1469C2×C511yescharacteristic, complemented
140C24×C5—22no—
141C24×C5C2×D412yes—
142C5×#41—82no—
143C5×#41—82no—
144C2×C5×Q8—21no—
145C2×C5×Q8—21no—
146C2×C5×Q8—21no—
147C2×C5×Q8—22no—
148C2×C5×Q8—22no—
149C2×C5×Q8—42no—
150C2×C5×Q8—82no—
151C2×C5×Q8C2×D411yescharacteristic
152C2×C5×D4—22no—
153C2×C5×D4—21no—
154C2×C5×D4—22no—
155C2×C5×D4—22no—
156C2×C5×D4—22no—
157C2×C5×D4—22no—
158C2×C5×D4—22no—
159C2×C5×D4—21no—
160C2×C5×D4—21no—
161C2×C5×D4—21no—
162C2×C5×D4—21no—
163C2×C5×D4—42no—
164C2×C5×D4—82no—
165C2×C5×D4—84no—
166C2×C5×D4C2×D411yescharacteristic
167C2×C5×D4C2×D412yes—
168C22×C4×C5—22no—
169C22×C4×C5—22no—
170C22×C4×C5—21no—
171C22×C4×C5—22no—
172C22×C4×C5—22no—
173C22×C4×C5—22no—
174C22×C4×C5C2×D411yescharacteristic
175C22×C4×C5C2×D411yescharacteristic
176C22×C4×C5C2×D412yes—
177C22×C4×C5C2411yescharacteristic
178C5×Q16—82no—
179C5×SD16—84no—
180C5×SD16—82no—
181C2×C5×C8—22no—
182C2×C5×C8—22no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—22no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—21no—
188C5×C4⋊C4—21no—
189C5×C4⋊C4—21no—
190C5×C4⋊C4—41no—
191C5×C4⋊C4—41no—
192C5×C4⋊C4—41no—
193C5×C4⋊C4—41no—
194C5×C4⋊C4C2×D412yes—
195C5×C4⋊C4C2×D411yescharacteristic
196C5×C4⋊C4C2×D411yescharacteristic
197C5×C22⋊C4—22no—
198C5×C22⋊C4—22no—
199C5×C22⋊C4—22no—
200C5×C22⋊C4—22no—
201C5×C22⋊C4—22no—
202C5×C22⋊C4—21no—
203C5×C22⋊C4—21no—
204C5×C22⋊C4—22no—
205C5×C22⋊C4—22no—
206C5×C22⋊C4—21no—
207C5×C22⋊C4—21no—
208C5×C22⋊C4—22no—
209C5×C22⋊C4—22no—
210C5×C22⋊C4—22no—
211C5×C22⋊C4—21no—
212C5×C22⋊C4—22no—
213C5×C22⋊C4—84no—
214C42×C5—22no—
215C42×C5—21no—
216C42×C5—42no—
217C42×C5C2×D411yescharacteristic
218C42×C5C2×D411yescharacteristic
219C42×C5C2×D411yescharacteristic
220C42×C5C2×D411yescharacteristic
221#562—21no—
222H(3,𝔽4)—21no—
223#559—21no—
224#548—21no—
225#546—21no—
226#546—22no—
227D42—21no—
228#542—21no—
229#538—22no—
230#538C22×C512yes—
231#535—21no—
232#534C22×C512yes—
233#532C22×C511yescharacteristic
234C2×C4⋊Q8C22×C511yescharacteristic, complemented
235#529C22×C511yescharacteristic
236C2×#124C22×C511yescharacteristic
237#497—21no—
238#492—21no—
239#491—21no—
240#487—21no—
241#462—21no—
242#461—22no—
243#461—21no—
244#451C22×C512yescomplemented
245#448C22×C512yes—
246#448C22×C514yescomplemented
247#432C22×C511yescharacteristic, complemented
248#431C22×C511yescharacteristic, complemented
249#397—22no—
250#393C22×C514yescomplemented
251#391—21no—
252#390—21no—
253#376C22×C511yescharacteristic, complemented
254#328C22×C512yescomplemented
255#327C22×C512yes—
256#327C22×C514yescomplemented
257#323C22×C514yescomplemented
258C23×C5—22no—
259C23×C5—21no—
260C23×C5—22no—
261C23×C5—24no—
262C23×C5—42no—
263C23×C5—42no—
264C23×C5—82no—
265C23×C5—82no—
266C23×C5—82no—
267C23×C5C22×D411yescharacteristic
268C5×Q8—41no—
269C5×Q8—44no—
270C5×Q8—42no—
271C5×Q8—42no—
272C5×Q8—42no—
273C5×Q8—81no—
274C5×Q8—81no—
275C5×D4—42no—
276C5×D4—41no—
277C5×D4—42no—
278C5×D4—44no—
279C5×D4—42no—
280C5×D4—42no—
281C5×D4—81no—
282C5×D4—81no—
283C5×D4—82no—
284C5×D4—82no—
285C5×D4—82no—
286C5×D4—82no—
287C5×D4—82no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—21no—
291C2×C4×C5—22no—
292C2×C4×C5—22no—
293C2×C4×C5—21no—
294C2×C4×C5—21no—
295C2×C4×C5—41no—
296C2×C4×C5—42no—
297C2×C4×C5—42no—
298C2×C4×C5—42no—
299C2×C4×C5—42no—
300C2×C4×C5—44no—
301C2×C4×C5—42no—
302C2×C4×C5—42no—
303C2×C4×C5—82no—
304C2×C4×C5—82no—
305C2×C4×C5—82no—
306C2×C4×C5#12011yescharacteristic
307C2×C4×C5#12011yescharacteristic
308C2×C4×C5#12011yescharacteristic
309C2×C4×C5#12011yescharacteristic
310C2×C4×C5C22×D411yescharacteristic
311C2×C4×C5C22×D411yescharacteristic
312C5×C8—42no—
313C5×C8—42no—
314C2×#41—21no—
315C2×#41—21no—
316C22×Q8C5×D411yescharacteristic
317C22×Q8C23×C512yes—
318C22×D4—21no—
319C22×D4—22no—
320C22×D4C5×D411yescharacteristic
321C2×Q16—22no—
322C2×SD16—22no—
323C2×SD16—24no—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8C5×D411yescharacteristic
329#127—21no—
330#127C5×D411yescharacteristic
331#126—22no—
332#126—21no—
333#125C5×D411yescharacteristic
334#124—21no—
335#124—22no—
336#124—22no—
337#124—22no—
338#124—21no—
339#124—21no—
340#124—21no—
341#124—21no—
342#124—21no—
343#124C5×D411yescharacteristic
344#123—22no—
345#123—21no—
346#123—22no—
347#123—22no—
348#123—22no—
349C22⋊Q8—22no—
350C22⋊Q8—21no—
351C22⋊Q8—22no—
352C22⋊Q8C23×C511yescharacteristic
353#121—22no—
354#121—22no—
355#121—22no—
356#121—21no—
357#121—22no—
358#121—21no—
359#121—22no—
360#121C23×C511yescharacteristic
361#121C23×C512yes—
362#120—22no—
363#120—22no—
364#120—22no—
365#120—22no—
366#120—22no—
367C4×Q8—21no—
368C4×Q8C5×D411yescharacteristic
369C4×D4—22no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4—22no—
373C4×D4—21no—
374C4×D4C5×D411yescharacteristic
375C4×D4C5×D412yes—
376#117—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4C5×D412yes—
382C2×C22⋊C4C23×C512yes—
383C2×C42—21no—
384C2×C42C23×C511yescharacteristic
385C4⋊C8—21no—
386C4⋊C8—21no—
387#103—22no—
388#103—24no—
389#103—22no—
390C22⋊C8C23×C512yes—
391C22⋊C8C23×C512yes—
392#97—21no—
393C4×C8—21no—
394#95—21no—
395#95—24no—
396#95C23×C512yes—
397C22×C5—21no—
398C22×C5—21no—
399C22×C5—42no—
400C22×C5—42no—
401C22×C5—41no—
402C22×C5—41no—
403C22×C5—42no—
404C22×C5—42no—
405C22×C5—42no—
406C22×C5—41no—
407C22×C5—42no—
408C22×C5—84no—
409C22×C5—164no—
410C22×C5C2×#12011yescharacteristic
411C22×C5#57712yes—
412C4×C5—21no—
413C4×C5—21no—
414C4×C5—21no—
415C4×C5—21no—
416C4×C5—21no—
417C4×C5—41no—
418C4×C5—42no—
419C4×C5—41no—
420C4×C5—42no—
421C4×C5—41no—
422C4×C5—82no—
423C4×C5—82no—
424C4×C5#45512yes—
425C24—22no—
426C24C2×C5×D412yes—
427#41—82no—
428#41—82no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—42no—
435C2×Q8—82no—
436C2×Q8C2×C5×D411yescharacteristic
437C2×D4—22no—
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—42no—
449C2×D4—82no—
450C2×D4—84no—
451C2×D4C2×C5×D411yescharacteristic
452C2×D4C2×C5×D412yes—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4C2×C5×D411yescharacteristic
460C22×C4C2×C5×D411yescharacteristic
461C22×C4C2×C5×D412yes—
462C22×C4C24×C511yescharacteristic
463Q16—82no—
464SD16—84no—
465SD16—82no—
466C2×C8—22no—
467C2×C8—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4C2×C5×D412yes—
480C4⋊C4C2×C5×D411yescharacteristic
481C4⋊C4C2×C5×D411yescharacteristic
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—21no—
497C22⋊C4—22no—
498C22⋊C4—84no—
499C42—22no—
500C42—21no—
501C42—42no—
502C42C2×C5×D411yescharacteristic
503C42C2×C5×D411yescharacteristic
504C42C2×C5×D411yescharacteristic
505C42C2×C5×D411yescharacteristic
506C2×C5—22no—
507C2×C5—82no—
508C2×C5—82no—
509C2×C5—81no—
510C2×C5#300911yescharacteristic
511C2×C5C2×#45511yescharacteristic
512C2×C5#302711yescharacteristic
513C23—22no—
514C23—21no—
515C23—22no—
516C23—24no—
517C23—42no—
518C23—42no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23C22×C5×D411yescharacteristic
523Q8—41no—
524Q8—44no—
525Q8—42no—
526Q8—42no—
527Q8—42no—
528Q8—81no—
529Q8—81no—
530D4—42no—
531D4—41no—
532D4—42no—
533D4—44no—
534D4—42no—
535D4—42no—
536D4—81no—
537D4—81no—
538D4—82no—
539D4—82no—
540D4—82no—
541D4—82no—
542D4—82no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4C5×#12011yescharacteristic
562C2×C4C5×#12011yescharacteristic
563C2×C4C5×#12011yescharacteristic
564C2×C4C5×#12011yescharacteristic
565C2×C4C22×C5×D411yescharacteristic
566C2×C4C22×C5×D411yescharacteristic
567C8—42no—
568C8—42no—
569C5#1842711yescharacteristic, direct product factor, Sylow
570C22—21no—
571C22—21no—
572C22—42no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—41no—
580C22—42no—
581C22—84no—
582C22—164no—
583C22C2×C5×#12011yescharacteristic
584C22C5×#57712yes—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—42no—
594C4—41no—
595C4—82no—
596C4—82no—
597C4C5×#45512yes—
598C2—22no—
599C2—82no—
600C2—81no—
601C2—82no—
602C2C5×#300911yescharacteristic
603C2C2×C5×#45511yescharacteristic
604C2C5×#302711yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC5×#3558C2
LCSG ▷ C22×C4 ▷ 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×#3218C2semidirect
3. C5×#3206C2semidirect
4. C5×#2640C2semidirect
5. C5×#1606C2semidirect
6. C5×#1601C2semidirect
7. C5×#1505C2semidirect
8. C5×#1500C2semidirect
9. C5×#1469C2semidirect
23. C2×C5×C4⋊Q8C22semidirect
33. C5×#451C22semidirect
35. C5×#448C22semidirect
36. C5×#432C22semidirect
37. C5×#431C22semidirect
39. C5×#393C22semidirect
42. C5×#376C22semidirect
43. C5×#328C22semidirect
45. C5×#327C22semidirect
46. C5×#323C22semidirect
47. #18427C5direct (N × H)
132. #3218C2×C5semidirect
133. #3206C2×C5semidirect
134. #2640C2×C5semidirect
135. #1606C2×C5semidirect
136. #1601C2×C5semidirect
137. #1505C2×C5semidirect
138. #1500C2×C5semidirect
139. #1469C2×C5semidirect
234. C2×C4⋊Q8C22×C5semidirect
244. #451C22×C5semidirect
246. #448C22×C5semidirect
247. #432C22×C5semidirect
248. #431C22×C5semidirect
250. #393C22×C5semidirect
253. #376C22×C5semidirect
254. #328C22×C5semidirect
256. #327C22×C5semidirect
257. #323C22×C5semidirect
569. C5#18427direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35582
C5×#32182
C5×#32062
C5×#26402
C5×#16062
C5×#16012
C5×#15052
C5×#15002
C5×#14692
#184275