Finite Group Databasebeta

← C5×#18268 #11940988

Subgroups — C5×#18268 #11940988

Order 1280.

940 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3538C211yesmaximal, characteristic
2C5×#3092C211yesmaximal, characteristic
3C2×C5×#483C211yesmaximal, characteristic, complemented
4C5×#2502C211yesmaximal, characteristic, complemented
5C5×#1611C214yesmaximal, complemented
6C5×#1609C214yesmaximal, complemented
7C5×#1508C211yesmaximal, characteristic, complemented
8C5×#1501C211yesmaximal, characteristic
9C5×#1472C211yesmaximal, characteristic, complemented
10C5×#563—22no—
11C5×#552—21no—
12C5×#552—21no—
13C5×#551—22no—
14C5×D4×Q8—21no—
15C5×D4×Q8—21no—
16C5×#544—21no—
17C5×#544C2212yes—
18C5×#541—21no—
19C5×#539—21no—
20C5×#539C2212yes—
21C5×#536—21no—
22C5×#525C2211yescharacteristic
23C2×C5×C22⋊Q8C2211yescharacteristic
24C2×C5×C22⋊Q8C2211yescharacteristic
25C22×C5×C4⋊C4C2211yescharacteristic
26C5×#484—21no—
27C5×#483—22no—
28C5×#483C2214yes—
29C5×#483C2212yes—
30C5×#481—21no—
31C5×#480C2212yes—
32C5×#428—21no—
33C2×C5×#106—21no—
34C5×#417—21no—
35C2×C5×#103—21no—
36C5×#407C2211yescharacteristic
37C2×C5×C22⋊C8C2211yescharacteristic
38C5×#398C2214yes—
39C5×#391—21no—
40C5×#385—22no—
41C5×#384—21no—
42C4×C5×C22⋊C4C2211yescharacteristic
43C5×#328C2212yes—
44C5×#328C2214yes—
45C5×#327C2212yes—
46C5×#324C2214yes—
47#18268C511yesmaximal, characteristic, direct product factor, Sylow
48C2×C5×#41D411yescharacteristic
49C22×C5×Q8—21no—
50C22×C5×Q8D411yescharacteristic
51C22×C5×Q8D411yescharacteristic
52C23×C4×C5—21no—
53C23×C4×C5C2311yescharacteristic
54C2×C5×M16—21no—
55C22×C5×C8—21no—
56C5×C4⋊Q8—21no—
57C5×C4⋊Q8—22no—
58C5×C4⋊Q8—22no—
59C5×C4⋊Q8—22no—
60C5×#126—22no—
61C5×#126—22no—
62C5×#126—22no—
63C5×#126—22no—
64C5×#125—21no—
65C5×#125—22no—
66C5×#125—22no—
67C5×#125—22no—
68C5×#125—22no—
69C5×#124—22no—
70C5×#123—22no—
71C5×#123—22no—
72C5×#123—22no—
73C5×#123—21no—
74C5×#123—22no—
75C5×C22⋊Q8—21no—
76C5×C22⋊Q8—21no—
77C5×C22⋊Q8—22no—
78C5×C22⋊Q8—22no—
79C5×C22⋊Q8—22no—
80C5×C22⋊Q8—21no—
81C5×C22⋊Q8—21no—
82C5×C22⋊Q8—21no—
83C5×C22⋊Q8—21no—
84C5×C22⋊Q8—21no—
85C5×C22⋊Q8—21no—
86C5×C22⋊Q8—22no—
87C5×C22⋊Q8D412yes—
88C5×C22⋊Q8D411yescharacteristic
89C5×C22⋊Q8C2312yes—
90C5×C22⋊Q8C2311yescharacteristic
91C5×#121D411yescharacteristic
92C5×#121C2311yescharacteristic
93C5×#120—21no—
94C4×C5×Q8—21no—
95C4×C5×Q8—21no—
96C4×C5×D4—22no—
97C4×C5×D4—21no—
98C4×C5×D4—21no—
99C4×C5×D4—22no—
100C5×#117—21no—
101C5×#117—22no—
102C5×#117—22no—
103C5×#117D411yescharacteristic
104C2×C5×C4⋊C4—21no—
105C2×C5×C4⋊C4—21no—
106C2×C5×C4⋊C4—21no—
107C2×C5×C4⋊C4—21no—
108C2×C5×C4⋊C4—21no—
109C2×C5×C4⋊C4—21no—
110C2×C5×C4⋊C4—24no—
111C2×C5×C4⋊C4D411yescharacteristic
112C2×C5×C4⋊C4D411yescharacteristic
113C2×C5×C4⋊C4C2312yes—
114C2×C5×C4⋊C4C2312yes—
115C2×C5×C22⋊C4—21no—
116C2×C5×C22⋊C4—21no—
117C2×C5×C22⋊C4D412yes—
118C2×C42×C5—21no—
119C5×#107—22no—
120C5×#106—24no—
121C5×#106—22no—
122C5×#103—24no—
123C5×#103—22no—
124C5×#102—22no—
125C5×C22⋊C8—21no—
126C5×C22⋊C8—21no—
127C5×C22⋊C8C2312yes—
128C5×C22⋊C8C2312yes—
129C5×#95—24no—
130C5×#95C2312yes—
131#3538C2×C511yescharacteristic
132#3092C2×C511yescharacteristic
133C2×#483C2×C511yescharacteristic, complemented
134#2502C2×C511yescharacteristic, complemented
135#1611C2×C514yescomplemented
136#1609C2×C514yescomplemented
137#1508C2×C511yescharacteristic, complemented
138#1501C2×C511yescharacteristic
139#1472C2×C511yescharacteristic, complemented
140C24×C5C2×D411yescharacteristic
141C5×#41—42no—
142C2×C5×Q8—22no—
143C2×C5×Q8—21no—
144C2×C5×Q8—21no—
145C2×C5×Q8—21no—
146C2×C5×Q8—42no—
147C2×C5×Q8—42no—
148C2×C5×Q8—82no—
149C2×C5×Q8C2×D412yes—
150C2×C5×Q8C2×D411yescharacteristic
151C2×C5×D4—21no—
152C2×C5×D4—21no—
153C2×C5×D4—21no—
154C2×C5×D4C2×D411yescharacteristic
155C22×C4×C5—21no—
156C22×C4×C5—22no—
157C22×C4×C5—22no—
158C22×C4×C5—21no—
159C22×C4×C5—21no—
160C22×C4×C5—22no—
161C22×C4×C5—42no—
162C22×C4×C5—41no—
163C22×C4×C5—41no—
164C22×C4×C5—82no—
165C22×C4×C5C2×D412yes—
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2×D412yes—
170C22×C4×C5C2×D411yescharacteristic
171C22×C4×C5C2×D411yescharacteristic
172C22×C4×C5C2411yescharacteristic
173C5×M16—42no—
174C2×C5×C8—22no—
175C2×C5×C8—22no—
176C2×C5×C8—42no—
177C5×C4⋊C4—22no—
178C5×C4⋊C4—22no—
179C5×C4⋊C4—22no—
180C5×C4⋊C4—22no—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—22no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—22no—
185C5×C4⋊C4—22no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—21no—
188C5×C4⋊C4—21no—
189C5×C4⋊C4—21no—
190C5×C4⋊C4—22no—
191C5×C4⋊C4—22no—
192C5×C4⋊C4—22no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—22no—
195C5×C4⋊C4—22no—
196C5×C4⋊C4—41no—
197C5×C4⋊C4—41no—
198C5×C4⋊C4—84no—
199C5×C4⋊C4C2×D412yes—
200C5×C4⋊C4C2×D411yescharacteristic
201C5×C4⋊C4C2×D411yescharacteristic
202C5×C22⋊C4—24no—
203C5×C22⋊C4—22no—
204C5×C22⋊C4—22no—
205C5×C22⋊C4—22no—
206C5×C22⋊C4—22no—
207C5×C22⋊C4—22no—
208C5×C22⋊C4—22no—
209C5×C22⋊C4—21no—
210C5×C22⋊C4—21no—
211C5×C22⋊C4—22no—
212C5×C22⋊C4—21no—
213C5×C22⋊C4—21no—
214C5×C22⋊C4—21no—
215C42×C5—22no—
216C42×C5—22no—
217C42×C5—22no—
218C42×C5—21no—
219C42×C5—42no—
220#563—22no—
221#552—21no—
222#552—21no—
223#551—22no—
224D4×Q8—21no—
225D4×Q8—21no—
226#544—21no—
227#544C22×C512yes—
228#541—21no—
229#539—21no—
230#539C22×C512yes—
231#536—21no—
232#525C22×C511yescharacteristic
233C2×C22⋊Q8C22×C511yescharacteristic
234C2×C22⋊Q8C22×C511yescharacteristic
235C22×C4⋊C4C22×C511yescharacteristic
236#484—21no—
237#483—22no—
238#483C22×C514yes—
239#483C22×C512yes—
240#481—21no—
241#480C22×C512yes—
242#428—21no—
243C2×#106—21no—
244#417—21no—
245C2×#103—21no—
246#407C22×C511yescharacteristic
247C2×C22⋊C8C22×C511yescharacteristic
248#398C22×C514yes—
249#391—21no—
250#385—22no—
251#384—21no—
252C4×C22⋊C4C22×C511yescharacteristic
253#328C22×C514yes—
254#328C22×C512yes—
255#327C22×C512yes—
256#324C22×C514yes—
257C23×C5—22no—
258C23×C5—22no—
259C23×C5—21no—
260C23×C5—41no—
261C23×C5#12011yescharacteristic
262C23×C5#12011yescharacteristic
263C23×C5C22×D411yescharacteristic
264C5×Q8—42no—
265C5×Q8—41no—
266C5×Q8—42no—
267C5×Q8—42no—
268C5×Q8—81no—
269C5×Q8—82no—
270C5×D4—41no—
271C5×D4—42no—
272C5×D4—42no—
273C5×D4—81no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—24no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—22no—
281C2×C4×C5—22no—
282C2×C4×C5—21no—
283C2×C4×C5—22no—
284C2×C4×C5—22no—
285C2×C4×C5—24no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—42no—
290C2×C4×C5—42no—
291C2×C4×C5—42no—
292C2×C4×C5—42no—
293C2×C4×C5—42no—
294C2×C4×C5—42no—
295C2×C4×C5—44no—
296C2×C4×C5—81no—
297C2×C4×C5—82no—
298C2×C4×C5—82no—
299C2×C4×C5—82no—
300C2×C4×C5#12011yescharacteristic
301C2×C4×C5#12011yescharacteristic
302C2×C4×C5C22×D411yescharacteristic
303C2×C4×C5C22×D411yescharacteristic
304C5×C8—42no—
305C5×C8—42no—
306C2×#41C5×D411yescharacteristic
307C22×Q8—21no—
308C22×Q8C5×D411yescharacteristic
309C22×Q8C5×D411yescharacteristic
310C23×C4—21no—
311C23×C4C23×C511yescharacteristic
312C2×M16—21no—
313C22×C8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317C4⋊Q8—22no—
318#126—22no—
319#126—22no—
320#126—22no—
321#126—22no—
322#125—21no—
323#125—22no—
324#125—22no—
325#125—22no—
326#125—22no—
327#124—22no—
328#123—22no—
329#123—22no—
330#123—22no—
331#123—21no—
332#123—22no—
333C22⋊Q8—21no—
334C22⋊Q8—21no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C22⋊Q8—21no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—21no—
344C22⋊Q8—22no—
345C22⋊Q8C5×D412yes—
346C22⋊Q8C5×D411yescharacteristic
347C22⋊Q8C23×C512yes—
348C22⋊Q8C23×C511yescharacteristic
349#121C5×D411yescharacteristic
350#121C23×C511yescharacteristic
351#120—21no—
352C4×Q8—21no—
353C4×Q8—21no—
354C4×D4—22no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—22no—
358#117—21no—
359#117—22no—
360#117—22no—
361#117C5×D411yescharacteristic
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4C5×D411yescharacteristic
370C2×C4⋊C4C5×D411yescharacteristic
371C2×C4⋊C4C23×C512yes—
372C2×C4⋊C4C23×C512yes—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4C5×D412yes—
376C2×C42—21no—
377#107—22no—
378#106—24no—
379#106—22no—
380#103—24no—
381#103—22no—
382#102—22no—
383C22⋊C8—21no—
384C22⋊C8—21no—
385C22⋊C8C23×C512yes—
386C22⋊C8C23×C512yes—
387#95—24no—
388#95C23×C512yes—
389C22×C5—21no—
390C22×C5—21no—
391C22×C5—21no—
392C22×C5—21no—
393C22×C5—21no—
394C22×C5—21no—
395C22×C5—21no—
396C22×C5—44no—
397C22×C5—41no—
398C22×C5—41no—
399C22×C5—41no—
400C22×C5#45612yes—
401C22×C5C2×#12011yescharacteristic
402C22×C5#57712yes—
403C4×C5—21no—
404C4×C5—21no—
405C4×C5—21no—
406C4×C5—44no—
407C4×C5—42no—
408C4×C5—41no—
409C4×C5—42no—
410C4×C5—41no—
411C4×C5—41no—
412C4×C5—42no—
413C4×C5—42no—
414C4×C5—82no—
415C4×C5#45812yes—
416C24C2×C5×D411yescharacteristic
417#41—42no—
418C2×Q8—22no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—82no—
425C2×Q8C2×C5×D412yes—
426C2×Q8C2×C5×D411yescharacteristic
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×C5×D411yescharacteristic
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—82no—
441C22×C4C2×C5×D412yes—
442C22×C4C2×C5×D411yescharacteristic
443C22×C4C2×C5×D411yescharacteristic
444C22×C4C2×C5×D411yescharacteristic
445C22×C4C2×C5×D412yes—
446C22×C4C2×C5×D411yescharacteristic
447C22×C4C2×C5×D411yescharacteristic
448C22×C4C24×C511yescharacteristic
449M16—42no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—42no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—84no—
475C4⋊C4C2×C5×D412yes—
476C4⋊C4C2×C5×D411yescharacteristic
477C4⋊C4C2×C5×D411yescharacteristic
478C22⋊C4—24no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—22no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—21no—
495C42—42no—
496C2×C5—22no—
497C2×C5—22no—
498C2×C5—41no—
499C2×C5—81no—
500C2×C5#301011yescharacteristic
501C2×C5C2×#45611yescharacteristic
502C2×C5C2×#45811yescharacteristic
503C23—22no—
504C23—22no—
505C23—21no—
506C23—41no—
507C23C5×#12011yescharacteristic
508C23C5×#12011yescharacteristic
509C23C22×C5×D411yescharacteristic
510Q8—42no—
511Q8—41no—
512Q8—42no—
513Q8—42no—
514Q8—81no—
515Q8—82no—
516D4—41no—
517D4—42no—
518D4—42no—
519D4—81no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4C5×#12011yescharacteristic
547C2×C4C5×#12011yescharacteristic
548C2×C4C22×C5×D411yescharacteristic
549C2×C4C22×C5×D411yescharacteristic
550C8—42no—
551C8—42no—
552C5#1826811yescharacteristic, direct product factor, Sylow
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—44no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22C5×#45612yes—
565C22C2×C5×#12011yescharacteristic
566C22C5×#57712yes—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—44no—
571C4—42no—
572C4—41no—
573C4—42no—
574C4—41no—
575C4—41no—
576C4—42no—
577C4—42no—
578C4—82no—
579C4C5×#45812yes—
580C2—22no—
581C2—22no—
582C2—41no—
583C2—81no—
584C2C5×#301011yescharacteristic
585C2C2×C5×#45611yescharacteristic
586C2C2×C5×#45811yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC5×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C5×#483C2semidirect
4. C5×#2502C2semidirect
5. C5×#1611C2semidirect
6. C5×#1609C2semidirect
7. C5×#1508C2semidirect
9. C5×#1472C2semidirect
47. #18268C5direct (N × H)
133. C2×#483C2×C5semidirect
134. #2502C2×C5semidirect
135. #1611C2×C5semidirect
136. #1609C2×C5semidirect
137. #1508C2×C5semidirect
139. #1472C2×C5semidirect
552. C5#18268direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35382
C5×#30922
C2×C5×#4832
C5×#25022
C5×#16112
C5×#16092
C5×#15082
C5×#15012
C5×#14722
#182685