Finite Group Databasebeta

← C5×#26416 #11949136

Subgroups — C5×#26416 #11949136

Order 1280.

774 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×#2442C211yesmaximal, characteristic
2C5×#2360C211yesmaximal, characteristic
3C5×#2101C211yesmaximal, characteristic, complemented
4C5×#2088C211yesmaximal, characteristic, complemented
5C5×#2066C211yesmaximal, characteristic, complemented
6C5×#2026C211yesmaximal, characteristic, complemented
7C5×#1989C211yesmaximal, characteristic, complemented
8C5×#1920C211yesmaximal, characteristic, complemented
9C5×#1892C211yesmaximal, characteristic
10C5×#1872C211yesmaximal, characteristic
11C5×#1865C211yesmaximal, characteristic, complemented
12C5×#1853C211yesmaximal, characteristic
13C5×#1850C211yesmaximal, characteristic
14C5×#1805C211yesmaximal, characteristic, complemented
15C2×C5×#340C211yesmaximal, characteristic
16C23×C5×C8C2211yescharacteristic
17C2×C5×#125C2211yescharacteristic
18C2×C5×C22⋊Q8C2211yescharacteristic
19C2×C4×C5×Q8C2211yescharacteristic
20C22×C5×C4⋊C4C2211yescharacteristic
21C22×C5×C4⋊C4C2211yescharacteristic
22C2×C5×#106C2211yescharacteristic
23C2×C5×#106C2211yescharacteristic
24C2×C5×C4⋊C8C2211yescharacteristic
25C2×C5×C4⋊C8C2211yescharacteristic
26C2×C5×#103C2211yescharacteristic
27C2×C5×#103C2211yescharacteristic
28C2×C5×#103C2211yescharacteristic
29C2×C5×#103C2211yescharacteristic
30C2×C5×C22⋊C8C2211yescharacteristic
31C5×#400—21no—
32C5×#398—21no—
33C5×#391C2211yescharacteristic
34C5×#389—21no—
35C5×#389C2211yescharacteristic
36C5×#389C2211yescharacteristic
37C5×#389C2211yescharacteristic
38C5×#387—21no—
39C5×#385—21no—
40C5×#385—21no—
41C5×#382C2211yescharacteristic
42C5×#380—21no—
43C5×#380C2211yescharacteristic
44C4×C5×C4⋊C4C2211yescharacteristic
45C4×C5×C22⋊C4—21no—
46C4×C5×C22⋊C4C2211yescharacteristic
47C5×#340C2211yescharacteristic
48C5×#340C2211yescharacteristic
49C5×#340C2211yescharacteristic
50C5×#340C2211yescharacteristic
51C5×#340C2211yescharacteristic
52C5×#340C2211yescharacteristic
53C5×#340C2211yescharacteristic
54C5×#340C2211yescharacteristic
55C5×#340C2211yescharacteristic
56C5×#340C2211yescharacteristic
57C5×#336C2211yescharacteristic
58C5×#336C2211yescharacteristic
59#26416C511yesmaximal, characteristic, direct product factor, Sylow
60C22×C5×Q8C2311yescharacteristic
61C23×C4×C5—21no—
62C23×C4×C5—21no—
63C23×C4×C5C2311yescharacteristic
64C22×C5×C8—22no—
65C22×C5×C8—41no—
66C22×C5×C8—41no—
67C22×C5×C8C2311yescharacteristic
68C22×C5×C8C2311yescharacteristic
69C22×C5×C8C2311yescharacteristic
70C22×C5×C8C2311yescharacteristic
71C5×#125—22no—
72C5×#125—41no—
73C5×C22⋊Q8—22no—
74C5×C22⋊Q8Q814yes—
75C4×C5×Q8—22no—
76C4×C5×Q8—41no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4—22no—
79C2×C5×C4⋊C4—22no—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4—21no—
82C2×C5×C4⋊C4—21no—
83C2×C5×C4⋊C4—21no—
84C2×C5×C4⋊C4—21no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4—21no—
87C2×C5×C4⋊C4—21no—
88C2×C5×C4⋊C4—41no—
89C2×C5×C4⋊C4—41no—
90C2×C5×C4⋊C4C2311yescharacteristic
91C2×C5×C4⋊C4C2311yescharacteristic
92C2×C5×C4⋊C4C2311yescharacteristic
93C2×C5×C4⋊C4C2311yescharacteristic
94C2×C5×C4⋊C4C2311yescharacteristic
95C2×C5×C4⋊C4C2311yescharacteristic
96C2×C5×C4⋊C4C2311yescharacteristic
97C2×C5×C22⋊C4—21no—
98C2×C5×C22⋊C4—21no—
99C2×C5×C22⋊C4D411yescharacteristic
100C2×C5×C22⋊C4D411yescharacteristic
101C2×C42×C5—21no—
102C2×C42×C5—21no—
103C2×C42×C5C2311yescharacteristic
104C2×C42×C5C2311yescharacteristic
105C5×#106—22no—
106C5×#106—41no—
107C5×C4⋊C8—41no—
108C5×C4⋊C8—41no—
109C5×#103—22no—
110C5×#103—41no—
111C5×#103—41no—
112C5×#103—41no—
113C5×C22⋊C8—22no—
114C5×#95—21no—
115C5×#95—21no—
116C5×#95—21no—
117C5×#95—21no—
118C5×#95—21no—
119C5×#95—21no—
120C5×#95—21no—
121C5×#95—21no—
122C5×#95—21no—
123C5×#95D411yescharacteristic
124C5×#95D411yescharacteristic
125#2442C2×C511yescharacteristic
126#2360C2×C511yescharacteristic
127#2101C2×C511yescharacteristic, complemented
128#2088C2×C511yescharacteristic, complemented
129#2066C2×C511yescharacteristic, complemented
130#2026C2×C511yescharacteristic, complemented
131#1989C2×C511yescharacteristic, complemented
132#1920C2×C511yescharacteristic, complemented
133#1892C2×C511yescharacteristic
134#1872C2×C511yescharacteristic
135#1865C2×C511yescharacteristic, complemented
136#1853C2×C511yescharacteristic
137#1850C2×C511yescharacteristic
138#1805C2×C511yescharacteristic, complemented
139C2×#340C2×C511yescharacteristic
140C24×C5C2×D411yescharacteristic
141C2×C5×Q8—21no—
142C2×C5×Q8—21no—
143C2×C5×Q8—21no—
144C2×C5×Q8—21no—
145C2×C5×Q8—21no—
146C2×C5×Q8C2×Q812yes—
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5—21no—
153C22×C4×C5—21no—
154C22×C4×C5—21no—
155C22×C4×C5—21no—
156C22×C4×C5—21no—
157C22×C4×C5—21no—
158C22×C4×C5—21no—
159C22×C4×C5—42no—
160C22×C4×C5—42no—
161C22×C4×C5—42no—
162C22×C4×C5—42no—
163C22×C4×C5—81no—
164C22×C4×C5—81no—
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5C2×Q812yes—
171C22×C4×C5#4112yes—
172C22×C4×C5C2411yescharacteristic
173C2×C5×C8—21no—
174C2×C5×C8—21no—
175C2×C5×C8—21no—
176C2×C5×C8—21no—
177C2×C5×C8—21no—
178C2×C5×C8—21no—
179C2×C5×C8—21no—
180C2×C5×C8—21no—
181C2×C5×C8—21no—
182C2×C5×C8—21no—
183C2×C5×C8—21no—
184C2×C5×C8—21no—
185C2×C5×C8—44no—
186C5×C4⋊C4—24no—
187C5×C4⋊C4—21no—
188C5×C4⋊C4—21no—
189C5×C4⋊C4—21no—
190C5×C4⋊C4—21no—
191C5×C4⋊C4—21no—
192C5×C4⋊C4—21no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—21no—
195C5×C4⋊C4—21no—
196C5×C4⋊C4—21no—
197C5×C4⋊C4—21no—
198C5×C4⋊C4—42no—
199C5×C4⋊C4—42no—
200C5×C4⋊C4—42no—
201C5×C4⋊C4—42no—
202C5×C4⋊C4—42no—
203C5×C4⋊C4—81no—
204C5×C4⋊C4—81no—
205C5×C4⋊C4—81no—
206C5×C4⋊C4C2×Q812yes—
207C5×C4⋊C4#4112yes—
208C5×C4⋊C4#4112yes—
209C5×C22⋊C4—24no—
210C5×C22⋊C4—22no—
211C5×C22⋊C4—22no—
212C5×C22⋊C4—42no—
213C42×C5—21no—
214C42×C5—21no—
215C42×C5—21no—
216C42×C5—21no—
217C42×C5—42no—
218C42×C5—81no—
219C23×C8C22×C511yescharacteristic
220C2×#125C22×C511yescharacteristic
221C2×C22⋊Q8C22×C511yescharacteristic
222C2×C4×Q8C22×C511yescharacteristic
223C22×C4⋊C4C22×C511yescharacteristic
224C22×C4⋊C4C22×C511yescharacteristic
225C2×#106C22×C511yescharacteristic
226C2×#106C22×C511yescharacteristic
227C2×C4⋊C8C22×C511yescharacteristic
228C2×C4⋊C8C22×C511yescharacteristic
229C2×#103C22×C511yescharacteristic
230C2×#103C22×C511yescharacteristic
231C2×#103C22×C511yescharacteristic
232C2×#103C22×C511yescharacteristic
233C2×C22⋊C8C22×C511yescharacteristic
234#400—21no—
235#398—21no—
236#391C22×C511yescharacteristic
237#389—21no—
238#389C22×C511yescharacteristic
239#389C22×C511yescharacteristic
240#389C22×C511yescharacteristic
241#387—21no—
242#385—21no—
243#385—21no—
244#382C22×C511yescharacteristic
245#380—21no—
246#380C22×C511yescharacteristic
247C4×C4⋊C4C22×C511yescharacteristic
248C4×C22⋊C4—21no—
249C4×C22⋊C4C22×C511yescharacteristic
250#340C22×C511yescharacteristic
251#340C22×C511yescharacteristic
252#340C22×C511yescharacteristic
253#340C22×C511yescharacteristic
254#340C22×C511yescharacteristic
255#340C22×C511yescharacteristic
256#340C22×C511yescharacteristic
257#340C22×C511yescharacteristic
258#340C22×C511yescharacteristic
259#340C22×C511yescharacteristic
260#336C22×C511yescharacteristic
261#336C22×C511yescharacteristic
262C23×C5—21no—
263C23×C5—21no—
264C23×C5—21no—
265C23×C5—21no—
266C23×C5C22⋊Q812yes—
267C23×C5#13512yes—
268C23×C5#13712yes—
269C23×C5C22×D411yescharacteristic
270C5×Q8—22no—
271C5×Q8—22no—
272C5×Q8—41no—
273C5×Q8—41no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—22no—
281C2×C4×C5—22no—
282C2×C4×C5—21no—
283C2×C4×C5—21no—
284C2×C4×C5—21no—
285C2×C4×C5—21no—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
291C2×C4×C5—21no—
292C2×C4×C5—21no—
293C2×C4×C5—21no—
294C2×C4×C5—21no—
295C2×C4×C5—21no—
296C2×C4×C5—41no—
297C2×C4×C5—41no—
298C2×C4×C5—41no—
299C2×C4×C5—41no—
300C2×C4×C5—41no—
301C2×C4×C5—41no—
302C2×C4×C5—41no—
303C2×C4×C5—41no—
304C2×C4×C5—41no—
305C2×C4×C5—41no—
306C2×C4×C5—41no—
307C2×C4×C5—41no—
308C2×C4×C5—41no—
309C2×C4×C5—41no—
310C2×C4×C5—41no—
311C2×C4×C5—41no—
312C2×C4×C5—41no—
313C2×C4×C5—41no—
314C2×C4×C5—41no—
315C2×C4×C5—41no—
316C2×C4×C5—84no—
317C2×C4×C5—84no—
318C2×C4×C5C22⋊Q812yes—
319C2×C4×C5C22×Q811yescharacteristic
320C2×C4×C5C2×#4111yescharacteristic
321C2×C4×C5C2×#4111yescharacteristic
322C2×C4×C5C2×#4111yescharacteristic
323C2×C4×C5H(5,𝔽2)11yescharacteristic
324C2×C4×C5H(5,𝔽2)11yescharacteristic
325C5×C8—41no—
326C5×C8—41no—
327C5×C8—41no—
328C5×C8—41no—
329C22×Q8C23×C511yescharacteristic
330C23×C4—21no—
331C23×C4—21no—
332C23×C4C23×C511yescharacteristic
333C22×C8—22no—
334C22×C8—41no—
335C22×C8—41no—
336C22×C8C23×C511yescharacteristic
337C22×C8C23×C511yescharacteristic
338C22×C8C23×C511yescharacteristic
339C22×C8C23×C511yescharacteristic
340#125—22no—
341#125—41no—
342C22⋊Q8—22no—
343C22⋊Q8C5×Q814yes—
344C4×Q8—22no—
345C4×Q8—41no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—41no—
358C2×C4⋊C4—41no—
359C2×C4⋊C4C23×C511yescharacteristic
360C2×C4⋊C4C23×C511yescharacteristic
361C2×C4⋊C4C23×C511yescharacteristic
362C2×C4⋊C4C23×C511yescharacteristic
363C2×C4⋊C4C23×C511yescharacteristic
364C2×C4⋊C4C23×C511yescharacteristic
365C2×C4⋊C4C23×C511yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4C5×D411yescharacteristic
369C2×C22⋊C4C5×D411yescharacteristic
370C2×C42—21no—
371C2×C42—21no—
372C2×C42C23×C511yescharacteristic
373C2×C42C23×C511yescharacteristic
374#106—22no—
375#106—41no—
376C4⋊C8—41no—
377C4⋊C8—41no—
378#103—22no—
379#103—41no—
380#103—41no—
381#103—41no—
382C22⋊C8—22no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95C5×D411yescharacteristic
393#95C5×D411yescharacteristic
394C22×C5—22no—
395C22×C5—22no—
396C22×C5—22no—
397C22×C5—22no—
398C22×C5—22no—
399C22×C5—22no—
400C22×C5#47914yes—
401C22×C5C2×C22⋊Q811yescharacteristic
402C22×C5#54611yescharacteristic
403C22×C5#54611yescharacteristic
404C22×C5C2×#13511yescharacteristic
405C22×C5C2×#13711yescharacteristic
406C22×C5#57711yescharacteristic
407C22×C5#57711yescharacteristic
408C4×C5—22no—
409C4×C5—42no—
410C4×C5—42no—
411C4×C5—41no—
412C4×C5—41no—
413C4×C5—42no—
414C4×C5—42no—
415C4×C5—41no—
416C4×C5—41no—
417C4×C5—81no—
418C4×C5—81no—
419C4×C5—81no—
420C4×C5—81no—
421C4×C5#52911yescharacteristic
422C4×C5#53811yescharacteristic
423C4×C5#55411yescharacteristic
424C4×C5#55411yescharacteristic
425C24C2×C5×D411yescharacteristic
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8C2×C5×Q812yes—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4C2×C5×D411yescharacteristic
451C22×C4C2×C5×D411yescharacteristic
452C22×C4C2×C5×D411yescharacteristic
453C22×C4C2×C5×D411yescharacteristic
454C22×C4C2×C5×D411yescharacteristic
455C22×C4C2×C5×Q812yes—
456C22×C4C5×#4112yes—
457C22×C4C24×C511yescharacteristic
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—44no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4C2×C5×Q812yes—
492C4⋊C4C5×#4112yes—
493C4⋊C4C5×#4112yes—
494C22⋊C4—24no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—42no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—42no—
503C42—81no—
504C2×C5—24no—
505C2×C5#245011yescharacteristic
506C2×C5C2×#47911yescharacteristic
507C2×C5#308411yescharacteristic
508C2×C5#329611yescharacteristic
509C2×C5#329611yescharacteristic
510C2×C5#331611yescharacteristic
511C2×C5#332011yescharacteristic
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23C5×C22⋊Q812yes—
517C23C5×#13512yes—
518C23C5×#13712yes—
519C23C22×C5×D411yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4C5×C22⋊Q812yes—
569C2×C4C22×C5×Q811yescharacteristic
570C2×C4C2×C5×#4111yescharacteristic
571C2×C4C2×C5×#4111yescharacteristic
572C2×C4C2×C5×#4111yescharacteristic
573C2×C4C5×H(5,𝔽2)11yescharacteristic
574C2×C4C5×H(5,𝔽2)11yescharacteristic
575C8—41no—
576C8—41no—
577C8—41no—
578C8—41no—
579C5#2641611yescharacteristic, direct product factor, Sylow
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22C5×#47914yes—
587C22C2×C5×C22⋊Q811yescharacteristic
588C22C5×#54611yescharacteristic
589C22C5×#54611yescharacteristic
590C22C2×C5×#13511yescharacteristic
591C22C2×C5×#13711yescharacteristic
592C22C5×#57711yescharacteristic
593C22C5×#57711yescharacteristic
594C4—22no—
595C4—41no—
596C4—41no—
597C4—42no—
598C4—42no—
599C4—41no—
600C4—41no—
601C4—42no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4C5×#52911yescharacteristic
608C4C5×#53811yescharacteristic
609C4C5×#55411yescharacteristic
610C4C5×#55411yescharacteristic
611C2—24no—
612C2C5×#245011yescharacteristic
613C2C2×C5×#47911yescharacteristic
614C2C5×#308411yescharacteristic
615C2C5×#329611yescharacteristic
616C2C5×#329611yescharacteristic
617C2C5×#331611yescharacteristic
618C2C5×#332011yescharacteristic
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC24×C5C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C4×C5×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C5×#2101C2semidirect
4. C5×#2088C2semidirect
5. C5×#2066C2semidirect
6. C5×#2026C2semidirect
7. C5×#1989C2semidirect
8. C5×#1920C2semidirect
11. C5×#1865C2semidirect
14. C5×#1805C2semidirect
59. #26416C5direct (N × H)
127. #2101C2×C5semidirect
128. #2088C2×C5semidirect
129. #2066C2×C5semidirect
130. #2026C2×C5semidirect
131. #1989C2×C5semidirect
132. #1920C2×C5semidirect
135. #1865C2×C5semidirect
138. #1805C2×C5semidirect
579. C5#26416direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#24422
C5×#23602
C5×#21012
C5×#20882
C5×#20662
C5×#20262
C5×#19892
C5×#19202
C5×#18922
C5×#18722
C5×#18652
C5×#18532
C5×#18502
C5×#18052
C2×C5×#3402
#264165