Finite Group Databasebeta

← C5×#26824 #11949544

Subgroups — C5×#26824 #11949544

Order 1280.

814 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#489C211yesmaximal, characteristic
2C2×C4×C5×SD16C211yesmaximal, characteristic
3C5×#2451C211yesmaximal, characteristic, complemented
4C5×#2447C211yesmaximal, characteristic
5C5×#2066C211yesmaximal, characteristic, complemented
6C5×#2058C211yesmaximal, characteristic, complemented
7C5×#2055C211yesmaximal, characteristic
8C5×#2015C211yesmaximal, characteristic
9C5×#1971C211yesmaximal, characteristic, complemented
10C5×#1936C211yesmaximal, characteristic
11C5×#1935C211yesmaximal, characteristic, complemented
12C5×#1933C211yesmaximal, characteristic, complemented
13C5×#1878C211yesmaximal, characteristic, complemented
14C5×#1864C211yesmaximal, characteristic
15C5×#1771C211yesmaximal, characteristic, complemented
16C22×C5×Q16C2211yescharacteristic
17C22×C5×SD16C2211yescharacteristic
18C2×C5×C4⋊Q8C2211yescharacteristic
19C2×C5×#124C2211yescharacteristic
20C2×C4×C5×Q8C2211yescharacteristic
21C2×C4×C5×D4C2211yescharacteristic
22C5×#489—24no—
23C4×C5×SD16—24no—
24C2×C5×#106C2211yescharacteristic
25C2×C5×#106C2211yescharacteristic
26C2×C5×C4⋊C8C2211yescharacteristic
27C2×C5×#103C2211yescharacteristic
28C2×C5×#103C2211yescharacteristic
29C2×C5×#103C2211yescharacteristic
30C2×C5×#103C2211yescharacteristic
31C2×C5×#103C2211yescharacteristic
32C2×C5×#102C2211yescharacteristic
33C2×C5×#102C2211yescharacteristic
34C2×C5×#102C2211yescharacteristic
35C2×C5×#97C2211yescharacteristic
36C2×C5×#97C2211yescharacteristic
37C2×C4×C5×C8C2211yescharacteristic
38C5×#398—21no—
39C5×#397—21no—
40C5×#397—21no—
41C5×#395—21no—
42C5×#391—21no—
43C5×#391C2211yescharacteristic
44C5×#391C2211yescharacteristic
45C5×#391C2211yescharacteristic
46C5×#391C2211yescharacteristic
47C5×#390C2211yescharacteristic
48C5×#389C2211yescharacteristic
49C5×#389C2211yescharacteristic
50C5×#389C2211yescharacteristic
51C5×#387—21no—
52C5×#385—21no—
53C5×#381—21no—
54C4×C5×C4⋊C4C2211yescharacteristic
55C5×#340C2211yescharacteristic
56C5×#340C2211yescharacteristic
57C5×#340C2211yescharacteristic
58C5×#340C2211yescharacteristic
59C5×#336C2211yescharacteristic
60C5×#336C2211yescharacteristic
61#26824C511yesmaximal, characteristic, direct product factor, Sylow
62C22×C5×Q8—21no—
63C22×C5×Q8C2311yescharacteristic
64C22×C5×Q8C2311yescharacteristic
65C22×C5×D4C2311yescharacteristic
66C23×C4×C5—21no—
67C2×C5×Q16—22no—
68C2×C5×Q16—22no—
69C2×C5×Q16—41no—
70C2×C5×SD16—22no—
71C2×C5×SD16—22no—
72C2×C5×SD16—22no—
73C22×C5×C8C2311yescharacteristic
74C22×C5×C8C2311yescharacteristic
75C22×C5×C8C2311yescharacteristic
76C22×C5×C8C2311yescharacteristic
77C5×C4⋊Q8—22no—
78C5×C4⋊Q8—41no—
79C5×#124—22no—
80C5×#124—41no—
81C4×C5×Q8—22no—
82C4×C5×Q8—41no—
83C4×C5×D4—22no—
84C4×C5×D4—41no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4—21no—
87C2×C5×C4⋊C4—21no—
88C2×C5×C4⋊C4—21no—
89C2×C5×C4⋊C4—21no—
90C2×C5×C4⋊C4D411yescharacteristic
91C2×C5×C4⋊C4D411yescharacteristic
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⋊C4—21no—
100C2×C5×C22⋊C4—21no—
101C2×C5×C22⋊C4—21no—
102C2×C42×C5—21no—
103C2×C42×C5C2311yescharacteristic
104C2×C42×C5C2311yescharacteristic
105C2×C42×C5C2311yescharacteristic
106C5×#106—22no—
107C5×#106—41no—
108C5×C4⋊C8—22no—
109C5×#103—22no—
110C5×#103—22no—
111C5×#103—22no—
112C5×#103—22no—
113C5×#103—22no—
114C5×#102—22no—
115C5×#102—22no—
116C5×#102D414yes—
117C5×#97—22no—
118C5×#97—22no—
119C4×C5×C8—22no—
120C5×#95—21no—
121C5×#95—21no—
122C5×#95—21no—
123C5×#95—21no—
124C5×#95—21no—
125C5×#95—21no—
126C5×#95—21no—
127C5×#95—21no—
128C5×#95—21no—
129C5×#95—21no—
130C5×#95—21no—
131C5×#95D411yescharacteristic
132C5×#95D411yescharacteristic
133C2×#489C2×C511yescharacteristic
134C2×C4×SD16C2×C511yescharacteristic
135#2451C2×C511yescharacteristic, complemented
136#2447C2×C511yescharacteristic
137#2066C2×C511yescharacteristic, complemented
138#2058C2×C511yescharacteristic, complemented
139#2055C2×C511yescharacteristic
140#2015C2×C511yescharacteristic
141#1971C2×C511yescharacteristic, complemented
142#1936C2×C511yescharacteristic
143#1935C2×C511yescharacteristic, complemented
144#1933C2×C511yescharacteristic, complemented
145#1878C2×C511yescharacteristic, complemented
146#1864C2×C511yescharacteristic
147#1771C2×C511yescharacteristic, complemented
148C24×C5—21no—
149C2×C5×Q8—21no—
150C2×C5×Q8—21no—
151C2×C5×Q8—21no—
152C2×C5×Q8—21no—
153C2×C5×Q8—21no—
154C2×C5×Q8—21no—
155C2×C5×Q8—21no—
156C2×C5×Q8—21no—
157C2×C5×Q8—21no—
158C2×C5×Q8—21no—
159C2×C5×Q8—21no—
160C2×C5×Q8—42no—
161C2×C5×Q8—42no—
162C2×C5×Q8—81no—
163C2×C5×Q8#4112yes—
164C2×C5×D4—21no—
165C2×C5×D4—21no—
166C2×C5×D4—21no—
167C2×C5×D4—21no—
168C2×C5×D4—21no—
169C2×C5×D4C2×D412yes—
170C22×C4×C5—21no—
171C22×C4×C5—21no—
172C22×C4×C5—21no—
173C22×C4×C5—21no—
174C22×C4×C5—21no—
175C22×C4×C5—21no—
176C22×C4×C5—21no—
177C22×C4×C5—42no—
178C22×C4×C5—42no—
179C22×C4×C5—81no—
180C22×C4×C5C2×D411yescharacteristic
181C22×C4×C5C2×D411yescharacteristic
182C22×C4×C5C2×D411yescharacteristic
183C22×C4×C5C2×D411yescharacteristic
184C22×C4×C5C2×D411yescharacteristic
185C22×C4×C5C2×D411yescharacteristic
186C22×C4×C5C2411yescharacteristic
187C5×Q16—44no—
188C5×SD16—28no—
189C2×C5×C8—21no—
190C2×C5×C8—21no—
191C2×C5×C8—21no—
192C2×C5×C8—21no—
193C2×C5×C8—21no—
194C2×C5×C8—21no—
195C2×C5×C8—21no—
196C2×C5×C8—21no—
197C2×C5×C8—21no—
198C2×C5×C8C2×D412yes—
199C2×C5×C8#4112yes—
200C2×C5×C8#4112yes—
201C5×C4⋊C4—22no—
202C5×C4⋊C4—21no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4—21no—
205C5×C4⋊C4—21no—
206C5×C4⋊C4—21no—
207C5×C4⋊C4—21no—
208C5×C4⋊C4—21no—
209C5×C4⋊C4—21no—
210C5×C4⋊C4—21no—
211C5×C4⋊C4—41no—
212C5×C4⋊C4—42no—
213C5×C4⋊C4—42no—
214C5×C4⋊C4—81no—
215C5×C4⋊C4—81no—
216C5×C4⋊C4—81no—
217C5×C4⋊C4C2×D412yes—
218C5×C22⋊C4—42no—
219C5×C22⋊C4—42no—
220C5×C22⋊C4—42no—
221C5×C22⋊C4—42no—
222C5×C22⋊C4—81no—
223C42×C5—21no—
224C42×C5—21no—
225C42×C5—21no—
226C42×C5—21no—
227C42×C5—21no—
228C42×C5—21no—
229C42×C5—42no—
230C22×Q16C22×C511yescharacteristic
231C22×SD16C22×C511yescharacteristic
232C2×C4⋊Q8C22×C511yescharacteristic
233C2×#124C22×C511yescharacteristic
234C2×C4×Q8C22×C511yescharacteristic
235C2×C4×D4C22×C511yescharacteristic
236#489—24no—
237C4×SD16—24no—
238C2×#106C22×C511yescharacteristic
239C2×#106C22×C511yescharacteristic
240C2×C4⋊C8C22×C511yescharacteristic
241C2×#103C22×C511yescharacteristic
242C2×#103C22×C511yescharacteristic
243C2×#103C22×C511yescharacteristic
244C2×#103C22×C511yescharacteristic
245C2×#103C22×C511yescharacteristic
246C2×#102C22×C511yescharacteristic
247C2×#102C22×C511yescharacteristic
248C2×#102C22×C511yescharacteristic
249C2×#97C22×C511yescharacteristic
250C2×#97C22×C511yescharacteristic
251C2×C4×C8C22×C511yescharacteristic
252#398—21no—
253#397—21no—
254#397—21no—
255#395—21no—
256#391—21no—
257#391C22×C511yescharacteristic
258#391C22×C511yescharacteristic
259#391C22×C511yescharacteristic
260#391C22×C511yescharacteristic
261#390C22×C511yescharacteristic
262#389C22×C511yescharacteristic
263#389C22×C511yescharacteristic
264#389C22×C511yescharacteristic
265#387—21no—
266#385—21no—
267#381—21no—
268C4×C4⋊C4C22×C511yescharacteristic
269#340C22×C511yescharacteristic
270#340C22×C511yescharacteristic
271#340C22×C511yescharacteristic
272#340C22×C511yescharacteristic
273#336C22×C511yescharacteristic
274#336C22×C511yescharacteristic
275C23×C5—22no—
276C23×C5—41no—
277C23×C5—41no—
278C23×C5—41no—
279C23×C5—41no—
280C23×C5—41no—
281C23×C5—41no—
282C23×C5C22×D411yescharacteristic
283C5×Q8—22no—
284C5×Q8—22no—
285C5×Q8—41no—
286C5×Q8—41no—
287C5×Q8—41no—
288C5×Q8—41no—
289C5×Q8—41no—
290C5×Q8—41no—
291C5×Q8—84no—
292C5×D4—22no—
293C5×D4—22no—
294C5×D4—41no—
295C5×D4—41no—
296C2×C4×C5—21no—
297C2×C4×C5—21no—
298C2×C4×C5—21no—
299C2×C4×C5—21no—
300C2×C4×C5—21no—
301C2×C4×C5—21no—
302C2×C4×C5—21no—
303C2×C4×C5—22no—
304C2×C4×C5—22no—
305C2×C4×C5—21no—
306C2×C4×C5—21no—
307C2×C4×C5—21no—
308C2×C4×C5—21no—
309C2×C4×C5—21no—
310C2×C4×C5—21no—
311C2×C4×C5—21no—
312C2×C4×C5—21no—
313C2×C4×C5—21no—
314C2×C4×C5—21no—
315C2×C4×C5—41no—
316C2×C4×C5—41no—
317C2×C4×C5—41no—
318C2×C4×C5—41no—
319C2×C4×C5—41no—
320C2×C4×C5—41no—
321C2×C4×C5—41no—
322C2×C4×C5—41no—
323C2×C4×C5—41no—
324C2×C4×C5—41no—
325C2×C4×C5—41no—
326C2×C4×C5—41no—
327C2×C4×C5—41no—
328C2×C4×C5—41no—
329C2×C4×C5—41no—
330C2×C4×C5—41no—
331C2×C4×C5—41no—
332C2×C4×C5—41no—
333C2×C4×C5—41no—
334C2×C4×C5—84no—
335C2×C4×C5#13712yes—
336C2×C4×C5C22×D411yescharacteristic
337C2×C4×C5C2×#4111yescharacteristic
338C2×C4×C5C2×#4111yescharacteristic
339C2×C4×C5C2×#4111yescharacteristic
340C2×C4×C5H(5,𝔽2)11yescharacteristic
341C2×C4×C52−1+411yescharacteristic
342C5×C8—22no—
343C5×C8—22no—
344C5×C8—22no—
345C5×C8—41no—
346C22×Q8—21no—
347C22×Q8C23×C511yescharacteristic
348C22×Q8C23×C511yescharacteristic
349C22×D4C23×C511yescharacteristic
350C23×C4—21no—
351C2×Q16—22no—
352C2×Q16—22no—
353C2×Q16—41no—
354C2×SD16—22no—
355C2×SD16—22no—
356C2×SD16—22no—
357C22×C8C23×C511yescharacteristic
358C22×C8C23×C511yescharacteristic
359C22×C8C23×C511yescharacteristic
360C22×C8C23×C511yescharacteristic
361C4⋊Q8—22no—
362C4⋊Q8—41no—
363#124—22no—
364#124—41no—
365C4×Q8—22no—
366C4×Q8—41no—
367C4×D4—22no—
368C4×D4—41no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4C5×D411yescharacteristic
375C2×C4⋊C4C5×D411yescharacteristic
376C2×C4⋊C4C23×C511yescharacteristic
377C2×C4⋊C4C23×C511yescharacteristic
378C2×C4⋊C4C23×C511yescharacteristic
379C2×C4⋊C4C23×C511yescharacteristic
380C2×C4⋊C4C23×C511yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C42—21no—
387C2×C42C23×C511yescharacteristic
388C2×C42C23×C511yescharacteristic
389C2×C42C23×C511yescharacteristic
390#106—22no—
391#106—41no—
392C4⋊C8—22no—
393#103—22no—
394#103—22no—
395#103—22no—
396#103—22no—
397#103—22no—
398#102—22no—
399#102—22no—
400#102C5×D414yes—
401#97—22no—
402#97—22no—
403C4×C8—22no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95C5×D411yescharacteristic
416#95C5×D411yescharacteristic
417C22×C5—42no—
418C22×C5—42no—
419C22×C5—42no—
420C22×C5—81no—
421C22×C5—81no—
422C22×C5—81no—
423C22×C5—81no—
424C22×C5#53211yescharacteristic
425C22×C5D4211yescharacteristic
426C22×C5#54811yescharacteristic
427C22×C5C2×#13711yescharacteristic
428C22×C5#57511yescharacteristic
429C22×C5#57711yescharacteristic
430C22×C5#57711yescharacteristic
431C4×C5—22no—
432C4×C5—41no—
433C4×C5—41no—
434C4×C5—42no—
435C4×C5—41no—
436C4×C5—41no—
437C4×C5—41no—
438C4×C5—42no—
439C4×C5—81no—
440C4×C5—81no—
441C4×C5—81no—
442C4×C5—81no—
443C4×C5—81no—
444C4×C5#52511yescharacteristic
445C4×C5#54211yescharacteristic
446C4×C5#54711yescharacteristic
447C4×C5#55111yescharacteristic
448C24—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—81no—
463C2×Q8C5×#4112yes—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4C2×C5×D412yes—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—81no—
480C22×C4C2×C5×D411yescharacteristic
481C22×C4C2×C5×D411yescharacteristic
482C22×C4C2×C5×D411yescharacteristic
483C22×C4C2×C5×D411yescharacteristic
484C22×C4C2×C5×D411yescharacteristic
485C22×C4C2×C5×D411yescharacteristic
486C22×C4C24×C511yescharacteristic
487Q16—44no—
488SD16—28no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8C2×C5×D412yes—
499C2×C8C5×#4112yes—
500C2×C8C5×#4112yes—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4C2×C5×D412yes—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—81no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—42no—
530C2×C5—82no—
531C2×C5#246111yescharacteristic
532C2×C5#317211yescharacteristic
533C2×C5#317311yescharacteristic
534C2×C5#326611yescharacteristic
535C2×C5#326711yescharacteristic
536C2×C5#336911yescharacteristic
537C2×C5#337111yescharacteristic
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C22×C5×D411yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—84no—
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—84no—
598C2×C4C5×#13712yes—
599C2×C4C22×C5×D411yescharacteristic
600C2×C4C2×C5×#4111yescharacteristic
601C2×C4C2×C5×#4111yescharacteristic
602C2×C4C2×C5×#4111yescharacteristic
603C2×C4C5×H(5,𝔽2)11yescharacteristic
604C2×C4C5×2−1+411yescharacteristic
605C8—22no—
606C8—22no—
607C8—22no—
608C8—41no—
609C5#2682411yescharacteristic, direct product factor, Sylow
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C5×#53211yescharacteristic
618C22C5×D4211yescharacteristic
619C22C5×#54811yescharacteristic
620C22C2×C5×#13711yescharacteristic
621C22C5×#57511yescharacteristic
622C22C5×#57711yescharacteristic
623C22C5×#57711yescharacteristic
624C4—22no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—81no—
637C4C5×#52511yescharacteristic
638C4C5×#54211yescharacteristic
639C4C5×#54711yescharacteristic
640C4C5×#55111yescharacteristic
641C2—82no—
642C2C5×#246111yescharacteristic
643C2C5×#317211yescharacteristic
644C2C5×#317311yescharacteristic
645C2C5×#326611yescharacteristic
646C2C5×#326711yescharacteristic
647C2C5×#336911yescharacteristic
648C2C5×#337111yescharacteristic
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC22×C5×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C4×C5×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C5×#2451C2semidirect
5. C5×#2066C2semidirect
6. C5×#2058C2semidirect
9. C5×#1971C2semidirect
11. C5×#1935C2semidirect
12. C5×#1933C2semidirect
13. C5×#1878C2semidirect
15. C5×#1771C2semidirect
61. #26824C5direct (N × H)
135. #2451C2×C5semidirect
137. #2066C2×C5semidirect
138. #2058C2×C5semidirect
141. #1971C2×C5semidirect
143. #1935C2×C5semidirect
144. #1933C2×C5semidirect
145. #1878C2×C5semidirect
147. #1771C2×C5semidirect
609. C5#26824direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#4892
C2×C4×C5×SD162
C5×#24512
C5×#24472
C5×#20662
C5×#20582
C5×#20552
C5×#20152
C5×#19712
C5×#19362
C5×#19352
C5×#19332
C5×#18782
C5×#18642
C5×#17712
#268245