Finite Group Databasebeta

← C5×#26809 #11949529

Subgroups — C5×#26809 #11949529

Order 1280.

776 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#485C211yesmaximal, characteristic
2C2×C5×#479C211yesmaximal, characteristic, complemented
3C5×#2489C211yesmaximal, characteristic
4C5×#2362C211yesmaximal, characteristic
5C5×#2068C211yesmaximal, characteristic, complemented
6C5×#2059C211yesmaximal, characteristic, complemented
7C5×#2036C211yesmaximal, characteristic
8C5×#2001C211yesmaximal, characteristic
9C5×#1971C211yesmaximal, characteristic, complemented
10C5×#1928C211yesmaximal, characteristic, complemented
11C5×#1928C211yesmaximal, characteristic, complemented
12C5×#1924C211yesmaximal, characteristic
13C5×#1878C211yesmaximal, characteristic, complemented
14C5×#1854C211yesmaximal, characteristic
15C5×#1776C211yesmaximal, characteristic, complemented
16C22×C5×Q16C2211yescharacteristic
17C2×C5×#125C2211yescharacteristic
18C2×C5×C22⋊Q8C2211yescharacteristic
19C2×C5×C22⋊Q8C2211yescharacteristic
20C2×C4×C5×Q8C2211yescharacteristic
21C2×C5×#117C2211yescharacteristic
22C5×#485—24no—
23C5×#479—24no—
24C2×C5×#107C2211yescharacteristic
25C2×C5×#107C2211yescharacteristic
26C2×C5×#107C2211yescharacteristic
27C2×C5×#106C2211yescharacteristic
28C2×C5×#106C2211yescharacteristic
29C2×C5×C4⋊C8C2211yescharacteristic
30C2×C5×#103C2211yescharacteristic
31C2×C5×#103C2211yescharacteristic
32C2×C5×#103C2211yescharacteristic
33C2×C5×#103C2211yescharacteristic
34C2×C5×#103C2211yescharacteristic
35C2×C5×#103C2211yescharacteristic
36C2×C5×C22⋊C8C2211yescharacteristic
37C2×C5×C22⋊C8C2211yescharacteristic
38C2×C5×#97C2211yescharacteristic
39C5×#400—21no—
40C5×#397—21no—
41C5×#391C2211yescharacteristic
42C5×#391C2211yescharacteristic
43C5×#391C2211yescharacteristic
44C5×#389C2211yescharacteristic
45C5×#388—21no—
46C5×#388—21no—
47C5×#388—21no—
48C5×#387—21no—
49C5×#382—21no—
50C5×#380C2211yescharacteristic
51C4×C5×C4⋊C4—21no—
52C4×C5×C4⋊C4C2211yescharacteristic
53C4×C5×C4⋊C4C2211yescharacteristic
54C4×C5×C22⋊C4C2211yescharacteristic
55C5×#340C2211yescharacteristic
56C5×#340C2211yescharacteristic
57C5×#340C2211yescharacteristic
58C5×#340C2211yescharacteristic
59C5×#336C2211yescharacteristic
60C5×#336C2211yescharacteristic
61#26809C511yesmaximal, characteristic, direct product factor, Sylow
62C22×C5×Q8C2311yescharacteristic
63C22×C5×Q8C2311yescharacteristic
64C23×C4×C5C2311yescharacteristic
65C2×C5×Q16—22no—
66C2×C5×Q16—22no—
67C2×C5×Q16—41no—
68C22×C5×C8C2311yescharacteristic
69C22×C5×C8C2311yescharacteristic
70C22×C5×C8C2311yescharacteristic
71C22×C5×C8C2311yescharacteristic
72C5×#125—22no—
73C5×#125—41no—
74C5×C22⋊Q8—22no—
75C5×C22⋊Q8—22no—
76C5×C22⋊Q8—22no—
77C5×C22⋊Q8—41no—
78C4×C5×Q8—22no—
79C4×C5×Q8—41no—
80C5×#117—22no—
81C5×#117—41no—
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⋊C4C2311yescharacteristic
89C2×C5×C4⋊C4C2311yescharacteristic
90C2×C5×C4⋊C4C2311yescharacteristic
91C2×C5×C4⋊C4C2311yescharacteristic
92C2×C5×C4⋊C4C2311yescharacteristic
93C2×C5×C4⋊C4C2311yescharacteristic
94C2×C5×C22⋊C4—21no—
95C2×C5×C22⋊C4—21no—
96C2×C5×C22⋊C4—21no—
97C2×C5×C22⋊C4—21no—
98C2×C5×C22⋊C4D411yescharacteristic
99C2×C5×C22⋊C4D411yescharacteristic
100C2×C42×C5—21no—
101C2×C42×C5—21no—
102C2×C42×C5—21no—
103C2×C42×C5—21no—
104C2×C42×C5C2311yescharacteristic
105C2×C42×C5C2311yescharacteristic
106C5×#107—22no—
107C5×#107—22no—
108C5×#107—22no—
109C5×#106—41no—
110C5×#106D414yes—
111C5×C4⋊C8—22no—
112C5×#103—22no—
113C5×#103—22no—
114C5×#103—22no—
115C5×#103—22no—
116C5×#103—22no—
117C5×#103—22no—
118C5×C22⋊C8—22no—
119C5×C22⋊C8—22no—
120C5×#97—22no—
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×#485C2×C511yescharacteristic
134C2×#479C2×C511yescharacteristic, complemented
135#2489C2×C511yescharacteristic
136#2362C2×C511yescharacteristic
137#2068C2×C511yescharacteristic, complemented
138#2059C2×C511yescharacteristic, complemented
139#2036C2×C511yescharacteristic
140#2001C2×C511yescharacteristic
141#1971C2×C511yescharacteristic, complemented
142#1928C2×C511yescharacteristic, complemented
143#1928C2×C511yescharacteristic, complemented
144#1924C2×C511yescharacteristic
145#1878C2×C511yescharacteristic, complemented
146#1854C2×C511yescharacteristic
147#1776C2×C511yescharacteristic, complemented
148C24×C5C2×D411yescharacteristic
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#4112yes—
161C22×C4×C5—21no—
162C22×C4×C5—21no—
163C22×C4×C5—21no—
164C22×C4×C5—21no—
165C22×C4×C5—21no—
166C22×C4×C5—21no—
167C22×C4×C5—21no—
168C22×C4×C5—21no—
169C22×C4×C5—21no—
170C22×C4×C5—21no—
171C22×C4×C5—21no—
172C22×C4×C5—21no—
173C22×C4×C5—21no—
174C22×C4×C5—21no—
175C22×C4×C5C2×D411yescharacteristic
176C22×C4×C5C2×D411yescharacteristic
177C22×C4×C5C2×D411yescharacteristic
178C22×C4×C5C2×D411yescharacteristic
179C22×C4×C5C2×D411yescharacteristic
180C22×C4×C5C2411yescharacteristic
181C5×Q16—44no—
182C2×C5×C8—21no—
183C2×C5×C8—21no—
184C2×C5×C8—21no—
185C2×C5×C8—21no—
186C2×C5×C8—21no—
187C2×C5×C8—21no—
188C2×C5×C8—21no—
189C2×C5×C8—21no—
190C2×C5×C8—21no—
191C2×C5×C8—21no—
192C2×C5×C8C2×D412yes—
193C2×C5×C8#4112yes—
194C5×C4⋊C4—24no—
195C5×C4⋊C4—21no—
196C5×C4⋊C4—21no—
197C5×C4⋊C4—21no—
198C5×C4⋊C4—21no—
199C5×C4⋊C4—21no—
200C5×C4⋊C4—21no—
201C5×C4⋊C4—21no—
202C5×C4⋊C4—21no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4—42no—
205C5×C4⋊C4—42no—
206C5×C4⋊C4—42no—
207C5×C4⋊C4—42no—
208C5×C4⋊C4—42no—
209C5×C4⋊C4C2×D412yes—
210C5×C4⋊C4C2×D412yes—
211C5×C4⋊C4#4112yes—
212C5×C22⋊C4—24no—
213C5×C22⋊C4—41no—
214C5×C22⋊C4—41no—
215C5×C22⋊C4—42no—
216C5×C22⋊C4—42no—
217C5×C22⋊C4—42no—
218C42×C5—21no—
219C42×C5—21no—
220C42×C5—21no—
221C42×C5—21no—
222C42×C5—42no—
223C42×C5—42no—
224C42×C5—42no—
225C42×C5—42no—
226C22×Q16C22×C511yescharacteristic
227C2×#125C22×C511yescharacteristic
228C2×C22⋊Q8C22×C511yescharacteristic
229C2×C22⋊Q8C22×C511yescharacteristic
230C2×C4×Q8C22×C511yescharacteristic
231C2×#117C22×C511yescharacteristic
232#485—24no—
233#479—24no—
234C2×#107C22×C511yescharacteristic
235C2×#107C22×C511yescharacteristic
236C2×#107C22×C511yescharacteristic
237C2×#106C22×C511yescharacteristic
238C2×#106C22×C511yescharacteristic
239C2×C4⋊C8C22×C511yescharacteristic
240C2×#103C22×C511yescharacteristic
241C2×#103C22×C511yescharacteristic
242C2×#103C22×C511yescharacteristic
243C2×#103C22×C511yescharacteristic
244C2×#103C22×C511yescharacteristic
245C2×#103C22×C511yescharacteristic
246C2×C22⋊C8C22×C511yescharacteristic
247C2×C22⋊C8C22×C511yescharacteristic
248C2×#97C22×C511yescharacteristic
249#400—21no—
250#397—21no—
251#391C22×C511yescharacteristic
252#391C22×C511yescharacteristic
253#391C22×C511yescharacteristic
254#389C22×C511yescharacteristic
255#388—21no—
256#388—21no—
257#388—21no—
258#387—21no—
259#382—21no—
260#380C22×C511yescharacteristic
261C4×C4⋊C4—21no—
262C4×C4⋊C4C22×C511yescharacteristic
263C4×C4⋊C4C22×C511yescharacteristic
264C4×C22⋊C4C22×C511yescharacteristic
265#340C22×C511yescharacteristic
266#340C22×C511yescharacteristic
267#340C22×C511yescharacteristic
268#340C22×C511yescharacteristic
269#336C22×C511yescharacteristic
270#336C22×C511yescharacteristic
271C23×C5—21no—
272C23×C5—21no—
273C23×C5—21no—
274C23×C5—21no—
275C23×C5—21no—
276C23×C5—21no—
277C23×C5—21no—
278C23×C5C22×D411yescharacteristic
279C5×Q8—22no—
280C5×Q8—22no—
281C5×Q8—41no—
282C5×Q8—41no—
283C5×Q8—41no—
284C5×Q8—41no—
285C5×Q8—41no—
286C5×Q8—41no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
291C2×C4×C5—21no—
292C2×C4×C5—22no—
293C2×C4×C5—21no—
294C2×C4×C5—22no—
295C2×C4×C5—21no—
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—21no—
304C2×C4×C5—21no—
305C2×C4×C5—21no—
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—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×C5C22×D411yescharacteristic
331C2×C4×C5C2×#4111yescharacteristic
332C2×C4×C5C2×#4111yescharacteristic
333C2×C4×C5C2×#4111yescharacteristic
334C2×C4×C5H(5,𝔽2)11yescharacteristic
335C2×C4×C52−1+411yescharacteristic
336C5×C8—22no—
337C5×C8—22no—
338C5×C8—41no—
339C5×C8—41no—
340C22×Q8C23×C511yescharacteristic
341C22×Q8C23×C511yescharacteristic
342C23×C4C23×C511yescharacteristic
343C2×Q16—22no—
344C2×Q16—22no—
345C2×Q16—41no—
346C22×C8C23×C511yescharacteristic
347C22×C8C23×C511yescharacteristic
348C22×C8C23×C511yescharacteristic
349C22×C8C23×C511yescharacteristic
350#125—22no—
351#125—41no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8—41no—
356C4×Q8—22no—
357C4×Q8—41no—
358#117—22no—
359#117—41no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4C23×C511yescharacteristic
367C2×C4⋊C4C23×C511yescharacteristic
368C2×C4⋊C4C23×C511yescharacteristic
369C2×C4⋊C4C23×C511yescharacteristic
370C2×C4⋊C4C23×C511yescharacteristic
371C2×C4⋊C4C23×C511yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4C5×D411yescharacteristic
377C2×C22⋊C4C5×D411yescharacteristic
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42C23×C511yescharacteristic
383C2×C42C23×C511yescharacteristic
384#107—22no—
385#107—22no—
386#107—22no—
387#106—41no—
388#106C5×D414yes—
389C4⋊C8—22no—
390#103—22no—
391#103—22no—
392#103—22no—
393#103—22no—
394#103—22no—
395#103—22no—
396C22⋊C8—22no—
397C22⋊C8—22no—
398#97—22no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95C5×D411yescharacteristic
410#95C5×D411yescharacteristic
411C22×C5—41no—
412C22×C5—41no—
413C22×C5—41no—
414C22×C5—41no—
415C22×C5—41no—
416C22×C5—41no—
417C22×C5—41no—
418C22×C5#52511yescharacteristic
419C22×C5D4211yescharacteristic
420C22×C5#54711yescharacteristic
421C22×C5#57511yescharacteristic
422C22×C5#57511yescharacteristic
423C22×C5#57811yescharacteristic
424C22×C5#57811yescharacteristic
425C4×C5—41no—
426C4×C5—41no—
427C4×C5—42no—
428C4×C5—41no—
429C4×C5—42no—
430C4×C5—41no—
431C4×C5—41no—
432C4×C5—42no—
433C4×C5—42no—
434C4×C5—81no—
435C4×C5—81no—
436C4×C5—81no—
437C4×C5—81no—
438C4×C5#53211yescharacteristic
439C4×C5#54211yescharacteristic
440C4×C5#54811yescharacteristic
441C4×C5#55111yescharacteristic
442C24C2×C5×D411yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8C5×#4112yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4C2×C5×D411yescharacteristic
470C22×C4C2×C5×D411yescharacteristic
471C22×C4C2×C5×D411yescharacteristic
472C22×C4C2×C5×D411yescharacteristic
473C22×C4C2×C5×D411yescharacteristic
474C22×C4C24×C511yescharacteristic
475Q16—44no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8C2×C5×D412yes—
487C2×C8C5×#4112yes—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4C2×C5×D412yes—
504C4⋊C4C2×C5×D412yes—
505C4⋊C4C5×#4112yes—
506C22⋊C4—24no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—42no—
520C2×C5—81no—
521C2×C5#246111yescharacteristic
522C2×C5#311111yescharacteristic
523C2×C5#312711yescharacteristic
524C2×C5#326911yescharacteristic
525C2×C5#326911yescharacteristic
526C2×C5#334911yescharacteristic
527C2×C5#335211yescharacteristic
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23C22×C5×D411yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
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×C4C22×C5×D411yescharacteristic
588C2×C4C2×C5×#4111yescharacteristic
589C2×C4C2×C5×#4111yescharacteristic
590C2×C4C2×C5×#4111yescharacteristic
591C2×C4C5×H(5,𝔽2)11yescharacteristic
592C2×C4C5×2−1+411yescharacteristic
593C8—22no—
594C8—22no—
595C8—41no—
596C8—41no—
597C5#2680911yescharacteristic, direct product factor, Sylow
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22C5×#52511yescharacteristic
606C22C5×D4211yescharacteristic
607C22C5×#54711yescharacteristic
608C22C5×#57511yescharacteristic
609C22C5×#57511yescharacteristic
610C22C5×#57811yescharacteristic
611C22C5×#57811yescharacteristic
612C4—41no—
613C4—41no—
614C4—42no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—42no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4C5×#53211yescharacteristic
626C4C5×#54211yescharacteristic
627C4C5×#54811yescharacteristic
628C4C5×#55111yescharacteristic
629C2—81no—
630C2C5×#246111yescharacteristic
631C2C5×#311111yescharacteristic
632C2C5×#312711yescharacteristic
633C2C5×#326911yescharacteristic
634C2C5×#326911yescharacteristic
635C2C5×#334911yescharacteristic
636C2C5×#335211yescharacteristic
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
2. C2×C5×#479C2semidirect
5. C5×#2068C2semidirect
6. C5×#2059C2semidirect
9. C5×#1971C2semidirect
10. C5×#1928C2semidirect
11. C5×#1928C2semidirect
13. C5×#1878C2semidirect
15. C5×#1776C2semidirect
61. #26809C5direct (N × H)
134. C2×#479C2×C5semidirect
137. #2068C2×C5semidirect
138. #2059C2×C5semidirect
141. #1971C2×C5semidirect
142. #1928C2×C5semidirect
143. #1928C2×C5semidirect
145. #1878C2×C5semidirect
147. #1776C2×C5semidirect
597. C5#26809direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#4852
C2×C5×#4792
C5×#24892
C5×#23622
C5×#20682
C5×#20592
C5×#20362
C5×#20012
C5×#19712
C5×#19282
C5×#19242
C5×#18782
C5×#18542
C5×#17762
#268095