Finite Group Databasebeta

← C5×#14632 #11937352

Subgroups — C5×#14632 #11937352

Order 1280.

1586 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#394C211yesmaximal, characteristic
2C2×C5×#388C212yesmaximal, complemented
3C5×#1442C212yesmaximal, complemented
4C5×#1441C211yesmaximal, characteristic, complemented
5C5×#1439C212yesmaximal, complemented
6C5×#1439C212yesmaximal, complemented
7C5×#1439C212yesmaximal, complemented
8C5×#1438C212yesmaximal
9C5×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C5×D4C2211yescharacteristic
11C22×C5×C4⋊C4C2212yes—
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C5×C4⋊C4C2211yescharacteristic
15C22×C5×C22⋊C4C2212yes—
16C22×C5×C22⋊C4C2212yes—
17C22×C5×C22⋊C4C2212yes—
18C22×C5×C22⋊C4C2212yes—
19C22×C5×C22⋊C4C2212yes—
20C22×C5×C22⋊C4C2212yes—
21C22×C42×C5C2211yescharacteristic
22C5×#394C418yescomplemented
23C5×#388—28no—
24C2×C5×#95C2212yes—
25C2×C5×#95C2212yes—
26C2×C5×#95C2212yes—
27C2×C5×#95C2212yes—
28C2×C5×#95C2212yes—
29C2×C5×#95C2211yescharacteristic
30C2×C5×#95C2212yes—
31C2×C5×#95C2212yes—
32C2×C5×#95C2211yescharacteristic
33#14632C511yesmaximal, characteristic, direct product factor, Sylow
34C25×C5C2312yes—
35C22×C5×D4—41no—
36C22×C5×D4—41no—
37C22×C5×D4—42no—
38C22×C5×D4—42no—
39C22×C5×D4C2×C414yes—
40C23×C4×C5C2312yes—
41C23×C4×C5C2312yes—
42C23×C4×C5C2312yes—
43C23×C4×C5C2311yescharacteristic
44C23×C4×C5C2312yes—
45C23×C4×C5C2311yescharacteristic
46C23×C4×C5C2312yes—
47C23×C4×C5C2311yescharacteristic
48C2×C5×C4⋊C4—24no—
49C2×C5×C4⋊C4—22no—
50C2×C5×C4⋊C4—41no—
51C2×C5×C4⋊C4—41no—
52C2×C5×C4⋊C4—41no—
53C2×C5×C4⋊C4—41no—
54C2×C5×C4⋊C4—42no—
55C2×C5×C4⋊C4—42no—
56C2×C5×C4⋊C4—42no—
57C2×C5×C4⋊C4—41no—
58C2×C5×C4⋊C4C2×C414yes—
59C2×C5×C22⋊C4—24no—
60C2×C5×C22⋊C4—24no—
61C2×C5×C22⋊C4—24no—
62C2×C5×C22⋊C4—24no—
63C2×C5×C22⋊C4—42no—
64C2×C5×C22⋊C4—42no—
65C2×C5×C22⋊C4—42no—
66C2×C5×C22⋊C4—42no—
67C2×C5×C22⋊C4—42no—
68C2×C5×C22⋊C4—42no—
69C2×C5×C22⋊C4—42no—
70C2×C5×C22⋊C4—42no—
71C2×C5×C22⋊C4—42no—
72C2×C5×C22⋊C4—42no—
73C2×C5×C22⋊C4—42no—
74C2×C5×C22⋊C4—42no—
75C2×C5×C22⋊C4C2×C418yes—
76C2×C5×C22⋊C4C2×C418yes—
77C2×C42×C5—41no—
78C2×C42×C5—41no—
79C2×C42×C5D414yescomplemented
80C5×#95—24no—
81C5×#95—24no—
82C5×#95—42no—
83C5×#95—42no—
84C5×#95—41no—
85C5×#95—42no—
86C5×#95—42no—
87C5×#95—42no—
88C5×#95C2×C414yes—
89C2×#394C2×C511yescharacteristic
90C2×#388C2×C512yescomplemented
91#1442C2×C512yescomplemented
92#1441C2×C511yescharacteristic, complemented
93#1439C2×C512yescomplemented
94#1439C2×C512yescomplemented
95#1439C2×C512yescomplemented
96#1438C2×C512yes—
97C4⋊#95C2×C511yescharacteristic, complemented
98C24×C5—22no—
99C24×C5—22no—
100C24×C5—22no—
101C24×C5—22no—
102C24×C5—22no—
103C24×C5—22no—
104C24×C5—22no—
105C24×C5—22no—
106C24×C5—22no—
107C24×C5—22no—
108C24×C5—22no—
109C24×C5—22no—
110C24×C5—22no—
111C24×C5—22no—
112C24×C5C22×C414yes—
113C24×C5C2411yescharacteristic
114C2×C5×D4—44no—
115C2×C5×D4—48no—
116C2×C5×D4—44no—
117C2×C5×D4—48no—
118C2×C5×D4—44no—
119C22×C4×C5—21no—
120C22×C4×C5—21no—
121C22×C4×C5—21no—
122C22×C4×C5—21no—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—22no—
132C22×C4×C5—22no—
133C22×C4×C5—21no—
134C22×C4×C5—21no—
135C22×C4×C5—21no—
136C22×C4×C5—21no—
137C22×C4×C5—22no—
138C22×C4×C5—22no—
139C22×C4×C5—22no—
140C22×C4×C5—22no—
141C22×C4×C5—22no—
142C22×C4×C5—22no—
143C22×C4×C5—22no—
144C22×C4×C5—22no—
145C22×C4×C5—22no—
146C22×C4×C5—22no—
147C22×C4×C5—22no—
148C22×C4×C5—22no—
149C22×C4×C5—22no—
150C22×C4×C5—22no—
151C22×C4×C5—22no—
152C22×C4×C5—22no—
153C22×C4×C5—22no—
154C22×C4×C5—22no—
155C22×C4×C5—22no—
156C22×C4×C5—22no—
157C22×C4×C5—22no—
158C22×C4×C5—22no—
159C22×C4×C5—22no—
160C22×C4×C5—22no—
161C22×C4×C5—22no—
162C22×C4×C5—22no—
163C22×C4×C5—22no—
164C22×C4×C5C22×C412yes—
165C22×C4×C5C22×C414yes—
166C22×C4×C5C22×C412yes—
167C22×C4×C5C22×C412yes—
168C22×C4×C5C2×D414yes—
169C22×C4×C5C2×D412yes—
170C22×C4×C5#4112yes—
171C5×C4⋊C4—44no—
172C5×C4⋊C4—44no—
173C5×C4⋊C4—48no—
174C5×C4⋊C4—44no—
175C5×C22⋊C4—216no—
176C5×C22⋊C4—48no—
177C5×C22⋊C4—48no—
178C5×C22⋊C4—48no—
179C5×C22⋊C4—48no—
180C5×C22⋊C4—48no—
181C42×C5—44no—
182C23×D4C22×C511yescharacteristic
183C22×C4⋊C4C22×C511yescharacteristic
184C22×C4⋊C4C22×C512yes—
185C22×C4⋊C4C22×C511yescharacteristic
186C22×C4⋊C4C22×C511yescharacteristic
187C22×C22⋊C4C22×C512yes—
188C22×C22⋊C4C22×C512yes—
189C22×C22⋊C4C22×C512yes—
190C22×C22⋊C4C22×C512yes—
191C22×C22⋊C4C22×C512yes—
192C22×C22⋊C4C22×C512yes—
193C22×C42C22×C511yescharacteristic
194#394C4×C518yescomplemented
195#388—28no—
196C2×#95C22×C512yes—
197C2×#95C22×C512yes—
198C2×#95C22×C512yes—
199C2×#95C22×C512yes—
200C2×#95C22×C512yes—
201C2×#95C22×C511yescharacteristic
202C2×#95C22×C512yes—
203C2×#95C22×C512yes—
204C2×#95C22×C511yescharacteristic
205C23×C5—24no—
206C23×C5—24no—
207C23×C5—24no—
208C23×C5—24no—
209C23×C5—24no—
210C23×C5—24no—
211C23×C5—24no—
212C23×C5—42no—
213C23×C5—42no—
214C23×C5—42no—
215C23×C5—42no—
216C23×C5—42no—
217C23×C5—42no—
218C23×C5—42no—
219C23×C5—42no—
220C23×C5—42no—
221C23×C5—42no—
222C23×C5—42no—
223C23×C5—42no—
224C23×C5—42no—
225C23×C5—42no—
226C23×C5—42no—
227C23×C5—42no—
228C23×C5—42no—
229C23×C5—42no—
230C23×C5—42no—
231C23×C5—42no—
232C23×C5—42no—
233C23×C5—42no—
234C23×C5—42no—
235C23×C5—42no—
236C23×C5—42no—
237C23×C5—42no—
238C23×C5—42no—
239C23×C5—42no—
240C23×C5C23×C411yescharacteristic
241C23×C5C22×D411yescharacteristic
242C23×C5C2×#4111yescharacteristic
243C23×C5H(5,𝔽2)11yescharacteristic
244C23×C5H(5,𝔽2)12yes—
245C23×C5H(5,𝔽2)11yescharacteristic
246C23×C5H(5,𝔽2)12yes—
247C23×C5H(5,𝔽2)11yescharacteristic
248C23×C5H(5,𝔽2)12yes—
249C23×C52−1+411yescharacteristic
250C23×C52−1+412yes—
251C5×D4—416no—
252C2×C4×C5—22no—
253C2×C4×C5—22no—
254C2×C4×C5—22no—
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—24no—
259C2×C4×C5—24no—
260C2×C4×C5—24no—
261C2×C4×C5—22no—
262C2×C4×C5—22no—
263C2×C4×C5—24no—
264C2×C4×C5—24no—
265C2×C4×C5—24no—
266C2×C4×C5—24no—
267C2×C4×C5—24no—
268C2×C4×C5—42no—
269C2×C4×C5—42no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5—41no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5—41no—
277C2×C4×C5—42no—
278C2×C4×C5—41no—
279C2×C4×C5—41no—
280C2×C4×C5—42no—
281C2×C4×C5—42no—
282C2×C4×C5—42no—
283C2×C4×C5—42no—
284C2×C4×C5—42no—
285C2×C4×C5—42no—
286C2×C4×C5—42no—
287C2×C4×C5—42no—
288C2×C4×C5—42no—
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—42no—
296C2×C4×C5—42no—
297C2×C4×C5—42no—
298C2×C4×C5—42no—
299C2×C4×C5—42no—
300C2×C4×C5—42no—
301C2×C4×C5—42no—
302C2×C4×C5—42no—
303C2×C4×C5C4×D414yes—
304C25C23×C512yes—
305C22×D4—42no—
306C22×D4—42no—
307C22×D4—41no—
308C22×D4—41no—
309C22×D4C2×C4×C514yes—
310C23×C4C23×C512yes—
311C23×C4C23×C512yes—
312C23×C4C23×C512yes—
313C23×C4C23×C511yescharacteristic
314C23×C4C23×C512yes—
315C23×C4C23×C511yescharacteristic
316C23×C4C23×C512yes—
317C23×C4C23×C511yescharacteristic
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—41no—
321C2×C4⋊C4—41no—
322C2×C4⋊C4—41no—
323C2×C4⋊C4—41no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—41no—
328C2×C4⋊C4C2×C4×C514yes—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—42no—
334C2×C22⋊C4—42no—
335C2×C22⋊C4—42no—
336C2×C22⋊C4—42no—
337C2×C22⋊C4—42no—
338C2×C22⋊C4—42no—
339C2×C22⋊C4—42no—
340C2×C22⋊C4—42no—
341C2×C22⋊C4—42no—
342C2×C22⋊C4—42no—
343C2×C22⋊C4—42no—
344C2×C22⋊C4—42no—
345C2×C22⋊C4C2×C4×C518yes—
346C2×C22⋊C4C2×C4×C518yes—
347C2×C42—41no—
348C2×C42—41no—
349C2×C42C5×D414yescomplemented
350#95—24no—
351#95—24no—
352#95—42no—
353#95—42no—
354#95—42no—
355#95—41no—
356#95—42no—
357#95—42no—
358#95C2×C4×C514yes—
359C22×C5—44no—
360C22×C5—44no—
361C22×C5—44no—
362C22×C5—44no—
363C22×C5—44no—
364C22×C5—44no—
365C22×C5—44no—
366C22×C5—82no—
367C22×C5—82no—
368C22×C5—82no—
369C22×C5—82no—
370C22×C5—82no—
371C22×C5—82no—
372C22×C5—82no—
373C22×C5—82no—
374C22×C5C2×C4×D411yescharacteristic
375C22×C5#51812yes—
376C22×C5#51811yescharacteristic
377C22×C5#52012yes—
378C22×C5#52011yescharacteristic
379C22×C5#53411yescharacteristic
380C22×C5#53512yes—
381C22×C5#53711yescharacteristic
382C22×C5#53712yes—
383C22×C5#53811yescharacteristic
384C22×C5#53911yescharacteristic
385C22×C5#54012yes—
386C22×C5#54212yes—
387C22×C5#56011yescharacteristic
388C22×C5#56012yes—
389C22×C5#56012yes—
390C22×C5H(3,𝔽4)11yescharacteristic
391C22×C5H(3,𝔽4)11yescharacteristic
392C22×C5#56212yes—
393C22×C5#56212yes—
394C22×C5#56212yes—
395C22×C5#56311yescharacteristic
396C22×C5#56312yes—
397C4×C5—24no—
398C4×C5—44no—
399C4×C5—42no—
400C4×C5—42no—
401C4×C5—44no—
402C4×C5—82no—
403C4×C5—82no—
404C4×C5—82no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24C22×C4×C514yes—
420C24C24×C511yescharacteristic
421C2×D4—44no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—44no—
425C2×D4—44no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4C22×C4×C514yes—
472C22×C4C22×C4×C512yes—
473C22×C4C22×C4×C512yes—
474C22×C4C22×C4×C512yes—
475C22×C4C2×C5×D414yes—
476C22×C4C2×C5×D412yes—
477C22×C4C5×#4112yes—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C22⋊C4—216no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C42—44no—
489C2×C5—84no—
490C2×C5#232412yes—
491C2×C5#232511yescharacteristic
492C2×C5#237511yescharacteristic
493C2×C5#237612yes—
494C2×C5#237911yescharacteristic
495C2×C5#281312yes—
496C2×C5#281911yescharacteristic
497C2×C5#282311yescharacteristic
498C2×C5#282811yescharacteristic
499C2×C5#283211yescharacteristic
500C2×C5#283712yes—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C23×C4×C511yescharacteristic
537C23C22×C5×D411yescharacteristic
538C23C2×C5×#4111yescharacteristic
539C23C5×H(5,𝔽2)11yescharacteristic
540C23C5×H(5,𝔽2)12yes—
541C23C5×H(5,𝔽2)11yescharacteristic
542C23C5×H(5,𝔽2)12yes—
543C23C5×H(5,𝔽2)11yescharacteristic
544C23C5×H(5,𝔽2)12yes—
545C23C5×2−1+411yescharacteristic
546C23C5×2−1+412yes—
547D4—416no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4C4×C5×D414yes—
600C5#1463211yescharacteristic, direct product factor, Sylow
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22C2×C4×C5×D411yescharacteristic
617C22C5×#51812yes—
618C22C5×#51811yescharacteristic
619C22C5×#52012yes—
620C22C5×#52011yescharacteristic
621C22C5×#53411yescharacteristic
622C22C5×#53512yes—
623C22C5×#53712yes—
624C22C5×#53711yescharacteristic
625C22C5×#53811yescharacteristic
626C22C5×#53911yescharacteristic
627C22C5×#54012yes—
628C22C5×#54212yes—
629C22C5×#56012yes—
630C22C5×#56012yes—
631C22C5×#56011yescharacteristic
632C22C5×H(3,𝔽4)11yescharacteristic
633C22C5×H(3,𝔽4)11yescharacteristic
634C22C5×#56212yes—
635C22C5×#56212yes—
636C22C5×#56212yes—
637C22C5×#56311yescharacteristic
638C22C5×#56312yes—
639C4—24no—
640C4—44no—
641C4—44no—
642C4—42no—
643C4—42no—
644C4—82no—
645C4—82no—
646C4—82no—
647C2—84no—
648C2C5×#232412yes—
649C2C5×#232511yescharacteristic
650C2C5×#237511yescharacteristic
651C2C5×#237612yes—
652C2C5×#237911yescharacteristic
653C2C5×#281312yes—
654C2C5×#281911yescharacteristic
655C2C5×#282311yescharacteristic
656C2C5×#282811yescharacteristic
657C2C5×#283211yescharacteristic
658C2C5×#283712yes—
subgroupG/H
center, socleC24×C5C24
commutatorC23C23×C4×C5
Frattini, cosocleC24C24×C5
prime radicalC23×C5×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#388C2semidirect
3. C5×#1442C2semidirect
4. C5×#1441C2semidirect
5. C5×#1439C2semidirect
6. C5×#1439C2semidirect
7. C5×#1439C2semidirect
9. C5×C4⋊#95C2semidirect
22. C5×#394C4semidirect
33. #14632C5direct (N × H)
79. C2×C42×C5D4semidirect
90. C2×#388C2×C5semidirect
91. #1442C2×C5semidirect
92. #1441C2×C5semidirect
93. #1439C2×C5semidirect
94. #1439C2×C5semidirect
95. #1439C2×C5semidirect
97. C4⋊#95C2×C5semidirect
194. #394C4×C5semidirect
349. C2×C42C5×D4semidirect
600. C5#14632direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#3942
C2×C5×#3882
C5×#14422
C5×#14412
C5×#14392
C5×#14382
C5×C4⋊#952
#146325