Finite Group Databasebeta

← C5×#16518 #11939238

Subgroups — C5×#16518 #11939238

Order 1280.

1430 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×#397C211yesmaximal, characteristic
2C2×C5×#396C212yesmaximal
3C2×C5×#389C211yesmaximal, characteristic, complemented
4C2×C5×#388C211yesmaximal, characteristic
5C2×C5×#382C211yesmaximal, characteristic, complemented
6C5×#1445C211yesmaximal, characteristic, complemented
7C5×#1444C211yesmaximal, characteristic, complemented
8C5×#1441C212yesmaximal, complemented
9C5×#1441C212yesmaximal, complemented
10C5×#1440C212yesmaximal
11C5×#1439C211yesmaximal, characteristic
12C22×C5×C4⋊C4C2212yes—
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C5×C4⋊C4C2211yescharacteristic
15C22×C5×C4⋊C4C2212yes—
16C22×C5×C4⋊C4C2212yes—
17C22×C5×C4⋊C4C2211yescharacteristic
18C22×C5×C22⋊C4C2211yescharacteristic
19C22×C5×C22⋊C4C2212yes—
20C22×C5×C22⋊C4C2212yes—
21C22×C5×C22⋊C4C2211yescharacteristic
22C22×C5×C22⋊C4C2211yescharacteristic
23C22×C42×C5C2211yescharacteristic
24C5×#397—24no—
25C5×#396—28no—
26C5×#389—24no—
27C5×#388—24no—
28C5×#382—24no—
29C2×C5×#95C2211yescharacteristic
30C2×C5×#95C2212yes—
31C2×C5×#95C2211yescharacteristic
32C2×C5×#95C2211yescharacteristic
33C2×C5×#95C2212yes—
34C2×C5×#95C2212yes—
35C2×C5×#95C2212yes—
36C2×C5×#95C2212yes—
37C2×C5×#95C2212yes—
38C2×C5×#95C2212yes—
39C2×C5×#95C2211yescharacteristic
40#16518C511yesmaximal, characteristic, direct product factor, Sylow
41C25×C5C2311yescharacteristic
42C23×C4×C5C2311yescharacteristic
43C23×C4×C5C2311yescharacteristic
44C23×C4×C5C2312yes—
45C23×C4×C5C2312yes—
46C23×C4×C5C2311yescharacteristic
47C23×C4×C5C2312yes—
48C23×C4×C5C2311yescharacteristic
49C23×C4×C5C2312yes—
50C23×C4×C5C2311yescharacteristic
51C23×C4×C5C2311yescharacteristic
52C2×C5×C4⋊C4—22no—
53C2×C5×C4⋊C4—24no—
54C2×C5×C4⋊C4—24no—
55C2×C5×C4⋊C4—22no—
56C2×C5×C4⋊C4—24no—
57C2×C5×C4⋊C4—22no—
58C2×C5×C4⋊C4—24no—
59C2×C5×C4⋊C4—24no—
60C2×C5×C4⋊C4—24no—
61C2×C5×C4⋊C4—22no—
62C2×C5×C4⋊C4—24no—
63C2×C5×C4⋊C4—24no—
64C2×C5×C4⋊C4—22no—
65C2×C5×C4⋊C4—42no—
66C2×C5×C4⋊C4—42no—
67C2×C5×C4⋊C4—42no—
68C2×C5×C22⋊C4—22no—
69C2×C5×C22⋊C4—22no—
70C2×C5×C22⋊C4—24no—
71C2×C5×C22⋊C4—24no—
72C2×C5×C22⋊C4—22no—
73C2×C5×C22⋊C4—22no—
74C2×C5×C22⋊C4—24no—
75C2×C5×C22⋊C4—22no—
76C2×C5×C22⋊C4—24no—
77C2×C5×C22⋊C4—22no—
78C2×C5×C22⋊C4—42no—
79C2×C5×C22⋊C4—42no—
80C2×C5×C22⋊C4—42no—
81C2×C5×C22⋊C4D414yes—
82C2×C42×C5—22no—
83C2×C42×C5—22no—
84C2×C42×C5D414yes—
85C5×#95—24no—
86C5×#95—22no—
87C5×#95—22no—
88C5×#95—22no—
89C5×#95—24no—
90C5×#95—24no—
91C5×#95—24no—
92C5×#95—42no—
93C5×#95—42no—
94C5×#95—42no—
95C5×#95D414yes—
96C2×#397C2×C511yescharacteristic
97C2×#396C2×C512yes—
98C2×#389C2×C511yescharacteristic, complemented
99C2×#388C2×C511yescharacteristic
100C2×#382C2×C511yescharacteristic, complemented
101#1445C2×C511yescharacteristic, complemented
102#1444C2×C511yescharacteristic, complemented
103#1441C2×C512yescomplemented
104#1441C2×C512yescomplemented
105#1440C2×C512yes—
106#1439C2×C511yescharacteristic
107C24×C5—21no—
108C24×C5—21no—
109C24×C5—21no—
110C24×C5—22no—
111C24×C5—21no—
112C24×C5—21no—
113C24×C5—22no—
114C24×C5—22no—
115C24×C5—22no—
116C24×C5—21no—
117C24×C5C2×D412yes—
118C24×C5C2411yescharacteristic
119C22×C4×C5—21no—
120C22×C4×C5—21no—
121C22×C4×C5—22no—
122C22×C4×C5—21no—
123C22×C4×C5—22no—
124C22×C4×C5—21no—
125C22×C4×C5—21no—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—21no—
129C22×C4×C5—22no—
130C22×C4×C5—21no—
131C22×C4×C5—21no—
132C22×C4×C5—22no—
133C22×C4×C5—22no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
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—21no—
146C22×C4×C5—22no—
147C22×C4×C5—22no—
148C22×C4×C5—22no—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—22no—
152C22×C4×C5—22no—
153C22×C4×C5—21no—
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—21no—
163C22×C4×C5—22no—
164C22×C4×C5—22no—
165C22×C4×C5—22no—
166C22×C4×C5—22no—
167C22×C4×C5—22no—
168C22×C4×C5—21no—
169C22×C4×C5C2×D412yes—
170C22×C4×C5C2×D414yes—
171C22×C4×C5C2×D412yes—
172C22×C4×C5C2×D412yes—
173C22×C4×C5C2×D412yes—
174C22×C4×C5C2×D412yes—
175C22×C4×C5C2×D412yes—
176C22×C4×C5#4114yes—
177C22×C4×C5#4112yes—
178C22×C4×C5#4112yes—
179C5×C4⋊C4—28no—
180C5×C4⋊C4—48no—
181C5×C4⋊C4—48no—
182C5×C4⋊C4—44no—
183C5×C4⋊C4—48no—
184C5×C4⋊C4—44no—
185C5×C22⋊C4—28no—
186C5×C22⋊C4—48no—
187C5×C22⋊C4—48no—
188C5×C22⋊C4—44no—
189C5×C22⋊C4—44no—
190C42×C5—28no—
191C22×C4⋊C4C22×C512yes—
192C22×C4⋊C4C22×C511yescharacteristic
193C22×C4⋊C4C22×C511yescharacteristic
194C22×C4⋊C4C22×C512yes—
195C22×C4⋊C4C22×C512yes—
196C22×C4⋊C4C22×C511yescharacteristic
197C22×C22⋊C4C22×C511yescharacteristic
198C22×C22⋊C4C22×C512yes—
199C22×C22⋊C4C22×C512yes—
200C22×C22⋊C4C22×C511yescharacteristic
201C22×C22⋊C4C22×C511yescharacteristic
202C22×C42C22×C511yescharacteristic
203#397—24no—
204#396—28no—
205#389—24no—
206#388—24no—
207#382—24no—
208C2×#95C22×C512yes—
209C2×#95C22×C512yes—
210C2×#95C22×C511yescharacteristic
211C2×#95C22×C512yes—
212C2×#95C22×C512yes—
213C2×#95C22×C511yescharacteristic
214C2×#95C22×C511yescharacteristic
215C2×#95C22×C512yes—
216C2×#95C22×C512yes—
217C2×#95C22×C512yes—
218C2×#95C22×C511yescharacteristic
219C23×C5—22no—
220C23×C5—24no—
221C23×C5—22no—
222C23×C5—24no—
223C23×C5—22no—
224C23×C5—41no—
225C23×C5—41no—
226C23×C5—41no—
227C23×C5—41no—
228C23×C5—41no—
229C23×C5—41no—
230C23×C5—42no—
231C23×C5—42no—
232C23×C5—42no—
233C23×C5—42no—
234C23×C5—42no—
235C23×C5—42no—
236C23×C5—41no—
237C23×C5—42no—
238C23×C5—41no—
239C23×C5—42no—
240C23×C5—42no—
241C23×C5—42no—
242C23×C5C22×D411yescharacteristic
243C23×C5C22×D411yescharacteristic
244C23×C5C22×D411yescharacteristic
245C23×C5C2×#4111yescharacteristic
246C23×C5C2×#4111yescharacteristic
247C23×C5C2×#4112yes—
248C23×C5H(5,𝔽2)12yes—
249C23×C5H(5,𝔽2)11yescharacteristic
250C23×C52−1+412yes—
251C23×C52−1+412yes—
252C23×C52−1+411yescharacteristic
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—22no—
260C2×C4×C5—22no—
261C2×C4×C5—24no—
262C2×C4×C5—22no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—24no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—24no—
269C2×C4×C5—24no—
270C2×C4×C5—24no—
271C2×C4×C5—24no—
272C2×C4×C5—24no—
273C2×C4×C5—24no—
274C2×C4×C5—42no—
275C2×C4×C5—41no—
276C2×C4×C5—41no—
277C2×C4×C5—42no—
278C2×C4×C5—42no—
279C2×C4×C5—42no—
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—41no—
288C2×C4×C5—42no—
289C2×C4×C5—42no—
290C2×C4×C5—41no—
291C2×C4×C5—41no—
292C2×C4×C5—41no—
293C2×C4×C5—41no—
294C2×C4×C5—42no—
295C2×C4×C5—41no—
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×C5—42no—
304C2×C4×C5—42no—
305C2×C4×C5—42no—
306C2×C4×C5—42no—
307C2×C4×C5—42no—
308C2×C4×C5—42no—
309C2×C4×C5—42no—
310C2×C4×C5—42no—
311C2×C4×C5#12114yes—
312C2×C4×C5#12114yes—
313C25C23×C511yescharacteristic
314C23×C4C23×C512yes—
315C23×C4C23×C511yescharacteristic
316C23×C4C23×C512yes—
317C23×C4C23×C511yescharacteristic
318C23×C4C23×C511yescharacteristic
319C23×C4C23×C512yes—
320C23×C4C23×C511yescharacteristic
321C23×C4C23×C512yes—
322C23×C4C23×C511yescharacteristic
323C23×C4C23×C511yescharacteristic
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4—42no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—42no—
351C2×C22⋊C4—42no—
352C2×C22⋊C4—42no—
353C2×C22⋊C4C5×D414yes—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42C5×D414yes—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—42no—
365#95—42no—
366#95—42no—
367#95C5×D414yes—
368C22×C5—42no—
369C22×C5—42no—
370C22×C5—44no—
371C22×C5—44no—
372C22×C5—42no—
373C22×C5—81no—
374C22×C5—81no—
375C22×C5—81no—
376C22×C5—81no—
377C22×C5—82no—
378C22×C5—82no—
379C22×C5C2×#12111yescharacteristic
380C22×C5C2×#12111yescharacteristic
381C22×C5#52511yescharacteristic
382C22×C5#53511yescharacteristic
383C22×C5#53611yescharacteristic
384C22×C5#53611yescharacteristic
385C22×C5#53612yes—
386C22×C5#53711yescharacteristic
387C22×C5#53712yes—
388C22×C5#54112yes—
389C22×C5#54112yes—
390C22×C5#54212yes—
391C22×C5#54212yes—
392C22×C5D4211yescharacteristic
393C22×C5#54612yes—
394C22×C5#54712yes—
395C22×C5#54711yescharacteristic
396C22×C5#54811yescharacteristic
397C22×C5#55212yes—
398C22×C5#55311yescharacteristic
399C22×C5#55312yes—
400C22×C5#56212yes—
401C22×C5#56312yes—
402C4×C5—24no—
403C4×C5—24no—
404C4×C5—42no—
405C4×C5—44no—
406C4×C5—42no—
407C4×C5—44no—
408C4×C5—81no—
409C4×C5—81no—
410C4×C5—82no—
411C4×C5—82no—
412C24—21no—
413C24—21no—
414C24—21no—
415C24—22no—
416C24—21no—
417C24—21no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—21no—
422C24C2×C5×D412yes—
423C24C24×C511yescharacteristic
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
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—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
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×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4C2×C5×D414yes—
475C22×C4C2×C5×D412yes—
476C22×C4C2×C5×D412yes—
477C22×C4C2×C5×D412yes—
478C22×C4C2×C5×D412yes—
479C22×C4C2×C5×D412yes—
480C22×C4C2×C5×D412yes—
481C22×C4C5×#4114yes—
482C22×C4C5×#4112yes—
483C22×C4C5×#4112yes—
484C4⋊C4—28no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C22⋊C4—28no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C42—28no—
496C2×C5—82no—
497C2×C5#241311yescharacteristic
498C2×C5#241611yescharacteristic
499C2×C5#253511yescharacteristic
500C2×C5#262011yescharacteristic
501C2×C5#262111yescharacteristic
502C2×C5#263711yescharacteristic
503C2×C5#267212yes—
504C2×C5#269812yes—
505C2×C5#272512yes—
506C2×C5#277212yes—
507C2×C5#282011yescharacteristic
508C23—22no—
509C23—24no—
510C23—22no—
511C23—24no—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23C22×C5×D411yescharacteristic
532C23C22×C5×D411yescharacteristic
533C23C22×C5×D411yescharacteristic
534C23C2×C5×#4111yescharacteristic
535C23C2×C5×#4111yescharacteristic
536C23C2×C5×#4112yes—
537C23C5×H(5,𝔽2)12yes—
538C23C5×H(5,𝔽2)11yescharacteristic
539C23C5×2−1+412yes—
540C23C5×2−1+412yes—
541C23C5×2−1+411yescharacteristic
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
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×C4—42no—
600C2×C4C5×#12114yes—
601C2×C4C5×#12114yes—
602C5#1651811yescharacteristic, direct product factor, Sylow
603C22—42no—
604C22—42no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22—82no—
609C22—82no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22C2×C5×#12111yescharacteristic
615C22C2×C5×#12111yescharacteristic
616C22C5×#52511yescharacteristic
617C22C5×#53511yescharacteristic
618C22C5×#53611yescharacteristic
619C22C5×#53611yescharacteristic
620C22C5×#53612yes—
621C22C5×#53712yes—
622C22C5×#53711yescharacteristic
623C22C5×#54112yes—
624C22C5×#54112yes—
625C22C5×#54212yes—
626C22C5×#54212yes—
627C22C5×D4211yescharacteristic
628C22C5×#54612yes—
629C22C5×#54712yes—
630C22C5×#54711yescharacteristic
631C22C5×#54811yescharacteristic
632C22C5×#55212yes—
633C22C5×#55312yes—
634C22C5×#55311yescharacteristic
635C22C5×#56212yes—
636C22C5×#56312yes—
637C4—24no—
638C4—24no—
639C4—42no—
640C4—44no—
641C4—42no—
642C4—44no—
643C4—81no—
644C4—81no—
645C4—82no—
646C4—82no—
647C2—82no—
648C2C5×#241311yescharacteristic
649C2C5×#241611yescharacteristic
650C2C5×#253511yescharacteristic
651C2C5×#262011yescharacteristic
652C2C5×#262111yescharacteristic
653C2C5×#263711yescharacteristic
654C2C5×#267212yes—
655C2C5×#269812yes—
656C2C5×#272512yes—
657C2C5×#277212yes—
658C2C5×#282011yescharacteristic
subgroupG/H
center, socleC24×C5C24
commutator, Frattini, cosocleC24C24×C5
prime radicalC25×C5C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C5 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C5×#389C2semidirect
5. C2×C5×#382C2semidirect
6. C5×#1445C2semidirect
7. C5×#1444C2semidirect
8. C5×#1441C2semidirect
9. C5×#1441C2semidirect
40. #16518C5direct (N × H)
98. C2×#389C2×C5semidirect
100. C2×#382C2×C5semidirect
101. #1445C2×C5semidirect
102. #1444C2×C5semidirect
103. #1441C2×C5semidirect
104. #1441C2×C5semidirect
602. C5#16518direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#3972
C2×C5×#3962
C2×C5×#3892
C2×C5×#3882
C2×C5×#3822
C5×#14452
C5×#14442
C5×#14412
C5×#14402
C5×#14392
#165185