Finite Group Databasebeta

← C2×C5×#1468 #11940093

Subgroups — C2×C5×#1468 #11940093

Order 1280.

1490 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×#123C211yesmaximal, characteristic
2C2×C5×#407C211yesmaximal, characteristic, complemented
3C22×C5×C22⋊C8C211yesmaximal, characteristic, complemented
4C5×#1468C218yesmaximal, direct product factor
5C2×C5×#324C212yesmaximal, complemented
6C2×C5×#323C212yesmaximal, complemented
7C23×C5×D4C411yescharacteristic
8C24×C4×C5C2211yescharacteristic
9C22×C5×M16—21no—
10C23×C5×C8—21no—
11C2×C5×#123—21no—
12C2×C5×#123—21no—
13C2×C5×#123—22no—
14C2×C5×#123—21no—
15C2×C5×#123—21no—
16C2×C5×#123—22no—
17C2×C5×#123C414yes—
18C2×C5×#123C2214yes—
19C22×C5×C4⋊C4C411yescharacteristic
20C22×C5×C4⋊C4C2211yescharacteristic
21C22×C5×C22⋊C4C411yescharacteristic
22C22×C5×C22⋊C4C411yescharacteristic
23C22×C5×C22⋊C4C2211yescharacteristic
24C5×#407—22no—
25C5×#407C2214yescomplemented
26C2×C5×C22⋊C8—21no—
27C2×C5×C22⋊C8—22no—
28C2×C5×C22⋊C8—21no—
29C2×C5×C22⋊C8C2212yes—
30C2×C5×C22⋊C8C2212yes—
31C2×C5×C22⋊C8C2214yescomplemented
32C5×#324C2218yescomplemented
33C5×#323C2218yescomplemented
34C2×#1468C511yesmaximal, characteristic, direct product factor, Sylow
35C25×C5C2×C411yescharacteristic
36C25×C5C2×C411yescharacteristic
37C22×C5×D4—22no—
38C22×C5×D4—21no—
39C22×C5×D4—21no—
40C22×C5×D4—44no—
41C22×C5×D4C2×C414yes—
42C23×C4×C5—44no—
43C23×C4×C5C2×C411yescharacteristic
44C23×C4×C5C2×C411yescharacteristic
45C23×C4×C5C2×C411yescharacteristic
46C23×C4×C5C2×C411yescharacteristic
47C23×C4×C5C2×C412yes—
48C23×C4×C5D411yescharacteristic
49C23×C4×C5D411yescharacteristic
50C23×C4×C5D412yes—
51C23×C4×C5D411yescharacteristic
52C23×C4×C5D411yescharacteristic
53C23×C4×C5D412yes—
54C23×C4×C5C2311yescharacteristic
55C23×C4×C5C2312yes—
56C2×C5×M16—24no—
57C2×C5×M16—44no—
58C22×C5×C8—22no—
59C22×C5×C8—22no—
60C22×C5×C8—24no—
61C22×C5×C8—44no—
62C5×#123—24no—
63C5×#123—24no—
64C5×#123—24no—
65C5×#123—24no—
66C5×#123—24no—
67C5×#123—24no—
68C5×#123—24no—
69C5×#123—24no—
70C2×C5×C4⋊C4—22no—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4—21no—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4C2×C414yes—
78C2×C5×C4⋊C4C2×C412yes—
79C2×C5×C4⋊C4C2312yes—
80C2×C5×C22⋊C4—22no—
81C2×C5×C22⋊C4—21no—
82C2×C5×C22⋊C4—21no—
83C2×C5×C22⋊C4—22no—
84C2×C5×C22⋊C4—21no—
85C2×C5×C22⋊C4—21no—
86C2×C5×C22⋊C4—21no—
87C2×C5×C22⋊C4—21no—
88C2×C5×C22⋊C4—21no—
89C2×C5×C22⋊C4—21no—
90C2×C5×C22⋊C4C2×C414yes—
91C2×C5×C22⋊C4C2×C414yes—
92C2×C5×C22⋊C4C2×C412yes—
93C2×C5×C22⋊C4C2312yes—
94C5×C22⋊C8—22no—
95C5×C22⋊C8—24no—
96C5×C22⋊C8—24no—
97C5×C22⋊C8—22no—
98C5×C22⋊C8C2314yes—
99C5×C22⋊C8C2314yes—
100C22×#123C2×C511yescharacteristic
101C2×#407C2×C511yescharacteristic, complemented
102C22×C22⋊C8C2×C511yescharacteristic, complemented
103#1468C2×C518yesdirect product factor
104C2×#324C2×C512yescomplemented
105C2×#323C2×C512yescomplemented
106C24×C5—24no—
107C24×C5—24no—
108C24×C5—21no—
109C24×C5—21no—
110C24×C5—24no—
111C24×C5—42no—
112C24×C5—42no—
113C24×C5—42no—
114C24×C5C22⋊C411yescharacteristic
115C24×C5C22⋊C412yes—
116C24×C5C22⋊C411yescharacteristic
117C24×C5C22×C412yes—
118C24×C5C22×C411yescharacteristic
119C24×C5C22×C412yes—
120C2×C5×D4—24no—
121C2×C5×D4—24no—
122C2×C5×D4—48no—
123C2×C5×D4—44no—
124C2×C5×D4—44no—
125C2×C5×D4—48no—
126C22×C4×C5—24no—
127C22×C4×C5—24no—
128C22×C4×C5—21no—
129C22×C4×C5—21no—
130C22×C4×C5—21no—
131C22×C4×C5—21no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5—24no—
135C22×C4×C5—42no—
136C22×C4×C5—44no—
137C22×C4×C5—44no—
138C22×C4×C5—42no—
139C22×C4×C5—42no—
140C22×C4×C5—44no—
141C22×C4×C5—42no—
142C22×C4×C5—44no—
143C22×C4×C5—42no—
144C22×C4×C5—42no—
145C22×C4×C5C22⋊C412yes—
146C22×C4×C5C22⋊C412yes—
147C22×C4×C5C22⋊C411yescharacteristic
148C22×C4×C5C22⋊C411yescharacteristic
149C22×C4×C5C22⋊C412yes—
150C22×C4×C5C22⋊C412yes—
151C22×C4×C5C22⋊C412yes—
152C22×C4×C5C22×C412yes—
153C22×C4×C5C22×C412yes—
154C22×C4×C5C22×C412yes—
155C22×C4×C5C22×C412yes—
156C22×C4×C5C22×C411yescharacteristic
157C22×C4×C5C2×D412yes—
158C22×C4×C5C2×D412yes—
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2×D412yes—
164C22×C4×C5C2×D412yes—
165C22×C4×C5C2411yescharacteristic
166C5×M16—48no—
167C2×C5×C8—24no—
168C2×C5×C8—24no—
169C2×C5×C8—44no—
170C2×C5×C8—48no—
171C2×C5×C8—44no—
172C5×C4⋊C4—24no—
173C5×C4⋊C4—24no—
174C5×C4⋊C4—22no—
175C5×C4⋊C4—22no—
176C5×C4⋊C4—22no—
177C5×C4⋊C4—22no—
178C5×C22⋊C4—24no—
179C5×C22⋊C4—24no—
180C5×C22⋊C4—24no—
181C5×C22⋊C4—24no—
182C5×C22⋊C4—22no—
183C5×C22⋊C4—22no—
184C5×C22⋊C4—22no—
185C5×C22⋊C4—22no—
186C23×D4C4×C511yescharacteristic
187C24×C4C22×C511yescharacteristic
188C22×M16—21no—
189C23×C8—21no—
190C2×#123—21no—
191C2×#123—21no—
192C2×#123—21no—
193C2×#123—21no—
194C2×#123—22no—
195C2×#123—22no—
196C2×#123C4×C514yes—
197C2×#123C22×C514yes—
198C22×C4⋊C4C4×C511yescharacteristic
199C22×C4⋊C4C22×C511yescharacteristic
200C22×C22⋊C4C4×C511yescharacteristic
201C22×C22⋊C4C4×C511yescharacteristic
202C22×C22⋊C4C22×C511yescharacteristic
203#407—22no—
204#407C22×C514yescomplemented
205C2×C22⋊C8—22no—
206C2×C22⋊C8—21no—
207C2×C22⋊C8—21no—
208C2×C22⋊C8C22×C512yes—
209C2×C22⋊C8C22×C512yes—
210C2×C22⋊C8C22×C514yescomplemented
211#324C22×C518yescomplemented
212#323C22×C518yescomplemented
213C23×C5—22no—
214C23×C5—22no—
215C23×C5—22no—
216C23×C5—22no—
217C23×C5—24no—
218C23×C5—24no—
219C23×C5—22no—
220C23×C5—24no—
221C23×C5—22no—
222C23×C5—24no—
223C23×C5—22no—
224C23×C5—22no—
225C23×C5—24no—
226C23×C5—42no—
227C23×C5—42no—
228C23×C5—416no—
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—88no—
238C23×C5#9914yes—
239C23×C5C2×C22⋊C412yes—
240C23×C5C2×C22⋊C411yescharacteristic
241C23×C5C2×C22⋊C412yes—
242C23×C5C2×C22⋊C411yescharacteristic
243C23×C5#13114yes—
244C23×C5C23×C411yescharacteristic
245C5×D4—48no—
246C5×D4—48no—
247C2×C4×C5—24no—
248C2×C4×C5—22no—
249C2×C4×C5—22no—
250C2×C4×C5—24no—
251C2×C4×C5—24no—
252C2×C4×C5—24no—
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—42no—
260C2×C4×C5—42no—
261C2×C4×C5—42no—
262C2×C4×C5—41no—
263C2×C4×C5—42no—
264C2×C4×C5—41no—
265C2×C4×C5—41no—
266C2×C4×C5—42no—
267C2×C4×C5—42no—
268C2×C4×C5—41no—
269C2×C4×C5—42no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5—42no—
277C2×C4×C5C2×C22⋊C411yescharacteristic
278C2×C4×C5C2×C22⋊C411yescharacteristic
279C2×C4×C5C2×C22⋊C412yes—
280C2×C4×C5C2×C22⋊C412yes—
281C2×C4×C5C22×D411yescharacteristic
282C2×C4×C5C22×D411yescharacteristic
283C5×C8—44no—
284C5×C8—44no—
285C25C2×C4×C511yescharacteristic
286C25C2×C4×C511yescharacteristic
287C22×D4—21no—
288C22×D4—22no—
289C22×D4—21no—
290C22×D4—44no—
291C22×D4C2×C4×C514yes—
292C23×C4—44no—
293C23×C4C2×C4×C511yescharacteristic
294C23×C4C2×C4×C511yescharacteristic
295C23×C4C2×C4×C511yescharacteristic
296C23×C4C2×C4×C511yescharacteristic
297C23×C4C2×C4×C512yes—
298C23×C4C5×D411yescharacteristic
299C23×C4C5×D411yescharacteristic
300C23×C4C5×D411yescharacteristic
301C23×C4C5×D411yescharacteristic
302C23×C4C5×D412yes—
303C23×C4C5×D412yes—
304C23×C4C23×C511yescharacteristic
305C23×C4C23×C512yes—
306C2×M16—24no—
307C2×M16—44no—
308C22×C8—22no—
309C22×C8—24no—
310C22×C8—22no—
311C22×C8—44no—
312#123—24no—
313#123—24no—
314#123—24no—
315#123—24no—
316#123—24no—
317#123—24no—
318#123—24no—
319#123—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4C2×C4×C514yes—
328C2×C4⋊C4C2×C4×C512yes—
329C2×C4⋊C4C23×C512yes—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4C2×C4×C514yes—
341C2×C22⋊C4C2×C4×C514yes—
342C2×C22⋊C4C2×C4×C512yes—
343C2×C22⋊C4C23×C512yes—
344C22⋊C8—22no—
345C22⋊C8—24no—
346C22⋊C8—24no—
347C22⋊C8—22no—
348C22⋊C8C23×C514yes—
349C22⋊C8C23×C514yes—
350C22×C5—22no—
351C22×C5—21no—
352C22×C5—21no—
353C22×C5—21no—
354C22×C5—21no—
355C22×C5—22no—
356C22×C5—28no—
357C22×C5—28no—
358C22×C5—22no—
359C22×C5—22no—
360C22×C5—22no—
361C22×C5—48no—
362C22×C5—48no—
363C22×C5—44no—
364C22×C5—42no—
365C22×C5—42no—
366C22×C5—42no—
367C22×C5—84no—
368C22×C5—84no—
369C22×C5—84no—
370C22×C5#40812yes—
371C22×C5#40812yes—
372C22×C5C2×#9914yes—
373C22×C5C2×#9912yes—
374C22×C5#41312yes—
375C22×C5C22×C22⋊C411yescharacteristic
376C22×C5C2×#13112yes—
377C4×C5—21no—
378C4×C5—21no—
379C4×C5—22no—
380C4×C5—21no—
381C4×C5—21no—
382C4×C5—22no—
383C4×C5—42no—
384C4×C5—42no—
385C4×C5—42no—
386C24—24no—
387C24—24no—
388C24—21no—
389C24—21no—
390C24—24no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24C5×C22⋊C411yescharacteristic
395C24C5×C22⋊C412yes—
396C24C5×C22⋊C411yescharacteristic
397C24C22×C4×C512yes—
398C24C22×C4×C511yescharacteristic
399C24C22×C4×C512yes—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—48no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—48no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4C5×C22⋊C411yescharacteristic
426C22×C4C5×C22⋊C411yescharacteristic
427C22×C4C5×C22⋊C412yes—
428C22×C4C5×C22⋊C412yes—
429C22×C4C5×C22⋊C412yes—
430C22×C4C5×C22⋊C412yes—
431C22×C4C5×C22⋊C412yes—
432C22×C4C22×C4×C512yes—
433C22×C4C22×C4×C512yes—
434C22×C4C22×C4×C512yes—
435C22×C4C22×C4×C512yes—
436C22×C4C22×C4×C511yescharacteristic
437C22×C4C2×C5×D411yescharacteristic
438C22×C4C2×C5×D411yescharacteristic
439C22×C4C2×C5×D412yes—
440C22×C4C2×C5×D412yes—
441C22×C4C2×C5×D411yescharacteristic
442C22×C4C2×C5×D411yescharacteristic
443C22×C4C2×C5×D412yes—
444C22×C4C2×C5×D412yes—
445C22×C4C24×C511yescharacteristic
446M16—48no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—44no—
450C2×C8—48no—
451C2×C8—44no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C2×C5—22no—
467C2×C5—22no—
468C2×C5—24no—
469C2×C5—42no—
470C2×C5—82no—
471C2×C5#146814yesdirect product factor
472C2×C5C2×#40811yescharacteristic
473C2×C5C22×#9911yescharacteristic
474C2×C5C2×#41311yescharacteristic
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—24no—
482C23—22no—
483C23—24no—
484C23—22no—
485C23—24no—
486C23—22no—
487C23—24no—
488C23—42no—
489C23—42no—
490C23—416no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—88no—
500C23C5×#9914yes—
501C23C2×C5×C22⋊C412yes—
502C23C2×C5×C22⋊C411yescharacteristic
503C23C2×C5×C22⋊C412yes—
504C23C2×C5×C22⋊C411yescharacteristic
505C23C5×#13114yes—
506C23C23×C4×C511yescharacteristic
507D4—48no—
508D4—48no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4C2×C5×C22⋊C412yes—
540C2×C4C2×C5×C22⋊C412yes—
541C2×C4C2×C5×C22⋊C411yescharacteristic
542C2×C4C2×C5×C22⋊C411yescharacteristic
543C2×C4C22×C5×D411yescharacteristic
544C2×C4C22×C5×D411yescharacteristic
545C8—44no—
546C8—44no—
547C5C2×#146811yescharacteristic, direct product factor, Sylow
548C22—22no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—28no—
555C22—28no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—48no—
560C22—48no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—84no—
566C22—84no—
567C22—84no—
568C22C5×#40812yes—
569C22C5×#40812yes—
570C22C2×C5×#9914yes—
571C22C2×C5×#9912yes—
572C22C5×#41312yes—
573C22C22×C5×C22⋊C411yescharacteristic
574C22C2×C5×#13112yes—
575C4—21no—
576C4—21no—
577C4—22no—
578C4—21no—
579C4—21no—
580C4—22no—
581C4—42no—
582C4—42no—
583C4—42no—
584C2—22no—
585C2—22no—
586C2—24no—
587C2—42no—
588C2—82no—
589C2C5×#146814yesdirect product factor
590C2C2×C5×#40811yescharacteristic
591C2C22×C5×#9911yescharacteristic
592C2C2×C5×#41311yescharacteristic
subgroupG/H
center, socleC23×C5C2×C22⋊C4
commutatorC23C23×C4×C5
Frattini, cosocleC22×C4C24×C5
prime radicalC23×C5×D4C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C24×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#407C2semidirect
3. C22×C5×C22⋊C8C2semidirect
4. C5×#1468C2direct (N × H)
5. C2×C5×#324C2semidirect
6. C2×C5×#323C2semidirect
25. C5×#407C22semidirect
31. C2×C5×C22⋊C8C22semidirect
32. C5×#324C22semidirect
33. C5×#323C22semidirect
34. C2×#1468C5direct (N × H)
101. C2×#407C2×C5semidirect
102. C22×C22⋊C8C2×C5semidirect
103. #1468C2×C5direct (N × H)
104. C2×#324C2×C5semidirect
105. C2×#323C2×C5semidirect
204. #407C22×C5semidirect
210. C2×C22⋊C8C22×C5semidirect
211. #324C22×C5semidirect
212. #323C22×C5semidirect
471. C2×C5#1468direct (N × H)
547. C5C2×#1468direct (N × H)
589. C2C5×#1468direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×#1232
C2×C5×#4072
C22×C5×C22⋊C82
C5×#14682
C2×C5×#3242
C2×C5×#3232
C2×#14685