Finite Group Databasebeta

← C2×C5×#1860 #11943582

Subgroups — C2×C5×#1860 #11943582

Order 1280.

1478 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
1C2×C5×#520C211yesmaximal, characteristic
2C2×C5×#428C211yesmaximal, characteristic, complemented
3C22×C5×#104C211yesmaximal, characteristic
4C2×C5×#417C211yesmaximal, characteristic
5C2×C5×#382C211yesmaximal, characteristic, complemented
6C5×#1860C218yesmaximal, direct product factor
7C2×C5×#343C211yesmaximal, characteristic, complemented
8C2×C5×#339C211yesmaximal, characteristic, complemented
9C22×C5×#41C2211yescharacteristic
10C22×C5×M16C2211yescharacteristic
11C22×C5×M16C2211yescharacteristic
12C5×#520—22no—
13C5×#520—22no—
14C5×#520—22no—
15C5×#520C2214yes—
16C2×C4×C5×Q8—21no—
17C2×C4×C5×D4—21no—
18C2×C4×C5×D4—22no—
19C2×C5×#117—21no—
20C2×C5×#117C2211yescharacteristic
21C22×C5×C4⋊C4—21no—
22C22×C5×C4⋊C4C2211yescharacteristic
23C22×C5×C4⋊C4C2211yescharacteristic
24C22×C42×C5C2211yescharacteristic
25C5×#428—22no—
26C5×#428C2214yescomplemented
27C2×C5×#107—21no—
28C2×C5×#106—21no—
29C2×C5×#104C418yescomplemented
30C2×C5×#104C2214yes—
31C5×#417—22no—
32C5×#417C2214yes—
33C2×C5×#103—21no—
34C2×C5×#102—21no—
35C5×#382—22no—
36C5×#382C2214yescomplemented
37C2×C5×#95—22no—
38C5×#343C2214yescomplemented
39C5×#339C2214yescomplemented
40C2×#1860C511yesmaximal, characteristic, direct product factor, Sylow
41C2×C5×#41—21no—
42C2×C5×#41—21no—
43C2×C5×#41—22no—
44C2×C5×#41—22no—
45C2×C5×#41C2×C414yes—
46C2×C5×#41D412yes—
47C2×C5×#41Q814yes—
48C2×C5×#41C2312yes—
49C22×C5×Q8D411yescharacteristic
50C22×C5×D4—21no—
51C22×C5×D4D411yescharacteristic
52C23×C4×C5—21no—
53C23×C4×C5—21no—
54C23×C4×C5—22no—
55C23×C4×C5—21no—
56C23×C4×C5D412yes—
57C23×C4×C5C2311yescharacteristic
58C2×C5×M16—21no—
59C2×C5×M16—22no—
60C2×C5×M16—21no—
61C2×C5×M16—22no—
62C2×C5×M16—22no—
63C2×C5×M16C2×C414yescomplemented
64C2×C5×M16C2312yes—
65C2×C5×M16C2312yes—
66C22×C5×C8—21no—
67C22×C5×C8—21no—
68C4×C5×Q8—24no—
69C4×C5×Q8—24no—
70C4×C5×D4—28no—
71C4×C5×D4—24no—
72C4×C5×D4—24no—
73C4×C5×D4—28no—
74C5×#117—24no—
75C5×#117—21no—
76C5×#117—21no—
77C5×#117—24no—
78C5×#117D412yes—
79C5×#117C2312yes—
80C2×C5×C4⋊C4—22no—
81C2×C5×C4⋊C4—24no—
82C2×C5×C4⋊C4—21no—
83C2×C5×C4⋊C4—21no—
84C2×C5×C4⋊C4—21no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4—24no—
87C2×C5×C4⋊C4—21no—
88C2×C5×C4⋊C4—21no—
89C2×C5×C4⋊C4—21no—
90C2×C5×C4⋊C4—21no—
91C2×C5×C4⋊C4—21no—
92C2×C5×C4⋊C4D412yes—
93C2×C5×C4⋊C4D411yescharacteristic
94C2×C5×C4⋊C4D411yescharacteristic
95C2×C5×C4⋊C4C2312yes—
96C2×C5×C4⋊C4C2312yes—
97C2×C5×C22⋊C4—22no—
98C2×C5×C22⋊C4—21no—
99C2×C42×C5—22no—
100C2×C42×C5—21no—
101C2×C42×C5—21no—
102C2×C42×C5—22no—
103C2×C42×C5C2×C414yescomplemented
104C2×C42×C5C2312yes—
105C5×#107—24no—
106C5×#106—24no—
107C5×#104C2×C4116yescomplemented
108C5×#103—24no—
109C5×#102—24no—
110C5×#95—28no—
111C2×#520C2×C511yescharacteristic
112C2×#428C2×C511yescharacteristic, complemented
113C22×#104C2×C511yescharacteristic
114C2×#417C2×C511yescharacteristic
115C2×#382C2×C511yescharacteristic, complemented
116#1860C2×C518yesdirect product factor
117C2×#343C2×C511yescharacteristic, complemented
118C2×#339C2×C511yescharacteristic, complemented
119C24×C5—21no—
120C24×C5C2×D411yescharacteristic
121C5×#41—24no—
122C5×#41—24no—
123C5×#41—24no—
124C5×#41—24no—
125C5×#41—24no—
126C5×#41—22no—
127C5×#41—22no—
128C5×#41C4⋊C418yes—
129C5×#41C22×C414yes—
130C5×#41C2×Q814yes—
131C2×C5×Q8—21no—
132C2×C5×Q8—22no—
133C2×C5×Q8C4⋊C414yes—
134C2×C5×Q8C2×D412yes—
135C2×C5×D4—24no—
136C2×C5×D4—22no—
137C2×C5×D4—24no—
138C2×C5×D4—22no—
139C2×C5×D4—21no—
140C2×C5×D4—22no—
141C2×C5×D4C4⋊C414yes—
142C2×C5×D4C2×D412yes—
143C22×C4×C5—21no—
144C22×C4×C5—24no—
145C22×C4×C5—22no—
146C22×C4×C5—24no—
147C22×C4×C5—22no—
148C22×C4×C5—24no—
149C22×C4×C5—21no—
150C22×C4×C5—22no—
151C22×C4×C5—21no—
152C22×C4×C5—22no—
153C22×C4×C5—28no—
154C22×C4×C5—22no—
155C22×C4×C5—22no—
156C22×C4×C5—42no—
157C22×C4×C5—42no—
158C22×C4×C5—44no—
159C22×C4×C5—44no—
160C22×C4×C5—41no—
161C22×C4×C5—41no—
162C22×C4×C5—84no—
163C22×C4×C5C22×C412yes—
164C22×C4×C5C2×D414yes—
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5C2×Q812yes—
171C22×C4×C5#4112yes—
172C22×C4×C5#4112yes—
173C22×C4×C5C2411yescharacteristic
174C5×M16—24no—
175C5×M16—22no—
176C5×M16—22no—
177C5×M16—22no—
178C5×M16—42no—
179C5×M16C22×C414yescomplemented
180C2×C5×C8—24no—
181C2×C5×C8—22no—
182C2×C5×C8—22no—
183C2×C5×C8—42no—
184C5×C4⋊C4—28no—
185C5×C4⋊C4—22no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—22no—
188C5×C4⋊C4—21no—
189C5×C4⋊C4—22no—
190C5×C4⋊C4—22no—
191C5×C4⋊C4—22no—
192C5×C4⋊C4—22no—
193C5×C4⋊C4—41no—
194C5×C4⋊C4—41no—
195C5×C4⋊C4C2×D412yes—
196C5×C4⋊C4C2×D412yes—
197C5×C22⋊C4—28no—
198C5×C22⋊C4—22no—
199C5×C22⋊C4—22no—
200C42×C5—28no—
201C42×C5—24no—
202C42×C5—22no—
203C42×C5—22no—
204C42×C5—22no—
205C42×C5C22×C414yescomplemented
206C22×#41C22×C511yescharacteristic
207C22×M16C22×C511yescharacteristic
208C22×M16C22×C511yescharacteristic
209#520—22no—
210#520—22no—
211#520—22no—
212#520C22×C514yes—
213C2×C4×Q8—21no—
214C2×C4×D4—21no—
215C2×C4×D4—22no—
216C2×#117—21no—
217C2×#117C22×C511yescharacteristic
218C22×C4⋊C4—21no—
219C22×C4⋊C4C22×C511yescharacteristic
220C22×C4⋊C4C22×C511yescharacteristic
221C22×C42C22×C511yescharacteristic
222#428—22no—
223#428C22×C514yescomplemented
224C2×#107—21no—
225C2×#106—21no—
226C2×#104C4×C518yescomplemented
227C2×#104C22×C514yes—
228#417—22no—
229#417C22×C514yes—
230C2×#103—21no—
231C2×#102—21no—
232#382—22no—
233#382C22×C514yescomplemented
234C2×#95—22no—
235#343C22×C514yescomplemented
236#339C22×C514yescomplemented
237C23×C5—22no—
238C23×C5—22no—
239C23×C5—22no—
240C23×C5—24no—
241C23×C5—42no—
242C23×C5—42no—
243C23×C5C4×D412yes—
244C23×C5#12011yescharacteristic
245C23×C5#12011yescharacteristic
246C23×C5C22⋊Q812yes—
247C23×C5C22×D411yescharacteristic
248C5×Q8—22no—
249C5×Q8—24no—
250C5×Q8C2×C4⋊C414yes—
251C5×D4—24no—
252C5×D4—24no—
253C5×D4—28no—
254C5×D4—22no—
255C5×D4—24no—
256C5×D4C2×C4⋊C414yes—
257C2×C4×C5—22no—
258C2×C4×C5—24no—
259C2×C4×C5—24no—
260C2×C4×C5—24no—
261C2×C4×C5—28no—
262C2×C4×C5—24no—
263C2×C4×C5—22no—
264C2×C4×C5—24no—
265C2×C4×C5—22no—
266C2×C4×C5—48no—
267C2×C4×C5—42no—
268C2×C4×C5—42no—
269C2×C4×C5—42no—
270C2×C4×C5—44no—
271C2×C4×C5—42no—
272C2×C4×C5—44no—
273C2×C4×C5—44no—
274C2×C4×C5—42no—
275C2×C4×C5—82no—
276C2×C4×C5—88no—
277C2×C4×C5—82no—
278C2×C4×C5—82no—
279C2×C4×C5C2×C4⋊C412yes—
280C2×C4×C5C2×C4⋊C412yes—
281C2×C4×C5C4×D412yes—
282C2×C4×C5C4×D412yes—
283C2×C4×C5C4×D412yes—
284C2×C4×C5#12011yescharacteristic
285C2×C4×C5#12011yescharacteristic
286C2×C4×C5C22⋊Q812yes—
287C2×C4×C5C22⋊Q812yes—
288C2×C4×C5C22⋊Q812yes—
289C2×C4×C5#12312yes—
290C2×C4×C5#12312yes—
291C2×C4×C5C23×C411yescharacteristic
292C2×C4×C5C22×D411yescharacteristic
293C2×C4×C5C22×D411yescharacteristic
294C2×C4×C5C22×Q811yescharacteristic
295C2×C4×C5C2×#4111yescharacteristic
296C2×C4×C5C2×#4111yescharacteristic
297C5×C8—24no—
298C5×C8—42no—
299C2×#41—21no—
300C2×#41—21no—
301C2×#41—22no—
302C2×#41—22no—
303C2×#41C2×C4×C514yes—
304C2×#41C5×D412yes—
305C2×#41C5×Q814yes—
306C2×#41C23×C512yes—
307C22×Q8C5×D411yescharacteristic
308C22×D4—21no—
309C22×D4C5×D411yescharacteristic
310C23×C4—21no—
311C23×C4—21no—
312C23×C4—22no—
313C23×C4—21no—
314C23×C4C5×D412yes—
315C23×C4C23×C511yescharacteristic
316C2×M16—21no—
317C2×M16—22no—
318C2×M16—21no—
319C2×M16—22no—
320C2×M16—22no—
321C2×M16C2×C4×C514yescomplemented
322C2×M16C23×C512yes—
323C2×M16C23×C512yes—
324C22×C8—21no—
325C22×C8—21no—
326C4×Q8—24no—
327C4×Q8—24no—
328C4×D4—28no—
329C4×D4—24no—
330C4×D4—24no—
331C4×D4—28no—
332#117—24no—
333#117—21no—
334#117—21no—
335#117—24no—
336#117C5×D412yes—
337#117C23×C512yes—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4C5×D412yes—
351C2×C4⋊C4C5×D411yescharacteristic
352C2×C4⋊C4C5×D411yescharacteristic
353C2×C4⋊C4C23×C512yes—
354C2×C4⋊C4C23×C512yes—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42C2×C4×C514yescomplemented
362C2×C42C23×C512yes—
363#107—24no—
364#106—24no—
365#104C2×C4×C5116yescomplemented
366#103—24no—
367#102—24no—
368#95—28no—
369C22×C5—24no—
370C22×C5—24no—
371C22×C5—21no—
372C22×C5—21no—
373C22×C5—21no—
374C22×C5—21no—
375C22×C5—24no—
376C22×C5—48no—
377C22×C5—42no—
378C22×C5—42no—
379C22×C5#38512yes—
380C22×C5#38512yes—
381C22×C5#45412yes—
382C22×C5#45512yes—
383C22×C5C2×C4×D411yescharacteristic
384C22×C5C2×#12011yescharacteristic
385C22×C5C2×C22⋊Q811yescharacteristic
386C4×C5—24no—
387C4×C5—44no—
388C4×C5—44no—
389C4×C5—42no—
390C4×C5—42no—
391C4×C5—82no—
392C4×C5#38512yes—
393C4×C5#38512yes—
394C4×C5C22×C4⋊C411yescharacteristic
395C4×C5C2×C4×D411yescharacteristic
396C4×C5C2×C22⋊Q811yescharacteristic
397C4×C5C2×#12311yescharacteristic
398C24—21no—
399C24C2×C5×D411yescharacteristic
400#41—24no—
401#41—24no—
402#41—24no—
403#41—24no—
404#41—24no—
405#41—22no—
406#41—22no—
407#41C5×C4⋊C418yes—
408#41C22×C4×C514yes—
409#41C2×C5×Q814yes—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8C5×C4⋊C414yes—
413C2×Q8C2×C5×D412yes—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4C5×C4⋊C414yes—
421C2×D4C2×C5×D412yes—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—84no—
442C22×C4C22×C4×C512yes—
443C22×C4C2×C5×D414yes—
444C22×C4C2×C5×D411yescharacteristic
445C22×C4C2×C5×D411yescharacteristic
446C22×C4C2×C5×D411yescharacteristic
447C22×C4C2×C5×D411yescharacteristic
448C22×C4C2×C5×D411yescharacteristic
449C22×C4C2×C5×Q812yes—
450C22×C4C5×#4112yes—
451C22×C4C5×#4112yes—
452C22×C4C24×C511yescharacteristic
453M16—24no—
454M16—22no—
455M16—22no—
456M16—22no—
457M16—42no—
458M16C22×C4×C514yescomplemented
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—42no—
463C4⋊C4—28no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4C2×C5×D412yes—
475C4⋊C4C2×C5×D412yes—
476C22⋊C4—28no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C42—28no—
480C42—24no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42C22×C4×C514yescomplemented
485C2×C5—21no—
486C2×C5—21no—
487C2×C5—22no—
488C2×C5—44no—
489C2×C5#186014yesdirect product factor
490C2×C5C2×#38511yescharacteristic
491C2×C5C2×#45511yescharacteristic
492C2×C5C2×#45411yescharacteristic
493C23—22no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—42no—
498C23—42no—
499C23C4×C5×D412yes—
500C23C5×#12011yescharacteristic
501C23C5×#12011yescharacteristic
502C23C5×C22⋊Q812yes—
503C23C22×C5×D411yescharacteristic
504Q8—24no—
505Q8—22no—
506Q8C2×C5×C4⋊C414yes—
507D4—28no—
508D4—24no—
509D4—24no—
510D4—24no—
511D4—22no—
512D4C2×C5×C4⋊C414yes—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—48no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—82no—
532C2×C4—88no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4C2×C5×C4⋊C412yes—
536C2×C4C2×C5×C4⋊C412yes—
537C2×C4C4×C5×D412yes—
538C2×C4C4×C5×D412yes—
539C2×C4C4×C5×D412yes—
540C2×C4C5×#12011yescharacteristic
541C2×C4C5×#12011yescharacteristic
542C2×C4C5×C22⋊Q812yes—
543C2×C4C5×C22⋊Q812yes—
544C2×C4C5×C22⋊Q812yes—
545C2×C4C5×#12312yes—
546C2×C4C5×#12312yes—
547C2×C4C23×C4×C511yescharacteristic
548C2×C4C22×C5×D411yescharacteristic
549C2×C4C22×C5×D411yescharacteristic
550C2×C4C22×C5×Q811yescharacteristic
551C2×C4C2×C5×#4111yescharacteristic
552C2×C4C2×C5×#4111yescharacteristic
553C8—24no—
554C8—42no—
555C5C2×#186011yescharacteristic, direct product factor, Sylow
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—48no—
566C22C5×#38512yes—
567C22C5×#38512yes—
568C22C5×#45412yes—
569C22C5×#45512yes—
570C22C2×C4×C5×D411yescharacteristic
571C22C2×C5×#12011yescharacteristic
572C22C2×C5×C22⋊Q811yescharacteristic
573C4—24no—
574C4—44no—
575C4—44no—
576C4—42no—
577C4—42no—
578C4—82no—
579C4C5×#38512yes—
580C4C5×#38512yes—
581C4C22×C5×C4⋊C411yescharacteristic
582C4C2×C4×C5×D411yescharacteristic
583C4C2×C5×C22⋊Q811yescharacteristic
584C4C2×C5×#12311yescharacteristic
585C2—21no—
586C2—22no—
587C2—21no—
588C2—44no—
589C2C5×#186014yesdirect product factor
590C2C2×C5×#38511yescharacteristic
591C2C2×C5×#45511yescharacteristic
592C2C2×C5×#45411yescharacteristic
subgroupG/H
center, socleC23×C5#120
commutatorC2×C4C23×C4×C5
Frattini, cosocleC22×C4C24×C5
prime radicalC22×C5×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C23×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#428C2semidirect
5. C2×C5×#382C2semidirect
6. C5×#1860C2direct (N × H)
7. C2×C5×#343C2semidirect
8. C2×C5×#339C2semidirect
26. C5×#428C22semidirect
29. C2×C5×#104C4semidirect
36. C5×#382C22semidirect
38. C5×#343C22semidirect
39. C5×#339C22semidirect
40. C2×#1860C5direct (N × H)
63. C2×C5×M16C2×C4semidirect
103. C2×C42×C5C2×C4semidirect
107. C5×#104C2×C4semidirect
112. C2×#428C2×C5semidirect
115. C2×#382C2×C5semidirect
116. #1860C2×C5direct (N × H)
117. C2×#343C2×C5semidirect
118. C2×#339C2×C5semidirect
179. C5×M16C22×C4semidirect
205. C42×C5C22×C4semidirect
223. #428C22×C5semidirect
226. C2×#104C4×C5semidirect
233. #382C22×C5semidirect
235. #343C22×C5semidirect
236. #339C22×C5semidirect
321. C2×M16C2×C4×C5semidirect
361. C2×C42C2×C4×C5semidirect
365. #104C2×C4×C5semidirect
458. M16C22×C4×C5semidirect
484. C42C22×C4×C5semidirect
489. C2×C5#1860direct (N × H)
555. C5C2×#1860direct (N × H)
589. C2C5×#1860direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5202
C2×C5×#4282
C22×C5×#1042
C2×C5×#4172
C2×C5×#3822
C5×#18602
C2×C5×#3432
C2×C5×#3392
C2×#18605