Finite Group Databasebeta

← C2×C5×#1859 #11943581

Subgroups — C2×C5×#1859 #11943581

Order 1280.

1566 conjugacy classes of subgroups.

The subgroup lattice has 598 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×#384C211yesmaximal, characteristic, complemented
6C5×#1859C218yesmaximal, 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⋊C4—22no—
23C22×C5×C4⋊C4C2211yescharacteristic
24C22×C5×C4⋊C4C2211yescharacteristic
25C22×C42×C5C2211yescharacteristic
26C5×#428—22no—
27C5×#428C2214yescomplemented
28C2×C5×#107—21no—
29C2×C5×#106—21no—
30C2×C5×#104C418yescomplemented
31C2×C5×#104C2214yes—
32C5×#417—22no—
33C5×#417C2214yes—
34C2×C5×#103—21no—
35C2×C5×#102—21no—
36C5×#384—22no—
37C5×#384C2214yescomplemented
38C5×#343C2214yescomplemented
39C5×#339C2214yescomplemented
40C2×#1859C511yesmaximal, 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⋊C4—28no—
93C2×C5×C4⋊C4—48no—
94C2×C5×C4⋊C4D412yes—
95C2×C5×C4⋊C4D411yescharacteristic
96C2×C5×C4⋊C4D411yescharacteristic
97C2×C5×C4⋊C4C2312yes—
98C2×C5×C4⋊C4C2312yes—
99C2×C5×C22⋊C4—22no—
100C2×C5×C22⋊C4—21no—
101C2×C42×C5—22no—
102C2×C42×C5—21no—
103C2×C42×C5—21no—
104C2×C42×C5—22no—
105C2×C42×C5C2×C414yescomplemented
106C2×C42×C5C2312yes—
107C5×#107—24no—
108C5×#106—24no—
109C5×#104C2×C4116yescomplemented
110C5×#103—24no—
111C5×#102—24no—
112C2×#520C2×C511yescharacteristic
113C2×#428C2×C511yescharacteristic, complemented
114C22×#104C2×C511yescharacteristic
115C2×#417C2×C511yescharacteristic
116C2×#384C2×C511yescharacteristic, complemented
117#1859C2×C518yesdirect product factor
118C2×#343C2×C511yescharacteristic, complemented
119C2×#339C2×C511yescharacteristic, complemented
120C24×C5—21no—
121C24×C5C2×D411yescharacteristic
122C5×#41—24no—
123C5×#41—24no—
124C5×#41—24no—
125C5×#41—24no—
126C5×#41—24no—
127C5×#41—22no—
128C5×#41—22no—
129C5×#41C4⋊C418yescomplemented
130C5×#41C22×C414yes—
131C5×#41C2×Q814yes—
132C2×C5×Q8—21no—
133C2×C5×Q8—22no—
134C2×C5×Q8C4⋊C414yescomplemented
135C2×C5×Q8C2×D412yes—
136C2×C5×D4—24no—
137C2×C5×D4—22no—
138C2×C5×D4—24no—
139C2×C5×D4—22no—
140C2×C5×D4—21no—
141C2×C5×D4—22no—
142C2×C5×D4C4⋊C414yescomplemented
143C2×C5×D4C2×D412yes—
144C22×C4×C5—24no—
145C22×C4×C5—24no—
146C22×C4×C5—24no—
147C22×C4×C5—21no—
148C22×C4×C5—22no—
149C22×C4×C5—24no—
150C22×C4×C5—22no—
151C22×C4×C5—24no—
152C22×C4×C5—22no—
153C22×C4×C5—21no—
154C22×C4×C5—22no—
155C22×C4×C5—21no—
156C22×C4×C5—28no—
157C22×C4×C5—22no—
158C22×C4×C5—22no—
159C22×C4×C5—44no—
160C22×C4×C5—44no—
161C22×C4×C5—41no—
162C22×C4×C5—41no—
163C22×C4×C5—84no—
164C22×C4×C5C22×C412yes—
165C22×C4×C5C2×D414yes—
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5C2×D411yescharacteristic
171C22×C4×C5C2×Q812yes—
172C22×C4×C5#4112yes—
173C22×C4×C5#4112yes—
174C22×C4×C5C2411yescharacteristic
175C5×M16—24no—
176C5×M16—22no—
177C5×M16—22no—
178C5×M16—22no—
179C5×M16—42no—
180C5×M16C22×C414yescomplemented
181C2×C5×C8—24no—
182C2×C5×C8—22no—
183C2×C5×C8—22no—
184C2×C5×C8—42no—
185C5×C4⋊C4—28no—
186C5×C4⋊C4—22no—
187C5×C4⋊C4—21no—
188C5×C4⋊C4—22no—
189C5×C4⋊C4—21no—
190C5×C4⋊C4—22no—
191C5×C4⋊C4—22no—
192C5×C4⋊C4—22no—
193C5×C4⋊C4—22no—
194C5×C4⋊C4—416no—
195C5×C4⋊C4—41no—
196C5×C4⋊C4—41no—
197C5×C4⋊C4C2×D412yes—
198C5×C4⋊C4C2×D412yes—
199C5×C22⋊C4—28no—
200C5×C22⋊C4—22no—
201C5×C22⋊C4—22no—
202C42×C5—28no—
203C42×C5—24no—
204C42×C5—22no—
205C42×C5—22no—
206C42×C5—22no—
207C42×C5C22×C414yescomplemented
208C22×#41C22×C511yescharacteristic
209C22×M16C22×C511yescharacteristic
210C22×M16C22×C511yescharacteristic
211#520—22no—
212#520—22no—
213#520—22no—
214#520C22×C514yes—
215C2×C4×Q8—21no—
216C2×C4×D4—21no—
217C2×C4×D4—22no—
218C2×#117—21no—
219C2×#117C22×C511yescharacteristic
220C22×C4⋊C4—21no—
221C22×C4⋊C4—22no—
222C22×C4⋊C4C22×C511yescharacteristic
223C22×C4⋊C4C22×C511yescharacteristic
224C22×C42C22×C511yescharacteristic
225#428—22no—
226#428C22×C514yescomplemented
227C2×#107—21no—
228C2×#106—21no—
229C2×#104C4×C518yescomplemented
230C2×#104C22×C514yes—
231#417—22no—
232#417C22×C514yes—
233C2×#103—21no—
234C2×#102—21no—
235#384—22no—
236#384C22×C514yescomplemented
237#343C22×C514yescomplemented
238#339C22×C514yescomplemented
239C23×C5—22no—
240C23×C5—22no—
241C23×C5—22no—
242C23×C5—24no—
243C23×C5—42no—
244C23×C5—42no—
245C23×C5C4×D412yes—
246C23×C5#12011yescharacteristic
247C23×C5#12011yescharacteristic
248C23×C5C22⋊Q812yes—
249C23×C5C22×D411yescharacteristic
250C5×Q8—22no—
251C5×Q8—24no—
252C5×Q8C2×C4⋊C414yescomplemented
253C5×D4—24no—
254C5×D4—24no—
255C5×D4—28no—
256C5×D4—22no—
257C5×D4—24no—
258C5×D4C2×C4⋊C414yescomplemented
259C2×C4×C5—24no—
260C2×C4×C5—216no—
261C2×C4×C5—24no—
262C2×C4×C5—24no—
263C2×C4×C5—22no—
264C2×C4×C5—24no—
265C2×C4×C5—24no—
266C2×C4×C5—24no—
267C2×C4×C5—28no—
268C2×C4×C5—22no—
269C2×C4×C5—24no—
270C2×C4×C5—22no—
271C2×C4×C5—44no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5—44no—
275C2×C4×C5—44no—
276C2×C4×C5—42no—
277C2×C4×C5—82no—
278C2×C4×C5—88no—
279C2×C4×C5—82no—
280C2×C4×C5—82no—
281C2×C4×C5C2×C4⋊C412yes—
282C2×C4×C5C2×C4⋊C412yes—
283C2×C4×C5C4×D412yes—
284C2×C4×C5C4×D412yes—
285C2×C4×C5C4×D412yes—
286C2×C4×C5#12011yescharacteristic
287C2×C4×C5#12011yescharacteristic
288C2×C4×C5C22⋊Q812yes—
289C2×C4×C5C22⋊Q812yes—
290C2×C4×C5C22⋊Q812yes—
291C2×C4×C5#12312yes—
292C2×C4×C5#12312yes—
293C2×C4×C5C23×C411yescharacteristic
294C2×C4×C5C22×D411yescharacteristic
295C2×C4×C5C22×D411yescharacteristic
296C2×C4×C5C22×Q811yescharacteristic
297C2×C4×C5C2×#4111yescharacteristic
298C2×C4×C5C2×#4111yescharacteristic
299C5×C8—24no—
300C5×C8—42no—
301C2×#41—21no—
302C2×#41—21no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41C2×C4×C514yes—
306C2×#41C5×D412yes—
307C2×#41C5×Q814yes—
308C2×#41C23×C512yes—
309C22×Q8C5×D411yescharacteristic
310C22×D4—21no—
311C22×D4C5×D411yescharacteristic
312C23×C4—21no—
313C23×C4—21no—
314C23×C4—22no—
315C23×C4—21no—
316C23×C4C5×D412yes—
317C23×C4C23×C511yescharacteristic
318C2×M16—21no—
319C2×M16—22no—
320C2×M16—21no—
321C2×M16—22no—
322C2×M16—22no—
323C2×M16C2×C4×C514yescomplemented
324C2×M16C23×C512yes—
325C2×M16C23×C512yes—
326C22×C8—21no—
327C22×C8—21no—
328C4×Q8—24no—
329C4×Q8—24no—
330C4×D4—28no—
331C4×D4—24no—
332C4×D4—24no—
333C4×D4—28no—
334#117—24no—
335#117—21no—
336#117—21no—
337#117—24no—
338#117C5×D412yes—
339#117C23×C512yes—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—28no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—48no—
354C2×C4⋊C4C5×D412yes—
355C2×C4⋊C4C5×D411yescharacteristic
356C2×C4⋊C4C5×D411yescharacteristic
357C2×C4⋊C4C23×C512yes—
358C2×C4⋊C4C23×C512yes—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42C2×C4×C514yescomplemented
366C2×C42C23×C512yes—
367#107—24no—
368#106—24no—
369#104C2×C4×C5116yescomplemented
370#103—24no—
371#102—24no—
372C22×C5—24no—
373C22×C5—24no—
374C22×C5—21no—
375C22×C5—21no—
376C22×C5—21no—
377C22×C5—21no—
378C22×C5—24no—
379C22×C5—48no—
380C22×C5—42no—
381C22×C5—42no—
382C22×C5#38512yes—
383C22×C5#38512yes—
384C22×C5#45312yes—
385C22×C5#45612yes—
386C22×C5C2×C4×D411yescharacteristic
387C22×C5C2×#12011yescharacteristic
388C22×C5C2×C22⋊Q811yescharacteristic
389C4×C5—28no—
390C4×C5—24no—
391C4×C5—44no—
392C4×C5—42no—
393C4×C5—42no—
394C4×C5—82no—
395C4×C5#38512yes—
396C4×C5#38512yes—
397C4×C5C22×C4⋊C411yescharacteristic
398C4×C5C2×C4×D411yescharacteristic
399C4×C5C2×C22⋊Q811yescharacteristic
400C4×C5C2×#12311yescharacteristic
401C24—21no—
402C24C2×C5×D411yescharacteristic
403#41—24no—
404#41—24no—
405#41—24no—
406#41—24no—
407#41—24no—
408#41—22no—
409#41—22no—
410#41C5×C4⋊C418yescomplemented
411#41C22×C4×C514yes—
412#41C2×C5×Q814yes—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8C5×C4⋊C414yescomplemented
416C2×Q8C2×C5×D412yes—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4C5×C4⋊C414yescomplemented
424C2×D4C2×C5×D412yes—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—28no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—84no—
445C22×C4C22×C4×C512yes—
446C22×C4C2×C5×D414yes—
447C22×C4C2×C5×D411yescharacteristic
448C22×C4C2×C5×D411yescharacteristic
449C22×C4C2×C5×D411yescharacteristic
450C22×C4C2×C5×D411yescharacteristic
451C22×C4C2×C5×D411yescharacteristic
452C22×C4C2×C5×Q812yes—
453C22×C4C5×#4112yes—
454C22×C4C5×#4112yes—
455C22×C4C24×C511yescharacteristic
456M16—24no—
457M16—22no—
458M16—22no—
459M16—22no—
460M16—42no—
461M16C22×C4×C514yescomplemented
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—42no—
466C4⋊C4—28no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—416no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4C2×C5×D412yes—
479C4⋊C4C2×C5×D412yes—
480C22⋊C4—28no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C42—28no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42C22×C4×C514yescomplemented
489C2×C5—21no—
490C2×C5—21no—
491C2×C5—22no—
492C2×C5—44no—
493C2×C5#185914yesdirect product factor
494C2×C5C2×#38511yescharacteristic
495C2×C5C2×#45311yescharacteristic
496C2×C5C2×#45611yescharacteristic
497C23—22no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—42no—
502C23—42no—
503C23C4×C5×D412yes—
504C23C5×#12011yescharacteristic
505C23C5×#12011yescharacteristic
506C23C5×C22⋊Q812yes—
507C23C22×C5×D411yescharacteristic
508Q8—24no—
509Q8—22no—
510Q8C2×C5×C4⋊C414yescomplemented
511D4—28no—
512D4—24no—
513D4—24no—
514D4—24no—
515D4—22no—
516D4C2×C5×C4⋊C414yescomplemented
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—216no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—82no—
536C2×C4—88no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4C2×C5×C4⋊C412yes—
540C2×C4C2×C5×C4⋊C412yes—
541C2×C4C4×C5×D412yes—
542C2×C4C4×C5×D412yes—
543C2×C4C4×C5×D412yes—
544C2×C4C5×#12011yescharacteristic
545C2×C4C5×#12011yescharacteristic
546C2×C4C5×C22⋊Q812yes—
547C2×C4C5×C22⋊Q812yes—
548C2×C4C5×C22⋊Q812yes—
549C2×C4C5×#12312yes—
550C2×C4C5×#12312yes—
551C2×C4C23×C4×C511yescharacteristic
552C2×C4C22×C5×D411yescharacteristic
553C2×C4C22×C5×D411yescharacteristic
554C2×C4C22×C5×Q811yescharacteristic
555C2×C4C2×C5×#4111yescharacteristic
556C2×C4C2×C5×#4111yescharacteristic
557C8—24no—
558C8—42no—
559C5C2×#185911yescharacteristic, direct product factor, Sylow
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—48no—
570C22C5×#38512yes—
571C22C5×#38512yes—
572C22C5×#45312yes—
573C22C5×#45612yes—
574C22C2×C4×C5×D411yescharacteristic
575C22C2×C5×#12011yescharacteristic
576C22C2×C5×C22⋊Q811yescharacteristic
577C4—28no—
578C4—24no—
579C4—44no—
580C4—42no—
581C4—42no—
582C4—82no—
583C4C5×#38512yes—
584C4C5×#38512yes—
585C4C22×C5×C4⋊C411yescharacteristic
586C4C2×C4×C5×D411yescharacteristic
587C4C2×C5×C22⋊Q811yescharacteristic
588C4C2×C5×#12311yescharacteristic
589C2—21no—
590C2—22no—
591C2—21no—
592C2—44no—
593C2C5×#185914yesdirect product factor
594C2C2×C5×#38511yescharacteristic
595C2C2×C5×#45311yescharacteristic
596C2C2×C5×#45611yescharacteristic
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×#384C2semidirect
6. C5×#1859C2direct (N × H)
7. C2×C5×#343C2semidirect
8. C2×C5×#339C2semidirect
27. C5×#428C22semidirect
30. C2×C5×#104C4semidirect
37. C5×#384C22semidirect
38. C5×#343C22semidirect
39. C5×#339C22semidirect
40. C2×#1859C5direct (N × H)
63. C2×C5×M16C2×C4semidirect
105. C2×C42×C5C2×C4semidirect
109. C5×#104C2×C4semidirect
113. C2×#428C2×C5semidirect
116. C2×#384C2×C5semidirect
117. #1859C2×C5direct (N × H)
118. C2×#343C2×C5semidirect
119. C2×#339C2×C5semidirect
129. C5×#41C4⋊C4semidirect
134. C2×C5×Q8C4⋊C4semidirect
142. C2×C5×D4C4⋊C4semidirect
180. C5×M16C22×C4semidirect
207. C42×C5C22×C4semidirect
226. #428C22×C5semidirect
229. C2×#104C4×C5semidirect
236. #384C22×C5semidirect
237. #343C22×C5semidirect
238. #339C22×C5semidirect
252. C5×Q8C2×C4⋊C4semidirect
258. C5×D4C2×C4⋊C4semidirect
323. C2×M16C2×C4×C5semidirect
365. C2×C42C2×C4×C5semidirect
369. #104C2×C4×C5semidirect
410. #41C5×C4⋊C4semidirect
415. C2×Q8C5×C4⋊C4semidirect
423. C2×D4C5×C4⋊C4semidirect
461. M16C22×C4×C5semidirect
488. C42C22×C4×C5semidirect
493. C2×C5#1859direct (N × H)
510. Q8C2×C5×C4⋊C4semidirect
516. D4C2×C5×C4⋊C4semidirect
559. C5C2×#1859direct (N × H)
593. C2C5×#1859direct (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×#3842
C5×#18592
C2×C5×#3432
C2×C5×#3392
C2×#18595