Finite Group Databasebeta

← C5×#23961 #11946681

Subgroups — C5×#23961 #11946681

Order 1280.

1494 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#547C211yesmaximal, characteristic
2C2×C5×#448C211yesmaximal, characteristic, complemented
3C22×C5×#103C211yesmaximal, characteristic, complemented
4C5×#2409C211yesmaximal, characteristic, complemented
5C5×#2024C212yesmaximal, complemented
6C5×#2000C212yesmaximal, complemented
7C5×#1893C211yesmaximal, characteristic, complemented
8C5×#1877C212yesmaximal, complemented
9C5×#1872C212yesmaximal, complemented
10C5×#1869C211yesmaximal, characteristic, complemented
11C5×#1823C211yesmaximal, characteristic, complemented
12C22×C5×#41—21no—
13C23×C5×Q8C2211yescharacteristic
14C22×C5×SD16C2212yes—
15C23×C5×C8C2211yescharacteristic
16C5×#547—24no—
17C5×#547—24no—
18C2×C5×C4⋊Q8C2211yescharacteristic
19C2×C5×#123—21no—
20C2×C5×#123—21no—
21C2×C5×C22⋊Q8—21no—
22C2×C5×C22⋊Q8C2212yes—
23C2×C5×#121C2211yescharacteristic
24C2×C5×#121C2211yescharacteristic
25C2×C4×C5×D4C2211yescharacteristic
26C2×C4×C5×D4C2211yescharacteristic
27C22×C5×C4⋊C4C2211yescharacteristic
28C22×C5×C4⋊C4C2211yescharacteristic
29C5×#448—24no—
30C2×C5×#103—24no—
31C2×C5×#103C2212yes—
32C2×C5×#103C2212yes—
33C2×C5×#103C2212yes—
34C2×C5×#102C2212yes—
35C2×C5×C22⋊C8C2211yescharacteristic
36C2×C5×C22⋊C8C2212yes—
37C5×#395—21no—
38C5×#393—21no—
39C5×#391C2212yes—
40C5×#386—21no—
41C5×#385—21no—
42C5×#380C2211yescharacteristic
43C5×#340C2212yes—
44C5×#340C2212yes—
45C5×#340C2212yes—
46C5×#336C2212yes—
47#23961C511yesmaximal, characteristic, direct product factor, Sylow
48C2×C5×#41—28no—
49C2×C5×#41—44no—
50C2×C5×#41—44no—
51C22×C5×Q8—21no—
52C22×C5×Q8—21no—
53C22×C5×Q8—24no—
54C22×C5×Q8—22no—
55C22×C5×Q8—22no—
56C22×C5×Q8—42no—
57C22×C5×Q8C2312yes—
58C22×C5×D4—21no—
59C22×C5×D4—21no—
60C22×C5×D4D411yescharacteristic
61C22×C5×D4C2311yescharacteristic
62C23×C4×C5—21no—
63C23×C4×C5—21no—
64C23×C4×C5—21no—
65C23×C4×C5—21no—
66C23×C4×C5C2311yescharacteristic
67C23×C4×C5C2311yescharacteristic
68C2×C5×SD16—24no—
69C2×C5×SD16—24no—
70C2×C5×SD16—42no—
71C22×C5×C8—22no—
72C22×C5×C8—42no—
73C22×C5×C8C2312yes—
74C22×C5×C8C2312yes—
75C5×C4⋊Q8—22no—
76C5×C4⋊Q8—22no—
77C5×#123—28no—
78C5×#123—28no—
79C5×C22⋊Q8—28no—
80C5×C22⋊Q8—24no—
81C5×C22⋊Q8—42no—
82C5×#121—22no—
83C5×#121—22no—
84C5×#121—22no—
85C5×#121D414yes—
86C4×C5×D4—22no—
87C4×C5×D4—22no—
88C4×C5×D4—22no—
89C4×C5×D4—22no—
90C2×C5×C4⋊C4—21no—
91C2×C5×C4⋊C4—21no—
92C2×C5×C4⋊C4—22no—
93C2×C5×C4⋊C4—21no—
94C2×C5×C4⋊C4—22no—
95C2×C5×C4⋊C4—22no—
96C2×C5×C4⋊C4—22no—
97C2×C5×C4⋊C4D411yescharacteristic
98C2×C5×C4⋊C4D414yes—
99C2×C5×C4⋊C4C2312yes—
100C2×C5×C4⋊C4C2311yescharacteristic
101C2×C5×C4⋊C4C2311yescharacteristic
102C2×C5×C4⋊C4C2311yescharacteristic
103C2×C5×C22⋊C4—21no—
104C2×C5×C22⋊C4—21no—
105C2×C5×C22⋊C4—21no—
106C2×C5×C22⋊C4—21no—
107C2×C5×C22⋊C4—22no—
108C2×C5×C22⋊C4D411yescharacteristic
109C2×C5×C22⋊C4D411yescharacteristic
110C2×C42×C5C2311yescharacteristic
111C5×#103—24no—
112C5×#103—24no—
113C5×#103—24no—
114C5×#102—42no—
115C5×C22⋊C8—22no—
116C5×C22⋊C8—42no—
117C5×#95—21no—
118C5×#95—22no—
119C5×#95—22no—
120C2×#547C2×C511yescharacteristic
121C2×#448C2×C511yescharacteristic, complemented
122C22×#103C2×C511yescharacteristic, complemented
123#2409C2×C511yescharacteristic, complemented
124#2024C2×C512yescomplemented
125#2000C2×C512yescomplemented
126#1893C2×C511yescharacteristic, complemented
127#1877C2×C512yescomplemented
128#1872C2×C512yescomplemented
129#1869C2×C511yescharacteristic, complemented
130#1823C2×C511yescharacteristic, complemented
131C24×C5—21no—
132C24×C5C2×D411yescharacteristic
133C24×C5C2×D411yescharacteristic
134C5×#41—432no—
135C2×C5×Q8—24no—
136C2×C5×Q8—22no—
137C2×C5×Q8—28no—
138C2×C5×Q8—22no—
139C2×C5×Q8—22no—
140C2×C5×Q8—22no—
141C2×C5×Q8—22no—
142C2×C5×Q8—44no—
143C2×C5×Q8—48no—
144C2×C5×Q8—44no—
145C2×C5×Q8—42no—
146C2×C5×Q8—42no—
147C2×C5×Q8—42no—
148C2×C5×Q8#4114yescomplemented
149C2×C5×D4—22no—
150C2×C5×D4—22no—
151C2×C5×D4—24no—
152C2×C5×D4—24no—
153C2×C5×D4—21no—
154C2×C5×D4—22no—
155C2×C5×D4—22no—
156C2×C5×D4—42no—
157C2×C5×D4—42no—
158C2×C5×D4—42no—
159C2×C5×D4—42no—
160C2×C5×D4SD1614yescomplemented
161C2×C5×D4C2×D412yes—
162C22×C4×C5—22no—
163C22×C4×C5—22no—
164C22×C4×C5—24no—
165C22×C4×C5—24no—
166C22×C4×C5—24no—
167C22×C4×C5—21no—
168C22×C4×C5—21no—
169C22×C4×C5—21no—
170C22×C4×C5—22no—
171C22×C4×C5—21no—
172C22×C4×C5—21no—
173C22×C4×C5—22no—
174C22×C4×C5—22no—
175C22×C4×C5—22no—
176C22×C4×C5—44no—
177C22×C4×C5—42no—
178C22×C4×C5—42no—
179C22×C4×C5—42no—
180C22×C4×C5—42no—
181C22×C4×C5—42no—
182C22×C4×C5—42no—
183C22×C4×C5—42no—
184C22×C4×C5C2×D411yescharacteristic
185C22×C4×C5C2×D411yescharacteristic
186C22×C4×C5C2×D411yescharacteristic
187C22×C4×C5C2×D412yes—
188C22×C4×C5C2×D412yes—
189C22×C4×C5C2×D411yescharacteristic
190C22×C4×C5C2411yescharacteristic
191C5×SD16—48no—
192C2×C5×C8—22no—
193C2×C5×C8—22no—
194C2×C5×C8—22no—
195C2×C5×C8—22no—
196C2×C5×C8—22no—
197C2×C5×C8—22no—
198C2×C5×C8—44no—
199C5×C4⋊C4—24no—
200C5×C4⋊C4—24no—
201C5×C4⋊C4—24no—
202C5×C4⋊C4—22no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4—21no—
205C5×C4⋊C4—21no—
206C5×C4⋊C4—22no—
207C5×C4⋊C4—22no—
208C5×C4⋊C4—44no—
209C5×C4⋊C4C2×D412yes—
210C5×C4⋊C4C2×D412yes—
211C5×C4⋊C4C2×D412yes—
212C5×C22⋊C4—24no—
213C5×C22⋊C4—24no—
214C5×C22⋊C4—22no—
215C5×C22⋊C4—24no—
216C5×C22⋊C4—22no—
217C5×C22⋊C4—44no—
218C5×C22⋊C4—42no—
219C42×C5—21no—
220C42×C5—21no—
221C22×#41—21no—
222C23×Q8C22×C511yescharacteristic
223C22×SD16C22×C512yes—
224C23×C8C22×C511yescharacteristic
225#547—24no—
226#547—24no—
227C2×C4⋊Q8C22×C511yescharacteristic
228C2×#123—21no—
229C2×#123—21no—
230C2×C22⋊Q8—21no—
231C2×C22⋊Q8C22×C512yes—
232C2×#121C22×C511yescharacteristic
233C2×#121C22×C511yescharacteristic
234C2×C4×D4C22×C511yescharacteristic
235C2×C4×D4C22×C511yescharacteristic
236C22×C4⋊C4C22×C511yescharacteristic
237C22×C4⋊C4C22×C511yescharacteristic
238#448—24no—
239C2×#103—24no—
240C2×#103C22×C512yes—
241C2×#103C22×C512yes—
242C2×#103C22×C512yes—
243C2×#102C22×C512yes—
244C2×C22⋊C8C22×C511yescharacteristic
245C2×C22⋊C8C22×C512yes—
246#395—21no—
247#393—21no—
248#391C22×C512yes—
249#386—21no—
250#385—21no—
251#380C22×C511yescharacteristic
252#340C22×C512yes—
253#340C22×C512yes—
254#340C22×C512yes—
255#336C22×C512yes—
256C23×C5—22no—
257C23×C5—22no—
258C23×C5—21no—
259C23×C5—22no—
260C23×C5—21no—
261C23×C5—22no—
262C23×C5—22no—
263C23×C5—42no—
264C23×C5—41no—
265C23×C5—41no—
266C23×C5—42no—
267C23×C5#12012yes—
268C23×C5C2×SD1612yes—
269C23×C5C2×SD1612yes—
270C23×C5AGL(1,ℤ/8)12yes—
271C23×C5C22×D411yescharacteristic
272C5×Q8—24no—
273C5×Q8—24no—
274C5×Q8—48no—
275C5×Q8—48no—
276C5×Q8—42no—
277C5×Q8—42no—
278C5×D4—28no—
279C5×D4—24no—
280C5×D4—48no—
281C5×D4—48no—
282C5×D4—41no—
283C5×D4—41no—
284C2×C4×C5—21no—
285C2×C4×C5—22no—
286C2×C4×C5—21no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—24no—
290C2×C4×C5—21no—
291C2×C4×C5—21no—
292C2×C4×C5—22no—
293C2×C4×C5—22no—
294C2×C4×C5—22no—
295C2×C4×C5—24no—
296C2×C4×C5—22no—
297C2×C4×C5—22no—
298C2×C4×C5—48no—
299C2×C4×C5—48no—
300C2×C4×C5—48no—
301C2×C4×C5—42no—
302C2×C4×C5—42no—
303C2×C4×C5—41no—
304C2×C4×C5—41no—
305C2×C4×C5—41no—
306C2×C4×C5—41no—
307C2×C4×C5—41no—
308C2×C4×C5—41no—
309C2×C4×C5—48no—
310C2×C4×C5—41no—
311C2×C4×C5—41no—
312C2×C4×C5—42no—
313C2×C4×C5—42no—
314C2×C4×C5—42no—
315C2×C4×C5—42no—
316C2×C4×C5#12012yes—
317C2×C4×C5C2×SD1612yes—
318C2×C4×C5C22×D411yescharacteristic
319C2×C4×C5C22×D411yescharacteristic
320C2×C4×C5C2×#4112yes—
321C2×C4×C5H(5,𝔽2)12yes—
322C5×C8—42no—
323C5×C8—42no—
324C2×#41—28no—
325C2×#41—44no—
326C2×#41—44no—
327C22×Q8—21no—
328C22×Q8—21no—
329C22×Q8—24no—
330C22×Q8—22no—
331C22×Q8—22no—
332C22×Q8—42no—
333C22×Q8C23×C512yes—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4C5×D411yescharacteristic
337C22×D4C23×C511yescharacteristic
338C23×C4—21no—
339C23×C4—21no—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4C23×C511yescharacteristic
343C23×C4C23×C511yescharacteristic
344C2×SD16—24no—
345C2×SD16—24no—
346C2×SD16—42no—
347C22×C8—22no—
348C22×C8—42no—
349C22×C8C23×C512yes—
350C22×C8C23×C512yes—
351C4⋊Q8—22no—
352C4⋊Q8—22no—
353#123—28no—
354#123—28no—
355C22⋊Q8—28no—
356C22⋊Q8—24no—
357C22⋊Q8—42no—
358#121—22no—
359#121—22no—
360#121—22no—
361#121C5×D414yes—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4C5×D411yescharacteristic
374C2×C4⋊C4C5×D414yes—
375C2×C4⋊C4C23×C512yes—
376C2×C4⋊C4C23×C511yescharacteristic
377C2×C4⋊C4C23×C511yescharacteristic
378C2×C4⋊C4C23×C511yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4C5×D411yescharacteristic
385C2×C22⋊C4C5×D411yescharacteristic
386C2×C42C23×C511yescharacteristic
387#103—24no—
388#103—24no—
389#103—24no—
390#102—42no—
391C22⋊C8—22no—
392C22⋊C8—42no—
393#95—21no—
394#95—22no—
395#95—22no—
396C22×C5—22no—
397C22×C5—22no—
398C22×C5—24no—
399C22×C5—24no—
400C22×C5—22no—
401C22×C5—24no—
402C22×C5—41no—
403C22×C5—41no—
404C22×C5—42no—
405C22×C5—42no—
406C22×C5—42no—
407C22×C5—42no—
408C22×C5—82no—
409C22×C5—82no—
410C22×C5#45014yes—
411C22×C5C2×#12011yescharacteristic
412C22×C5#54612yes—
413C22×C5C22×SD1611yescharacteristic
414C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
415C22×C5#57812yes—
416C4×C5—22no—
417C4×C5—22no—
418C4×C5—41no—
419C4×C5—44no—
420C4×C5—42no—
421C4×C5—42no—
422C4×C5—42no—
423C4×C5—41no—
424C4×C5—82no—
425C4×C5#52511yescharacteristic
426C4×C5#53511yescharacteristic
427C4×C5#54612yes—
428C24—21no—
429C24C2×C5×D411yescharacteristic
430C24C2×C5×D411yescharacteristic
431#41—432no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—28no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—44no—
440C2×Q8—48no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8C5×#4114yescomplemented
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4C5×SD1614yescomplemented
458C2×D4C2×C5×D412yes—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4C2×C5×D411yescharacteristic
482C22×C4C2×C5×D411yescharacteristic
483C22×C4C2×C5×D411yescharacteristic
484C22×C4C2×C5×D412yes—
485C22×C4C2×C5×D412yes—
486C22×C4C2×C5×D411yescharacteristic
487C22×C4C24×C511yescharacteristic
488SD16—48no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—44no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—44no—
506C4⋊C4C2×C5×D412yes—
507C4⋊C4C2×C5×D412yes—
508C4⋊C4C2×C5×D412yes—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—22no—
512C22⋊C4—24no—
513C22⋊C4—22no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C42—21no—
517C42—21no—
518C2×C5—24no—
519C2×C5—42no—
520C2×C5—82no—
521C2×C5#240411yescharacteristic
522C2×C5C2×#45011yescharacteristic
523C2×C5#301111yescharacteristic
524C2×C5#329812yes—
525C2×C5#330812yes—
526C23—22no—
527C23—22no—
528C23—21no—
529C23—22no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23C5×#12012yes—
538C23C2×C5×SD1612yes—
539C23C2×C5×SD1612yes—
540C23C5×AGL(1,ℤ/8)12yes—
541C23C22×C5×D411yescharacteristic
542Q8—24no—
543Q8—24no—
544Q8—48no—
545Q8—48no—
546Q8—42no—
547Q8—42no—
548D4—28no—
549D4—24no—
550D4—48no—
551D4—48no—
552D4—41no—
553D4—41no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4C5×#12012yes—
587C2×C4C2×C5×SD1612yes—
588C2×C4C22×C5×D411yescharacteristic
589C2×C4C22×C5×D411yescharacteristic
590C2×C4C2×C5×#4112yes—
591C2×C4C5×H(5,𝔽2)12yes—
592C8—42no—
593C8—42no—
594C5#2396111yescharacteristic, direct product factor, Sylow
595C22—22no—
596C22—24no—
597C22—24no—
598C22—22no—
599C22—22no—
600C22—24no—
601C22—41no—
602C22—42no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—82no—
608C22—82no—
609C22C5×#45014yes—
610C22C2×C5×#12011yescharacteristic
611C22C5×#54612yes—
612C22C22×C5×SD1611yescharacteristic
613C22C2×C5×AGL(1,ℤ/8)11yescharacteristic
614C22C5×#57812yes—
615C4—22no—
616C4—22no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—44no—
623C4—82no—
624C4C5×#52511yescharacteristic
625C4C5×#53511yescharacteristic
626C4C5×#54612yes—
627C2—24no—
628C2—42no—
629C2—82no—
630C2C5×#240411yescharacteristic
631C2C2×C5×#45011yescharacteristic
632C2C5×#301111yescharacteristic
633C2C5×#329812yes—
634C2C5×#330812yes—
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC2×C5×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C2×C4×C5×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#448C2semidirect
3. C22×C5×#103C2semidirect
4. C5×#2409C2semidirect
5. C5×#2024C2semidirect
6. C5×#2000C2semidirect
7. C5×#1893C2semidirect
8. C5×#1877C2semidirect
9. C5×#1872C2semidirect
10. C5×#1869C2semidirect
11. C5×#1823C2semidirect
47. #23961C5direct (N × H)
121. C2×#448C2×C5semidirect
122. C22×#103C2×C5semidirect
123. #2409C2×C5semidirect
124. #2024C2×C5semidirect
125. #2000C2×C5semidirect
126. #1893C2×C5semidirect
127. #1877C2×C5semidirect
128. #1872C2×C5semidirect
129. #1869C2×C5semidirect
130. #1823C2×C5semidirect
148. C2×C5×Q8#41semidirect
160. C2×C5×D4SD16semidirect
445. C2×Q8C5×#41semidirect
457. C2×D4C5×SD16semidirect
594. C5#23961direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5472
C2×C5×#4482
C22×C5×#1032
C5×#24092
C5×#20242
C5×#20002
C5×#18932
C5×#18772
C5×#18722
C5×#18692
C5×#18232
#239615