Finite Group Databasebeta

← C5×#23970 #11946690

Subgroups — C5×#23970 #11946690

Order 1280.

1278 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#554C211yesmaximal, characteristic
2C2×C5×#451C211yesmaximal, characteristic, complemented
3C22×C5×#103C211yesmaximal, characteristic, complemented
4C5×#2409C211yesmaximal, characteristic
5C5×#2024C212yesmaximal, complemented
6C5×#2001C212yesmaximal, complemented
7C5×#1892C211yesmaximal, characteristic, complemented
8C5×#1878C212yesmaximal, complemented
9C5×#1872C212yesmaximal, complemented
10C5×#1869C211yesmaximal, characteristic
11C5×#1824C211yesmaximal, characteristic, complemented
12C23×C5×Q8C2211yescharacteristic
13C22×C5×Q16C2212yes—
14C23×C5×C8C2211yescharacteristic
15C5×#554—24no—
16C5×#554—24no—
17C2×C5×C4⋊Q8C2211yescharacteristic
18C2×C5×#125—21no—
19C2×C5×C22⋊Q8—21no—
20C2×C5×C22⋊Q8—21no—
21C2×C5×C22⋊Q8C2211yescharacteristic
22C2×C5×C22⋊Q8C2212yes—
23C2×C5×C22⋊Q8C2211yescharacteristic
24C2×C4×C5×Q8C2211yescharacteristic
25C2×C4×C5×D4—21no—
26C2×C4×C5×D4C2211yescharacteristic
27C22×C5×C4⋊C4C2211yescharacteristic
28C22×C5×C4⋊C4C2211yescharacteristic
29C5×#451—24no—
30C2×C5×#103—24no—
31C2×C5×#103C2212yes—
32C2×C5×#103C2212yes—
33C2×C5×#103C2212yes—
34C2×C5×#103C2212yes—
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#23970C511yesmaximal, characteristic, direct product factor, Sylow
48C22×C5×Q8—21no—
49C22×C5×Q8—21no—
50C22×C5×Q8—24no—
51C22×C5×Q8—22no—
52C22×C5×Q8—22no—
53C22×C5×Q8—42no—
54C22×C5×Q8C2311yescharacteristic
55C22×C5×Q8C2312yes—
56C22×C5×D4D411yescharacteristic
57C23×C4×C5—21no—
58C23×C4×C5—21no—
59C23×C4×C5—21no—
60C23×C4×C5C2311yescharacteristic
61C23×C4×C5C2311yescharacteristic
62C2×C5×Q16—24no—
63C2×C5×Q16—24no—
64C2×C5×Q16—42no—
65C22×C5×C8—22no—
66C22×C5×C8—42no—
67C22×C5×C8C2312yes—
68C22×C5×C8C2312yes—
69C5×C4⋊Q8—22no—
70C5×C4⋊Q8—22no—
71C5×#125—28no—
72C5×C22⋊Q8—28no—
73C5×C22⋊Q8—28no—
74C5×C22⋊Q8—22no—
75C5×C22⋊Q8—22no—
76C5×C22⋊Q8—22no—
77C5×C22⋊Q8—24no—
78C5×C22⋊Q8—42no—
79C5×C22⋊Q8D414yes—
80C4×C5×Q8—22no—
81C4×C5×Q8—22no—
82C4×C5×D4—28no—
83C4×C5×D4—22no—
84C4×C5×D4—22no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4—21no—
87C2×C5×C4⋊C4—22no—
88C2×C5×C4⋊C4—21no—
89C2×C5×C4⋊C4—21no—
90C2×C5×C4⋊C4—21no—
91C2×C5×C4⋊C4—21no—
92C2×C5×C4⋊C4—22no—
93C2×C5×C4⋊C4—22no—
94C2×C5×C4⋊C4—22no—
95C2×C5×C4⋊C4D411yescharacteristic
96C2×C5×C4⋊C4D414yes—
97C2×C5×C4⋊C4C2312yes—
98C2×C5×C4⋊C4C2311yescharacteristic
99C2×C5×C4⋊C4C2311yescharacteristic
100C2×C5×C4⋊C4C2311yescharacteristic
101C2×C5×C22⋊C4—21no—
102C2×C5×C22⋊C4—21no—
103C2×C5×C22⋊C4—21no—
104C2×C5×C22⋊C4—22no—
105C2×C5×C22⋊C4D411yescharacteristic
106C2×C5×C22⋊C4D411yescharacteristic
107C2×C42×C5—21no—
108C2×C42×C5C2311yescharacteristic
109C5×#103—24no—
110C5×#103—24no—
111C5×#103—24no—
112C5×#103—42no—
113C5×C22⋊C8—22no—
114C5×C22⋊C8—42no—
115C5×#95—21no—
116C5×#95—22no—
117C5×#95—22no—
118C2×#554C2×C511yescharacteristic
119C2×#451C2×C511yescharacteristic, complemented
120C22×#103C2×C511yescharacteristic, complemented
121#2409C2×C511yescharacteristic
122#2024C2×C512yescomplemented
123#2001C2×C512yescomplemented
124#1892C2×C511yescharacteristic, complemented
125#1878C2×C512yescomplemented
126#1872C2×C512yescomplemented
127#1869C2×C511yescharacteristic
128#1824C2×C511yescharacteristic, complemented
129C24×C5C2×D411yescharacteristic
130C24×C5C2×D411yescharacteristic
131C2×C5×Q8—24no—
132C2×C5×Q8—21no—
133C2×C5×Q8—22no—
134C2×C5×Q8—28no—
135C2×C5×Q8—22no—
136C2×C5×Q8—22no—
137C2×C5×Q8—22no—
138C2×C5×Q8—22no—
139C2×C5×Q8—22no—
140C2×C5×Q8—22no—
141C2×C5×Q8—44no—
142C2×C5×Q8—48no—
143C2×C5×Q8—44no—
144C2×C5×Q8—42no—
145C2×C5×Q8—82no—
146C2×C5×Q8C2×D412yes—
147C2×C5×Q8#4114yes—
148C2×C5×D4—22no—
149C2×C5×D4—22no—
150C2×C5×D4Q1614yescomplemented
151C22×C4×C5—22no—
152C22×C4×C5—22no—
153C22×C4×C5—21no—
154C22×C4×C5—21no—
155C22×C4×C5—21no—
156C22×C4×C5—21no—
157C22×C4×C5—22no—
158C22×C4×C5—24no—
159C22×C4×C5—24no—
160C22×C4×C5—21no—
161C22×C4×C5—21no—
162C22×C4×C5—22no—
163C22×C4×C5—22no—
164C22×C4×C5—22no—
165C22×C4×C5—44no—
166C22×C4×C5—42no—
167C22×C4×C5—82no—
168C22×C4×C5—82no—
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5C2×D411yescharacteristic
171C22×C4×C5C2×D411yescharacteristic
172C22×C4×C5C2×D412yes—
173C22×C4×C5C2×D412yes—
174C22×C4×C5C2×D411yescharacteristic
175C22×C4×C5C2411yescharacteristic
176C5×Q16—48no—
177C2×C5×C8—22no—
178C2×C5×C8—22no—
179C2×C5×C8—22no—
180C2×C5×C8—22no—
181C2×C5×C8—22no—
182C2×C5×C8—22no—
183C2×C5×C8—44no—
184C5×C4⋊C4—24no—
185C5×C4⋊C4—24no—
186C5×C4⋊C4—24no—
187C5×C4⋊C4—24no—
188C5×C4⋊C4—24no—
189C5×C4⋊C4—24no—
190C5×C4⋊C4—22no—
191C5×C4⋊C4—21no—
192C5×C4⋊C4—21no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—22no—
195C5×C4⋊C4—22no—
196C5×C4⋊C4—44no—
197C5×C4⋊C4C2×D412yes—
198C5×C4⋊C4C2×D412yes—
199C5×C4⋊C4C2×D412yes—
200C5×C22⋊C4—24no—
201C5×C22⋊C4—24no—
202C5×C22⋊C4—22no—
203C5×C22⋊C4—22no—
204C5×C22⋊C4—44no—
205C5×C22⋊C4—42no—
206C42×C5—24no—
207C42×C5—21no—
208C42×C5—21no—
209C23×Q8C22×C511yescharacteristic
210C22×Q16C22×C512yes—
211C23×C8C22×C511yescharacteristic
212#554—24no—
213#554—24no—
214C2×C4⋊Q8C22×C511yescharacteristic
215C2×#125—21no—
216C2×C22⋊Q8—21no—
217C2×C22⋊Q8—21no—
218C2×C22⋊Q8C22×C511yescharacteristic
219C2×C22⋊Q8C22×C512yes—
220C2×C22⋊Q8C22×C511yescharacteristic
221C2×C4×Q8C22×C511yescharacteristic
222C2×C4×D4—21no—
223C2×C4×D4C22×C511yescharacteristic
224C22×C4⋊C4C22×C511yescharacteristic
225C22×C4⋊C4C22×C511yescharacteristic
226#451—24no—
227C2×#103—24no—
228C2×#103C22×C512yes—
229C2×#103C22×C512yes—
230C2×#103C22×C512yes—
231C2×#103C22×C512yes—
232C2×C22⋊C8C22×C511yescharacteristic
233C2×C22⋊C8C22×C512yes—
234#395—21no—
235#393—21no—
236#391C22×C512yes—
237#386—21no—
238#385—21no—
239#380C22×C511yescharacteristic
240#340C22×C512yes—
241#340C22×C512yes—
242#340C22×C512yes—
243#336C22×C512yes—
244C23×C5—22no—
245C23×C5—22no—
246C23×C5—21no—
247C23×C5—22no—
248C23×C5—21no—
249C23×C5—22no—
250C23×C5#12012yes—
251C23×C5C2×Q1612yes—
252C23×C5C2×Q1612yes—
253C23×C5#13712yes—
254C23×C5C22×D411yescharacteristic
255C5×Q8—24no—
256C5×Q8—24no—
257C5×Q8—24no—
258C5×Q8—48no—
259C5×Q8—41no—
260C5×Q8—41no—
261C5×Q8—42no—
262C5×Q8—42no—
263C5×Q8—84no—
264C5×D4—28no—
265C2×C4×C5—21no—
266C2×C4×C5—22no—
267C2×C4×C5—21no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—24no—
271C2×C4×C5—22no—
272C2×C4×C5—21no—
273C2×C4×C5—22no—
274C2×C4×C5—21no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—24no—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—48no—
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—41no—
289C2×C4×C5—42no—
290C2×C4×C5—42no—
291C2×C4×C5—42no—
292C2×C4×C5—42no—
293C2×C4×C5—84no—
294C2×C4×C5—84no—
295C2×C4×C5#12012yes—
296C2×C4×C5C2×Q1612yes—
297C2×C4×C5C22×D411yescharacteristic
298C2×C4×C5C22×D411yescharacteristic
299C2×C4×C5C2×#4112yes—
300C2×C4×C5H(5,𝔽2)12yes—
301C5×C8—42no—
302C5×C8—42no—
303C22×Q8—21no—
304C22×Q8—21no—
305C22×Q8—24no—
306C22×Q8—22no—
307C22×Q8—22no—
308C22×Q8—42no—
309C22×Q8C23×C511yescharacteristic
310C22×Q8C23×C512yes—
311C22×D4C5×D411yescharacteristic
312C23×C4—21no—
313C23×C4—21no—
314C23×C4—21no—
315C23×C4C23×C511yescharacteristic
316C23×C4C23×C511yescharacteristic
317C2×Q16—24no—
318C2×Q16—24no—
319C2×Q16—42no—
320C22×C8—22no—
321C22×C8—42no—
322C22×C8C23×C512yes—
323C22×C8C23×C512yes—
324C4⋊Q8—22no—
325C4⋊Q8—22no—
326#125—28no—
327C22⋊Q8—28no—
328C22⋊Q8—28no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8—24no—
333C22⋊Q8—42no—
334C22⋊Q8C5×D414yes—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×D4—28no—
338C4×D4—22no—
339C4×D4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4C5×D411yescharacteristic
351C2×C4⋊C4C5×D414yes—
352C2×C4⋊C4C23×C512yes—
353C2×C4⋊C4C23×C511yescharacteristic
354C2×C4⋊C4C23×C511yescharacteristic
355C2×C4⋊C4C23×C511yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4C5×D411yescharacteristic
361C2×C22⋊C4C5×D411yescharacteristic
362C2×C42—21no—
363C2×C42C23×C511yescharacteristic
364#103—24no—
365#103—24no—
366#103—24no—
367#103—42no—
368C22⋊C8—22no—
369C22⋊C8—42no—
370#95—21no—
371#95—22no—
372#95—22no—
373C22×C5—22no—
374C22×C5—22no—
375C22×C5—24no—
376C22×C5—24no—
377C22×C5—22no—
378C22×C5—24no—
379C22×C5—41no—
380C22×C5—41no—
381C22×C5—42no—
382C22×C5#45114yes—
383C22×C5C2×#12011yescharacteristic
384C22×C5#54612yes—
385C22×C5C22×Q1611yescharacteristic
386C22×C5C2×#13711yescharacteristic
387C22×C5#57712yes—
388C4×C5—22no—
389C4×C5—22no—
390C4×C5—41no—
391C4×C5—44no—
392C4×C5—42no—
393C4×C5—41no—
394C4×C5—42no—
395C4×C5—42no—
396C4×C5—42no—
397C4×C5—82no—
398C4×C5#52511yescharacteristic
399C4×C5#53511yescharacteristic
400C4×C5#54612yes—
401C24C2×C5×D411yescharacteristic
402C24C2×C5×D411yescharacteristic
403C2×Q8—24no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—28no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—44no—
414C2×Q8—48no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8—82no—
418C2×Q8C2×C5×D412yes—
419C2×Q8C5×#4114yes—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4C5×Q1614yescomplemented
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4C2×C5×D411yescharacteristic
442C22×C4C2×C5×D411yescharacteristic
443C22×C4C2×C5×D411yescharacteristic
444C22×C4C2×C5×D412yes—
445C22×C4C2×C5×D412yes—
446C22×C4C2×C5×D411yescharacteristic
447C22×C4C24×C511yescharacteristic
448Q16—48no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—44no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—44no—
469C4⋊C4C2×C5×D412yes—
470C4⋊C4C2×C5×D412yes—
471C4⋊C4C2×C5×D412yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—44no—
477C22⋊C4—42no—
478C42—24no—
479C42—21no—
480C42—21no—
481C2×C5—24no—
482C2×C5—42no—
483C2×C5#240411yescharacteristic
484C2×C5C2×#45111yescharacteristic
485C2×C5#300911yescharacteristic
486C2×C5#329912yes—
487C2×C5#331612yes—
488C23—22no—
489C23—22no—
490C23—21no—
491C23—22no—
492C23—21no—
493C23—22no—
494C23C5×#12012yes—
495C23C2×C5×Q1612yes—
496C23C2×C5×Q1612yes—
497C23C5×#13712yes—
498C23C22×C5×D411yescharacteristic
499Q8—24no—
500Q8—24no—
501Q8—24no—
502Q8—48no—
503Q8—41no—
504Q8—42no—
505Q8—41no—
506Q8—42no—
507Q8—84no—
508D4—28no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—48no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4C5×#12012yes—
540C2×C4C2×C5×Q1612yes—
541C2×C4C22×C5×D411yescharacteristic
542C2×C4C22×C5×D411yescharacteristic
543C2×C4C2×C5×#4112yes—
544C2×C4C5×H(5,𝔽2)12yes—
545C8—42no—
546C8—42no—
547C5#2397011yescharacteristic, direct product factor, Sylow
548C22—22no—
549C22—24no—
550C22—24no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—41no—
555C22—42no—
556C22—41no—
557C22C5×#45114yes—
558C22C2×C5×#12011yescharacteristic
559C22C5×#54612yes—
560C22C22×C5×Q1611yescharacteristic
561C22C2×C5×#13711yescharacteristic
562C22C5×#57712yes—
563C4—22no—
564C4—22no—
565C4—41no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—44no—
571C4—42no—
572C4—82no—
573C4C5×#52511yescharacteristic
574C4C5×#53511yescharacteristic
575C4C5×#54612yes—
576C2—24no—
577C2—42no—
578C2C5×#240411yescharacteristic
579C2C2×C5×#45111yescharacteristic
580C2C5×#300911yescharacteristic
581C2C5×#329912yes—
582C2C5×#331612yes—
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC22×C5×D4D4
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×#451C2semidirect
3. C22×C5×#103C2semidirect
5. C5×#2024C2semidirect
6. C5×#2001C2semidirect
7. C5×#1892C2semidirect
8. C5×#1878C2semidirect
9. C5×#1872C2semidirect
11. C5×#1824C2semidirect
47. #23970C5direct (N × H)
119. C2×#451C2×C5semidirect
120. C22×#103C2×C5semidirect
122. #2024C2×C5semidirect
123. #2001C2×C5semidirect
124. #1892C2×C5semidirect
125. #1878C2×C5semidirect
126. #1872C2×C5semidirect
128. #1824C2×C5semidirect
150. C2×C5×D4Q16semidirect
422. C2×D4C5×Q16semidirect
547. C5#23970direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5542
C2×C5×#4512
C22×C5×#1032
C5×#24092
C5×#20242
C5×#20012
C5×#18922
C5×#18782
C5×#18722
C5×#18692
C5×#18242
#239705