Finite Group Databasebeta

← C5×#17387 #11940107

Subgroups — C5×#17387 #11940107

Order 1280.

984 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#539C211yesmaximal, characteristic
2C5×#2973C211yesmaximal, characteristic, complemented
3C5×#2972C211yesmaximal, characteristic, complemented
4C5×#1473C212yesmaximal, complemented
5C5×#1472C212yesmaximal, complemented
6C5×#1470C212yesmaximal, complemented
7C5×#1468C212yesmaximal, complemented
8C5×#1459C212yesmaximal, complemented
9C2×C5×#324C212yesmaximal, complemented
10C5×#539—24no—
11C5×#539—24no—
12C2×C5×#126C411yescharacteristic
13C2×C5×#126C2211yescharacteristic
14C2×C5×#125C411yescharacteristic
15C2×C5×#125C2211yescharacteristic
16C2×C5×#123C411yescharacteristic
17C2×C5×#123C411yescharacteristic
18C2×C5×#123C411yescharacteristic
19C2×C5×#123C2211yescharacteristic
20C2×C5×C22⋊Q8C411yescharacteristic
21C2×C5×C22⋊Q8C411yescharacteristic
22C2×C5×C22⋊Q8C2211yescharacteristic
23C2×C5×#121C411yescharacteristic
24C2×C4×C5×D4C2211yescharacteristic
25C2×C4×C5×D4C2211yescharacteristic
26C22×C5×C4⋊C4C2211yescharacteristic
27C5×#435—24no—
28C5×C8×D4—24no—
29C5×#432C2212yes—
30C5×#431C2212yes—
31C5×#407C2212yes—
32C2×C5×C22⋊C8C2212yes—
33C2×C5×C22⋊C8C2212yes—
34C2×C5×C22⋊C8C2212yes—
35C5×#324C2214yes—
36C5×#324C2214yes—
37C5×#324C2214yes—
38C5×#323C2214yes—
39#17387C511yesmaximal, characteristic, direct product factor, Sylow
40C22×C5×Q8C2×C411yescharacteristic
41C22×C5×D4C2×C411yescharacteristic
42C22×C5×D4C2×C411yescharacteristic
43C22×C5×D4C2×C411yescharacteristic
44C23×C4×C5C2×C411yescharacteristic
45C23×C4×C5C2×C411yescharacteristic
46C23×C4×C5C2×C411yescharacteristic
47C23×C4×C5C2311yescharacteristic
48C23×C4×C5C2311yescharacteristic
49C2×C5×M16—22no—
50C22×C5×C8—22no—
51C22×C5×C8—22no—
52C22×C5×C8—22no—
53C5×#126—24no—
54C5×#126—22no—
55C5×#126—22no—
56C5×#125—24no—
57C5×#125—22no—
58C5×#125—22no—
59C5×#123—24no—
60C5×#123—22no—
61C5×#123—22no—
62C5×#123—24no—
63C5×#123—24no—
64C5×C22⋊Q8—22no—
65C5×C22⋊Q8—22no—
66C5×C22⋊Q8—24no—
67C5×C22⋊Q8—24no—
68C5×#121—24no—
69C4×C5×D4—22no—
70C4×C5×D4—22no—
71C4×C5×D4D414yes—
72C4×C5×D4D414yes—
73C2×C5×C4⋊C4—22no—
74C2×C5×C4⋊C4—22no—
75C2×C5×C4⋊C4C2×C411yescharacteristic
76C2×C5×C4⋊C4C2×C411yescharacteristic
77C2×C5×C4⋊C4C2×C411yescharacteristic
78C2×C5×C4⋊C4C2×C411yescharacteristic
79C2×C5×C4⋊C4C2×C411yescharacteristic
80C2×C5×C4⋊C4C2×C411yescharacteristic
81C2×C5×C4⋊C4C2×C411yescharacteristic
82C2×C5×C4⋊C4C2×C411yescharacteristic
83C2×C5×C4⋊C4C2×C411yescharacteristic
84C2×C5×C4⋊C4C2×C411yescharacteristic
85C2×C5×C4⋊C4C2×C411yescharacteristic
86C2×C5×C4⋊C4C2311yescharacteristic
87C2×C5×C4⋊C4C2311yescharacteristic
88C2×C5×C4⋊C4C2311yescharacteristic
89C2×C5×C22⋊C4C2×C411yescharacteristic
90C2×C5×C22⋊C4C2×C411yescharacteristic
91C2×C5×C22⋊C4C2×C411yescharacteristic
92C2×C5×C22⋊C4C2×C411yescharacteristic
93C2×C5×C22⋊C4C2×C411yescharacteristic
94C2×C5×C22⋊C4C2×C411yescharacteristic
95C2×C5×C22⋊C4C2×C411yescharacteristic
96C2×C5×C22⋊C4C2×C411yescharacteristic
97C2×C5×C22⋊C4C2×C411yescharacteristic
98C2×C5×C22⋊C4C2311yescharacteristic
99C2×C42×C5C2×C411yescharacteristic
100C2×C42×C5C2311yescharacteristic
101C5×C4⋊C8—22no—
102C5×C4⋊C8—22no—
103C5×C22⋊C8—22no—
104C5×C22⋊C8—22no—
105C5×C22⋊C8—22no—
106C5×C22⋊C8—22no—
107C5×C22⋊C8C2314yes—
108C5×C22⋊C8C2314yes—
109C5×#97—22no—
110C4×C5×C8—22no—
111C2×#539C2×C511yescharacteristic
112#2973C2×C511yescharacteristic, complemented
113#2972C2×C511yescharacteristic, complemented
114#1473C2×C512yescomplemented
115#1472C2×C512yescomplemented
116#1470C2×C512yescomplemented
117#1468C2×C512yescomplemented
118#1459C2×C512yescomplemented
119C2×#324C2×C512yescomplemented
120C24×C5C22×C411yescharacteristic
121C24×C5C22×C411yescharacteristic
122C24×C5C22×C411yescharacteristic
123C2×C5×Q8—22no—
124C2×C5×Q8—81no—
125C2×C5×D4—24no—
126C2×C5×D4—22no—
127C2×C5×D4—24no—
128C2×C5×D4—22no—
129C2×C5×D4—81no—
130C2×C5×D4C22⋊C414yes—
131C22×C4×C5—22no—
132C22×C4×C5—24no—
133C22×C4×C5—22no—
134C22×C4×C5—22no—
135C22×C4×C5—81no—
136C22×C4×C5—81no—
137C22×C4×C5—81no—
138C22×C4×C5—81no—
139C22×C4×C5C22×C411yescharacteristic
140C22×C4×C5C22×C411yescharacteristic
141C22×C4×C5C22×C411yescharacteristic
142C22×C4×C5C22×C411yescharacteristic
143C22×C4×C5C22×C411yescharacteristic
144C22×C4×C5C22×C411yescharacteristic
145C22×C4×C5C22×C411yescharacteristic
146C22×C4×C5C22×C411yescharacteristic
147C22×C4×C5C22×C411yescharacteristic
148C22×C4×C5C22×C411yescharacteristic
149C22×C4×C5C22×C411yescharacteristic
150C22×C4×C5C2×D412yes—
151C22×C4×C5C2×D412yes—
152C22×C4×C5C2×D412yes—
153C22×C4×C5C2×D412yes—
154C22×C4×C5C2411yescharacteristic
155C5×M16—82no—
156C2×C5×C8—24no—
157C2×C5×C8—24no—
158C2×C5×C8—82no—
159C2×C5×C8—82no—
160C2×C5×C8—82no—
161C5×C4⋊C4—22no—
162C5×C4⋊C4—22no—
163C5×C4⋊C4—22no—
164C5×C4⋊C4—22no—
165C5×C4⋊C4—22no—
166C5×C4⋊C4—22no—
167C5×C4⋊C4—22no—
168C5×C4⋊C4—22no—
169C5×C4⋊C4—22no—
170C5×C4⋊C4—22no—
171C5×C4⋊C4—21no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—21no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4C22⋊C414yes—
178C5×C22⋊C4—22no—
179C5×C22⋊C4—22no—
180C5×C22⋊C4—22no—
181C5×C22⋊C4—22no—
182C5×C22⋊C4—22no—
183C5×C22⋊C4—22no—
184C5×C22⋊C4—22no—
185C5×C22⋊C4—21no—
186C5×C22⋊C4—21no—
187C5×C22⋊C4C22⋊C414yes—
188C5×C22⋊C4C22⋊C414yes—
189C42×C5—22no—
190C42×C5C2×D412yes—
191C42×C5C2×D412yes—
192#539—24no—
193#539—24no—
194C2×#126C4×C511yescharacteristic
195C2×#126C22×C511yescharacteristic
196C2×#125C4×C511yescharacteristic
197C2×#125C22×C511yescharacteristic
198C2×#123C4×C511yescharacteristic
199C2×#123C4×C511yescharacteristic
200C2×#123C4×C511yescharacteristic
201C2×#123C22×C511yescharacteristic
202C2×C22⋊Q8C4×C511yescharacteristic
203C2×C22⋊Q8C4×C511yescharacteristic
204C2×C22⋊Q8C22×C511yescharacteristic
205C2×#121C4×C511yescharacteristic
206C2×C4×D4C22×C511yescharacteristic
207C2×C4×D4C22×C511yescharacteristic
208C22×C4⋊C4C22×C511yescharacteristic
209#435—24no—
210C8×D4—24no—
211#432C22×C512yes—
212#431C22×C512yes—
213#407C22×C512yes—
214C2×C22⋊C8C22×C512yes—
215C2×C22⋊C8C22×C512yes—
216C2×C22⋊C8C22×C512yes—
217#324C22×C514yes—
218#324C22×C514yes—
219#324C22×C514yes—
220#323C22×C514yes—
221C23×C5—22no—
222C23×C5—21no—
223C23×C5—22no—
224C23×C5—41no—
225C23×C5—41no—
226C23×C5—41no—
227C23×C5—41no—
228C23×C5—41no—
229C23×C5—41no—
230C23×C5C2×C22⋊C412yes—
231C23×C5C2×C22⋊C412yes—
232C23×C5#13114yes—
233C23×C5C23×C411yescharacteristic
234C5×Q8—82no—
235C5×D4—28no—
236C5×D4—44no—
237C5×D4—82no—
238C2×C4×C5—22no—
239C2×C4×C5—21no—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—21no—
244C2×C4×C5—21no—
245C2×C4×C5—21no—
246C2×C4×C5—22no—
247C2×C4×C5—22no—
248C2×C4×C5—22no—
249C2×C4×C5—42no—
250C2×C4×C5—42no—
251C2×C4×C5—42no—
252C2×C4×C5—42no—
253C2×C4×C5—41no—
254C2×C4×C5—41no—
255C2×C4×C5—41no—
256C2×C4×C5—41no—
257C2×C4×C5—44no—
258C2×C4×C5—41no—
259C2×C4×C5—41no—
260C2×C4×C5—41no—
261C2×C4×C5—41no—
262C2×C4×C5—82no—
263C2×C4×C5—82no—
264C2×C4×C5C2×C22⋊C412yes—
265C2×C4×C5C2×C22⋊C412yes—
266C2×C4×C5C2×C22⋊C412yes—
267C2×C4×C5C2×C22⋊C412yes—
268C2×C4×C5C22×D411yescharacteristic
269C2×C4×C5C22×D411yescharacteristic
270C5×C8—44no—
271C5×C8—44no—
272C22×Q8C2×C4×C511yescharacteristic
273C22×D4C2×C4×C511yescharacteristic
274C22×D4C2×C4×C511yescharacteristic
275C22×D4C2×C4×C511yescharacteristic
276C23×C4C2×C4×C511yescharacteristic
277C23×C4C2×C4×C511yescharacteristic
278C23×C4C2×C4×C511yescharacteristic
279C23×C4C23×C511yescharacteristic
280C23×C4C23×C511yescharacteristic
281C2×M16—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285#126—24no—
286#126—22no—
287#126—22no—
288#125—24no—
289#125—22no—
290#125—22no—
291#123—24no—
292#123—22no—
293#123—22no—
294#123—24no—
295#123—24no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300#121—24no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4C5×D414yes—
304C4×D4C5×D414yes—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4C2×C4×C511yescharacteristic
308C2×C4⋊C4C2×C4×C511yescharacteristic
309C2×C4⋊C4C2×C4×C511yescharacteristic
310C2×C4⋊C4C2×C4×C511yescharacteristic
311C2×C4⋊C4C2×C4×C511yescharacteristic
312C2×C4⋊C4C2×C4×C511yescharacteristic
313C2×C4⋊C4C2×C4×C511yescharacteristic
314C2×C4⋊C4C2×C4×C511yescharacteristic
315C2×C4⋊C4C2×C4×C511yescharacteristic
316C2×C4⋊C4C2×C4×C511yescharacteristic
317C2×C4⋊C4C2×C4×C511yescharacteristic
318C2×C4⋊C4C23×C511yescharacteristic
319C2×C4⋊C4C23×C511yescharacteristic
320C2×C4⋊C4C23×C511yescharacteristic
321C2×C22⋊C4C2×C4×C511yescharacteristic
322C2×C22⋊C4C2×C4×C511yescharacteristic
323C2×C22⋊C4C2×C4×C511yescharacteristic
324C2×C22⋊C4C2×C4×C511yescharacteristic
325C2×C22⋊C4C2×C4×C511yescharacteristic
326C2×C22⋊C4C2×C4×C511yescharacteristic
327C2×C22⋊C4C2×C4×C511yescharacteristic
328C2×C22⋊C4C2×C4×C511yescharacteristic
329C2×C22⋊C4C2×C4×C511yescharacteristic
330C2×C22⋊C4C23×C511yescharacteristic
331C2×C42C2×C4×C511yescharacteristic
332C2×C42C23×C511yescharacteristic
333C4⋊C8—22no—
334C4⋊C8—22no—
335C22⋊C8—22no—
336C22⋊C8—22no—
337C22⋊C8—22no—
338C22⋊C8—22no—
339C22⋊C8C23×C514yes—
340C22⋊C8C23×C514yes—
341#97—22no—
342C4×C8—22no—
343C22×C5—21no—
344C22×C5—22no—
345C22×C5—21no—
346C22×C5—22no—
347C22×C5—22no—
348C22×C5—22no—
349C22×C5—22no—
350C22×C5—44no—
351C22×C5—41no—
352C22×C5—41no—
353C22×C5—41no—
354C22×C5—82no—
355C22×C5—82no—
356C22×C5#40814yes—
357C22×C5C22×C22⋊C411yescharacteristic
358C22×C5C2×#13112yes—
359C4×C5—22no—
360C4×C5—21no—
361C4×C5—21no—
362C4×C5—22no—
363C4×C5—22no—
364C4×C5—22no—
365C4×C5—41no—
366C4×C5—41no—
367C4×C5—41no—
368C4×C5—41no—
369C4×C5—41no—
370C4×C5—41no—
371C4×C5—41no—
372C24C22×C4×C511yescharacteristic
373C24C22×C4×C511yescharacteristic
374C24C22×C4×C511yescharacteristic
375C2×Q8—22no—
376C2×Q8—81no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—24no—
380C2×D4—22no—
381C2×D4—81no—
382C2×D4C5×C22⋊C414yes—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4C22×C4×C511yescharacteristic
392C22×C4C22×C4×C511yescharacteristic
393C22×C4C22×C4×C511yescharacteristic
394C22×C4C22×C4×C511yescharacteristic
395C22×C4C22×C4×C511yescharacteristic
396C22×C4C22×C4×C511yescharacteristic
397C22×C4C22×C4×C511yescharacteristic
398C22×C4C22×C4×C511yescharacteristic
399C22×C4C22×C4×C511yescharacteristic
400C22×C4C22×C4×C511yescharacteristic
401C22×C4C22×C4×C511yescharacteristic
402C22×C4C2×C5×D412yes—
403C22×C4C2×C5×D412yes—
404C22×C4C2×C5×D412yes—
405C22×C4C2×C5×D412yes—
406C22×C4C24×C511yescharacteristic
407M16—82no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—82no—
411C2×C8—82no—
412C2×C8—82no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4C5×C22⋊C414yes—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4C5×C22⋊C414yes—
440C22⋊C4C5×C22⋊C414yes—
441C42—22no—
442C42C2×C5×D412yes—
443C42C2×C5×D412yes—
444C2×C5—22no—
445C2×C5—24no—
446C2×C5—42no—
447C2×C5—81no—
448C2×C5C2×#40811yescharacteristic
449C2×C5#288311yescharacteristic
450C2×C5#288911yescharacteristic
451C23—22no—
452C23—21no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23C2×C5×C22⋊C412yes—
461C23C2×C5×C22⋊C412yes—
462C23C5×#13114yes—
463C23C23×C4×C511yescharacteristic
464Q8—82no—
465D4—28no—
466D4—44no—
467D4—82no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4C2×C5×C22⋊C412yes—
495C2×C4C2×C5×C22⋊C412yes—
496C2×C4C2×C5×C22⋊C412yes—
497C2×C4C2×C5×C22⋊C412yes—
498C2×C4C22×C5×D411yescharacteristic
499C2×C4C22×C5×D411yescharacteristic
500C8—44no—
501C8—44no—
502C5#1738711yescharacteristic, direct product factor, Sylow
503C22—21no—
504C22—22no—
505C22—21no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—44no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—82no—
515C22—82no—
516C22C5×#40814yes—
517C22C22×C5×C22⋊C411yescharacteristic
518C22C2×C5×#13112yes—
519C4—22no—
520C4—21no—
521C4—21no—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—41no—
532C2—22no—
533C2—24no—
534C2—42no—
535C2—81no—
536C2C2×C5×#40811yescharacteristic
537C2C5×#288311yescharacteristic
538C2C5×#288911yescharacteristic
subgroupG/H
center, socleC22×C5C22×C22⋊C4
commutatorC23C23×C4×C5
Frattini, cosocleC22×C4C24×C5
prime radicalC2×C5×#121C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C4×C5×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2973C2semidirect
3. C5×#2972C2semidirect
4. C5×#1473C2semidirect
5. C5×#1472C2semidirect
6. C5×#1470C2semidirect
7. C5×#1468C2semidirect
8. C5×#1459C2semidirect
9. C2×C5×#324C2semidirect
39. #17387C5direct (N × H)
112. #2973C2×C5semidirect
113. #2972C2×C5semidirect
114. #1473C2×C5semidirect
115. #1472C2×C5semidirect
116. #1470C2×C5semidirect
117. #1468C2×C5semidirect
118. #1459C2×C5semidirect
119. C2×#324C2×C5semidirect
502. C5#17387direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5392
C5×#29732
C5×#29722
C5×#14732
C5×#14722
C5×#14702
C5×#14682
C5×#14592
C2×C5×#3242
#173875