Finite Group Databasebeta

← C5×#24157 #11946877

Subgroups — C5×#24157 #11946877

Order 1280.

1114 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#533C211yesmaximal, characteristic
2C2×C5×#475C211yesmaximal, characteristic, complemented
3C2×C5×#462C211yesmaximal, characteristic, complemented
4C4×C5×C22⋊Q8C211yesmaximal, characteristic
5C5×#2035C212yesmaximal
6C5×#2001C212yesmaximal
7C5×#1988C211yesmaximal, characteristic, complemented
8C5×#1981C211yesmaximal, characteristic, complemented
9C5×#1878C212yesmaximal, complemented
10C5×#1865C212yesmaximal, complemented
11C5×#1797C211yesmaximal, characteristic
12C22×C5×Q16C2212yes—
13C5×#533—24no—
14C5×#533—24no—
15C2×C5×C4⋊Q8C2211yescharacteristic
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C5×#125—21no—
18C2×C5×C22⋊Q8—21no—
19C2×C5×C22⋊Q8C2212yes—
20C2×C5×C22⋊Q8C2211yescharacteristic
21C2×C4×C5×Q8—21no—
22C2×C4×C5×Q8C2211yescharacteristic
23C2×C4×C5×Q8C2212yes—
24C2×C5×#117—21no—
25C22×C42×C5C2211yescharacteristic
26C5×#475—24no—
27C5×#462—24no—
28C2×C5×#106C2212yes—
29C2×C5×C4⋊C8C2211yescharacteristic
30C2×C5×C4⋊C8C2211yescharacteristic
31C2×C5×#103C2212yes—
32C2×C5×#103C2212yes—
33C2×C5×#103C2212yes—
34C2×C5×#103C2212yes—
35C2×C5×C22⋊C8C2212yes—
36C5×#391C2211yescharacteristic
37C5×#389C2211yescharacteristic
38C5×#387—21no—
39C5×#385—21no—
40C5×#380C2211yescharacteristic
41C4×C5×C4⋊C4—21no—
42C4×C5×C4⋊C4C2211yescharacteristic
43C4×C5×C22⋊C4—21no—
44C5×#340C2212yes—
45C5×#340C2212yes—
46C5×#336C2212yes—
47#24157C511yesmaximal, characteristic, direct product factor, Sylow
48C22×C5×Q8—21no—
49C22×C5×Q8C2312yes—
50C22×C5×Q8C2311yescharacteristic
51C23×C4×C5D411yescharacteristic
52C23×C4×C5D411yescharacteristic
53C23×C4×C5C2311yescharacteristic
54C2×C5×Q16—24no—
55C2×C5×Q16—24no—
56C2×C5×Q16—42no—
57C22×C5×C8C2312yes—
58C22×C5×C8C2312yes—
59C5×C4⋊Q8—22no—
60C5×C4⋊Q8—22no—
61C5×C4⋊Q8D414yes—
62C5×C4⋊Q8D414yes—
63C5×#125—28no—
64C5×C22⋊Q8—28no—
65C5×C22⋊Q8—24no—
66C5×C22⋊Q8—22no—
67C5×C22⋊Q8—24no—
68C5×C22⋊Q8—22no—
69C4×C5×Q8—28no—
70C4×C5×Q8—24no—
71C4×C5×Q8—24no—
72C4×C5×Q8—22no—
73C4×C5×Q8—22no—
74C5×#117—28no—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—22no—
80C2×C5×C4⋊C4—22no—
81C2×C5×C4⋊C4—21no—
82C2×C5×C4⋊C4—21no—
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C5×C4⋊C4C2311yescharacteristic
85C2×C5×C4⋊C4C2312yes—
86C2×C5×C4⋊C4C2311yescharacteristic
87C2×C5×C22⋊C4—21no—
88C2×C5×C22⋊C4—22no—
89C2×C5×C22⋊C4—21no—
90C2×C42×C5—22no—
91C2×C42×C5—22no—
92C2×C42×C5—22no—
93C2×C42×C5—21no—
94C2×C42×C5—21no—
95C2×C42×C5D411yescharacteristic
96C2×C42×C5D411yescharacteristic
97C2×C42×C5C2311yescharacteristic
98C2×C42×C5C2311yescharacteristic
99C5×#106—24no—
100C5×C4⋊C8—22no—
101C5×C4⋊C8—22no—
102C5×#103—24no—
103C5×#103—24no—
104C5×#103—24no—
105C5×#103—42no—
106C5×C22⋊C8—42no—
107C5×#95—21no—
108C5×#95—21no—
109C5×#95—21no—
110C5×#95—21no—
111C5×#95—21no—
112C2×#533C2×C511yescharacteristic
113C2×#475C2×C511yescharacteristic, complemented
114C2×#462C2×C511yescharacteristic, complemented
115C4×C22⋊Q8C2×C511yescharacteristic
116#2035C2×C512yes—
117#2001C2×C512yes—
118#1988C2×C511yescharacteristic, complemented
119#1981C2×C511yescharacteristic, complemented
120#1878C2×C512yescomplemented
121#1865C2×C512yescomplemented
122#1797C2×C511yescharacteristic
123C24×C5C2×D411yescharacteristic
124C2×C5×Q8—24no—
125C2×C5×Q8—22no—
126C2×C5×Q8—21no—
127C2×C5×Q8—21no—
128C2×C5×Q8—21no—
129C2×C5×Q8—22no—
130C2×C5×Q8—22no—
131C2×C5×Q8—21no—
132C2×C5×Q8—22no—
133C2×C5×Q8—22no—
134C2×C5×Q8—21no—
135C2×C5×Q8—42no—
136C2×C5×Q8—42no—
137C2×C5×Q8C2×D414yes—
138C2×C5×Q8#4112yes—
139C22×C4×C5—22no—
140C22×C4×C5—22no—
141C22×C4×C5—22no—
142C22×C4×C5—22no—
143C22×C4×C5—22no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—22no—
147C22×C4×C5—22no—
148C22×C4×C5—22no—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5—21no—
153C22×C4×C5—21no—
154C22×C4×C5—21no—
155C22×C4×C5—21no—
156C22×C4×C5—21no—
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2411yescharacteristic
163C5×Q16—48no—
164C2×C5×C8—22no—
165C2×C5×C8—22no—
166C2×C5×C8—22no—
167C2×C5×C8—22no—
168C2×C5×C8—22no—
169C2×C5×C8—22no—
170C5×C4⋊C4—28no—
171C5×C4⋊C4—28no—
172C5×C4⋊C4—24no—
173C5×C4⋊C4—24no—
174C5×C4⋊C4—24no—
175C5×C4⋊C4—24no—
176C5×C4⋊C4—22no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4—42no—
183C5×C4⋊C4—42no—
184C5×C4⋊C4C2×D414yes—
185C5×C4⋊C4#4112yes—
186C5×C22⋊C4—28no—
187C5×C22⋊C4—24no—
188C5×C22⋊C4—42no—
189C42×C5—28no—
190C42×C5—22no—
191C42×C5—22no—
192C42×C5—21no—
193C42×C5—21no—
194C42×C5—42no—
195C42×C5—42no—
196C42×C5C2×D412yes—
197C42×C5C2×D412yes—
198C22×Q16C22×C512yes—
199#533—24no—
200#533—24no—
201C2×C4⋊Q8C22×C511yescharacteristic
202C2×C4⋊Q8C22×C511yescharacteristic
203C2×#125—21no—
204C2×C22⋊Q8—21no—
205C2×C22⋊Q8C22×C512yes—
206C2×C22⋊Q8C22×C511yescharacteristic
207C2×C4×Q8—21no—
208C2×C4×Q8C22×C511yescharacteristic
209C2×C4×Q8C22×C512yes—
210C2×#117—21no—
211C22×C42C22×C511yescharacteristic
212#475—24no—
213#462—24no—
214C2×#106C22×C512yes—
215C2×C4⋊C8C22×C511yescharacteristic
216C2×C4⋊C8C22×C511yescharacteristic
217C2×#103C22×C512yes—
218C2×#103C22×C512yes—
219C2×#103C22×C512yes—
220C2×#103C22×C512yes—
221C2×C22⋊C8C22×C512yes—
222#391C22×C511yescharacteristic
223#389C22×C511yescharacteristic
224#387—21no—
225#385—21no—
226#380C22×C511yescharacteristic
227C4×C4⋊C4—21no—
228C4×C4⋊C4C22×C511yescharacteristic
229C4×C22⋊C4—21no—
230#340C22×C512yes—
231#340C22×C512yes—
232#336C22×C512yes—
233C23×C5—21no—
234C23×C5—21no—
235C23×C5—21no—
236C23×C5—21no—
237C23×C5—21no—
238C23×C5—21no—
239C23×C5#13712yes—
240C23×C5C22×D411yescharacteristic
241C5×Q8—22no—
242C5×Q8—24no—
243C5×Q8—24no—
244C5×Q8—22no—
245C5×Q8—41no—
246C5×Q8—42no—
247C5×Q8—42no—
248C5×Q8—41no—
249C5×Q8—48no—
250C2×C4×C5—28no—
251C2×C4×C5—21no—
252C2×C4×C5—21no—
253C2×C4×C5—21no—
254C2×C4×C5—21no—
255C2×C4×C5—21no—
256C2×C4×C5—21no—
257C2×C4×C5—24no—
258C2×C4×C5—24no—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—41no—
266C2×C4×C5—42no—
267C2×C4×C5—42no—
268C2×C4×C5—41no—
269C2×C4×C5—41no—
270C2×C4×C5—42no—
271C2×C4×C5—41no—
272C2×C4×C5—42no—
273C2×C4×C5—41no—
274C2×C4×C5—41no—
275C2×C4×C5—41no—
276C2×C4×C5—41no—
277C2×C4×C5—44no—
278C2×C4×C5—41no—
279C2×C4×C5—41no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5#12012yes—
285C2×C4×C5#12012yes—
286C2×C4×C5#13512yes—
287C2×C4×C5#13712yes—
288C2×C4×C5#13712yes—
289C2×C4×C5C22×D411yescharacteristic
290C2×C4×C5C22×D411yescharacteristic
291C2×C4×C5C2×#4111yescharacteristic
292C2×C4×C5C2×#4111yescharacteristic
293C2×C4×C5H(5,𝔽2)11yescharacteristic
294C2×C4×C52−1+411yescharacteristic
295C5×C8—42no—
296C5×C8—42no—
297C22×Q8—21no—
298C22×Q8C23×C512yes—
299C22×Q8C23×C511yescharacteristic
300C23×C4C5×D411yescharacteristic
301C23×C4C5×D411yescharacteristic
302C23×C4C23×C511yescharacteristic
303C2×Q16—24no—
304C2×Q16—24no—
305C2×Q16—42no—
306C22×C8C23×C512yes—
307C22×C8C23×C512yes—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310C4⋊Q8C5×D414yes—
311C4⋊Q8C5×D414yes—
312#125—28no—
313C22⋊Q8—28no—
314C22⋊Q8—24no—
315C22⋊Q8—22no—
316C22⋊Q8—24no—
317C22⋊Q8—22no—
318C4×Q8—28no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—22no—
322C4×Q8—22no—
323#117—28no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4C23×C511yescharacteristic
333C2×C4⋊C4C23×C511yescharacteristic
334C2×C4⋊C4C23×C512yes—
335C2×C4⋊C4C23×C511yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42C5×D411yescharacteristic
345C2×C42C5×D411yescharacteristic
346C2×C42C23×C511yescharacteristic
347C2×C42C23×C511yescharacteristic
348#106—24no—
349C4⋊C8—22no—
350C4⋊C8—22no—
351#103—24no—
352#103—24no—
353#103—24no—
354#103—42no—
355C22⋊C8—42no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361C22×C5—22no—
362C22×C5—22no—
363C22×C5—22no—
364C22×C5—41no—
365C22×C5—41no—
366C22×C5—41no—
367C22×C5—41no—
368C22×C5C2×#12011yescharacteristic
369C22×C5#52511yescharacteristic
370C22×C5#53711yescharacteristic
371C22×C5C2×#13511yescharacteristic
372C22×C5C2×#13711yescharacteristic
373C22×C5C2×#13711yescharacteristic
374C22×C5C2×#13711yescharacteristic
375C4×C5—22no—
376C4×C5—22no—
377C4×C5—42no—
378C4×C5—41no—
379C4×C5—44no—
380C4×C5—42no—
381C4×C5—41no—
382C4×C5—44no—
383C4×C5—81no—
384C4×C5—81no—
385C4×C5#45214yes—
386C4×C5#54611yescharacteristic
387C4×C5#54711yescharacteristic
388C4×C5#54811yescharacteristic
389C4×C5#55011yescharacteristic
390C24C2×C5×D411yescharacteristic
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8C2×C5×D414yes—
405C2×Q8C5×#4112yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4C2×C5×D411yescharacteristic
425C22×C4C2×C5×D411yescharacteristic
426C22×C4C2×C5×D411yescharacteristic
427C22×C4C2×C5×D411yescharacteristic
428C22×C4C2×C5×D411yescharacteristic
429C22×C4C24×C511yescharacteristic
430Q16—48no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4C2×C5×D414yes—
452C4⋊C4C5×#4112yes—
453C22⋊C4—28no—
454C22⋊C4—24no—
455C22⋊C4—42no—
456C42—28no—
457C42—22no—
458C42—22no—
459C42—21no—
460C42—21no—
461C42—42no—
462C42—42no—
463C42C2×C5×D412yes—
464C42C2×C5×D412yes—
465C2×C5—42no—
466C2×C5#240711yescharacteristic
467C2×C5C2×#45211yescharacteristic
468C2×C5#300411yescharacteristic
469C2×C5#310711yescharacteristic
470C2×C5#311311yescharacteristic
471C2×C5#324811yescharacteristic
472C2×C5#325511yescharacteristic
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23C5×#13712yes—
480C23C22×C5×D411yescharacteristic
481Q8—22no—
482Q8—24no—
483Q8—24no—
484Q8—22no—
485Q8—48no—
486Q8—41no—
487Q8—42no—
488Q8—42no—
489Q8—41no—
490C2×C4—28no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4C5×#12012yes—
525C2×C4C5×#12012yes—
526C2×C4C5×#13512yes—
527C2×C4C5×#13712yes—
528C2×C4C5×#13712yes—
529C2×C4C22×C5×D411yescharacteristic
530C2×C4C22×C5×D411yescharacteristic
531C2×C4C2×C5×#4111yescharacteristic
532C2×C4C2×C5×#4111yescharacteristic
533C2×C4C5×H(5,𝔽2)11yescharacteristic
534C2×C4C5×2−1+411yescharacteristic
535C8—42no—
536C8—42no—
537C5#2415711yescharacteristic, direct product factor, Sylow
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22C2×C5×#12011yescharacteristic
546C22C5×#52511yescharacteristic
547C22C5×#53711yescharacteristic
548C22C2×C5×#13511yescharacteristic
549C22C2×C5×#13711yescharacteristic
550C22C2×C5×#13711yescharacteristic
551C22C2×C5×#13711yescharacteristic
552C4—22no—
553C4—22no—
554C4—41no—
555C4—44no—
556C4—42no—
557C4—41no—
558C4—44no—
559C4—42no—
560C4—81no—
561C4—81no—
562C4C5×#45214yes—
563C4C5×#54611yescharacteristic
564C4C5×#54711yescharacteristic
565C4C5×#54811yescharacteristic
566C4C5×#55011yescharacteristic
567C2—42no—
568C2C5×#240711yescharacteristic
569C2C2×C5×#45211yescharacteristic
570C2C5×#300411yescharacteristic
571C2C5×#310711yescharacteristic
572C2C5×#311311yescharacteristic
573C2C5×#324811yescharacteristic
574C2C5×#325511yescharacteristic
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC24×C5C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#475C2semidirect
3. C2×C5×#462C2semidirect
7. C5×#1988C2semidirect
8. C5×#1981C2semidirect
9. C5×#1878C2semidirect
10. C5×#1865C2semidirect
47. #24157C5direct (N × H)
113. C2×#475C2×C5semidirect
114. C2×#462C2×C5semidirect
118. #1988C2×C5semidirect
119. #1981C2×C5semidirect
120. #1878C2×C5semidirect
121. #1865C2×C5semidirect
537. C5#24157direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5332
C2×C5×#4752
C2×C5×#4622
C4×C5×C22⋊Q82
C5×#20352
C5×#20012
C5×#19882
C5×#19812
C5×#18782
C5×#18652
C5×#17972
#241575