Finite Group Databasebeta

← C5×#24091 #11946811

Subgroups — C5×#24091 #11946811

Order 1280.

1164 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#553C211yesmaximal, characteristic
2C2×C5×#480C211yesmaximal, characteristic, complemented
3C22×C5×#106C211yesmaximal, characteristic, complemented
4C5×#2486C211yesmaximal, characteristic, complemented
5C5×#2059C212yesmaximal, complemented
6C5×#2034C212yesmaximal, complemented
7C5×#1893C211yesmaximal, characteristic, complemented
8C5×#1869C211yesmaximal, characteristic, complemented
9C5×#1864C212yesmaximal, complemented
10C5×#1854C212yesmaximal, complemented
11C5×#1824C211yesmaximal, characteristic, complemented
12C23×C5×C8C2211yescharacteristic
13C5×#553—24no—
14C5×#553—24no—
15C2×C5×#126—21no—
16C2×C5×#125C2211yescharacteristic
17C2×C5×#123—21no—
18C2×C5×#123—21no—
19C2×C5×#121—21no—
20C2×C5×#121C2211yescharacteristic
21C2×C5×#121C2211yescharacteristic
22C2×C4×C5×D4—21no—
23C2×C4×C5×D4C2211yescharacteristic
24C2×C4×C5×D4C2211yescharacteristic
25C2×C5×#117C2211yescharacteristic
26C22×C5×C4⋊C4C2211yescharacteristic
27C22×C5×C4⋊C4C2211yescharacteristic
28C5×#480—24no—
29C2×C5×#107C2212yes—
30C2×C5×#106—24no—
31C2×C5×#106C2212yes—
32C2×C5×#106C2212yes—
33C2×C5×#102C2212yes—
34C2×C5×#102C2212yes—
35C2×C5×C22⋊C8C2212yes—
36C2×C5×C22⋊C8C2211yescharacteristic
37C5×#398—21no—
38C5×#397—21no—
39C5×#389C2212yes—
40C5×#381—21no—
41C5×#380C2212yes—
42C5×#380C2211yescharacteristic
43C5×#340C2212yes—
44C5×#340C2212yes—
45C5×#340C2212yes—
46C5×#336C2212yes—
47#24091C511yesmaximal, characteristic, direct product factor, Sylow
48C22×C5×D4—21no—
49C22×C5×D4—21no—
50C22×C5×D4D411yescharacteristic
51C22×C5×D4C2311yescharacteristic
52C23×C4×C5—21no—
53C23×C4×C5—21no—
54C23×C4×C5—21no—
55C23×C4×C5C2311yescharacteristic
56C23×C4×C5C2311yescharacteristic
57C22×C5×C8—22no—
58C22×C5×C8—24no—
59C22×C5×C8C2312yes—
60C22×C5×C8C2312yes—
61C5×#126—28no—
62C5×#125—22no—
63C5×#125—22no—
64C5×#123—28no—
65C5×#123—44no—
66C5×#121—28no—
67C5×#121—22no—
68C5×#121—22no—
69C5×#121—22no—
70C5×#121D414yes—
71C4×C5×D4—28no—
72C4×C5×D4—22no—
73C4×C5×D4—22no—
74C4×C5×D4—22no—
75C4×C5×D4—22no—
76C5×#117—22no—
77C5×#117—22no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—22no—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4—21no—
82C2×C5×C4⋊C4—22no—
83C2×C5×C4⋊C4—22no—
84C2×C5×C4⋊C4—22no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4D411yescharacteristic
87C2×C5×C4⋊C4D414yes—
88C2×C5×C4⋊C4C2312yes—
89C2×C5×C4⋊C4C2312yes—
90C2×C5×C4⋊C4C2311yescharacteristic
91C2×C5×C4⋊C4C2311yescharacteristic
92C2×C5×C4⋊C4C2311yescharacteristic
93C2×C5×C22⋊C4—21no—
94C2×C5×C22⋊C4—21no—
95C2×C5×C22⋊C4—21no—
96C2×C5×C22⋊C4—21no—
97C2×C5×C22⋊C4—21no—
98C2×C5×C22⋊C4—22no—
99C2×C5×C22⋊C4D411yescharacteristic
100C2×C5×C22⋊C4D411yescharacteristic
101C2×C42×C5—21no—
102C2×C42×C5C2311yescharacteristic
103C5×#107—24no—
104C5×#106—24no—
105C5×#106—24no—
106C5×#102—24no—
107C5×#102—24no—
108C5×C22⋊C8—22no—
109C5×C22⋊C8—24no—
110C5×#95—22no—
111C5×#95—21no—
112C5×#95—22no—
113C2×#553C2×C511yescharacteristic
114C2×#480C2×C511yescharacteristic, complemented
115C22×#106C2×C511yescharacteristic, complemented
116#2486C2×C511yescharacteristic, complemented
117#2059C2×C512yescomplemented
118#2034C2×C512yescomplemented
119#1893C2×C511yescharacteristic, complemented
120#1869C2×C511yescharacteristic, complemented
121#1864C2×C512yescomplemented
122#1854C2×C512yescomplemented
123#1824C2×C511yescharacteristic, complemented
124C24×C5—21no—
125C24×C5C2×D411yescharacteristic
126C24×C5C2×D411yescharacteristic
127C2×C5×D4—22no—
128C2×C5×D4—24no—
129C2×C5×D4—22no—
130C2×C5×D4—22no—
131C2×C5×D4—24no—
132C2×C5×D4—21no—
133C2×C5×D4—22no—
134C2×C5×D4—22no—
135C2×C5×D4—42no—
136C2×C5×D4—42no—
137C2×C5×D4—82no—
138C2×C5×D4C2×D412yes—
139C22×C4×C5—22no—
140C22×C4×C5—22no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—24no—
145C22×C4×C5—24no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—22no—
149C22×C4×C5—22no—
150C22×C4×C5—22no—
151C22×C4×C5—22no—
152C22×C4×C5—42no—
153C22×C4×C5—42no—
154C22×C4×C5—42no—
155C22×C4×C5—82no—
156C22×C4×C5—82no—
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2×D412yes—
162C22×C4×C5C2×D412yes—
163C22×C4×C5C2411yescharacteristic
164C2×C5×C8—28no—
165C2×C5×C8—22no—
166C2×C5×C8—22no—
167C2×C5×C8—22no—
168C2×C5×C8—22no—
169C2×C5×C8—22no—
170C2×C5×C8#4114yes—
171C5×C4⋊C4—24no—
172C5×C4⋊C4—24no—
173C5×C4⋊C4—24no—
174C5×C4⋊C4—22no—
175C5×C4⋊C4—21no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—22no—
179C5×C4⋊C4—22no—
180C5×C4⋊C4—22no—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—44no—
183C5×C4⋊C4—42no—
184C5×C4⋊C4C2×D412yes—
185C5×C4⋊C4C2×D412yes—
186C5×C4⋊C4C2×D412yes—
187C5×C22⋊C4—24no—
188C5×C22⋊C4—24no—
189C5×C22⋊C4—24no—
190C5×C22⋊C4—22no—
191C5×C22⋊C4—24no—
192C5×C22⋊C4—22no—
193C5×C22⋊C4—44no—
194C5×C22⋊C4—42no—
195C42×C5—24no—
196C42×C5—21no—
197C42×C5—21no—
198C23×C8C22×C511yescharacteristic
199#553—24no—
200#553—24no—
201C2×#126—21no—
202C2×#125C22×C511yescharacteristic
203C2×#123—21no—
204C2×#123—21no—
205C2×#121—21no—
206C2×#121C22×C511yescharacteristic
207C2×#121C22×C511yescharacteristic
208C2×C4×D4—21no—
209C2×C4×D4C22×C511yescharacteristic
210C2×C4×D4C22×C511yescharacteristic
211C2×#117C22×C511yescharacteristic
212C22×C4⋊C4C22×C511yescharacteristic
213C22×C4⋊C4C22×C511yescharacteristic
214#480—24no—
215C2×#107C22×C512yes—
216C2×#106—24no—
217C2×#106C22×C512yes—
218C2×#106C22×C512yes—
219C2×#102C22×C512yes—
220C2×#102C22×C512yes—
221C2×C22⋊C8C22×C511yescharacteristic
222C2×C22⋊C8C22×C512yes—
223#398—21no—
224#397—21no—
225#389C22×C512yes—
226#381—21no—
227#380C22×C511yescharacteristic
228#380C22×C512yes—
229#340C22×C512yes—
230#340C22×C512yes—
231#340C22×C512yes—
232#336C22×C512yes—
233C23×C5—22no—
234C23×C5—22no—
235C23×C5—21no—
236C23×C5—22no—
237C23×C5—22no—
238C23×C5—21no—
239C23×C5—21no—
240C23×C5—22no—
241C23×C5—42no—
242C23×C5—41no—
243C23×C5—41no—
244C23×C5—42no—
245C23×C5#12012yes—
246C23×C5#13512yes—
247C23×C5AGL(1,ℤ/8)12yes—
248C23×C5C22×D411yescharacteristic
249C5×D4—24no—
250C5×D4—44no—
251C5×D4—48no—
252C5×D4—41no—
253C5×D4—41no—
254C5×D4—84no—
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—24no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—22no—
262C2×C4×C5—21no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—22no—
266C2×C4×C5—21no—
267C2×C4×C5—21no—
268C2×C4×C5—22no—
269C2×C4×C5—41no—
270C2×C4×C5—41no—
271C2×C4×C5—42no—
272C2×C4×C5—42no—
273C2×C4×C5—41no—
274C2×C4×C5—41no—
275C2×C4×C5—42no—
276C2×C4×C5—42no—
277C2×C4×C5—42no—
278C2×C4×C5—42no—
279C2×C4×C5—42no—
280C2×C4×C5—42no—
281C2×C4×C5—42no—
282C2×C4×C5—84no—
283C2×C4×C5—84no—
284C2×C4×C5—84no—
285C2×C4×C5#12012yes—
286C2×C4×C5C22×D411yescharacteristic
287C2×C4×C5C22×D411yescharacteristic
288C2×C4×C5C2×#4112yes—
289C2×C4×C52−1+412yes—
290C5×C8—24no—
291C5×C8—42no—
292C22×D4—21no—
293C22×D4—21no—
294C22×D4C5×D411yescharacteristic
295C22×D4C23×C511yescharacteristic
296C23×C4—21no—
297C23×C4—21no—
298C23×C4—21no—
299C23×C4C23×C511yescharacteristic
300C23×C4C23×C511yescharacteristic
301C22×C8—22no—
302C22×C8—24no—
303C22×C8C23×C512yes—
304C22×C8C23×C512yes—
305#126—28no—
306#125—22no—
307#125—22no—
308#123—28no—
309#123—44no—
310#121—28no—
311#121—22no—
312#121—22no—
313#121—22no—
314#121C5×D414yes—
315C4×D4—28no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320#117—22no—
321#117—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4C5×D411yescharacteristic
331C2×C4⋊C4C5×D414yes—
332C2×C4⋊C4C23×C512yes—
333C2×C4⋊C4C23×C512yes—
334C2×C4⋊C4C23×C511yescharacteristic
335C2×C4⋊C4C23×C511yescharacteristic
336C2×C4⋊C4C23×C511yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4C5×D411yescharacteristic
344C2×C22⋊C4C5×D411yescharacteristic
345C2×C42—21no—
346C2×C42C23×C511yescharacteristic
347#107—24no—
348#106—24no—
349#106—24no—
350#102—24no—
351#102—24no—
352C22⋊C8—22no—
353C22⋊C8—24no—
354#95—22no—
355#95—21no—
356#95—22no—
357C22×C5—24no—
358C22×C5—24no—
359C22×C5—22no—
360C22×C5—22no—
361C22×C5—42no—
362C22×C5—42no—
363C22×C5—41no—
364C22×C5—42no—
365C22×C5—41no—
366C22×C5—42no—
367C22×C5—42no—
368C22×C5—41no—
369C22×C5—82no—
370C22×C5—82no—
371C22×C5#44914yes—
372C22×C5C2×#12011yescharacteristic
373C22×C5#54712yes—
374C22×C5C2×#13511yescharacteristic
375C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
376C22×C5#57712yes—
377C4×C5—22no—
378C4×C5—41no—
379C4×C5—41no—
380C4×C5—42no—
381C4×C5—41no—
382C4×C5—42no—
383C4×C5—42no—
384C4×C5—82no—
385C4×C5—82no—
386C4×C5#52511yescharacteristic
387C4×C5#53611yescharacteristic
388C4×C5#54712yes—
389C24—21no—
390C24C2×C5×D411yescharacteristic
391C24C2×C5×D411yescharacteristic
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—82no—
403C2×D4C2×C5×D412yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4C2×C5×D411yescharacteristic
423C22×C4C2×C5×D411yescharacteristic
424C22×C4C2×C5×D411yescharacteristic
425C22×C4C2×C5×D411yescharacteristic
426C22×C4C2×C5×D412yes—
427C22×C4C2×C5×D412yes—
428C22×C4C24×C511yescharacteristic
429C2×C8—28no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8C5×#4114yes—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4C2×C5×D412yes—
450C4⋊C4C2×C5×D412yes—
451C4⋊C4C2×C5×D412yes—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—22no—
456C22⋊C4—24no—
457C22⋊C4—22no—
458C22⋊C4—44no—
459C22⋊C4—42no—
460C42—24no—
461C42—21no—
462C42—21no—
463C2×C5—24no—
464C2×C5—81no—
465C2×C5—82no—
466C2×C5#240611yescharacteristic
467C2×C5C2×#44911yescharacteristic
468C2×C5#300911yescharacteristic
469C2×C5#333712yes—
470C2×C5#334512yes—
471C23—22no—
472C23—22no—
473C23—21no—
474C23—22no—
475C23—22no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23C5×#12012yes—
484C23C5×#13512yes—
485C23C5×AGL(1,ℤ/8)12yes—
486C23C22×C5×D411yescharacteristic
487D4—24no—
488D4—44no—
489D4—48no—
490D4—41no—
491D4—41no—
492D4—84no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—84no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4C5×#12012yes—
524C2×C4C22×C5×D411yescharacteristic
525C2×C4C22×C5×D411yescharacteristic
526C2×C4C2×C5×#4112yes—
527C2×C4C5×2−1+412yes—
528C8—24no—
529C8—42no—
530C5#2409111yescharacteristic, direct product factor, Sylow
531C22—22no—
532C22—24no—
533C22—24no—
534C22—22no—
535C22—41no—
536C22—42no—
537C22—42no—
538C22—41no—
539C22—42no—
540C22—42no—
541C22—41no—
542C22—42no—
543C22—82no—
544C22—82no—
545C22C5×#44914yes—
546C22C2×C5×#12011yescharacteristic
547C22C5×#54712yes—
548C22C2×C5×#13511yescharacteristic
549C22C2×C5×AGL(1,ℤ/8)11yescharacteristic
550C22C5×#57712yes—
551C4—22no—
552C4—41no—
553C4—42no—
554C4—41no—
555C4—42no—
556C4—42no—
557C4—41no—
558C4—82no—
559C4—82no—
560C4C5×#52511yescharacteristic
561C4C5×#53611yescharacteristic
562C4C5×#54712yes—
563C2—24no—
564C2—81no—
565C2—82no—
566C2C5×#240611yescharacteristic
567C2C2×C5×#44911yescharacteristic
568C2C5×#300911yescharacteristic
569C2C5×#333712yes—
570C2C5×#334512yes—
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC2×C5×#553C2
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×#480C2semidirect
3. C22×C5×#106C2semidirect
4. C5×#2486C2semidirect
5. C5×#2059C2semidirect
6. C5×#2034C2semidirect
7. C5×#1893C2semidirect
8. C5×#1869C2semidirect
9. C5×#1864C2semidirect
10. C5×#1854C2semidirect
11. C5×#1824C2semidirect
47. #24091C5direct (N × H)
114. C2×#480C2×C5semidirect
115. C22×#106C2×C5semidirect
116. #2486C2×C5semidirect
117. #2059C2×C5semidirect
118. #2034C2×C5semidirect
119. #1893C2×C5semidirect
120. #1869C2×C5semidirect
121. #1864C2×C5semidirect
122. #1854C2×C5semidirect
123. #1824C2×C5semidirect
530. C5#24091direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5532
C2×C5×#4802
C22×C5×#1062
C5×#24862
C5×#20592
C5×#20342
C5×#18932
C5×#18692
C5×#18642
C5×#18542
C5×#18242
#240915