Finite Group Databasebeta

← C5×#24305 #11947025

Subgroups — C5×#24305 #11947025

Order 1280.

1162 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×#125C211yesmaximal, characteristic, complemented
2C2×C5×#478C212yesmaximal, complemented
3C4×C5×#121C211yesmaximal, characteristic
4C5×#2059C212yesmaximal, complemented
5C5×#2025C212yesmaximal, complemented
6C5×#1989C212yesmaximal, complemented
7C5×#1874C211yesmaximal, characteristic
8C5×#1874C211yesmaximal, characteristic
9C5×#1853C211yesmaximal, characteristic, complemented
10C5×#1852C211yesmaximal, characteristic, complemented
11C5×#1797C211yesmaximal, characteristic, complemented
12C2×C5×#125—28no—
13C2×C5×#125—21no—
14C2×C5×#125—21no—
15C2×C5×#125C2212yes—
16C2×C5×#125C2212yes—
17C2×C5×#121C2211yescharacteristic
18C2×C4×C5×D4—21no—
19C2×C4×C5×D4C2211yescharacteristic
20C22×C5×C4⋊C4—21no—
21C22×C5×C4⋊C4—21no—
22C22×C5×C4⋊C4C2211yescharacteristic
23C22×C5×C4⋊C4C2211yescharacteristic
24C22×C42×C5C2211yescharacteristic
25C5×#478—28no—
26C2×C5×#107C2212yes—
27C2×C5×#106C2212yes—
28C2×C5×C4⋊C8C2212yes—
29C2×C5×#102C2212yes—
30C2×C5×#102C2212yes—
31C2×C5×C22⋊C8C2211yescharacteristic
32C2×C5×C22⋊C8C2211yescharacteristic
33C5×#390C2211yescharacteristic
34C5×#389C2211yescharacteristic
35C5×#388—21no—
36C5×#386—21no—
37C5×#380C2211yescharacteristic
38C4×C5×C4⋊C4C2211yescharacteristic
39C4×C5×C22⋊C4—21no—
40C5×#340C2212yes—
41C5×#340C2212yes—
42C5×#340C2212yes—
43C5×#340C2212yes—
44C5×#336C2212yes—
45#24305C511yesmaximal, characteristic, direct product factor, Sylow
46C22×C5×D4—21no—
47C22×C5×D4C2311yescharacteristic
48C23×C4×C5—21no—
49C23×C4×C5—21no—
50C23×C4×C5—21no—
51C23×C4×C5D411yescharacteristic
52C23×C4×C5D411yescharacteristic
53C23×C4×C5C2311yescharacteristic
54C22×C5×C8C2312yes—
55C22×C5×C8C2312yes—
56C5×#125—28no—
57C5×#125—28no—
58C5×#125—24no—
59C5×#125—24no—
60C5×#125—24no—
61C5×#125D418yescomplemented
62C5×#121—22no—
63C5×#121Q814yes—
64C4×C5×D4—24no—
65C4×C5×D4—44no—
66C2×C5×C4⋊C4—28no—
67C2×C5×C4⋊C4—28no—
68C2×C5×C4⋊C4—22no—
69C2×C5×C4⋊C4—22no—
70C2×C5×C4⋊C4—22no—
71C2×C5×C4⋊C4—22no—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4—21no—
74C2×C5×C4⋊C4—24no—
75C2×C5×C4⋊C4—24no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4C2311yescharacteristic
78C2×C5×C4⋊C4C2311yescharacteristic
79C2×C5×C4⋊C4C2312yes—
80C2×C5×C4⋊C4C2312yes—
81C2×C5×C4⋊C4C2311yescharacteristic
82C2×C5×C22⋊C4—21no—
83C2×C5×C22⋊C4—21no—
84C2×C5×C22⋊C4—21no—
85C2×C5×C22⋊C4—21no—
86C2×C5×C22⋊C4—21no—
87C2×C42×C5—24no—
88C2×C42×C5—21no—
89C2×C42×C5D411yescharacteristic
90C2×C42×C5D411yescharacteristic
91C2×C42×C5C2311yescharacteristic
92C2×C42×C5C2311yescharacteristic
93C5×#107—24no—
94C5×#106—24no—
95C5×C4⋊C8—24no—
96C5×#102—24no—
97C5×#102—24no—
98C5×C22⋊C8—22no—
99C5×C22⋊C8—22no—
100C5×#95—21no—
101C5×#95—21no—
102C5×#95—21no—
103C22×#125C2×C511yescharacteristic, complemented
104C2×#478C2×C512yescomplemented
105C4×#121C2×C511yescharacteristic
106#2059C2×C512yescomplemented
107#2025C2×C512yescomplemented
108#1989C2×C512yescomplemented
109#1874C2×C511yescharacteristic
110#1874C2×C511yescharacteristic
111#1853C2×C511yescharacteristic, complemented
112#1852C2×C511yescharacteristic, complemented
113#1797C2×C511yescharacteristic, complemented
114C24×C5—21no—
115C24×C5C2×D411yescharacteristic
116C2×C5×D4—24no—
117C2×C5×D4—22no—
118C2×C5×D4—21no—
119C2×C5×D4—22no—
120C2×C5×D4—42no—
121C2×C5×D4—42no—
122C2×C5×D4C2×Q812yes—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—24no—
126C22×C4×C5—24no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—21no—
130C22×C4×C5—22no—
131C22×C4×C5—22no—
132C22×C4×C5—22no—
133C22×C4×C5—22no—
134C22×C4×C5—21no—
135C22×C4×C5—21no—
136C22×C4×C5—41no—
137C22×C4×C5—41no—
138C22×C4×C5—42no—
139C22×C4×C5—42no—
140C22×C4×C5—42no—
141C22×C4×C5—42no—
142C22×C4×C5—42no—
143C22×C4×C5—42no—
144C22×C4×C5—42no—
145C22×C4×C5C2×D411yescharacteristic
146C22×C4×C5C2×D411yescharacteristic
147C22×C4×C5C2×D411yescharacteristic
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×Q812yes—
151C22×C4×C5#4112yes—
152C22×C4×C5C2411yescharacteristic
153C2×C5×C8—22no—
154C2×C5×C8—22no—
155C2×C5×C8—22no—
156C2×C5×C8—22no—
157C2×C5×C8—22no—
158C2×C5×C8—22no—
159C5×C4⋊C4—24no—
160C5×C4⋊C4—24no—
161C5×C4⋊C4—28no—
162C5×C4⋊C4—28no—
163C5×C4⋊C4—28no—
164C5×C4⋊C4—28no—
165C5×C4⋊C4—22no—
166C5×C4⋊C4—22no—
167C5×C4⋊C4—21no—
168C5×C4⋊C4—22no—
169C5×C4⋊C4—22no—
170C5×C4⋊C4—42no—
171C5×C4⋊C4C2×D414yes—
172C5×C4⋊C4C2×D414yes—
173C5×C4⋊C4C2×Q812yes—
174C5×C22⋊C4—24no—
175C5×C22⋊C4—42no—
176C5×C22⋊C4—42no—
177C5×C22⋊C4—42no—
178C5×C22⋊C4—42no—
179C42×C5—22no—
180C42×C5—22no—
181C42×C5—22no—
182C42×C5—42no—
183C42×C5C2×D414yes—
184C2×#125—28no—
185C2×#125—21no—
186C2×#125—21no—
187C2×#125C22×C512yes—
188C2×#125C22×C512yes—
189C2×#121C22×C511yescharacteristic
190C2×C4×D4—21no—
191C2×C4×D4C22×C511yescharacteristic
192C22×C4⋊C4—21no—
193C22×C4⋊C4—21no—
194C22×C4⋊C4C22×C511yescharacteristic
195C22×C4⋊C4C22×C511yescharacteristic
196C22×C42C22×C511yescharacteristic
197#478—28no—
198C2×#107C22×C512yes—
199C2×#106C22×C512yes—
200C2×C4⋊C8C22×C512yes—
201C2×#102C22×C512yes—
202C2×#102C22×C512yes—
203C2×C22⋊C8C22×C511yescharacteristic
204C2×C22⋊C8C22×C511yescharacteristic
205#390C22×C511yescharacteristic
206#389C22×C511yescharacteristic
207#388—21no—
208#386—21no—
209#380C22×C511yescharacteristic
210C4×C4⋊C4C22×C511yescharacteristic
211C4×C22⋊C4—21no—
212#340C22×C512yes—
213#340C22×C512yes—
214#340C22×C512yes—
215#340C22×C512yes—
216#336C22×C512yes—
217C23×C5—21no—
218C23×C5—21no—
219C23×C5—21no—
220C23×C5—21no—
221C23×C5—22no—
222C23×C5—41no—
223C23×C5—41no—
224C23×C5—41no—
225C23×C5—41no—
226C23×C5—41no—
227C23×C5—41no—
228C23×C5C22⋊Q812yes—
229C23×C5#13512yes—
230C23×C5AGL(1,ℤ/8)12yes—
231C23×C5C22×D411yescharacteristic
232C5×D4—24no—
233C5×D4—48no—
234C5×D4—42no—
235C2×C4×C5—21no—
236C2×C4×C5—21no—
237C2×C4×C5—21no—
238C2×C4×C5—21no—
239C2×C4×C5—21no—
240C2×C4×C5—21no—
241C2×C4×C5—24no—
242C2×C4×C5—24no—
243C2×C4×C5—21no—
244C2×C4×C5—21no—
245C2×C4×C5—24no—
246C2×C4×C5—24no—
247C2×C4×C5—22no—
248C2×C4×C5—21no—
249C2×C4×C5—21no—
250C2×C4×C5—21no—
251C2×C4×C5—21no—
252C2×C4×C5—44no—
253C2×C4×C5—44no—
254C2×C4×C5—48no—
255C2×C4×C5—48no—
256C2×C4×C5—42no—
257C2×C4×C5—41no—
258C2×C4×C5—41no—
259C2×C4×C5—42no—
260C2×C4×C5—42no—
261C2×C4×C5—42no—
262C2×C4×C5—41no—
263C2×C4×C5—41no—
264C2×C4×C5—41no—
265C2×C4×C5—41no—
266C2×C4×C5—41no—
267C2×C4×C5—41no—
268C2×C4×C5—84no—
269C2×C4×C5#12012yes—
270C2×C4×C5#12012yes—
271C2×C4×C5C22⋊Q812yes—
272C2×C4×C5C22×D412yes—
273C2×C4×C5C22×Q811yescharacteristic
274C2×C4×C5C2×#4111yescharacteristic
275C2×C4×C52−1+412yes—
276C5×C8—42no—
277C5×C8—42no—
278C22×D4—21no—
279C22×D4C23×C511yescharacteristic
280C23×C4—21no—
281C23×C4—21no—
282C23×C4—21no—
283C23×C4C5×D411yescharacteristic
284C23×C4C5×D411yescharacteristic
285C23×C4C23×C511yescharacteristic
286C22×C8C23×C512yes—
287C22×C8C23×C512yes—
288#125—28no—
289#125—28no—
290#125—24no—
291#125—24no—
292#125—24no—
293#125C5×D418yescomplemented
294#121—22no—
295#121C5×Q814yes—
296C4×D4—24no—
297C4×D4—44no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4C23×C511yescharacteristic
310C2×C4⋊C4C23×C511yescharacteristic
311C2×C4⋊C4C23×C512yes—
312C2×C4⋊C4C23×C512yes—
313C2×C4⋊C4C23×C511yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C42—24no—
320C2×C42—21no—
321C2×C42C5×D411yescharacteristic
322C2×C42C5×D411yescharacteristic
323C2×C42C23×C511yescharacteristic
324C2×C42C23×C511yescharacteristic
325#107—24no—
326#106—24no—
327C4⋊C8—24no—
328#102—24no—
329#102—24no—
330C22⋊C8—22no—
331C22⋊C8—22no—
332#95—21no—
333#95—21no—
334#95—21no—
335C22×C5—22no—
336C22×C5—22no—
337C22×C5—22no—
338C22×C5—22no—
339C22×C5—22no—
340C22×C5—22no—
341C22×C5—42no—
342C22×C5—42no—
343C22×C5—42no—
344C22×C5—81no—
345C22×C5—81no—
346C22×C5—81no—
347C22×C5—81no—
348C22×C5#47814yescomplemented
349C22×C5C2×#12011yescharacteristic
350C22×C5C2×C22⋊Q811yescharacteristic
351C22×C5#53611yescharacteristic
352C22×C5C2×#13511yescharacteristic
353C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
354C22×C5#57511yescharacteristic
355C22×C5#57511yescharacteristic
356C4×C5—22no—
357C4×C5—22no—
358C4×C5—22no—
359C4×C5—42no—
360C4×C5—41no—
361C4×C5—44no—
362C4×C5—41no—
363C4×C5—44no—
364C4×C5—81no—
365C4×C5—81no—
366C4×C5#54812yes—
367C4×C5D4×Q812yes—
368C24—21no—
369C24C2×C5×D411yescharacteristic
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4C2×C5×Q812yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4C2×C5×D411yescharacteristic
400C22×C4C2×C5×D411yescharacteristic
401C22×C4C2×C5×D411yescharacteristic
402C22×C4C2×C5×D411yescharacteristic
403C22×C4C2×C5×D411yescharacteristic
404C22×C4C2×C5×Q812yes—
405C22×C4C5×#4112yes—
406C22×C4C24×C511yescharacteristic
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—28no—
416C4⋊C4—28no—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—42no—
425C4⋊C4C2×C5×D414yes—
426C4⋊C4C2×C5×D414yes—
427C4⋊C4C2×C5×Q812yes—
428C22⋊C4—24no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—42no—
437C42C2×C5×D414yes—
438C2×C5—24no—
439C2×C5—82no—
440C2×C5#240911yescharacteristic
441C2×C5#300211yescharacteristic
442C2×C5#300611yescharacteristic
443C2×C5C2×#47811yescharacteristic
444C2×C5#308011yescharacteristic
445C2×C5#323211yescharacteristic
446C2×C5#323411yescharacteristic
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23C5×C22⋊Q812yes—
459C23C5×#13512yes—
460C23C5×AGL(1,ℤ/8)12yes—
461C23C22×C5×D411yescharacteristic
462D4—24no—
463D4—48no—
464D4—42no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—84no—
499C2×C4C5×#12012yes—
500C2×C4C5×#12012yes—
501C2×C4C5×C22⋊Q812yes—
502C2×C4C22×C5×D412yes—
503C2×C4C22×C5×Q811yescharacteristic
504C2×C4C2×C5×#4111yescharacteristic
505C2×C4C5×2−1+412yes—
506C8—42no—
507C8—42no—
508C5#2430511yescharacteristic, direct product factor, Sylow
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22C5×#47814yescomplemented
523C22C2×C5×#12011yescharacteristic
524C22C2×C5×C22⋊Q811yescharacteristic
525C22C5×#53611yescharacteristic
526C22C2×C5×#13511yescharacteristic
527C22C2×C5×AGL(1,ℤ/8)11yescharacteristic
528C22C5×#57511yescharacteristic
529C22C5×#57511yescharacteristic
530C4—22no—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—44no—
535C4—41no—
536C4—44no—
537C4—42no—
538C4—81no—
539C4—81no—
540C4C5×#54812yes—
541C4C5×D4×Q812yes—
542C2—24no—
543C2—82no—
544C2C5×#240911yescharacteristic
545C2C5×#300211yescharacteristic
546C2C5×#300611yescharacteristic
547C2C2×C5×#47811yescharacteristic
548C2C5×#308011yescharacteristic
549C2C5×#323211yescharacteristic
550C2C5×#323411yescharacteristic
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC2×C5×#121C22
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
1. C22×C5×#125C2semidirect
2. C2×C5×#478C2semidirect
4. C5×#2059C2semidirect
5. C5×#2025C2semidirect
6. C5×#1989C2semidirect
9. C5×#1853C2semidirect
10. C5×#1852C2semidirect
11. C5×#1797C2semidirect
45. #24305C5direct (N × H)
61. C5×#125D4semidirect
103. C22×#125C2×C5semidirect
104. C2×#478C2×C5semidirect
106. #2059C2×C5semidirect
107. #2025C2×C5semidirect
108. #1989C2×C5semidirect
111. #1853C2×C5semidirect
112. #1852C2×C5semidirect
113. #1797C2×C5semidirect
293. #125C5×D4semidirect
348. C22×C5#478semidirect
508. C5#24305direct (N × H)
522. C22C5×#478semidirect

Maximal subgroups

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

subgroupindex
C22×C5×#1252
C2×C5×#4782
C4×C5×#1212
C5×#20592
C5×#20252
C5×#19892
C5×#18742
C5×#18532
C5×#18522
C5×#17972
#243055