Finite Group Databasebeta

← C5×#29734 #11952454

Subgroups — C5×#29734 #11952454

Order 1280.

1200 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#465C211yesmaximal, characteristic, complemented
2C22×C5×C22⋊C8C211yesmaximal, characteristic, complemented
3C5×#2848C211yesmaximal, characteristic, complemented
4C5×#2847C211yesmaximal, characteristic
5C5×#2077C212yesmaximal, complemented
6C5×#2035C212yesmaximal, complemented
7C5×#2034C212yesmaximal, complemented
8C5×#2000C212yesmaximal, complemented
9C5×#1893C211yesmaximal, characteristic, complemented
10C5×#1892C211yesmaximal, characteristic, complemented
11C5×#1807C211yesmaximal, characteristic, complemented
12C24×C4×C5C2211yescharacteristic
13C22×C5×SD16C2211yescharacteristic, complemented
14C23×C5×C8C2211yescharacteristic
15C23×C5×C8C2211yescharacteristic
16C2×C5×C22⋊Q8C2211yescharacteristic
17C2×C5×C22⋊Q8C2212yes—
18C2×C5×#121C2211yescharacteristic
19C2×C5×#121C2212yes—
20C2×C5×#120—21no—
21C5×#465—24no—
22C2×C5×#106C2211yescharacteristic, complemented
23C2×C5×#106C2212yescomplemented
24C2×C5×#103C2211yescharacteristic, complemented
25C2×C5×#103C2212yescomplemented
26C2×C5×#102C2211yescharacteristic, complemented
27C2×C5×#102C2212yescomplemented
28C2×C5×C22⋊C8—24no—
29C2×C5×C22⋊C8C2212yes—
30C2×C5×C22⋊C8C2212yes—
31C5×#386—22no—
32C5×#386—21no—
33C5×#385—22no—
34C5×#385—21no—
35C5×#380C2212yes—
36C5×#380C2212yes—
37C5×#380C2211yescharacteristic
38C5×#380C2211yescharacteristic
39C5×#379—21no—
40C5×#340C2212yescomplemented
41C5×#340C2212yescomplemented
42C5×#340C2212yescomplemented
43#29734C511yesmaximal, characteristic, direct product factor, Sylow
44C25×C5D411yescharacteristic
45C22×C5×Q8C2311yescharacteristic
46C22×C5×D4—22no—
47C22×C5×D4—21no—
48C22×C5×D4C2311yescharacteristic
49C23×C4×C5—22no—
50C23×C4×C5—22no—
51C23×C4×C5—42no—
52C23×C4×C5—42no—
53C23×C4×C5D411yescharacteristic
54C23×C4×C5D412yes—
55C23×C4×C5C2312yes—
56C23×C4×C5C2311yescharacteristic
57C2×C5×SD16—22no—
58C2×C5×SD16—42no—
59C22×C5×C8—22no—
60C22×C5×C8—42no—
61C22×C5×C8—42no—
62C22×C5×C8D414yescomplemented
63C22×C5×C8C2312yes—
64C22×C5×C8C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C5×C22⋊Q8—22no—
67C5×C22⋊Q8—22no—
68C5×C22⋊Q8—42no—
69C5×C22⋊Q8—42no—
70C5×#121—22no—
71C5×#121—22no—
72C5×#121—42no—
73C5×#121—42no—
74C5×#120—44no—
75C2×C5×C4⋊C4—22no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4C2311yescharacteristic
78C2×C5×C4⋊C4C2312yes—
79C2×C5×C4⋊C4C2311yescharacteristic
80C2×C5×C4⋊C4C2312yes—
81C2×C5×C22⋊C4—22no—
82C2×C5×C22⋊C4—21no—
83C2×C5×C22⋊C4—22no—
84C2×C5×C22⋊C4—21no—
85C2×C5×C22⋊C4—22no—
86C2×C5×C22⋊C4—21no—
87C2×C5×C22⋊C4—22no—
88C2×C5×C22⋊C4—21no—
89C5×#106—22no—
90C5×#106—42no—
91C5×#103—22no—
92C5×#103—42no—
93C5×#102—22no—
94C5×#102—42no—
95C5×C22⋊C8—24no—
96C5×C22⋊C8—24no—
97C5×#95—22no—
98C5×#95—21no—
99C5×#95—22no—
100C5×#95—21no—
101C2×#465C2×C511yescharacteristic, complemented
102C22×C22⋊C8C2×C511yescharacteristic, complemented
103#2848C2×C511yescharacteristic, complemented
104#2847C2×C511yescharacteristic
105#2077C2×C512yescomplemented
106#2035C2×C512yescomplemented
107#2034C2×C512yescomplemented
108#2000C2×C512yescomplemented
109#1893C2×C511yescharacteristic, complemented
110#1892C2×C511yescharacteristic, complemented
111#1807C2×C511yescharacteristic, complemented
112C24×C5—22no—
113C24×C5—22no—
114C24×C5—21no—
115C24×C5—42no—
116C24×C5—42no—
117C24×C5SD1614yescomplemented
118C24×C5C2×D412yes—
119C24×C5C2×D411yescharacteristic
120C2×C5×Q8—21no—
121C2×C5×Q8—21no—
122C2×C5×Q8—22no—
123C2×C5×Q8—22no—
124C2×C5×D4—21no—
125C2×C5×D4—21no—
126C2×C5×D4—22no—
127C2×C5×D4—22no—
128C2×C5×D4—44no—
129C2×C5×D4—44no—
130C2×C5×D4—42no—
131C2×C5×D4—44no—
132C2×C5×D4—42no—
133C2×C5×D4—42no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
137C22×C4×C5—24no—
138C22×C4×C5—22no—
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—21no—
148C22×C4×C5—22no—
149C22×C4×C5—42no—
150C22×C4×C5—42no—
151C22×C4×C5—42no—
152C22×C4×C5—44no—
153C22×C4×C5—44no—
154C22×C4×C5—48no—
155C22×C4×C5C2×D412yes—
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2×D412yes—
158C22×C4×C5#4112yes—
159C22×C4×C5C2411yescharacteristic
160C5×SD16—44no—
161C2×C5×C8—22no—
162C2×C5×C8—22no—
163C2×C5×C8—22no—
164C2×C5×C8—22no—
165C2×C5×C8—22no—
166C2×C5×C8—44no—
167C2×C5×C8—44no—
168C2×C5×C8C2×D412yes—
169C2×C5×C8C2×D412yes—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—21no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—22no—
175C5×C4⋊C4—22no—
176C5×C4⋊C4—22no—
177C5×C4⋊C4—22no—
178C5×C4⋊C4—44no—
179C5×C4⋊C4—42no—
180C5×C22⋊C4—44no—
181C5×C22⋊C4—44no—
182C5×C22⋊C4—44no—
183C5×C22⋊C4—42no—
184C5×C22⋊C4—44no—
185C5×C22⋊C4—42no—
186C5×C22⋊C4—42no—
187C5×C22⋊C4—42no—
188C24×C4C22×C511yescharacteristic
189C22×SD16C22×C511yescharacteristic, complemented
190C23×C8C22×C511yescharacteristic
191C23×C8C22×C511yescharacteristic
192C2×C22⋊Q8C22×C511yescharacteristic
193C2×C22⋊Q8C22×C512yes—
194C2×#121C22×C511yescharacteristic
195C2×#121C22×C512yes—
196C2×#120—21no—
197#465—24no—
198C2×#106C22×C511yescharacteristic, complemented
199C2×#106C22×C512yescomplemented
200C2×#103C22×C511yescharacteristic, complemented
201C2×#103C22×C512yescomplemented
202C2×#102C22×C511yescharacteristic, complemented
203C2×#102C22×C512yescomplemented
204C2×C22⋊C8—24no—
205C2×C22⋊C8C22×C512yes—
206C2×C22⋊C8C22×C512yes—
207#386—21no—
208#386—22no—
209#385—21no—
210#385—22no—
211#380C22×C511yescharacteristic
212#380C22×C511yescharacteristic
213#380C22×C512yes—
214#380C22×C512yes—
215#379—21no—
216#340C22×C512yescomplemented
217#340C22×C512yescomplemented
218#340C22×C512yescomplemented
219C23×C5—22no—
220C23×C5—22no—
221C23×C5—22no—
222C23×C5—22no—
223C23×C5—22no—
224C23×C5—22no—
225C23×C5—22no—
226C23×C5—22no—
227C23×C5—28no—
228C23×C5—48no—
229C23×C5—44no—
230C23×C5—48no—
231C23×C5—44no—
232C23×C5—41no—
233C23×C5—42no—
234C23×C5—41no—
235C23×C5—41no—
236C23×C5—42no—
237C23×C5#12112yes—
238C23×C5C2×SD1614yes—
239C23×C5C2×SD1612yes—
240C23×C5#13512yes—
241C23×C5C22×D411yescharacteristic
242C5×Q8—41no—
243C5×Q8—41no—
244C5×Q8—42no—
245C5×D4—41no—
246C5×D4—41no—
247C5×D4—42no—
248C5×D4—88no—
249C5×D4—84no—
250C2×C4×C5—22no—
251C2×C4×C5—22no—
252C2×C4×C5—22no—
253C2×C4×C5—24no—
254C2×C4×C5—22no—
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5—48no—
258C2×C4×C5—44no—
259C2×C4×C5—42no—
260C2×C4×C5—42no—
261C2×C4×C5—42no—
262C2×C4×C5—42no—
263C2×C4×C5—41no—
264C2×C4×C5—41no—
265C2×C4×C5—41no—
266C2×C4×C5—42no—
267C2×C4×C5—42no—
268C2×C4×C5—42no—
269C2×C4×C5—42no—
270C2×C4×C5—41no—
271C2×C4×C5—42no—
272C2×C4×C5—41no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5#12112yes—
277C2×C4×C5C22×D411yescharacteristic
278C2×C4×C5C2×#4111yescharacteristic
279C2×C4×C5H(5,𝔽2)12yes—
280C2×C4×C5H(5,𝔽2)12yes—
281C5×C8—22no—
282C5×C8—22no—
283C5×C8—42no—
284C25C5×D411yescharacteristic
285C22×Q8C23×C511yescharacteristic
286C22×D4—22no—
287C22×D4—21no—
288C22×D4C23×C511yescharacteristic
289C23×C4—22no—
290C23×C4—22no—
291C23×C4—42no—
292C23×C4—42no—
293C23×C4C5×D411yescharacteristic
294C23×C4C5×D412yes—
295C23×C4C23×C512yes—
296C23×C4C23×C511yescharacteristic
297C2×SD16—22no—
298C2×SD16—42no—
299C22×C8—22no—
300C22×C8—42no—
301C22×C8—42no—
302C22×C8C5×D414yescomplemented
303C22×C8C23×C512yes—
304C22×C8C23×C511yescharacteristic
305C22×C8C23×C511yescharacteristic
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308C22⋊Q8—42no—
309C22⋊Q8—42no—
310#121—22no—
311#121—22no—
312#121—42no—
313#121—42no—
314#120—44no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4C23×C511yescharacteristic
318C2×C4⋊C4C23×C512yes—
319C2×C4⋊C4C23×C511yescharacteristic
320C2×C4⋊C4C23×C512yes—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329#106—22no—
330#106—42no—
331#103—22no—
332#103—42no—
333#102—22no—
334#102—42no—
335C22⋊C8—24no—
336C22⋊C8—24no—
337#95—22no—
338#95—21no—
339#95—22no—
340#95—21no—
341C22×C5—22no—
342C22×C5—24no—
343C22×C5—24no—
344C22×C5—22no—
345C22×C5—24no—
346C22×C5—24no—
347C22×C5—24no—
348C22×C5—416no—
349C22×C5—42no—
350C22×C5—42no—
351C22×C5—42no—
352C22×C5—42no—
353C22×C5—81no—
354C22×C5—82no—
355C22×C5—82no—
356C22×C5—81no—
357C22×C5—81no—
358C22×C5#46514yes—
359C22×C5C2×#12111yescharacteristic
360C22×C5#53412yes—
361C22×C5C22×SD1611yescharacteristic
362C22×C5C2×#13511yescharacteristic
363C22×C5#57512yes—
364C4×C5—22no—
365C4×C5—42no—
366C4×C5—81no—
367C4×C5—82no—
368C4×C5—82no—
369C4×C5—81no—
370C4×C5—81no—
371C4×C5#53411yescharacteristic
372C4×C5#53511yescharacteristic
373C4×C5#53812yes—
374C24—22no—
375C24—22no—
376C24—21no—
377C24—42no—
378C24—42no—
379C24C5×SD1614yescomplemented
380C24C2×C5×D412yes—
381C24C2×C5×D411yescharacteristic
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—42no—
393C2×D4—44no—
394C2×D4—42no—
395C2×D4—42no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4C2×C5×D412yes—
418C22×C4C2×C5×D411yescharacteristic
419C22×C4C2×C5×D412yes—
420C22×C4C5×#4112yes—
421C22×C4C24×C511yescharacteristic
422SD16—44no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—44no—
429C2×C8—44no—
430C2×C8C2×C5×D412yes—
431C2×C8C2×C5×D412yes—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—42no—
446C22⋊C4—44no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C2×C5—24no—
451C2×C5—44no—
452C2×C5—161no—
453C2×C5#261711yescharacteristic
454C2×C5C2×#46511yescharacteristic
455C2×C5#305411yescharacteristic
456C2×C5#318712yes—
457C2×C5#319112yes—
458C23—22no—
459C23—22no—
460C23—28no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—48no—
468C23—44no—
469C23—44no—
470C23—48no—
471C23—41no—
472C23—42no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23C5×#12112yes—
477C23C2×C5×SD1614yes—
478C23C2×C5×SD1612yes—
479C23C5×#13512yes—
480C23C22×C5×D411yescharacteristic
481Q8—41no—
482Q8—41no—
483Q8—42no—
484D4—41no—
485D4—41no—
486D4—42no—
487D4—88no—
488D4—84no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—48no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4C5×#12112yes—
516C2×C4C22×C5×D411yescharacteristic
517C2×C4C2×C5×#4111yescharacteristic
518C2×C4C5×H(5,𝔽2)12yes—
519C2×C4C5×H(5,𝔽2)12yes—
520C8—22no—
521C8—22no—
522C8—42no—
523C5#2973411yescharacteristic, direct product factor, Sylow
524C22—22no—
525C22—22no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—416no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—82no—
540C22—82no—
541C22C5×#46514yes—
542C22C2×C5×#12111yescharacteristic
543C22C5×#53412yes—
544C22C22×C5×SD1611yescharacteristic
545C22C2×C5×#13511yescharacteristic
546C22C5×#57512yes—
547C4—22no—
548C4—42no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—82no—
553C4—82no—
554C4C5×#53411yescharacteristic
555C4C5×#53511yescharacteristic
556C4C5×#53812yes—
557C2—24no—
558C2—44no—
559C2—161no—
560C2C5×#261711yescharacteristic
561C2C2×C5×#46511yescharacteristic
562C2C5×#305411yescharacteristic
563C2C5×#318712yes—
564C2C5×#319112yes—
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC5×#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C24×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C5×#465C2semidirect
2. C22×C5×C22⋊C8C2semidirect
3. C5×#2848C2semidirect
5. C5×#2077C2semidirect
6. C5×#2035C2semidirect
7. C5×#2034C2semidirect
8. C5×#2000C2semidirect
9. C5×#1893C2semidirect
10. C5×#1892C2semidirect
11. C5×#1807C2semidirect
13. C22×C5×SD16C22semidirect
22. C2×C5×#106C22semidirect
23. C2×C5×#106C22semidirect
24. C2×C5×#103C22semidirect
25. C2×C5×#103C22semidirect
26. C2×C5×#102C22semidirect
27. C2×C5×#102C22semidirect
40. C5×#340C22semidirect
41. C5×#340C22semidirect
42. C5×#340C22semidirect
43. #29734C5direct (N × H)
62. C22×C5×C8D4semidirect
101. C2×#465C2×C5semidirect
102. C22×C22⋊C8C2×C5semidirect
103. #2848C2×C5semidirect
105. #2077C2×C5semidirect
106. #2035C2×C5semidirect
107. #2034C2×C5semidirect
108. #2000C2×C5semidirect
109. #1893C2×C5semidirect
110. #1892C2×C5semidirect
111. #1807C2×C5semidirect
117. C24×C5SD16semidirect
189. C22×SD16C22×C5semidirect
198. C2×#106C22×C5semidirect
199. C2×#106C22×C5semidirect
200. C2×#103C22×C5semidirect
201. C2×#103C22×C5semidirect
202. C2×#102C22×C5semidirect
203. C2×#102C22×C5semidirect
216. #340C22×C5semidirect
217. #340C22×C5semidirect
218. #340C22×C5semidirect
302. C22×C8C5×D4semidirect
379. C24C5×SD16semidirect
523. C5#29734direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#4652
C22×C5×C22⋊C82
C5×#28482
C5×#28472
C5×#20772
C5×#20352
C5×#20342
C5×#20002
C5×#18932
C5×#18922
C5×#18072
#297345