Finite Group Databasebeta

← C5×#29634 #11952354

Subgroups — C5×#29634 #11952354

Order 1280.

1908 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#534C211yesmaximal, characteristic, complemented
2C5×#2405C211yesmaximal, characteristic, complemented
3C5×#2403C212yesmaximal, complemented
4C5×#2044C214yesmaximal, complemented
5C5×#2021C214yesmaximal, complemented
6C5×#1819C211yesmaximal, characteristic, complemented
7C5×#1781C212yesmaximal, complemented
8C23×C5×D4C2212yes—
9C23×C5×D4C2211yescharacteristic
10C5×#534—24no—
11C5×#534—22no—
12C5×#534C2214yes—
13C2×C5×#123—22no—
14C2×C5×#123—21no—
15C2×C5×#121—21no—
16C2×C5×#121—21no—
17C2×C5×#120—22no—
18C2×C5×#120—22no—
19C2×C5×#120C2214yescomplemented
20C22×C5×C22⋊C4C2211yescharacteristic
21C2×C5×#99C2218yescomplemented
22C5×#394—22no—
23C5×#392—24no—
24C5×#386—22no—
25C5×#386—21no—
26C5×#386C2214yescomplemented
27C5×#385C2212yescomplemented
28C5×#379C2211yescharacteristic
29C5×#379C2212yes—
30C5×#379C2212yescomplemented
31C5×#342C2214yescomplemented
32#29634C511yesmaximal, characteristic, direct product factor, Sylow
33C25×C5D412yescomplemented
34C25×C5C2311yescharacteristic
35C25×C5C2311yescharacteristic
36C22×C5×D4—22no—
37C22×C5×D4—21no—
38C22×C5×D4—22no—
39C22×C5×D4—22no—
40C22×C5×D4—22no—
41C22×C5×D4—22no—
42C22×C5×D4—21no—
43C22×C5×D4—21no—
44C22×C5×D4—22no—
45C22×C5×D4—21no—
46C22×C5×D4—24no—
47C22×C5×D4—44no—
48C22×C5×D4—48no—
49C22×C5×D4D414yescomplemented
50C22×C5×D4C2314yes—
51C23×C4×C5D412yescomplemented
52C23×C4×C5D411yescharacteristic, complemented
53C23×C4×C5D411yescharacteristic, complemented
54C23×C4×C5C2311yescharacteristic
55C5×#123—28no—
56C5×#123—28no—
57C5×#123—44no—
58C5×#121—24no—
59C5×#121—24no—
60C5×#121—24no—
61C5×#121—24no—
62C5×#120—28no—
63C5×#120—28no—
64C5×#120—48no—
65C5×#120—48no—
66C2×C5×C4⋊C4—21no—
67C2×C5×C4⋊C4—21no—
68C2×C5×C4⋊C4—22no—
69C2×C5×C22⋊C4—22no—
70C2×C5×C22⋊C4—22no—
71C2×C5×C22⋊C4—22no—
72C2×C5×C22⋊C4—22no—
73C2×C5×C22⋊C4—22no—
74C2×C5×C22⋊C4—21no—
75C2×C5×C22⋊C4—24no—
76C2×C5×C22⋊C4—22no—
77C2×C5×C22⋊C4—24no—
78C2×C5×C22⋊C4—22no—
79C2×C5×C22⋊C4—21no—
80C2×C5×C22⋊C4—22no—
81C2×C5×C22⋊C4D412yescomplemented
82C2×C5×C22⋊C4C2312yes—
83C2×C5×C22⋊C4C2314yes—
84C2×C5×C22⋊C4C2312yes—
85C5×#99—48no—
86C5×#95—22no—
87C5×#95—24no—
88C2×#534C2×C511yescharacteristic, complemented
89#2405C2×C511yescharacteristic, complemented
90#2403C2×C512yescomplemented
91#2044C2×C514yescomplemented
92#2021C2×C514yescomplemented
93#1819C2×C511yescharacteristic, complemented
94#1781C2×C512yescomplemented
95C24×C5—24no—
96C24×C5—22no—
97C24×C5—21no—
98C24×C5—22no—
99C24×C5—22no—
100C24×C5—24no—
101C24×C5—44no—
102C24×C5—42no—
103C24×C5—42no—
104C24×C5—44no—
105C24×C5—81no—
106C24×C5—81no—
107C24×C5—82no—
108C24×C5C2×D412yes—
109C24×C5C2×D411yescharacteristic
110C24×C5C2×D414yes—
111C24×C5C2×D412yes—
112C24×C5C2×D411yescharacteristic
113C24×C5C2411yescharacteristic
114C2×C5×D4—28no—
115C2×C5×D4—24no—
116C2×C5×D4—24no—
117C2×C5×D4—24no—
118C2×C5×D4—24no—
119C2×C5×D4—24no—
120C2×C5×D4—28no—
121C2×C5×D4—28no—
122C2×C5×D4—24no—
123C2×C5×D4—24no—
124C2×C5×D4—44no—
125C2×C5×D4—44no—
126C2×C5×D4—44no—
127C2×C5×D4—42no—
128C2×C5×D4—42no—
129C2×C5×D4—44no—
130C2×C5×D4—48no—
131C2×C5×D4—48no—
132C2×C5×D4—48no—
133C2×C5×D4—48no—
134C2×C5×D4—42no—
135C2×C5×D4—44no—
136C2×C5×D4—42no—
137C2×C5×D4—41no—
138C2×C5×D4—41no—
139C2×C5×D4—48no—
140C2×C5×D4—48no—
141C2×C5×D4—82no—
142C2×C5×D4—82no—
143C2×C5×D4—84no—
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—22no—
153C22×C4×C5—24no—
154C22×C4×C5—42no—
155C22×C4×C5—44no—
156C22×C4×C5—42no—
157C22×C4×C5—81no—
158C22×C4×C5C2×D414yes—
159C22×C4×C5C2×D412yes—
160C22×C4×C5C2×D412yes—
161C5×C4⋊C4—24no—
162C5×C4⋊C4—24no—
163C5×C4⋊C4—44no—
164C5×C22⋊C4—28no—
165C5×C22⋊C4—28no—
166C5×C22⋊C4—24no—
167C5×C22⋊C4—24no—
168C5×C22⋊C4—28no—
169C5×C22⋊C4—24no—
170C5×C22⋊C4—48no—
171C5×C22⋊C4—48no—
172C5×C22⋊C4—44no—
173C5×C22⋊C4—44no—
174C5×C22⋊C4—42no—
175C5×C22⋊C4—82no—
176C5×C22⋊C4—81no—
177C5×C22⋊C4—82no—
178C23×D4C22×C512yes—
179C23×D4C22×C511yescharacteristic
180#534—24no—
181#534—22no—
182#534C22×C514yes—
183C2×#123—22no—
184C2×#123—21no—
185C2×#121—21no—
186C2×#121—21no—
187C2×#120—22no—
188C2×#120—22no—
189C2×#120C22×C514yescomplemented
190C22×C22⋊C4C22×C511yescharacteristic
191C2×#99C22×C518yescomplemented
192#394—22no—
193#392—24no—
194#386—22no—
195#386—21no—
196#386C22×C514yescomplemented
197#385C22×C512yescomplemented
198#379C22×C512yes—
199#379C22×C511yescharacteristic
200#379C22×C512yescomplemented
201#342C22×C514yescomplemented
202C23×C5—24no—
203C23×C5—22no—
204C23×C5—22no—
205C23×C5—24no—
206C23×C5—24no—
207C23×C5—22no—
208C23×C5—22no—
209C23×C5—24no—
210C23×C5—24no—
211C23×C5—24no—
212C23×C5—24no—
213C23×C5—41no—
214C23×C5—41no—
215C23×C5—42no—
216C23×C5—41no—
217C23×C5—41no—
218C23×C5—42no—
219C23×C5—41no—
220C23×C5—42no—
221C23×C5—44no—
222C23×C5—44no—
223C23×C5—42no—
224C23×C5—44no—
225C23×C5—41no—
226C23×C5—42no—
227C23×C5—44no—
228C23×C5—44no—
229C23×C5—44no—
230C23×C5—41no—
231C23×C5—44no—
232C23×C5—42no—
233C23×C5—44no—
234C23×C5—82no—
235C23×C5—82no—
236C23×C5—88no—
237C23×C5—88no—
238C23×C5—816no—
239C23×C5#12012yes—
240C23×C5#12011yescharacteristic
241C23×C5#12011yescharacteristic
242C23×C5C22×D412yes—
243C23×C5C22×D411yescharacteristic
244C23×C5H(5,𝔽2)14yes—
245C5×D4—48no—
246C5×D4—48no—
247C5×D4—44no—
248C5×D4—48no—
249C5×D4—48no—
250C5×D4—48no—
251C5×D4—44no—
252C5×D4—84no—
253C5×D4—84no—
254C5×D4—816no—
255C2×C4×C5—24no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—24no—
260C2×C4×C5—24no—
261C2×C4×C5—24no—
262C2×C4×C5—44no—
263C2×C4×C5—44no—
264C2×C4×C5—41no—
265C2×C4×C5—42no—
266C2×C4×C5—42no—
267C2×C4×C5—41no—
268C2×C4×C5—42no—
269C2×C4×C5—42no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5—44no—
273C2×C4×C5—42no—
274C2×C4×C5—44no—
275C2×C4×C5—44no—
276C2×C4×C5—42no—
277C2×C4×C5—81no—
278C2×C4×C5—81no—
279C25C5×D412yescomplemented
280C25C23×C511yescharacteristic
281C25C23×C511yescharacteristic
282C22×D4—22no—
283C22×D4—22no—
284C22×D4—22no—
285C22×D4—21no—
286C22×D4—22no—
287C22×D4—22no—
288C22×D4—21no—
289C22×D4—21no—
290C22×D4—22no—
291C22×D4—21no—
292C22×D4—24no—
293C22×D4—44no—
294C22×D4—48no—
295C22×D4C5×D414yescomplemented
296C22×D4C23×C514yes—
297C23×C4C5×D411yescharacteristic, complemented
298C23×C4C5×D412yescomplemented
299C23×C4C5×D411yescharacteristic, complemented
300C23×C4C23×C511yescharacteristic
301#123—28no—
302#123—28no—
303#123—44no—
304#121—24no—
305#121—24no—
306#121—24no—
307#121—24no—
308#120—28no—
309#120—28no—
310#120—48no—
311#120—48no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4C5×D412yescomplemented
328C2×C22⋊C4C23×C514yes—
329C2×C22⋊C4C23×C512yes—
330C2×C22⋊C4C23×C512yes—
331#99—48no—
332#95—22no—
333#95—24no—
334C22×C5—21no—
335C22×C5—21no—
336C22×C5—22no—
337C22×C5—21no—
338C22×C5—21no—
339C22×C5—22no—
340C22×C5—22no—
341C22×C5—21no—
342C22×C5—21no—
343C22×C5—22no—
344C22×C5—21no—
345C22×C5—22no—
346C22×C5—44no—
347C22×C5—44no—
348C22×C5—48no—
349C22×C5—48no—
350C22×C5—48no—
351C22×C5—44no—
352C22×C5—44no—
353C22×C5—42no—
354C22×C5—44no—
355C22×C5—44no—
356C22×C5—84no—
357C22×C5—82no—
358C22×C5—84no—
359C22×C5—84no—
360C22×C5—82no—
361C22×C5—88no—
362C22×C5—88no—
363C22×C5—81no—
364C22×C5—81no—
365C22×C5B(4,𝔽2)12yes—
366C22×C5C2×#12011yescharacteristic
367C22×C5#53414yes—
368C22×C5#53512yes—
369C4×C5—44no—
370C4×C5—42no—
371C4×C5—42no—
372C4×C5—42no—
373C4×C5—82no—
374C4×C5—84no—
375C24—24no—
376C24—22no—
377C24—21no—
378C24—22no—
379C24—22no—
380C24—24no—
381C24—44no—
382C24—42no—
383C24—42no—
384C24—44no—
385C24—81no—
386C24—81no—
387C24—82no—
388C24C2×C5×D412yes—
389C24C2×C5×D411yescharacteristic
390C24C2×C5×D414yes—
391C24C2×C5×D412yes—
392C24C2×C5×D411yescharacteristic
393C24C24×C511yescharacteristic
394C2×D4—28no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—44no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—42no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—84no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—81no—
438C22×C4C2×C5×D414yes—
439C22×C4C2×C5×D412yes—
440C22×C4C2×C5×D412yes—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—44no—
444C22⋊C4—28no—
445C22⋊C4—28no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—28no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—42no—
455C22⋊C4—82no—
456C22⋊C4—81no—
457C22⋊C4—82no—
458C2×C5—24no—
459C2×C5—22no—
460C2×C5—22no—
461C2×C5—41no—
462C2×C5—42no—
463C2×C5—82no—
464C2×C5—84no—
465C2×C5#261311yescharacteristic
466C2×C5C2×B(4,𝔽2)11yescharacteristic
467C2×C5#302611yescharacteristic
468C23—22no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—41no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—41no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—41no—
498C23—44no—
499C23—42no—
500C23—82no—
501C23—82no—
502C23—88no—
503C23—88no—
504C23—816no—
505C23C5×#12012yes—
506C23C5×#12011yescharacteristic
507C23C5×#12011yescharacteristic
508C23C22×C5×D412yes—
509C23C22×C5×D411yescharacteristic
510C23C5×H(5,𝔽2)14yes—
511D4—44no—
512D4—48no—
513D4—48no—
514D4—48no—
515D4—44no—
516D4—48no—
517D4—48no—
518D4—84no—
519D4—84no—
520D4—816no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—81no—
544C2×C4—81no—
545C5#2963411yescharacteristic, direct product factor, Sylow
546C22—21no—
547C22—22no—
548C22—21no—
549C22—21no—
550C22—22no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—21no—
555C22—22no—
556C22—22no—
557C22—21no—
558C22—44no—
559C22—48no—
560C22—44no—
561C22—48no—
562C22—48no—
563C22—44no—
564C22—42no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—84no—
569C22—82no—
570C22—84no—
571C22—84no—
572C22—82no—
573C22—88no—
574C22—88no—
575C22—81no—
576C22—81no—
577C22C5×B(4,𝔽2)12yes—
578C22C2×C5×#12011yescharacteristic
579C22C5×#53414yes—
580C22C5×#53512yes—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—44no—
585C4—84no—
586C4—82no—
587C2—24no—
588C2—22no—
589C2—22no—
590C2—42no—
591C2—41no—
592C2—82no—
593C2—84no—
594C2C5×#261311yescharacteristic
595C2C2×C5×B(4,𝔽2)11yescharacteristic
596C2C5×#302611yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC24C24×C5
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C25×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C5×#534C2semidirect
2. C5×#2405C2semidirect
3. C5×#2403C2semidirect
4. C5×#2044C2semidirect
5. C5×#2021C2semidirect
6. C5×#1819C2semidirect
7. C5×#1781C2semidirect
19. C2×C5×#120C22semidirect
21. C2×C5×#99C22semidirect
26. C5×#386C22semidirect
27. C5×#385C22semidirect
30. C5×#379C22semidirect
31. C5×#342C22semidirect
32. #29634C5direct (N × H)
33. C25×C5D4semidirect
49. C22×C5×D4D4semidirect
51. C23×C4×C5D4semidirect
52. C23×C4×C5D4semidirect
53. C23×C4×C5D4semidirect
81. C2×C5×C22⋊C4D4semidirect
88. C2×#534C2×C5semidirect
89. #2405C2×C5semidirect
90. #2403C2×C5semidirect
91. #2044C2×C5semidirect
92. #2021C2×C5semidirect
93. #1819C2×C5semidirect
94. #1781C2×C5semidirect
189. C2×#120C22×C5semidirect
191. C2×#99C22×C5semidirect
196. #386C22×C5semidirect
197. #385C22×C5semidirect
200. #379C22×C5semidirect
201. #342C22×C5semidirect
279. C25C5×D4semidirect
295. C22×D4C5×D4semidirect
297. C23×C4C5×D4semidirect
298. C23×C4C5×D4semidirect
299. C23×C4C5×D4semidirect
327. C2×C22⋊C4C5×D4semidirect
545. C5#29634direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5342
C5×#24052
C5×#24032
C5×#20442
C5×#20212
C5×#18192
C5×#17812
#296345