Finite Group Databasebeta

← C5×#25901 #11948621

Subgroups — C5×#25901 #11948621

Order 1280.

1406 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3496C211yesmaximal, characteristic
2C5×#2591C212yesmaximal, complemented
3C5×#2579C211yesmaximal, characteristic, complemented
4C5×#2044C212yesmaximal, complemented
5C5×#2044C212yesmaximal, complemented
6C5×#1964C212yesmaximal, complemented
7C5×#1790C212yesmaximal, complemented
8C5×#1790C212yesmaximal, complemented
9C4×C5×#99C211yesmaximal, characteristic, complemented
10C2×C5×H(5,𝔽2)C2212yes—
11C5×#547—22no—
12C5×#546—24no—
13C5×D42—22no—
14C5×#537—21no—
15C5×#534—21no—
16C5×#534C2212yes—
17C5×#532—21no—
18C2×C5×#121—21no—
19C2×C5×#121C2212yes—
20C2×C4×C5×D4C2211yescharacteristic
21C22×C5×C4⋊C4C2211yescharacteristic, complemented
22C2×C5×#99C2212yes—
23C2×C5×#99C2214yescomplemented
24C2×C5×#99C2214yescomplemented
25C5×#400—22no—
26C5×#398—21no—
27C5×#396—21no—
28C5×#394—22no—
29C5×#388—24no—
30C5×#387—21no—
31C5×#386C2211yescharacteristic
32C5×#386C2211yescharacteristic
33C5×#386C2214yescomplemented
34C5×#386C2214yescomplemented
35C5×#385—21no—
36C5×#385C2211yescharacteristic, complemented
37C5×#385C2211yescharacteristic, complemented
38C5×#380C2211yescharacteristic, complemented
39C4×C5×C22⋊C4C2212yescomplemented
40C5×#342C2212yescomplemented
41#25901C511yesmaximal, characteristic, direct product factor, Sylow
42C5×H(5,𝔽2)—24no—
43C5×H(5,𝔽2)—24no—
44C5×H(5,𝔽2)—44no—
45C2×C5×#41—22no—
46C2×C5×#41—24no—
47C22×C5×D4—22no—
48C22×C5×D4—24no—
49C22×C5×D4—21no—
50C22×C5×D4D412yes—
51C22×C5×D4C2314yes—
52C22×C5×D4C2311yescharacteristic
53C23×C4×C5D411yescharacteristic
54C23×C4×C5C2311yescharacteristic
55C23×C4×C5C2311yescharacteristic
56C23×C4×C5C2311yescharacteristic
57C5×C4⋊Q8—21no—
58C5×#127—21no—
59C5×#124—22no—
60C5×#123—24no—
61C5×#123—24no—
62C5×C22⋊Q8—24no—
63C5×#121—22no—
64C5×#121—22no—
65C5×#121—24no—
66C5×#121—22no—
67C5×#121—22no—
68C5×#121—22no—
69C5×#121—22no—
70C5×#121—22no—
71C5×#121—22no—
72C5×#120—24no—
73C5×#120—24no—
74C4×C5×D4—22no—
75C4×C5×D4—22no—
76C4×C5×D4—22no—
77C4×C5×D4—22no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4—22no—
82C2×C5×C4⋊C4—22no—
83C2×C5×C4⋊C4—21no—
84C2×C5×C4⋊C4—21no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4D411yescharacteristic
87C2×C5×C4⋊C4D412yes—
88C2×C5×C22⋊C4—22no—
89C2×C5×C22⋊C4—21no—
90C2×C5×C22⋊C4—24no—
91C2×C5×C22⋊C4—21no—
92C2×C5×C22⋊C4—24no—
93C2×C5×C22⋊C4D414yes—
94C2×C5×C22⋊C4D411yescharacteristic
95C2×C5×C22⋊C4C2314yes—
96C2×C5×C22⋊C4C2311yescharacteristic
97C2×C5×C22⋊C4C2311yescharacteristic
98C2×C5×C22⋊C4C2311yescharacteristic
99C2×C42×C5—22no—
100C2×C42×C5D411yescharacteristic
101C5×#99—24no—
102C5×#99—44no—
103C5×#99—44no—
104C5×#95—22no—
105C5×#95—21no—
106C5×#95—21no—
107C5×#95—21no—
108C5×#95—21no—
109C5×#95—21no—
110C5×#95—24no—
111C5×#95—24no—
112#3496C2×C511yescharacteristic
113#2591C2×C512yescomplemented
114#2579C2×C511yescharacteristic, complemented
115#2044C2×C512yescomplemented
116#2044C2×C512yescomplemented
117#1964C2×C512yescomplemented
118#1790C2×C512yescomplemented
119#1790C2×C512yescomplemented
120C4×#99C2×C511yescharacteristic, complemented
121C24×C5—22no—
122C24×C5C2×D414yes—
123C24×C5C2×D411yescharacteristic
124C24×C5C2411yescharacteristic
125C5×#41—48no—
126C5×#41—48no—
127C5×#41—44no—
128C5×#41—44no—
129C2×C5×Q8—22no—
130C2×C5×D4—22no—
131C2×C5×D4—22no—
132C2×C5×D4—24no—
133C2×C5×D4—22no—
134C2×C5×D4—24no—
135C2×C5×D4—24no—
136C2×C5×D4—24no—
137C2×C5×D4—24no—
138C2×C5×D4—24no—
139C2×C5×D4—24no—
140C2×C5×D4—22no—
141C2×C5×D4—22no—
142C2×C5×D4—22no—
143C2×C5×D4—22no—
144C2×C5×D4—48no—
145C2×C5×D4—48no—
146C2×C5×D4—44no—
147C2×C5×D4—44no—
148C2×C5×D4—82no—
149C2×C5×D4—82no—
150C2×C5×D4#4114yes—
151C22×C4×C5—22no—
152C22×C4×C5—24no—
153C22×C4×C5—22no—
154C22×C4×C5—21no—
155C22×C4×C5—21no—
156C22×C4×C5—21no—
157C22×C4×C5—24no—
158C22×C4×C5—22no—
159C22×C4×C5—21no—
160C22×C4×C5—21no—
161C22×C4×C5—21no—
162C22×C4×C5—42no—
163C22×C4×C5—81no—
164C22×C4×C5—81no—
165C22×C4×C5—81no—
166C22×C4×C5C2×D414yes—
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5C2×D412yes—
171C22×C4×C5C2×D411yescharacteristic
172C22×C4×C5C2×D411yescharacteristic
173C22×C4×C5C2×D412yes—
174C5×C4⋊C4—22no—
175C5×C4⋊C4—22no—
176C5×C4⋊C4—22no—
177C5×C4⋊C4—22no—
178C5×C4⋊C4—22no—
179C5×C4⋊C4—42no—
180C5×C4⋊C4—41no—
181C5×C4⋊C4—41no—
182C5×C4⋊C4—42no—
183C5×C4⋊C4—81no—
184C5×C4⋊C4—81no—
185C5×C22⋊C4—24no—
186C5×C22⋊C4—24no—
187C5×C22⋊C4—22no—
188C5×C22⋊C4—24no—
189C5×C22⋊C4—22no—
190C5×C22⋊C4—24no—
191C5×C22⋊C4—24no—
192C5×C22⋊C4—22no—
193C5×C22⋊C4—22no—
194C5×C22⋊C4—24no—
195C5×C22⋊C4—48no—
196C5×C22⋊C4—42no—
197C5×C22⋊C4—84no—
198C5×C22⋊C4—81no—
199C42×C5—21no—
200C42×C5—21no—
201C42×C5—82no—
202C2×H(5,𝔽2)C22×C512yes—
203#547—22no—
204#546—24no—
205D42—22no—
206#537—21no—
207#534—21no—
208#534C22×C512yes—
209#532—21no—
210C2×#121—21no—
211C2×#121C22×C512yes—
212C2×C4×D4C22×C511yescharacteristic
213C22×C4⋊C4C22×C511yescharacteristic, complemented
214C2×#99C22×C512yes—
215C2×#99C22×C514yescomplemented
216C2×#99C22×C514yescomplemented
217#400—22no—
218#398—21no—
219#396—21no—
220#394—22no—
221#388—24no—
222#387—21no—
223#386C22×C511yescharacteristic
224#386C22×C511yescharacteristic
225#386C22×C514yescomplemented
226#386C22×C514yescomplemented
227#385—21no—
228#385C22×C511yescharacteristic, complemented
229#385C22×C511yescharacteristic, complemented
230#380C22×C511yescharacteristic, complemented
231C4×C22⋊C4C22×C512yescomplemented
232#342C22×C512yescomplemented
233C23×C5—21no—
234C23×C5—21no—
235C23×C5—24no—
236C23×C5—24no—
237C23×C5—24no—
238C23×C5—41no—
239C23×C5—41no—
240C23×C5—41no—
241C23×C5—41no—
242C23×C5—44no—
243C23×C5—44no—
244C23×C5—44no—
245C23×C5—82no—
246C23×C5—82no—
247C23×C5#12412yes—
248C23×C5C22×D411yescharacteristic
249C23×C5C22×D412yes—
250C23×C5C2×#4112yes—
251C23×C5H(5,𝔽2)12yes—
252C5×Q8—44no—
253C5×D4—28no—
254C5×D4—44no—
255C5×D4—44no—
256C5×D4—48no—
257C5×D4—44no—
258C5×D4—44no—
259C5×D4—44no—
260C5×D4—44no—
261C5×D4—82no—
262C5×D4—82no—
263C5×D4—82no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—24no—
267C2×C4×C5—22no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—24no—
271C2×C4×C5—24no—
272C2×C4×C5—24no—
273C2×C4×C5—42no—
274C2×C4×C5—41no—
275C2×C4×C5—41no—
276C2×C4×C5—42no—
277C2×C4×C5—42no—
278C2×C4×C5—48no—
279C2×C4×C5—41no—
280C2×C4×C5—42no—
281C2×C4×C5—44no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—44no—
286C2×C4×C5—41no—
287C2×C4×C5—44no—
288C2×C4×C5—41no—
289C2×C4×C5—41no—
290C2×C4×C5—44no—
291C2×C4×C5—41no—
292C2×C4×C5—41no—
293C2×C4×C5—41no—
294C2×C4×C5—41no—
295C2×C4×C5—41no—
296C2×C4×C5—41no—
297C2×C4×C5—44no—
298C2×C4×C5—81no—
299C2×C4×C5#12011yescharacteristic
300C2×C4×C5#12011yescharacteristic
301C2×C4×C5#12012yes—
302C2×C4×C5#12412yes—
303H(5,𝔽2)—24no—
304H(5,𝔽2)—24no—
305H(5,𝔽2)—44no—
306C2×#41—22no—
307C2×#41—24no—
308C22×D4—22no—
309C22×D4—24no—
310C22×D4—21no—
311C22×D4C5×D412yes—
312C22×D4C23×C514yes—
313C22×D4C23×C511yescharacteristic
314C23×C4C5×D411yescharacteristic
315C23×C4C23×C511yescharacteristic
316C23×C4C23×C511yescharacteristic
317C23×C4C23×C511yescharacteristic
318C4⋊Q8—21no—
319#127—21no—
320#124—22no—
321#123—24no—
322#123—24no—
323C22⋊Q8—24no—
324#121—22no—
325#121—22no—
326#121—24no—
327#121—22no—
328#121—22no—
329#121—22no—
330#121—22no—
331#121—22no—
332#121—22no—
333#120—24no—
334#120—24no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4C5×D411yescharacteristic
348C2×C4⋊C4C5×D412yes—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4C5×D411yescharacteristic
355C2×C22⋊C4C5×D414yes—
356C2×C22⋊C4C23×C514yes—
357C2×C22⋊C4C23×C511yescharacteristic
358C2×C22⋊C4C23×C511yescharacteristic
359C2×C22⋊C4C23×C511yescharacteristic
360C2×C42—22no—
361C2×C42C5×D411yescharacteristic
362#99—24no—
363#99—44no—
364#99—44no—
365#95—22no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—24no—
372#95—24no—
373C22×C5—21no—
374C22×C5—22no—
375C22×C5—21no—
376C22×C5—22no—
377C22×C5—22no—
378C22×C5—42no—
379C22×C5—42no—
380C22×C5—42no—
381C22×C5—41no—
382C22×C5—41no—
383C22×C5—44no—
384C22×C5—44no—
385C22×C5—44no—
386C22×C5—81no—
387C22×C5—84no—
388C22×C5—84no—
389C22×C5—164no—
390C22×C5C2×#12011yescharacteristic
391C22×C5C2×#12411yescharacteristic
392C22×C5#53511yescharacteristic
393C22×C5#54614yes—
394C4×C5—22no—
395C4×C5—21no—
396C4×C5—21no—
397C4×C5—22no—
398C4×C5—44no—
399C4×C5—42no—
400C4×C5—41no—
401C4×C5—41no—
402C4×C5—81no—
403C4×C5—81no—
404C4×C5—84no—
405C4×C5—81no—
406C24—22no—
407C24C2×C5×D414yes—
408C24C2×C5×D411yescharacteristic
409C24C24×C511yescharacteristic
410#41—48no—
411#41—48no—
412#41—44no—
413#41—44no—
414C2×Q8—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4C5×#4114yes—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4C2×C5×D414yes—
452C22×C4C2×C5×D411yescharacteristic
453C22×C4C2×C5×D411yescharacteristic
454C22×C4C2×C5×D411yescharacteristic
455C22×C4C2×C5×D411yescharacteristic
456C22×C4C2×C5×D412yes—
457C22×C4C2×C5×D412yes—
458C22×C4C2×C5×D411yescharacteristic
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—42no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—42no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—22no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—24no—
480C22⋊C4—48no—
481C22⋊C4—42no—
482C22⋊C4—84no—
483C22⋊C4—81no—
484C42—21no—
485C42—21no—
486C42—82no—
487C2×C5—21no—
488C2×C5—24no—
489C2×C5—21no—
490C2×C5—42no—
491C2×C5—84no—
492C2×C5#243511yescharacteristic
493C2×C5#302511yescharacteristic
494C2×C5#302611yescharacteristic
495C23—21no—
496C23—21no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—82no—
508C23—82no—
509C23C5×#12412yes—
510C23C22×C5×D411yescharacteristic
511C23C22×C5×D412yes—
512C23C2×C5×#4112yes—
513C23C5×H(5,𝔽2)12yes—
514Q8—44no—
515D4—28no—
516D4—44no—
517D4—44no—
518D4—48no—
519D4—44no—
520D4—44no—
521D4—44no—
522D4—44no—
523D4—82no—
524D4—82no—
525D4—82no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—48no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—81no—
561C2×C4C5×#12011yescharacteristic
562C2×C4C5×#12011yescharacteristic
563C2×C4C5×#12012yes—
564C2×C4C5×#12412yes—
565C5#2590111yescharacteristic, direct product factor, Sylow
566C22—21no—
567C22—22no—
568C22—21no—
569C22—22no—
570C22—22no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—81no—
580C22—84no—
581C22—84no—
582C22—164no—
583C22C2×C5×#12011yescharacteristic
584C22C2×C5×#12411yescharacteristic
585C22C5×#53511yescharacteristic
586C22C5×#54614yes—
587C4—21no—
588C4—21no—
589C4—22no—
590C4—22no—
591C4—42no—
592C4—41no—
593C4—41no—
594C4—44no—
595C4—81no—
596C4—81no—
597C4—84no—
598C4—81no—
599C2—21no—
600C2—24no—
601C2—21no—
602C2—42no—
603C2—84no—
604C2C5×#243511yescharacteristic
605C2C5×#302511yescharacteristic
606C2C5×#302611yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC24C24×C5
prime radicalC5×#3496C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2591C2semidirect
3. C5×#2579C2semidirect
4. C5×#2044C2semidirect
5. C5×#2044C2semidirect
6. C5×#1964C2semidirect
7. C5×#1790C2semidirect
8. C5×#1790C2semidirect
9. C4×C5×#99C2semidirect
21. C22×C5×C4⋊C4C22semidirect
23. C2×C5×#99C22semidirect
24. C2×C5×#99C22semidirect
33. C5×#386C22semidirect
34. C5×#386C22semidirect
36. C5×#385C22semidirect
37. C5×#385C22semidirect
38. C5×#380C22semidirect
39. C4×C5×C22⋊C4C22semidirect
40. C5×#342C22semidirect
41. #25901C5direct (N × H)
113. #2591C2×C5semidirect
114. #2579C2×C5semidirect
115. #2044C2×C5semidirect
116. #2044C2×C5semidirect
117. #1964C2×C5semidirect
118. #1790C2×C5semidirect
119. #1790C2×C5semidirect
120. C4×#99C2×C5semidirect
213. C22×C4⋊C4C22×C5semidirect
215. C2×#99C22×C5semidirect
216. C2×#99C22×C5semidirect
225. #386C22×C5semidirect
226. #386C22×C5semidirect
228. #385C22×C5semidirect
229. #385C22×C5semidirect
230. #380C22×C5semidirect
231. C4×C22⋊C4C22×C5semidirect
232. #342C22×C5semidirect
565. C5#25901direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#34962
C5×#25912
C5×#25792
C5×#20442
C5×#19642
C5×#17902
C4×C5×#992
#259015