Finite Group Databasebeta

← C5×#14543 #11937263

Subgroups — C5×#14543 #11937263

Order 1280.

1490 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#399C211yesmaximal, characteristic
2C2×C5×#388C212yesmaximal
3C2×C5×#387C212yesmaximal, complemented
4C5×#1441C212yesmaximal, complemented
5C5×#1440C211yesmaximal, characteristic
6C5×#1439C212yesmaximal
7C5×#1437C211yesmaximal, characteristic
8C5×#1436C212yesmaximal, complemented
9C5×C4⋊#95C212yesmaximal, complemented
10C22×C5×C4⋊C4C2212yes—
11C22×C5×C4⋊C4C2212yes—
12C22×C5×C4⋊C4C2212yes—
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C5×C22⋊C4C2211yescharacteristic
15C22×C5×C22⋊C4C2212yes—
16C22×C5×C22⋊C4C2212yes—
17C22×C5×C22⋊C4C2211yescharacteristic
18C22×C5×C22⋊C4C2211yescharacteristic
19C22×C42×C5C2211yescharacteristic
20C22×C42×C5C2212yes—
21C5×#399C418yescomplemented
22C5×#388—28no—
23C5×#387—28no—
24C2×C5×#95C2212yes—
25C2×C5×#95C2212yes—
26C2×C5×#95C2212yes—
27C2×C5×#95C2212yes—
28C2×C5×#95C2211yescharacteristic
29C2×C5×#95C2212yes—
30C2×C5×#95C2212yes—
31C2×C5×#95C2212yes—
32C2×C5×#95C2211yescharacteristic
33C2×C5×#95C2212yes—
34#14543C511yesmaximal, characteristic, direct product factor, Sylow
35C25×C5C2311yescharacteristic
36C23×C4×C5C2312yes—
37C23×C4×C5C2311yescharacteristic
38C23×C4×C5C2311yescharacteristic
39C23×C4×C5C2312yes—
40C23×C4×C5C2312yes—
41C23×C4×C5C2312yes—
42C23×C4×C5C2312yes—
43C23×C4×C5C2312yes—
44C2×C5×C4⋊C4—24no—
45C2×C5×C4⋊C4—24no—
46C2×C5×C4⋊C4—24no—
47C2×C5×C4⋊C4—24no—
48C2×C5×C4⋊C4—24no—
49C2×C5×C4⋊C4—24no—
50C2×C5×C4⋊C4—41no—
51C2×C5×C4⋊C4—41no—
52C2×C5×C4⋊C4—42no—
53C2×C5×C4⋊C4—42no—
54C2×C5×C4⋊C4C2×C418yes—
55C2×C5×C4⋊C4C2×C414yes—
56C2×C5×C22⋊C4—24no—
57C2×C5×C22⋊C4—24no—
58C2×C5×C22⋊C4—24no—
59C2×C5×C22⋊C4—24no—
60C2×C5×C22⋊C4—24no—
61C2×C5×C22⋊C4—41no—
62C2×C5×C22⋊C4—41no—
63C2×C5×C22⋊C4—42no—
64C2×C5×C22⋊C4—41no—
65C2×C5×C22⋊C4—42no—
66C2×C5×C22⋊C4—41no—
67C2×C5×C22⋊C4C2×C418yes—
68C2×C5×C22⋊C4C2×C414yes—
69C2×C42×C5—24no—
70C2×C42×C5—24no—
71C2×C42×C5—24no—
72C2×C42×C5—42no—
73C2×C42×C5—41no—
74C5×#95—24no—
75C5×#95—24no—
76C5×#95—24no—
77C5×#95—24no—
78C5×#95—42no—
79C5×#95—42no—
80C5×#95—41no—
81C5×#95—42no—
82C5×#95C2×C414yes—
83C5×#95D418yes—
84C2×#399C2×C511yescharacteristic
85C2×#388C2×C512yes—
86C2×#387C2×C512yescomplemented
87#1441C2×C512yescomplemented
88#1440C2×C511yescharacteristic
89#1439C2×C512yes—
90#1437C2×C511yescharacteristic
91#1436C2×C512yescomplemented
92C4⋊#95C2×C512yescomplemented
93C24×C5—22no—
94C24×C5—22no—
95C24×C5—22no—
96C24×C5—21no—
97C24×C5—21no—
98C24×C5—22no—
99C24×C5—22no—
100C24×C5—22no—
101C24×C5C22×C412yes—
102C24×C5C2411yescharacteristic
103C22×C4×C5—22no—
104C22×C4×C5—22no—
105C22×C4×C5—22no—
106C22×C4×C5—22no—
107C22×C4×C5—22no—
108C22×C4×C5—22no—
109C22×C4×C5—22no—
110C22×C4×C5—22no—
111C22×C4×C5—22no—
112C22×C4×C5—22no—
113C22×C4×C5—22no—
114C22×C4×C5—22no—
115C22×C4×C5—22no—
116C22×C4×C5—22no—
117C22×C4×C5—22no—
118C22×C4×C5—22no—
119C22×C4×C5—21no—
120C22×C4×C5—22no—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—21no—
124C22×C4×C5—21no—
125C22×C4×C5—22no—
126C22×C4×C5—21no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—22no—
132C22×C4×C5—22no—
133C22×C4×C5—22no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
137C22×C4×C5—22no—
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—22no—
145C22×C4×C5—22no—
146C22×C4×C5C22×C414yes—
147C22×C4×C5C22×C412yes—
148C22×C4×C5C22×C412yes—
149C22×C4×C5C22×C414yes—
150C22×C4×C5C2×D414yes—
151C22×C4×C5C2×D414yes—
152C22×C4×C5C2×D414yes—
153C22×C4×C5#4114yes—
154C22×C4×C5#4114yes—
155C5×C4⋊C4—216no—
156C5×C4⋊C4—44no—
157C5×C4⋊C4—48no—
158C5×C4⋊C4—48no—
159C5×C22⋊C4—216no—
160C5×C22⋊C4—44no—
161C5×C22⋊C4—44no—
162C5×C22⋊C4—48no—
163C5×C22⋊C4—44no—
164C42×C5—48no—
165C42×C5—44no—
166C22×C4⋊C4C22×C512yes—
167C22×C4⋊C4C22×C512yes—
168C22×C4⋊C4C22×C512yes—
169C22×C4⋊C4C22×C511yescharacteristic
170C22×C22⋊C4C22×C512yes—
171C22×C22⋊C4C22×C511yescharacteristic
172C22×C22⋊C4C22×C512yes—
173C22×C22⋊C4C22×C511yescharacteristic
174C22×C22⋊C4C22×C511yescharacteristic
175C22×C42C22×C512yes—
176C22×C42C22×C511yescharacteristic
177#399C4×C518yescomplemented
178#388—28no—
179#387—28no—
180C2×#95C22×C512yes—
181C2×#95C22×C512yes—
182C2×#95C22×C512yes—
183C2×#95C22×C512yes—
184C2×#95C22×C511yescharacteristic
185C2×#95C22×C512yes—
186C2×#95C22×C512yes—
187C2×#95C22×C512yes—
188C2×#95C22×C511yescharacteristic
189C2×#95C22×C512yes—
190C23×C5—24no—
191C23×C5—22no—
192C23×C5—22no—
193C23×C5—22no—
194C23×C5—24no—
195C23×C5—42no—
196C23×C5—42no—
197C23×C5—42no—
198C23×C5—42no—
199C23×C5—42no—
200C23×C5—42no—
201C23×C5—41no—
202C23×C5—42no—
203C23×C5—42no—
204C23×C5—42no—
205C23×C5—42no—
206C23×C5—42no—
207C23×C5—41no—
208C23×C5—41no—
209C23×C5—42no—
210C23×C5—41no—
211C23×C5C23×C411yescharacteristic
212C23×C5C22×D412yes—
213C23×C5C2×#4112yes—
214C23×C5C2×#4112yes—
215C23×C5H(5,𝔽2)11yescharacteristic
216C23×C5H(5,𝔽2)11yescharacteristic
217C23×C5H(5,𝔽2)12yes—
218C23×C52−1+412yes—
219C23×C52−1+412yes—
220C2×C4×C5—24no—
221C2×C4×C5—24no—
222C2×C4×C5—24no—
223C2×C4×C5—24no—
224C2×C4×C5—24no—
225C2×C4×C5—24no—
226C2×C4×C5—24no—
227C2×C4×C5—24no—
228C2×C4×C5—24no—
229C2×C4×C5—22no—
230C2×C4×C5—22no—
231C2×C4×C5—24no—
232C2×C4×C5—24no—
233C2×C4×C5—24no—
234C2×C4×C5—24no—
235C2×C4×C5—24no—
236C2×C4×C5—24no—
237C2×C4×C5—24no—
238C2×C4×C5—24no—
239C2×C4×C5—24no—
240C2×C4×C5—24no—
241C2×C4×C5—24no—
242C2×C4×C5—42no—
243C2×C4×C5—42no—
244C2×C4×C5—42no—
245C2×C4×C5—42no—
246C2×C4×C5—42no—
247C2×C4×C5—42no—
248C2×C4×C5—41no—
249C2×C4×C5—42no—
250C2×C4×C5—42no—
251C2×C4×C5—42no—
252C2×C4×C5—42no—
253C2×C4×C5—42no—
254C2×C4×C5—42no—
255C2×C4×C5—42no—
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—42no—
263C2×C4×C5—41no—
264C2×C4×C5—42no—
265C2×C4×C5—42no—
266C2×C4×C5—42no—
267C2×C4×C5—42no—
268C2×C4×C5—42no—
269C2×C4×C5—42no—
270C2×C4×C5—42no—
271C2×C4×C5C4×D418yes—
272C25C23×C511yescharacteristic
273C23×C4C23×C512yes—
274C23×C4C23×C512yes—
275C23×C4C23×C511yescharacteristic
276C23×C4C23×C511yescharacteristic
277C23×C4C23×C512yes—
278C23×C4C23×C512yes—
279C23×C4C23×C512yes—
280C23×C4C23×C512yes—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—41no—
288C2×C4⋊C4—41no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4C2×C4×C518yes—
292C2×C4⋊C4C2×C4×C514yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4C2×C4×C518yes—
305C2×C22⋊C4C2×C4×C514yes—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—42no—
310C2×C42—41no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—41no—
316#95—42no—
317#95—42no—
318#95—42no—
319#95C2×C4×C514yes—
320#95C5×D418yes—
321C22×C5—44no—
322C22×C5—42no—
323C22×C5—44no—
324C22×C5—42no—
325C22×C5—42no—
326C22×C5—82no—
327C22×C5—82no—
328C22×C5—82no—
329C22×C5—82no—
330C22×C5C2×C4×D412yes—
331C22×C5C4×#4111yescharacteristic
332C22×C5#51811yescharacteristic
333C22×C5#51911yescharacteristic
334C22×C5#52012yes—
335C22×C5#53612yes—
336C22×C5#53712yes—
337C22×C5#53912yes—
338C22×C5#54112yes—
339C22×C5#54212yes—
340C22×C5#54212yes—
341C22×C5D4211yescharacteristic
342C22×C5#54612yes—
343C22×C5#54712yes—
344C22×C5#55012yes—
345C22×C5#55112yes—
346C22×C5#55212yes—
347C22×C5#55911yescharacteristic
348C22×C5#56011yescharacteristic
349C22×C5#56212yes—
350C22×C5#56311yescharacteristic
351C4×C5—28no—
352C4×C5—44no—
353C4×C5—42no—
354C4×C5—42no—
355C4×C5—44no—
356C4×C5—44no—
357C4×C5—44no—
358C4×C5—82no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24C22×C4×C512yes—
368C24C24×C511yescharacteristic
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4C22×C4×C512yes—
413C22×C4C22×C4×C514yes—
414C22×C4C22×C4×C512yes—
415C22×C4C22×C4×C514yes—
416C22×C4C2×C5×D414yes—
417C22×C4C2×C5×D414yes—
418C22×C4C2×C5×D414yes—
419C22×C4C5×#4114yes—
420C22×C4C5×#4114yes—
421C4⋊C4—216no—
422C4⋊C4—44no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C22⋊C4—216no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—48no—
429C22⋊C4—44no—
430C42—48no—
431C42—44no—
432C2×C5—82no—
433C2×C5#232312yes—
434C2×C5#233211yescharacteristic
435C2×C5#235912yes—
436C2×C5#236611yescharacteristic
437C2×C5#237611yescharacteristic
438C2×C5#267212yes—
439C2×C5#270512yes—
440C2×C5#272112yes—
441C2×C5#275412yes—
442C23—24no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—41no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—41no—
463C23C23×C4×C511yescharacteristic
464C23C22×C5×D412yes—
465C23C2×C5×#4112yes—
466C23C2×C5×#4112yes—
467C23C5×H(5,𝔽2)11yescharacteristic
468C23C5×H(5,𝔽2)11yescharacteristic
469C23C5×H(5,𝔽2)12yes—
470C23C5×2−1+412yes—
471C23C5×2−1+412yes—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4C4×C5×D418yes—
524C5#1454311yescharacteristic, direct product factor, Sylow
525C22—44no—
526C22—42no—
527C22—44no—
528C22—42no—
529C22—42no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22C2×C4×C5×D412yes—
535C22C4×C5×#4111yescharacteristic
536C22C5×#51811yescharacteristic
537C22C5×#51911yescharacteristic
538C22C5×#52012yes—
539C22C5×#53612yes—
540C22C5×#53712yes—
541C22C5×#53912yes—
542C22C5×#54112yes—
543C22C5×#54212yes—
544C22C5×#54212yes—
545C22C5×D4211yescharacteristic
546C22C5×#54612yes—
547C22C5×#54712yes—
548C22C5×#55012yes—
549C22C5×#55112yes—
550C22C5×#55212yes—
551C22C5×#55911yescharacteristic
552C22C5×#56011yescharacteristic
553C22C5×#56212yes—
554C22C5×#56311yescharacteristic
555C4—28no—
556C4—44no—
557C4—42no—
558C4—42no—
559C4—44no—
560C4—44no—
561C4—44no—
562C4—82no—
563C2—82no—
564C2C5×#232312yes—
565C2C5×#233211yescharacteristic
566C2C5×#235912yes—
567C2C5×#236611yescharacteristic
568C2C5×#237611yescharacteristic
569C2C5×#267212yes—
570C2C5×#270512yes—
571C2C5×#272112yes—
572C2C5×#275412yes—
subgroupG/H
center, socleC24×C5C24
commutatorC23C23×C4×C5
Frattini, cosocleC24C24×C5
prime radicalC25×C5C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C5 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C5×#387C2semidirect
4. C5×#1441C2semidirect
8. C5×#1436C2semidirect
9. C5×C4⋊#95C2semidirect
21. C5×#399C4semidirect
34. #14543C5direct (N × H)
86. C2×#387C2×C5semidirect
87. #1441C2×C5semidirect
91. #1436C2×C5semidirect
92. C4⋊#95C2×C5semidirect
177. #399C4×C5semidirect
524. C5#14543direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#3992
C2×C5×#3882
C2×C5×#3872
C5×#14412
C5×#14402
C5×#14392
C5×#14372
C5×#14362
C5×C4⋊#952
#145435