Finite Group Databasebeta

← C5×#14669 #11937389

Subgroups — C5×#14669 #11937389

Order 1280.

1342 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#385C211yesmaximal, characteristic
2C5×#1445C212yesmaximal, complemented
3C5×#1443C212yesmaximal, complemented
4C5×#1442C212yesmaximal, complemented
5C5×#1441C212yesmaximal, complemented
6C5×#1440C212yesmaximal
7C5×#1439C211yesmaximal, characteristic
8C5×#1439C211yesmaximal, characteristic
9C5×C22⋊#95C212yesmaximal
10C24×C4×C5C2211yescharacteristic
11C22×C5×C4⋊C4C2211yescharacteristic
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C5×C4⋊C4C2212yes—
14C22×C5×C22⋊C4C2211yescharacteristic
15C22×C5×C22⋊C4C2212yes—
16C22×C5×C22⋊C4C2212yes—
17C22×C5×C22⋊C4C2211yescharacteristic
18C5×#385C418yescomplemented
19C2×C5×#95C2212yes—
20C2×C5×#95C2212yes—
21C2×C5×#95C2212yes—
22C2×C5×#95C2212yes—
23C2×C5×#95C2212yes—
24C2×C5×#95C2212yes—
25C2×C5×#95C2211yescharacteristic
26C2×C5×#95C2211yescharacteristic
27C2×C5×#95C2212yes—
28C2×C5×#95C2212yes—
29C2×C5×#95C2212yes—
30C2×C5×#95C2212yes—
31C2×C5×#95C2212yes—
32#14669C511yesmaximal, characteristic, direct product factor, Sylow
33C25×C5C2311yescharacteristic
34C23×C4×C5—42no—
35C23×C4×C5—42no—
36C23×C4×C5—41no—
37C23×C4×C5—41no—
38C23×C4×C5C2×C414yes—
39C23×C4×C5C2312yes—
40C23×C4×C5C2312yes—
41C23×C4×C5C2312yes—
42C23×C4×C5C2312yes—
43C23×C4×C5C2312yes—
44C23×C4×C5C2311yescharacteristic
45C23×C4×C5C2312yes—
46C23×C4×C5C2311yescharacteristic
47C2×C5×C4⋊C4—42no—
48C2×C5×C4⋊C4—42no—
49C2×C5×C4⋊C4—41no—
50C2×C5×C4⋊C4—41no—
51C2×C5×C4⋊C4—42no—
52C2×C5×C4⋊C4—42no—
53C2×C5×C4⋊C4C2×C418yes—
54C2×C5×C22⋊C4—22no—
55C2×C5×C22⋊C4—22no—
56C2×C5×C22⋊C4—24no—
57C2×C5×C22⋊C4—24no—
58C2×C5×C22⋊C4—41no—
59C2×C5×C22⋊C4—41no—
60C2×C5×C22⋊C4—42no—
61C2×C5×C22⋊C4—42no—
62C2×C5×C22⋊C4—42no—
63C2×C5×C22⋊C4—42no—
64C2×C5×C22⋊C4C2×C414yes—
65C2×C5×C22⋊C4C2×C414yes—
66C5×#95—41no—
67C5×#95—41no—
68C5×#95—42no—
69C5×#95—42no—
70C5×#95—42no—
71C5×#95—42no—
72C5×#95—42no—
73C5×#95—42no—
74C5×#95—42no—
75C5×#95—42no—
76C5×#95—42no—
77C5×#95—42no—
78C5×#95C2×C418yes—
79C2×#385C2×C511yescharacteristic
80#1445C2×C512yescomplemented
81#1443C2×C512yescomplemented
82#1442C2×C512yescomplemented
83#1441C2×C512yescomplemented
84#1440C2×C512yes—
85#1439C2×C511yescharacteristic
86#1439C2×C511yescharacteristic
87C22⋊#95C2×C512yes—
88C24×C5—22no—
89C24×C5—21no—
90C24×C5—21no—
91C24×C5—22no—
92C24×C5—22no—
93C24×C5—21no—
94C24×C5—21no—
95C24×C5—22no—
96C24×C5C22×C412yes—
97C24×C5#4112yes—
98C24×C5#4112yes—
99C24×C5C2411yescharacteristic
100C22×C4×C5—22no—
101C22×C4×C5—22no—
102C22×C4×C5—21no—
103C22×C4×C5—21no—
104C22×C4×C5—21no—
105C22×C4×C5—21no—
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—22no—
120C22×C4×C5—22no—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
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×C5—22no—
147C22×C4×C5—22no—
148C22×C4×C5—48no—
149C22×C4×C5—44no—
150C22×C4×C5—44no—
151C22×C4×C5—48no—
152C22×C4×C5—44no—
153C22×C4×C5C22×C412yes—
154C22×C4×C5C22×C414yes—
155C22×C4×C5C22×C412yes—
156C22×C4×C5C22×C414yes—
157C5×C4⋊C4—48no—
158C5×C4⋊C4—44no—
159C5×C4⋊C4—44no—
160C5×C22⋊C4—28no—
161C5×C22⋊C4—44no—
162C5×C22⋊C4—48no—
163C5×C22⋊C4—48no—
164C24×C4C22×C511yescharacteristic
165C22×C4⋊C4C22×C511yescharacteristic
166C22×C4⋊C4C22×C511yescharacteristic
167C22×C4⋊C4C22×C512yes—
168C22×C22⋊C4C22×C511yescharacteristic
169C22×C22⋊C4C22×C512yes—
170C22×C22⋊C4C22×C512yes—
171C22×C22⋊C4C22×C511yescharacteristic
172#385C4×C518yescomplemented
173C2×#95C22×C512yes—
174C2×#95C22×C512yes—
175C2×#95C22×C512yes—
176C2×#95C22×C512yes—
177C2×#95C22×C512yes—
178C2×#95C22×C512yes—
179C2×#95C22×C511yescharacteristic
180C2×#95C22×C511yescharacteristic
181C2×#95C22×C512yes—
182C2×#95C22×C512yes—
183C2×#95C22×C512yes—
184C2×#95C22×C512yes—
185C2×#95C22×C512yes—
186C23×C5—22no—
187C23×C5—24no—
188C23×C5—22no—
189C23×C5—24no—
190C23×C5—24no—
191C23×C5—24no—
192C23×C5—22no—
193C23×C5—22no—
194C23×C5—22no—
195C23×C5—22no—
196C23×C5—24no—
197C23×C5—24no—
198C23×C5—41no—
199C23×C5—42no—
200C23×C5—41no—
201C23×C5—41no—
202C23×C5—42no—
203C23×C5—42no—
204C23×C5—42no—
205C23×C5—42no—
206C23×C5—42no—
207C23×C5—41no—
208C23×C5#11714yes—
209C23×C5C23×C411yescharacteristic
210C23×C5C2×#4111yescharacteristic
211C23×C5C2×#4111yescharacteristic
212C23×C5H(5,𝔽2)12yes—
213C23×C5H(5,𝔽2)12yes—
214C23×C5H(5,𝔽2)11yescharacteristic
215C23×C5H(5,𝔽2)11yescharacteristic
216C23×C5H(5,𝔽2)12yes—
217C23×C52−1+411yescharacteristic
218C23×C52−1+411yescharacteristic
219C23×C52−1+412yes—
220C2×C4×C5—24no—
221C2×C4×C5—22no—
222C2×C4×C5—22no—
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—24no—
230C2×C4×C5—416no—
231C2×C4×C5—42no—
232C2×C4×C5—42no—
233C2×C4×C5—42no—
234C2×C4×C5—42no—
235C2×C4×C5—41no—
236C2×C4×C5—41no—
237C2×C4×C5—42no—
238C2×C4×C5—42no—
239C2×C4×C5—42no—
240C2×C4×C5—42no—
241C2×C4×C5—41no—
242C2×C4×C5—42no—
243C2×C4×C5—42no—
244C2×C4×C5—41no—
245C2×C4×C5—42no—
246C2×C4×C5—42no—
247C2×C4×C5—42no—
248C2×C4×C5—42no—
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—42no—
258C2×C4×C5—42no—
259C2×C4×C5—42no—
260C2×C4×C5—42no—
261C2×C4×C5—42no—
262C2×C4×C5—42no—
263C2×C4×C5—42no—
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×C5—42no—
272C2×C4×C5—42no—
273C25C23×C511yescharacteristic
274C23×C4—41no—
275C23×C4—41no—
276C23×C4—42no—
277C23×C4—42no—
278C23×C4C2×C4×C514yes—
279C23×C4C23×C512yes—
280C23×C4C23×C512yes—
281C23×C4C23×C512yes—
282C23×C4C23×C512yes—
283C23×C4C23×C511yescharacteristic
284C23×C4C23×C512yes—
285C23×C4C23×C511yescharacteristic
286C23×C4C23×C512yes—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4C2×C4×C518yes—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4C2×C4×C514yes—
305C2×C22⋊C4C2×C4×C514yes—
306#95—42no—
307#95—42no—
308#95—41no—
309#95—41no—
310#95—42no—
311#95—42no—
312#95—42no—
313#95—42no—
314#95—42no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95C2×C4×C518yes—
319C22×C5—24no—
320C22×C5—28no—
321C22×C5—42no—
322C22×C5—42no—
323C22×C5—44no—
324C22×C5—44no—
325C22×C5—42no—
326C22×C5—42no—
327C22×C5—42no—
328C22×C5—42no—
329C22×C5—44no—
330C22×C5C2×#11711yescharacteristic
331C22×C5#51811yescharacteristic
332C22×C5#51812yes—
333C22×C5#51911yescharacteristic
334C22×C5#52012yes—
335C22×C5#53812yes—
336C22×C5#53812yes—
337C22×C5#53911yescharacteristic
338C22×C5#53911yescharacteristic
339C22×C5#53911yescharacteristic
340C22×C5#53911yescharacteristic
341C22×C5#53912yes—
342C22×C5#53912yes—
343C22×C5#56011yescharacteristic
344C22×C5#56011yescharacteristic
345C22×C5#56012yes—
346C22×C5H(3,𝔽4)11yescharacteristic
347C22×C5#56212yes—
348C22×C5#56212yes—
349C22×C5#56212yes—
350C22×C5#56312yes—
351C22×C5#56312yes—
352C22×C5#56311yescharacteristic
353C4×C5—44no—
354C4×C5—42no—
355C4×C5—44no—
356C4×C5—42no—
357C4×C5—82no—
358C4×C5—82no—
359C4×C5—82no—
360C4×C5—82no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—21no—
367C24—21no—
368C24—22no—
369C24C22×C4×C512yes—
370C24C5×#4112yes—
371C24C5×#4112yes—
372C24C24×C511yescharacteristic
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
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—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
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×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4C22×C4×C512yes—
427C22×C4C22×C4×C514yes—
428C22×C4C22×C4×C514yes—
429C22×C4C22×C4×C512yes—
430C4⋊C4—48no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C22⋊C4—28no—
434C22⋊C4—44no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C2×C5—44no—
438C2×C5#231311yescharacteristic
439C2×C5#231812yes—
440C2×C5#237511yescharacteristic
441C2×C5#237611yescharacteristic
442C2×C5#237812yes—
443C2×C5#282411yescharacteristic
444C2×C5#282611yescharacteristic
445C2×C5#282712yes—
446C2×C5#282712yes—
447C2×C5#283411yescharacteristic
448C2×C5#283511yescharacteristic
449C23—22no—
450C23—24no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—41no—
462C23—42no—
463C23—41no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23C5×#11714yes—
472C23C23×C4×C511yescharacteristic
473C23C2×C5×#4111yescharacteristic
474C23C2×C5×#4111yescharacteristic
475C23C5×H(5,𝔽2)12yes—
476C23C5×H(5,𝔽2)12yes—
477C23C5×H(5,𝔽2)11yescharacteristic
478C23C5×H(5,𝔽2)11yescharacteristic
479C23C5×H(5,𝔽2)12yes—
480C23C5×2−1+411yescharacteristic
481C23C5×2−1+411yescharacteristic
482C23C5×2−1+412yes—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—416no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C5#1466911yescharacteristic, direct product factor, Sylow
537C22—24no—
538C22—28no—
539C22—42no—
540C22—42no—
541C22—44no—
542C22—44no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—44no—
548C22C2×C5×#11711yescharacteristic
549C22C5×#51811yescharacteristic
550C22C5×#51812yes—
551C22C5×#51911yescharacteristic
552C22C5×#52012yes—
553C22C5×#53812yes—
554C22C5×#53812yes—
555C22C5×#53911yescharacteristic
556C22C5×#53911yescharacteristic
557C22C5×#53911yescharacteristic
558C22C5×#53911yescharacteristic
559C22C5×#53912yes—
560C22C5×#53912yes—
561C22C5×#56011yescharacteristic
562C22C5×#56011yescharacteristic
563C22C5×#56012yes—
564C22C5×H(3,𝔽4)11yescharacteristic
565C22C5×#56212yes—
566C22C5×#56212yes—
567C22C5×#56212yes—
568C22C5×#56312yes—
569C22C5×#56312yes—
570C22C5×#56311yescharacteristic
571C4—44no—
572C4—44no—
573C4—42no—
574C4—42no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C2—44no—
580C2C5×#231311yescharacteristic
581C2C5×#231812yes—
582C2C5×#237511yescharacteristic
583C2C5×#237611yescharacteristic
584C2C5×#237812yes—
585C2C5×#282411yescharacteristic
586C2C5×#282611yescharacteristic
587C2C5×#282712yes—
588C2C5×#282712yes—
589C2C5×#283411yescharacteristic
590C2C5×#283511yescharacteristic
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
2. C5×#1445C2semidirect
3. C5×#1443C2semidirect
4. C5×#1442C2semidirect
5. C5×#1441C2semidirect
18. C5×#385C4semidirect
32. #14669C5direct (N × H)
80. #1445C2×C5semidirect
81. #1443C2×C5semidirect
82. #1442C2×C5semidirect
83. #1441C2×C5semidirect
172. #385C4×C5semidirect
536. C5#14669direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#3852
C5×#14452
C5×#14432
C5×#14422
C5×#14412
C5×#14402
C5×#14392
C5×C22⋊#952
#146695