Finite Group Databasebeta

← C5×#14327 #11937047

Subgroups — C5×#14327 #11937047

Order 1280.

1346 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#395C211yesmaximal, characteristic
2C2×C5×#391C211yesmaximal, characteristic
3C2×C5×#387C211yesmaximal, characteristic
4C5×#1446C211yesmaximal, characteristic
5C5×#1444C211yesmaximal, characteristic
6C5×#1442C212yesmaximal
7C5×#1441C212yesmaximal
8C5×#1441C211yesmaximal, characteristic
9C5×C4⋊#95C212yesmaximal
10C5×#1434C212yesmaximal
11C4×C5×#95C211yesmaximal, characteristic
12C23×C5×Q8C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C5×C4⋊C4C2211yescharacteristic
15C22×C5×C4⋊C4C2211yescharacteristic
16C22×C5×C4⋊C4C2212yes—
17C22×C5×C4⋊C4C2212yes—
18C22×C42×C5C2211yescharacteristic
19C22×C42×C5C2212yes—
20C5×#395C418yescomplemented
21C5×#391—24no—
22C5×#387—24no—
23C2×C5×#95C2211yescharacteristic
24C2×C5×#95C2211yescharacteristic
25C2×C5×#95C2211yescharacteristic
26C2×C5×#95C2211yescharacteristic
27C2×C5×#95C2212yes—
28C2×C5×#95C2212yes—
29C2×C5×#95C2212yes—
30C2×C5×#95C2211yescharacteristic
31C2×C5×#95C2212yes—
32C2×C5×#95C2212yes—
33C2×C5×#95C2212yes—
34C2×C5×#95C2212yes—
35C2×C5×#95C2212yes—
36C2×C5×#95C2211yescharacteristic
37C2×C5×#95C2212yes—
38#14327C511yesmaximal, characteristic, direct product factor, Sylow
39C22×C5×Q8—22no—
40C22×C5×Q8—22no—
41C22×C5×Q8—42no—
42C22×C5×Q8—42no—
43C22×C5×Q8C2×C414yes—
44C23×C4×C5C2311yescharacteristic
45C23×C4×C5C2312yes—
46C23×C4×C5C2311yescharacteristic
47C23×C4×C5C2311yescharacteristic
48C23×C4×C5C2312yes—
49C23×C4×C5C2311yescharacteristic
50C23×C4×C5C2311yescharacteristic
51C23×C4×C5C2312yes—
52C23×C4×C5C2311yescharacteristic
53C23×C4×C5C2312yes—
54C23×C4×C5C2311yescharacteristic
55C2×C5×C4⋊C4—22no—
56C2×C5×C4⋊C4—22no—
57C2×C5×C4⋊C4—22no—
58C2×C5×C4⋊C4—22no—
59C2×C5×C4⋊C4—24no—
60C2×C5×C4⋊C4—42no—
61C2×C5×C4⋊C4—42no—
62C2×C5×C4⋊C4—42no—
63C2×C5×C4⋊C4—42no—
64C2×C5×C4⋊C4—42no—
65C2×C5×C4⋊C4—42no—
66C2×C5×C4⋊C4C2×C414yes—
67C2×C5×C4⋊C4C2×C418yes—
68C2×C42×C5—42no—
69C2×C42×C5—42no—
70C2×C42×C5—42no—
71C2×C42×C5D414yes—
72C2×C42×C5Q818yescomplemented
73C5×#95—22no—
74C5×#95—22no—
75C5×#95—22no—
76C5×#95—22no—
77C5×#95—24no—
78C5×#95—24no—
79C5×#95—22no—
80C5×#95—24no—
81C5×#95—24no—
82C5×#95—42no—
83C5×#95—42no—
84C5×#95—42no—
85C5×#95—42no—
86C5×#95C2×C418yes—
87C5×#95C2×C414yes—
88C2×#395C2×C511yescharacteristic
89C2×#391C2×C511yescharacteristic
90C2×#387C2×C511yescharacteristic
91#1446C2×C511yescharacteristic
92#1444C2×C511yescharacteristic
93#1442C2×C512yes—
94#1441C2×C511yescharacteristic
95#1441C2×C512yes—
96C4⋊#95C2×C512yes—
97#1434C2×C512yes—
98C4×#95C2×C511yescharacteristic
99C24×C5C2411yescharacteristic
100C2×C5×Q8—28no—
101C2×C5×Q8—48no—
102C2×C5×Q8—44no—
103C2×C5×Q8—44no—
104C2×C5×Q8—48no—
105C22×C4×C5—22no—
106C22×C4×C5—21no—
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—21no—
114C22×C4×C5—21no—
115C22×C4×C5—21no—
116C22×C4×C5—22no—
117C22×C4×C5—21no—
118C22×C4×C5—21no—
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—21no—
133C22×C4×C5—22no—
134C22×C4×C5—21no—
135C22×C4×C5—21no—
136C22×C4×C5—21no—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—22no—
140C22×C4×C5—21no—
141C22×C4×C5—22no—
142C22×C4×C5—21no—
143C22×C4×C5—22no—
144C22×C4×C5—22no—
145C22×C4×C5—22no—
146C22×C4×C5—22no—
147C22×C4×C5—22no—
148C22×C4×C5—21no—
149C22×C4×C5—22no—
150C22×C4×C5—22no—
151C22×C4×C5—22no—
152C22×C4×C5—22no—
153C22×C4×C5—21no—
154C22×C4×C5—21no—
155C22×C4×C5—22no—
156C22×C4×C5—21no—
157C22×C4×C5C22×C412yes—
158C22×C4×C5C22×C414yes—
159C22×C4×C5C22×C412yes—
160C22×C4×C5C22×C412yes—
161C22×C4×C5C22×C414yes—
162C22×C4×C5C2×D412yes—
163C22×C4×C5C2×D412yes—
164C22×C4×C5C2×D412yes—
165C22×C4×C5C2×Q814yes—
166C22×C4×C5C2×Q814yes—
167C22×C4×C5C2×Q814yes—
168C22×C4×C5#4112yes—
169C22×C4×C5#4114yes—
170C5×C4⋊C4—28no—
171C5×C4⋊C4—48no—
172C5×C4⋊C4—44no—
173C5×C4⋊C4—48no—
174C5×C4⋊C4—44no—
175C42×C5—44no—
176C42×C5—48no—
177C23×Q8C22×C511yescharacteristic
178C22×C4⋊C4C22×C511yescharacteristic
179C22×C4⋊C4C22×C512yes—
180C22×C4⋊C4C22×C511yescharacteristic
181C22×C4⋊C4C22×C511yescharacteristic
182C22×C4⋊C4C22×C512yes—
183C22×C42C22×C511yescharacteristic
184C22×C42C22×C512yes—
185#395C4×C518yescomplemented
186#391—24no—
187#387—24no—
188C2×#95C22×C512yes—
189C2×#95C22×C511yescharacteristic
190C2×#95C22×C511yescharacteristic
191C2×#95C22×C512yes—
192C2×#95C22×C512yes—
193C2×#95C22×C511yescharacteristic
194C2×#95C22×C512yes—
195C2×#95C22×C511yescharacteristic
196C2×#95C22×C511yescharacteristic
197C2×#95C22×C512yes—
198C2×#95C22×C512yes—
199C2×#95C22×C512yes—
200C2×#95C22×C512yes—
201C2×#95C22×C511yescharacteristic
202C2×#95C22×C512yes—
203C23×C5C23×C411yescharacteristic
204C23×C5C22×D411yescharacteristic
205C23×C5C22×Q812yes—
206C23×C5C2×#4111yescharacteristic
207C23×C5C2×#4112yes—
208C23×C5H(5,𝔽2)12yes—
209C23×C5H(5,𝔽2)12yes—
210C23×C5H(5,𝔽2)11yescharacteristic
211C23×C52−1+411yescharacteristic
212C23×C52−1+411yescharacteristic
213C23×C52−1+411yescharacteristic
214C5×Q8—416no—
215C2×C4×C5—24no—
216C2×C4×C5—24no—
217C2×C4×C5—24no—
218C2×C4×C5—22no—
219C2×C4×C5—22no—
220C2×C4×C5—22no—
221C2×C4×C5—22no—
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—22no—
236C2×C4×C5—22no—
237C2×C4×C5—22no—
238C2×C4×C5—24no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—24no—
242C2×C4×C5—24no—
243C2×C4×C5—41no—
244C2×C4×C5—42no—
245C2×C4×C5—41no—
246C2×C4×C5—42no—
247C2×C4×C5—41no—
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—41no—
256C2×C4×C5—41no—
257C2×C4×C5—42no—
258C2×C4×C5—42no—
259C2×C4×C5—42no—
260C2×C4×C5—42no—
261C2×C4×C5—41no—
262C2×C4×C5—41no—
263C2×C4×C5—41no—
264C2×C4×C5—42no—
265C2×C4×C5—41no—
266C2×C4×C5—41no—
267C2×C4×C5—42no—
268C2×C4×C5—41no—
269C2×C4×C5—42no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5—42no—
277C2×C4×C5C4×D414yes—
278C2×C4×C5C4×Q818yes—
279C2×C4×C5C4⋊Q814yes—
280C22×Q8—22no—
281C22×Q8—22no—
282C22×Q8—42no—
283C22×Q8—42no—
284C22×Q8C2×C4×C514yes—
285C23×C4C23×C511yescharacteristic
286C23×C4C23×C512yes—
287C23×C4C23×C511yescharacteristic
288C23×C4C23×C511yescharacteristic
289C23×C4C23×C512yes—
290C23×C4C23×C511yescharacteristic
291C23×C4C23×C511yescharacteristic
292C23×C4C23×C512yes—
293C23×C4C23×C511yescharacteristic
294C23×C4C23×C512yes—
295C23×C4C23×C511yescharacteristic
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4C2×C4×C514yes—
308C2×C4⋊C4C2×C4×C518yes—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42C5×D414yes—
313C2×C42C5×Q818yescomplemented
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—42no—
324#95—42no—
325#95—42no—
326#95—42no—
327#95C2×C4×C518yes—
328#95C2×C4×C514yes—
329C22×C5C2×C4×D411yescharacteristic
330C22×C5C2×C4×Q812yes—
331C22×C5#51812yes—
332C22×C5#51911yescharacteristic
333C22×C5#52011yescharacteristic
334C22×C5C2×C4⋊Q811yescharacteristic
335C22×C5#53211yescharacteristic
336C22×C5#53312yes—
337C22×C5#53411yescharacteristic
338C22×C5#53511yescharacteristic
339C22×C5#53611yescharacteristic
340C22×C5#53711yescharacteristic
341C22×C5#53812yes—
342C22×C5#53812yes—
343C22×C5#53912yes—
344C22×C5#54111yescharacteristic
345C22×C5#54211yescharacteristic
346C22×C5#54212yes—
347C22×C5#54212yes—
348C22×C5#54312yes—
349C22×C5#54412yes—
350C22×C5#54412yes—
351C22×C5#54412yes—
352C4×C5—24no—
353C4×C5—28no—
354C4×C5—24no—
355C4×C5—42no—
356C4×C5—44no—
357C4×C5—42no—
358C4×C5—42no—
359C4×C5—44no—
360C4×C5—82no—
361C4×C5—81no—
362C4×C5—81no—
363C24C24×C511yescharacteristic
364C2×Q8—28no—
365C2×Q8—48no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—48no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
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—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
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—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4C22×C4×C512yes—
422C22×C4C22×C4×C514yes—
423C22×C4C22×C4×C512yes—
424C22×C4C22×C4×C512yes—
425C22×C4C22×C4×C514yes—
426C22×C4C2×C5×D412yes—
427C22×C4C2×C5×D412yes—
428C22×C4C2×C5×D412yes—
429C22×C4C2×C5×Q814yes—
430C22×C4C2×C5×Q814yes—
431C22×C4C2×C5×Q814yes—
432C22×C4C5×#4112yes—
433C22×C4C5×#4114yes—
434C4⋊C4—28no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C42—44no—
440C42—48no—
441C2×C5C4×C4⋊Q811yescharacteristic
442C2×C5#232111yescharacteristic
443C2×C5#232311yescharacteristic
444C2×C5#232912yes—
445C2×C5#233112yes—
446C2×C5#265911yescharacteristic
447C2×C5#266011yescharacteristic
448C2×C5#266211yescharacteristic
449C2×C5#266311yescharacteristic
450C2×C5#266512yes—
451C2×C5#266612yes—
452C23C23×C4×C511yescharacteristic
453C23C22×C5×D411yescharacteristic
454C23C22×C5×Q812yes—
455C23C2×C5×#4111yescharacteristic
456C23C2×C5×#4112yes—
457C23C5×H(5,𝔽2)12yes—
458C23C5×H(5,𝔽2)12yes—
459C23C5×H(5,𝔽2)11yescharacteristic
460C23C5×2−1+411yescharacteristic
461C23C5×2−1+411yescharacteristic
462C23C5×2−1+411yescharacteristic
463Q8—416no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
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×C4C4×C5×D414yes—
527C2×C4C4×C5×Q818yes—
528C2×C4C5×C4⋊Q814yes—
529C5#1432711yescharacteristic, direct product factor, Sylow
530C22C2×C4×C5×D411yescharacteristic
531C22C2×C4×C5×Q812yes—
532C22C5×#51812yes—
533C22C5×#51911yescharacteristic
534C22C5×#52011yescharacteristic
535C22C2×C5×C4⋊Q811yescharacteristic
536C22C5×#53211yescharacteristic
537C22C5×#53312yes—
538C22C5×#53411yescharacteristic
539C22C5×#53511yescharacteristic
540C22C5×#53611yescharacteristic
541C22C5×#53711yescharacteristic
542C22C5×#53812yes—
543C22C5×#53812yes—
544C22C5×#53912yes—
545C22C5×#54111yescharacteristic
546C22C5×#54211yescharacteristic
547C22C5×#54212yes—
548C22C5×#54212yes—
549C22C5×#54312yes—
550C22C5×#54412yes—
551C22C5×#54412yes—
552C22C5×#54412yes—
553C4—24no—
554C4—28no—
555C4—24no—
556C4—44no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—44no—
561C4—82no—
562C4—81no—
563C4—81no—
564C2C4×C5×C4⋊Q811yescharacteristic
565C2C5×#232111yescharacteristic
566C2C5×#232311yescharacteristic
567C2C5×#232912yes—
568C2C5×#233112yes—
569C2C5×#265911yescharacteristic
570C2C5×#266011yescharacteristic
571C2C5×#266211yescharacteristic
572C2C5×#266311yescharacteristic
573C2C5×#266512yes—
574C2C5×#266612yes—
subgroupG/H
center, socle, prime radicalC24×C5C24
commutatorC23C23×C4×C5
Frattini, cosocleC24C24×C5
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C5 ◁ G

Complements

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

NH ≅ G/Ntype
20. C5×#395C4semidirect
38. #14327C5direct (N × H)
72. C2×C42×C5Q8semidirect
185. #395C4×C5semidirect
313. C2×C42C5×Q8semidirect
529. C5#14327direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#3952
C2×C5×#3912
C2×C5×#3872
C5×#14462
C5×#14442
C5×#14422
C5×#14412
C5×C4⋊#952
C5×#14342
C4×C5×#952
#143275