Finite Group Databasebeta

← C5×#24352 #11947072

Subgroups — C5×#24352 #11947072

Order 1280.

1198 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×#3501C211yesmaximal, characteristic, complemented
2C5×#2626C212yesmaximal, complemented
3C5×#2321C211yesmaximal, characteristic
4C5×#2062C212yesmaximal, complemented
5C5×#2029C212yesmaximal, complemented
6C5×#1996C212yesmaximal, complemented
7C5×#1889C212yesmaximal
8C5×#1867C212yesmaximal, complemented
9C5×#1799C211yesmaximal, characteristic, complemented
10C5×#554—24no—
11C5×#554—22no—
12C5×D4×Q8—22no—
13C5×#544—21no—
14C5×#543—21no—
15C5×#543C2212yes—
16C5×#533—21no—
17C2×C5×C22⋊Q8—21no—
18C2×C5×C22⋊Q8C2212yes—
19C2×C5×#120C2211yescharacteristic
20C5×#518C2212yes—
21C2×C4×C5×D4C2211yescharacteristic
22C22×C5×C22⋊C4C2211yescharacteristic
23C2×C5×#99C2212yes—
24C5×#399—22no—
25C5×#396—24no—
26C5×#393—22no—
27C5×#390—21no—
28C5×#388—21no—
29C5×#388—21no—
30C5×#386—21no—
31C5×#386C2211yescharacteristic
32C5×#385C2211yescharacteristic
33C5×#385C2214yes—
34C5×#380C2212yes—
35C5×#379C2211yescharacteristic
36C5×#379C2214yes—
37C4×C5×C22⋊C4C2211yescharacteristic
38C5×#342C2212yes—
39C5×#342C2214yes—
40C5×#342C2214yes—
41#24352C511yesmaximal, characteristic, direct product factor, Sylow
42C25×C5C2311yescharacteristic
43C22×C5×Q8—21no—
44C22×C5×Q8D412yescomplemented
45C22×C5×D4—21no—
46C22×C5×D4C2311yescharacteristic
47C23×C4×C5D411yescharacteristic
48C23×C4×C5C2311yescharacteristic
49C23×C4×C5C2311yescharacteristic
50C5×C4⋊Q8—21no—
51C5×C4⋊Q8—21no—
52C5×C4⋊Q8—24no—
53C5×#125—22no—
54C5×#125—24no—
55C5×C22⋊Q8—22no—
56C5×C22⋊Q8—22no—
57C5×C22⋊Q8—24no—
58C5×C22⋊Q8—24no—
59C5×C22⋊Q8—22no—
60C5×C22⋊Q8—22no—
61C5×C22⋊Q8—22no—
62C5×C22⋊Q8—22no—
63C5×C22⋊Q8—22no—
64C5×C22⋊Q8—22no—
65C5×C22⋊Q8—24no—
66C5×C22⋊Q8—24no—
67C5×#120—22no—
68C5×#120Q814yescomplemented
69C4×C5×Q8—22no—
70C4×C5×Q8—22no—
71C4×C5×D4—22no—
72C4×C5×D4—22no—
73C4×C5×D4—24no—
74C4×C5×D4—24no—
75C5×#117—22no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4—22no—
78C2×C5×C4⋊C4—22no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4—24no—
81C2×C5×C4⋊C4D411yescharacteristic
82C2×C5×C4⋊C4D414yescomplemented
83C2×C5×C4⋊C4D412yescomplemented
84C2×C5×C22⋊C4—22no—
85C2×C5×C22⋊C4—21no—
86C2×C5×C22⋊C4—21no—
87C2×C5×C22⋊C4—21no—
88C2×C5×C22⋊C4—22no—
89C2×C5×C22⋊C4—22no—
90C2×C5×C22⋊C4—21no—
91C2×C5×C22⋊C4—21no—
92C2×C5×C22⋊C4—21no—
93C2×C5×C22⋊C4—24no—
94C2×C5×C22⋊C4—24no—
95C2×C5×C22⋊C4D411yescharacteristic
96C2×C5×C22⋊C4C2311yescharacteristic
97C2×C5×C22⋊C4C2314yes—
98C2×C5×C22⋊C4C2311yescharacteristic
99C2×C5×C22⋊C4C2314yes—
100C2×C5×C22⋊C4C2311yescharacteristic
101C2×C42×C5—21no—
102C2×C42×C5D411yescharacteristic
103C5×#99—24no—
104C5×#95—21no—
105C5×#95—22no—
106C5×#95—21no—
107C5×#95—24no—
108C5×#95—21no—
109#3501C2×C511yescharacteristic, complemented
110#2626C2×C512yescomplemented
111#2321C2×C511yescharacteristic
112#2062C2×C512yescomplemented
113#2029C2×C512yescomplemented
114#1996C2×C512yescomplemented
115#1889C2×C512yes—
116#1867C2×C512yescomplemented
117#1799C2×C511yescharacteristic, complemented
118C24×C5—22no—
119C24×C5—21no—
120C24×C5—21no—
121C24×C5—22no—
122C24×C5—81no—
123C24×C5—81no—
124C24×C5C2×D411yescharacteristic
125C24×C5C2×Q812yescomplemented
126C24×C5C2411yescharacteristic
127C2×C5×Q8—22no—
128C2×C5×Q8—22no—
129C2×C5×Q8—22no—
130C2×C5×Q8—22no—
131C2×C5×Q8—82no—
132C2×C5×Q8—82no—
133C2×C5×D4—22no—
134C2×C5×D4—22no—
135C2×C5×D4—24no—
136C2×C5×D4—81no—
137C2×C5×D4—81no—
138C2×C5×D4C2×Q812yes—
139C2×C5×D4#4112yes—
140C22×C4×C5—21no—
141C22×C4×C5—22no—
142C22×C4×C5—22no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—24no—
148C22×C4×C5—22no—
149C22×C4×C5—81no—
150C22×C4×C5—81no—
151C22×C4×C5—81no—
152C22×C4×C5C2×D411yescharacteristic
153C22×C4×C5C2×D411yescharacteristic
154C22×C4×C5C2×D414yes—
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2×D414yes—
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×D412yes—
160C22×C4×C5C2×D412yes—
161C5×C4⋊C4—22no—
162C5×C4⋊C4—24no—
163C5×C4⋊C4—22no—
164C5×C4⋊C4—22no—
165C5×C4⋊C4—24no—
166C5×C4⋊C4—22no—
167C5×C4⋊C4—24no—
168C5×C4⋊C4—22no—
169C5×C4⋊C4—22no—
170C5×C4⋊C4—22no—
171C5×C4⋊C4—24no—
172C5×C4⋊C4—82no—
173C5×C4⋊C4—81no—
174C5×C4⋊C4—84no—
175C5×C22⋊C4—24no—
176C5×C22⋊C4—24no—
177C5×C22⋊C4—22no—
178C5×C22⋊C4—24no—
179C5×C22⋊C4—21no—
180C5×C22⋊C4—22no—
181C5×C22⋊C4—24no—
182C5×C22⋊C4—24no—
183C5×C22⋊C4—22no—
184C5×C22⋊C4—24no—
185C5×C22⋊C4—41no—
186C5×C22⋊C4—41no—
187C5×C22⋊C4—42no—
188C5×C22⋊C4—81no—
189C5×C22⋊C4—84no—
190C5×C22⋊C4—81no—
191C5×C22⋊C4—84no—
192C5×C22⋊C4—81no—
193C5×C22⋊C4C2×Q812yescomplemented
194C42×C5—24no—
195C42×C5—21no—
196C42×C5—21no—
197C42×C5—81no—
198#554—24no—
199#554—22no—
200D4×Q8—22no—
201#544—21no—
202#543—21no—
203#543C22×C512yes—
204#533—21no—
205C2×C22⋊Q8—21no—
206C2×C22⋊Q8C22×C512yes—
207C2×#120C22×C511yescharacteristic
208#518C22×C512yes—
209C2×C4×D4C22×C511yescharacteristic
210C22×C22⋊C4C22×C511yescharacteristic
211C2×#99C22×C512yes—
212#399—22no—
213#396—24no—
214#393—22no—
215#390—21no—
216#388—21no—
217#388—21no—
218#386—21no—
219#386C22×C511yescharacteristic
220#385C22×C511yescharacteristic
221#385C22×C514yes—
222#380C22×C512yes—
223#379C22×C511yescharacteristic
224#379C22×C514yes—
225C4×C22⋊C4C22×C511yescharacteristic
226#342C22×C512yes—
227#342C22×C514yes—
228#342C22×C514yes—
229C23×C5—21no—
230C23×C5—21no—
231C23×C5—22no—
232C23×C5—24no—
233C23×C5—41no—
234C23×C5—41no—
235C23×C5—41no—
236C23×C5—41no—
237C23×C5—44no—
238C23×C5—41no—
239C23×C5—44no—
240C23×C5—41no—
241C23×C5—41no—
242C23×C5—41no—
243C23×C5—41no—
244C23×C5—41no—
245C23×C5—42no—
246C23×C5—82no—
247C23×C5—82no—
248C23×C5—84no—
249C23×C5C22⋊Q812yes—
250C23×C5C22×D411yescharacteristic
251C23×C5C22×D412yes—
252C23×C5C22×Q811yescharacteristic
253C23×C5C2×#4111yescharacteristic
254C23×C52−1+412yes—
255C5×Q8—44no—
256C5×Q8—82no—
257C5×Q8—82no—
258C5×Q8—82no—
259C5×D4—24no—
260C5×D4—24no—
261C5×D4—84no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—24no—
266C2×C4×C5—24no—
267C2×C4×C5—21no—
268C2×C4×C5—22no—
269C2×C4×C5—41no—
270C2×C4×C5—42no—
271C2×C4×C5—41no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5—41no—
275C2×C4×C5—41no—
276C2×C4×C5—44no—
277C2×C4×C5—41no—
278C2×C4×C5—41no—
279C2×C4×C5—44no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—44no—
283C2×C4×C5—44no—
284C2×C4×C5—41no—
285C2×C4×C5—44no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—44no—
290C2×C4×C5—81no—
291C2×C4×C5—82no—
292C2×C4×C5—82no—
293C2×C4×C5#12011yescharacteristic
294C2×C4×C5#12011yescharacteristic
295C2×C4×C5#12012yes—
296C2×C4×C5C22⋊Q812yes—
297C25C23×C511yescharacteristic
298C22×Q8—21no—
299C22×Q8C5×D412yescomplemented
300C22×D4—21no—
301C22×D4C23×C511yescharacteristic
302C23×C4C5×D411yescharacteristic
303C23×C4C23×C511yescharacteristic
304C23×C4C23×C511yescharacteristic
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8—24no—
308#125—22no—
309#125—24no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—24no—
313C22⋊Q8—24no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8—22no—
320C22⋊Q8—24no—
321C22⋊Q8—24no—
322#120—22no—
323#120C5×Q814yescomplemented
324C4×Q8—22no—
325C4×Q8—22no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—24no—
329C4×D4—24no—
330#117—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4C5×D411yescharacteristic
337C2×C4⋊C4C5×D414yescomplemented
338C2×C4⋊C4C5×D412yescomplemented
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4C5×D411yescharacteristic
351C2×C22⋊C4C23×C511yescharacteristic
352C2×C22⋊C4C23×C514yes—
353C2×C22⋊C4C23×C511yescharacteristic
354C2×C22⋊C4C23×C514yes—
355C2×C22⋊C4C23×C511yescharacteristic
356C2×C42—21no—
357C2×C42C5×D411yescharacteristic
358#99—24no—
359#95—21no—
360#95—22no—
361#95—21no—
362#95—24no—
363#95—21no—
364C22×C5—22no—
365C22×C5—21no—
366C22×C5—22no—
367C22×C5—21no—
368C22×C5—22no—
369C22×C5—41no—
370C22×C5—41no—
371C22×C5—42no—
372C22×C5—42no—
373C22×C5—42no—
374C22×C5—42no—
375C22×C5—48no—
376C22×C5—41no—
377C22×C5—41no—
378C22×C5—42no—
379C22×C5—81no—
380C22×C5—84no—
381C22×C5—81no—
382C22×C5—81no—
383C22×C5—81no—
384C22×C5—81no—
385C22×C5C2×#12011yescharacteristic
386C22×C5C2×C22⋊Q811yescharacteristic
387C22×C5#53611yescharacteristic
388C22×C5#54812yes—
389C22×C5D4×Q812yes—
390C4×C5—22no—
391C4×C5—21no—
392C4×C5—21no—
393C4×C5—22no—
394C4×C5—42no—
395C4×C5—41no—
396C4×C5—44no—
397C4×C5—44no—
398C4×C5—41no—
399C4×C5—84no—
400C4×C5—81no—
401C4×C5—81no—
402C24—22no—
403C24—21no—
404C24—21no—
405C24—22no—
406C24—81no—
407C24—81no—
408C24C2×C5×D411yescharacteristic
409C24C2×C5×Q812yescomplemented
410C24C24×C511yescharacteristic
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4C2×C5×Q812yes—
423C2×D4C5×#4112yes—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—81no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4C2×C5×D411yescharacteristic
437C22×C4C2×C5×D411yescharacteristic
438C22×C4C2×C5×D414yes—
439C22×C4C2×C5×D411yescharacteristic
440C22×C4C2×C5×D414yes—
441C22×C4C2×C5×D411yescharacteristic
442C22×C4C2×C5×D412yes—
443C22×C4C2×C5×D411yescharacteristic
444C22×C4C2×C5×D412yes—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—82no—
457C4⋊C4—81no—
458C4⋊C4—84no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—22no—
462C22⋊C4—24no—
463C22⋊C4—21no—
464C22⋊C4—22no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—22no—
468C22⋊C4—24no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—42no—
472C22⋊C4—81no—
473C22⋊C4—84no—
474C22⋊C4—81no—
475C22⋊C4—84no—
476C22⋊C4—81no—
477C22⋊C4C2×C5×Q812yescomplemented
478C42—24no—
479C42—21no—
480C42—21no—
481C42—81no—
482C2×C5—21no—
483C2×C5—22no—
484C2×C5—21no—
485C2×C5—22no—
486C2×C5—42no—
487C2×C5—82no—
488C2×C5#240911yescharacteristic
489C2×C5#302611yescharacteristic
490C2×C5#302711yescharacteristic
491C23—21no—
492C23—21no—
493C23—22no—
494C23—24no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—44no—
500C23—41no—
501C23—44no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—82no—
509C23—82no—
510C23—84no—
511C23C5×C22⋊Q812yes—
512C23C22×C5×D411yescharacteristic
513C23C22×C5×D412yes—
514C23C22×C5×Q811yescharacteristic
515C23C2×C5×#4111yescharacteristic
516C23C5×2−1+412yes—
517Q8—44no—
518Q8—82no—
519Q8—82no—
520Q8—82no—
521D4—24no—
522D4—24no—
523D4—84no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4C5×#12011yescharacteristic
556C2×C4C5×#12011yescharacteristic
557C2×C4C5×#12012yes—
558C2×C4C5×C22⋊Q812yes—
559C5#2435211yescharacteristic, direct product factor, Sylow
560C22—22no—
561C22—21no—
562C22—22no—
563C22—21no—
564C22—22no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—48no—
572C22—41no—
573C22—41no—
574C22—42no—
575C22—81no—
576C22—84no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×C5×#12011yescharacteristic
582C22C2×C5×C22⋊Q811yescharacteristic
583C22C5×#53611yescharacteristic
584C22C5×#54812yes—
585C22C5×D4×Q812yes—
586C4—21no—
587C4—21no—
588C4—22no—
589C4—22no—
590C4—42no—
591C4—41no—
592C4—44no—
593C4—44no—
594C4—41no—
595C4—84no—
596C4—81no—
597C4—81no—
598C2—21no—
599C2—22no—
600C2—21no—
601C2—22no—
602C2—42no—
603C2—82no—
604C2C5×#240911yescharacteristic
605C2C5×#302611yescharacteristic
606C2C5×#302711yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC24C24×C5
prime radicalC2×C5×#120C22
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
1. C5×#3501C2semidirect
2. C5×#2626C2semidirect
4. C5×#2062C2semidirect
5. C5×#2029C2semidirect
6. C5×#1996C2semidirect
8. C5×#1867C2semidirect
9. C5×#1799C2semidirect
41. #24352C5direct (N × H)
44. C22×C5×Q8D4semidirect
68. C5×#120Q8semidirect
82. C2×C5×C4⋊C4D4semidirect
83. C2×C5×C4⋊C4D4semidirect
109. #3501C2×C5semidirect
110. #2626C2×C5semidirect
112. #2062C2×C5semidirect
113. #2029C2×C5semidirect
114. #1996C2×C5semidirect
116. #1867C2×C5semidirect
117. #1799C2×C5semidirect
125. C24×C5C2×Q8semidirect
193. C5×C22⋊C4C2×Q8semidirect
299. C22×Q8C5×D4semidirect
323. #120C5×Q8semidirect
337. C2×C4⋊C4C5×D4semidirect
338. C2×C4⋊C4C5×D4semidirect
409. C24C2×C5×Q8semidirect
477. C22⋊C4C2×C5×Q8semidirect
559. C5#24352direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35012
C5×#26262
C5×#23212
C5×#20622
C5×#20292
C5×#19962
C5×#18892
C5×#18672
C5×#17992
#243525