Finite Group Databasebeta

← #12262073

Subgroups — #12262073

Order 1280.

2714 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65858C211yesmaximal, characteristic
2C5×#2314C211yesmaximal, characteristic, complemented
3#10613962C211yesmaximal, characteristic, complemented
4#10607567C214yesmaximal, complemented
5#10607220C214yesmaximal, complemented
6#10607031C214yesmaximal, complemented
7#10607029C214yesmaximal, complemented
8#10606091C212yesmaximal, complemented
9#10605960C212yesmaximal, complemented
10#10605734C212yesmaximal, complemented
11D5×#380C212yesmaximal, complemented
12#10605502C212yesmaximal, complemented
13C4⋊C4×D20C212yesmaximal, complemented
14C22×#709C2211yescharacteristic
15C23×C4×D5C2211yescharacteristic
16C2×C5×#117C2211yescharacteristic
17#65858C4116yescomplemented
18C2×#4043C2212yes—
19C2×#4041C2212yes—
20C2×#4039C2214yes—
21C2×#4038C2212yescomplemented
22C2×D5×C4⋊C4C2212yes—
23C2×D5×C4⋊C4C2212yes—
24C2×D5×C4⋊C4C2212yescomplemented
25C2×#4032C2212yes—
26C2×#4028C2212yes—
27C2×D5×C22⋊C4C2214yes—
28C2×C4×D20C2214yes—
29C2×C4×D20C2212yes—
30C2×C4×D20C2212yes—
31C5×#388C2214yes—
32C5×#387C2214yescomplemented
33C5×#384C2211yescharacteristic, complemented
34C5×#382C2211yescharacteristic, complemented
35C5×#380C2212yes—
36C4×C5×C4⋊C4C2212yescomplemented
37#65501C2212yes—
38#65278C2214yes—
39#65278C2214yescomplemented
40#65276C2212yes—
41#65276C2212yes—
42#65276C2212yescomplemented
43#65276C2212yescomplemented
44#65270C2214yescomplemented
45#65262C2214yescomplemented
46#65246C2214yes—
47#65245C2214yes—
48#65240C2214yes—
49#65231C2214yes—
50#65229C2214yescomplemented
51C4×#685C2214yes—
52#65225C2211yescharacteristic, complemented
53#65224C2211yescharacteristic, complemented
54C4×C4⋊Dic5C2212yescomplemented
55#65217C2214yescomplemented
56#64958C2214yescomplemented
57#64957C2214yes—
58#64957C2214yescomplemented
59#64956C2214yes—
60#64955C2214yes—
61#64954C2214yes—
62#64954C2214yes—
63#64954C2214yescomplemented
64D5×#95C2214yes—
65#64943C2214yescomplemented
66#64942C2214yescomplemented
67#64935C2214yescomplemented
68#34360—51nomaximal, Sylow
69C24×D5C2311yescharacteristic
70C23×C4×C5C2311yescharacteristic
71C22×C22⋊D5C2312yes—
72C23×Dic5C2311yescharacteristic
73C2×#709—42no—
74C2×#709—42no—
75C2×#709C2×C418yescomplemented
76C22×D20C2312yes—
77C22×D20C2312yes—
78C22×D20C2311yescharacteristic
79C22×C4×D5—42no—
80C22×C4×D5—42no—
81C22×C4×D5C2×C418yescomplemented
82C22×C4×D5C2312yes—
83C22×C4×D5C2312yes—
84C22×C4×D5C2314yes—
85C22×C4×D5C2314yes—
86C22×C4×D5C2314yes—
87C22×C4×D5C2312yes—
88C22×Dic10C2311yescharacteristic
89C5×#117C2×C418yescomplemented
90C2×C5×C4⋊C4C2312yes—
91C2×C5×C4⋊C4C2312yes—
92C2×C5×C4⋊C4C2314yes—
93C2×C5×C4⋊C4C2312yes—
94C2×C5×C22⋊C4C2314yes—
95C2×C5×C22⋊C4C2312yes—
96C2×C42×C5C2314yes—
97C2×C42×C5C2311yescharacteristic
98C2×C42×C5C2311yescharacteristic
99C2×C22⋊Dic5C2314yes—
100C2×#685C2314yes—
101C2×#685C2314yes—
102C2×#685C2314yes—
103C2×#685C2314yes—
104C2×#685C2314yes—
105C2×#685C2312yes—
106C2×#685C2312yes—
107C2×#685C2312yes—
108C2×#685C2312yes—
109C2×C4⋊Dic5C2312yes—
110C2×C4⋊Dic5C2312yes—
111C2×C4⋊Dic5C2314yes—
112C2×C4⋊Dic5C2312yes—
113C2×C4⋊Dic5C2312yes—
114C2×#683C2314yes—
115C2×#683C2314yes—
116C2×#683C2314yes—
117C2×C4×Dic5C2314yes—
118C2×C4×Dic5C2314yes—
119#4043C2×C4116yescomplemented
120#4041C2×C4116yescomplemented
121#4039—48no—
122#4038—44no—
123D5×C4⋊C4—44no—
124D5×C4⋊C4—44no—
125D5×C4⋊C4—44no—
126#4032C2×C4116yescomplemented
127#4028C2×C4116yescomplemented
128D5×C22⋊C4—48no—
129C4×D20—48no—
130C4×D20C2×C4116yescomplemented
131C4×D20C2×C4116yescomplemented
132C5×#95C2314yes—
133C5×#95C2314yes—
134C5×#95C2314yes—
135#3963C2314yes—
136#3963C2314yes—
137#3963C2314yes—
138#3963C2314yes—
139#3963C2314yes—
140#3963C2314yes—
141#3963C2314yes—
142C2×#525—51no—
143#2376—54no—
144#2358—54no—
145#2352—52no—
146#2352—52no—
147D4×C4⋊C4—52no—
148D4×C4⋊C4—52no—
149#2329—54no—
150#2325—54no—
151#2314—51no—
152#2314D511yescharacteristic, complemented
153#2311—52no—
154C2×#380—52no—
155C24×C5C2411yescharacteristic
156C23×D5—24no—
157C23×D5—24no—
158C23×D5—41no—
159C23×D5—41no—
160C23×D5C22×C414yes—
161C23×D5C2412yes—
162C23×D5C2412yes—
163C22×C4×C5—22no—
164C22×C4×C5—22no—
165C22×C4×C5C22×C414yes—
166C22×C4×C5C2414yes—
167C22×C4×C5C2414yes—
168C22×C4×C5C2412yes—
169C22×C4×C5C2412yes—
170C22×C4×C5C2411yescharacteristic
171C22×C4×C5C2411yescharacteristic
172C2×C22⋊D5—42no—
173C2×C22⋊D5—42no—
174C2×C22⋊D5C22×C418yes—
175C22×Dic5—41no—
176C22×Dic5—41no—
177C22×Dic5C22×C414yes—
178C22×Dic5C2414yes—
179C22×Dic5C2414yes—
180C22×Dic5C2412yes—
181C22×Dic5C2412yes—
182#709—416no—
183C2×D20—24no—
184C2×D20—24no—
185C2×D20—24no—
186C2×D20—24no—
187C2×D20—41no—
188C2×D20—41no—
189C2×D20C22×C418yes—
190C2×D20C22×C418yes—
191C2×D20C22×C414yes—
192C2×C4×D5—24no—
193C2×C4×D5—24no—
194C2×C4×D5—24no—
195C2×C4×D5—24no—
196C2×C4×D5—28no—
197C2×C4×D5—28no—
198C2×C4×D5—416no—
199C2×C4×D5—42no—
200C2×C4×D5—42no—
201C2×C4×D5—44no—
202C2×C4×D5—44no—
203C2×C4×D5—44no—
204C2×C4×D5—44no—
205C2×C4×D5—44no—
206C2×C4×D5—44no—
207C2×C4×D5—44no—
208C2×C4×D5C22×C418yes—
209C2×C4×D5C22×C418yes—
210C2×C4×D5C22×C418yes—
211C2×Dic10—41no—
212C2×Dic10—41no—
213C2×Dic10C22×C414yes—
214C5×C4⋊C4—24no—
215C5×C4⋊C4—24no—
216C5×C4⋊C4—44no—
217C5×C4⋊C4C22×C418yes—
218C5×C22⋊C4—44no—
219C5×C22⋊C4C22×C418yes—
220C42×C5—44no—
221C42×C5C22×C414yes—
222C42×C5C22×C414yes—
223C22⋊Dic5—44no—
224#685—28no—
225#685—28no—
226#685—44no—
227#685—44no—
228#685—44no—
229#685C22×C418yes—
230#685C22×C418yes—
231#685C22×C418yes—
232#685C22×C418yes—
233C4⋊Dic5—24no—
234C4⋊Dic5—24no—
235C4⋊Dic5—44no—
236C4⋊Dic5C22×C418yes—
237C4⋊Dic5C22×C418yes—
238#683—44no—
239#683—44no—
240#683—44no—
241C4×Dic5—44no—
242C4×Dic5—44no—
243C22×#41—51no—
244C24×C4—51no—
245#525—516no—
246C2×#123—52no—
247C2×C22⋊Q8—52no—
248C2×#121—52no—
249C2×#120—52no—
250C2×C4×D4—54no—
251C2×C4×D4—52no—
252C2×C4×D4—52no—
253C2×C4×D4—54no—
254C2×#117—52no—
255C2×#117C2×D511yescharacteristic
256C22×C4⋊C4—52no—
257C22×C4⋊C4—52no—
258C22×C4⋊C4—52no—
259C22×C22⋊C4—54no—
260#391—54no—
261#390—54no—
262#389—54no—
263#389—54no—
264#388—54no—
265#388—54no—
266#388—54no—
267#388C2×D514yes—
268#387—54no—
269#387—54no—
270#387—54no—
271#387C2×D514yescomplemented
272#386—54no—
273#386—54no—
274#386—54no—
275#386—54no—
276#385—54no—
277#385—54no—
278#385—54no—
279#385—54no—
280#384—51no—
281#384C2×D511yescharacteristic, complemented
282#382—51no—
283#382C2×D511yescharacteristic, complemented
284#380—52no—
285#380—52no—
286#380—52no—
287#380—52no—
288#380—52no—
289#380—54no—
290#380—54no—
291#380C2×D512yes—
292C4×C4⋊C4—52no—
293C4×C4⋊C4C2×D512yescomplemented
294C4×C22⋊C4—54no—
295C4×C22⋊C4—54no—
296C2×#95—54no—
297C23×C5—22no—
298C23×C5—21no—
299C23×C5—21no—
300C23×C5—22no—
301C23×C5C23×C412yes—
302C23×C5C2511yescharacteristic
303C22×D5—22no—
304C22×D5—22no—
305C22×D5—22no—
306C22×D5—22no—
307C22×D5—22no—
308C22×D5—22no—
309C22×D5—22no—
310C22×D5—22no—
311C22×D5—22no—
312C22×D5—22no—
313C22×D5—216no—
314C22×D5—44no—
315C22×D5—48no—
316C22×D5—44no—
317C22×D5—44no—
318C22×D5C23×C414yes—
319C22×D5C23×C414yes—
320C22×D5H(5,𝔽2)14yes—
321C22×D52−1+414yes—
322C2×C4×C5—22no—
323C2×C4×C5—22no—
324C2×C4×C5—21no—
325C2×C4×C5—21no—
326C2×C4×C5—22no—
327C2×C4×C5—22no—
328C2×C4×C5—28no—
329C2×C4×C5—24no—
330C2×C4×C5—24no—
331C2×C4×C5—24no—
332C2×C4×C5—24no—
333C2×C4×C5—24no—
334C2×C4×C5—24no—
335C2×C4×C5C23×C414yes—
336C2×C4×C5C23×C414yes—
337C2×C4×C5C23×C412yes—
338C2×C4×C5C23×C412yes—
339C2×C4×C5H(5,𝔽2)12yes—
340C2×C4×C5H(5,𝔽2)12yes—
341C22⋊D5—48no—
342C2×Dic5—22no—
343C2×Dic5—22no—
344C2×Dic5—22no—
345C2×Dic5—22no—
346C2×Dic5—24no—
347C2×Dic5—24no—
348C2×Dic5—24no—
349C2×Dic5—24no—
350C2×Dic5—24no—
351C2×Dic5—24no—
352C2×Dic5—44no—
353C2×Dic5C23×C414yes—
354C2×Dic5C23×C414yes—
355D20—216no—
356D20—216no—
357D20—44no—
358C4×D5—216no—
359C4×D5—216no—
360C4×D5—48no—
361C4×D5—88no—
362C4×D5—88no—
363C4×D5—88no—
364Dic10—44no—
365C25—51no—
366C2×#41—58no—
367C2×#41—202no—
368C2×#41—202no—
369C22×Q8—51no—
370C22×D4—52no—
371C22×D4—52no—
372C22×D4—52no—
373C22×D4—51no—
374C23×C4—51no—
375C23×C4—52no—
376C23×C4—52no—
377C23×C4—58no—
378C23×C4—52no—
379C23×C4—54no—
380C23×C4—54no—
381C23×C4—54no—
382C23×C4—202no—
383C23×C4—202no—
384C23×C4C22×D511yescharacteristic
385#123—516no—
386C22⋊Q8—516no—
387#121—516no—
388#120—516no—
389C4×D4—516no—
390C4×D4—516no—
391C4×D4—208no—
392C4×D4—208no—
393#117—204no—
394#117C4×D518yescomplemented
395C2×C4⋊C4—52no—
396C2×C4⋊C4—52no—
397C2×C4⋊C4—52no—
398C2×C4⋊C4—52no—
399C2×C4⋊C4—54no—
400C2×C4⋊C4—54no—
401C2×C4⋊C4—54no—
402C2×C4⋊C4—54no—
403C2×C4⋊C4—204no—
404C2×C4⋊C4—204no—
405C2×C4⋊C4—204no—
406C2×C4⋊C4C22×D512yes—
407C2×C4⋊C4C22×D512yes—
408C2×C4⋊C4C22×D514yes—
409C2×C4⋊C4C22×D512yes—
410C2×C22⋊C4—52no—
411C2×C22⋊C4—52no—
412C2×C22⋊C4—52no—
413C2×C22⋊C4—52no—
414C2×C22⋊C4—54no—
415C2×C22⋊C4—54no—
416C2×C22⋊C4—54no—
417C2×C22⋊C4—54no—
418C2×C22⋊C4—54no—
419C2×C22⋊C4—54no—
420C2×C22⋊C4—208no—
421C2×C22⋊C4C22×D514yes—
422C2×C22⋊C4C22×D512yes—
423C2×C42—54no—
424C2×C42—54no—
425C2×C42C22×D514yes—
426C2×C42C22×D511yescharacteristic
427C2×C42C22×D511yescharacteristic
428#95—54no—
429#95—54no—
430#95—54no—
431#95—54no—
432#95—54no—
433#95—54no—
434#95—54no—
435#95C22×D514yes—
436#95C22×D514yes—
437#95C22×D514yes—
438C22×C5—22no—
439C22×C5—24no—
440C22×C5—42no—
441C22×C5—41no—
442C22×C5—41no—
443C22×C5C24×C411yescharacteristic
444C22×C5C2×H(5,𝔽2)11yescharacteristic
445C22×C5C2×H(5,𝔽2)11yescharacteristic
446C22×C5C2×H(5,𝔽2)12yes—
447C22×C5C2×2−1+412yes—
448C2×D5—24no—
449C2×D5—24no—
450C2×D5—24no—
451C2×D5—24no—
452C2×D5—24no—
453C2×D5—24no—
454C2×D5—24no—
455C2×D5—24no—
456C2×D5—24no—
457C2×D5—416no—
458C2×D5—42no—
459C2×D5—42no—
460C2×D5—42no—
461C2×D5—42no—
462C2×D5#52018yes—
463C4×C5—22no—
464C4×C5—24no—
465C4×C5—24no—
466C4×C5—44no—
467C4×C5—44no—
468C4×C5#51814yes—
469Dic5—24no—
470Dic5—24no—
471Dic5—44no—
472Dic5—44no—
473C24—52no—
474C24—52no—
475C24—54no—
476C24—104no—
477C24—104no—
478C24—201no—
479C24—201no—
480C24C23×D511yescharacteristic
481#41—2016no—
482C2×Q8—54no—
483C2×Q8—201no—
484C2×Q8—201no—
485C2×D4—58no—
486C2×D4—58no—
487C2×D4—58no—
488C2×D4—54no—
489C2×D4—104no—
490C2×D4—104no—
491C2×D4—104no—
492C2×D4—104no—
493C2×D4—201no—
494C2×D4—201no—
495C2×D4—202no—
496C2×D4—202no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—52no—
500C22×C4—52no—
501C22×C4—54no—
502C22×C4—58no—
503C22×C4—58no—
504C22×C4—58no—
505C22×C4—54no—
506C22×C4—54no—
507C22×C4—108no—
508C22×C4—108no—
509C22×C4—104no—
510C22×C4—104no—
511C22×C4—104no—
512C22×C4—104no—
513C22×C4—2016no—
514C22×C4—202no—
515C22×C4—202no—
516C22×C4—204no—
517C22×C4—204no—
518C22×C4—204no—
519C22×C4—204no—
520C22×C4—204no—
521C22×C4—204no—
522C22×C4—204no—
523C22×C4—201no—
524C22×C4—201no—
525C22×C4C2×C4×D514yes—
526C22×C4C23×D514yes—
527C22×C4C23×D514yes—
528C22×C4C23×D511yescharacteristic
529C22×C4C23×D512yes—
530C22×C4C23×D512yes—
531C22×C4C23×D511yescharacteristic
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—44no—
535C4⋊C4—58no—
536C4⋊C4—58no—
537C4⋊C4—104no—
538C4⋊C4—104no—
539C4⋊C4—204no—
540C4⋊C4—204no—
541C4⋊C4—204no—
542C4⋊C4—204no—
543C4⋊C4C2×C4×D518yes—
544C22⋊C4—44no—
545C22⋊C4—58no—
546C22⋊C4—58no—
547C22⋊C4—58no—
548C22⋊C4—58no—
549C22⋊C4—108no—
550C22⋊C4—108no—
551C22⋊C4—204no—
552C22⋊C4—204no—
553C22⋊C4—204no—
554C22⋊C4—204no—
555C22⋊C4C2×C4×D518yes—
556C42—44no—
557C42—204no—
558C42—204no—
559C42C2×C4×D514yes—
560C42C2×C4×D514yes—
561C2×C5—42no—
562C2×C5C2×#51811yescharacteristic
563C2×C5C2×#52012yes—
564C2×C5#353612yes—
565C2×C5#353711yescharacteristic
566C2×C5#353811yescharacteristic
567D5—28no—
568D5—44no—
569C23—21no—
570C23—21no—
571C23—22no—
572C23—22no—
573C23—54no—
574C23—54no—
575C23—54no—
576C23—54no—
577C23—1016no—
578C23—102no—
579C23—102no—
580C23—102no—
581C23—102no—
582C23—102no—
583C23—102no—
584C23—102no—
585C23—102no—
586C23—102no—
587C23—102no—
588C23—204no—
589C23—204no—
590C23—204no—
591C23—208no—
592C23C22×C4×D512yes—
593C23C24×D511yescharacteristic
594Q8—204no—
595D4—1016no—
596D4—1016no—
597D4—204no—
598D4—208no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—54no—
613C2×C4—54no—
614C2×C4—102no—
615C2×C4—102no—
616C2×C4—102no—
617C2×C4—102no—
618C2×C4—1016no—
619C2×C4—1016no—
620C2×C4—104no—
621C2×C4—104no—
622C2×C4—104no—
623C2×C4—104no—
624C2×C4—104no—
625C2×C4—104no—
626C2×C4—208no—
627C2×C4—204no—
628C2×C4—408no—
629C2×C4—408no—
630C2×C4—408no—
631C2×C4C22×C4×D514yes—
632C2×C4C22×C4×D514yes—
633C2×C4C22×C4×D512yes—
634C2×C4C22×C4×D512yes—
635C2×C4#414912yes—
636C2×C4#414912yes—
637C5#3436011yescharacteristic, complemented, Sylow
638C22—22no—
639C22—24no—
640C22—41no—
641C22—41no—
642C22—42no—
643C22—58no—
644C22—104no—
645C22—104no—
646C22—104no—
647C22—104no—
648C22—104no—
649C22—104no—
650C22—104no—
651C22—104no—
652C22—104no—
653C22—2016no—
654C22—202no—
655C22—202no—
656C22—202no—
657C22—202no—
658C22C23×C4×D511yescharacteristic
659C22C2×#414911yescharacteristic
660C22C2×#414911yescharacteristic
661C22D5×H(5,𝔽2)12yes—
662C22D5×2−1+412yes—
663C4—22no—
664C4—24no—
665C4—24no—
666C4—44no—
667C4—44no—
668C4—104no—
669C4—104no—
670C4—204no—
671C4—204no—
672C4#6585514yes—
673C2—42no—
674C2—108no—
675C2—204no—
676C2C2×#6585511yescharacteristic
677C2D5×#52012yes—
678C2#1061839211yescharacteristic
679C2#1061841611yescharacteristic
680C2#1061846212yes—
subgroupG/H
center, FrattiniC23C24×D5
commutatorC22×C5C24×C4
FittingC5×#2314C2
socle, cosocleC23×C5C25
prime radicalC2×#65858C2
hypercenter#2314D5
hypocenterC5#34360
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. C5×#2314C2semidirect
3. #10613962C2semidirect
4. #10607567C2semidirect
5. #10607220C2semidirect
6. #10607031C2semidirect
7. #10607029C2semidirect
8. #10606091C2semidirect
9. #10605960C2semidirect
10. #10605734C2semidirect
11. D5×#380C2semidirect
12. #10605502C2semidirect
13. C4⋊C4×D20C2semidirect
17. #65858C4semidirect
21. C2×#4038C22semidirect
24. C2×D5×C4⋊C4C22semidirect
32. C5×#387C22semidirect
33. C5×#384C22semidirect
34. C5×#382C22semidirect
36. C4×C5×C4⋊C4C22semidirect
39. #65278C22semidirect
42. #65276C22semidirect
43. #65276C22semidirect
44. #65270C22semidirect
45. #65262C22semidirect
50. #65229C22semidirect
52. #65225C22semidirect
53. #65224C22semidirect
54. C4×C4⋊Dic5C22semidirect
55. #65217C22semidirect
56. #64958C22semidirect
58. #64957C22semidirect
63. #64954C22semidirect
65. #64943C22semidirect
66. #64942C22semidirect
67. #64935C22semidirect
75. C2×#709C2×C4semidirect
81. C22×C4×D5C2×C4semidirect
89. C5×#117C2×C4semidirect
119. #4043C2×C4semidirect
120. #4041C2×C4semidirect
126. #4032C2×C4semidirect
127. #4028C2×C4semidirect
130. C4×D20C2×C4semidirect
131. C4×D20C2×C4semidirect
152. #2314D5semidirect
271. #387C2×D5semidirect
281. #384C2×D5semidirect
283. #382C2×D5semidirect
293. C4×C4⋊C4C2×D5semidirect
394. #117C4×D5semidirect
637. C5#34360semidirect

Maximal subgroups

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

subgroupindex
C2×#658582
C5×#23142
#106139622
#106075672
#106072202
#106070312
#106070292
#106060912
#106059602
#106057342
D5×#3802
#106055022
C4⋊C4×D202
#343605