Finite Group Databasebeta

← #12265910

Subgroups — #12265910

Order 1280.

2002 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2659C211yesmaximal, characteristic, complemented
2#10614553C212yesmaximal
3Dic5×C22⋊Q8C212yesmaximal, complemented
4#10609576C214yesmaximal, complemented
5#10609470C214yesmaximal, complemented
6#10608942C214yesmaximal, complemented
7#10608901C214yesmaximal, complemented
8#10605687C212yesmaximal, complemented
9#10605683C212yesmaximal
10#10605597C212yesmaximal, complemented
11#10605593C212yesmaximal
12#10605092C211yesmaximal, characteristic, complemented
13C2×#64955C211yesmaximal, characteristic
14C23×C4×D5C2211yescharacteristic
15C2×C5×C22⋊Q8C2212yes—
16C2×Q8×Dic5C2212yescomplemented
17C22×#685C2211yescharacteristic
18C22×#685C2211yescharacteristic
19C2×#4070C2212yes—
20C22×C4×Dic5C2211yescharacteristic
21C22×C4×Dic5C2211yescharacteristic
22C2×#4038C2212yes—
23C2×#4038C2212yes—
24C2×#4033C2212yescomplemented
25C5×#398C2214yescomplemented
26C5×#395C2214yescomplemented
27C5×#380C2212yes—
28C5×#380C2212yes—
29C2×C5×#95C2211yescharacteristic
30#65520C2214yes—
31#65519C2212yescomplemented
32#65501C2212yes—
33#65501C2212yes—
34C2×#3963C2211yescharacteristic
35#65283C2214yes—
36#65283C2214yes—
37#65283C2214yes—
38#65283C2214yes—
39#65281C2214yes—
40#65276C2212yes—
41#65276C2212yes—
42#65275C2214yescomplemented
43#65274C2212yescomplemented
44#65273C2214yescomplemented
45#65270C2214yescomplemented
46C4⋊C4×Dic5C2212yescomplemented
47C4⋊C4×Dic5C2214yescomplemented
48C4⋊C4×Dic5C2212yescomplemented
49#65263C2212yescomplemented
50#65262C2212yescomplemented
51#65244C2214yes—
52#65239C2214yes—
53#65235C2214yes—
54C22⋊C4×Dic5C2214yes—
55#64966C2214yes—
56#64962C2214yes—
57#64955C2214yes—
58#64955C2214yes—
59#64955C2214yes—
60#64953C2214yes—
61#64950C2214yescomplemented
62#64950C2214yescomplemented
63#64949C2214yescomplemented
64#64945C2214yescomplemented
65#64941C2214yescomplemented
66#64938C2214yescomplemented
67#34578—51nomaximal, Sylow
68C24×D5C2311yescharacteristic
69C22×C5×Q8C2312yes—
70C23×C4×C5C2311yescharacteristic
71C23×C4×C5C2311yescharacteristic
72C23×C4×C5C2311yescharacteristic
73C23×Dic5C2311yescharacteristic
74C23×Dic5C2311yescharacteristic
75C23×Dic5C2311yescharacteristic
76C22×C4×D5—42no—
77C22×C4×D5—42no—
78C22×C4×D5Q818yescomplemented
79C22×C4×D5C2312yes—
80C22×C4×D5C2312yes—
81C22×Dic10C2312yes—
82C5×C22⋊Q8—28no—
83C2×C5×C4⋊C4C2312yes—
84C2×C5×C4⋊C4C2314yes—
85C2×C5×C4⋊C4C2312yes—
86C2×C5×C4⋊C4C2312yes—
87C2×C5×C4⋊C4C2312yes—
88C2×C5×C4⋊C4C2312yes—
89C2×C5×C22⋊C4C2314yes—
90C2×C22⋊Dic5C2314yes—
91Q8×Dic5—44no—
92C2×#685—24no—
93C2×#685—24no—
94C2×#685C2314yes—
95C2×#685C2314yes—
96C2×#685C2314yes—
97C2×#685C2314yes—
98C2×#685C2314yes—
99C2×#685C2314yes—
100C2×C4⋊Dic5C2312yes—
101C2×C4⋊Dic5C2312yes—
102C2×C4⋊Dic5C2312yes—
103C2×C4⋊Dic5C2314yes—
104C2×C4⋊Dic5C2314yes—
105C2×C4⋊Dic5C2312yes—
106C2×C4⋊Dic5C2312yes—
107#4070—28no—
108C2×#683C2314yes—
109C2×#683C2314yes—
110C2×#683C2314yes—
111C2×C4×Dic5—42no—
112C2×C4×Dic5—42no—
113C2×C4×Dic5C2314yes—
114C2×C4×Dic5C2312yes—
115C2×C4×Dic5C2314yes—
116C2×C4×Dic5C2312yes—
117C2×C4×Dic5C2312yes—
118C2×C4×Dic5C2312yes—
119#4038—44no—
120#4038—44no—
121#4033—44no—
122C5×#95C2314yes—
123C5×#95C2314yes—
124C5×#95C2314yes—
125#3963C2314yes—
126#3963C2314yes—
127#3963C2314yes—
128#3963C2314yes—
129#3963C2314yes—
130#3963C2314yes—
131#3963C2314yes—
132#3963C2314yes—
133#3963C2314yes—
134#2667—54no—
135#2665—54no—
136#2659—51no—
137#2659D511yescharacteristic, complemented
138#2646—52no—
139#2626—52no—
140#2626—52no—
141#2396—54no—
142#2396—54no—
143C4×C22⋊Q8—52no—
144C4×C22⋊Q8—52no—
145#2280—52no—
146C2×C4×C22⋊C4—51no—
147C24×C5C2411yescharacteristic
148C23×D5—22no—
149C23×D5—22no—
150C23×D5—41no—
151C23×D5—41no—
152C23×D5—41no—
153C23×D5—41no—
154C23×D5C2×Q814yes—
155C23×D5C2412yes—
156C2×C5×Q8—24no—
157C2×C5×Q8—42no—
158C2×C5×Q8—42no—
159C22×C4×C5—41no—
160C22×C4×C5—41no—
161C22×C4×C5—41no—
162C22×C4×C5—41no—
163C22×C4×C5—41no—
164C22×C4×C5—41no—
165C22×C4×C5C2×Q814yes—
166C22×C4×C5#4114yes—
167C22×C4×C5#4114yes—
168C22×C4×C5C2414yes—
169C22×C4×C5C2414yes—
170C22×C4×C5C2412yes—
171C22×C4×C5C2412yes—
172C22×C4×C5C2412yes—
173C22×Dic5—22no—
174C22×Dic5—22no—
175C22×Dic5—22no—
176C22×Dic5—22no—
177C22×Dic5—41no—
178C22×Dic5—41no—
179C22×Dic5—41no—
180C22×Dic5—41no—
181C22×Dic5C2×Q814yes—
182C22×Dic5C2414yes—
183C22×Dic5C2414yes—
184C22×Dic5C2412yes—
185C22×Dic5C2412yes—
186C22×Dic5C2411yescharacteristic
187C22×Dic5C2411yescharacteristic
188C2×C4×D5—24no—
189C2×C4×D5—24no—
190C2×C4×D5—416no—
191C2×C4×D5—42no—
192C2×C4×D5—42no—
193C2×C4×D5C2×Q818yes—
194C2×C4×D5C2×Q818yes—
195C2×Dic10—24no—
196C2×Dic10—42no—
197C2×Dic10—42no—
198C5×C4⋊C4—24no—
199C5×C4⋊C4—28no—
200C5×C4⋊C4—42no—
201C5×C4⋊C4—42no—
202C5×C4⋊C4—42no—
203C5×C4⋊C4—42no—
204C5×C22⋊C4—28no—
205C22⋊Dic5—28no—
206#685—28no—
207#685—28no—
208#685—44no—
209#685—44no—
210#685—44no—
211#685—44no—
212C4⋊Dic5—24no—
213C4⋊Dic5—42no—
214C4⋊Dic5—42no—
215C4⋊Dic5—44no—
216C4⋊Dic5—42no—
217C4⋊Dic5—44no—
218C4⋊Dic5—42no—
219#683—28no—
220#683—44no—
221#683—44no—
222C4×Dic5—28no—
223C4×Dic5—24no—
224C4×Dic5—28no—
225C4×Dic5—24no—
226C4×Dic5—42no—
227C4×Dic5—42no—
228C24×C4—51no—
229C2×C22⋊Q8—52no—
230C2×C22⋊Q8C2×D512yes—
231C2×C4×Q8—52no—
232C2×C4×Q8—52no—
233C2×#117—52no—
234C2×#117—52no—
235C22×C22⋊C4—51no—
236C22×C22⋊C4—51no—
237C22×C42—51no—
238C22×C42—51no—
239#400—54no—
240#400—54no—
241#399—54no—
242#399—54no—
243#398—54no—
244#398—54no—
245#398—54no—
246#398C2×D514yescomplemented
247#396—54no—
248#396—54no—
249#396—54no—
250#396—54no—
251#395—54no—
252#395C2×D514yescomplemented
253#393—54no—
254#393—54no—
255#391—52no—
256#391—52no—
257#389—52no—
258#389—52no—
259#387—54no—
260#387—54no—
261#385—54no—
262#385—54no—
263#381—54no—
264#380—52no—
265#380—52no—
266#380—52no—
267#380—52no—
268#380C2×D512yes—
269#380C2×D512yes—
270#379—54no—
271C4×C4⋊C4—54no—
272C4×C4⋊C4—52no—
273C4×C4⋊C4—52no—
274C4×C4⋊C4—54no—
275C4×C22⋊C4—54no—
276C4×C22⋊C4—54no—
277C4×C22⋊C4—54no—
278C4×C22⋊C4—54no—
279C2×#95—51no—
280C2×#95C2×D511yescharacteristic
281C23×C5—21no—
282C23×C5—21no—
283C23×C5—21no—
284C23×C5—21no—
285C23×C5C22×Q812yes—
286C23×C5C2×#4112yes—
287C23×C5C2×#4112yes—
288C23×C5C2511yescharacteristic
289C22×D5—28no—
290C22×D5—22no—
291C22×D5—22no—
292C22×D5—22no—
293C22×D5—22no—
294C22×D5—22no—
295C22×D5—22no—
296C22×D5—44no—
297C22×D5—44no—
298C22×D5—44no—
299C22×D5—44no—
300C22×D5—44no—
301C22×D5—44no—
302C22×D5C22×Q814yes—
303C5×Q8—48no—
304C2×C4×C5—22no—
305C2×C4×C5—22no—
306C2×C4×C5—22no—
307C2×C4×C5—22no—
308C2×C4×C5—22no—
309C2×C4×C5—22no—
310C2×C4×C5—24no—
311C2×C4×C5—24no—
312C2×C4×C5—24no—
313C2×C4×C5—24no—
314C2×C4×C5—24no—
315C2×C4×C5—24no—
316C2×C4×C5—44no—
317C2×C4×C5—44no—
318C2×C4×C5—44no—
319C2×C4×C5C22×Q814yes—
320C2×C4×C5C2×#4114yes—
321C2×C4×C5C2×#4114yes—
322C2×Dic5—21no—
323C2×Dic5—21no—
324C2×Dic5—28no—
325C2×Dic5—24no—
326C2×Dic5—24no—
327C2×Dic5—22no—
328C2×Dic5—22no—
329C2×Dic5—22no—
330C2×Dic5—22no—
331C2×Dic5—24no—
332C2×Dic5—24no—
333C2×Dic5—22no—
334C2×Dic5—24no—
335C2×Dic5—22no—
336C2×Dic5—24no—
337C2×Dic5—44no—
338C2×Dic5—44no—
339C2×Dic5C22×Q812yes—
340C2×Dic5C22×Q812yes—
341C2×Dic5H(5,𝔽2)12yes—
342C2×Dic5H(5,𝔽2)12yes—
343C4×D5—216no—
344C4×D5—48no—
345Dic10—48no—
346C25—51no—
347C22×Q8—52no—
348C22×Q8C22×D512yes—
349C23×C4—51no—
350C23×C4—52no—
351C23×C4—58no—
352C23×C4—52no—
353C23×C4—51no—
354C23×C4—51no—
355C23×C4—202no—
356C23×C4—202no—
357C23×C4C22×D511yescharacteristic
358C23×C4C22×D511yescharacteristic
359C23×C4C22×D511yescharacteristic
360C22⋊Q8—28no—
361C22⋊Q8—108no—
362C4×Q8—204no—
363C4×Q8—204no—
364#117—204no—
365#117—204no—
366C2×C4⋊C4—54no—
367C2×C4⋊C4—52no—
368C2×C4⋊C4—52no—
369C2×C4⋊C4—52no—
370C2×C4⋊C4—52no—
371C2×C4⋊C4—54no—
372C2×C4⋊C4—54no—
373C2×C4⋊C4—52no—
374C2×C4⋊C4—54no—
375C2×C4⋊C4—54no—
376C2×C4⋊C4C22×D514yes—
377C2×C4⋊C4C22×D512yes—
378C2×C4⋊C4C22×D512yes—
379C2×C4⋊C4C22×D512yes—
380C2×C4⋊C4C22×D512yes—
381C2×C4⋊C4C22×D512yes—
382C2×C22⋊C4—54no—
383C2×C22⋊C4—54no—
384C2×C22⋊C4—54no—
385C2×C22⋊C4—54no—
386C2×C22⋊C4—54no—
387C2×C22⋊C4—54no—
388C2×C22⋊C4—54no—
389C2×C22⋊C4—104no—
390C2×C22⋊C4—104no—
391C2×C22⋊C4C22×D514yes—
392C2×C42—54no—
393C2×C42—54no—
394C2×C42—52no—
395C2×C42—52no—
396C2×C42—52no—
397C2×C42—52no—
398C2×C42—202no—
399C2×C42—202no—
400#95—54no—
401#95—54no—
402#95—54no—
403#95—54no—
404#95—54no—
405#95—54no—
406#95—54no—
407#95—54no—
408#95—54no—
409#95C22×D514yes—
410#95C22×D514yes—
411#95C22×D514yes—
412C22×C5—22no—
413C22×C5—22no—
414C22×C5—22no—
415C22×C5—22no—
416C22×C5—22no—
417C22×C5—22no—
418C22×C5#53314yes—
419C22×C5C23×Q811yescharacteristic
420C22×C5C22×#4111yescharacteristic
421C22×C5C22×#4111yescharacteristic
422C22×C5C2×H(5,𝔽2)11yescharacteristic
423C22×C5C2×H(5,𝔽2)11yescharacteristic
424C22×C5#58511yescharacteristic
425C22×C5#58511yescharacteristic
426C2×D5—24no—
427C2×D5—24no—
428C2×D5—24no—
429C2×D5—416no—
430C2×D5—42no—
431C2×D5—42no—
432C2×D5—42no—
433C2×D5—42no—
434C4×C5—24no—
435C4×C5—24no—
436C4×C5—24no—
437C4×C5—44no—
438C4×C5—44no—
439Dic5—22no—
440Dic5—44no—
441Dic5—44no—
442Dic5—42no—
443Dic5—42no—
444Dic5#54314yes—
445C24—52no—
446C24—54no—
447C24—102no—
448C24—102no—
449C24—201no—
450C24—201no—
451C24—201no—
452C24—201no—
453C24C23×D511yescharacteristic
454C2×Q8—24no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—104no—
458C2×Q8—202no—
459C2×Q8—202no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—51no—
467C22×C4—54no—
468C22×C4—51no—
469C22×C4—58no—
470C22×C4—58no—
471C22×C4—52no—
472C22×C4—52no—
473C22×C4—54no—
474C22×C4—54no—
475C22×C4—104no—
476C22×C4—104no—
477C22×C4—102no—
478C22×C4—102no—
479C22×C4—102no—
480C22×C4—102no—
481C22×C4—202no—
482C22×C4—202no—
483C22×C4—2016no—
484C22×C4—201no—
485C22×C4—201no—
486C22×C4—201no—
487C22×C4—201no—
488C22×C4#71114yes—
489C22×C4Q8×D514yes—
490C22×C4#71314yes—
491C22×C4C23×D514yes—
492C22×C4C23×D514yes—
493C22×C4C23×D512yes—
494C22×C4C23×D512yes—
495C22×C4C23×D512yes—
496C4⋊C4—24no—
497C4⋊C4—28no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—108no—
503C4⋊C4—104no—
504C4⋊C4—202no—
505C4⋊C4—202no—
506C4⋊C4—204no—
507C4⋊C4—202no—
508C4⋊C4—204no—
509C4⋊C4—204no—
510C4⋊C4—202no—
511C4⋊C4—204no—
512C22⋊C4—28no—
513C22⋊C4—108no—
514C22⋊C4—108no—
515C22⋊C4—108no—
516C22⋊C4—204no—
517C22⋊C4—204no—
518C22⋊C4—204no—
519C22⋊C4—204no—
520C42—104no—
521C42—104no—
522C42—108no—
523C42—108no—
524C42—202no—
525C42—202no—
526C2×C5—24no—
527C2×C5C2×#53311yescharacteristic
528C2×C5C2×#54311yescharacteristic
529C2×C5#345811yescharacteristic
530C2×C5#345911yescharacteristic
531C2×C5#345911yescharacteristic
532C2×C5#346011yescharacteristic
533C2×C5#346011yescharacteristic
534D5—44no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—54no—
540C23—108no—
541C23—102no—
542C23—102no—
543C23—102no—
544C23—102no—
545C23—102no—
546C23—102no—
547C23—204no—
548C23—204no—
549C23—204no—
550C23—204no—
551C23—204no—
552C23—204no—
553C23C2×#71112yes—
554C23C2×Q8×D512yes—
555C23C2×#71312yes—
556C23C24×D511yescharacteristic
557Q8—48no—
558Q8—208no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—52no—
575C2×C4—52no—
576C2×C4—52no—
577C2×C4—52no—
578C2×C4—1016no—
579C2×C4—108no—
580C2×C4—101no—
581C2×C4—101no—
582C2×C4—102no—
583C2×C4—102no—
584C2×C4—104no—
585C2×C4—104no—
586C2×C4—102no—
587C2×C4—102no—
588C2×C4—104no—
589C2×C4—104no—
590C2×C4—104no—
591C2×C4—104no—
592C2×C4—102no—
593C2×C4—102no—
594C2×C4—208no—
595C2×C4—204no—
596C2×C4—204no—
597C2×C4C2×#71114yes—
598C2×C4C2×Q8×D514yes—
599C2×C4C2×#71314yes—
600C5#3457811yescharacteristic, complemented, Sylow
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—104no—
608C22—104no—
609C22—104no—
610C22—2016no—
611C22—202no—
612C22—202no—
613C22—202no—
614C22—202no—
615C22#6606214yes—
616C22C22×#71111yescharacteristic
617C22C22×Q8×D511yescharacteristic
618C22C22×#71311yescharacteristic
619C22D5×H(5,𝔽2)11yescharacteristic
620C22D5×H(5,𝔽2)11yescharacteristic
621C22#6628511yescharacteristic
622C22#6628711yescharacteristic
623C4—24no—
624C4—24no—
625C4—24no—
626C4—44no—
627C4—44no—
628C4—54no—
629C4—102no—
630C4—204no—
631C4—202no—
632C4—204no—
633C4—202no—
634C2—24no—
635C2—204no—
636C2C2×#6606211yescharacteristic
637C2#1061819611yescharacteristic
638C2#1061826511yescharacteristic
639C2#1061847311yescharacteristic
640C2D5×#54311yescharacteristic
641C2#1061848211yescharacteristic
642C2#1061848311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D5
commutator, socle, cosocleC23×C5C25
FittingC5×#2659C2
prime radicalC24×D5C23
hypercenter#2659D5
hypocenterC5#34578
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #2659

Complements

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

NH ≅ G/Ntype
1. C5×#2659C2semidirect
3. Dic5×C22⋊Q8C2semidirect
4. #10609576C2semidirect
5. #10609470C2semidirect
6. #10608942C2semidirect
7. #10608901C2semidirect
8. #10605687C2semidirect
10. #10605597C2semidirect
12. #10605092C2semidirect
16. C2×Q8×Dic5C22semidirect
24. C2×#4033C22semidirect
25. C5×#398C22semidirect
26. C5×#395C22semidirect
31. #65519C22semidirect
42. #65275C22semidirect
43. #65274C22semidirect
44. #65273C22semidirect
45. #65270C22semidirect
46. C4⋊C4×Dic5C22semidirect
47. C4⋊C4×Dic5C22semidirect
48. C4⋊C4×Dic5C22semidirect
49. #65263C22semidirect
50. #65262C22semidirect
61. #64950C22semidirect
62. #64950C22semidirect
63. #64949C22semidirect
64. #64945C22semidirect
65. #64941C22semidirect
66. #64938C22semidirect
78. C22×C4×D5Q8semidirect
137. #2659D5semidirect
246. #398C2×D5semidirect
252. #395C2×D5semidirect
600. C5#34578semidirect

Maximal subgroups

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

subgroupindex
C5×#26592
#106145532
Dic5×C22⋊Q82
#106095762
#106094702
#106089422
#106089012
#106056872
#106056832
#106055972
#106055932
#106050922
C2×#649552
#345785