Finite Group Databasebeta

← #12966940

Subgroups — #12966940

Order 1280.

2030 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10619161C211yesmaximal, characteristic
2C2×#65834C211yesmaximal, characteristic, complemented
3C2×C5×#417C211yesmaximal, characteristic
4#10615803C211yesmaximal, characteristic
5#10615483C211yesmaximal, characteristic, complemented
6#10615481C211yesmaximal, characteristic, complemented
7C2×#65253C211yesmaximal, characteristic, complemented
8#10610604C214yesmaximal, complemented
9#10610603C214yesmaximal, complemented
10#10610602C214yesmaximal, complemented
11#10610601C214yesmaximal, complemented
12#10610510C214yesmaximal, complemented
13#10610462C214yesmaximal, complemented
14C22×C5×#41C2211yescharacteristic
15C22×C5×M16C2211yescharacteristic
16C22×C5×C4⋊C4C2211yescharacteristic
17#66161—21no—
18#66161—21no—
19#66161C2212yes—
20#66161C2212yes—
21#41×Dic5—22no—
22#41×Dic5C2214yes—
23#66146C2212yes—
24#66132—22no—
25#66132C2212yes—
26C2×#4072—22no—
27C2×#4072C2211yescharacteristic
28C2×#4072C2211yescharacteristic
29C2×#4072C2211yescharacteristic
30C2×#4070—22no—
31C22×#681C2211yescharacteristic
32#65834—22no—
33#65834—22no—
34#65834C2214yes—
35#65834C2214yescomplemented
36C2×#4036—21no—
37C2×#4036C2212yescomplemented
38C2×#4034—21no—
39C2×#4034C2212yescomplemented
40C5×#417C2214yes—
41C5×#417C2214yes—
42C2×C5×#103C2212yes—
43C2×C5×#102C2212yes—
44#65522C2214yes—
45#65521C2212yes—
46#65521C2212yes—
47#65514C2212yes—
48#65504C2212yes—
49#65410C2212yescomplemented
50#65410C2212yescomplemented
51#65410C2212yescomplemented
52#65410C2212yescomplemented
53#65408C2214yescomplemented
54#65407C2214yescomplemented
55#65395C2214yes—
56#65391C2214yes—
57#65253C2214yes—
58#65253C2214yescomplemented
59C2×#3940C2212yescomplemented
60C2×#3939C2212yescomplemented
61#65088C2218yescomplemented
62#65085C2218yescomplemented
63#65083C2218yescomplemented
64#65080C2218yescomplemented
65#65055C2218yescomplemented
66#65052C2218yescomplemented
67#65050C2218yescomplemented
68#65048C2218yescomplemented
69#60493—51nomaximal, Sylow
70C2×C5×#41—21no—
71C2×C5×#41—21no—
72C2×C5×#41—24no—
73C2×C5×#41Q818yescomplemented
74C2×C5×#41C2312yes—
75C2×C5×#41C2312yes—
76C22×C5×Q8C2311yescharacteristic
77C22×C5×D4—21no—
78C22×C5×D4C2311yescharacteristic
79C23×C4×C5—21no—
80C23×C4×C5—21no—
81C23×C4×C5C2311yescharacteristic
82C22×Dic10—22no—
83C2×C5×M16—24no—
84C2×C5×M16C2312yes—
85C2×C5×M16C2312yes—
86C22×C5×C8C2312yes—
87C2×C5×C4⋊C4—21no—
88C2×C5×C4⋊C4—21no—
89C2×C5×C4⋊C4—21no—
90C2×C5×C4⋊C4C2312yes—
91C2×C5×C4⋊C4C2312yes—
92C2×C5×C4⋊C4C2312yes—
93C2×C22⋊Dic5—21no—
94C2×C22⋊Dic5—21no—
95C2×C22⋊Dic5—24no—
96C2×C22⋊Dic5D412yes—
97Q8×Dic5—22no—
98Q8×Dic5—22no—
99Q8×Dic5C2314yes—
100Q8×Dic5C2314yes—
101D4×Dic5—24no—
102D4×Dic5—24no—
103D4×Dic5—24no—
104D4×Dic5—24no—
105D4×Dic5—22no—
106D4×Dic5—22no—
107D4×Dic5C2314yes—
108D4×Dic5C2314yes—
109#4072—21no—
110#4072—21no—
111#4072—21no—
112#4072—21no—
113#4072—24no—
114#4072—22no—
115#4072—22no—
116#4072—24no—
117#4072D412yes—
118#4072D412yes—
119#4072C2312yes—
120#4072C2312yes—
121#4072C2312yes—
122#4072C2312yes—
123#4072C2312yes—
124#4072C2312yes—
125C2×C4⋊Dic5—22no—
126C2×C4⋊Dic5—22no—
127C2×C4⋊Dic5—22no—
128C2×C4⋊Dic5D411yescharacteristic
129C2×C4⋊Dic5D411yescharacteristic
130C2×C4⋊Dic5C2311yescharacteristic
131C2×C4⋊Dic5C2311yescharacteristic
132C2×C4⋊Dic5C2311yescharacteristic
133C2×C4⋊Dic5C2311yescharacteristic
134#4070—28no—
135#4070—28no—
136C2×#683—22no—
137C2×#683—22no—
138C2×C4×Dic5—21no—
139C2×C4×Dic5—21no—
140C2×C4×Dic5—24no—
141C2×C4×Dic5C2312yes—
142C2×#681—24no—
143C2×#681C2312yes—
144C2×#681C2312yes—
145C22×#168C2312yes—
146#4036—24no—
147#4036—24no—
148#4036—24no—
149#4036C2318yes—
150#4034—24no—
151#4034—24no—
152#4034—24no—
153#4034C2318yes—
154C5×#103C2318yes—
155C5×#102C2318yes—
156#3967C2314yes—
157#3967C2314yes—
158#3964C2314yes—
159#3964C2314yes—
160#3950C2314yes—
161#3950C2314yes—
162#3949C2314yes—
163#3949C2314yes—
164#3946C2318yes—
165#3946C2318yes—
166#3940C2318yes—
167#3939C2318yes—
168C2×#544—51no—
169#3428—51no—
170#3084—54no—
171#3083—54no—
172#3082—54no—
173#3081—54no—
174#3080—54no—
175#3079—54no—
176#2911—51no—
177C2×#428—51no—
178#2905—51no—
179#2896—51no—
180C2×#417D511yescharacteristic
181C24×C5—21no—
182C24×C5C2×D411yescharacteristic
183C5×#41—28no—
184C5×#41—28no—
185C5×#41—24no—
186C5×#41—24no—
187C5×#41C2×Q818yes—
188C5×#41C2×Q818yescomplemented
189C2×C5×Q8—21no—
190C2×C5×Q8—22no—
191C2×C5×Q8C2×Q814yescomplemented
192C2×C5×Q8C2412yes—
193C2×C5×D4—28no—
194C2×C5×D4—22no—
195C2×C5×D4—22no—
196C2×C5×D4—21no—
197C2×C5×D4—22no—
198C2×C5×D4C2×Q814yescomplemented
199C2×C5×D4C2412yes—
200C22×C4×C5—21no—
201C22×C4×C5—28no—
202C22×C4×C5—22no—
203C22×C4×C5—22no—
204C22×C4×C5—22no—
205C22×C4×C5—22no—
206C22×C4×C5—22no—
207C22×C4×C5—82no—
208C22×C4×C5C2×D411yescharacteristic
209C22×C4×C5C2×D411yescharacteristic
210C22×C4×C5C2×D411yescharacteristic
211C22×C4×C5C2×Q814yes—
212C22×C4×C5#4114yes—
213C22×C4×C5C2411yescharacteristic
214C22×C4×C5C2411yescharacteristic
215C22×C4×C5C2411yescharacteristic
216C22×Dic5—21no—
217C22×Dic5—21no—
218C22×Dic5—21no—
219C22×Dic5—21no—
220C22×Dic5—24no—
221C22×Dic5—24no—
222C22×Dic5C2×D412yes—
223C22×Dic5C2×D412yes—
224C2×Dic10—28no—
225C2×Dic10—84no—
226C5×M16—24no—
227C5×M16—24no—
228C2×C5×C8—24no—
229C2×C5×C8C2414yes—
230C5×C4⋊C4—22no—
231C5×C4⋊C4—22no—
232C5×C4⋊C4—22no—
233C5×C4⋊C4C2414yes—
234C22⋊Dic5—22no—
235C22⋊Dic5—22no—
236C22⋊Dic5—22no—
237C22⋊Dic5—22no—
238C22⋊Dic5—24no—
239C22⋊Dic5—24no—
240C22⋊Dic5C2×D414yes—
241C22⋊Dic5C2×D414yes—
242C4⋊Dic5—21no—
243C4⋊Dic5—21no—
244C4⋊Dic5—21no—
245C4⋊Dic5—21no—
246C4⋊Dic5—28no—
247C4⋊Dic5—24no—
248C4⋊Dic5—24no—
249C4⋊Dic5C2×D412yes—
250C4⋊Dic5C2×D412yes—
251C4⋊Dic5C2×D412yes—
252C4⋊Dic5C2×D412yes—
253C4⋊Dic5C2412yes—
254C4⋊Dic5C2412yes—
255C4⋊Dic5C2412yes—
256C4⋊Dic5C2412yes—
257#683—28no—
258#683—28no—
259C4×Dic5—22no—
260C4×Dic5—22no—
261C4×Dic5—22no—
262C4×Dic5—22no—
263C4×Dic5—24no—
264C4×Dic5—24no—
265C4×Dic5C2×D414yes—
266C4×Dic5C2414yes—
267#681—24no—
268#681—24no—
269C2×#168—24no—
270C2×#168C2414yes—
271C22×#41C2×D511yescharacteristic
272C22×M16—51no—
273C22×M16C2×D511yescharacteristic
274#544—54no—
275#544—54no—
276#544—102no—
277#544—102no—
278C2×C4⋊Q8—52no—
279C2×C4⋊Q8—101no—
280C2×#125—52no—
281C2×#125—101no—
282C2×C22⋊Q8—102no—
283#520—52no—
284#520—52no—
285#520—101no—
286#520—101no—
287#519—52no—
288#518—52no—
289#518—102no—
290C4×#41—54no—
291C4×#41—102no—
292C2×#117—51no—
293C2×#117—51no—
294C2×#117—51no—
295C2×#117—102no—
296C22×C4⋊C4C2×D511yescharacteristic
297#479—58no—
298#479—58no—
299#478—58no—
300#478—58no—
301#477—58no—
302#476—58no—
303#475—58no—
304#474—58no—
305#428—52no—
306#428—52no—
307#428—52no—
308#428—52no—
309#428—54no—
310#428—54no—
311#427—54no—
312C2×#107—52no—
313C2×#106—52no—
314#424—54no—
315#424—54no—
316#423—54no—
317#419—52no—
318#418—52no—
319#417—52no—
320#417—52no—
321#417C2×D514yes—
322#417C2×D514yes—
323#416—54no—
324C2×#103C2×D512yes—
325C2×#102C2×D512yes—
326C23×C5—22no—
327C23×C5—24no—
328C23×C5—41no—
329C23×C5—41no—
330C23×C5—81no—
331C23×C5—81no—
332C23×C5C22⋊Q814yes—
333C23×C5C22×D411yescharacteristic
334C23×C5C22×D411yescharacteristic
335C23×C5C22×D411yescharacteristic
336C5×Q8—24no—
337C5×Q8—22no—
338C5×Q8C22×Q814yes—
339C5×D4—28no—
340C5×D4—28no—
341C5×D4—24no—
342C5×D4—22no—
343C5×D4C22×Q814yes—
344C2×C4×C5—24no—
345C2×C4×C5—22no—
346C2×C4×C5—28no—
347C2×C4×C5—28no—
348C2×C4×C5—24no—
349C2×C4×C5—44no—
350C2×C4×C5—82no—
351C2×C4×C5—82no—
352C2×C4×C5C22⋊Q814yes—
353C2×C4×C5C22⋊Q814yes—
354C2×C4×C5C22⋊Q814yes—
355C2×C4×C5C22×D411yescharacteristic
356C2×C4×C5C22×D411yescharacteristic
357C2×C4×C5C22×D411yescharacteristic
358C2×C4×C5C22×Q812yes—
359C2×C4×C5C22×Q812yes—
360C2×C4×C5C22×Q812yes—
361C2×C4×C5C2×#4112yes—
362C2×C4×C5C2×#4112yes—
363C2×C4×C5C2×#4112yes—
364C2×C4×C5C2511yescharacteristic
365C2×Dic5—22no—
366C2×Dic5—22no—
367C2×Dic5—22no—
368C2×Dic5—22no—
369C2×Dic5—42no—
370C2×Dic5—42no—
371C2×Dic5—42no—
372C2×Dic5—42no—
373C2×Dic5—84no—
374C2×Dic5—84no—
375C2×Dic5—81no—
376C2×Dic5—81no—
377C2×Dic5C22×D414yes—
378C2×Dic5C22×D414yes—
379Dic10—88no—
380C5×C8—24no—
381#168—24no—
382C2×#41—21no—
383C2×#41—21no—
384C2×#41—24no—
385C2×#41Dic1018yescomplemented
386C2×#41C22×D512yes—
387C2×#41C22×D512yes—
388C22×Q8—102no—
389C22×Q8C22×D511yescharacteristic
390C22×D4—21no—
391C22×D4C22×D511yescharacteristic
392C23×C4—21no—
393C23×C4—21no—
394C23×C4C22×D511yescharacteristic
395C2×M16—24no—
396C2×M16—52no—
397C2×M16—52no—
398C2×M16—104no—
399C2×M16C22×D512yes—
400C2×M16C22×D512yes—
401C22×C8—52no—
402C22×C8C22×D512yes—
403C4⋊Q8—58no—
404C4⋊Q8—104no—
405C4⋊Q8—104no—
406C4⋊Q8—104no—
407#125—58no—
408#125—104no—
409#125—104no—
410#125—104no—
411C22⋊Q8—108no—
412C22⋊Q8—108no—
413C4×Q8—54no—
414C4×Q8—54no—
415C4×Q8—102no—
416C4×Q8—102no—
417C4×D4—54no—
418C4×D4—54no—
419C4×D4—104no—
420C4×D4—104no—
421C4×D4—102no—
422C4×D4—102no—
423C4×D4—104no—
424C4×D4—104no—
425#117—52no—
426#117—52no—
427#117—52no—
428#117—52no—
429#117—52no—
430#117—52no—
431#117—52no—
432#117—52no—
433#117—104no—
434#117—104no—
435#117—102no—
436#117—102no—
437#117—101no—
438#117—101no—
439#117—101no—
440#117—101no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—51no—
445C2×C4⋊C4—51no—
446C2×C4⋊C4—51no—
447C2×C4⋊C4—51no—
448C2×C4⋊C4—51no—
449C2×C4⋊C4—51no—
450C2×C4⋊C4—102no—
451C2×C4⋊C4—102no—
452C2×C4⋊C4—102no—
453C2×C4⋊C4—102no—
454C2×C4⋊C4—102no—
455C2×C4⋊C4C22×D512yes—
456C2×C4⋊C4C22×D512yes—
457C2×C4⋊C4C22×D512yes—
458C2×C22⋊C4—52no—
459C2×C22⋊C4—104no—
460C2×C22⋊C4—101no—
461C2×C22⋊C4—101no—
462C2×C42—52no—
463C2×C42—104no—
464C2×C42—101no—
465C2×C42—101no—
466#107—54no—
467#107—54no—
468#107—58no—
469#106—54no—
470#106—54no—
471#106—58no—
472C4⋊C8—58no—
473C4⋊C8—58no—
474#103—54no—
475#103—54no—
476#103C22×D518yes—
477#102—54no—
478#102—54no—
479#102C22×D518yes—
480C22×C5—21no—
481C22×C5—21no—
482C22×C5—21no—
483C22×C5—21no—
484C22×C5—21no—
485C22×C5—21no—
486C22×C5—24no—
487C22×C5—44no—
488C22×C5—42no—
489C22×C5—42no—
490C22×C5—84no—
491C22×C5C2×C22⋊Q812yes—
492C22×C5C2×C22⋊Q812yes—
493C22×C5C2×C22⋊Q812yes—
494C22×C5C23×D411yescharacteristic
495C4×C5—24no—
496C4×C5—44no—
497C4×C5C2×C22⋊Q812yes—
498C4×C5C2×C22⋊Q812yes—
499C4×C5C2×C22⋊Q812yes—
500C4×C5C23×Q811yescharacteristic
501C4×C5C22×#4111yescharacteristic
502Dic5—24no—
503Dic5—24no—
504Dic5—42no—
505Dic5—42no—
506Dic5—42no—
507Dic5—42no—
508C24—21no—
509C24D4×D511yescharacteristic
510#41—24no—
511#41—24no—
512#41—28no—
513#41—28no—
514#41C2×Dic1018yes—
515#41C2×Dic1018yescomplemented
516C2×Q8—21no—
517C2×Q8—22no—
518C2×Q8—108no—
519C2×Q8—404no—
520C2×Q8C2×Dic1014yescomplemented
521C2×Q8C23×D512yes—
522C2×D4—22no—
523C2×D4—22no—
524C2×D4—28no—
525C2×D4—21no—
526C2×D4—22no—
527C2×D4C2×Dic1014yescomplemented
528C2×D4C23×D512yes—
529C22×C4—21no—
530C22×C4—22no—
531C22×C4—22no—
532C22×C4—28no—
533C22×C4—22no—
534C22×C4—22no—
535C22×C4—22no—
536C22×C4—52no—
537C22×C4—52no—
538C22×C4—82no—
539C22×C4—104no—
540C22×C4—104no—
541C22×C4—101no—
542C22×C4—101no—
543C22×C4—101no—
544C22×C4—101no—
545C22×C4C2×Dic1014yes—
546C22×C4D4×D511yescharacteristic
547C22×C4D4×D511yescharacteristic
548C22×C4D4×D511yescharacteristic
549C22×C4#71114yes—
550C22×C4C23×D511yescharacteristic
551C22×C4C23×D511yescharacteristic
552C22×C4C23×D511yescharacteristic
553M16—24no—
554M16—24no—
555M16—104no—
556M16—104no—
557C2×C8—24no—
558C2×C8—54no—
559C2×C8—104no—
560C2×C8C23×D514yes—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—52no—
565C4⋊C4—52no—
566C4⋊C4—52no—
567C4⋊C4—52no—
568C4⋊C4—52no—
569C4⋊C4—52no—
570C4⋊C4—52no—
571C4⋊C4—52no—
572C4⋊C4—104no—
573C4⋊C4—104no—
574C4⋊C4—108no—
575C4⋊C4—108no—
576C4⋊C4—108no—
577C4⋊C4—101no—
578C4⋊C4—101no—
579C4⋊C4—101no—
580C4⋊C4—101no—
581C4⋊C4C23×D514yes—
582C22⋊C4—54no—
583C22⋊C4—54no—
584C22⋊C4—104no—
585C22⋊C4—104no—
586C22⋊C4—102no—
587C22⋊C4—102no—
588C22⋊C4—102no—
589C22⋊C4—102no—
590C42—54no—
591C42—54no—
592C42—104no—
593C42—104no—
594C42—102no—
595C42—102no—
596C42—102no—
597C42—102no—
598C2×C5—22no—
599C2×C5—22no—
600C2×C5—22no—
601C2×C5—44no—
602C2×C5C22×C22⋊Q811yescharacteristic
603C2×C5#358511yescharacteristic
604C2×C5#358611yescharacteristic
605C23—22no—
606C23—24no—
607C23—41no—
608C23—41no—
609C23—81no—
610C23—81no—
611C23#402414yes—
612C23C2×D4×D511yescharacteristic
613C23C2×D4×D511yescharacteristic
614C23C2×D4×D511yescharacteristic
615Q8—24no—
616Q8—22no—
617Q8—408no—
618Q8C22×Dic1014yes—
619D4—28no—
620D4—28no—
621D4—24no—
622D4—22no—
623D4C22×Dic1014yes—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—24no—
629C2×C4—44no—
630C2×C4—54no—
631C2×C4—54no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—102no—
635C2×C4—102no—
636C2×C4—102no—
637C2×C4—102no—
638C2×C4—202no—
639C2×C4—202no—
640C2×C4—202no—
641C2×C4—202no—
642C2×C4—404no—
643C2×C4—404no—
644C2×C4—401no—
645C2×C4—401no—
646C2×C4#402414yes—
647C2×C4#402414yes—
648C2×C4#402414yes—
649C2×C4C22×Dic1012yes—
650C2×C4C22×Dic1012yes—
651C2×C4C22×Dic1012yes—
652C2×C4C2×D4×D511yescharacteristic
653C2×C4C2×D4×D511yescharacteristic
654C2×C4C2×D4×D511yescharacteristic
655C2×C4C2×#71112yes—
656C2×C4C2×#71112yes—
657C2×C4C2×#71112yes—
658C2×C4C24×D511yescharacteristic
659C8—24no—
660C8—104no—
661C5#6049311yescharacteristic, complemented, Sylow
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—24no—
669C22—44no—
670C22—42no—
671C22—42no—
672C22—84no—
673C22C2×#402412yes—
674C22C2×#402412yes—
675C22C2×#402412yes—
676C22C22×D4×D511yescharacteristic
677C4—24no—
678C4—44no—
679C4—104no—
680C4—104no—
681C4—202no—
682C4—202no—
683C4—202no—
684C4—202no—
685C4C2×#402412yes—
686C4C2×#402412yes—
687C4C2×#402412yes—
688C4C23×Dic1011yescharacteristic
689C4C22×#71111yescharacteristic
690C2—22no—
691C2—22no—
692C2—22no—
693C2—44no—
694C2C22×#402411yescharacteristic
695C2#1061909311yescharacteristic
696C2#1061909811yescharacteristic
subgroupG/H
centerC22C22×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#417C2
socleC22×C5C23×D4
prime radicalC22×C5×#41C22
hypercenterC2×#417D5
hypocenterC5#60493
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#65834C2semidirect
5. #10615483C2semidirect
6. #10615481C2semidirect
7. C2×#65253C2semidirect
8. #10610604C2semidirect
9. #10610603C2semidirect
10. #10610602C2semidirect
11. #10610601C2semidirect
12. #10610510C2semidirect
13. #10610462C2semidirect
35. #65834C22semidirect
37. C2×#4036C22semidirect
39. C2×#4034C22semidirect
49. #65410C22semidirect
50. #65410C22semidirect
51. #65410C22semidirect
52. #65410C22semidirect
53. #65408C22semidirect
54. #65407C22semidirect
58. #65253C22semidirect
59. C2×#3940C22semidirect
60. C2×#3939C22semidirect
61. #65088C22semidirect
62. #65085C22semidirect
63. #65083C22semidirect
64. #65080C22semidirect
65. #65055C22semidirect
66. #65052C22semidirect
67. #65050C22semidirect
68. #65048C22semidirect
73. C2×C5×#41Q8semidirect
188. C5×#41C2×Q8semidirect
191. C2×C5×Q8C2×Q8semidirect
198. C2×C5×D4C2×Q8semidirect
385. C2×#41Dic10semidirect
515. #41C2×Dic10semidirect
520. C2×Q8C2×Dic10semidirect
527. C2×D4C2×Dic10semidirect
661. C5#60493semidirect

Maximal subgroups

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

subgroupindex
#106191612
C2×#658342
C2×C5×#4172
#106158032
#106154832
#106154812
C2×#652532
#106106042
#106106032
#106106022
#106106012
#106105102
#106104622
#604935