Finite Group Databasebeta

← #12966945

Subgroups — #12966945

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×#65853C211yesmaximal, characteristic, complemented
3C5×#2896C211yesmaximal, characteristic
4C2×#65521C211yesmaximal, characteristic
5C2×#65410C211yesmaximal, characteristic, complemented
6#10615481C211yesmaximal, characteristic, complemented
7#10613938C211yesmaximal, characteristic, complemented
8#10610681C214yesmaximal, complemented
9#10610649C214yesmaximal, complemented
10#10610602C214yesmaximal, complemented
11#10610601C214yesmaximal, complemented
12#10610555C214yesmaximal, complemented
13#10610554C214yesmaximal, complemented
14C22×C5×#41C2211yescharacteristic
15C22×C5×M16C2211yescharacteristic
16C2×C5×#117C2211yescharacteristic
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
29C22×C4⋊Dic5C2211yescharacteristic
30C22×#681C2211yescharacteristic
31#65853—22no—
32#65853—22no—
33#65853C2214yes—
34#65853C2214yescomplemented
35C2×#4036C2212yescomplemented
36C2×#4034C2212yescomplemented
37C2×#4024—22no—
38C2×#4016—21no—
39C2×#4015—21no—
40C5×#419C2212yes—
41C5×#418C2212yes—
42C5×#417C2212yes—
43C5×#417C2212yes—
44C5×#416C2214yes—
45#65521C2214yes—
46#65521C2214yes—
47C2×#3967C2212yes—
48C2×#3964C2212yes—
49#65410C2214yes—
50#65410C2214yescomplemented
51#65408C2214yes—
52#65407C2214yescomplemented
53C2×#3950C2212yescomplemented
54C2×#3949C2212yescomplemented
55#65391C2214yescomplemented
56#65287C2214yes—
57#65284C2212yescomplemented
58#65284C2212yescomplemented
59#65253C2212yescomplemented
60#65253C2212yescomplemented
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×C5C2311yescharacteristic
81C23×Dic5—21no—
82C22×Dic10—22no—
83C2×C5×M16—24no—
84C2×C5×M16C2312yes—
85C2×C5×M16C2312yes—
86C22×C5×C8C2312yes—
87C5×#117—21no—
88C5×#117—21no—
89C5×#117C2312yes—
90C5×#117C2312yes—
91C2×C5×C4⋊C4C2311yescharacteristic
92C2×C5×C4⋊C4C2311yescharacteristic
93C2×C5×C22⋊C4—21no—
94C2×C42×C5—21no—
95C2×C22⋊Dic5—21no—
96C2×C22⋊Dic5—24no—
97C2×C22⋊Dic5D412yes—
98Q8×Dic5—22no—
99Q8×Dic5—22no—
100Q8×Dic5C2314yes—
101Q8×Dic5C2314yes—
102D4×Dic5—24no—
103D4×Dic5—24no—
104D4×Dic5—24no—
105D4×Dic5—24no—
106D4×Dic5—22no—
107D4×Dic5—22no—
108D4×Dic5C2314yes—
109D4×Dic5C2314yes—
110#4072—21no—
111#4072—21no—
112#4072—24no—
113#4072—22no—
114#4072—22no—
115#4072—24no—
116#4072D412yes—
117#4072D412yes—
118#4072C2312yes—
119#4072C2312yes—
120#4072C2312yes—
121#4072C2312yes—
122C2×C4⋊Dic5—21no—
123C2×C4⋊Dic5—21no—
124C2×C4⋊Dic5—21no—
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⋊Dic5C2312yes—
133C2×C4⋊Dic5C2312yes—
134C2×C4⋊Dic5C2312yes—
135C2×#683—22no—
136C2×#683—22no—
137C2×C4×Dic5—21no—
138C2×C4×Dic5—24no—
139C2×C4×Dic5C2312yes—
140C2×#681—24no—
141C2×#681C2312yes—
142C2×#681C2312yes—
143C22×#168C2312yes—
144#4036—24no—
145#4036C2318yes—
146#4034—24no—
147#4034C2318yes—
148#4024—28no—
149#4024—28no—
150#4016—24no—
151#4016—24no—
152#4015—24no—
153#4015—24no—
154C5×#103C2314yes—
155C5×#103C2314yes—
156C5×#102C2314yes—
157C5×#102C2314yes—
158#3967C2318yes—
159#3964C2318yes—
160#3950C2318yes—
161#3949C2318yes—
162#3946C2318yes—
163#3946C2318yes—
164#3940C2314yes—
165#3940C2314yes—
166#3939C2314yes—
167#3939C2314yes—
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#2896D511yescharacteristic
180C2×#417—51no—
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—21no—
205C22×C4×C5—21no—
206C22×C4×C5C2×D411yescharacteristic
207C22×C4×C5C2×D411yescharacteristic
208C22×C4×C5C2×D411yescharacteristic
209C22×C4×C5C2×Q814yes—
210C22×C4×C5#4114yes—
211C22×C4×C5C2411yescharacteristic
212C22×C4×C5C2411yescharacteristic
213C22×C4×C5C2411yescharacteristic
214C22×Dic5—22no—
215C22×Dic5—21no—
216C22×Dic5—21no—
217C22×Dic5—22no—
218C22×Dic5—22no—
219C22×Dic5—24no—
220C22×Dic5—24no—
221C22×Dic5—82no—
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—21no—
231C5×C4⋊C4—21no—
232C5×C4⋊C4C2412yes—
233C5×C4⋊C4C2412yes—
234C5×C22⋊C4—22no—
235C5×C22⋊C4—22no—
236C42×C5—22no—
237C42×C5—22no—
238C22⋊Dic5—22no—
239C22⋊Dic5—22no—
240C22⋊Dic5—24no—
241C22⋊Dic5—24no—
242C22⋊Dic5C2×D414yes—
243C22⋊Dic5C2×D414yes—
244C4⋊Dic5—22no—
245C4⋊Dic5—22no—
246C4⋊Dic5—22no—
247C4⋊Dic5—21no—
248C4⋊Dic5—21no—
249C4⋊Dic5—28no—
250C4⋊Dic5—24no—
251C4⋊Dic5—24no—
252C4⋊Dic5C2×D412yes—
253C4⋊Dic5C2×D412yes—
254C4⋊Dic5C2×D412yes—
255C4⋊Dic5C2×D412yes—
256C4⋊Dic5C2412yes—
257C4⋊Dic5C2412yes—
258C4⋊Dic5C2414yes—
259#683—28no—
260#683—28no—
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—102no—
295C2×#117C2×D511yescharacteristic
296C22×C4⋊C4—51no—
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—54no—
306#428—54no—
307#428—52no—
308#428—52no—
309#428—52no—
310#428—52no—
311#427—54no—
312C2×#107—52no—
313C2×#106—52no—
314#424—54no—
315#424—54no—
316#423—54no—
317#419C2×D512yes—
318#418C2×D512yes—
319#417—54no—
320#417—54no—
321#417C2×D512yes—
322#417C2×D512yes—
323#416C2×D514yes—
324C2×#103—52no—
325C2×#102—52no—
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—22no—
349C2×C4×C5—22no—
350C2×C4×C5—42no—
351C2×C4×C5—42no—
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—24no—
368C2×Dic5—42no—
369C2×Dic5—42no—
370C2×Dic5—44no—
371C2×Dic5—84no—
372C2×Dic5—84no—
373C2×Dic5—81no—
374C2×Dic5—81no—
375C2×Dic5—82no—
376C2×Dic5—82no—
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—101no—
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—21no—
426#117—21no—
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#117C22×D512yes—
440#117C22×D512yes—
441C2×C4⋊C4—51no—
442C2×C4⋊C4—51no—
443C2×C4⋊C4—52no—
444C2×C4⋊C4—52no—
445C2×C4⋊C4—52no—
446C2×C4⋊C4—51no—
447C2×C4⋊C4—51no—
448C2×C4⋊C4—102no—
449C2×C4⋊C4—102no—
450C2×C4⋊C4—102no—
451C2×C4⋊C4—102no—
452C2×C4⋊C4—102no—
453C2×C4⋊C4—101no—
454C2×C4⋊C4—101no—
455C2×C4⋊C4—101no—
456C2×C4⋊C4C22×D511yescharacteristic
457C2×C4⋊C4C22×D511yescharacteristic
458C2×C22⋊C4—21no—
459C2×C22⋊C4—52no—
460C2×C22⋊C4—104no—
461C2×C22⋊C4—101no—
462C2×C42—21no—
463C2×C42—52no—
464C2×C42—104no—
465C2×C42—101no—
466#107—58no—
467#107—54no—
468#107—54no—
469#106—58no—
470#106—54no—
471#106—54no—
472C4⋊C8—58no—
473C4⋊C8—58no—
474#103—58no—
475#103C22×D514yes—
476#103C22×D514yes—
477#102—58no—
478#102C22×D514yes—
479#102C22×D514yes—
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—42no—
497C4×C5—42no—
498C4×C5C2×C22⋊Q812yes—
499C4×C5C2×C22⋊Q812yes—
500C4×C5C2×C22⋊Q812yes—
501C4×C5C23×Q811yescharacteristic
502C4×C5C22×#4111yescharacteristic
503Dic5—24no—
504Dic5—24no—
505Dic5—44no—
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—21no—
534C22×C4—21no—
535C22×C4—52no—
536C22×C4—52no—
537C22×C4—104no—
538C22×C4—104no—
539C22×C4—102no—
540C22×C4—101no—
541C22×C4—101no—
542C22×C4—102no—
543C22×C4—102no—
544C22×C4—402no—
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—21no—
562C4⋊C4—21no—
563C4⋊C4—52no—
564C4⋊C4—52no—
565C4⋊C4—52no—
566C4⋊C4—52no—
567C4⋊C4—54no—
568C4⋊C4—52no—
569C4⋊C4—52no—
570C4⋊C4—104no—
571C4⋊C4—104no—
572C4⋊C4—108no—
573C4⋊C4—108no—
574C4⋊C4—108no—
575C4⋊C4—101no—
576C4⋊C4—101no—
577C4⋊C4—102no—
578C4⋊C4—102no—
579C4⋊C4—102no—
580C4⋊C4C23×D512yes—
581C4⋊C4C23×D512yes—
582C22⋊C4—22no—
583C22⋊C4—22no—
584C22⋊C4—54no—
585C22⋊C4—54no—
586C22⋊C4—104no—
587C22⋊C4—104no—
588C22⋊C4—102no—
589C22⋊C4—102no—
590C42—22no—
591C42—22no—
592C42—54no—
593C42—54no—
594C42—104no—
595C42—104no—
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—22no—
629C2×C4—22no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—54no—
633C2×C4—54no—
634C2×C4—102no—
635C2×C4—102no—
636C2×C4—104no—
637C2×C4—202no—
638C2×C4—202no—
639C2×C4—204no—
640C2×C4—404no—
641C2×C4—404no—
642C2×C4—402no—
643C2×C4—402no—
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—42no—
679C4—42no—
680C4—104no—
681C4—104no—
682C4—204no—
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#1061910111yescharacteristic
696C2#1061910211yescharacteristic
subgroupG/H
centerC22C22×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC5×#2896C2
socleC22×C5C23×D4
prime radicalC22×C5×#41C22
hypercenter#2896D5
hypocenterC5#60493
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. C2×#65853C2semidirect
5. C2×#65410C2semidirect
6. #10615481C2semidirect
7. #10613938C2semidirect
8. #10610681C2semidirect
9. #10610649C2semidirect
10. #10610602C2semidirect
11. #10610601C2semidirect
12. #10610555C2semidirect
13. #10610554C2semidirect
34. #65853C22semidirect
35. C2×#4036C22semidirect
36. C2×#4034C22semidirect
50. #65410C22semidirect
52. #65407C22semidirect
53. C2×#3950C22semidirect
54. C2×#3949C22semidirect
55. #65391C22semidirect
57. #65284C22semidirect
58. #65284C22semidirect
59. #65253C22semidirect
60. #65253C22semidirect
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×#658532
C5×#28962
C2×#655212
C2×#654102
#106154812
#106139382
#106106812
#106106492
#106106022
#106106012
#106105552
#106105542
#604935