Finite Group Databasebeta

← #12966929

Subgroups — #12966929

Order 1280.

2038 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
1C2×#66161C211yesmaximal, characteristic
2#10617626C211yesmaximal, characteristic, complemented
3C5×#2896C211yesmaximal, characteristic
4#10615803C211yesmaximal, characteristic
5C2×#65410C211yesmaximal, characteristic, complemented
6#10615481C211yesmaximal, characteristic, complemented
7C2×#65284C211yesmaximal, characteristic, complemented
8#10610680C214yesmaximal, complemented
9#10610648C214yesmaximal, complemented
10#10610604C214yesmaximal, complemented
11#10610603C214yesmaximal, complemented
12#10610555C214yesmaximal, complemented
13#10610554C214yesmaximal, complemented
14C22×C5×#41C2211yescharacteristic
15C22×C5×M16C2211yescharacteristic
16C2×C5×#117C2211yescharacteristic
17#66161—22no—
18#66161—22no—
19#66161C2214yes—
20#66161C2214yes—
21C2×Q8×Dic5C2212yes—
22C2×D4×Dic5—22no—
23C2×D4×Dic5C2212yes—
24C2×#4072—21no—
25C2×#4072C2211yescharacteristic
26C2×#4072C2211yescharacteristic
27C22×C4⋊Dic5—21no—
28C22×C4⋊Dic5C2211yescharacteristic
29C22×#681C2211yescharacteristic
30#65853—21no—
31#65853—21no—
32#65853C2212yescomplemented
33#65853C2212yescomplemented
34#65851—22no—
35#65851C2214yes—
36#65834C2212yescomplemented
37#65834C2212yescomplemented
38#65817—22no—
39#65804—22no—
40C5×#419C2212yes—
41C5×#418C2212yes—
42C5×#417C2212yes—
43C5×#417C2212yes—
44C5×#416C2214yes—
45#65522C2214yes—
46#65521C2212yes—
47#65521C2212yes—
48#65514C2212yes—
49#65504C2212yes—
50#65410C2214yes—
51#65410C2214yescomplemented
52#65408C2214yescomplemented
53#65407C2214yes—
54C2×#3950C2212yescomplemented
55C2×#3949C2212yescomplemented
56#65391C2214yescomplemented
57#65284C2214yes—
58#65284C2214yescomplemented
59C2×#3940C2212yescomplemented
60C2×#3939C2212yescomplemented
61#65088C2218yescomplemented
62#65085C2218yescomplemented
63#65083C2218yescomplemented
64#65080C2218yescomplemented
65#65055C2218yescomplemented
66#65052C2218yescomplemented
67#65050C2218yescomplemented
68#65048C2218yescomplemented
69#60492—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—
82C23×Dic5—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—22no—
97C2×C22⋊Dic5D412yes—
98Q8×Dic5—24no—
99Q8×Dic5C2318yes—
100D4×Dic5—28no—
101D4×Dic5—28no—
102D4×Dic5—24no—
103D4×Dic5C2318yes—
104#4072—21no—
105#4072—21no—
106#4072—24no—
107#4072—24no—
108#4072D412yes—
109#4072D412yes—
110#4072C2312yes—
111#4072C2312yes—
112#4072C2312yes—
113#4072C2312yes—
114C2×C4⋊Dic5—21no—
115C2×C4⋊Dic5—21no—
116C2×C4⋊Dic5—21no—
117C2×C4⋊Dic5—22no—
118C2×C4⋊Dic5—24no—
119C2×C4⋊Dic5—24no—
120C2×C4⋊Dic5D411yescharacteristic
121C2×C4⋊Dic5D411yescharacteristic
122C2×C4⋊Dic5C2312yes—
123C2×C4⋊Dic5C2312yes—
124C2×C4⋊Dic5C2311yescharacteristic
125C2×C4⋊Dic5C2311yescharacteristic
126C2×C4⋊Dic5C2312yes—
127#4070—24no—
128#4070—24no—
129C2×C4×Dic5—21no—
130C2×C4×Dic5—22no—
131C2×C4×Dic5C2312yes—
132C2×#681—24no—
133C2×#681C2312yes—
134C2×#681C2312yes—
135C22×#168C2312yes—
136#4036—22no—
137#4036—22no—
138#4036C2314yes—
139#4036C2314yes—
140#4034—22no—
141#4034—22no—
142#4034C2314yes—
143#4034C2314yes—
144#4024—24no—
145#4024—24no—
146#4024—24no—
147#4024—24no—
148#4016—22no—
149#4016—22no—
150#4015—22no—
151#4015—22no—
152C4×Dic10—24no—
153C4×Dic10—24no—
154C5×#103C2314yes—
155C5×#103C2314yes—
156C5×#102C2314yes—
157C5×#102C2314yes—
158#3967C2314yes—
159#3967C2314yes—
160#3964C2314yes—
161#3964C2314yes—
162#3950C2318yes—
163#3949C2318yes—
164#3946C2318yes—
165#3946C2318yes—
166#3940C2318yes—
167#3939C2318yes—
168#3458—51no—
169C2×#520—51no—
170#3084—54no—
171#3084—54no—
172#3083—54no—
173#3083—54no—
174#3078—54no—
175#3077—54no—
176C2×#428—51no—
177C2×#428—51no—
178#2905—51no—
179#2896—51no—
180#2896D511yescharacteristic
181C24×C5—21no—
182C24×C5C2×D411yescharacteristic
183C5×#41—28no—
184C5×#41—28no—
185C5×#41—24no—
186C5×#41—24no—
187C5×#41C2×Q818yes—
188C5×#41C2×Q818yes—
189C2×C5×Q8—21no—
190C2×C5×Q8—22no—
191C2×C5×Q8C2×Q814yes—
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×Q814yes—
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—28no—
220C22×Dic5—84no—
221C22×Dic5—82no—
222C22×Dic5C2×D412yes—
223C22×Dic5C2×D412yes—
224C2×Dic10—24no—
225C2×Dic10—24no—
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—28no—
241C22⋊Dic5C2×D414yes—
242C22⋊Dic5C2×D414yes—
243C4⋊Dic5—21no—
244C4⋊Dic5—21no—
245C4⋊Dic5—22no—
246C4⋊Dic5—22no—
247C4⋊Dic5—22no—
248C4⋊Dic5—24no—
249C4⋊Dic5—24no—
250C4⋊Dic5—28no—
251C4⋊Dic5C2×D412yes—
252C4⋊Dic5C2×D412yes—
253C4⋊Dic5C2×D412yes—
254C4⋊Dic5C2×D412yes—
255C4⋊Dic5C2414yes—
256C4⋊Dic5C2412yes—
257C4⋊Dic5C2412yes—
258#683—24no—
259#683—24no—
260#683—24no—
261#683—24no—
262C4×Dic5—22no—
263C4×Dic5—22no—
264C4×Dic5—28no—
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—52no—
275#544—52no—
276#544—52no—
277#544—52no—
278#544—101no—
279#544—101no—
280#543—102no—
281#533—54no—
282#533—102no—
283#533—102no—
284#520—54no—
285#520—54no—
286#520—102no—
287#520—102no—
288C2×C4×Q8—52no—
289C2×C4×D4—52no—
290C2×C4×D4—102no—
291C2×#117—51no—
292C2×#117—51no—
293C2×#117—101no—
294C2×#117C2×D511yescharacteristic
295C22×C4⋊C4—51no—
296C22×C4⋊C4—101no—
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—54no—
308#428—54no—
309C2×#107—52no—
310C2×#107—52no—
311C2×#106—52no—
312C2×#106—52no—
313#424—54no—
314#424—54no—
315#423—54no—
316#419—52no—
317#419C2×D512yes—
318#418—52no—
319#418C2×D512yes—
320#417—52no—
321#417—52no—
322#417C2×D512yes—
323#417C2×D512yes—
324#416—54no—
325#416C2×D514yes—
326C23×C5—24no—
327C23×C5—22no—
328C23×C5—41no—
329C23×C5—41no—
330C23×C5—42no—
331C23×C5—42no—
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—81no—
351C2×C4×C5—81no—
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—88no—
372C2×Dic5—81no—
373C2×Dic5—81no—
374C2×Dic5—82no—
375C2×Dic5—82no—
376C2×Dic5C22×D414yes—
377C2×Dic5C22×D414yes—
378Dic10—84no—
379Dic10—84no—
380C5×C8—24no—
381#168—24no—
382C2×#41—21no—
383C2×#41—21no—
384C2×#41—24no—
385C2×#41Dic1018yescomplemented
386C2×#41C22×D512yes—
387C2×#41C22×D512yes—
388C22×Q8C22×D511yescharacteristic
389C22×D4—21no—
390C22×D4C22×D511yescharacteristic
391C23×C4—21no—
392C23×C4—102no—
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—54no—
404C4⋊Q8—54no—
405C4⋊Q8—102no—
406C4⋊Q8—102no—
407C4⋊Q8—102no—
408C4⋊Q8—102no—
409#125—54no—
410#125—54no—
411#125—102no—
412#125—102no—
413#125—102no—
414#125—102no—
415C22⋊Q8—104no—
416C22⋊Q8—104no—
417C22⋊Q8—104no—
418C22⋊Q8—104no—
419C22⋊Q8—104no—
420C22⋊Q8—104no—
421C4×Q8—58no—
422C4×Q8—104no—
423C4×Q8—104no—
424C4×Q8—104no—
425C4×D4—58no—
426C4×D4—104no—
427C4×D4—108no—
428C4×D4—108no—
429#117—21no—
430#117—21no—
431#117—52no—
432#117—52no—
433#117—52no—
434#117—52no—
435#117—52no—
436#117—52no—
437#117—104no—
438#117—104no—
439#117—101no—
440#117—101no—
441#117C22×D512yes—
442#117C22×D512yes—
443C2×C4⋊C4—51no—
444C2×C4⋊C4—51no—
445C2×C4⋊C4—52no—
446C2×C4⋊C4—52no—
447C2×C4⋊C4—52no—
448C2×C4⋊C4—51no—
449C2×C4⋊C4—51no—
450C2×C4⋊C4—104no—
451C2×C4⋊C4—104no—
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—102no—
461C2×C22⋊C4—101no—
462C2×C42—21no—
463C2×C42—52no—
464C2×C42—102no—
465C2×C42—101no—
466#107—58no—
467#107—58no—
468#106—58no—
469#106—58no—
470C4⋊C8—58no—
471C4⋊C8—58no—
472#103—54no—
473#103—54no—
474#103C22×D514yes—
475#103C22×D514yes—
476#102—54no—
477#102—54no—
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—48no—
489C22×C5—42no—
490C22×C5—42no—
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—42no—
506Dic5—44no—
507Dic5—42no—
508C24—21no—
509C24D4×D511yescharacteristic
510#41—24no—
511#41—24no—
512#41—28no—
513#41—28no—
514#41C2×Dic1018yes—
515#41C2×Dic1018yes—
516C2×Q8—21no—
517C2×Q8—22no—
518C2×Q8—104no—
519C2×Q8—104no—
520C2×Q8C2×Dic1014yes—
521C2×Q8C23×D512yes—
522C2×D4—22no—
523C2×D4—22no—
524C2×D4—28no—
525C2×D4—21no—
526C2×D4—22no—
527C2×D4C2×Dic1014yes—
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—108no—
538C22×C4—102no—
539C22×C4—101no—
540C22×C4—101no—
541C22×C4—102no—
542C22×C4—102no—
543C22×C4—404no—
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—52no—
568C4⋊C4—52no—
569C4⋊C4—54no—
570C4⋊C4—108no—
571C4⋊C4—104no—
572C4⋊C4—104no—
573C4⋊C4—104no—
574C4⋊C4—104no—
575C4⋊C4—104no—
576C4⋊C4—104no—
577C4⋊C4—102no—
578C4⋊C4—102no—
579C4⋊C4—102no—
580C4⋊C4—101no—
581C4⋊C4—101no—
582C4⋊C4C23×D512yes—
583C4⋊C4C23×D512yes—
584C22⋊C4—22no—
585C22⋊C4—22no—
586C22⋊C4—54no—
587C22⋊C4—54no—
588C22⋊C4—108no—
589C22⋊C4—102no—
590C22⋊C4—102no—
591C42—22no—
592C42—22no—
593C42—54no—
594C42—54no—
595C42—108no—
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—24no—
606C23—22no—
607C23—41no—
608C23—41no—
609C23—42no—
610C23—42no—
611C23#402414yes—
612C23C2×D4×D511yescharacteristic
613C23C2×D4×D511yescharacteristic
614C23C2×D4×D511yescharacteristic
615Q8—24no—
616Q8—22no—
617Q8—404no—
618Q8—404no—
619Q8C22×Dic1014yes—
620D4—28no—
621D4—28no—
622D4—24no—
623D4—22no—
624D4C22×Dic1014yes—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—54no—
632C2×C4—54no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—102no—
636C2×C4—102no—
637C2×C4—104no—
638C2×C4—202no—
639C2×C4—202no—
640C2×C4—204no—
641C2×C4—408no—
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#6049211yescharacteristic, complemented, Sylow
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—24no—
669C22—44no—
670C22—48no—
671C22—42no—
672C22—42no—
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#60492
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. #10617626C2semidirect
5. C2×#65410C2semidirect
6. #10615481C2semidirect
7. C2×#65284C2semidirect
8. #10610680C2semidirect
9. #10610648C2semidirect
10. #10610604C2semidirect
11. #10610603C2semidirect
12. #10610555C2semidirect
13. #10610554C2semidirect
32. #65853C22semidirect
33. #65853C22semidirect
36. #65834C22semidirect
37. #65834C22semidirect
51. #65410C22semidirect
52. #65408C22semidirect
54. C2×#3950C22semidirect
55. C2×#3949C22semidirect
56. #65391C22semidirect
58. #65284C22semidirect
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
385. C2×#41Dic10semidirect
661. C5#60492semidirect

Maximal subgroups

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

subgroupindex
C2×#661612
#106176262
C5×#28962
#106158032
C2×#654102
#106154812
C2×#652842
#106106802
#106106482
#106106042
#106106032
#106105552
#106105542
#604925