Finite Group Databasebeta

← #12966942

Subgroups — #12966942

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#10617632C211yesmaximal, characteristic
2C2×#65834C211yesmaximal, characteristic, complemented
3C2×C5×#428C211yesmaximal, characteristic, complemented
4#10615499C211yesmaximal, characteristic
5#10615481C211yesmaximal, characteristic, complemented
6#10613938C211yesmaximal, characteristic, complemented
7C2×#65255C211yesmaximal, characteristic
8#10610977C214yesmaximal, complemented
9#10610973C214yesmaximal, complemented
10#10610972C214yesmaximal, complemented
11#10610971C214yesmaximal, complemented
12#10610893C214yesmaximal, complemented
13#10610861C214yesmaximal, complemented
14C22×#709C2211yescharacteristic
15C22×C5×M16C2211yescharacteristic
16C2×C5×#117C2211yescharacteristic
17C22×C5×C4⋊C4C2211yescharacteristic
18C2×#4072C2211yescharacteristic
19C2×#4072C2211yescharacteristic
20C2×#4070—22no—
21C22×#681C2211yescharacteristic
22#65857—21no—
23#65857—21no—
24#65856C2214yes—
25#65855C2212yes—
26D5×#117—22no—
27#65850C2212yes—
28#65838C2212yes—
29#65838C2212yes—
30#65834—22no—
31#65834—22no—
32#65834C2214yes—
33#65834C2214yescomplemented
34C2×#4036—21no—
35C2×#4036C2212yescomplemented
36C2×#4034—21no—
37C2×#4034C2212yescomplemented
38#65820—22no—
39C4×#709—22no—
40C5×#428C2214yes—
41C5×#428C2214yescomplemented
42C2×C5×#107C2212yescomplemented
43C2×C5×#106C2212yescomplemented
44#65422C2212yes—
45#65422C2212yes—
46#65420C2212yes—
47#65409C2212yes—
48#65408C2214yes—
49#65407C2214yescomplemented
50#65402C2214yes—
51#65391C2214yescomplemented
52#65287C2214yes—
53#65284C2212yescomplemented
54#65284C2212yescomplemented
55C2×#3942C2212yes—
56#65255C2214yes—
57#65255C2214yes—
58C2×#3941C2212yes—
59#65253C2212yescomplemented
60#65253C2212yescomplemented
61#65180C2218yescomplemented
62#65179C2218yescomplemented
63#65166C2218yescomplemented
64#65164C2218yescomplemented
65#65160C2218yescomplemented
66#65159C2218yescomplemented
67#65146C2218yescomplemented
68#65143C2218yescomplemented
69#60493—51nomaximal, Sylow
70C23×C4×C5—21no—
71C23×C4×C5C2311yescharacteristic
72C22×C22⋊D5—21no—
73C2×#709—21no—
74C2×#709—21no—
75C2×#709—24no—
76C2×#709Q818yescomplemented
77C2×#709C2312yes—
78C2×#709C2312yes—
79C22×D20C2311yescharacteristic
80C22×C4×D5—21no—
81C22×Dic10—22no—
82C22×Dic10C2311yescharacteristic
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⋊C4—21no—
92C2×C5×C4⋊C4—21no—
93C2×C5×C4⋊C4—21no—
94C2×C5×C4⋊C4C2312yes—
95C2×C5×C4⋊C4C2312yes—
96C2×C5×C4⋊C4C2312yes—
97C2×C5×C4⋊C4C2311yescharacteristic
98C2×C5×C4⋊C4C2311yescharacteristic
99C2×C5×C22⋊C4—21no—
100C2×C42×C5—21no—
101C2×C22⋊Dic5—21no—
102C2×C22⋊Dic5D412yes—
103C4×C22⋊D5—24no—
104C4×C22⋊D5—24no—
105#4072—21no—
106#4072—21no—
107#4072D412yes—
108#4072D412yes—
109#4072C2312yes—
110#4072C2312yes—
111#4072C2312yes—
112#4072C2312yes—
113C2×C4⋊Dic5—22no—
114C2×C4⋊Dic5D411yescharacteristic
115C2×C4⋊Dic5D411yescharacteristic
116C2×C4⋊Dic5C2311yescharacteristic
117C2×C4⋊Dic5C2311yescharacteristic
118#4070—28no—
119#4070—28no—
120C2×#683—22no—
121C2×#683—22no—
122C2×C4×Dic5—21no—
123C2×C4×Dic5C2312yes—
124C2×#681—24no—
125C2×#681C2312yes—
126C2×#681C2312yes—
127C22×#168C2312yes—
128#4039C2314yes—
129#4039C2314yes—
130#4038—22no—
131#4038—22no—
132D5×C4⋊C4—22no—
133D5×C4⋊C4—22no—
134#4036—24no—
135#4036—24no—
136#4036—24no—
137#4036C2318yes—
138#4034—24no—
139#4034—24no—
140#4034—24no—
141#4034C2318yes—
142#4033C2314yes—
143#4033C2314yes—
144#4027—24no—
145#4027—24no—
146D5×C22⋊C4—24no—
147#4023—24no—
148C4×D20—22no—
149C4×D20—22no—
150#4018—24no—
151C42×D5—24no—
152C4×Dic10—22no—
153C4×Dic10—22no—
154C5×#107C2318yes—
155C5×#106C2318yes—
156#3953C2314yes—
157#3953C2314yes—
158#3948C2314yes—
159#3948C2314yes—
160#3946C2318yes—
161#3946C2318yes—
162#3942C2318yes—
163#3941C2318yes—
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×#428D511yescharacteristic, complemented
178#2905—51no—
179#2896—51no—
180C2×#417—51no—
181C24×C5C2×D411yescharacteristic
182C23×D5—21no—
183C22×C4×C5—22no—
184C22×C4×C5—21no—
185C22×C4×C5—21no—
186C22×C4×C5—22no—
187C22×C4×C5—22no—
188C22×C4×C5—82no—
189C22×C4×C5C2×D411yescharacteristic
190C22×C4×C5C2×D411yescharacteristic
191C22×C4×C5C2×D411yescharacteristic
192C22×C4×C5C2×Q814yes—
193C22×C4×C5#4114yes—
194C22×C4×C5C2411yescharacteristic
195C22×C4×C5C2411yescharacteristic
196C22×C4×C5C2411yescharacteristic
197C2×C22⋊D5—28no—
198C2×C22⋊D5—22no—
199C2×C22⋊D5—22no—
200C22×Dic5—21no—
201C22×Dic5—21no—
202C22×Dic5—21no—
203C22×Dic5C2×D412yes—
204C22×Dic5C2×D412yes—
205#709—28no—
206#709—28no—
207#709—24no—
208#709—24no—
209#709C2×Q818yes—
210#709C2×Q818yescomplemented
211C2×D20—21no—
212C2×D20—22no—
213C2×D20C2×Q814yescomplemented
214C2×D20C2412yes—
215C2×C4×D5—28no—
216C2×C4×D5—22no—
217C2×C4×D5—22no—
218C2×C4×D5—24no—
219C2×C4×D5—24no—
220C2×Dic10—21no—
221C2×Dic10—22no—
222C2×Dic10—28no—
223C2×Dic10—84no—
224C2×Dic10C2×Q814yescomplemented
225C2×Dic10C2412yes—
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⋊C4—21no—
234C5×C4⋊C4—21no—
235C5×C4⋊C4C2414yes—
236C5×C4⋊C4C2412yes—
237C5×C4⋊C4C2412yes—
238C5×C22⋊C4—22no—
239C5×C22⋊C4—22no—
240C42×C5—22no—
241C42×C5—22no—
242C22⋊Dic5—22no—
243C22⋊Dic5—22no—
244C22⋊Dic5C2×D414yes—
245C22⋊Dic5C2×D414yes—
246#685—24no—
247#685—24no—
248C4⋊Dic5—21no—
249C4⋊Dic5—21no—
250C4⋊Dic5—28no—
251C4⋊Dic5C2×D412yes—
252C4⋊Dic5C2×D412yes—
253C4⋊Dic5C2×D412yes—
254C4⋊Dic5C2×D412yes—
255C4⋊Dic5C2412yes—
256C4⋊Dic5C2412yes—
257#683—24no—
258#683—24no—
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×#41—51no—
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⋊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#428C2×D514yes—
310#428C2×D514yescomplemented
311#427—54no—
312C2×#107C2×D512yescomplemented
313C2×#106C2×D512yescomplemented
314#424—54no—
315#424—54no—
316#423—54no—
317#419—52no—
318#418—52no—
319#417—52no—
320#417—52no—
321#417—54no—
322#417—54no—
323#416—54no—
324C2×#103—52no—
325C2×#102—52no—
326C23×C5—41no—
327C23×C5—41no—
328C23×C5C22⋊Q814yes—
329C23×C5C22×D411yescharacteristic
330C23×C5C22×D411yescharacteristic
331C23×C5C22×D411yescharacteristic
332C22×D5—22no—
333C22×D5—24no—
334C22×D5—81no—
335C22×D5—81no—
336C2×C4×C5—22no—
337C2×C4×C5—22no—
338C2×C4×C5—24no—
339C2×C4×C5—44no—
340C2×C4×C5—81no—
341C2×C4×C5—81no—
342C2×C4×C5—82no—
343C2×C4×C5—82no—
344C2×C4×C5C22⋊Q814yes—
345C2×C4×C5C22⋊Q814yes—
346C2×C4×C5C22⋊Q814yes—
347C2×C4×C5C22×D411yescharacteristic
348C2×C4×C5C22×D411yescharacteristic
349C2×C4×C5C22×D411yescharacteristic
350C2×C4×C5C22×Q812yes—
351C2×C4×C5C22×Q812yes—
352C2×C4×C5C22×Q812yes—
353C2×C4×C5C2×#4112yes—
354C2×C4×C5C2×#4112yes—
355C2×C4×C5C2×#4112yes—
356C2×C4×C5C2511yescharacteristic
357C22⋊D5—28no—
358C22⋊D5—28no—
359C2×Dic5—24no—
360C2×Dic5—22no—
361C2×Dic5—22no—
362C2×Dic5—22no—
363C2×Dic5—42no—
364C2×Dic5—42no—
365C2×Dic5—42no—
366C2×Dic5—42no—
367C2×Dic5C22×D414yes—
368C2×Dic5C22×D414yes—
369D20—24no—
370D20—22no—
371D20C22×Q814yes—
372C4×D5—28no—
373C4×D5—28no—
374C4×D5—84no—
375C4×D5—84no—
376Dic10—24no—
377Dic10—22no—
378Dic10—88no—
379Dic10C22×Q814yes—
380C5×C8—24no—
381#168—24no—
382C2×#41—58no—
383C2×#41—52no—
384C2×#41—52no—
385C2×#41—104no—
386C2×#41—101no—
387C2×#41—101no—
388C22×Q8—51no—
389C22×Q8—102no—
390C22×D4—51no—
391C22×D4—101no—
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—102no—
420C4×D4—102no—
421C4×D4—104no—
422C4×D4—104no—
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—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—102no—
449C2×C4⋊C4—102no—
450C2×C4⋊C4—102no—
451C2×C4⋊C4—102no—
452C2×C4⋊C4—102no—
453C2×C4⋊C4C22×D512yes—
454C2×C4⋊C4C22×D512yes—
455C2×C4⋊C4C22×D512yes—
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—54no—
467#107—54no—
468#107C22×D518yes—
469#106—54no—
470#106—54no—
471#106C22×D518yes—
472C4⋊C8—58no—
473C4⋊C8—58no—
474#103—54no—
475#103—54no—
476#103—58no—
477#102—54no—
478#102—54no—
479#102—58no—
480C22×C5—21no—
481C22×C5—21no—
482C22×C5—21no—
483C22×C5—21no—
484C22×C5—21no—
485C22×C5—21no—
486C22×C5—44no—
487C22×C5C2×C22⋊Q812yes—
488C22×C5C2×C22⋊Q812yes—
489C22×C5C2×C22⋊Q812yes—
490C22×C5C23×D411yescharacteristic
491C2×D5—24no—
492C2×D5—42no—
493C2×D5—42no—
494C2×D5—84no—
495C4×C5—44no—
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—24no—
506Dic5—42no—
507Dic5—42no—
508C24—101no—
509C24D4×D511yescharacteristic
510#41—58no—
511#41—58no—
512#41—104no—
513#41—104no—
514#41—108no—
515#41—108no—
516C2×Q8—54no—
517C2×Q8—52no—
518C2×Q8—108no—
519C2×Q8—102no—
520C2×Q8—101no—
521C2×Q8—404no—
522C2×D4—54no—
523C2×D4—52no—
524C2×D4—102no—
525C2×D4—102no—
526C2×D4—108no—
527C2×D4—102no—
528C2×D4—101no—
529C22×C4—22no—
530C22×C4—21no—
531C22×C4—21no—
532C22×C4—22no—
533C22×C4—22no—
534C22×C4—52no—
535C22×C4—52no—
536C22×C4—82no—
537C22×C4—104no—
538C22×C4—104no—
539C22×C4—101no—
540C22×C4—102no—
541C22×C4—102no—
542C22×C4—108no—
543C22×C4—101no—
544C22×C4—101no—
545C22×C4#70914yes—
546C22×C4D4×D511yescharacteristic
547C22×C4D4×D511yescharacteristic
548C22×C4D4×D511yescharacteristic
549C22×C4Q8×D514yes—
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—21no—
565C4⋊C4—21no—
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⋊C4C23×D514yes—
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×C5C22×C22⋊Q811yescharacteristic
602C2×C5#358511yescharacteristic
603C2×C5#358611yescharacteristic
604D5—44no—
605C23—41no—
606C23—41no—
607C23—102no—
608C23—104no—
609C23—401no—
610C23—401no—
611C23#404214yes—
612C23C2×D4×D511yescharacteristic
613C23C2×D4×D511yescharacteristic
614C23C2×D4×D511yescharacteristic
615Q8—54no—
616Q8—104no—
617Q8—102no—
618Q8—408no—
619D4—54no—
620D4—108no—
621D4—108no—
622D4—104no—
623D4—102no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—44no—
628C2×C4—54no—
629C2×C4—54no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—104no—
635C2×C4—102no—
636C2×C4—108no—
637C2×C4—108no—
638C2×C4—102no—
639C2×C4—102no—
640C2×C4—202no—
641C2×C4—202no—
642C2×C4—202no—
643C2×C4—202no—
644C2×C4—404no—
645C2×C4—404no—
646C2×C4#404214yes—
647C2×C4#404214yes—
648C2×C4#404214yes—
649C2×C4C2×#70912yes—
650C2×C4C2×#70912yes—
651C2×C4C2×#70912yes—
652C2×C4C2×D4×D511yescharacteristic
653C2×C4C2×D4×D511yescharacteristic
654C2×C4C2×D4×D511yescharacteristic
655C2×C4C2×Q8×D512yes—
656C2×C4C2×Q8×D512yes—
657C2×C4C2×Q8×D512yes—
658C2×C4C24×D511yescharacteristic
659C8—24no—
660C8—104no—
661C5#6049311yescharacteristic, complemented, Sylow
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—44no—
669C22—104no—
670C22—202no—
671C22—202no—
672C22—404no—
673C22C2×#404212yes—
674C22C2×#404212yes—
675C22C2×#404212yes—
676C22C22×D4×D511yescharacteristic
677C4—44no—
678C4—42no—
679C4—42no—
680C4—104no—
681C4—104no—
682C4—104no—
683C4—202no—
684C4—202no—
685C4C2×#404212yes—
686C4C2×#404212yes—
687C4C2×#404212yes—
688C4C22×#70911yescharacteristic
689C4C22×Q8×D511yescharacteristic
690C2—22no—
691C2—22no—
692C2—22no—
693C2—204no—
694C2C22×#404211yescharacteristic
695C2#1061909311yescharacteristic
696C2#1061909811yescharacteristic
subgroupG/H
centerC22C22×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#428C2
socleC22×C5C23×D4
prime radicalC22×#709C22
hypercenterC2×#428D5
hypocenterC5#60493
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#65834C2semidirect
3. C2×C5×#428C2semidirect
5. #10615481C2semidirect
6. #10613938C2semidirect
8. #10610977C2semidirect
9. #10610973C2semidirect
10. #10610972C2semidirect
11. #10610971C2semidirect
12. #10610893C2semidirect
13. #10610861C2semidirect
33. #65834C22semidirect
35. C2×#4036C22semidirect
37. C2×#4034C22semidirect
41. C5×#428C22semidirect
42. C2×C5×#107C22semidirect
43. C2×C5×#106C22semidirect
49. #65407C22semidirect
51. #65391C22semidirect
53. #65284C22semidirect
54. #65284C22semidirect
59. #65253C22semidirect
60. #65253C22semidirect
61. #65180C22semidirect
62. #65179C22semidirect
63. #65166C22semidirect
64. #65164C22semidirect
65. #65160C22semidirect
66. #65159C22semidirect
67. #65146C22semidirect
68. #65143C22semidirect
76. C2×#709Q8semidirect
177. C2×#428D5semidirect
210. #709C2×Q8semidirect
213. C2×D20C2×Q8semidirect
224. C2×Dic10C2×Q8semidirect
310. #428C2×D5semidirect
312. C2×#107C2×D5semidirect
313. C2×#106C2×D5semidirect
661. C5#60493semidirect

Maximal subgroups

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

subgroupindex
#106176322
C2×#658342
C2×C5×#4282
#106154992
#106154812
#106139382
C2×#652552
#106109772
#106109732
#106109722
#106109712
#106108932
#106108612
#604935