Finite Group Databasebeta

← #12101252

Subgroups — #12101252

Order 1280.

1686 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66156C211yesmaximal, characteristic
2C2×#66079C211yesmaximal, characteristic, complemented
3C2×C5×#417C211yesmaximal, characteristic, complemented
4C2×#65526C211yesmaximal, characteristic, complemented
5C2×#65255C211yesmaximal, characteristic, complemented
6#10602791C211yesmaximal, characteristic, complemented
7C2×#64750C211yesmaximal, characteristic, complemented
8#10601079C214yesmaximal, complemented
9#10601011C214yesmaximal, complemented
10C22×C5×#41C2211yescharacteristic
11C22×#709C2211yescharacteristic
12C22×C5×M16C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C2×#4097—21no—
15#66156—24no—
16#66156C418yescomplemented
17C2×#4096—22no—
18C2×#4095—21no—
19C2×#4088C2212yes—
20C2×#4078C2212yes—
21C22×#681C2211yescharacteristic
22C22×#681C2211yescharacteristic
23C22×#681C2211yescharacteristic
24#66079—24no—
25#66079—24no—
26C2×#4053—22no—
27C2×D5×M16—22no—
28C2×#4047C2214yes—
29C5×#417—22no—
30C5×#417—22no—
31C2×C5×#103C2212yescomplemented
32C2×C5×#102C2212yescomplemented
33#65526—24no—
34C2×#3968C2212yescomplemented
35C2×#3965C2212yescomplemented
36C2×#3942—21no—
37#65255—22no—
38#65255C2214yes—
39C2×#3941—21no—
40#64772C2214yescomplemented
41#64750C2214yescomplemented
42#21004—51nomaximal, Sylow
43C2×C5×#41—22no—
44C2×C5×#41—21no—
45C2×C5×#41—21no—
46C2×C5×#41—21no—
47C2×C5×#41—21no—
48C2×C5×#41—22no—
49C2×C5×#41—22no—
50C2×C5×#41C2×C414yes—
51C22×C5×Q8C2311yescharacteristic
52C22×C5×D4—21no—
53C22×C5×D4C2311yescharacteristic
54C23×C4×C5—21no—
55C23×C4×C5—21no—
56C23×C4×C5C2311yescharacteristic
57C22×C22⋊D5—21no—
58C2×#709—21no—
59C2×#709—22no—
60C2×#709—22no—
61C2×#709—21no—
62C2×#709C2×C414yes—
63C2×#709D414yes—
64C2×#709D412yes—
65C2×#709C2312yes—
66C22×D20D411yescharacteristic
67C22×C4×D5—21no—
68C22×Dic10D411yescharacteristic
69C2×C5×M16—22no—
70C2×C5×M16—22no—
71C2×C5×M16D412yes—
72C2×C5×M16D412yes—
73C22×C5×C8C2312yes—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4C2312yes—
80#4097—28no—
81#4096—216no—
82#4095—28no—
83C2×#689—22no—
84#4088—24no—
85#4088C2×C418yescomplemented
86C2×#688—22no—
87C2×#687—22no—
88#4078—24no—
89#4078C2×C418yescomplemented
90C2×#686—22no—
91C2×#681—21no—
92C2×#681—22no—
93C2×#681—21no—
94C2×#681—22no—
95C2×#681—21no—
96C2×#681—22no—
97C2×#681—22no—
98C2×#681C2×C414yes—
99C2×#681D414yes—
100C2×#681C2312yes—
101C2×#681C2312yes—
102C2×#681C2312yes—
103C22×#168—21no—
104C22×#168—21no—
105C22×#168—21no—
106#4053—216no—
107D5×M16—216no—
108#4047—28no—
109#4047—28no—
110C2×#676—24no—
111C2×C8×D5—24no—
112C5×#103—24no—
113C5×#102—24no—
114#3968—24no—
115#3965—24no—
116#3942—24no—
117#3941—24no—
118C2×#575—51no—
119C2×#568—51no—
120C2×#417—51no—
121C2×#417D511yescharacteristic, complemented
122C2×#413—51no—
123#1908—54no—
124#1907—54no—
125C2×#343—51no—
126#1738—51no—
127C24×C5—21no—
128C24×C5#4111yescharacteristic
129C23×D5—21no—
130C5×#41—24no—
131C5×#41—24no—
132C5×#41—24no—
133C5×#41—24no—
134C5×#41—42no—
135C5×#41—42no—
136C5×#41—42no—
137C5×#41—42no—
138C2×C5×Q8—21no—
139C2×C5×Q8—21no—
140C2×C5×Q8—21no—
141C2×C5×Q8—21no—
142C2×C5×Q8—21no—
143C2×C5×Q8C22×C412yes—
144C2×C5×D4—24no—
145C2×C5×D4—21no—
146C2×C5×D4—21no—
147C2×C5×D4—21no—
148C2×C5×D4—21no—
149C2×C5×D4—21no—
150C2×C5×D4—41no—
151C2×C5×D4—41no—
152C2×C5×D4—42no—
153C2×C5×D4C22×C412yes—
154C22×C4×C5—21no—
155C22×C4×C5—24no—
156C22×C4×C5—22no—
157C22×C4×C5—41no—
158C22×C4×C5—41no—
159C22×C4×C5—42no—
160C22×C4×C5—41no—
161C22×C4×C5—41no—
162C22×C4×C5—82no—
163C22×C4×C5C22×C412yes—
164C22×C4×C5C2×D412yes—
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D412yes—
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5#4112yes—
171C22×C4×C5#4111yescharacteristic
172C22×C4×C5C2411yescharacteristic
173C2×C22⋊D5—24no—
174C2×C22⋊D5—22no—
175C2×C22⋊D5—24no—
176C2×C22⋊D5—22no—
177C22×Dic5—21no—
178#709—24no—
179#709—24no—
180#709—24no—
181#709—24no—
182#709—24no—
183#709—22no—
184#709—22no—
185#709C22⋊C418yes—
186#709C22×C414yes—
187#709C2×D414yes—
188C2×D20—22no—
189C2×D20—21no—
190C2×D20C22⋊C414yes—
191C2×D20C2×D412yes—
192C2×C4×D5—24no—
193C2×C4×D5—24no—
194C2×C4×D5—22no—
195C2×C4×D5—22no—
196C2×Dic10—22no—
197C2×Dic10—21no—
198C2×Dic10C22⋊C414yes—
199C2×Dic10C2×D412yes—
200C5×M16—24no—
201C5×M16—24no—
202C2×C5×C8—22no—
203C2×C5×C8—22no—
204C2×C5×C8C2×D414yes—
205C5×C4⋊C4—22no—
206C5×C4⋊C4—22no—
207C5×C4⋊C4—41no—
208C5×C4⋊C4—41no—
209#689—28no—
210#688—28no—
211#687—28no—
212#686—28no—
213#681—24no—
214#681—24no—
215#681—22no—
216#681—22no—
217#681—22no—
218#681—22no—
219#681—42no—
220#681C22×C414yes—
221#681C2×D414yes—
222C2×#168—24no—
223C2×#168—24no—
224C2×#168—22no—
225C2×#168—22no—
226C2×#168—22no—
227C2×#168—42no—
228#676—216no—
229C8×D5—216no—
230C22×#41—51no—
231C22×#41C2×D511yescharacteristic
232#575—58no—
233#575—104no—
234C2×#137—52no—
235C2×#137—101no—
236C2×AGL(1,ℤ/8)—52no—
237C2×AGL(1,ℤ/8)—101no—
238C2×#135—102no—
239#568—104no—
240#568—104no—
241C2×#131—54no—
242C2×#131—102no—
243C22×M16—51no—
244C22×M16—51no—
245C22×M16—51no—
246C22×M16—102no—
247C22×M16C2×D511yescharacteristic
248C22×C4⋊C4C2×D511yescharacteristic
249#417—22no—
250#417—22no—
251#417—54no—
252#417—102no—
253C2×#103—101no—
254C2×#103C2×D512yescomplemented
255C2×#102—101no—
256C2×#102C2×D512yescomplemented
257#413—104no—
258C2×#101—52no—
259C2×#100—52no—
260#343—54no—
261#341—54no—
262C23×C5—22no—
263C23×C5—41no—
264C23×C5—41no—
265C23×C5—41no—
266C23×C5—41no—
267C23×C5—81no—
268C23×C5—81no—
269C23×C5C4×D412yes—
270C23×C5#12111yescharacteristic
271C23×C5#12111yescharacteristic
272C23×C5#12312yes—
273C23×C5C2×#4111yescharacteristic
274C22×D5—22no—
275C22×D5—24no—
276C22×D5—81no—
277C22×D5—81no—
278C5×Q8—22no—
279C5×Q8—22no—
280C5×Q8—41no—
281C5×Q8—41no—
282C5×D4—22no—
283C5×D4—22no—
284C5×D4—44no—
285C5×D4—44no—
286C5×D4—41no—
287C5×D4—41no—
288C2×C4×C5—22no—
289C2×C4×C5—41no—
290C2×C4×C5—41no—
291C2×C4×C5—44no—
292C2×C4×C5—44no—
293C2×C4×C5—42no—
294C2×C4×C5—82no—
295C2×C4×C5—82no—
296C2×C4×C5—82no—
297C2×C4×C5C2×C22⋊C412yes—
298C2×C4×C5C2×C22⋊C412yes—
299C2×C4×C5C4×D412yes—
300C2×C4×C5C4×D412yes—
301C2×C4×C5C4×D412yes—
302C2×C4×C5#12012yes—
303C2×C4×C5#12012yes—
304C2×C4×C5#12112yes—
305C2×C4×C5#12112yes—
306C2×C4×C5#12111yescharacteristic
307C2×C4×C5#12111yescharacteristic
308C2×C4×C5#12312yes—
309C2×C4×C5C23×C411yescharacteristic
310C2×C4×C5C22×D411yescharacteristic
311C2×C4×C5C22×D411yescharacteristic
312C2×C4×C5C22×D411yescharacteristic
313C2×C4×C5C22×D411yescharacteristic
314C2×C4×C5C2×#4111yescharacteristic
315C22⋊D5—24no—
316C22⋊D5—24no—
317C22⋊D5—28no—
318C2×Dic5—22no—
319C2×Dic5—24no—
320D20—22no—
321D20—24no—
322D20C2×C22⋊C414yes—
323C4×D5—24no—
324C4×D5—24no—
325C4×D5—28no—
326Dic10—22no—
327Dic10—24no—
328Dic10C2×C22⋊C414yes—
329C5×C8—24no—
330#168—24no—
331#168—24no—
332#168—42no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41—21no—
336C2×#41—21no—
337C2×#41—22no—
338C2×#41—21no—
339C2×#41—21no—
340C2×#41—54no—
341C2×#41—54no—
342C2×#41—52no—
343C2×#41—52no—
344C2×#41—102no—
345C2×#41—102no—
346C2×#41—101no—
347C2×#41—101no—
348C2×#41C4×D514yes—
349C22×Q8—51no—
350C22×Q8C22×D511yescharacteristic
351C22×D4—21no—
352C22×D4—51no—
353C22×D4—101no—
354C22×D4C22×D511yescharacteristic
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—101no—
358C23×C4C22×D511yescharacteristic
359#137—58no—
360#137—104no—
361#137—108no—
362AGL(1,ℤ/8)—58no—
363AGL(1,ℤ/8)—104no—
364AGL(1,ℤ/8)—108no—
365#135—1016no—
366C2×Q16—102no—
367C2×SD16—102no—
368C2×SD16—102no—
369C2×D8—102no—
370#131—108no—
371#131—108no—
372#131—1016no—
373C2×M16—22no—
374C2×M16—22no—
375C2×M16—54no—
376C2×M16—54no—
377C2×M16—52no—
378C2×M16—52no—
379C2×M16—52no—
380C2×M16—1016no—
381C2×M16—102no—
382C2×M16—101no—
383C2×M16—102no—
384C2×M16—101no—
385C2×M16—101no—
386C2×M16—104no—
387C2×M16—102no—
388C2×M16—102no—
389C2×M16C22⋊D512yes—
390C2×M16C22⋊D512yes—
391C22×C8—101no—
392C22×C8—101no—
393C22×C8—104no—
394C22×C8—101no—
395C22×C8C22×D512yes—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4C22×D512yes—
402#103—24no—
403#103—104no—
404#102—24no—
405#102—104no—
406#101—104no—
407#100—104no—
408C22×C5—21no—
409C22×C5—21no—
410C22×C5—21no—
411C22×C5—21no—
412C22×C5—21no—
413C22×C5—21no—
414C22×C5—42no—
415C22×C5—42no—
416C22×C5—42no—
417C22×C5—82no—
418C22×C5—82no—
419C22×C5—81no—
420C22×C5—81no—
421C22×C5#38612yes—
422C22×C5#38612yes—
423C22×C5C2×C4×D411yescharacteristic
424C22×C5C2×#12111yescharacteristic
425C22×C5C2×#12311yescharacteristic
426C2×D5—24no—
427C2×D5—42no—
428C2×D5—42no—
429C2×D5—84no—
430C4×C5—42no—
431C4×C5—82no—
432C4×C5#38612yes—
433C4×C5#38612yes—
434C4×C5C22×C22⋊C411yescharacteristic
435C4×C5C2×C4×D411yescharacteristic
436C4×C5C2×#12011yescharacteristic
437C4×C5C2×#12111yescharacteristic
438Dic5—24no—
439C24—21no—
440C24—101no—
441C24#71111yescharacteristic
442#41—24no—
443#41—24no—
444#41—24no—
445#41—24no—
446#41—42no—
447#41—42no—
448#41—42no—
449#41—42no—
450#41—54no—
451#41—54no—
452#41—58no—
453#41—102no—
454#41—102no—
455#41—104no—
456#41—104no—
457#41—104no—
458#41—104no—
459#41—104no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—52no—
466C2×Q8—54no—
467C2×Q8—102no—
468C2×Q8—101no—
469C2×Q8C2×C4×D512yes—
470C2×D4—24no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—42no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—52no—
480C2×D4—54no—
481C2×D4—102no—
482C2×D4—104no—
483C2×D4—104no—
484C2×D4—102no—
485C2×D4—102no—
486C2×D4—101no—
487C2×D4C2×C4×D512yes—
488C22×C4—21no—
489C22×C4—24no—
490C22×C4—22no—
491C22×C4—42no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—82no—
497C22×C4—101no—
498C22×C4—102no—
499C22×C4—104no—
500C22×C4—104no—
501C22×C4—102no—
502C22×C4C2×C4×D512yes—
503C22×C4#70912yes—
504C22×C4D4×D512yes—
505C22×C4D4×D511yescharacteristic
506C22×C4D4×D511yescharacteristic
507C22×C4D4×D512yes—
508C22×C4#71111yescharacteristic
509C22×C4C2×C22⋊D511yescharacteristic
510C22×C4C2×C22⋊D511yescharacteristic
511C22×C4C23×D511yescharacteristic
512Q16—108no—
513SD16—108no—
514SD16—108no—
515D8—108no—
516M16—24no—
517M16—24no—
518M16—54no—
519M16—54no—
520M16—1016no—
521M16—102no—
522M16—102no—
523M16—104no—
524M16—104no—
525M16—102no—
526M16—102no—
527M16—202no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—1016no—
531C2×C8—102no—
532C2×C8—102no—
533C2×C8—104no—
534C2×C8—104no—
535C2×C8—102no—
536C2×C8—202no—
537C2×C8C2×C22⋊D514yes—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—41no—
541C4⋊C4—41no—
542C2×C5—22no—
543C2×C5—21no—
544C2×C5—21no—
545C2×C5—22no—
546C2×C5—82no—
547C2×C5C2×#38611yescharacteristic
548C2×C5#306811yescharacteristic
549C2×C5#306911yescharacteristic
550D5—44no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—81no—
557C23—81no—
558C23—102no—
559C23—104no—
560C23—401no—
561C23—401no—
562C23#402712yes—
563C23#402912yes—
564C23#408411yescharacteristic
565C23#408411yescharacteristic
566C23C2×#71111yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—41no—
570Q8—41no—
571Q8—54no—
572Q8—102no—
573Q8—104no—
574D4—22no—
575D4—22no—
576D4—44no—
577D4—44no—
578D4—41no—
579D4—41no—
580D4—54no—
581D4—104no—
582D4—104no—
583D4—108no—
584D4—102no—
585D4—104no—
586C2×C4—22no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—102no—
596C2×C4—104no—
597C2×C4—104no—
598C2×C4—104no—
599C2×C4—108no—
600C2×C4D5×C22⋊C412yes—
601C2×C4D5×C22⋊C412yes—
602C2×C4#402712yes—
603C2×C4#402912yes—
604C2×C4#403012yes—
605C2×C4#403012yes—
606C2×C4C4×C22⋊D512yes—
607C2×C4C4×C22⋊D512yes—
608C2×C4#408312yes—
609C2×C4#408312yes—
610C2×C4#408411yescharacteristic
611C2×C4#408411yescharacteristic
612C2×C4C22×C4×D511yescharacteristic
613C2×C4C2×#70911yescharacteristic
614C2×C4C2×D4×D511yescharacteristic
615C2×C4C2×D4×D511yescharacteristic
616C2×C4C2×D4×D511yescharacteristic
617C2×C4C22×C22⋊D511yescharacteristic
618C8—24no—
619C8—104no—
620C8—104no—
621C8—202no—
622C5#2100411yescharacteristic, complemented, Sylow
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—82no—
633C22—82no—
634C22—81no—
635C22—81no—
636C22—104no—
637C22—202no—
638C22—202no—
639C22—404no—
640C22#6524512yes—
641C22#6524512yes—
642C22C2×#402711yescharacteristic
643C22C2×#402911yescharacteristic
644C22C2×#408411yescharacteristic
645C4—42no—
646C4—82no—
647C4—104no—
648C4#6524512yes—
649C4#6524512yes—
650C4C2×D5×C22⋊C411yescharacteristic
651C4C2×#403011yescharacteristic
652C4C2×C4×C22⋊D511yescharacteristic
653C4C2×#408311yescharacteristic
654C2—21no—
655C2—21no—
656C2—22no—
657C2—22no—
658C2—82no—
659C2—204no—
660C2C2×#6524511yescharacteristic
661C2#1061569411yescharacteristic
662C2#1061572911yescharacteristic
subgroupG/H
centerC22C2×#4084
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#417C2
socleC22×C5C2×#121
prime radicalC2×#66156C2
cosocleC22×C4×C5C24
hypercenterC2×#417D5
hypocenterC5#21004
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×#66079C2semidirect
3. C2×C5×#417C2semidirect
4. C2×#65526C2semidirect
5. C2×#65255C2semidirect
6. #10602791C2semidirect
7. C2×#64750C2semidirect
8. #10601079C2semidirect
9. #10601011C2semidirect
16. #66156C4semidirect
31. C2×C5×#103C22semidirect
32. C2×C5×#102C22semidirect
34. C2×#3968C22semidirect
35. C2×#3965C22semidirect
40. #64772C22semidirect
41. #64750C22semidirect
85. #4088C2×C4semidirect
89. #4078C2×C4semidirect
121. C2×#417D5semidirect
254. C2×#103C2×D5semidirect
256. C2×#102C2×D5semidirect
622. C5#21004semidirect

Maximal subgroups

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

subgroupindex
C2×#661562
C2×#660792
C2×C5×#4172
C2×#655262
C2×#652552
#106027912
C2×#647502
#106010792
#106010112
#210045