Finite Group Databasebeta

← #12100690

Subgroups — #12100690

Order 1280.

2046 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4078C211yesmaximal, characteristic
2C2×C4×#709C211yesmaximal, characteristic, complemented
3C2×C5×#390C211yesmaximal, characteristic, complemented
4C2×#65504C211yesmaximal, characteristic, complemented
5C2×#65255C211yesmaximal, characteristic, complemented
6#10602351C211yesmaximal, characteristic, complemented
7#10602320C211yesmaximal, characteristic, complemented
8#10600453C214yesmaximal, complemented
9#10600450C214yesmaximal, complemented
10C23×C5×D4C2211yescharacteristic
11C22×#709C2211yescharacteristic
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C5×C22⋊C4—22no—
14C22×C42×C5C2211yescharacteristic
15C22×#687C2212yes—
16C2×#4078—24no—
17C2×#4078—24no—
18C2×#4078C418yescomplemented
19C22×#686C2212yes—
20C2×C4×C22⋊D5—22no—
21C2×#4072C2211yescharacteristic
22C22×#681C2211yescharacteristic
23C22×#681C2211yescharacteristic
24C4×#709—24no—
25C4×#709—24no—
26C2×C4×D20C2212yescomplemented
27C2×#4018—21no—
28C2×C42×D5—21no—
29C2×C4×Dic10C2212yescomplemented
30C5×#390—21no—
31C5×#390—21no—
32C5×#390C2214yes—
33#65504—24no—
34C2×#3964C2214yes—
35C2×#3942C2212yescomplemented
36#65255—24no—
37C2×#3941C2212yescomplemented
38#64749C2214yescomplemented
39#64744C2214yescomplemented
40#20928—51nomaximal, Sylow
41C25×C5—21no—
42C22×C5×D4—22no—
43C22×C5×D4—22no—
44C22×C5×D4—22no—
45C22×C5×D4—21no—
46C22×C5×D4—22no—
47C22×C5×D4—22no—
48C22×C5×D4C2×C414yescomplemented
49C22×C5×D4C2312yes—
50C23×C4×C5—21no—
51C23×C4×C5D412yes—
52C23×C4×C5C2311yescharacteristic
53C22×C22⋊D5—21no—
54C2×#709—22no—
55C2×#709—22no—
56C2×#709—22no—
57C2×#709C2×C414yes—
58C2×#709D414yes—
59C2×#709D414yes—
60C22×D20C2311yescharacteristic
61C22×C4×D5—21no—
62C22×C4×D5—22no—
63C22×Dic10C2311yescharacteristic
64C2×C5×C4⋊C4—22no—
65C2×C5×C4⋊C4—22no—
66C2×C5×C4⋊C4—21no—
67C2×C5×C4⋊C4C2312yes—
68C2×C5×C22⋊C4—24no—
69C2×C5×C22⋊C4—24no—
70C2×C5×C22⋊C4—48no—
71C2×C42×C5—22no—
72C2×C42×C5—22no—
73C2×C42×C5D411yescharacteristic
74C2×C42×C5D411yescharacteristic
75C2×C42×C5C2312yes—
76C2×C22⋊Dic5—21no—
77C2×#687—24no—
78C2×#687—24no—
79C2×#687C2×C418yescomplemented
80#4078—232no—
81C2×#686—24no—
82C2×#686—24no—
83C2×#686C2×C418yescomplemented
84C4×C22⋊D5—216no—
85C2×#685—22no—
86#4072—22no—
87#4072D414yes—
88C2×C4⋊Dic5C2312yes—
89C2×#683—22no—
90C2×C4×Dic5—22no—
91C2×C4×Dic5—21no—
92C2×#681—22no—
93C2×#681—22no—
94C2×#681—22no—
95C2×#681—22no—
96C2×#681—22no—
97C2×#681C2×C414yescomplemented
98C22×#168C2312yes—
99C22×#168C2312yes—
100C4×D20—24no—
101C4×D20—24no—
102#4018—28no—
103C42×D5—28no—
104C4×Dic10—24no—
105C4×Dic10—24no—
106#3964—28no—
107#3942—24no—
108#3941—24no—
109C22×AGL(1,ℤ/8)—51no—
110C2×C4×#41—51no—
111C2×#417—51no—
112C2×#418—51no—
113C2×#390D511yescharacteristic, complemented
114#1880—54no—
115#1879—54no—
116#1736—51no—
117#1728—51no—
118C24×C5—24no—
119C24×C5—22no—
120C24×C5—44no—
121C24×C5—44no—
122C24×C5—42no—
123C24×C5—42no—
124C24×C5C2×D411yescharacteristic
125C23×D5—21no—
126C2×C5×D4—28no—
127C2×C5×D4—28no—
128C2×C5×D4—24no—
129C2×C5×D4—22no—
130C2×C5×D4—22no—
131C2×C5×D4—22no—
132C2×C5×D4—22no—
133C2×C5×D4—22no—
134C2×C5×D4—42no—
135C2×C5×D4—44no—
136C2×C5×D4—42no—
137C2×C5×D4—44no—
138C2×C5×D4C22×C414yes—
139C22×C4×C5—24no—
140C22×C4×C5—24no—
141C22×C4×C5—22no—
142C22×C4×C5—42no—
143C22×C4×C5—44no—
144C22×C4×C5—42no—
145C22×C4×C5C22×C412yes—
146C22×C4×C5C2×D412yes—
147C22×C4×C5C2×D412yes—
148C22×C4×C5C2×D412yes—
149C22×C4×C5C2×D412yes—
150C22×C4×C5C2×D412yes—
151C22×C4×C5C2×D411yescharacteristic
152C22×C4×C5#4112yes—
153C22×C4×C5#4112yes—
154C22×C4×C5C2411yescharacteristic
155C2×C22⋊D5—24no—
156C2×C22⋊D5—24no—
157C2×C22⋊D5—24no—
158C22×Dic5—21no—
159C22×Dic5—22no—
160#709—28no—
161#709—28no—
162#709—28no—
163#709C22⋊C4116yescomplemented
164C2×D20—21no—
165C2×D20—21no—
166C2×D20—21no—
167C2×D20C22×C412yes—
168C2×D20C2×D412yes—
169C2×D20C2×D412yes—
170C2×C4×D5—24no—
171C2×C4×D5—24no—
172C2×C4×D5—24no—
173C2×C4×D5—28no—
174C2×C4×D5—44no—
175C2×C4×D5—44no—
176C2×Dic10—21no—
177C2×Dic10—21no—
178C2×Dic10—21no—
179C2×Dic10C22×C412yes—
180C2×Dic10C2×D412yes—
181C2×Dic10C2×D412yes—
182C5×C4⋊C4—22no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—42no—
185C5×C22⋊C4—48no—
186C5×C22⋊C4—48no—
187C42×C5—22no—
188C42×C5—22no—
189C42×C5—22no—
190C42×C5—22no—
191C22⋊Dic5—24no—
192#687—216no—
193#686—216no—
194#685—28no—
195C4⋊Dic5—22no—
196C4⋊Dic5C2×D414yes—
197#683—28no—
198C4×Dic5—28no—
199C4×Dic5—24no—
200#681—28no—
201#681—44no—
202C2×#168—22no—
203C2×#168—22no—
204C2×#168—22no—
205C2×#168—22no—
206C2×#168—22no—
207C2×#168C22×C414yes—
208C22×#41—51no—
209C23×D4C2×D511yescharacteristic
210C2×AGL(1,ℤ/8)—58no—
211C2×AGL(1,ℤ/8)—104no—
212C2×AGL(1,ℤ/8)—104no—
213C22×SD16—52no—
214C22×D8—52no—
215C22×M16—51no—
216C22×M16—51no—
217C4×#41—104no—
218C4×#41—104no—
219C2×C4×Q8—52no—
220C2×C4×D4—52no—
221C2×C4×D4—102no—
222C2×#117—51no—
223C2×#117—101no—
224C22×C4⋊C4C2×D511yescharacteristic
225C22×C22⋊C4—22no—
226C22×C42—101no—
227C22×C42C2×D511yescharacteristic
228#418—104no—
229#417—104no—
230C2×#103—52no—
231C2×#102—54no—
232C2×#102—52no—
233#390—21no—
234#390—21no—
235#390C2×D514yes—
236#340—54no—
237#336—54no—
238C23×C5—22no—
239C23×C5—22no—
240C23×C5—24no—
241C23×C5—416no—
242C23×C5—42no—
243C23×C5—42no—
244C23×C5—44no—
245C23×C5—44no—
246C23×C5—48no—
247C23×C5—88no—
248C23×C5—84no—
249C23×C5—84no—
250C23×C5C4×D412yes—
251C23×C5#12012yes—
252C23×C5#12112yes—
253C23×C5C22×D411yescharacteristic
254C22×D5—22no—
255C22×D5—22no—
256C22×D5—22no—
257C22×D5—42no—
258C22×D5—42no—
259C5×D4—24no—
260C5×D4—24no—
261C5×D4—42no—
262C5×D4—42no—
263C5×D4—48no—
264C2×C4×C5—22no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—48no—
270C2×C4×C5—42no—
271C2×C4×C5—44no—
272C2×C4×C5—84no—
273C2×C4×C5C2×C22⋊C414yes—
274C2×C4×C5C4×D414yes—
275C2×C4×C5C4×D412yes—
276C2×C4×C5#12012yes—
277C2×C4×C5#12114yes—
278C2×C4×C5#12112yes—
279C2×C4×C5#12314yes—
280C2×C4×C5AGL(1,ℤ/8)14yescomplemented
281C2×C4×C5C23×C411yescharacteristic
282C2×C4×C5C22×D411yescharacteristic
283C2×C4×C5C22×D411yescharacteristic
284C2×C4×C5C22×D411yescharacteristic
285C2×C4×C5C2×#4111yescharacteristic
286C2×C4×C5C2×#4111yescharacteristic
287C22⋊D5—216no—
288C2×Dic5—22no—
289C2×Dic5—22no—
290C2×Dic5—22no—
291C2×Dic5—24no—
292C2×Dic5—42no—
293C2×Dic5—42no—
294D20—22no—
295D20—22no—
296D20—22no—
297D20C2×C22⋊C414yescomplemented
298C4×D5—216no—
299C4×D5—416no—
300Dic10—22no—
301Dic10—22no—
302Dic10—22no—
303Dic10C2×C22⋊C414yescomplemented
304#168—24no—
305#168—42no—
306C25—21no—
307C2×#41—54no—
308C2×#41—54no—
309C2×#41—54no—
310C2×#41—102no—
311C2×#41—102no—
312C2×#41—102no—
313C22×Q8—51no—
314C22×D4—22no—
315C22×D4—22no—
316C22×D4—22no—
317C22×D4—22no—
318C22×D4—22no—
319C22×D4—21no—
320C22×D4—51no—
321C22×D4—101no—
322C22×D4C4×D514yescomplemented
323C22×D4C22×D512yes—
324C23×C4—21no—
325C23×C4—101no—
326C23×C4—102no—
327C23×C4C22⋊D512yes—
328C23×C4C22×D511yescharacteristic
329AGL(1,ℤ/8)—1032no—
330C2×SD16—58no—
331C2×SD16—104no—
332C2×SD16—104no—
333C2×D8—58no—
334C2×D8—104no—
335C2×D8—104no—
336C2×M16—54no—
337C2×M16—102no—
338C2×M16—102no—
339C2×M16—102no—
340C2×M16—102no—
341C2×M16—102no—
342C22×C8—52no—
343C22×C8—52no—
344C4×Q8—104no—
345C4×Q8—104no—
346C4×D4—104no—
347C4×D4—104no—
348C4×D4—1016no—
349#117—54no—
350#117—108no—
351#117—102no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—52no—
356C2×C4⋊C4—102no—
357C2×C4⋊C4C22×D512yes—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—48no—
361C2×C22⋊C4—101no—
362C2×C22⋊C4—102no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—108no—
366C2×C42—101no—
367C2×C42—102no—
368C2×C42C22⋊D511yescharacteristic
369C2×C42C22⋊D511yescharacteristic
370C2×C42C22×D512yes—
371#103—104no—
372#102—104no—
373#102—108no—
374C22×C5—24no—
375C22×C5—24no—
376C22×C5—22no—
377C22×C5—22no—
378C22×C5—48no—
379C22×C5—44no—
380C22×C5—816no—
381C22×C5—84no—
382C22×C5—82no—
383C22×C5—82no—
384C22×C5#38614yes—
385C22×C5C2×C4×D411yescharacteristic
386C22×C5C2×#12011yescharacteristic
387C22×C5C2×#12111yescharacteristic
388C22×C5C2×AGL(1,ℤ/8)12yes—
389C22×C5#57512yes—
390C2×D5—24no—
391C2×D5—44no—
392C2×D5—44no—
393C2×D5—42no—
394C2×D5—42no—
395C4×C5—24no—
396C4×C5—42no—
397C4×C5—82no—
398C4×C5#38614yes—
399C4×C5C22×C22⋊C411yescharacteristic
400C4×C5C2×C4×D411yescharacteristic
401C4×C5C2×#12111yescharacteristic
402C4×C5C2×#12311yescharacteristic
403Dic5—24no—
404Dic5—44no—
405C24—22no—
406C24—24no—
407C24—44no—
408C24—44no—
409C24—42no—
410C24—42no—
411C24—101no—
412C24C2×C22⋊D511yescharacteristic
413#41—516no—
414#41—108no—
415#41—108no—
416#41—108no—
417C2×Q8—52no—
418C2×Q8—52no—
419C2×Q8—52no—
420C2×Q8—101no—
421C2×Q8—101no—
422C2×Q8—101no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—28no—
429C2×D4—28no—
430C2×D4—24no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—52no—
436C2×D4—52no—
437C2×D4—52no—
438C2×D4—104no—
439C2×D4—104no—
440C2×D4—104no—
441C2×D4—101no—
442C2×D4—101no—
443C2×D4—101no—
444C2×D4C2×C4×D514yes—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—108no—
452C22×C4—101no—
453C22×C4—104no—
454C22×C4—104no—
455C22×C4—104no—
456C22×C4—102no—
457C22×C4—204no—
458C22×C4—204no—
459C22×C4C2×C4×D512yes—
460C22×C4#70912yes—
461C22×C4D4×D512yes—
462C22×C4D4×D512yes—
463C22×C4D4×D512yes—
464C22×C4#71112yes—
465C22×C4C2×C22⋊D512yes—
466C22×C4C2×C22⋊D512yes—
467C22×C4C2×C22⋊D511yescharacteristic
468C22×C4C23×D511yescharacteristic
469SD16—1016no—
470D8—1016no—
471M16—108no—
472M16—204no—
473C2×C8—54no—
474C2×C8—102no—
475C2×C8—102no—
476C2×C8—102no—
477C2×C8—102no—
478C2×C8—102no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—42no—
482C4⋊C4—54no—
483C4⋊C4—108no—
484C4⋊C4—102no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—108no—
488C22⋊C4—104no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—108no—
494C42—104no—
495C2×C5—24no—
496C2×C5—84no—
497C2×C5C2×#38611yescharacteristic
498C2×C5C4×AGL(1,ℤ/8)12yes—
499C2×C5#300612yes—
500C2×C5#303811yescharacteristic
501C2×C5#304011yescharacteristic
502D5—44no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—416no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—48no—
512C23—88no—
513C23—84no—
514C23—84no—
515C23—102no—
516C23—102no—
517C23—102no—
518C23—202no—
519C23—202no—
520C23C4×C22⋊D512yes—
521C23#408312yes—
522C23#408412yes—
523C23C22×C22⋊D511yescharacteristic
524Q8—54no—
525Q8—102no—
526Q8—102no—
527Q8—102no—
528D4—24no—
529D4—24no—
530D4—42no—
531D4—42no—
532D4—48no—
533D4—54no—
534D4—1016no—
535D4—102no—
536D4—102no—
537D4—102no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—84no—
547C2×C4—102no—
548C2×C4—102no—
549C2×C4—102no—
550C2×C4—1016no—
551C2×C4—104no—
552C2×C4—2016no—
553C2×C4—202no—
554C2×C4—202no—
555C2×C4D5×C22⋊C414yes—
556C2×C4#402714yes—
557C2×C4#402914yes—
558C2×C4#403014yes—
559C2×C4C4×C22⋊D512yes—
560C2×C4#407814yescomplemented
561C2×C4#408312yes—
562C2×C4#408412yes—
563C2×C4C22×C4×D511yescharacteristic
564C2×C4C2×#70911yescharacteristic
565C2×C4C2×D4×D511yescharacteristic
566C2×C4C2×D4×D511yescharacteristic
567C2×C4C2×D4×D511yescharacteristic
568C2×C4C2×#71111yescharacteristic
569C8—104no—
570C8—202no—
571C5#2092811yescharacteristic, complemented, Sylow
572C22—24no—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—48no—
577C22—44no—
578C22—816no—
579C22—84no—
580C22—82no—
581C22—82no—
582C22—104no—
583C22—204no—
584C22—204no—
585C22—202no—
586C22—202no—
587C22#6524514yes—
588C22C2×C4×C22⋊D511yescharacteristic
589C22C2×#407812yes—
590C22C2×#408311yescharacteristic
591C22C2×#408411yescharacteristic
592C22#6615612yes—
593C4—24no—
594C4—42no—
595C4—82no—
596C4—104no—
597C4—204no—
598C4#6524514yes—
599C4C2×D5×C22⋊C411yescharacteristic
600C4C2×#402711yescharacteristic
601C4C2×#402911yescharacteristic
602C4C2×#403011yescharacteristic
603C2—24no—
604C2—84no—
605C2—204no—
606C2C2×#6524511yescharacteristic
607C2C4×#407812yes—
608C2#1061435512yes—
609C2#1061467611yescharacteristic
610C2#1061489611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#390C2
socleC23×C5C22×D4
prime radicalC22×#4078C2
cosocleC22×C4×C5C24
hypercenterC2×#390D5
hypocenterC5#20928
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×C4×#709C2semidirect
3. C2×C5×#390C2semidirect
4. C2×#65504C2semidirect
5. C2×#65255C2semidirect
6. #10602351C2semidirect
7. #10602320C2semidirect
8. #10600453C2semidirect
9. #10600450C2semidirect
18. C2×#4078C4semidirect
26. C2×C4×D20C22semidirect
29. C2×C4×Dic10C22semidirect
35. C2×#3942C22semidirect
37. C2×#3941C22semidirect
38. #64749C22semidirect
39. #64744C22semidirect
48. C22×C5×D4C2×C4semidirect
79. C2×#687C2×C4semidirect
83. C2×#686C2×C4semidirect
97. C2×#681C2×C4semidirect
113. C2×#390D5semidirect
163. #709C22⋊C4semidirect
280. C2×C4×C5AGL(1,ℤ/8)semidirect
297. D20C2×C22⋊C4semidirect
303. Dic10C2×C22⋊C4semidirect
322. C22×D4C4×D5semidirect
560. C2×C4#4078semidirect
571. C5#20928semidirect

Maximal subgroups

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

subgroupindex
C22×#40782
C2×C4×#7092
C2×C5×#3902
C2×#655042
C2×#652552
#106023512
#106023202
#106004532
#106004502
#209285