Finite Group Databasebeta

← #12100719

Subgroups — #12100719

Order 1280.

1926 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4078C211yesmaximal, characteristic
2C2×#65838C211yesmaximal, characteristic, complemented
3C22×C5×#102C211yesmaximal, characteristic, complemented
4#10615775C211yesmaximal, characteristic, complemented
5C2×#65402C211yesmaximal, characteristic, complemented
6#10602758C211yesmaximal, characteristic, complemented
7#10602351C211yesmaximal, characteristic, complemented
8#10601006C214yesmaximal, complemented
9#10601001C214yesmaximal, complemented
10C23×C5×D4C2211yescharacteristic
11C22×#709C2211yescharacteristic
12C23×C5×C8C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×#687C2212yes—
15C2×#4078—24no—
16C2×#4078—24no—
17C2×#4078C418yescomplemented
18C22×#686C2212yes—
19C2×C4×C22⋊D5—22no—
20C2×#4072C2211yescharacteristic
21C2×#4072C2211yescharacteristic
22C22×#681C2211yescharacteristic
23#65838—24no—
24#65838—24no—
25C2×#4039C2212yescomplemented
26C2×#4038—21no—
27C2×D5×C4⋊C4—21no—
28C2×#4033C2212yescomplemented
29C2×C5×#102—24no—
30C2×C5×#102C2214yes—
31#65536—22no—
32#65510—21no—
33#65510—21no—
34#65510C2214yes—
35#65402—24no—
36C2×#3953C2212yescomplemented
37C2×#3948C2212yescomplemented
38#64770C2214yescomplemented
39#64749C2214yescomplemented
40#20932—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×C5C2311yescharacteristic
52C22×C22⋊D5—21no—
53C2×#709—22no—
54C2×#709—22no—
55C2×#709—22no—
56C2×#709C2×C414yes—
57C2×#709D414yes—
58C2×#709D414yes—
59C22×D20C2311yescharacteristic
60C22×C4×D5—21no—
61C22×C4×D5—22no—
62C22×Dic10C2311yescharacteristic
63C22×C5×C8—22no—
64C22×C5×C8—22no—
65C22×C5×C8—22no—
66C22×C5×C8C2312yes—
67C2×C5×C4⋊C4—21no—
68C2×C5×C4⋊C4—22no—
69C2×C5×C4⋊C4D414yes—
70C2×C5×C4⋊C4C2312yes—
71C2×C22⋊Dic5—21no—
72C2×C22⋊Dic5—24no—
73C2×C22⋊Dic5—24no—
74C2×C22⋊Dic5D412yes—
75C2×#687—24no—
76C2×#687—24no—
77C2×#687C2×C418yescomplemented
78#4078—232no—
79C2×#686—24no—
80C2×#686—24no—
81C2×#686C2×C418yescomplemented
82C4×C22⋊D5—216no—
83C2×#685—22no—
84#4072—22no—
85#4072—22no—
86#4072—22no—
87#4072—22no—
88C2×C4⋊Dic5D411yescharacteristic
89C2×C4⋊Dic5D411yescharacteristic
90C2×C4⋊Dic5C2312yes—
91C2×#683—22no—
92C2×C4×Dic5—22no—
93C2×C4×Dic5—21no—
94C2×C4×Dic5C2312yes—
95C2×#681—22no—
96C2×#681—22no—
97C2×#681C2×C414yescomplemented
98C22×#168C2312yes—
99#4039—24no—
100#4039—24no—
101#4038—28no—
102D5×C4⋊C4—28no—
103#4033—24no—
104#4033—24no—
105C5×#102—28no—
106#3953—24no—
107#3948—24no—
108C22×AGL(1,ℤ/8)—51no—
109C2×#520—51no—
110C2×#416—51no—
111C22×#102D511yescharacteristic, complemented
112#2309—51no—
113#1880—54no—
114#1879—54no—
115#1736—51no—
116#1729—51no—
117C24×C5—22no—
118C24×C5—24no—
119C24×C5—42no—
120C24×C5—42no—
121C24×C5—82no—
122C24×C5—82no—
123C24×C5C2×D411yescharacteristic
124C23×D5—21no—
125C2×C5×D4—28no—
126C2×C5×D4—28no—
127C2×C5×D4—24no—
128C2×C5×D4—22no—
129C2×C5×D4—22no—
130C2×C5×D4—22no—
131C2×C5×D4—22no—
132C2×C5×D4—22no—
133C2×C5×D4—42no—
134C2×C5×D4—42no—
135C2×C5×D4—44no—
136C2×C5×D4—44no—
137C2×C5×D4C22×C414yes—
138C22×C4×C5—22no—
139C22×C4×C5—24no—
140C22×C4×C5—82no—
141C22×C4×C5C22×C412yes—
142C22×C4×C5C2×D412yes—
143C22×C4×C5C2×D412yes—
144C22×C4×C5C2×D411yescharacteristic
145C22×C4×C5C2×D412yes—
146C22×C4×C5#4112yes—
147C22×C4×C5#4112yes—
148C22×C4×C5C2411yescharacteristic
149C2×C22⋊D5—24no—
150C2×C22⋊D5—24no—
151C2×C22⋊D5—24no—
152C22×Dic5—21no—
153C22×Dic5—22no—
154C22×Dic5C2×D412yes—
155C22×Dic5C2×D412yes—
156#709—28no—
157#709—28no—
158#709—28no—
159#709C22⋊C4116yescomplemented
160C2×D20—21no—
161C2×D20—21no—
162C2×D20—21no—
163C2×D20C22×C412yes—
164C2×D20C2×D412yes—
165C2×D20C2×D412yes—
166C2×C4×D5—24no—
167C2×C4×D5—24no—
168C2×C4×D5—24no—
169C2×C4×D5—28no—
170C2×C4×D5—84no—
171C2×Dic10—21no—
172C2×Dic10—21no—
173C2×Dic10—21no—
174C2×Dic10C22×C412yes—
175C2×Dic10C2×D412yes—
176C2×Dic10C2×D412yes—
177C2×C5×C8—22no—
178C2×C5×C8—22no—
179C2×C5×C8—22no—
180C2×C5×C8—44no—
181C5×C4⋊C4—24no—
182C5×C4⋊C4—22no—
183C5×C4⋊C4C2×D414yes—
184C22⋊Dic5—24no—
185C22⋊Dic5—42no—
186C22⋊Dic5—84no—
187C22⋊Dic5—84no—
188#687—216no—
189#686—216no—
190#685—28no—
191C4⋊Dic5—22no—
192C4⋊Dic5—22no—
193C4⋊Dic5—22no—
194C4⋊Dic5—22no—
195#683—28no—
196C4×Dic5—28no—
197C4×Dic5—22no—
198C4×Dic5—22no—
199C4×Dic5—42no—
200#681—28no—
201C2×#168—22no—
202C2×#168—22no—
203C2×#168C22×C414yes—
204C22×#41—51no—
205C23×D4C2×D511yescharacteristic
206C2×AGL(1,ℤ/8)—58no—
207C2×AGL(1,ℤ/8)—104no—
208C2×AGL(1,ℤ/8)—104no—
209C22×SD16—52no—
210C22×D8—52no—
211C22×M16—51no—
212C23×C8C2×D511yescharacteristic
213#520—104no—
214#520—104no—
215C2×C4×Q8—52no—
216C2×C4×D4—52no—
217C2×C4×D4—102no—
218C2×#117—51no—
219C2×#117—51no—
220C2×#117—101no—
221C22×C4⋊C4—101no—
222C22×C4⋊C4C2×D511yescharacteristic
223#416—104no—
224C2×#103—52no—
225C2×#102—24no—
226C2×#102—52no—
227C2×#102C2×D514yes—
228#390—54no—
229#390—101no—
230#390—101no—
231#379—102no—
232#340—54no—
233#336—54no—
234C23×C5—22no—
235C23×C5—22no—
236C23×C5—24no—
237C23×C5—42no—
238C23×C5—42no—
239C23×C5—44no—
240C23×C5—44no—
241C23×C5—48no—
242C23×C5—88no—
243C23×C5—84no—
244C23×C5—84no—
245C23×C5—88no—
246C23×C5C4×D412yes—
247C23×C5#12012yes—
248C23×C5#12112yes—
249C23×C5C22×D411yescharacteristic
250C22×D5—22no—
251C22×D5—22no—
252C22×D5—22no—
253C22×D5—42no—
254C22×D5—42no—
255C5×D4—24no—
256C5×D4—24no—
257C5×D4—42no—
258C5×D4—42no—
259C5×D4—48no—
260C2×C4×C5—24no—
261C2×C4×C5—44no—
262C2×C4×C5—84no—
263C2×C4×C5C2×C22⋊C414yes—
264C2×C4×C5C4×D412yes—
265C2×C4×C5C4×D414yes—
266C2×C4×C5#12012yes—
267C2×C4×C5#12112yes—
268C2×C4×C5#12114yes—
269C2×C4×C5#12314yes—
270C2×C4×C5C23×C411yescharacteristic
271C2×C4×C5C22×D411yescharacteristic
272C2×C4×C5C22×D411yescharacteristic
273C2×C4×C5C22×D411yescharacteristic
274C2×C4×C5C2×#4111yescharacteristic
275C2×C4×C5C2×#4111yescharacteristic
276C22⋊D5—216no—
277C2×Dic5—22no—
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—22no—
281C2×Dic5—22no—
282C2×Dic5—24no—
283C2×Dic5—24no—
284C2×Dic5—42no—
285C2×Dic5—42no—
286C2×Dic5—42no—
287D20—22no—
288D20—22no—
289D20—22no—
290D20C2×C22⋊C414yes—
291C4×D5—216no—
292C4×D5—88no—
293Dic10—22no—
294Dic10—22no—
295Dic10—22no—
296Dic10C2×C22⋊C414yes—
297C5×C8—42no—
298#168—24no—
299C25—21no—
300C2×#41—54no—
301C2×#41—54no—
302C2×#41—54no—
303C2×#41—102no—
304C2×#41—102no—
305C2×#41—102no—
306C22×Q8—51no—
307C22×D4—22no—
308C22×D4—22no—
309C22×D4—22no—
310C22×D4—21no—
311C22×D4—22no—
312C22×D4—22no—
313C22×D4—51no—
314C22×D4—101no—
315C22×D4C4×D514yescomplemented
316C22×D4C22×D512yes—
317C23×C4—21no—
318C23×C4—102no—
319C23×C4—101no—
320C23×C4C22×D511yescharacteristic
321AGL(1,ℤ/8)—1032no—
322C2×SD16—58no—
323C2×SD16—104no—
324C2×SD16—104no—
325C2×D8—58no—
326C2×D8—104no—
327C2×D8—104no—
328C2×M16—54no—
329C2×M16—102no—
330C2×M16—102no—
331C22×C8—22no—
332C22×C8—22no—
333C22×C8—22no—
334C22×C8—52no—
335C22×C8C22×D512yes—
336C4×Q8—104no—
337C4×Q8—104no—
338C4×D4—104no—
339C4×D4—1016no—
340C4×D4—104no—
341#117—108no—
342#117—102no—
343#117—102no—
344#117—102no—
345#117—102no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—51no—
349C2×C4⋊C4—51no—
350C2×C4⋊C4—52no—
351C2×C4⋊C4—108no—
352C2×C4⋊C4—102no—
353C2×C4⋊C4C22⋊D514yes—
354C2×C4⋊C4C22×D512yes—
355C2×C22⋊C4—52no—
356C2×C22⋊C4—102no—
357C2×C22⋊C4—101no—
358C2×C22⋊C4—104no—
359C2×C22⋊C4—104no—
360C2×C42—52no—
361C2×C42—102no—
362C2×C42—101no—
363#103—104no—
364#102—28no—
365#102—104no—
366C22×C5—24no—
367C22×C5—22no—
368C22×C5—22no—
369C22×C5—24no—
370C22×C5—48no—
371C22×C5—44no—
372C22×C5—816no—
373C22×C5—82no—
374C22×C5—82no—
375C22×C5—84no—
376C22×C5#38614yes—
377C22×C5C2×C4×D411yescharacteristic
378C22×C5C2×#12011yescharacteristic
379C22×C5C2×#12111yescharacteristic
380C22×C5#57612yes—
381C22×C5#57712yes—
382C2×D5—24no—
383C2×D5—44no—
384C2×D5—42no—
385C2×D5—42no—
386C2×D5—44no—
387C4×C5—44no—
388C4×C5#38614yes—
389C4×C5C22×C22⋊C411yescharacteristic
390C4×C5C2×C4×D411yescharacteristic
391C4×C5C2×#12111yescharacteristic
392C4×C5C2×#12311yescharacteristic
393Dic5—24no—
394Dic5—42no—
395Dic5—42no—
396Dic5—82no—
397C24—22no—
398C24—24no—
399C24—42no—
400C24—42no—
401C24—82no—
402C24—82no—
403C24—101no—
404C24D4×D511yescharacteristic
405#41—516no—
406#41—108no—
407#41—108no—
408#41—108no—
409C2×Q8—52no—
410C2×Q8—52no—
411C2×Q8—52no—
412C2×Q8—101no—
413C2×Q8—101no—
414C2×Q8—101no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—28no—
421C2×D4—24no—
422C2×D4—28no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4—52no—
428C2×D4—52no—
429C2×D4—52no—
430C2×D4—104no—
431C2×D4—104no—
432C2×D4—104no—
433C2×D4—101no—
434C2×D4—101no—
435C2×D4—101no—
436C2×D4C2×C4×D514yes—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—52no—
440C22×C4—52no—
441C22×C4—82no—
442C22×C4—108no—
443C22×C4—101no—
444C22×C4—104no—
445C22×C4—104no—
446C22×C4—104no—
447C22×C4—102no—
448C22×C4—404no—
449C22×C4C2×C4×D512yes—
450C22×C4#70912yes—
451C22×C4D4×D512yes—
452C22×C4D4×D512yes—
453C22×C4D4×D511yescharacteristic
454C22×C4#71112yes—
455C22×C4C2×C22⋊D512yes—
456C22×C4C23×D511yescharacteristic
457SD16—1016no—
458D8—1016no—
459M16—108no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—44no—
464C2×C8—54no—
465C2×C8—102no—
466C2×C8—102no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—108no—
470C4⋊C4—102no—
471C4⋊C4—102no—
472C4⋊C4—102no—
473C4⋊C4—102no—
474C4⋊C4C2×C22⋊D514yes—
475C22⋊C4—108no—
476C22⋊C4—104no—
477C22⋊C4—202no—
478C22⋊C4—404no—
479C22⋊C4—404no—
480C42—108no—
481C42—102no—
482C42—102no—
483C42—202no—
484C2×C5—24no—
485C2×C5—84no—
486C2×C5C2×#38611yescharacteristic
487C2×C5#294612yes—
488C2×C5#300811yescharacteristic
489C2×C5#301011yescharacteristic
490C2×C5#304512yes—
491D5—44no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—48no—
500C23—88no—
501C23—84no—
502C23—84no—
503C23—88no—
504C23—102no—
505C23—102no—
506C23—102no—
507C23—202no—
508C23—202no—
509C23#402712yes—
510C23#403012yes—
511C23#408312yes—
512C23C2×D4×D511yescharacteristic
513Q8—54no—
514Q8—102no—
515Q8—102no—
516Q8—102no—
517D4—24no—
518D4—24no—
519D4—42no—
520D4—42no—
521D4—48no—
522D4—54no—
523D4—1016no—
524D4—102no—
525D4—102no—
526D4—102no—
527C2×C4—24no—
528C2×C4—44no—
529C2×C4—84no—
530C2×C4—102no—
531C2×C4—102no—
532C2×C4—102no—
533C2×C4—1016no—
534C2×C4—104no—
535C2×C4—104no—
536C2×C4—102no—
537C2×C4—102no—
538C2×C4—202no—
539C2×C4—202no—
540C2×C4—202no—
541C2×C4—408no—
542C2×C4D5×C22⋊C414yes—
543C2×C4#402712yes—
544C2×C4#402914yes—
545C2×C4#403012yes—
546C2×C4C4×C22⋊D514yes—
547C2×C4#408312yes—
548C2×C4#408414yes—
549C2×C4C22×C4×D511yescharacteristic
550C2×C4C2×#70911yescharacteristic
551C2×C4C2×D4×D511yescharacteristic
552C2×C4C2×D4×D511yescharacteristic
553C2×C4C2×#71111yescharacteristic
554C2×C4C22×C22⋊D511yescharacteristic
555C8—42no—
556C8—104no—
557C5#2093211yescharacteristic, complemented, Sylow
558C22—22no—
559C22—24no—
560C22—22no—
561C22—24no—
562C22—48no—
563C22—44no—
564C22—816no—
565C22—82no—
566C22—82no—
567C22—84no—
568C22—104no—
569C22—202no—
570C22—204no—
571C22—202no—
572C22—204no—
573C22#6524514yes—
574C22C2×#402711yescharacteristic
575C22C2×#403011yescharacteristic
576C22#6609112yes—
577C22#6609612yes—
578C22C2×#408311yescharacteristic
579C4—44no—
580C4—104no—
581C4—202no—
582C4—202no—
583C4—402no—
584C4#6524514yes—
585C4C2×D5×C22⋊C411yescharacteristic
586C4C2×#402911yescharacteristic
587C4C2×C4×C22⋊D511yescharacteristic
588C4C2×#408411yescharacteristic
589C2—24no—
590C2—84no—
591C2—204no—
592C2#1061045712yes—
593C2#1061049812yes—
594C2C2×#6524511yescharacteristic
595C2#1061556411yescharacteristic
596C2#1061559111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC22×C5×#102C2
socleC23×C5C22×D4
prime radicalC22×#4078C2
cosocleC22×C4×C5C24
hypercenterC22×#102D5
hypocenterC5#20932
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C2×#65838C2semidirect
3. C22×C5×#102C2semidirect
4. #10615775C2semidirect
5. C2×#65402C2semidirect
6. #10602758C2semidirect
7. #10602351C2semidirect
8. #10601006C2semidirect
9. #10601001C2semidirect
17. C2×#4078C4semidirect
25. C2×#4039C22semidirect
28. C2×#4033C22semidirect
36. C2×#3953C22semidirect
37. C2×#3948C22semidirect
38. #64770C22semidirect
39. #64749C22semidirect
48. C22×C5×D4C2×C4semidirect
77. C2×#687C2×C4semidirect
81. C2×#686C2×C4semidirect
97. C2×#681C2×C4semidirect
111. C22×#102D5semidirect
159. #709C22⋊C4semidirect
315. C22×D4C4×D5semidirect
557. C5#20932semidirect

Maximal subgroups

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

subgroupindex
C22×#40782
C2×#658382
C22×C5×#1022
#106157752
C2×#654022
#106027582
#106023512
#106010062
#106010012
#209325