Finite Group Databasebeta

← #12100304

Subgroups — #12100304

Order 1280.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#520C211yesmaximal, characteristic
2C2×#65521C211yesmaximal, characteristic
3C2×#65522C211yesmaximal, characteristic
4#10613963C211yesmaximal, characteristic, complemented
5C2×#65253C211yesmaximal, characteristic, complemented
6#10602422C212yesmaximal, complemented
7#10602421C212yesmaximal, complemented
8#10602400C212yesmaximal, complemented
9#10602399C212yesmaximal, complemented
10#10602372C211yesmaximal, characteristic, complemented
11C2×#64749C211yesmaximal, characteristic, complemented
12C22×C5×#41C2211yescharacteristic
13C5×#520—24no—
14C5×#520—24no—
15C2×C4×C5×Q8C2212yes—
16C2×C4×C5×D4—22no—
17C2×C4×C5×D4C2212yes—
18C2×C5×#117—21no—
19C2×C5×#117C2211yescharacteristic
20C22×C5×C4⋊C4—21no—
21C22×C5×C4⋊C4C2211yescharacteristic
22C2×#4072C2211yescharacteristic
23C22×C4⋊Dic5C2211yescharacteristic
24C22×#681C2211yescharacteristic
25C23×#168C2211yescharacteristic
26#65522—24no—
27#65521C418yescomplemented
28C2×#3967C2211yescharacteristic
29C2×#3967C2211yescharacteristic
30C2×#3967C2212yes—
31C2×#3964C2211yescharacteristic
32C2×#3964C2211yescharacteristic
33C2×#3964C2212yes—
34#65262—21no—
35#65262—21no—
36#65253—24no—
37C2×#3940C2212yes—
38C2×#3939C2212yes—
39#65231—22no—
40#65217C2212yes—
41#65217C2212yes—
42#64749C2212yes—
43#64749C2212yes—
44#64744C2212yes—
45#64744C2212yes—
46#20878—51nomaximal, Sylow
47C2×C5×#41—22no—
48C2×C5×#41—22no—
49C2×C5×#41—22no—
50C2×C5×#41C2×C414yes—
51C2×C5×#41D414yes—
52C2×C5×#41Q814yes—
53C22×C5×Q8C2311yescharacteristic
54C22×C5×D4—21no—
55C22×C5×D4C2311yescharacteristic
56C23×C4×C5—21no—
57C23×C4×C5—21no—
58C23×C4×C5—22no—
59C23×C4×C5C2311yescharacteristic
60C23×Dic5—21no—
61C4×C5×Q8—24no—
62C4×C5×Q8—24no—
63C4×C5×D4—216no—
64C4×C5×D4—24no—
65C4×C5×D4—24no—
66C5×#117—28no—
67C5×#117—22no—
68C5×#117—22no—
69C2×C5×C4⋊C4—28no—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4—22no—
72C2×C5×C4⋊C4—22no—
73C2×C5×C4⋊C4D411yescharacteristic
74C2×C5×C4⋊C4D411yescharacteristic
75C2×C5×C4⋊C4D414yes—
76C2×C5×C4⋊C4C2312yes—
77C2×C5×C22⋊C4—22no—
78C2×C5×C22⋊C4D412yes—
79C2×C42×C5—22no—
80C2×C42×C5C2312yes—
81C2×C22⋊Dic5—21no—
82#4072—22no—
83#4072—22no—
84C2×C4⋊Dic5—21no—
85C2×C4⋊Dic5—22no—
86C2×C4⋊Dic5C2×C414yescomplemented
87C2×C4⋊Dic5C2311yescharacteristic
88C2×C4⋊Dic5C2311yescharacteristic
89C2×C4⋊Dic5C2312yes—
90C2×#683—22no—
91C2×#683—22no—
92C2×C4×Dic5—21no—
93C2×#681—22no—
94C2×#681—22no—
95C2×#681C2×C414yescomplemented
96C22×#168—22no—
97C22×#168—22no—
98C22×#168—22no—
99C22×#168C2311yescharacteristic
100C22×#168C2311yescharacteristic
101C22×#168C2312yes—
102#3967—22no—
103#3967—22no—
104#3967C2×C418yescomplemented
105#3964—22no—
106#3964—22no—
107#3964C2×C418yescomplemented
108#3963—22no—
109#3963—22no—
110#3940—24no—
111#3939—24no—
112C2×#520D511yescharacteristic
113C2×#428—51no—
114C2×#417—51no—
115C2×#416—51no—
116#2316—51no—
117#1865—52no—
118#1864—52no—
119#1863—52no—
120#1862—52no—
121C2×#340—51no—
122#1729—51no—
123C24×C5—21no—
124C24×C5C2×D411yescharacteristic
125C5×#41—28no—
126C5×#41—28no—
127C5×#41—28no—
128C5×#41C4⋊C4116yescomplemented
129C2×C5×Q8—21no—
130C2×C5×Q8—21no—
131C2×C5×Q8—21no—
132C2×C5×Q8C22×C412yes—
133C2×C5×Q8C2×D412yes—
134C2×C5×Q8C2×Q812yes—
135C2×C5×D4—24no—
136C2×C5×D4—24no—
137C2×C5×D4—24no—
138C2×C5×D4—21no—
139C2×C5×D4—21no—
140C2×C5×D4—21no—
141C2×C5×D4C22×C412yes—
142C2×C5×D4C2×D412yes—
143C2×C5×D4C2×Q812yes—
144C22×C4×C5—21no—
145C22×C4×C5—24no—
146C22×C4×C5—24no—
147C22×C4×C5—24no—
148C22×C4×C5—22no—
149C22×C4×C5—28no—
150C22×C4×C5—24no—
151C22×C4×C5—82no—
152C22×C4×C5—82no—
153C22×C4×C5—81no—
154C22×C4×C5—81no—
155C22×C4×C5C22×C412yes—
156C22×C4×C5C2×D412yes—
157C22×C4×C5C2×D412yes—
158C22×C4×C5C2×D412yes—
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D412yes—
161C22×C4×C5C2×Q812yes—
162C22×C4×C5#4112yes—
163C22×C4×C5#4112yes—
164C22×C4×C5C2411yescharacteristic
165C22×Dic5—22no—
166C22×Dic5—24no—
167C22×Dic5—21no—
168C22×Dic5—21no—
169C22×Dic5—81no—
170C22×Dic5—81no—
171C5×C4⋊C4—28no—
172C5×C4⋊C4—22no—
173C5×C4⋊C4—24no—
174C5×C4⋊C4—22no—
175C5×C4⋊C4—22no—
176C5×C4⋊C4C2×D414yes—
177C5×C22⋊C4—28no—
178C5×C22⋊C4—24no—
179C42×C5—28no—
180C42×C5—22no—
181C42×C5—22no—
182C22⋊Dic5—42no—
183C4⋊Dic5—24no—
184C4⋊Dic5—22no—
185C4⋊Dic5—21no—
186C4⋊Dic5—21no—
187C4⋊Dic5—21no—
188C4⋊Dic5—21no—
189C4⋊Dic5C22×C414yes—
190#683—82no—
191#683—82no—
192C4×Dic5—42no—
193#681—28no—
194C2×#168—22no—
195C2×#168—22no—
196C2×#168—21no—
197C2×#168—21no—
198C2×#168—21no—
199C2×#168—21no—
200C2×#168—21no—
201C2×#168—21no—
202C2×#168—44no—
203C2×#168C22×C414yes—
204C22×#41C2×D511yescharacteristic
205C22×M16—51no—
206C23×C8—51no—
207#520—24no—
208#520—24no—
209C2×C4×Q8C2×D512yes—
210C2×C4×D4—22no—
211C2×C4×D4C2×D512yes—
212C2×#117—21no—
213C2×#117—51no—
214C2×#117C2×D511yescharacteristic
215C22×C4⋊C4—21no—
216C22×C4⋊C4—51no—
217C22×C4⋊C4C2×D511yescharacteristic
218#428—104no—
219C2×#107—52no—
220C2×#106—52no—
221#417—58no—
222#416—104no—
223C2×#103—52no—
224C2×#103—51no—
225C2×#103—51no—
226C2×#102—52no—
227C2×#102—51no—
228C2×#102—51no—
229#389—52no—
230#389—52no—
231#389—101no—
232#389—101no—
233#380—102no—
234#340—52no—
235#340—52no—
236#336—52no—
237#336—52no—
238C23×C5—21no—
239C23×C5—21no—
240C23×C5—21no—
241C23×C5—21no—
242C23×C5—22no—
243C23×C5—22no—
244C23×C5—22no—
245C23×C5—41no—
246C23×C5—41no—
247C23×C5—41no—
248C23×C5—41no—
249C23×C5C4×D412yes—
250C23×C5#12012yes—
251C23×C5C22⋊Q812yes—
252C23×C5C22×D411yescharacteristic
253C5×Q8—22no—
254C5×Q8—22no—
255C5×Q8—22no—
256C5×Q8C2×C4⋊C414yes—
257C5×D4—216no—
258C5×D4—22no—
259C5×D4—22no—
260C5×D4—22no—
261C5×D4C2×C4⋊C414yes—
262C2×C4×C5—22no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—216no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—24no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—84no—
276C2×C4×C5—88no—
277C2×C4×C5C2×C4⋊C414yes—
278C2×C4×C5C4×D414yes—
279C2×C4×C5C4×D412yes—
280C2×C4×C5#12012yes—
281C2×C4×C5C22⋊Q814yes—
282C2×C4×C5C22⋊Q812yes—
283C2×C4×C5#12314yes—
284C2×C4×C5C23×C411yescharacteristic
285C2×C4×C5C22×D411yescharacteristic
286C2×C4×C5C22×D411yescharacteristic
287C2×C4×C5C22×Q811yescharacteristic
288C2×C4×C5C2×#4111yescharacteristic
289C2×C4×C5C2×#4111yescharacteristic
290C2×Dic5—24no—
291C2×Dic5—42no—
292C2×Dic5—42no—
293C2×Dic5—41no—
294C2×Dic5—41no—
295C2×Dic5—41no—
296C2×Dic5—41no—
297C2×Dic5—41no—
298C2×Dic5—41no—
299C2×Dic5—84no—
300#168—24no—
301#168—41no—
302#168—41no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41—22no—
306C2×#41Dic1014yes—
307C2×#41C4×D514yes—
308C2×#41C22⋊D514yes—
309C22×Q8C22×D511yescharacteristic
310C22×D4—21no—
311C22×D4C22×D511yescharacteristic
312C23×C4—21no—
313C23×C4—21no—
314C23×C4—22no—
315C23×C4—101no—
316C23×C4C22×D511yescharacteristic
317C2×M16—54no—
318C2×M16—102no—
319C2×M16—102no—
320C22×C8—52no—
321C22×C8—51no—
322C22×C8—51no—
323C22×C8—102no—
324C22×C8—102no—
325C22×C8—102no—
326C4×Q8—24no—
327C4×Q8—24no—
328C4×D4—216no—
329C4×D4—24no—
330C4×D4—24no—
331#117—28no—
332#117—22no—
333#117—22no—
334#117—102no—
335#117—102no—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—54no—
341C2×C4⋊C4—52no—
342C2×C4⋊C4—51no—
343C2×C4⋊C4—51no—
344C2×C4⋊C4—102no—
345C2×C4⋊C4—101no—
346C2×C4⋊C4—102no—
347C2×C4⋊C4—102no—
348C2×C4⋊C4C22⋊D511yescharacteristic
349C2×C4⋊C4C22⋊D511yescharacteristic
350C2×C4⋊C4C22⋊D514yes—
351C2×C4⋊C4C22×D512yes—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—101no—
354C2×C22⋊C4C22⋊D512yes—
355C2×C42—22no—
356C2×C42—101no—
357C2×C42C22×D512yes—
358#107—104no—
359#106—104no—
360#103—58no—
361#103—102no—
362#103—102no—
363#102—58no—
364#102—102no—
365#102—102no—
366#95—102no—
367#95—102no—
368C22×C5—22no—
369C22×C5—22no—
370C22×C5—22no—
371C22×C5—22no—
372C22×C5—22no—
373C22×C5—22no—
374C22×C5—24no—
375C22×C5—42no—
376C22×C5—42no—
377C22×C5—42no—
378C22×C5—42no—
379C22×C5—42no—
380C22×C5—42no—
381C22×C5#38514yes—
382C22×C5C2×C4×D411yescharacteristic
383C22×C5C2×#12011yescharacteristic
384C22×C5C2×C22⋊Q811yescharacteristic
385C22×C5#57611yescharacteristic
386C22×C5#57711yescharacteristic
387C22×C5#57711yescharacteristic
388C22×C5#57811yescharacteristic
389C4×C5—24no—
390C4×C5—42no—
391C4×C5—44no—
392C4×C5—42no—
393C4×C5#38514yes—
394C4×C5C22×C4⋊C411yescharacteristic
395C4×C5C2×C4×D411yescharacteristic
396C4×C5C2×C22⋊Q811yescharacteristic
397C4×C5C2×#12311yescharacteristic
398Dic5—44no—
399Dic5—81no—
400Dic5—81no—
401C24—21no—
402C24C2×C22⋊D511yescharacteristic
403#41—28no—
404#41—28no—
405#41—28no—
406#41#683116yescomplemented
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8C2×Dic1012yes—
411C2×Q8C2×C4×D512yes—
412C2×Q8C2×C22⋊D512yes—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4C2×Dic1012yes—
420C2×D4C2×C4×D512yes—
421C2×D4C2×C22⋊D512yes—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—102no—
434C22×C4—104no—
435C22×C4—101no—
436C22×C4—101no—
437C22×C4—401no—
438C22×C4—401no—
439C22×C4C2×Dic1012yes—
440C22×C4C2×C4×D512yes—
441C22×C4#70912yes—
442C22×C4#70912yes—
443C22×C4C2×C22⋊D512yes—
444C22×C4C2×C22⋊D512yes—
445C22×C4C2×C22⋊D511yescharacteristic
446C22×C4C2×C22⋊D512yes—
447C22×C4C2×C22⋊D512yes—
448C22×C4C23×D511yescharacteristic
449M16—108no—
450C2×C8—54no—
451C2×C8—102no—
452C2×C8—102no—
453C2×C8—101no—
454C2×C8—101no—
455C2×C8—101no—
456C2×C8—101no—
457C2×C8—101no—
458C2×C8—101no—
459C2×C8—204no—
460C4⋊C4—28no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—54no—
466C4⋊C4—102no—
467C4⋊C4—104no—
468C4⋊C4—101no—
469C4⋊C4—101no—
470C4⋊C4—101no—
471C4⋊C4—101no—
472C4⋊C4—402no—
473C4⋊C4—402no—
474C4⋊C4C2×C22⋊D514yes—
475C22⋊C4—28no—
476C22⋊C4—24no—
477C22⋊C4—202no—
478C42—28no—
479C42—22no—
480C42—22no—
481C42—202no—
482C2×C5—24no—
483C2×C5—44no—
484C2×C5C2×#38511yescharacteristic
485C2×C5#294611yescharacteristic
486C2×C5#294711yescharacteristic
487C2×C5#301211yescharacteristic
488C2×C5#301311yescharacteristic
489C2×C5#308311yescharacteristic
490C2×C5#308411yescharacteristic
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23#407012yes—
503C23C4×C22⋊D512yes—
504C23#409912yes—
505C23C22×C22⋊D511yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8C2×#68314yes—
510D4—216no—
511D4—22no—
512D4—22no—
513D4—22no—
514D4C2×#68314yes—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—216no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—84no—
529C2×C4—88no—
530C2×C4—104no—
531C2×C4—202no—
532C2×C4—202no—
533C2×C4—201no—
534C2×C4—201no—
535C2×C4—201no—
536C2×C4—201no—
537C2×C4—201no—
538C2×C4—201no—
539C2×C4—404no—
540C2×C4C2×#68314yes—
541C2×C4#407014yes—
542C2×C4#407012yes—
543C2×C4C4×C22⋊D514yes—
544C2×C4C4×C22⋊D512yes—
545C2×C4#407514yes—
546C2×C4#409912yes—
547C2×C4C22×Dic1011yescharacteristic
548C2×C4C22×C4×D511yescharacteristic
549C2×C4C2×#70911yescharacteristic
550C2×C4C2×#70911yescharacteristic
551C2×C4C22×C22⋊D511yescharacteristic
552C2×C4C22×C22⋊D511yescharacteristic
553C8—104no—
554C8—201no—
555C8—201no—
556C5#2087811yescharacteristic, complemented, Sylow
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22#6549914yes—
571C22C2×#407011yescharacteristic
572C22C2×C4×C22⋊D511yescharacteristic
573C22#6616911yescharacteristic
574C22#6617011yescharacteristic
575C22#6617111yescharacteristic
576C22#6617211yescharacteristic
577C22C2×#409911yescharacteristic
578C4—24no—
579C4—42no—
580C4—44no—
581C4—42no—
582C4—204no—
583C4—401no—
584C4—401no—
585C4#6549914yes—
586C4C22×#68311yescharacteristic
587C4C2×#407011yescharacteristic
588C4C2×C4×C22⋊D511yescharacteristic
589C4C2×#407511yescharacteristic
590C2—24no—
591C2—44no—
592C2#1061429011yescharacteristic
593C2#1061429111yescharacteristic
594C2#1061429511yescharacteristic
595C2#1061430211yescharacteristic
596C2C2×#6549911yescharacteristic
597C2#1061583311yescharacteristic
598C2#1061584511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#520C2
socleC23×C5C22×D4
prime radicalC22×C5×#41C22
cosocleC22×C4×C5C24
hypercenterC2×#520D5
hypocenterC5#20878
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
4. #10613963C2semidirect
5. C2×#65253C2semidirect
6. #10602422C2semidirect
7. #10602421C2semidirect
8. #10602400C2semidirect
9. #10602399C2semidirect
10. #10602372C2semidirect
11. C2×#64749C2semidirect
27. #65521C4semidirect
86. C2×C4⋊Dic5C2×C4semidirect
95. C2×#681C2×C4semidirect
104. #3967C2×C4semidirect
107. #3964C2×C4semidirect
128. C5×#41C4⋊C4semidirect
406. #41#683semidirect
556. C5#20878semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5202
C2×#655212
C2×#655222
#106139632
C2×#652532
#106024222
#106024212
#106024002
#106023992
#106023722
C2×#647492
#208785