Finite Group Databasebeta

← #12110542

Subgroups — #12110542

Order 1280.

1212 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#390C211yesmaximal, characteristic
2C2×#65507C211yesmaximal, characteristic
3C4×#4070C211yesmaximal, characteristic, complemented
4#10602921C211yesmaximal, characteristic
5#10602352C211yesmaximal, characteristic, complemented
6#10602348C211yesmaximal, characteristic, complemented
7#10602322C211yesmaximal, characteristic, complemented
8#10600453C212yesmaximal, complemented
9#10600442C212yesmaximal, complemented
10#10600438C212yesmaximal, complemented
11#10600429C212yesmaximal, complemented
12C23×C5×D4C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C5×C22⋊C4—21no—
15C22×C5×C22⋊C4—21no—
16C22×C42×C5C2211yescharacteristic
17C22×#687C2212yes—
18C2×#4070C2211yescharacteristic
19C2×C4×Dic10C2211yescharacteristic, complemented
20C5×#390—21no—
21C5×#390—21no—
22C5×#390C2212yes—
23C5×#390C2212yes—
24#65507C418yescomplemented
25C2×#3964C2212yes—
26C2×#3964C2212yes—
27C2×#3964C2212yes—
28#65501C2211yescharacteristic
29#65499—21no—
30C4×C22⋊Dic5—21no—
31C2×C22⋊#168C2211yescharacteristic
32C2×C22⋊#168C2211yescharacteristic
33C2×#3942C2212yescomplemented
34C2×#3939C2212yescomplemented
35C4×C4⋊Dic5C2211yescharacteristic, complemented
36#65222—21no—
37C4×#683—21no—
38#65218C2211yescharacteristic, complemented
39#65217C2211yescharacteristic, complemented
40C2×#3935C2212yescomplemented
41#64749C2212yescomplemented
42#64749C2212yescomplemented
43#64744C2212yescomplemented
44#22049—51nomaximal, Sylow
45C25×C5—21no—
46C22×C5×D4—21no—
47C22×C5×D4—22no—
48C22×C5×D4—42no—
49C22×C5×D4—42no—
50C22×C5×D4C2×C414yes—
51C22×C5×D4C2312yes—
52C23×C4×C5—21no—
53C23×C4×C5D411yescharacteristic
54C23×C4×C5D411yescharacteristic
55C23×C4×C5C2311yescharacteristic
56C22×Dic10C2311yescharacteristic
57C2×C5×C4⋊C4—21no—
58C2×C5×C4⋊C4—42no—
59C2×C5×C4⋊C4C2312yes—
60C2×C5×C22⋊C4—22no—
61C2×C5×C22⋊C4—22no—
62C2×C5×C22⋊C4—22no—
63C2×C5×C22⋊C4—22no—
64C2×C5×C22⋊C4—44no—
65C2×C5×C22⋊C4—44no—
66C2×C42×C5—42no—
67C2×C42×C5D411yescharacteristic
68C2×C42×C5D411yescharacteristic
69C2×C42×C5C2311yescharacteristic
70C2×C42×C5C2311yescharacteristic
71C2×C22⋊Dic5—21no—
72C2×C22⋊Dic5—21no—
73C2×#687—42no—
74C2×#687—42no—
75C2×#687C2×C418yescomplemented
76C2×C4⋊Dic5C2311yescharacteristic
77C2×C4⋊Dic5C2311yescharacteristic
78C2×C4⋊Dic5C2311yescharacteristic
79#4070—22no—
80#4070C2×C414yes—
81C2×#683—21no—
82C2×#683—21no—
83C2×#683—21no—
84C2×C4×Dic5—21no—
85C2×C4×Dic5—21no—
86C22×#168C2312yes—
87C22×#168C2312yes—
88C4×Dic10—42no—
89#3964—42no—
90#3964—42no—
91#3964C2×C418yescomplemented
92#3963—21no—
93#3963—21no—
94#3963—21no—
95#3963—21no—
96#3963—21no—
97C22⋊#168—22no—
98C22⋊#168C2×C414yes—
99#3942—42no—
100#3939—42no—
101#3935—42no—
102C2×#450—51no—
103C4×C22⋊Q8—51no—
104C2×#390D511yescharacteristic
105#1973—52no—
106#1936—52no—
107#1925—52no—
108#1880—52no—
109#1872—51no—
110#1852—51no—
111#1786—51no—
112#1749—51no—
113C24×C5—24no—
114C24×C5—22no—
115C24×C5—42no—
116C24×C5—42no—
117C24×C5—42no—
118C24×C5—42no—
119C24×C5—42no—
120C24×C5—42no—
121C24×C5C2×D411yescharacteristic
122C2×C5×D4—24no—
123C2×C5×D4—22no—
124C2×C5×D4—22no—
125C2×C5×D4—22no—
126C2×C5×D4—22no—
127C2×C5×D4—22no—
128C2×C5×D4—48no—
129C2×C5×D4—42no—
130C2×C5×D4—42no—
131C2×C5×D4—48no—
132C2×C5×D4C22×C414yes—
133C22×C4×C5—22no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
137C22×C4×C5—22no—
138C22×C4×C5—22no—
139C22×C4×C5—41no—
140C22×C4×C5—41no—
141C22×C4×C5—41no—
142C22×C4×C5—42no—
143C22×C4×C5—42no—
144C22×C4×C5—42no—
145C22×C4×C5C22×C412yes—
146C22×C4×C5C2×D411yescharacteristic
147C22×C4×C5C2×D411yescharacteristic
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×D411yescharacteristic
151C22×C4×C5#4112yes—
152C22×C4×C5C2411yescharacteristic
153C22×Dic5—21no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—21no—
157C2×Dic10—22no—
158C2×Dic10—22no—
159C2×Dic10—21no—
160C2×Dic10C22×C412yes—
161C5×C4⋊C4—22no—
162C5×C4⋊C4—22no—
163C5×C4⋊C4—42no—
164C5×C22⋊C4—44no—
165C5×C22⋊C4—44no—
166C5×C22⋊C4—44no—
167C5×C22⋊C4—44no—
168C42×C5—22no—
169C42×C5—22no—
170C42×C5—41no—
171C42×C5—41no—
172C22⋊Dic5—24no—
173C22⋊Dic5—42no—
174#687—48no—
175C4⋊Dic5—21no—
176C4⋊Dic5—22no—
177C4⋊Dic5—22no—
178C4⋊Dic5C22×C412yes—
179#683—24no—
180#683—42no—
181#683—42no—
182C4×Dic5—42no—
183C4×Dic5—42no—
184C2×#168—22no—
185C2×#168—22no—
186C2×#168—22no—
187C2×#168—22no—
188C2×#168—22no—
189C2×#168C22×C414yes—
190C23×D4C2×D511yescharacteristic
191C22×SD16—52no—
192C2×C22⋊Q8—51no—
193C2×C4×Q8—51no—
194C22×C4⋊C4C2×D511yescharacteristic
195C22×C22⋊C4—21no—
196C22×C22⋊C4—21no—
197C22×C42C2×D511yescharacteristic
198#450—58no—
199C2×#107—52no—
200C2×#103—52no—
201C2×#102—52no—
202C2×#102—52no—
203C2×#102—52no—
204C2×C22⋊C8—51no—
205C2×C22⋊C8—51no—
206C2×#97—52no—
207#391—51no—
208#390—21no—
209#390—21no—
210#390C2×D512yes—
211#390C2×D512yes—
212#389—51no—
213#387—101no—
214#385—101no—
215#380—51no—
216C4×C4⋊C4—51no—
217C4×C4⋊C4—101no—
218C4×C22⋊C4—101no—
219#340—52no—
220#340—52no—
221#336—52no—
222C23×C5—21no—
223C23×C5—21no—
224C23×C5—21no—
225C23×C5—21no—
226C23×C5—24no—
227C23×C5—48no—
228C23×C5—48no—
229C23×C5—42no—
230C23×C5—42no—
231C23×C5—42no—
232C23×C5—42no—
233C23×C5—42no—
234C23×C5—42no—
235C23×C5—48no—
236C23×C5—84no—
237C23×C5—84no—
238C23×C5—84no—
239C23×C5—84no—
240C23×C5C4×D412yes—
241C23×C5AGL(1,ℤ/8)12yes—
242C23×C5#13712yes—
243C23×C5C22×D411yescharacteristic
244C5×D4—24no—
245C5×D4—24no—
246C5×D4—48no—
247C5×D4—42no—
248C5×D4—42no—
249C2×C4×C5—21no—
250C2×C4×C5—21no—
251C2×C4×C5—21no—
252C2×C4×C5—21no—
253C2×C4×C5—24no—
254C2×C4×C5—24no—
255C2×C4×C5—21no—
256C2×C4×C5—21no—
257C2×C4×C5—21no—
258C2×C4×C5—21no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—44no—
265C2×C4×C5—44no—
266C2×C4×C5—42no—
267C2×C4×C5—42no—
268C2×C4×C5—42no—
269C2×C4×C5—84no—
270C2×C4×C5C4×D412yes—
271C2×C4×C5AGL(1,ℤ/8)12yes—
272C2×C4×C5C23×C411yescharacteristic
273C2×C4×C5C2×#4111yescharacteristic
274C2×C4×C5H(5,𝔽2)12yes—
275C2×C4×C5H(5,𝔽2)12yes—
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5—41no—
279C2×Dic5—41no—
280C2×Dic5—41no—
281C2×Dic5—41no—
282C2×Dic5—41no—
283C2×Dic5—41no—
284C2×Dic5—41no—
285C2×Dic5—41no—
286C2×Dic5—41no—
287C2×Dic5—41no—
288Dic10—24no—
289Dic10—42no—
290#168—24no—
291#168—42no—
292C25—21no—
293C22×Q8—51no—
294C22×D4—22no—
295C22×D4—21no—
296C22×D4—42no—
297C22×D4—42no—
298C22×D4C4×D514yes—
299C22×D4C22×D512yes—
300C23×C4—21no—
301C23×C4C22⋊D511yescharacteristic
302C23×C4C22⋊D511yescharacteristic
303C23×C4C22×D511yescharacteristic
304C2×SD16—58no—
305C2×SD16—202no—
306C2×SD16—202no—
307C22×C8—52no—
308C22×C8—52no—
309C22⋊Q8—54no—
310C22⋊Q8—102no—
311C4×Q8—202no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—51no—
315C2×C4⋊C4—51no—
316C2×C4⋊C4—51no—
317C2×C4⋊C4—101no—
318C2×C4⋊C4—101no—
319C2×C4⋊C4—101no—
320C2×C4⋊C4C22×D512yes—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—101no—
328C2×C22⋊C4—101no—
329C2×C42—42no—
330C2×C42—101no—
331C2×C42—101no—
332C2×C42C22⋊D511yescharacteristic
333C2×C42C22⋊D511yescharacteristic
334C2×C42C22×D511yescharacteristic
335C2×C42C22×D511yescharacteristic
336#107—202no—
337#103—202no—
338#102—58no—
339#102—202no—
340#102—202no—
341C22⋊C8—54no—
342C22⋊C8—102no—
343#97—202no—
344#95—101no—
345#95—101no—
346#95—101no—
347#95—101no—
348#95—101no—
349C22×C5—22no—
350C22×C5—22no—
351C22×C5—22no—
352C22×C5—22no—
353C22×C5—22no—
354C22×C5—22no—
355C22×C5—44no—
356C22×C5—44no—
357C22×C5—44no—
358C22×C5—816no—
359C22×C5—82no—
360C22×C5—82no—
361C22×C5—82no—
362C22×C5—82no—
363C22×C5#44014yes—
364C22×C5C2×C4×D411yescharacteristic
365C22×C5#53411yescharacteristic
366C22×C5#53511yescharacteristic
367C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
368C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
369C22×C5C2×#13711yescharacteristic
370C22×C5#57511yescharacteristic
371C4×C5—22no—
372C4×C5—22no—
373C4×C5—41no—
374C4×C5—41no—
375C4×C5—41no—
376C4×C5—82no—
377C4×C5#51812yes—
378C4×C5#53812yes—
379Dic5—42no—
380Dic5—42no—
381Dic5—81no—
382Dic5—81no—
383C24—22no—
384C24—24no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24C2×C22⋊D511yescharacteristic
392C2×Q8—52no—
393C2×Q8—101no—
394C2×Q8—102no—
395C2×Q8—102no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—48no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—48no—
406C2×D4C2×C4×D514yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—101no—
420C22×C4—101no—
421C22×C4—101no—
422C22×C4—101no—
423C22×C4C2×C4×D512yes—
424C22×C4#70912yes—
425C22×C4C2×C22⋊D511yescharacteristic
426C22×C4C2×C22⋊D511yescharacteristic
427C22×C4C2×C22⋊D511yescharacteristic
428C22×C4C2×C22⋊D511yescharacteristic
429C22×C4C2×C22⋊D511yescharacteristic
430C22×C4C23×D511yescharacteristic
431SD16—208no—
432C2×C8—54no—
433C2×C8—102no—
434C2×C8—102no—
435C2×C8—102no—
436C2×C8—102no—
437C2×C8—102no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—52no—
442C4⋊C4—101no—
443C4⋊C4—104no—
444C4⋊C4—102no—
445C4⋊C4—102no—
446C4⋊C4—202no—
447C4⋊C4—202no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—104no—
453C22⋊C4—202no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—202no—
459C42—202no—
460C2×C5—24no—
461C2×C5—84no—
462C2×C5#232111yescharacteristic
463C2×C5C2×#44011yescharacteristic
464C2×C5C4×AGL(1,ℤ/8)11yescharacteristic
465C2×C5#319311yescharacteristic
466C2×C5#319411yescharacteristic
467C2×C5#321511yescharacteristic
468C2×C5#321711yescharacteristic
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—24no—
474C23—48no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—84no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23C4×C22⋊D512yes—
488C23#407812yes—
489C23#409712yes—
490C23C22×C22⋊D511yescharacteristic
491Q8—104no—
492Q8—202no—
493D4—24no—
494D4—24no—
495D4—48no—
496D4—42no—
497D4—42no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—84no—
519C2×C4—102no—
520C2×C4—102no—
521C2×C4—201no—
522C2×C4—201no—
523C2×C4—201no—
524C2×C4—201no—
525C2×C4—201no—
526C2×C4—201no—
527C2×C4—201no—
528C2×C4—201no—
529C2×C4—201no—
530C2×C4—201no—
531C2×C4C4×C22⋊D512yes—
532C2×C4#407812yes—
533C2×C4C22×C4×D511yescharacteristic
534C2×C4C2×#70911yescharacteristic
535C2×C4#414412yes—
536C2×C4#414412yes—
537C8—104no—
538C8—202no—
539C5#2204911yescharacteristic, complemented, Sylow
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22#6530714yes—
555C22C2×C4×C22⋊D511yescharacteristic
556C22C2×#407811yescharacteristic
557C22C2×#407811yescharacteristic
558C22#6613811yescharacteristic
559C22#6613911yescharacteristic
560C22#6615611yescharacteristic
561C22C2×#409711yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—82no—
568C4—202no—
569C4—202no—
570C4—401no—
571C4—401no—
572C4#6582012yes—
573C4#6582812yes—
574C2—24no—
575C2—84no—
576C2#1061382711yescharacteristic
577C2C2×#6530711yescharacteristic
578C2C4×#407811yescharacteristic
579C2#1061435911yescharacteristic
580C2#1061436211yescharacteristic
581C2#1061468011yescharacteristic
582C2#1061489811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#390C2
socleC23×C5C22×D4
prime radicalC23×C5×D4C22
cosocleC22×C4×C5C24
hypercenterC2×#390D5
hypocenterC5#22049
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
3. C4×#4070C2semidirect
5. #10602352C2semidirect
6. #10602348C2semidirect
7. #10602322C2semidirect
8. #10600453C2semidirect
9. #10600442C2semidirect
10. #10600438C2semidirect
11. #10600429C2semidirect
19. C2×C4×Dic10C22semidirect
24. #65507C4semidirect
33. C2×#3942C22semidirect
34. C2×#3939C22semidirect
35. C4×C4⋊Dic5C22semidirect
38. #65218C22semidirect
39. #65217C22semidirect
40. C2×#3935C22semidirect
41. #64749C22semidirect
42. #64749C22semidirect
43. #64744C22semidirect
75. C2×#687C2×C4semidirect
91. #3964C2×C4semidirect
539. C5#22049semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3902
C2×#655072
C4×#40702
#106029212
#106023522
#106023482
#106023222
#106004532
#106004422
#106004382
#106004292
#220495