Finite Group Databasebeta

← #12101950

Subgroups — #12101950

Order 1280.

1122 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#405C211yesmaximal, characteristic, complemented
2C2×#65422C211yesmaximal, characteristic
3C2×#65402C211yesmaximal, characteristic
4#10614001C211yesmaximal, characteristic
5#10613965C211yesmaximal, characteristic, complemented
6#10602778C211yesmaximal, characteristic, complemented
7C2×#64771C211yesmaximal, characteristic, complemented
8#10600611C212yesmaximal, complemented
9#10600610C212yesmaximal, complemented
10#10600561C212yesmaximal, complemented
11#10600560C212yesmaximal, complemented
12C22×#709C2211yescharacteristic
13C22×C5×M16C2211yescharacteristic
14C23×C5×C8C2211yescharacteristic
15C2×C5×#117C2211yescharacteristic
16C2×#4072C2211yescharacteristic
17C22×C4⋊Dic5C2211yescharacteristic
18C22×C4⋊Dic5C2211yescharacteristic
19C5×#405—24no—
20C2×C5×#97C2212yes—
21C2×C4×C5×C8C2212yes—
22#65422—24no—
23#65402C418yescomplemented
24C2×#3953C2212yes—
25C2×#3953C2211yescharacteristic
26C2×#3953C2211yescharacteristic
27C2×#3948C2212yes—
28C2×#3948C2211yescharacteristic
29C2×#3948C2211yescharacteristic
30#65276—21no—
31#65276—21no—
32#65266—21no—
33#65265—21no—
34#65247—22no—
35#65235—22no—
36#65228C2212yes—
37#65218C2212yes—
38#65217C2212yes—
39#65217C2212yes—
40#64771C2212yes—
41#64771C2212yes—
42#64771C2212yes—
43#64771C2212yes—
44#21087—51nomaximal, Sylow
45C23×C4×C5C2311yescharacteristic
46C22×C22⋊D5—21no—
47C23×Dic5—21no—
48C23×Dic5—21no—
49C2×#709—22no—
50C2×#709—22no—
51C2×#709—22no—
52C2×#709—22no—
53C2×#709—22no—
54C2×#709C2×C414yes—
55C22×D20C2311yescharacteristic
56C22×C4×D5—21no—
57C22×Dic10C2311yescharacteristic
58C2×C5×M16—22no—
59C2×C5×M16—22no—
60C2×C5×M16D414yes—
61C22×C5×C8—22no—
62C22×C5×C8—22no—
63C22×C5×C8C2×C414yes—
64C22×C5×C8C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C22×C5×C8C2312yes—
67C5×#117—22no—
68C5×#117—22no—
69C2×C5×C4⋊C4D411yescharacteristic
70C2×C5×C4⋊C4D411yescharacteristic
71C2×C5×C22⋊C4D412yes—
72C2×C42×C5C2312yes—
73C2×C22⋊Dic5—21no—
74C2×#685—22no—
75C2×#685—22no—
76#4072—22no—
77#4072C2×C414yes—
78C2×C4⋊Dic5—22no—
79C2×C4⋊Dic5—22no—
80C2×C4⋊Dic5—22no—
81C2×C4⋊Dic5—22no—
82C2×C4⋊Dic5—21no—
83C2×C4⋊Dic5—21no—
84C2×C4⋊Dic5C2312yes—
85C2×C4⋊Dic5C2312yes—
86C2×C4⋊Dic5C2311yescharacteristic
87C2×C4⋊Dic5C2311yescharacteristic
88C2×#683—22no—
89C2×#683—22no—
90C2×C4×Dic5—21no—
91C5×#97—24no—
92C4×C5×C8—24no—
93#3963—22no—
94#3963—22no—
95#3963—22no—
96#3963—22no—
97#3953—24no—
98#3953C2×C414yescomplemented
99#3953C2×C414yescomplemented
100#3948—24no—
101#3948C2×C414yescomplemented
102#3948C2×C414yescomplemented
103C2×#417—51no—
104C2×#416—51no—
105C2×#405D511yescharacteristic, complemented
106#2317—51no—
107#2311—51no—
108#1936—52no—
109#1935—52no—
110#1934—52no—
111#1933—52no—
112#1736—51no—
113C2×#340—51no—
114C24×C5C2×D411yescharacteristic
115C23×D5—21no—
116C22×C4×C5C22×C412yes—
117C22×C4×C5C2×D412yes—
118C22×C4×C5C2×D412yes—
119C22×C4×C5C2×D411yescharacteristic
120C22×C4×C5C2×D412yes—
121C22×C4×C5#4112yes—
122C22×C4×C5#4112yes—
123C22×C4×C5#4112yes—
124C22×C4×C5#4112yes—
125C22×C4×C5C2411yescharacteristic
126C2×C22⋊D5—24no—
127C2×C22⋊D5—42no—
128C2×C22⋊D5—42no—
129C22×Dic5—21no—
130C22×Dic5—21no—
131C22×Dic5—21no—
132C22×Dic5—22no—
133C22×Dic5—22no—
134C22×Dic5—42no—
135C22×Dic5—42no—
136C22×Dic5—81no—
137C22×Dic5—81no—
138C22×Dic5—81no—
139C22×Dic5—81no—
140#709—28no—
141#709—28no—
142#709—44no—
143#709—44no—
144C2×D20—21no—
145C2×D20—21no—
146C2×D20—21no—
147C2×D20—21no—
148C2×D20—21no—
149C2×D20C22×C412yes—
150C2×C4×D5—24no—
151C2×C4×D5—42no—
152C2×C4×D5—42no—
153C2×Dic10—21no—
154C2×Dic10—21no—
155C2×Dic10—21no—
156C2×Dic10—21no—
157C2×Dic10—21no—
158C2×Dic10C22×C412yes—
159C5×M16—28no—
160C2×C5×C8—28no—
161C2×C5×C8—21no—
162C2×C5×C8—21no—
163C2×C5×C8—21no—
164C2×C5×C8—21no—
165C2×C5×C8—22no—
166C2×C5×C8—22no—
167C2×C5×C8C22×C412yes—
168C2×C5×C8C22×C412yes—
169C2×C5×C8C2×D414yes—
170C5×C4⋊C4—22no—
171C5×C4⋊C4—22no—
172C5×C22⋊C4—24no—
173C42×C5—22no—
174C42×C5—22no—
175C22⋊Dic5—24no—
176#685—82no—
177#685—82no—
178C4⋊Dic5—21no—
179C4⋊Dic5—21no—
180C4⋊Dic5—22no—
181C4⋊Dic5—22no—
182C4⋊Dic5—22no—
183C4⋊Dic5—22no—
184C4⋊Dic5—42no—
185C4⋊Dic5—42no—
186C4⋊Dic5C22×C412yes—
187C4⋊Dic5C22×C412yes—
188#683—82no—
189#683—82no—
190C4×Dic5—24no—
191C22×#41—51no—
192C22×M16C2×D511yescharacteristic
193C23×C8C2×D511yescharacteristic
194C2×#117—51no—
195C2×#117C2×D511yescharacteristic
196C22×C4⋊C4—51no—
197C22×C4⋊C4—51no—
198#417—104no—
199#416—58no—
200C2×#103—51no—
201C2×#103—51no—
202C2×#103—52no—
203C2×#102—51no—
204C2×#102—51no—
205C2×#102—52no—
206#405—24no—
207C2×#97C2×D512yes—
208C2×C4×C8C2×D512yes—
209#391—52no—
210#390—52no—
211#389—52no—
212#389—52no—
213#386—102no—
214#385—102no—
215#384—101no—
216#382—101no—
217#380—101no—
218#380—101no—
219#340—52no—
220#340—52no—
221#340—52no—
222#340—52no—
223C23×C5—21no—
224C23×C5—21no—
225C23×C5—21no—
226C23×C5—21no—
227C23×C5C4×D412yes—
228C23×C5#12112yes—
229C23×C5#12412yes—
230C23×C5C22×D411yescharacteristic
231C22×D5—22no—
232C22×D5—41no—
233C22×D5—41no—
234C22×D5—41no—
235C22×D5—41no—
236C22×D5—41no—
237C22×D5—41no—
238C2×C4×C5—22no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—22no—
242C2×C4×C5—22no—
243C2×C4×C5—22no—
244C2×C4×C5#11714yes—
245C2×C4×C5C4×D412yes—
246C2×C4×C5C4×D414yes—
247C2×C4×C5#12112yes—
248C2×C4×C5#12314yes—
249C2×C4×C5#12412yes—
250C2×C4×C5#12614yes—
251C2×C4×C5C23×C411yescharacteristic
252C2×C4×C5C22×D411yescharacteristic
253C2×C4×C5C2×#4111yescharacteristic
254C2×C4×C5C2×#4111yescharacteristic
255C2×C4×C5C2×#4111yescharacteristic
256C2×C4×C5C2×#4111yescharacteristic
257C22⋊D5—48no—
258C2×Dic5—22no—
259C2×Dic5—22no—
260C2×Dic5—22no—
261C2×Dic5—41no—
262C2×Dic5—41no—
263C2×Dic5—41no—
264C2×Dic5—41no—
265C2×Dic5—41no—
266C2×Dic5—41no—
267C2×Dic5—42no—
268C2×Dic5—42no—
269C2×Dic5—42no—
270C2×Dic5—42no—
271C2×Dic5—42no—
272C2×Dic5—42no—
273C2×Dic5—84no—
274C2×Dic5—84no—
275D20—22no—
276D20—22no—
277D20—41no—
278D20—41no—
279C4×D5—48no—
280Dic10—22no—
281Dic10—22no—
282Dic10—41no—
283Dic10—41no—
284C5×C8—22no—
285C5×C8—22no—
286C5×C8—24no—
287C2×#41—54no—
288C2×#41—102no—
289C2×#41—102no—
290C2×#41—102no—
291C2×#41—102no—
292C2×#41—102no—
293C22×Q8—51no—
294C22×D4—51no—
295C22×D4—101no—
296C23×C4—101no—
297C23×C4—101no—
298C23×C4—101no—
299C23×C4C22×D511yescharacteristic
300C2×M16—22no—
301C2×M16—22no—
302C2×M16C22⋊D514yes—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8C4×D514yes—
306C22×C8C22×D511yescharacteristic
307C22×C8C22×D511yescharacteristic
308C22×C8C22×D512yes—
309#117—22no—
310#117—22no—
311#117—54no—
312#117—102no—
313C2×C4⋊C4—51no—
314C2×C4⋊C4—51no—
315C2×C4⋊C4—52no—
316C2×C4⋊C4—52no—
317C2×C4⋊C4—102no—
318C2×C4⋊C4—102no—
319C2×C4⋊C4—102no—
320C2×C4⋊C4—102no—
321C2×C4⋊C4—101no—
322C2×C4⋊C4—101no—
323C2×C4⋊C4—102no—
324C2×C4⋊C4—102no—
325C2×C4⋊C4D2011yescharacteristic
326C2×C4⋊C4D2011yescharacteristic
327C2×C22⋊C4—101no—
328C2×C22⋊C4—102no—
329C2×C22⋊C4—102no—
330C2×C22⋊C4D2012yes—
331C2×C42—101no—
332C2×C42C22×D512yes—
333#103—54no—
334#103—54no—
335#103—104no—
336#102—54no—
337#102—54no—
338#102—104no—
339#97—24no—
340C4×C8—24no—
341#95—102no—
342#95—102no—
343#95—102no—
344#95—102no—
345C22×C5—22no—
346C22×C5—22no—
347C22×C5—22no—
348C22×C5—22no—
349C22×C5—22no—
350C22×C5—22no—
351C22×C5#38814yes—
352C22×C5C2×C4×D411yescharacteristic
353C22×C5C2×#12111yescharacteristic
354C22×C5C2×#12411yescharacteristic
355C22×C5#57611yescharacteristic
356C22×C5#57711yescharacteristic
357C22×C5#57711yescharacteristic
358C22×C5#57811yescharacteristic
359C2×D5—42no—
360C2×D5—42no—
361C2×D5—42no—
362C2×D5—81no—
363C2×D5—81no—
364C2×D5—81no—
365C2×D5—81no—
366C4×C5—42no—
367C4×C5—42no—
368C4×C5#38814yes—
369C4×C5C2×#11711yescharacteristic
370C4×C5C2×C4×D411yescharacteristic
371C4×C5C2×#12311yescharacteristic
372C4×C5C2×#12611yescharacteristic
373Dic5—42no—
374Dic5—42no—
375Dic5—42no—
376Dic5—82no—
377Dic5—82no—
378C24—101no—
379C24C2×D2011yescharacteristic
380#41—108no—
381#41—108no—
382#41—204no—
383#41—204no—
384C2×Q8—52no—
385C2×Q8—101no—
386C2×Q8—101no—
387C2×Q8—101no—
388C2×Q8—101no—
389C2×Q8—101no—
390C2×D4—52no—
391C2×D4—104no—
392C2×D4—101no—
393C2×D4—101no—
394C2×D4—101no—
395C2×D4—101no—
396C2×D4—101no—
397C2×D4—202no—
398C2×D4—202no—
399C22×C4—101no—
400C22×C4—101no—
401C22×C4—101no—
402C22×C4—104no—
403C22×C4—102no—
404C22×C4—102no—
405C22×C4—202no—
406C22×C4—202no—
407C22×C4—202no—
408C22×C4—202no—
409C22×C4—401no—
410C22×C4—401no—
411C22×C4—401no—
412C22×C4—401no—
413C22×C4C2×C4×D512yes—
414C22×C4C2×D2012yes—
415C22×C4C2×D2012yes—
416C22×C4C2×D2011yescharacteristic
417C22×C4#70912yes—
418C22×C4#70912yes—
419C22×C4#70912yes—
420C22×C4#70912yes—
421C22×C4C2×C22⋊D512yes—
422C22×C4C23×D511yescharacteristic
423M16—28no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—28no—
431C2×C8C2×C4×D512yes—
432C2×C8C2×C4×D512yes—
433C2×C8C2×C22⋊D514yes—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—52no—
437C4⋊C4—52no—
438C4⋊C4—101no—
439C4⋊C4—101no—
440C4⋊C4—102no—
441C4⋊C4—102no—
442C4⋊C4—102no—
443C4⋊C4—102no—
444C4⋊C4—202no—
445C4⋊C4—202no—
446C4⋊C4—402no—
447C4⋊C4—402no—
448C22⋊C4—24no—
449C22⋊C4—104no—
450C22⋊C4—402no—
451C22⋊C4—402no—
452C42—22no—
453C42—22no—
454C42—104no—
455C2×C5—24no—
456C2×C5C2×#38811yescharacteristic
457C2×C5#295111yescharacteristic
458C2×C5#295211yescharacteristic
459C2×C5#306211yescharacteristic
460C2×C5#306311yescharacteristic
461C2×C5#313611yescharacteristic
462C2×C5#313711yescharacteristic
463D5—82no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—102no—
469C23—201no—
470C23—201no—
471C23—201no—
472C23—201no—
473C23—201no—
474C23—201no—
475C23C4×D2012yes—
476C23#402112yes—
477C23#407612yes—
478C23C22×D2011yescharacteristic
479Q8—102no—
480Q8—102no—
481Q8—201no—
482Q8—201no—
483D4—102no—
484D4—102no—
485D4—208no—
486D4—201no—
487D4—201no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—102no—
495C2×C4—102no—
496C2×C4—102no—
497C2×C4—201no—
498C2×C4—201no—
499C2×C4—201no—
500C2×C4—201no—
501C2×C4—201no—
502C2×C4—201no—
503C2×C4—208no—
504C2×C4—202no—
505C2×C4—202no—
506C2×C4—202no—
507C2×C4—202no—
508C2×C4—202no—
509C2×C4—202no—
510C2×C4—404no—
511C2×C4—404no—
512C2×C4#401814yes—
513C2×C4C4×D2012yes—
514C2×C4#402112yes—
515C2×C4#402214yes—
516C2×C4C4×C22⋊D514yes—
517C2×C4#407514yes—
518C2×C4#407612yes—
519C2×C4C22×C4×D511yescharacteristic
520C2×C4C2×#70911yescharacteristic
521C2×C4C2×#70911yescharacteristic
522C2×C4C2×#70911yescharacteristic
523C2×C4C2×#70911yescharacteristic
524C2×C4C22×C22⋊D511yescharacteristic
525C8—22no—
526C8—22no—
527C8—24no—
528C5#2108711yescharacteristic, complemented, Sylow
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—202no—
536C22—202no—
537C22—202no—
538C22—401no—
539C22—401no—
540C22—401no—
541C22—401no—
542C22#6522914yes—
543C22C2×C4×D2011yescharacteristic
544C22C2×#402111yescharacteristic
545C22#6608511yescharacteristic
546C22#6608511yescharacteristic
547C22#6608611yescharacteristic
548C22#6608711yescharacteristic
549C22C2×#407611yescharacteristic
550C4—42no—
551C4—42no—
552C4—202no—
553C4—202no—
554C4—202no—
555C4—402no—
556C4—402no—
557C4#6522914yes—
558C4C2×#401811yescharacteristic
559C4C2×#402211yescharacteristic
560C4C2×C4×C22⋊D511yescharacteristic
561C4C2×#407511yescharacteristic
562C2—24no—
563C2—402no—
564C2#1061029311yescharacteristic
565C2#1061029411yescharacteristic
566C2#1061030511yescharacteristic
567C2#1061030611yescharacteristic
568C2C2×#6522911yescharacteristic
569C2#1061553311yescharacteristic
570C2#1061553911yescharacteristic
subgroupG/H
centerC23C22×D20
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#405C2
socleC23×C5C22×D4
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenterC2×#405D5
hypocenterC5#21087
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
1. C2×C5×#405C2semidirect
5. #10613965C2semidirect
6. #10602778C2semidirect
7. C2×#64771C2semidirect
8. #10600611C2semidirect
9. #10600610C2semidirect
10. #10600561C2semidirect
11. #10600560C2semidirect
23. #65402C4semidirect
98. #3953C2×C4semidirect
99. #3953C2×C4semidirect
101. #3948C2×C4semidirect
102. #3948C2×C4semidirect
105. C2×#405D5semidirect
528. C5#21087semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#4052
C2×#654222
C2×#654022
#106140012
#106139652
#106027782
C2×#647712
#106006112
#106006102
#106005612
#106005602
#210875