Finite Group Databasebeta

← #12100568

Subgroups — #12100568

Order 1280.

1970 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66139C211yesmaximal, characteristic
2C2×#65857C211yesmaximal, characteristic, complemented
3C2×C5×#418C211yesmaximal, characteristic, complemented
4C22×#3964C211yesmaximal, characteristic, complemented
5C2×#65418C211yesmaximal, characteristic, complemented
6#10602778C211yesmaximal, characteristic, complemented
7#10602373C211yesmaximal, characteristic, complemented
8#10600994C214yesmaximal, complemented
9#10600993C214yesmaximal, complemented
10C23×C5×D4C2211yescharacteristic
11C22×#709C2211yescharacteristic
12C22×C5×M16C2211yescharacteristic
13C2×C5×#117C2211yescharacteristic
14C2×#4099—22no—
15#66139—24no—
16#66139C418yescomplemented
17C2×#4086—21no—
18C2×#4084C2214yes—
19C2×#4082—21no—
20C2×#4072C2211yescharacteristic
21C22×C4⋊Dic5C2211yescharacteristic
22C23×#168C2211yescharacteristic
23#65857—24no—
24#65857—24no—
25C2×#4038C2212yescomplemented
26C2×D5×C4⋊C4C2212yescomplemented
27C2×#4027—22no—
28C2×C4×D20—21no—
29C2×C4×Dic10—21no—
30C5×#418—24no—
31C2×C5×#102C2212yes—
32C2×C5×#102C2212yes—
33C2×#3964—24no—
34C2×#3964C2214yes—
35#65418—24no—
36C2×#3952C2214yescomplemented
37#64771C2214yescomplemented
38#64749C2212yescomplemented
39#64749C2212yescomplemented
40#20914—51nomaximal, Sylow
41C25×C5—21no—
42C22×C5×D4—22no—
43C22×C5×D4—22no—
44C22×C5×D4—21no—
45C22×C5×D4—22no—
46C22×C5×D4—22no—
47C22×C5×D4—22no—
48C22×C5×D4C2×C414yescomplemented
49C22×C5×D4C2312yes—
50C23×C4×C5C2311yescharacteristic
51C22×C22⋊D5—24no—
52C22×C22⋊D5D412yes—
53C23×Dic5—21no—
54C2×#709—22no—
55C2×#709—22no—
56C2×#709—22no—
57C2×#709—22no—
58C2×#709C2×C414yes—
59C2×#709D414yes—
60C22×D20D411yescharacteristic
61C22×C4×D5—22no—
62C22×C4×D5C2312yes—
63C22×Dic10D411yescharacteristic
64C2×C5×M16—22no—
65C2×C5×M16—22no—
66C2×C5×M16—22no—
67C22×C5×C8C2312yes—
68C5×#117—22no—
69C5×#117D414yescomplemented
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C22⋊C4—21no—
73C2×C42×C5—21no—
74#4099—216no—
75C2×C22⋊Dic5—21no—
76C2×C22⋊Dic5—24no—
77#4086—28no—
78#4084—28no—
79#4084C2×C4116yescomplemented
80#4082—28no—
81C2×#685—22no—
82#4072—22no—
83#4072C2×C414yescomplemented
84C2×C4⋊Dic5—21no—
85C2×C4⋊Dic5—22no—
86C2×C4⋊Dic5D414yescomplemented
87C2×C4⋊Dic5C2312yes—
88C2×C4⋊Dic5C2312yes—
89C2×#683—22no—
90C2×C4×Dic5—21no—
91C2×C4×Dic5—22no—
92C22×#168—22no—
93C22×#168—22no—
94C22×#168—22no—
95C22×#168C2312yes—
96#4038—24no—
97#4038—24no—
98D5×C4⋊C4—24no—
99D5×C4⋊C4—24no—
100#4027—216no—
101C4×D20—28no—
102C4×Dic10—28no—
103C5×#102—24no—
104C5×#102—24no—
105#3964—28no—
106#3952—28no—
107C2×#535—51no—
108C2×#520—51no—
109C2×#418D511yescharacteristic, complemented
110C22×#102—51no—
111C2×#408—51no—
112#1875—54no—
113#1874—54no—
114#1736—51no—
115#1735—51no—
116C24×C5—22no—
117C24×C5—24no—
118C24×C5—42no—
119C24×C5—42no—
120C24×C5—82no—
121C24×C5—82no—
122C24×C5C2×D411yescharacteristic
123C23×D5C2×D412yes—
124C2×C5×D4—24no—
125C2×C5×D4—28no—
126C2×C5×D4—28no—
127C2×C5×D4—22no—
128C2×C5×D4—22no—
129C2×C5×D4—22no—
130C2×C5×D4—22no—
131C2×C5×D4—22no—
132C2×C5×D4—42no—
133C2×C5×D4—44no—
134C2×C5×D4—44no—
135C2×C5×D4—42no—
136C2×C5×D4C22×C414yes—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5C22×C412yes—
140C22×C4×C5C2×D411yescharacteristic
141C22×C4×C5C2×D412yes—
142C22×C4×C5C2×D412yes—
143C22×C4×C5C2×D412yes—
144C22×C4×C5#4112yes—
145C22×C4×C5#4112yes—
146C22×C4×C5C2411yescharacteristic
147C2×C22⋊D5—24no—
148C2×C22⋊D5—24no—
149C2×C22⋊D5—216no—
150C2×C22⋊D5—84no—
151C2×C22⋊D5—84no—
152C2×C22⋊D5C22⋊C418yes—
153C22×Dic5—22no—
154C22×Dic5—22no—
155C22×Dic5—24no—
156C22×Dic5—82no—
157C22×Dic5C2×D412yes—
158#709—28no—
159#709—28no—
160#709—28no—
161#709—28no—
162C2×D20—22no—
163C2×D20—22no—
164C2×D20C22⋊C414yes—
165C2×C4×D5—22no—
166C2×C4×D5—22no—
167C2×C4×D5—22no—
168C2×C4×D5—22no—
169C2×C4×D5—28no—
170C2×C4×D5—84no—
171C2×C4×D5C22×C414yes—
172C2×C4×D5C2×D414yes—
173C2×Dic10—22no—
174C2×Dic10—22no—
175C2×Dic10C22⋊C414yes—
176C5×M16—44no—
177C2×C5×C8—22no—
178C2×C5×C8—22no—
179C2×C5×C8—22no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4C2×D412yes—
183C5×C4⋊C4C2×D412yes—
184C5×C22⋊C4—24no—
185C42×C5—24no—
186C22⋊Dic5—24no—
187C22⋊Dic5—216no—
188#685—28no—
189C4⋊Dic5—24no—
190C4⋊Dic5—22no—
191C4⋊Dic5—22no—
192C4⋊Dic5C22×C414yes—
193C4⋊Dic5C2×D414yes—
194#683—28no—
195C4×Dic5—28no—
196C4×Dic5—24no—
197C2×#168—22no—
198C2×#168—22no—
199C2×#168—22no—
200C2×#168—44no—
201C22×#41—51no—
202C23×D4C2×D511yescharacteristic
203C22×M16C2×D511yescharacteristic
204C23×C8—51no—
205#535—58no—
206#535—104no—
207C2×#127—101no—
208C2×#124—101no—
209C2×#121—54no—
210C2×#120—102no—
211#520—104no—
212#520—104no—
213C2×C4×Q8—101no—
214C2×C4×D4—101no—
215C2×C4×D4—102no—
216C2×#117—51no—
217C2×#117—52no—
218C2×#117C2×D511yescharacteristic
219C22×C4⋊C4—51no—
220C22×C4⋊C4—52no—
221#418—24no—
222C2×#102—54no—
223C2×#102—104no—
224C2×#102C2×D512yes—
225C2×#102C2×D512yes—
226#408—104no—
227C2×C22⋊C8—54no—
228#340—54no—
229#340—52no—
230#340—52no—
231C23×C5—22no—
232C23×C5—22no—
233C23×C5—24no—
234C23×C5—42no—
235C23×C5—42no—
236C23×C5—44no—
237C23×C5—44no—
238C23×C5—48no—
239C23×C5—88no—
240C23×C5—88no—
241C23×C5—84no—
242C23×C5—84no—
243C23×C5C2×C22⋊C412yes—
244C23×C5#12012yes—
245C23×C5#12312yes—
246C23×C5C22×D411yescharacteristic
247C22×D5—22no—
248C22×D5—22no—
249C22×D5—24no—
250C22×D5—24no—
251C22×D5C2×C22⋊C414yes—
252C5×D4—24no—
253C5×D4—24no—
254C5×D4—42no—
255C5×D4—42no—
256C5×D4—48no—
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—42no—
260C2×C4×C5—42no—
261C2×C4×C5C2×C22⋊C412yes—
262C2×C4×C5C4×D414yes—
263C2×C4×C5C4×D414yes—
264C2×C4×C5#12012yes—
265C2×C4×C5#12114yes—
266C2×C4×C5#12114yes—
267C2×C4×C5#12312yes—
268C2×C4×C5C23×C411yescharacteristic
269C2×C4×C5C22×D411yescharacteristic
270C2×C4×C5C22×D411yescharacteristic
271C2×C4×C5C22×D411yescharacteristic
272C2×C4×C5C2×#4111yescharacteristic
273C2×C4×C5C2×#4111yescharacteristic
274C22⋊D5—216no—
275C22⋊D5—816no—
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5—24no—
279C2×Dic5—24no—
280C2×Dic5—44no—
281C2×Dic5—42no—
282C2×Dic5—42no—
283C2×Dic5—84no—
284C2×Dic5C2×C22⋊C414yes—
285D20—28no—
286C4×D5—24no—
287C4×D5—24no—
288C4×D5—24no—
289C4×D5—24no—
290C4×D5—88no—
291Dic10—28no—
292C5×C8—42no—
293#168—42no—
294C25—21no—
295C2×#41—54no—
296C2×#41—54no—
297C2×#41—102no—
298C2×#41—102no—
299C2×#41—102no—
300C2×#41—102no—
301C22×Q8—51no—
302C22×D4—22no—
303C22×D4—22no—
304C22×D4—21no—
305C22×D4—22no—
306C22×D4—22no—
307C22×D4—22no—
308C22×D4—52no—
309C22×D4—51no—
310C22×D4—104no—
311C22×D4C4×D514yescomplemented
312C22×D4C22×D512yes—
313C23×C4—52no—
314C23×C4—102no—
315C23×C4—101no—
316C23×C4C22×D511yescharacteristic
317C2×M16—22no—
318C2×M16—22no—
319C2×M16—22no—
320C22×C8—52no—
321C22×C8—102no—
322C22×C8—102no—
323C22×C8—102no—
324C22×C8C22×D512yes—
325#127—108no—
326#124—108no—
327#121—516no—
328#121—108no—
329#120—1016no—
330C4×Q8—108no—
331C4×D4—108no—
332C4×D4—1016no—
333#117—22no—
334#117—54no—
335#117—104no—
336#117—102no—
337#117—104no—
338#117C22⋊D514yescomplemented
339C2×C4⋊C4—54no—
340C2×C4⋊C4—52no—
341C2×C4⋊C4—52no—
342C2×C4⋊C4—104no—
343C2×C4⋊C4—102no—
344C2×C4⋊C4—101no—
345C2×C4⋊C4—102no—
346C2×C4⋊C4—104no—
347C2×C4⋊C4C22×D511yescharacteristic
348C2×C4⋊C4C22×D511yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—102no—
351C2×C22⋊C4—104no—
352C2×C22⋊C4—101no—
353C2×C42—21no—
354C2×C42—102no—
355C2×C42—101no—
356#102—24no—
357#102—24no—
358#102—108no—
359C22⋊C8—108no—
360C22×C5—24no—
361C22×C5—24no—
362C22×C5—22no—
363C22×C5—22no—
364C22×C5—48no—
365C22×C5—44no—
366C22×C5—816no—
367C22×C5—84no—
368C22×C5—82no—
369C22×C5—82no—
370C22×C5#38614yes—
371C22×C5C22×C22⋊C411yescharacteristic
372C22×C5C2×#12011yescharacteristic
373C22×C5C2×#12311yescharacteristic
374C22×C5#57612yes—
375C22×C5#57712yes—
376C2×D5—28no—
377C2×D5—24no—
378C2×D5—44no—
379C2×D5—42no—
380C2×D5—42no—
381C4×C5—42no—
382C4×C5—42no—
383C4×C5#38614yes—
384C4×C5C2×C4×D411yescharacteristic
385C4×C5C2×C4×D411yescharacteristic
386C4×C5C2×#12111yescharacteristic
387C4×C5C2×#12111yescharacteristic
388Dic5—24no—
389Dic5—44no—
390Dic5—44no—
391C24—22no—
392C24—24no—
393C24—42no—
394C24—42no—
395C24—52no—
396C24—82no—
397C24—82no—
398C24D4×D511yescharacteristic
399#41—108no—
400#41—108no—
401#41—108no—
402#41—108no—
403C2×Q8—54no—
404C2×Q8—102no—
405C2×Q8—102no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—58no—
419C2×D4—54no—
420C2×D4—104no—
421C2×D4—104no—
422C2×D4—1016no—
423C2×D4—102no—
424C2×D4—102no—
425C2×D4—404no—
426C2×D4—404no—
427C2×D4C2×C4×D514yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—52no—
431C22×C4—54no—
432C22×C4—54no—
433C22×C4—108no—
434C22×C4—102no—
435C22×C4—102no—
436C22×C4—102no—
437C22×C4—102no—
438C22×C4—102no—
439C22×C4—102no—
440C22×C4—104no—
441C22×C4—404no—
442C22×C4—402no—
443C22×C4C2×C4×D512yes—
444C22×C4#70912yes—
445C22×C4D4×D512yes—
446C22×C4D4×D512yes—
447C22×C4D4×D511yescharacteristic
448C22×C4#71112yes—
449C22×C4C2×C22⋊D512yes—
450C22×C4C23×D511yescharacteristic
451M16—44no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—102no—
456C2×C8—102no—
457C2×C8—102no—
458C2×C8—204no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—54no—
462C4⋊C4—54no—
463C4⋊C4—108no—
464C4⋊C4—102no—
465C4⋊C4—102no—
466C4⋊C4—104no—
467C4⋊C4C2×C22⋊D512yes—
468C4⋊C4C2×C22⋊D512yes—
469C22⋊C4—24no—
470C22⋊C4—108no—
471C22⋊C4—1016no—
472C22⋊C4—104no—
473C42—24no—
474C42—108no—
475C42—104no—
476C2×C5—24no—
477C2×C5—84no—
478C2×C5C2×#38611yescharacteristic
479C2×C5#289712yes—
480C2×C5#300811yescharacteristic
481C2×C5#301011yescharacteristic
482C2×C5#309612yes—
483D5—44no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—42no—
491C23—48no—
492C23—54no—
493C23—88no—
494C23—84no—
495C23—84no—
496C23—88no—
497C23—102no—
498C23—104no—
499C23—104no—
500C23—102no—
501C23D5×C22⋊C412yes—
502C23#402912yes—
503C23#408312yes—
504C23C2×D4×D511yescharacteristic
505Q8—108no—
506D4—24no—
507D4—24no—
508D4—42no—
509D4—42no—
510D4—48no—
511D4—1016no—
512D4—108no—
513D4—4016no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—54no—
519C2×C4—102no—
520C2×C4—102no—
521C2×C4—104no—
522C2×C4—104no—
523C2×C4—104no—
524C2×C4—104no—
525C2×C4—104no—
526C2×C4—104no—
527C2×C4—204no—
528C2×C4—202no—
529C2×C4—202no—
530C2×C4—408no—
531C2×C4—404no—
532C2×C4D5×C22⋊C412yes—
533C2×C4#402714yes—
534C2×C4#402912yes—
535C2×C4#403014yes—
536C2×C4C4×C22⋊D514yes—
537C2×C4#408312yes—
538C2×C4#408414yes—
539C2×C4C22×C4×D511yescharacteristic
540C2×C4C2×#70911yescharacteristic
541C2×C4C2×D4×D511yescharacteristic
542C2×C4C2×D4×D511yescharacteristic
543C2×C4C2×#71111yescharacteristic
544C2×C4C22×C22⋊D511yescharacteristic
545C8—42no—
546C8—202no—
547C5#2091411yescharacteristic, complemented, Sylow
548C22—22no—
549C22—22no—
550C22—24no—
551C22—24no—
552C22—44no—
553C22—48no—
554C22—816no—
555C22—82no—
556C22—82no—
557C22—84no—
558C22—104no—
559C22—108no—
560C22—202no—
561C22—202no—
562C22—204no—
563C22#6524514yes—
564C22C2×D5×C22⋊C411yescharacteristic
565C22C2×#402911yescharacteristic
566C22#6610812yes—
567C22#6610912yes—
568C22C2×#408311yescharacteristic
569C4—42no—
570C4—42no—
571C4—104no—
572C4—204no—
573C4—204no—
574C4#6524514yes—
575C4C2×#402711yescharacteristic
576C4C2×#403011yescharacteristic
577C4C2×C4×C22⋊D511yescharacteristic
578C4C2×#408411yescharacteristic
579C2—24no—
580C2—84no—
581C2—204no—
582C2#1061065012yes—
583C2#1061065612yes—
584C2C2×#6524511yescharacteristic
585C2#1061567811yescharacteristic
586C2#1061567911yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#418C2
socleC23×C5C22×D4
prime radicalC2×#66139C2
cosocleC22×C4×C5C24
hypercenterC2×#418D5
hypocenterC5#20914
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×#65857C2semidirect
3. C2×C5×#418C2semidirect
4. C22×#3964C2semidirect
5. C2×#65418C2semidirect
6. #10602778C2semidirect
7. #10602373C2semidirect
8. #10600994C2semidirect
9. #10600993C2semidirect
16. #66139C4semidirect
25. C2×#4038C22semidirect
26. C2×D5×C4⋊C4C22semidirect
36. C2×#3952C22semidirect
37. #64771C22semidirect
38. #64749C22semidirect
39. #64749C22semidirect
48. C22×C5×D4C2×C4semidirect
69. C5×#117D4semidirect
79. #4084C2×C4semidirect
83. #4072C2×C4semidirect
86. C2×C4⋊Dic5D4semidirect
109. C2×#418D5semidirect
311. C22×D4C4×D5semidirect
338. #117C22⋊D5semidirect
547. C5#20914semidirect

Maximal subgroups

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

subgroupindex
C2×#661392
C2×#658572
C2×C5×#4182
C22×#39642
C2×#654182
#106027782
#106023732
#106009942
#106009932
#209145