Finite Group Databasebeta

← #12070536

Subgroups — #12070536

Order 1280.

642 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10614872C211yesmaximal, characteristic
2#10614863C211yesmaximal, characteristic, complemented
3#10611067C211yesmaximal, characteristic, complemented
4C5×#1494C211yesmaximal, characteristic, complemented
5#10601590C211yesmaximal, characteristic, complemented
6C2×#64679C211yesmaximal, characteristic, complemented
7#10600519C211yesmaximal, characteristic, complemented
8#10599019C211yesmaximal, characteristic, complemented
9#10599010C211yesmaximal, characteristic, complemented
10#10599009C211yesmaximal, characteristic, complemented
11#10599007C211yesmaximal, characteristic, complemented
12#10598956C211yesmaximal, characteristic, complemented
13#10598946C211yesmaximal, characteristic, complemented
14#10598945C211yesmaximal, characteristic, complemented
15#10598944C211yesmaximal, characteristic, complemented
16C5×#529C2211yescharacteristic
17C2×C4×D20C2211yescharacteristic
18C5×#432C2211yescharacteristic
19C2×C4×C5×C8C2211yescharacteristic
20#65355C411yescharacteristic
21#65355C411yescharacteristic
22#65346C411yescharacteristic
23#65346C411yescharacteristic
24#65335C411yescharacteristic
25#65333C411yescharacteristic
26#65325C411yescharacteristic
27#65323C411yescharacteristic
28C4×#688C2211yescharacteristic
29C4×#688C2211yescharacteristic
30#65313C2211yescharacteristic, complemented
31Q8×#168C2211yescharacteristic, complemented
32C4×#686C2211yescharacteristic
33C4×#686C2211yescharacteristic
34#65301C2211yescharacteristic, complemented
35#65300—21no—
36D4×#168—21no—
37D4×#168C2211yescharacteristic, complemented
38#65289—21no—
39#65288—21no—
40#65214C2211yescharacteristic
41#65214C2211yescharacteristic
42C2×C4⋊#168C2211yescharacteristic
43#65127C2211yescharacteristic
44#65127C2211yescharacteristic
45#65123C2211yescharacteristic
46#65123C2211yescharacteristic
47#65015—21no—
48D5×C22⋊C8—21no—
49#64995—21no—
50#64995—21no—
51C5×Q8⋊C8C2211yescharacteristic, complemented
52C5×Q8⋊C8C2211yescharacteristic, complemented
53C5×D4⋊C8C2211yescharacteristic, complemented
54C5×D4⋊C8C2211yescharacteristic, complemented
55#64700C2211yescharacteristic, complemented
56#64700C2211yescharacteristic, complemented
57#64699C2211yescharacteristic, complemented
58#64699C2211yescharacteristic, complemented
59#64679C2211yescharacteristic
60#64679C2211yescharacteristic
61#64679C2211yescharacteristic
62#64679C2211yescharacteristic
63#64674C2211yescharacteristic, complemented
64#64674C2211yescharacteristic, complemented
65#64674C2211yescharacteristic, complemented
66#64674C2211yescharacteristic, complemented
67#18020—51nomaximal, Sylow
68C22×D20C2×C411yescharacteristic
69C22×C4×D5—21no—
70C22×C5×C8—21no—
71C5×#126—21no—
72C5×#125C2×C411yescharacteristic
73C5×#124C2×C411yescharacteristic
74C5×#123—21no—
75C5×C22⋊Q8C2×C411yescharacteristic
76C5×#121C2×C411yescharacteristic
77C4×C5×Q8C2311yescharacteristic
78C4×C5×D4—21no—
79C4×C5×D4C2311yescharacteristic
80C5×#117—21no—
81C2×C42×C5C2311yescharacteristic
82C2×#688C2×C411yescharacteristic
83C2×#688C2×C411yescharacteristic
84C2×#686C2×C411yescharacteristic
85C2×#686C2×C411yescharacteristic
86C2×#685—21no—
87C2×C4⋊Dic5C2×C411yescharacteristic
88C2×#681—21no—
89C22×#168—21no—
90C22×#168—21no—
91C2×#676—21no—
92C2×#676—21no—
93C2×C8×D5—21no—
94C2×C8×D5—21no—
95C4×D20—21no—
96C4×D20—21no—
97C4×D20C2×C411yescharacteristic
98C4×D20C2×C411yescharacteristic
99C4×D20C2311yescharacteristic
100C4×D20C2311yescharacteristic
101C5×C4⋊C8C2311yescharacteristic
102C5×C4⋊C8C2311yescharacteristic
103C5×C22⋊C8—21no—
104C5×#97—21no—
105C4×C5×C8—21no—
106C4×C5×C8C2311yescharacteristic
107C4×C5×C8C2311yescharacteristic
108#3967C2×C411yescharacteristic
109#3967C2×C411yescharacteristic
110#3964C2×C411yescharacteristic
111#3964C2×C411yescharacteristic
112C22⋊#168—21no—
113C22⋊#168—21no—
114C22⋊#168C2×C411yescharacteristic
115C22⋊#168C2×C411yescharacteristic
116C22⋊#168D411yescharacteristic
117C22⋊#168D411yescharacteristic
118#3952—21no—
119#3952—21no—
120#3952—21no—
121#3952—21no—
122#3941C2×C411yescharacteristic
123#3941C2×C411yescharacteristic
124#3941C2×C411yescharacteristic
125#3941C2×C411yescharacteristic
126#3940C2×C411yescharacteristic
127#3940C2×C411yescharacteristic
128#3939C2×C411yescharacteristic
129#3939C2×C411yescharacteristic
130C4⋊#168—21no—
131C4⋊#168—21no—
132C4⋊#168—21no—
133C4⋊#168C2×C411yescharacteristic
134C4⋊#168C2×C411yescharacteristic
135C4⋊#168D411yescharacteristic
136C4⋊#168D411yescharacteristic
137C4⋊#168C2311yescharacteristic
138C4⋊#168C2311yescharacteristic
139#3935—21no—
140C4×#168—21no—
141C4×#168C2311yescharacteristic
142C4×#168C2311yescharacteristic
143C4×#168C2311yescharacteristic
144C4×#168C2311yescharacteristic
145#3105—51no—
146#2994—51no—
147#2974—51no—
148#1594—51no—
149#1592—51no—
150#1583—51no—
151#1582—51no—
152#1581—51no—
153#1581—51no—
154#1580—51no—
155SD16⋊C8—51no—
156#1573—51no—
157#1494D511yescharacteristic, complemented
158C2×D4⋊C8—51no—
159#1452—51no—
160C23×D5—21no—
161C2×C5×Q8C22×C411yescharacteristic
162C2×C5×D4—21no—
163C2×C5×D4C22×C411yescharacteristic
164C22×C4×C5—21no—
165C22×C4×C5C22×C411yescharacteristic
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5#4111yescharacteristic
168C22×Dic5—21no—
169C2×D20—22no—
170C2×D20—21no—
171C2×D20—41no—
172C2×D20C22×C411yescharacteristic
173C2×D20C22×C411yescharacteristic
174C2×C4×D5—21no—
175C2×C4×D5—21no—
176C2×C4×D5—21no—
177C2×C4×D5—21no—
178C2×C4×D5—81no—
179C2×C4×D5—81no—
180C2×C5×C8—21no—
181C2×C5×C8—21no—
182C2×C5×C8—21no—
183C2×C5×C8—21no—
184C2×C5×C8—42no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—21no—
188C5×C4⋊C4—21no—
189C5×C4⋊C4C22×C411yescharacteristic
190C5×C4⋊C4C22×C411yescharacteristic
191C5×C22⋊C4—21no—
192C5×C22⋊C4—21no—
193C5×C22⋊C4—21no—
194C5×C22⋊C4—21no—
195C5×C22⋊C4—21no—
196C42×C5—21no—
197C42×C5C22×C411yescharacteristic
198C42×C5C2×D411yescharacteristic
199C42×C5#4111yescharacteristic
200C42×C5C2411yescharacteristic
201#688—22no—
202#688—41no—
203#686—22no—
204#686—41no—
205#685—21no—
206#685—21no—
207#685—21no—
208#685—21no—
209C4⋊Dic5—21no—
210C4⋊Dic5C22×C411yescharacteristic
211C4⋊Dic5C22×C411yescharacteristic
212#681—81no—
213C2×#168—21no—
214C2×#168—21no—
215C2×#168—81no—
216C2×#168—81no—
217C2×#168C22×C411yescharacteristic
218C2×#168C22×C411yescharacteristic
219C2×#168C22×C411yescharacteristic
220C2×#168C22×C411yescharacteristic
221C2×#168C2×D411yescharacteristic
222C2×#168C2×D411yescharacteristic
223C2×#168C2×D411yescharacteristic
224C2×#168C2×D411yescharacteristic
225#676—81no—
226#676—81no—
227C8×D5—81no—
228C8×D5—81no—
229#529C2×D511yescharacteristic
230C2×C4×D4—51no—
231#484—51no—
232#481—51no—
233#478—51no—
234#478—51no—
235#463—51no—
236#463—51no—
237#450—51no—
238#447—51no—
239#446—51no—
240C8×Q8—51no—
241C4×SD16—51no—
242C4×SD16—51no—
243C4×D8—51no—
244C4×D8—51no—
245#436—51no—
246#436—51no—
247#436—51no—
248#435—101no—
249#435—101no—
250#435—101no—
251C8×D4—51no—
252C8×D4—51no—
253C8×D4—51no—
254C8×D4—101no—
255#433—101no—
256#432C2×D511yescharacteristic
257#431—51no—
258#431—51no—
259#424—101no—
260C2×C4⋊C8—51no—
261#407—101no—
262C2×C22⋊C8—101no—
263C2×C4×C8C2×D511yescharacteristic
264#335—51no—
265#335—51no—
266#334—51no—
267#334—51no—
268Q8⋊C8C2×D511yescharacteristic, complemented
269Q8⋊C8C2×D511yescharacteristic, complemented
270D4⋊C8—51no—
271D4⋊C8—51no—
272D4⋊C8—51no—
273D4⋊C8—51no—
274D4⋊C8C2×D511yescharacteristic, complemented
275D4⋊C8C2×D511yescharacteristic, complemented
276#322—51no—
277#322—51no—
278#322—51no—
279#322—51no—
280C23×C5—21no—
281C23×C5C4×D411yescharacteristic
282C22×D5—21no—
283C22×D5—21no—
284C22×D5—42no—
285C22×D5—42no—
286C22×D5—81no—
287C5×Q8—21no—
288C5×Q8—21no—
289C5×D4—21no—
290C5×D4—21no—
291C5×D4—42no—
292C2×C4×C5—22no—
293C2×C4×C5—21no—
294C2×C4×C5—21no—
295C2×C4×C5—21no—
296C2×C4×C5—21no—
297C2×C4×C5—21no—
298C2×C4×C5—21no—
299C2×C4×C5—21no—
300C2×C4×C5—41no—
301C2×C4×C5—41no—
302C2×C4×C5C4×D411yescharacteristic
303C2×C4×C5C4×D411yescharacteristic
304C2×C4×C5C4×D411yescharacteristic
305C2×C4×C5C23×C411yescharacteristic
306C2×C4×C5C22×D411yescharacteristic
307C2×C4×C5C2×#4111yescharacteristic
308C2×Dic5—21no—
309C2×Dic5—21no—
310C2×Dic5—81no—
311D20—21no—
312D20—21no—
313D20—21no—
314D20—42no—
315D20#13112yes—
316C4×D5—41no—
317C4×D5—41no—
318C4×D5—41no—
319C4×D5—41no—
320C4×D5—81no—
321C4×D5—81no—
322C5×C8—41no—
323C5×C8—41no—
324C5×C8—41no—
325C5×C8—41no—
326#168—21no—
327#168—21no—
328#168—21no—
329#168—21no—
330#168—21no—
331#168—21no—
332#168—21no—
333#168—21no—
334#168—41no—
335#168—41no—
336C22×D4—51no—
337C23×C4—101no—
338C2×SD16—51no—
339C2×SD16—51no—
340C2×D8—51no—
341C2×D8—51no—
342C2×M16—101no—
343C2×M16—101no—
344C2×M16—101no—
345C22×C8—21no—
346C22×C8—101no—
347C22×C8—101no—
348C22×C8—101no—
349C22×C8—101no—
350#126—21no—
351#125C4×D511yescharacteristic
352#124C4×D511yescharacteristic
353#123—21no—
354C22⋊Q8C4×D511yescharacteristic
355#121C4×D511yescharacteristic
356C4×Q8C22×D511yescharacteristic
357C4×D4—21no—
358C4×D4—51no—
359C4×D4—51no—
360C4×D4—51no—
361C4×D4—51no—
362C4×D4—101no—
363C4×D4—101no—
364C4×D4C22×D511yescharacteristic
365#117—21no—
366C2×C4⋊C4—51no—
367C2×C22⋊C4—101no—
368C2×C42C22×D511yescharacteristic
369#107—51no—
370#107—51no—
371#106—51no—
372#106—51no—
373C4⋊C8—51no—
374C4⋊C8—51no—
375C4⋊C8—51no—
376C4⋊C8—51no—
377C4⋊C8—51no—
378C4⋊C8—51no—
379C4⋊C8—101no—
380C4⋊C8—101no—
381C4⋊C8—101no—
382C4⋊C8C22×D511yescharacteristic
383C4⋊C8C22×D511yescharacteristic
384#103—51no—
385#103—51no—
386#102—51no—
387#102—51no—
388#102—51no—
389#102—51no—
390#102—51no—
391#102—51no—
392C22⋊C8—21no—
393C22⋊C8—51no—
394C22⋊C8—51no—
395C22⋊C8—51no—
396C22⋊C8—51no—
397C22⋊C8—101no—
398C22⋊C8—101no—
399C22⋊C8—101no—
400C22⋊C8—101no—
401C22⋊C8—101no—
402C22⋊C8—101no—
403#97—21no—
404#97—101no—
405C4×C8—21no—
406C4×C8—51no—
407C4×C8—51no—
408C4×C8—51no—
409C4×C8—51no—
410C4×C8—101no—
411C4×C8C22×D511yescharacteristic
412C4×C8C22×D511yescharacteristic
413C22×C5—21no—
414C22×C5—21no—
415C22×C5—41no—
416C22×C5—41no—
417C22×C5—41no—
418C22×C5#44412yes—
419C22×C5C2×C4×D411yescharacteristic
420C2×D5—22no—
421C2×D5—41no—
422C2×D5—41no—
423C2×D5—41no—
424C2×D5—41no—
425C2×D5—41no—
426C2×D5—84no—
427C4×C5—21no—
428C4×C5—21no—
429C4×C5—21no—
430C4×C5—21no—
431C4×C5—41no—
432C4×C5—41no—
433C4×C5—41no—
434C4×C5C2×#13111yescharacteristic
435C4×C5#56811yescharacteristic
436C4×C5#57611yescharacteristic
437C4×C5#57711yescharacteristic
438Dic5—41no—
439Dic5—41no—
440C24—101no—
441C2×Q8C2×C4×D511yescharacteristic
442C2×D4—21no—
443C2×D4—51no—
444C2×D4—51no—
445C2×D4—102no—
446C2×D4—101no—
447C2×D4—201no—
448C2×D4C2×C4×D511yescharacteristic
449C22×C4—21no—
450C22×C4—101no—
451C22×C4—101no—
452C22×C4—101no—
453C22×C4—101no—
454C22×C4—101no—
455C22×C4—401no—
456C22×C4—401no—
457C22×C4C2×C4×D511yescharacteristic
458C22×C4D4×D511yescharacteristic
459C22×C4#71111yescharacteristic
460SD16—102no—
461SD16—201no—
462D8—102no—
463D8—201no—
464M16—401no—
465M16—401no—
466M16—401no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—42no—
472C2×C8—51no—
473C2×C8—51no—
474C2×C8—51no—
475C2×C8—51no—
476C2×C8—51no—
477C2×C8—51no—
478C2×C8—51no—
479C2×C8—51no—
480C2×C8—101no—
481C2×C8—101no—
482C2×C8—401no—
483C2×C8—401no—
484C2×C8—401no—
485C2×C8—401no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—51no—
491C4⋊C4—51no—
492C4⋊C4—101no—
493C4⋊C4C2×C4×D511yescharacteristic
494C4⋊C4C2×C4×D511yescharacteristic
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—101no—
501C22⋊C4—101no—
502C22⋊C4—101no—
503C22⋊C4—101no—
504C42—21no—
505C42C2×C4×D511yescharacteristic
506C42D4×D511yescharacteristic
507C42#71111yescharacteristic
508C42C23×D511yescharacteristic
509C2×C5—22no—
510C2×C5—81no—
511C2×C5#293311yescharacteristic
512C2×C5#294611yescharacteristic
513C2×C5C2×#44411yescharacteristic
514D5—42no—
515D5—81no—
516C23—21no—
517C23—101no—
518C23—101no—
519C23—202no—
520C23—202no—
521C23—401no—
522C23#402711yescharacteristic
523Q8—21no—
524Q8—21no—
525D4—21no—
526D4—21no—
527D4—42no—
528D4—52no—
529D4—101no—
530D4—101no—
531D4—101no—
532D4—202no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—101no—
544C2×C4—101no—
545C2×C4—201no—
546C2×C4—201no—
547C2×C4—201no—
548C2×C4—201no—
549C2×C4—401no—
550C2×C4—401no—
551C2×C4—401no—
552C2×C4#402711yescharacteristic
553C2×C4#402711yescharacteristic
554C2×C4#402711yescharacteristic
555C2×C4C22×C4×D511yescharacteristic
556C2×C4C2×D4×D511yescharacteristic
557C2×C4C2×#71111yescharacteristic
558C8—41no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—101no—
563C8—101no—
564C8—101no—
565C8—101no—
566C8—101no—
567C8—101no—
568C8—101no—
569C8—101no—
570C8—201no—
571C8—201no—
572C5#1802011yescharacteristic, complemented, Sylow
573C22—21no—
574C22—21no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—102no—
579C22—201no—
580C22—201no—
581C22—201no—
582C22—201no—
583C22—201no—
584C22—404no—
585C22#6511212yes—
586C22C2×#402711yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—201no—
595C4—201no—
596C4D5×#13111yescharacteristic
597C4#6608411yescharacteristic
598C4#6610311yescharacteristic
599C4#6610411yescharacteristic
600C2—22no—
601C2—81no—
602C2—202no—
603C2—401no—
604C2#1061039011yescharacteristic
605C2#1061055111yescharacteristic
606C2C2×#6511211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
FittingC5×#1494C2
socleC22×C5C2×C4×D4
prime radical#65325C4
cosocleC42×C5C24
hypercenter#1494D5
hypocenterC5#18020
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1494

Complements

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

NH ≅ G/Ntype
2. #10614863C2semidirect
3. #10611067C2semidirect
4. C5×#1494C2semidirect
5. #10601590C2semidirect
6. C2×#64679C2semidirect
7. #10600519C2semidirect
8. #10599019C2semidirect
9. #10599010C2semidirect
10. #10599009C2semidirect
11. #10599007C2semidirect
12. #10598956C2semidirect
13. #10598946C2semidirect
14. #10598945C2semidirect
15. #10598944C2semidirect
30. #65313C22semidirect
31. Q8×#168C22semidirect
34. #65301C22semidirect
37. D4×#168C22semidirect
51. C5×Q8⋊C8C22semidirect
52. C5×Q8⋊C8C22semidirect
53. C5×D4⋊C8C22semidirect
54. C5×D4⋊C8C22semidirect
55. #64700C22semidirect
56. #64700C22semidirect
57. #64699C22semidirect
58. #64699C22semidirect
63. #64674C22semidirect
64. #64674C22semidirect
65. #64674C22semidirect
66. #64674C22semidirect
157. #1494D5semidirect
268. Q8⋊C8C2×D5semidirect
269. Q8⋊C8C2×D5semidirect
274. D4⋊C8C2×D5semidirect
275. D4⋊C8C2×D5semidirect
572. C5#18020semidirect

Maximal subgroups

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

subgroupindex
#106148722
#106148632
#106110672
C5×#14942
#106015902
C2×#646792
#106005192
#105990192
#105990102
#105990092
#105990072
#105989562
#105989462
#105989452
#105989442
#180205