Finite Group Databasebeta

← #12071040

Subgroups — #12071040

Order 1280.

718 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2974C211yesmaximal, characteristic, complemented
2C2×C4×#686C211yesmaximal, characteristic
3#10611018C211yesmaximal, characteristic, complemented
4#10602392C211yesmaximal, characteristic, complemented
5#10601655C212yesmaximal, complemented
6#10601654C212yesmaximal, complemented
7#10601634C212yesmaximal, complemented
8#10601633C212yesmaximal, complemented
9#10601592C211yesmaximal, characteristic, complemented
10#10601587C211yesmaximal, characteristic, complemented
11#10601534C211yesmaximal, characteristic, complemented
12C2×C4×C5×D4C2211yescharacteristic
13C22×#686C411yescharacteristic
14C2×C4×D20C2211yescharacteristic
15C5×#436C2212yescomplemented
16C5×#435—22no—
17C5×C8×D4C2212yescomplemented
18C5×#432C2211yescharacteristic
19C5×#431C2211yescharacteristic
20C5×#407—21no—
21C2×C5×C22⋊C8—21no—
22C2×#3964C411yescharacteristic
23C4×#686C411yescharacteristic
24C4×#686C411yescharacteristic
25C4×#686C411yescharacteristic
26C4×#686C411yescharacteristic
27C4×#686C2211yescharacteristic
28C4×#686C2211yescharacteristic
29C4×#686C2211yescharacteristic
30C4×#686C2211yescharacteristic
31C2×#3941C411yescharacteristic
32C2×#3939C411yescharacteristic
33#65215C2211yescharacteristic
34C4×#681C2211yescharacteristic
35C2×C4×#168C2211yescharacteristic
36#65125—22no—
37#65015—21no—
38D5×C22⋊C8—21no—
39#64977C2212yescomplemented
40C8×D20C2212yescomplemented
41D4⋊#168C2212yescomplemented
42D4⋊#168C2212yescomplemented
43#64703C2212yescomplemented
44#64703C2212yescomplemented
45#64700C2212yescomplemented
46#64699C2212yescomplemented
47#64675C2212yescomplemented
48#64674C2212yescomplemented
49#18064—51nomaximal, Sylow
50C22×C5×D4C2×C411yescharacteristic
51C23×C4×C5—21no—
52C22×D20C2×C411yescharacteristic
53C22×C4×D5—21no—
54C2×C5×M16—22no—
55C22×C5×C8—22no—
56C4×C5×D4—21no—
57C4×C5×D4—21no—
58C4×C5×D4C2×C411yescharacteristic
59C4×C5×D4C2×C411yescharacteristic
60C4×C5×D4C2311yescharacteristic
61C4×C5×D4C2311yescharacteristic
62C2×C5×C4⋊C4C2×C411yescharacteristic
63C2×C5×C22⋊C4—21no—
64C2×C42×C5C2311yescharacteristic
65C2×#686—24no—
66C2×#686C2×C411yescharacteristic
67C2×#686C2×C411yescharacteristic
68C2×#686C2×C411yescharacteristic
69C2×#686C2×C411yescharacteristic
70C2×#685—21no—
71C2×C4⋊Dic5C2×C411yescharacteristic
72C2×#681—21no—
73C22×#168C2×C411yescharacteristic
74C22×#168C2×C411yescharacteristic
75C2×#676—22no—
76C2×C8×D5—22no—
77C4×D20—21no—
78C4×D20—21no—
79C4×D20C2×C411yescharacteristic
80C4×D20C2×C411yescharacteristic
81C4×D20C2311yescharacteristic
82C4×D20C2311yescharacteristic
83C5×C4⋊C8D412yes—
84C5×C4⋊C8C2312yes—
85C5×C22⋊C8—21no—
86C5×C22⋊C8—22no—
87C5×C22⋊C8—22no—
88C5×C22⋊C8D411yescharacteristic
89C5×C22⋊C8D411yescharacteristic
90C5×#97—21no—
91C4×C5×C8C2312yes—
92#3964C2×C411yescharacteristic
93#3964C2×C411yescharacteristic
94#3964C2×C411yescharacteristic
95#3964C2×C411yescharacteristic
96C22⋊#168—21no—
97#3952—22no—
98#3952—22no—
99#3941C2×C411yescharacteristic
100#3941C2×C411yescharacteristic
101#3941C2×C411yescharacteristic
102#3941C2×C411yescharacteristic
103#3939C2×C411yescharacteristic
104#3939C2×C411yescharacteristic
105#3939C2×C411yescharacteristic
106#3939C2×C411yescharacteristic
107C4⋊#168C2312yes—
108#3935—21no—
109#3935—21no—
110C4×#168C2×C411yescharacteristic
111C4×#168C2×C411yescharacteristic
112C4×#168C2312yes—
113C4×#168C2311yescharacteristic
114C4×#168C2311yescharacteristic
115#2974—51no—
116#2974D511yescharacteristic, complemented
117C2×C4×D8—51no—
118#1594—52no—
119#1581—52no—
120#1581—52no—
121#1580—52no—
122#1568—51no—
123#1490—51no—
124#1490—51no—
125#1451—51no—
126C24×C5—21no—
127C23×D5—21no—
128C2×C5×D4—22no—
129C2×C5×D4—21no—
130C2×C5×D4—41no—
131C2×C5×D4C22×C411yescharacteristic
132C2×C5×D4C22×C411yescharacteristic
133C22×C4×C5—21no—
134C22×C4×C5—21no—
135C22×C4×C5—21no—
136C22×C4×C5—21no—
137C22×C4×C5—21no—
138C22×C4×C5—82no—
139C22×C4×C5C22×C411yescharacteristic
140C22×C4×C5C2×D411yescharacteristic
141C22×C4×C5#4111yescharacteristic
142C22×Dic5—21no—
143C2×D20—22no—
144C2×D20—21no—
145C2×D20—41no—
146C2×D20C22×C411yescharacteristic
147C2×D20C22×C411yescharacteristic
148C2×C4×D5—21no—
149C2×C4×D5—21no—
150C2×C4×D5—21no—
151C2×C4×D5—21no—
152C2×C4×D5—82no—
153C5×M16—82no—
154C2×C5×C8—22no—
155C2×C5×C8—82no—
156C2×C5×C8C2×D412yes—
157C2×C5×C8C2×D412yes—
158C5×C4⋊C4—21no—
159C5×C4⋊C4C22×C411yescharacteristic
160C5×C4⋊C4C22×C411yescharacteristic
161C5×C22⋊C4—21no—
162C5×C22⋊C4—21no—
163C5×C22⋊C4—21no—
164C5×C22⋊C4—21no—
165C42×C5C22×C411yescharacteristic
166C42×C5C2×D411yescharacteristic
167C42×C5#4111yescharacteristic
168C42×C5C2411yescharacteristic
169#686—22no—
170#686—22no—
171#686—22no—
172#686M1614yescomplemented
173#685—21no—
174#685—21no—
175#685—21no—
176#685—21no—
177C4⋊Dic5—21no—
178C4⋊Dic5C22×C411yescharacteristic
179C4⋊Dic5C22×C411yescharacteristic
180#681—42no—
181C2×#168—22no—
182C2×#168—22no—
183C2×#168—22no—
184C2×#168—41no—
185C2×#168C22×C411yescharacteristic
186C2×#168C22×C411yescharacteristic
187C2×#168C22×C411yescharacteristic
188C2×#168C22×C411yescharacteristic
189#676—82no—
190C8×D5—82no—
191C22×D8—51no—
192C2×C4×D4—51no—
193C2×C4×D4C2×D511yescharacteristic
194C4×D8—51no—
195C4×D8—51no—
196C4×D8—51no—
197C4×D8—51no—
198C4×D8—51no—
199C4×D8—51no—
200C4×D8—51no—
201C4×D8—51no—
202#436—52no—
203#436C2×D512yescomplemented
204#435—22no—
205#435—102no—
206C8×D4—52no—
207C8×D4C2×D512yescomplemented
208#432—51no—
209#432C2×D511yescharacteristic
210#431C2×D511yescharacteristic
211C2×#107—51no—
212C2×#102—51no—
213C2×#102—51no—
214#407—21no—
215#407—101no—
216C2×C22⋊C8—21no—
217C2×C22⋊C8—101no—
218C4×M16—51no—
219C2×C4×C8—51no—
220#335—52no—
221#334—52no—
222D4⋊C8—52no—
223D4⋊C8—52no—
224D4⋊C8—52no—
225D4⋊C8—52no—
226#322—52no—
227#322—52no—
228C23×C5—21no—
229C23×C5—21no—
230C23×C5—42no—
231C23×C5—42no—
232C23×C5—81no—
233C23×C5C4×D411yescharacteristic
234C22×D5—21no—
235C22×D5—21no—
236C22×D5—42no—
237C22×D5—42no—
238C22×D5—81no—
239C5×D4—21no—
240C5×D4—21no—
241C5×D4—21no—
242C5×D4—42no—
243C5×D4C2×M1612yes—
244C2×C4×C5—22no—
245C2×C4×C5—21no—
246C2×C4×C5—21no—
247C2×C4×C5—22no—
248C2×C4×C5—21no—
249C2×C4×C5—21no—
250C2×C4×C5—42no—
251C2×C4×C5—42no—
252C2×C4×C5—81no—
253C2×C4×C5—81no—
254C2×C4×C5—81no—
255C2×C4×C5C4×D411yescharacteristic
256C2×C4×C5C4×D411yescharacteristic
257C2×C4×C5C4×D411yescharacteristic
258C2×C4×C5C23×C411yescharacteristic
259C2×C4×C5C22×D411yescharacteristic
260C2×C4×C5C2×#4111yescharacteristic
261C2×Dic5—21no—
262C2×Dic5—21no—
263C2×Dic5—81no—
264D20—21no—
265D20—21no—
266D20—21no—
267D20—42no—
268D20C2×M1612yes—
269C4×D5—42no—
270C4×D5—42no—
271C4×D5—81no—
272C4×D5—81no—
273C5×C8—22no—
274C5×C8—22no—
275C5×C8—42no—
276#168—21no—
277#168—21no—
278#168—21no—
279#168—42no—
280#168—42no—
281#168C2×M1612yes—
282C22×D4—51no—
283C22×D4C4×D511yescharacteristic
284C23×C4—21no—
285C23×C4—101no—
286C2×D8—51no—
287C2×D8—51no—
288C2×D8—51no—
289C2×D8—51no—
290C2×D8—104no—
291C2×M16—22no—
292C2×M16—101no—
293C2×M16—102no—
294C22×C8—22no—
295C22×C8—51no—
296C22×C8—51no—
297C22×C8—102no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—51no—
301C4×D4—51no—
302C4×D4—51no—
303C4×D4—51no—
304C4×D4—101no—
305C4×D4—101no—
306C4×D4C4×D511yescharacteristic
307C4×D4C4×D511yescharacteristic
308C4×D4C22×D511yescharacteristic
309C4×D4C22×D511yescharacteristic
310C2×C4⋊C4—51no—
311C2×C4⋊C4C4×D511yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—101no—
314C2×C42C22×D511yescharacteristic
315#107—51no—
316#107—51no—
317#107—51no—
318#107—51no—
319C4⋊C8—52no—
320C4⋊C8C22⋊D512yes—
321C4⋊C8C22×D512yes—
322#102—51no—
323#102—51no—
324#102—51no—
325#102—51no—
326#102—51no—
327#102—51no—
328#102—51no—
329#102—51no—
330C22⋊C8—21no—
331C22⋊C8—22no—
332C22⋊C8—22no—
333C22⋊C8—101no—
334C22⋊C8—102no—
335C22⋊C8—102no—
336C22⋊C8C22⋊D511yescharacteristic
337C22⋊C8C22⋊D511yescharacteristic
338#97—21no—
339#97—101no—
340#97—101no—
341C4×C8—51no—
342C4×C8—51no—
343C4×C8—51no—
344C4×C8—51no—
345C4×C8—52no—
346C4×C8C22×D512yes—
347C22×C5—21no—
348C22×C5—21no—
349C22×C5—22no—
350C22×C5—41no—
351C22×C5—41no—
352C22×C5—41no—
353C22×C5—41no—
354C22×C5—41no—
355C22×C5—84no—
356C22×C5#44412yes—
357C22×C5C2×C4×D411yescharacteristic
358C2×D5—22no—
359C2×D5—41no—
360C2×D5—41no—
361C2×D5—41no—
362C2×D5—41no—
363C2×D5—41no—
364C2×D5—84no—
365C4×C5—21no—
366C4×C5—21no—
367C4×C5—21no—
368C4×C5—21no—
369C4×C5—41no—
370C4×C5—41no—
371C4×C5C22×M1611yescharacteristic
372C4×C5#56811yescharacteristic
373C4×C5#57612yes—
374Dic5—41no—
375Dic5—41no—
376C24—21no—
377C24—101no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—41no—
381C2×D4—51no—
382C2×D4—51no—
383C2×D4—102no—
384C2×D4—101no—
385C2×D4—201no—
386C2×D4C2×C4×D511yescharacteristic
387C2×D4C2×C4×D511yescharacteristic
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—82no—
394C22×C4—101no—
395C22×C4—101no—
396C22×C4—101no—
397C22×C4—101no—
398C22×C4—101no—
399C22×C4—402no—
400C22×C4C2×C4×D511yescharacteristic
401C22×C4#70911yescharacteristic
402C22×C4C2×C22⋊D511yescharacteristic
403D8—54no—
404D8—102no—
405D8—102no—
406D8—102no—
407M16—82no—
408M16—202no—
409M16—402no—
410C2×C8—22no—
411C2×C8—51no—
412C2×C8—51no—
413C2×C8—51no—
414C2×C8—51no—
415C2×C8—82no—
416C2×C8—102no—
417C2×C8—102no—
418C2×C8—102no—
419C2×C8—201no—
420C2×C8—402no—
421C2×C8C2×C22⋊D512yes—
422C2×C8C2×C22⋊D512yes—
423C4⋊C4—21no—
424C4⋊C4—51no—
425C4⋊C4—51no—
426C4⋊C4—101no—
427C4⋊C4C2×C4×D511yescharacteristic
428C4⋊C4C2×C4×D511yescharacteristic
429C22⋊C4—21no—
430C22⋊C4—21no—
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—101no—
434C22⋊C4—101no—
435C22⋊C4—101no—
436C22⋊C4—101no—
437C42C2×C4×D511yescharacteristic
438C42#70911yescharacteristic
439C42C2×C22⋊D511yescharacteristic
440C42C23×D511yescharacteristic
441C2×C5—22no—
442C2×C5—42no—
443C2×C5—81no—
444C2×C5D4×M1611yescharacteristic
445C2×C5#294811yescharacteristic
446C2×C5C2×#44411yescharacteristic
447D5—42no—
448D5—81no—
449C23—21no—
450C23—21no—
451C23—42no—
452C23—42no—
453C23—81no—
454C23—101no—
455C23—101no—
456C23—202no—
457C23—202no—
458C23—401no—
459C23C4×C22⋊D511yescharacteristic
460D4—21no—
461D4—21no—
462D4—21no—
463D4—42no—
464D4—52no—
465D4—101no—
466D4—101no—
467D4—101no—
468D4—202no—
469D4D5×M1612yes—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—101no—
482C2×C4—101no—
483C2×C4—202no—
484C2×C4—202no—
485C2×C4—401no—
486C2×C4—401no—
487C2×C4—401no—
488C2×C4C4×C22⋊D511yescharacteristic
489C2×C4C4×C22⋊D511yescharacteristic
490C2×C4C4×C22⋊D511yescharacteristic
491C2×C4C22×C4×D511yescharacteristic
492C2×C4C2×#70911yescharacteristic
493C2×C4C22×C22⋊D511yescharacteristic
494C8—22no—
495C8—22no—
496C8—42no—
497C8—52no—
498C8—101no—
499C8—101no—
500C8—101no—
501C8—202no—
502C8—202no—
503C5#1806411yescharacteristic, complemented, Sylow
504C22—21no—
505C22—21no—
506C22—22no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—84no—
513C22—102no—
514C22—201no—
515C22—201no—
516C22—201no—
517C22—201no—
518C22—201no—
519C22—404no—
520C22#6543412yes—
521C22C2×C4×C22⋊D511yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—41no—
527C4—41no—
528C4—201no—
529C4—201no—
530C4C2×D5×M1611yescharacteristic
531C4#6607911yescharacteristic
532C4#6616912yes—
533C2—22no—
534C2—42no—
535C2—81no—
536C2—202no—
537C2—401no—
538C2#1061426911yescharacteristic
539C2M16×C22⋊D511yescharacteristic
540C2C2×#6543411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
FittingC5×#2974C2
socleC22×C5C2×C4×D4
prime radicalC22×#686C4
cosocleC42×C5C24
hypercenter#2974D5
hypocenterC5#18064
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2974

Complements

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

NH ≅ G/Ntype
1. C5×#2974C2semidirect
3. #10611018C2semidirect
4. #10602392C2semidirect
5. #10601655C2semidirect
6. #10601654C2semidirect
7. #10601634C2semidirect
8. #10601633C2semidirect
9. #10601592C2semidirect
10. #10601587C2semidirect
11. #10601534C2semidirect
15. C5×#436C22semidirect
17. C5×C8×D4C22semidirect
39. #64977C22semidirect
40. C8×D20C22semidirect
41. D4⋊#168C22semidirect
42. D4⋊#168C22semidirect
43. #64703C22semidirect
44. #64703C22semidirect
45. #64700C22semidirect
46. #64699C22semidirect
47. #64675C22semidirect
48. #64674C22semidirect
116. #2974D5semidirect
172. #686M16semidirect
203. #436C2×D5semidirect
207. C8×D4C2×D5semidirect
503. C5#18064semidirect

Maximal subgroups

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

subgroupindex
C5×#29742
C2×C4×#6862
#106110182
#106023922
#106016552
#106016542
#106016342
#106016332
#106015922
#106015872
#106015342
#180645