Finite Group Databasebeta

← #12069407

Subgroups — #12069407

Order 1280.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3107C211yesmaximal, characteristic
2#10615062C211yesmaximal, characteristic, complemented
3#10614251C211yesmaximal, characteristic
4#10602617C211yesmaximal, characteristic, complemented
5#10602433C211yesmaximal, characteristic
6#10601697C211yesmaximal, characteristic, complemented
7#10601696C211yesmaximal, characteristic, complemented
8#10601695C211yesmaximal, characteristic, complemented
9Q16×#168C211yesmaximal, characteristic, complemented
10#10601690C211yesmaximal, characteristic, complemented
11#10601688C211yesmaximal, characteristic, complemented
12SD16⋊#168C211yesmaximal, characteristic, complemented
13SD16×#168C211yesmaximal, characteristic, complemented
14#10601545C211yesmaximal, characteristic, complemented
15#10601537C211yesmaximal, characteristic, complemented
16C5×#533C2211yescharacteristic
17C4×C5×#41C2211yescharacteristic
18C5×#485C411yescharacteristic
19C5×#485C411yescharacteristic
20C5×#475C411yescharacteristic
21C5×#474C411yescharacteristic
22C5×#462C411yescharacteristic
23C5×#461C411yescharacteristic
24C5×#452C411yescharacteristic
25C5×#452C411yescharacteristic
26C4×C5×Q16C2211yescharacteristic
27C4×C5×Q16C2211yescharacteristic
28C4×C5×SD16C2211yescharacteristic
29C4×C5×SD16C2211yescharacteristic
30C5×#431C2211yescharacteristic
31#65313—21no—
32#65313—21no—
33#65313C2211yescharacteristic
34#65313C2211yescharacteristic
35Q8×#168C2211yescharacteristic
36Q8×#168C2211yescharacteristic
37Q8×#168C2211yescharacteristic
38Q8×#168C2211yescharacteristic
39D4×#168—21no—
40D4×#168—21no—
41D4×#168C2211yescharacteristic
42D4×#168C2211yescharacteristic
43#65289—21no—
44#65289—21no—
45#65215C2211yescharacteristic
46#65214—21no—
47#65214C2211yescharacteristic
48#65214C2211yescharacteristic
49C2×C4⋊#168—21no—
50C2×C4⋊#168C2211yescharacteristic
51Q8⋊#168C2211yescharacteristic
52Q8⋊#168C2211yescharacteristic
53Q8⋊#168C2211yescharacteristic
54Q8⋊#168C2211yescharacteristic
55Q8⋊#168C2211yescharacteristic
56Q8⋊#168C2211yescharacteristic
57D4⋊#168C2211yescharacteristic
58D4⋊#168C2211yescharacteristic
59#64678C2211yescharacteristic
60#64678C2211yescharacteristic
61#64677C2211yescharacteristic
62#64677C2211yescharacteristic
63#64675C2211yescharacteristic
64#64674C2211yescharacteristic
65#64674C2211yescharacteristic
66C8×#168C2211yescharacteristic
67#17921—51nomaximal, Sylow
68C2×C5×#41C2×C411yescharacteristic
69C2×C5×Q16C2×C411yescharacteristic
70C2×C5×Q16C2×C411yescharacteristic
71C2×C5×SD16C2×C411yescharacteristic
72C2×C5×SD16C2×C411yescharacteristic
73C5×C4⋊Q8C2×C411yescharacteristic
74C5×C4⋊Q8C2×C411yescharacteristic
75C5×#125—21no—
76C5×C22⋊Q8—21no—
77C5×C22⋊Q8C2×C411yescharacteristic
78C5×C22⋊Q8C2×C411yescharacteristic
79C4×C5×Q8—21no—
80C4×C5×Q8C2×C411yescharacteristic
81C4×C5×Q8C2311yescharacteristic
82C4×C5×Q8C2311yescharacteristic
83C4×C5×Q8C2311yescharacteristic
84C4×C5×D4—21no—
85C4×C5×D4—21no—
86C4×C5×D4C2×C411yescharacteristic
87C4×C5×D4C2311yescharacteristic
88C5×#117—21no—
89C5×#117—21no—
90C5×#117C2×C411yescharacteristic
91C2×C42×C5—21no—
92C2×C42×C5C2311yescharacteristic
93C22×#168—21no—
94C22×#168—21no—
95C22×#168—21no—
96C5×#107C2×C411yescharacteristic
97C5×#107C2×C411yescharacteristic
98C5×#106C2×C411yescharacteristic
99C5×#106C2×C411yescharacteristic
100C5×C4⋊C8C2×C411yescharacteristic
101C5×C4⋊C8C2×C411yescharacteristic
102C5×#103C2×C411yescharacteristic
103C5×#103C2×C411yescharacteristic
104C5×#103C2×C411yescharacteristic
105C5×#103C2×C411yescharacteristic
106C5×#103C2×C411yescharacteristic
107C5×#103C2×C411yescharacteristic
108C5×#102C2×C411yescharacteristic
109C5×#102C2×C411yescharacteristic
110C5×C22⋊C8C2×C411yescharacteristic
111C5×C22⋊C8C2×C411yescharacteristic
112C4×C5×C8C2311yescharacteristic
113C4×C5×C8C2311yescharacteristic
114C22⋊#168—21no—
115C22⋊#168—21no—
116C22⋊#168—21no—
117C22⋊#168—21no—
118C22⋊#168D411yescharacteristic
119C22⋊#168D411yescharacteristic
120C4⋊#168—21no—
121C4⋊#168—21no—
122C4⋊#168—21no—
123C4⋊#168—21no—
124C4⋊#168—21no—
125C4⋊#168—21no—
126C4⋊#168—21no—
127C4⋊#168—21no—
128C4⋊#168D411yescharacteristic
129C4⋊#168D411yescharacteristic
130C4⋊#168C2311yescharacteristic
131C4⋊#168C2311yescharacteristic
132C4⋊#168C2311yescharacteristic
133C4⋊#168C2311yescharacteristic
134#3935—21no—
135#3935—21no—
136#3935—21no—
137C4×#168—21no—
138C4×#168—21no—
139C4×#168—21no—
140C4×#168—21no—
141C4×#168C2311yescharacteristic
142C4×#168C2311yescharacteristic
143C4×#168C2311yescharacteristic
144C4×#168C2311yescharacteristic
145#3107D511yescharacteristic
146#2988—51no—
147#2985—51no—
148#1587—51no—
149#1585—51no—
150#1584—51no—
151#1582—51no—
152#1579—51no—
153SD16⋊C8—51no—
154C8×Q16—51no—
155C8×SD16—51no—
156#1573—51no—
157#1488—51no—
158#1481—51no—
159#1454—51no—
160C5×#41—22no—
161C5×#41—22no—
162C2×C5×Q8—21no—
163C2×C5×Q8C22×C411yescharacteristic
164C2×C5×Q8C22×C411yescharacteristic
165C2×C5×Q8C22×C411yescharacteristic
166C2×C5×D4—21no—
167C2×C5×D4C22×C411yescharacteristic
168C22×C4×C5—21no—
169C22×C4×C5—21no—
170C22×C4×C5—21no—
171C22×C4×C5C22×C411yescharacteristic
172C22×C4×C5C2×D411yescharacteristic
173C22×C4×C5#4111yescharacteristic
174C5×Q16—22no—
175C5×Q16—22no—
176C5×SD16—22no—
177C5×SD16—22no—
178C2×C5×C8C22×C411yescharacteristic
179C2×C5×C8C22×C411yescharacteristic
180C2×C5×C8C22×C411yescharacteristic
181C2×C5×C8C22×C411yescharacteristic
182C5×C4⋊C4—21no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—21no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—21no—
188C5×C4⋊C4—21no—
189C5×C4⋊C4—21no—
190C5×C4⋊C4—21no—
191C5×C4⋊C4C22×C411yescharacteristic
192C5×C4⋊C4C22×C411yescharacteristic
193C5×C4⋊C4C22×C411yescharacteristic
194C5×C4⋊C4C22×C411yescharacteristic
195C5×C22⋊C4—21no—
196C5×C22⋊C4—21no—
197C5×C22⋊C4—21no—
198C5×C22⋊C4—21no—
199C5×C22⋊C4—21no—
200C42×C5—21no—
201C42×C5—21no—
202C42×C5—21no—
203C42×C5—21no—
204C42×C5—21no—
205C42×C5C22×C411yescharacteristic
206C42×C5C2×D411yescharacteristic
207C42×C5#4111yescharacteristic
208C42×C5C2411yescharacteristic
209C2×#168—21no—
210C2×#168—21no—
211C2×#168—21no—
212C2×#168—21no—
213C2×#168—21no—
214C2×#168—21no—
215C2×#168—81no—
216C2×#168—81no—
217C2×#168—81no—
218C2×#168C2×D411yescharacteristic
219C2×#168C2×D411yescharacteristic
220C2×#168C2×D411yescharacteristic
221C2×#168C2×D411yescharacteristic
222#533C2×D511yescharacteristic
223C4×#41C2×D511yescharacteristic
224#485Dic511yescharacteristic
225#485Dic511yescharacteristic
226#475Dic511yescharacteristic
227#474Dic511yescharacteristic
228#462Dic511yescharacteristic
229#461Dic511yescharacteristic
230#452Dic511yescharacteristic
231#452Dic511yescharacteristic
232#446—51no—
233#446—51no—
234#446—101no—
235#446—101no—
236C8×Q8—51no—
237C8×Q8—51no—
238C8×Q8—51no—
239C8×Q8—51no—
240C4×Q16C2×D511yescharacteristic
241C4×Q16C2×D511yescharacteristic
242C4×SD16C2×D511yescharacteristic
243C4×SD16C2×D511yescharacteristic
244C8×D4—51no—
245C8×D4—51no—
246C8×D4—101no—
247C8×D4—101no—
248#433—101no—
249#433—101no—
250#432—51no—
251#431—51no—
252#431—51no—
253#431—101no—
254#431C2×D511yescharacteristic
255C2×C4⋊C8—51no—
256C2×C4⋊C8—101no—
257#335—51no—
258#335—51no—
259#334—51no—
260#334—51no—
261Q8⋊C8—51no—
262Q8⋊C8—51no—
263Q8⋊C8—51no—
264Q8⋊C8—51no—
265Q8⋊C8—51no—
266Q8⋊C8—51no—
267D4⋊C8—51no—
268D4⋊C8—51no—
269#322—51no—
270#322—51no—
271#322—51no—
272C82—51no—
273C23×C5—21no—
274C23×C5C4×D411yescharacteristic
275C5×Q8—21no—
276C5×Q8—21no—
277C5×Q8—21no—
278C5×Q8—21no—
279C5×Q8—21no—
280C5×Q8—81no—
281C5×Q8#13112yes—
282C5×D4—24no—
283C5×D4—21no—
284C5×D4#13112yes—
285C2×C4×C5—24no—
286C2×C4×C5—21no—
287C2×C4×C5—21no—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
291C2×C4×C5—21no—
292C2×C4×C5—21no—
293C2×C4×C5—21no—
294C2×C4×C5—21no—
295C2×C4×C5—21no—
296C2×C4×C5—21no—
297C2×C4×C5—41no—
298C2×C4×C5—41no—
299C2×C4×C5—41no—
300C2×C4×C5—81no—
301C2×C4×C5C4×D411yescharacteristic
302C2×C4×C5C4×D411yescharacteristic
303C2×C4×C5C4×D411yescharacteristic
304C2×C4×C5C23×C411yescharacteristic
305C2×C4×C5C22×D411yescharacteristic
306C2×C4×C5C2×#4111yescharacteristic
307C5×C8—21no—
308C5×C8—21no—
309C5×C8—21no—
310C5×C8—21no—
311#168—21no—
312#168—21no—
313#168—21no—
314#168—21no—
315#168—41no—
316#168—41no—
317#168—41no—
318#168—41no—
319#168—41no—
320#168—41no—
321C2×#41C2×Dic511yescharacteristic
322C2×Q16C2×Dic511yescharacteristic
323C2×Q16C2×Dic511yescharacteristic
324C2×SD16C2×Dic511yescharacteristic
325C2×SD16C2×Dic511yescharacteristic
326C22×C8—101no—
327C22×C8—101no—
328C22×C8—101no—
329C4⋊Q8C2×Dic511yescharacteristic
330C4⋊Q8C2×Dic511yescharacteristic
331#125—21no—
332C22⋊Q8—21no—
333C22⋊Q8C2×Dic511yescharacteristic
334C22⋊Q8C2×Dic511yescharacteristic
335C4×Q8—21no—
336C4×Q8C2×Dic511yescharacteristic
337C4×Q8C22×D511yescharacteristic
338C4×Q8C22×D511yescharacteristic
339C4×Q8C22×D511yescharacteristic
340C4×D4—21no—
341C4×D4—21no—
342C4×D4C2×Dic511yescharacteristic
343C4×D4C22×D511yescharacteristic
344#117—21no—
345#117—21no—
346#117C2×Dic511yescharacteristic
347C2×C42—21no—
348C2×C42C22×D511yescharacteristic
349#107C2×Dic511yescharacteristic
350#107C2×Dic511yescharacteristic
351#106C2×Dic511yescharacteristic
352#106C2×Dic511yescharacteristic
353C4⋊C8—51no—
354C4⋊C8—51no—
355C4⋊C8—51no—
356C4⋊C8—51no—
357C4⋊C8—51no—
358C4⋊C8—51no—
359C4⋊C8—101no—
360C4⋊C8—101no—
361C4⋊C8—101no—
362C4⋊C8—101no—
363C4⋊C8—101no—
364C4⋊C8—101no—
365C4⋊C8—101no—
366C4⋊C8—101no—
367C4⋊C8C2×Dic511yescharacteristic
368C4⋊C8C2×Dic511yescharacteristic
369#103C2×Dic511yescharacteristic
370#103C2×Dic511yescharacteristic
371#103C2×Dic511yescharacteristic
372#103C2×Dic511yescharacteristic
373#103C2×Dic511yescharacteristic
374#103C2×Dic511yescharacteristic
375#102C2×Dic511yescharacteristic
376#102C2×Dic511yescharacteristic
377C22⋊C8—51no—
378C22⋊C8—51no—
379C22⋊C8—101no—
380C22⋊C8—101no—
381C22⋊C8—101no—
382C22⋊C8—101no—
383C22⋊C8C2×Dic511yescharacteristic
384C22⋊C8C2×Dic511yescharacteristic
385#97—101no—
386#97—101no—
387#97—101no—
388C4×C8—51no—
389C4×C8—51no—
390C4×C8—51no—
391C4×C8—51no—
392C4×C8—101no—
393C4×C8—101no—
394C4×C8—101no—
395C4×C8—101no—
396C4×C8C22×D511yescharacteristic
397C4×C8C22×D511yescharacteristic
398C22×C5—21no—
399C22×C5—22no—
400C22×C5—21no—
401C22×C5—21no—
402C22×C5—41no—
403C22×C5—41no—
404C22×C5C2×C4×D411yescharacteristic
405C4×C5—21no—
406C4×C5—22no—
407C4×C5—21no—
408C4×C5—21no—
409C4×C5—21no—
410C4×C5—41no—
411C4×C5—41no—
412C4×C5—41no—
413C4×C5—41no—
414C4×C5—41no—
415C4×C5—41no—
416C4×C5C2×#13111yescharacteristic
417C4×C5C2×#13111yescharacteristic
418C4×C5#57711yescharacteristic
419C4×C5#57811yescharacteristic
420#41—22no—
421#41—22no—
422C2×Q8—21no—
423C2×Q8C22×Dic511yescharacteristic
424C2×Q8C22×Dic511yescharacteristic
425C2×Q8C22×Dic511yescharacteristic
426C2×D4—21no—
427C2×D4C22×Dic511yescharacteristic
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4D4×D511yescharacteristic
432C22×C4#71111yescharacteristic
433C22×C4C22×Dic511yescharacteristic
434Q16—22no—
435Q16—22no—
436SD16—22no—
437SD16—22no—
438C2×C8—51no—
439C2×C8—51no—
440C2×C8—51no—
441C2×C8—51no—
442C2×C8—101no—
443C2×C8—101no—
444C2×C8—101no—
445C2×C8—101no—
446C2×C8—101no—
447C2×C8—101no—
448C2×C8—401no—
449C2×C8—401no—
450C2×C8—401no—
451C2×C8C22×Dic511yescharacteristic
452C2×C8C22×Dic511yescharacteristic
453C2×C8C22×Dic511yescharacteristic
454C2×C8C22×Dic511yescharacteristic
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4C22×Dic511yescharacteristic
465C4⋊C4C22×Dic511yescharacteristic
466C4⋊C4C22×Dic511yescharacteristic
467C4⋊C4C22×Dic511yescharacteristic
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42D4×D511yescharacteristic
479C42#71111yescharacteristic
480C42C22×Dic511yescharacteristic
481C42C23×D511yescharacteristic
482C2×C5—41no—
483C2×C5—42no—
484C2×C5#292911yescharacteristic
485C2×C5#294711yescharacteristic
486C2×C5#295711yescharacteristic
487C23—21no—
488C23D4×Dic511yescharacteristic
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—81no—
495Q8#409412yes—
496D4—24no—
497D4—21no—
498D4#409412yes—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—81no—
515C2×C4D4×Dic511yescharacteristic
516C2×C4D4×Dic511yescharacteristic
517C2×C4D4×Dic511yescharacteristic
518C2×C4C2×D4×D511yescharacteristic
519C2×C4C2×#71111yescharacteristic
520C2×C4C23×Dic511yescharacteristic
521C8—21no—
522C8—21no—
523C8—21no—
524C8—21no—
525C8—101no—
526C8—101no—
527C8—101no—
528C8—101no—
529C8—201no—
530C8—201no—
531C8—201no—
532C8—201no—
533C8—201no—
534C8—201no—
535C5#1792111yescharacteristic, complemented, Sylow
536C22—21no—
537C22—21no—
538C22—22no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22C2×D4×Dic511yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—21no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4#6611211yescharacteristic
555C4#6611311yescharacteristic
556C4C2×#409411yescharacteristic
557C4C2×#409411yescharacteristic
558C2—41no—
559C2—42no—
560C2#1061464711yescharacteristic
561C2#1061564111yescharacteristic
562C2#1061569611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
FittingC5×#3107C2
socleC22×C5C2×C4×D4
prime radicalC2×C5×#41C2×C4
cosocleC42×C5C24
hypercenter#3107D5
hypocenterC5#17921
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
2. #10615062C2semidirect
4. #10602617C2semidirect
6. #10601697C2semidirect
7. #10601696C2semidirect
8. #10601695C2semidirect
9. Q16×#168C2semidirect
10. #10601690C2semidirect
11. #10601688C2semidirect
12. SD16⋊#168C2semidirect
13. SD16×#168C2semidirect
14. #10601545C2semidirect
15. #10601537C2semidirect
535. C5#17921semidirect

Maximal subgroups

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

subgroupindex
C5×#31072
#106150622
#106142512
#106026172
#106024332
#106016972
#106016962
#106016952
Q16×#1682
#106016902
#106016882
SD16⋊#1682
SD16×#1682
#106015452
#106015372
#179215