Finite Group Databasebeta

← #12066494

Subgroups — #12066494

Order 1280.

1166 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10617691C211yesmaximal, characteristic
2C5×#2275C211yesmaximal, characteristic, complemented
3#10614034C211yesmaximal, characteristic, complemented
4#10600263C211yesmaximal, characteristic, complemented
5#10600262C211yesmaximal, characteristic, complemented
6#10600237C212yesmaximal, complemented
7#10600236C212yesmaximal, complemented
8#10600225C211yesmaximal, characteristic, complemented
9#10600224C211yesmaximal, characteristic, complemented
10#10600191C212yesmaximal, complemented
11C2×#64671C212yesmaximal, complemented
12C2×C4×C5×D4—21no—
13C2×C4×C5×D4C2211yescharacteristic
14D4×C22⋊D5—21no—
15#66125C2212yes—
16C2×#4076C2212yes—
17#65899C412yescomplemented
18#65897C411yescharacteristic, complemented
19#65896—21no—
20#65894—22no—
21#65887C411yescharacteristic, complemented
22#65884C412yescomplemented
23D4×D20C411yescharacteristic, complemented
24#65879—21no—
25#65877C411yescharacteristic, complemented
26#65848—21no—
27#65824—22no—
28#65813C2211yescharacteristic
29#65811C2211yescharacteristic
30C4×C5×C4⋊C4—21no—
31C4×C5×C4⋊C4C2211yescharacteristic, complemented
32C4×C5×C22⋊C4—22no—
33C4×C5×C22⋊C4C2212yescomplemented
34C5×#376C2211yescharacteristic, complemented
35C2×C5×#95C2212yescomplemented
36#65496C2212yescomplemented
37C2×C22⋊#168C2212yescomplemented
38#65301—21no—
39#65300—22no—
40D4×#168—21no—
41#65215C2211yescharacteristic, complemented
42#65214C2211yescharacteristic, complemented
43#64672C2211yescharacteristic, complemented
44#64672C2211yescharacteristic, complemented
45#64672C2212yescomplemented
46#64671C2211yescharacteristic
47#64671C2211yescharacteristic
48#64671C2212yes—
49#64671C2212yescomplemented
50#64671C2212yescomplemented
51#64671C2212yescomplemented
52#64671C2212yescomplemented
53#17547—51nomaximal, Sylow
54C22×C5×D4D411yescharacteristic
55C23×C4×C5—22no—
56C23×C4×C5C2312yes—
57C22×C22⋊D5—22no—
58C2×D4×D5—21no—
59C2×#709—21no—
60C2×#709C2×C412yescomplemented
61C22×D20C2×C412yescomplemented
62C4×C5×D4—28no—
63C4×C5×D4C2×C411yescharacteristic
64C4×C5×D4C2×C412yes—
65C4×C5×D4C2×C411yescharacteristic
66C4×C5×D4D411yescharacteristic
67C4×C5×D4D412yes—
68C4×C5×D4D411yescharacteristic
69C2×C5×C4⋊C4—21no—
70C2×C5×C4⋊C4D411yescharacteristic
71C2×C5×C22⋊C4—22no—
72C2×C5×C22⋊C4D412yes—
73C2×C42×C5—21no—
74C2×C42×C5—21no—
75C2×C42×C5—22no—
76C2×C42×C5C2311yescharacteristic
77#4099—22no—
78#4086—21no—
79#4085—22no—
80#4084—21no—
81#4083—22no—
82D4×Dic5—21no—
83#4076—21no—
84#4076—22no—
85#4076—22no—
86#4076C2×C412yescomplemented
87#4076C2×C412yescomplemented
88#4076C2×C412yescomplemented
89#4076C2312yes—
90#4076C2311yescharacteristic
91#4075—21no—
92#4075—22no—
93C4×C22⋊D5—21no—
94C4×C22⋊D5—22no—
95C4×C22⋊D5—22no—
96C2×#685—22no—
97#4072C2×C412yescomplemented
98C2×C4⋊Dic5C2×C412yescomplemented
99#4070C2×C412yescomplemented
100#4070C2311yescharacteristic
101C2×#681—22no—
102C22×#168—22no—
103#4044—22no—
104#4041—21no—
105#4040—22no—
106#4039—21no—
107#4038—21no—
108#4036—21no—
109#4031—22no—
110#4030—22no—
111#4030—22no—
112#4029—22no—
113#4028—22no—
114#4027—22no—
115D5×C22⋊C4—22no—
116#4024—22no—
117#4022—21no—
118#4021—21no—
119#4021C2×C411yescharacteristic, complemented
120#4020C2×C411yescharacteristic, complemented
121C4×D20—21no—
122C4×D20C2×C412yescomplemented
123C4×D20C2×C411yescharacteristic, complemented
124#4018—21no—
125#4016C2×C411yescharacteristic, complemented
126#4015C2×C411yescharacteristic, complemented
127C4×Dic10C2×C411yescharacteristic, complemented
128C5×#95—22no—
129C5×#95—21no—
130C5×#95C2311yescharacteristic
131C5×#95C2312yes—
132C5×#95C2311yescharacteristic
133C22⋊#168—22no—
134C22⋊#168—22no—
135C22⋊#168C2311yescharacteristic
136C22⋊#168C2311yescharacteristic
137C22⋊#168C2312yes—
138C4⋊#168—21no—
139C4⋊#168—21no—
140#3935—21no—
141C4×#168—21no—
142#3519—51no—
143#2974—51no—
144#2275D511yescharacteristic, complemented
145#1509—51no—
146#1507—52no—
147#1505—51no—
148#1504—52no—
149#1502—51no—
150#1501—52no—
151#1500—51no—
152C2×#327—52no—
153C24×C5C2×D412yes—
154C23×D5—22no—
155C2×C5×D4—24no—
156C2×C5×D4C22⋊C412yes—
157C2×C5×D4C22⋊C411yescharacteristic
158C2×C5×D4C22⋊C411yescharacteristic
159C22×C4×C5—21no—
160C22×C4×C5—22no—
161C22×C4×C5—21no—
162C22×C4×C5—28no—
163C22×C4×C5—82no—
164C22×C4×C5—82no—
165C22×C4×C5—82no—
166C22×C4×C5C22×C412yes—
167C22×C4×C5C22×C412yes—
168C22×C4×C5C2×D412yes—
169C22×C4×C5C2×D412yes—
170C22×C4×C5C2×D411yescharacteristic
171C22×C4×C5C2×D411yescharacteristic
172C22×C4×C5C2×D412yes—
173C22×C4×C5C2411yescharacteristic
174C2×C22⋊D5—22no—
175C2×C22⋊D5—22no—
176C2×C22⋊D5—21no—
177C2×C22⋊D5—22no—
178C2×C22⋊D5—22no—
179C2×C22⋊D5—22no—
180C2×C22⋊D5—82no—
181C2×C22⋊D5—82no—
182C22×Dic5—22no—
183D4×D5—44no—
184#709—82no—
185#709—81no—
186#709—81no—
187C2×D20—21no—
188C2×D20—22no—
189C2×D20—21no—
190C2×D20—82no—
191C2×D20C22×C411yescharacteristic
192C2×D20C22×C412yes—
193C2×C4×D5—21no—
194C2×C4×D5—22no—
195C2×C4×D5—22no—
196C2×Dic10—21no—
197C2×Dic10C22×C411yescharacteristic
198C5×C4⋊C4—24no—
199C5×C4⋊C4C22⋊C411yescharacteristic
200C5×C4⋊C4C22⋊C411yescharacteristic
201C5×C4⋊C4C22⋊C412yes—
202C5×C22⋊C4—28no—
203C5×C22⋊C4C22⋊C412yes—
204C5×C22⋊C4C22⋊C412yes—
205C5×C22⋊C4C22⋊C412yes—
206C5×C22⋊C4C22⋊C412yes—
207C42×C5—24no—
208C42×C5—81no—
209C42×C5—82no—
210C42×C5C22×C411yescharacteristic
211C42×C5C22×C411yescharacteristic
212C42×C5C2×D411yescharacteristic
213C42×C5C2×D411yescharacteristic
214C22⋊Dic5—22no—
215C22⋊Dic5—22no—
216C22⋊Dic5—21no—
217#685—21no—
218#685—22no—
219#685—21no—
220#685—21no—
221#685—22no—
222#685—22no—
223#685—22no—
224#685—21no—
225#685—22no—
226C4⋊Dic5—21no—
227C4⋊Dic5—22no—
228C4⋊Dic5C22×C411yescharacteristic
229C4⋊Dic5C22×C412yes—
230C4⋊Dic5C22×C411yescharacteristic
231#683—21no—
232#683—21no—
233#683—21no—
234#683—22no—
235#683—21no—
236C4×Dic5—21no—
237C4×Dic5—22no—
238#681—82no—
239C2×#168—21no—
240C2×#168—21no—
241C2×#168—22no—
242C2×#168—82no—
243#555—51no—
244#554—51no—
245#553—52no—
246#553—101no—
247#550—51no—
248#546—52no—
249#546—102no—
250D42—51no—
251D42—101no—
252#540—101no—
253#538—102no—
254#532—51no—
255#529—51no—
256#525—52no—
257C2×#121—52no—
258#518—101no—
259C2×C4×D4—21no—
260C2×C4×D4C2×D511yescharacteristic
261#436—101no—
262#435—102no—
263C8×D4—101no—
264#432—51no—
265#431—51no—
266#407—52no—
267C2×C22⋊C8—52no—
268C4×C4⋊C4—21no—
269C4×C4⋊C4C2×D511yescharacteristic, complemented
270C4×C22⋊C4—22no—
271C4×C22⋊C4C2×D512yescomplemented
272#376C2×D511yescharacteristic, complemented
273C2×#95C2×D512yescomplemented
274#328—51no—
275#328—51no—
276#328—52no—
277#327—51no—
278#327—52no—
279#327—51no—
280#327—52no—
281#327—52no—
282#327—52no—
283#327—52no—
284C23×C5—22no—
285C23×C5—22no—
286C23×C5—24no—
287C23×C5—42no—
288C23×C5C2×C22⋊C412yes—
289C23×C5C2×C22⋊C412yes—
290C23×C5C22×D411yescharacteristic
291C22×D5—21no—
292C22×D5—22no—
293C22×D5—44no—
294C22×D5—82no—
295C22×D5—82no—
296C5×D4—24no—
297C5×D4—22no—
298C5×D4#10414yes—
299C2×C4×C5—21no—
300C2×C4×C5—21no—
301C2×C4×C5—22no—
302C2×C4×C5—22no—
303C2×C4×C5—22no—
304C2×C4×C5—24no—
305C2×C4×C5—42no—
306C2×C4×C5—42no—
307C2×C4×C5—42no—
308C2×C4×C5—42no—
309C2×C4×C5—42no—
310C2×C4×C5—41no—
311C2×C4×C5—41no—
312C2×C4×C5—42no—
313C2×C4×C5—41no—
314C2×C4×C5—42no—
315C2×C4×C5—41no—
316C2×C4×C5—41no—
317C2×C4×C5—88no—
318C2×C4×C5C2×C22⋊C412yes—
319C2×C4×C5C2×C22⋊C412yes—
320C2×C4×C5C2×C22⋊C411yescharacteristic
321C2×C4×C5C2×C22⋊C411yescharacteristic
322C2×C4×C5C2×C22⋊C411yescharacteristic
323C2×C4×C5C2×C22⋊C411yescharacteristic
324C2×C4×C5C23×C411yescharacteristic
325C2×C4×C5C22×D411yescharacteristic
326C22⋊D5—42no—
327C22⋊D5—41no—
328C22⋊D5—42no—
329C22⋊D5—41no—
330C22⋊D5—44no—
331C22⋊D5—82no—
332C22⋊D5—82no—
333C22⋊D5—82no—
334C2×Dic5—21no—
335C2×Dic5—21no—
336C2×Dic5—21no—
337C2×Dic5—22no—
338C2×Dic5—82no—
339D20—41no—
340D20—42no—
341D20—42no—
342D20—82no—
343D20—81no—
344C4×D5—42no—
345C4×D5—82no—
346C4×D5—82no—
347Dic10—41no—
348Dic10—81no—
349#168—41no—
350#168—42no—
351#168—41no—
352C2×#41—52no—
353C2×#41—101no—
354C22×D4—52no—
355C22×D4—102no—
356C22×D4—101no—
357C22×D4C22⋊D511yescharacteristic
358C23×C4—22no—
359C23×C4C22×D512yes—
360C2×M16—102no—
361C22×C8—102no—
362C4⋊Q8—51no—
363#127—51no—
364#127—101no—
365#126—102no—
366#126—101no—
367#125—51no—
368#125—101no—
369#124—51no—
370#124—102no—
371#124—101no—
372#123—102no—
373#123—101no—
374#123—102no—
375#123—102no—
376C22⋊Q8—52no—
377C22⋊Q8—51no—
378C22⋊Q8—102no—
379#121—52no—
380#121—52no—
381#121—52no—
382#121—51no—
383#121—52no—
384#121—102no—
385#121—101no—
386#121—102no—
387#121—102no—
388#121—101no—
389#121—101no—
390#121—102no—
391#121—102no—
392#120—102no—
393#120—102no—
394#120—102no—
395C4×Q8—51no—
396C4×D4—28no—
397C4×D4—51no—
398C4×D4—52no—
399C4×D4—101no—
400C4×D4—102no—
401C4×D4—102no—
402C4×D4—101no—
403C4×D4—101no—
404C4×D4—102no—
405C4×D4—101no—
406C4×D4C4×D511yescharacteristic
407C4×D4C4×D512yes—
408C4×D4C4×D511yescharacteristic
409C4×D4D2011yescharacteristic
410C4×D4D2012yes—
411C4×D4D2011yescharacteristic
412#117—52no—
413#117—101no—
414#117—101no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—52no—
417C2×C4⋊C4C22⋊D511yescharacteristic
418C2×C22⋊C4—22no—
419C2×C22⋊C4—102no—
420C2×C22⋊C4—102no—
421C2×C22⋊C4C22⋊D512yes—
422C2×C42—21no—
423C2×C42—21no—
424C2×C42—22no—
425C2×C42C22×D511yescharacteristic
426C4⋊C8—101no—
427C4⋊C8—101no—
428C22⋊C8—51no—
429C22⋊C8—52no—
430C22⋊C8—51no—
431C22⋊C8—102no—
432C22⋊C8—102no—
433#97—101no—
434C4×C8—101no—
435#95—22no—
436#95—21no—
437#95C22×D511yescharacteristic
438#95C22×D512yes—
439#95C22×D511yescharacteristic
440C22×C5—21no—
441C22×C5—22no—
442C22×C5—22no—
443C22×C5—21no—
444C22×C5—22no—
445C22×C5—22no—
446C22×C5—22no—
447C22×C5—44no—
448C22×C5—44no—
449C22×C5C2×#10414yes—
450C22×C5C22×C22⋊C411yescharacteristic
451C22×C5#57611yescharacteristic
452C22×C5#57711yescharacteristic
453C2×D5—42no—
454C2×D5—41no—
455C2×D5—42no—
456C2×D5—41no—
457C2×D5—41no—
458C2×D5—42no—
459C2×D5—164no—
460C4×C5—21no—
461C4×C5—21no—
462C4×C5—22no—
463C4×C5—22no—
464C4×C5—21no—
465C4×C5—21no—
466C4×C5—21no—
467C4×C5—42no—
468C4×C5—44no—
469C4×C5—42no—
470C4×C5C2×#10412yes—
471Dic5—41no—
472Dic5—42no—
473Dic5—41no—
474Dic5—41no—
475C24—102no—
476C24C2×C22⋊D512yes—
477#41—401no—
478#41—401no—
479#41—402no—
480C2×Q8—51no—
481C2×Q8—101no—
482C2×D4—24no—
483C2×D4—51no—
484C2×D4—52no—
485C2×D4—102no—
486C2×D4—101no—
487C2×D4—102no—
488C2×D4—102no—
489C2×D4—102no—
490C2×D4—102no—
491C2×D4—101no—
492C2×D4—101no—
493C2×D4—102no—
494C2×D4—204no—
495C2×D4—402no—
496C2×D4—402no—
497C2×D4—402no—
498C2×D4#68512yes—
499C2×D4#68511yescharacteristic
500C2×D4#68511yescharacteristic
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—28no—
505C22×C4—82no—
506C22×C4—82no—
507C22×C4—82no—
508C22×C4—102no—
509C22×C4—101no—
510C22×C4—102no—
511C22×C4—102no—
512C22×C4C2×C4×D512yes—
513C22×C4C2×C4×D512yes—
514C22×C4C2×D2012yes—
515C22×C4C2×D2012yes—
516C22×C4C2×C22⋊D511yescharacteristic
517C22×C4C2×C22⋊D511yescharacteristic
518C22×C4C2×C22⋊D512yes—
519C22×C4C23×D511yescharacteristic
520M16—402no—
521C2×C8—101no—
522C2×C8—101no—
523C2×C8—102no—
524C2×C8—402no—
525C4⋊C4—24no—
526C4⋊C4—51no—
527C4⋊C4—52no—
528C4⋊C4—51no—
529C4⋊C4—101no—
530C4⋊C4—102no—
531C4⋊C4—101no—
532C4⋊C4—101no—
533C4⋊C4—101no—
534C4⋊C4—102no—
535C4⋊C4—101no—
536C4⋊C4#68511yescharacteristic
537C4⋊C4#68511yescharacteristic
538C4⋊C4#68512yes—
539C22⋊C4—28no—
540C22⋊C4—101no—
541C22⋊C4—102no—
542C22⋊C4—102no—
543C22⋊C4—101no—
544C22⋊C4—102no—
545C22⋊C4—102no—
546C22⋊C4—101no—
547C22⋊C4—102no—
548C22⋊C4—101no—
549C22⋊C4—102no—
550C22⋊C4—101no—
551C22⋊C4—102no—
552C22⋊C4#68512yes—
553C22⋊C4#68512yes—
554C22⋊C4#68512yes—
555C22⋊C4#68512yes—
556C42—24no—
557C42—81no—
558C42—82no—
559C42—102no—
560C42—101no—
561C42C2×C4×D511yescharacteristic
562C42C2×C4×D511yescharacteristic
563C42C2×D2011yescharacteristic
564C42C2×D2011yescharacteristic
565C2×C5—21no—
566C2×C5—21no—
567C2×C5—24no—
568C2×C5—42no—
569C2×C5#288311yescharacteristic
570C2×C5#289711yescharacteristic
571C2×C5C22×#10411yescharacteristic
572D5—82no—
573D5—81no—
574C23—22no—
575C23—22no—
576C23—24no—
577C23—42no—
578C23—101no—
579C23—102no—
580C23—204no—
581C23—402no—
582C23—402no—
583C23C2×#68512yes—
584C23C2×#68512yes—
585C23C22×C22⋊D511yescharacteristic
586Q8—201no—
587Q8—401no—
588D4—24no—
589D4—22no—
590D4—201no—
591D4—202no—
592D4—201no—
593D4—202no—
594D4—201no—
595D4—202no—
596D4—204no—
597D4—202no—
598D4—402no—
599D4—401no—
600D4—402no—
601D4—402no—
602D4—402no—
603D4#393714yes—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—88no—
623C2×C4—101no—
624C2×C4—102no—
625C2×C4—101no—
626C2×C4—101no—
627C2×C4—202no—
628C2×C4—402no—
629C2×C4—402no—
630C2×C4—402no—
631C2×C4C2×#68512yes—
632C2×C4C2×#68511yescharacteristic
633C2×C4C2×#68511yescharacteristic
634C2×C4C2×#68511yescharacteristic
635C2×C4C2×#68512yes—
636C2×C4C2×#68511yescharacteristic
637C2×C4C22×C4×D511yescharacteristic
638C2×C4C22×D2011yescharacteristic
639C8—201no—
640C8—202no—
641C8—201no—
642C5#1754711yescharacteristic, complemented, Sylow
643C22—21no—
644C22—22no—
645C22—21no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—44no—
651C22—44no—
652C22—202no—
653C22—201no—
654C22—201no—
655C22—202no—
656C22—202no—
657C22—201no—
658C22—804no—
659C22C2×#393714yes—
660C22C22×#68511yescharacteristic
661C22#6616911yescharacteristic
662C22#6617111yescharacteristic
663C4—21no—
664C4—21no—
665C4—22no—
666C4—22no—
667C4—21no—
668C4—21no—
669C4—21no—
670C4—42no—
671C4—44no—
672C4—42no—
673C4—201no—
674C4—201no—
675C4—201no—
676C4—202no—
677C4C2×#393712yes—
678C2—21no—
679C2—21no—
680C2—24no—
681C2—42no—
682C2—401no—
683C2—402no—
684C2C22×#393711yescharacteristic
685C2#1061427711yescharacteristic
686C2#1061429411yescharacteristic
subgroupG/H
centerC22C22×#685
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2275C2
socleC22×C5C22×C22⋊C4
prime radical#10617691C2
cosocleC22×C4×C5C24
hypercenter#2275D5
hypocenterC5#17547
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2275

Complements

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

NH ≅ G/Ntype
2. C5×#2275C2semidirect
3. #10614034C2semidirect
4. #10600263C2semidirect
5. #10600262C2semidirect
6. #10600237C2semidirect
7. #10600236C2semidirect
8. #10600225C2semidirect
9. #10600224C2semidirect
10. #10600191C2semidirect
11. C2×#64671C2semidirect
17. #65899C4semidirect
18. #65897C4semidirect
21. #65887C4semidirect
22. #65884C4semidirect
23. D4×D20C4semidirect
25. #65877C4semidirect
31. C4×C5×C4⋊C4C22semidirect
33. C4×C5×C22⋊C4C22semidirect
34. C5×#376C22semidirect
35. C2×C5×#95C22semidirect
36. #65496C22semidirect
37. C2×C22⋊#168C22semidirect
41. #65215C22semidirect
42. #65214C22semidirect
43. #64672C22semidirect
44. #64672C22semidirect
45. #64672C22semidirect
49. #64671C22semidirect
50. #64671C22semidirect
51. #64671C22semidirect
52. #64671C22semidirect
60. C2×#709C2×C4semidirect
61. C22×D20C2×C4semidirect
86. #4076C2×C4semidirect
87. #4076C2×C4semidirect
88. #4076C2×C4semidirect
97. #4072C2×C4semidirect
98. C2×C4⋊Dic5C2×C4semidirect
99. #4070C2×C4semidirect
119. #4021C2×C4semidirect
120. #4020C2×C4semidirect
122. C4×D20C2×C4semidirect
123. C4×D20C2×C4semidirect
125. #4016C2×C4semidirect
126. #4015C2×C4semidirect
127. C4×Dic10C2×C4semidirect
144. #2275D5semidirect
269. C4×C4⋊C4C2×D5semidirect
271. C4×C22⋊C4C2×D5semidirect
272. #376C2×D5semidirect
273. C2×#95C2×D5semidirect
642. C5#17547semidirect

Maximal subgroups

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

subgroupindex
#106176912
C5×#22752
#106140342
#106002632
#106002622
#106002372
#106002362
#106002252
#106002242
#106001912
C2×#646712
#175475