Finite Group Databasebeta

← #12066505

Subgroups — #12066505

Order 1280.

1102 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10617668C211yesmaximal, characteristic
2C5×#2275C211yesmaximal, characteristic, complemented
3#10614034C211yesmaximal, characteristic, complemented
4#10600264C211yesmaximal, characteristic, complemented
5#10600263C211yesmaximal, characteristic, complemented
6#10600238C212yesmaximal, complemented
7#10600236C212yesmaximal, complemented
8#10600226C211yesmaximal, characteristic, complemented
9#10600224C211yesmaximal, characteristic, complemented
10C2×#64672C212yesmaximal, complemented
11#10600191C212yesmaximal, complemented
12C2×C4×C5×D4—21no—
13C2×C4×C5×D4C2211yescharacteristic
14#66140—21no—
15#66125C2212yes—
16C2×#4070C2212yes—
17#65898—22no—
18#65895—21no—
19#65889C411yescharacteristic, complemented
20#65886C412yescomplemented
21#65885C411yescharacteristic, complemented
22#65880—21no—
23#65876C411yescharacteristic, complemented
24#65875C412yescomplemented
25D4×Dic10C411yescharacteristic, complemented
26#65845—21no—
27#65829—22no—
28#65813C2211yescharacteristic
29#65804C2211yescharacteristic
30C4×C5×C4⋊C4—21no—
31C4×C5×C4⋊C4C2211yescharacteristic
32C4×C5×C22⋊C4—22no—
33C4×C5×C22⋊C4C2212yes—
34C5×#376C2211yescharacteristic
35C2×C5×#95C2212yes—
36#65496C2212yes—
37C2×C22⋊#168C2212yes—
38#65301—21no—
39#65300—22no—
40D4×#168—21no—
41#65215C2211yescharacteristic
42#65214C2211yescharacteristic
43#64672C2211yescharacteristic
44#64672C2212yes—
45#64672C2212yes—
46#64672C2212yes—
47#64672C2211yescharacteristic
48#64672C2212yes—
49#64672C2212yes—
50#64671C2211yescharacteristic
51#64671C2211yescharacteristic
52#64671C2212yes—
53#17548—51nomaximal, Sylow
54C22×C5×D4D411yescharacteristic
55C23×C4×C5—22no—
56C23×C4×C5C2312yes—
57C2×#711—21no—
58C2×#709C2×C412yescomplemented
59C22×Dic10C2×C412yescomplemented
60C4×C5×D4—28no—
61C4×C5×D4C2×C411yescharacteristic
62C4×C5×D4C2×C412yes—
63C4×C5×D4C2×C411yescharacteristic
64C4×C5×D4D411yescharacteristic
65C4×C5×D4D412yes—
66C4×C5×D4D411yescharacteristic
67C2×C5×C4⋊C4—21no—
68C2×C5×C4⋊C4D411yescharacteristic
69C2×C5×C22⋊C4—22no—
70C2×C5×C22⋊C4D412yes—
71C2×C42×C5—21no—
72C2×C42×C5—21no—
73C2×C42×C5—22no—
74C2×C42×C5C2311yescharacteristic
75#4099—22no—
76C2×C22⋊Dic5—22no—
77#4085—22no—
78#4084—21no—
79#4082—21no—
80#4081—22no—
81D4×Dic5—21no—
82#4076C2×C412yescomplemented
83#4076C2311yescharacteristic
84#4075—21no—
85#4075—22no—
86C4×C22⋊D5—21no—
87C4×C22⋊D5—22no—
88C4×C22⋊D5—22no—
89#4072—21no—
90#4072C2×C412yescomplemented
91C2×C4⋊Dic5C2×C412yescomplemented
92#4070—21no—
93#4070—22no—
94#4070—22no—
95#4070C2×C412yescomplemented
96#4070C2×C412yescomplemented
97#4070C2×C412yescomplemented
98#4070C2312yes—
99#4070C2311yescharacteristic
100C2×#683—22no—
101C2×#681—22no—
102C22×#168—22no—
103#4043—21no—
104#4042—22no—
105D5×C4⋊C4—21no—
106#4035—22no—
107#4034—21no—
108#4033—21no—
109#4032—22no—
110#4031—22no—
111#4029—22no—
112#4027—22no—
113#4025—22no—
114#4025—22no—
115#4024—22no—
116#4023—22no—
117#4022—21no—
118#4021C2×C411yescharacteristic, complemented
119C4×D20C2×C411yescharacteristic, complemented
120#4018—21no—
121#4016—21no—
122#4016C2×C411yescharacteristic, complemented
123#4015C2×C411yescharacteristic, complemented
124#4015C2×C411yescharacteristic, complemented
125C4×Dic10—21no—
126C4×Dic10C2×C412yescomplemented
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#3520—51no—
143#2974—51no—
144#2275D511yescharacteristic, complemented
145#1509—51no—
146#1508—52no—
147#1506—51no—
148#1504—52no—
149#1503—51no—
150#1501—52no—
151#1500—51no—
152C2×#328—52no—
153C24×C5C2×D412yes—
154C2×C5×D4—24no—
155C2×C5×D4C22⋊C412yes—
156C2×C5×D4C22⋊C411yescharacteristic
157C2×C5×D4C22⋊C411yescharacteristic
158C22×C4×C5—21no—
159C22×C4×C5—22no—
160C22×C4×C5—21no—
161C22×C4×C5—28no—
162C22×C4×C5—82no—
163C22×C4×C5—82no—
164C22×C4×C5—82no—
165C22×C4×C5C22×C412yes—
166C22×C4×C5C22×C412yes—
167C22×C4×C5C2×D412yes—
168C22×C4×C5C2×D412yes—
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5C2×D411yescharacteristic
171C22×C4×C5C2×D412yes—
172C22×C4×C5C2411yescharacteristic
173C2×C22⋊D5—22no—
174C2×C22⋊D5—21no—
175C2×C22⋊D5—22no—
176C22×Dic5—22no—
177C22×Dic5—22no—
178#711—44no—
179#709—82no—
180C2×D20C22×C411yescharacteristic
181C2×C4×D5—21no—
182C2×C4×D5—22no—
183C2×Dic10—21no—
184C2×Dic10—22no—
185C2×Dic10—82no—
186C2×Dic10C22×C411yescharacteristic
187C2×Dic10C22×C412yes—
188C5×C4⋊C4—24no—
189C5×C4⋊C4C22⋊C411yescharacteristic
190C5×C4⋊C4C22⋊C411yescharacteristic
191C5×C4⋊C4C22⋊C412yes—
192C5×C22⋊C4—28no—
193C5×C22⋊C4C22⋊C412yes—
194C5×C22⋊C4C22⋊C412yes—
195C5×C22⋊C4C22⋊C412yes—
196C5×C22⋊C4C22⋊C412yes—
197C42×C5—24no—
198C42×C5—81no—
199C42×C5—82no—
200C42×C5C22×C411yescharacteristic
201C42×C5C22×C411yescharacteristic
202C42×C5C2×D411yescharacteristic
203C42×C5C2×D411yescharacteristic
204C22⋊Dic5—22no—
205C22⋊Dic5—22no—
206C22⋊Dic5—22no—
207C22⋊Dic5—22no—
208C22⋊Dic5—21no—
209C22⋊Dic5—22no—
210#685—21no—
211#685—21no—
212#685—21no—
213#685—22no—
214#685—21no—
215C4⋊Dic5—21no—
216C4⋊Dic5—21no—
217C4⋊Dic5—21no—
218C4⋊Dic5—22no—
219C4⋊Dic5C22×C412yes—
220C4⋊Dic5C22×C411yescharacteristic
221C4⋊Dic5C22×C411yescharacteristic
222#683—22no—
223#683—21no—
224#683—21no—
225#683—21no—
226#683—22no—
227#683—22no—
228#683—22no—
229#683—22no—
230#683—21no—
231C4×Dic5—22no—
232C4×Dic5—21no—
233C4×Dic5—22no—
234#681—82no—
235C2×#168—21no—
236C2×#168—21no—
237C2×#168—22no—
238C2×#168—82no—
239#556—51no—
240#554—101no—
241#552—52no—
242#551—102no—
243D4×Q8—51no—
244#548—52no—
245#547—51no—
246#547—51no—
247#546—101no—
248#542—101no—
249#539—102no—
250#533—51no—
251#529—51no—
252#525—52no—
253C2×C22⋊Q8—52no—
254#520—101no—
255C2×C4×D4—21no—
256C2×C4×D4C2×D511yescharacteristic
257#436—101no—
258#435—102no—
259C8×D4—101no—
260#432—51no—
261#431—51no—
262#407—52no—
263C2×C22⋊C8—52no—
264C4×C4⋊C4—21no—
265C4×C4⋊C4C2×D511yescharacteristic
266C4×C22⋊C4—22no—
267C4×C22⋊C4C2×D512yes—
268#376C2×D511yescharacteristic
269C2×#95C2×D512yes—
270#328—51no—
271#328—52no—
272#328—52no—
273#328—52no—
274#328—51no—
275#328—52no—
276#328—52no—
277#327—51no—
278#327—51no—
279#327—52no—
280C23×C5—22no—
281C23×C5—22no—
282C23×C5—24no—
283C23×C5—42no—
284C23×C5C2×C22⋊C412yes—
285C23×C5C2×C22⋊C412yes—
286C23×C5C22×D411yescharacteristic
287C22×D5—21no—
288C5×D4—24no—
289C5×D4—22no—
290C5×D4#10414yes—
291C2×C4×C5—21no—
292C2×C4×C5—21no—
293C2×C4×C5—22no—
294C2×C4×C5—22no—
295C2×C4×C5—22no—
296C2×C4×C5—24no—
297C2×C4×C5—42no—
298C2×C4×C5—42no—
299C2×C4×C5—42no—
300C2×C4×C5—42no—
301C2×C4×C5—42no—
302C2×C4×C5—41no—
303C2×C4×C5—41no—
304C2×C4×C5—42no—
305C2×C4×C5—41no—
306C2×C4×C5—42no—
307C2×C4×C5—41no—
308C2×C4×C5—41no—
309C2×C4×C5—88no—
310C2×C4×C5C2×C22⋊C411yescharacteristic
311C2×C4×C5C2×C22⋊C412yes—
312C2×C4×C5C2×C22⋊C411yescharacteristic
313C2×C4×C5C2×C22⋊C411yescharacteristic
314C2×C4×C5C2×C22⋊C412yes—
315C2×C4×C5C2×C22⋊C411yescharacteristic
316C2×C4×C5C23×C411yescharacteristic
317C2×C4×C5C22×D411yescharacteristic
318C22⋊D5—41no—
319C22⋊D5—41no—
320C22⋊D5—44no—
321C22⋊D5—82no—
322C2×Dic5—21no—
323C2×Dic5—22no—
324C2×Dic5—22no—
325C2×Dic5—21no—
326C2×Dic5—21no—
327C2×Dic5—44no—
328C2×Dic5—82no—
329D20—41no—
330C4×D5—42no—
331C4×D5—82no—
332Dic10—42no—
333Dic10—41no—
334Dic10—42no—
335Dic10—82no—
336#168—41no—
337#168—42no—
338#168—41no—
339C2×#41—52no—
340C2×#41—101no—
341C22×Q8—52no—
342C22×D4C22⋊D511yescharacteristic
343C23×C4—22no—
344C23×C4C22×D512yes—
345C2×M16—102no—
346C22×C8—102no—
347C4⋊Q8—51no—
348C4⋊Q8—51no—
349C4⋊Q8—101no—
350#126—102no—
351#126—102no—
352#126—101no—
353#125—51no—
354#125—102no—
355#125—101no—
356#124—51no—
357#124—101no—
358#124—102no—
359#123—102no—
360#123—102no—
361#123—101no—
362#123—102no—
363#123—102no—
364C22⋊Q8—52no—
365C22⋊Q8—52no—
366C22⋊Q8—52no—
367C22⋊Q8—51no—
368C22⋊Q8—52no—
369C22⋊Q8—102no—
370C22⋊Q8—102no—
371C22⋊Q8—101no—
372C22⋊Q8—102no—
373C22⋊Q8—101no—
374C22⋊Q8—102no—
375#121—52no—
376#121—51no—
377#121—102no—
378#121—101no—
379#120—102no—
380C4×Q8—51no—
381C4×Q8—52no—
382C4×Q8—101no—
383C4×Q8—101no—
384C4×D4—28no—
385C4×D4—51no—
386C4×D4—101no—
387C4×D4—102no—
388C4×D4—102no—
389C4×D4—101no—
390C4×D4—102no—
391C4×D4C4×D511yescharacteristic
392C4×D4C4×D511yescharacteristic
393C4×D4C4×D512yes—
394C4×D4D2011yescharacteristic
395C4×D4D2012yes—
396C4×D4D2011yescharacteristic
397#117—52no—
398#117—101no—
399#117—102no—
400#117—101no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—52no—
403C2×C4⋊C4—102no—
404C2×C4⋊C4—101no—
405C2×C4⋊C4C22⋊D511yescharacteristic
406C2×C22⋊C4—22no—
407C2×C22⋊C4—102no—
408C2×C22⋊C4C22⋊D512yes—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—22no—
412C2×C42C22×D511yescharacteristic
413C4⋊C8—101no—
414C4⋊C8—101no—
415C22⋊C8—51no—
416C22⋊C8—52no—
417C22⋊C8—51no—
418C22⋊C8—102no—
419C22⋊C8—102no—
420#97—101no—
421C4×C8—101no—
422#95—22no—
423#95—21no—
424#95C22×D511yescharacteristic
425#95C22×D512yes—
426#95C22×D511yescharacteristic
427C22×C5—22no—
428C22×C5—21no—
429C22×C5—22no—
430C22×C5—21no—
431C22×C5—22no—
432C22×C5—22no—
433C22×C5—22no—
434C22×C5—44no—
435C22×C5—44no—
436C22×C5C2×#10414yes—
437C22×C5C22×C22⋊C411yescharacteristic
438C22×C5#57711yescharacteristic
439C22×C5#57811yescharacteristic
440C2×D5—41no—
441C2×D5—41no—
442C2×D5—41no—
443C4×C5—21no—
444C4×C5—21no—
445C4×C5—22no—
446C4×C5—22no—
447C4×C5—21no—
448C4×C5—21no—
449C4×C5—21no—
450C4×C5—42no—
451C4×C5—44no—
452C4×C5—42no—
453C4×C5C2×#10412yes—
454Dic5—41no—
455Dic5—42no—
456Dic5—41no—
457Dic5—42no—
458Dic5—41no—
459C24C2×C22⋊D512yes—
460#41—204no—
461#41—402no—
462C2×Q8—51no—
463C2×Q8—52no—
464C2×Q8—101no—
465C2×Q8—102no—
466C2×Q8—402no—
467C2×D4—24no—
468C2×D4—51no—
469C2×D4—101no—
470C2×D4—102no—
471C2×D4—102no—
472C2×D4#68512yes—
473C2×D4#68511yescharacteristic
474C2×D4#68511yescharacteristic
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—28no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—102no—
483C22×C4—102no—
484C22×C4—101no—
485C22×C4—102no—
486C22×C4C2×C4×D512yes—
487C22×C4C2×C4×D512yes—
488C22×C4C2×D2012yes—
489C22×C4C2×D2012yes—
490C22×C4C2×C22⋊D511yescharacteristic
491C22×C4C2×C22⋊D511yescharacteristic
492C22×C4C2×C22⋊D512yes—
493C22×C4C23×D511yescharacteristic
494M16—402no—
495C2×C8—101no—
496C2×C8—101no—
497C2×C8—102no—
498C2×C8—402no—
499C4⋊C4—24no—
500C4⋊C4—51no—
501C4⋊C4—52no—
502C4⋊C4—51no—
503C4⋊C4—101no—
504C4⋊C4—102no—
505C4⋊C4—102no—
506C4⋊C4—102no—
507C4⋊C4—101no—
508C4⋊C4—102no—
509C4⋊C4—101no—
510C4⋊C4—101no—
511C4⋊C4—101no—
512C4⋊C4—102no—
513C4⋊C4—101no—
514C4⋊C4—102no—
515C4⋊C4—101no—
516C4⋊C4#68511yescharacteristic
517C4⋊C4#68512yes—
518C4⋊C4#68511yescharacteristic
519C22⋊C4—28no—
520C22⋊C4—101no—
521C22⋊C4—102no—
522C22⋊C4—102no—
523C22⋊C4—102no—
524C22⋊C4—102no—
525C22⋊C4—102no—
526C22⋊C4—101no—
527C22⋊C4—101no—
528C22⋊C4—101no—
529C22⋊C4—102no—
530C22⋊C4—101no—
531C22⋊C4#68512yes—
532C22⋊C4#68512yes—
533C22⋊C4#68512yes—
534C22⋊C4#68512yes—
535C42—24no—
536C42—81no—
537C42—82no—
538C42—102no—
539C42—102no—
540C42—101no—
541C42C2×C4×D511yescharacteristic
542C42C2×C4×D511yescharacteristic
543C42C2×D2011yescharacteristic
544C42C2×D2011yescharacteristic
545C2×C5—21no—
546C2×C5—21no—
547C2×C5—24no—
548C2×C5—42no—
549C2×C5#288311yescharacteristic
550C2×C5#289811yescharacteristic
551C2×C5C22×#10411yescharacteristic
552D5—81no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—42no—
557C23—101no—
558C23C2×#68512yes—
559C23C2×#68512yes—
560C23C22×C22⋊D511yescharacteristic
561Q8—201no—
562Q8—202no—
563Q8—202no—
564Q8—402no—
565D4—24no—
566D4—22no—
567D4—201no—
568D4—201no—
569D4—201no—
570D4—204no—
571D4—402no—
572D4#393714yes—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—88no—
592C2×C4—101no—
593C2×C4—102no—
594C2×C4—101no—
595C2×C4—102no—
596C2×C4—101no—
597C2×C4—204no—
598C2×C4—202no—
599C2×C4—402no—
600C2×C4—402no—
601C2×C4C2×#68511yescharacteristic
602C2×C4C2×#68512yes—
603C2×C4C2×#68511yescharacteristic
604C2×C4C2×#68511yescharacteristic
605C2×C4C2×#68512yes—
606C2×C4C2×#68511yescharacteristic
607C2×C4C22×C4×D511yescharacteristic
608C2×C4C22×D2011yescharacteristic
609C8—201no—
610C8—202no—
611C8—201no—
612C5#1754811yescharacteristic, complemented, Sylow
613C22—21no—
614C22—22no—
615C22—21no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—44no—
621C22—44no—
622C22—201no—
623C22—201no—
624C22—201no—
625C22C2×#393714yes—
626C22C22×#68511yescharacteristic
627C22#6617011yescharacteristic
628C22#6617211yescharacteristic
629C4—21no—
630C4—21no—
631C4—22no—
632C4—22no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—202no—
640C4—201no—
641C4—201no—
642C4—202no—
643C4—201no—
644C4C2×#393712yes—
645C2—21no—
646C2—21no—
647C2—24no—
648C2—42no—
649C2—401no—
650C2C22×#393711yescharacteristic
651C2#1061427711yescharacteristic
652C2#1061430111yescharacteristic
subgroupG/H
centerC22C22×#685
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2275C2
socleC22×C5C22×C22⋊C4
prime radical#10617668C2
cosocleC22×C4×C5C24
hypercenter#2275D5
hypocenterC5#17548
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. #10600264C2semidirect
5. #10600263C2semidirect
6. #10600238C2semidirect
7. #10600236C2semidirect
8. #10600226C2semidirect
9. #10600224C2semidirect
10. C2×#64672C2semidirect
11. #10600191C2semidirect
19. #65889C4semidirect
20. #65886C4semidirect
21. #65885C4semidirect
23. #65876C4semidirect
24. #65875C4semidirect
25. D4×Dic10C4semidirect
58. C2×#709C2×C4semidirect
59. C22×Dic10C2×C4semidirect
82. #4076C2×C4semidirect
90. #4072C2×C4semidirect
91. C2×C4⋊Dic5C2×C4semidirect
95. #4070C2×C4semidirect
96. #4070C2×C4semidirect
97. #4070C2×C4semidirect
118. #4021C2×C4semidirect
119. C4×D20C2×C4semidirect
122. #4016C2×C4semidirect
123. #4015C2×C4semidirect
124. #4015C2×C4semidirect
126. C4×Dic10C2×C4semidirect
127. C4×Dic10C2×C4semidirect
144. #2275D5semidirect
612. C5#17548semidirect

Maximal subgroups

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

subgroupindex
#106176682
C5×#22752
#106140342
#106002642
#106002632
#106002382
#106002362
#106002262
#106002242
C2×#646722
#106001912
#175485