Finite Group Databasebeta

← #12126742

Subgroups — #12126742

Order 1280.

1164 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10617626C211yesmaximal, characteristic, complemented
2C2×C5×#410C211yesmaximal, characteristic
3C2×#65529C211yesmaximal, characteristic
4C2×#65529C211yesmaximal, characteristic
5#10602386C211yesmaximal, characteristic, complemented
6#10602386C211yesmaximal, characteristic, complemented
7#10602385C211yesmaximal, characteristic, complemented
8#10600818C218yesmaximal, complemented
9C22×C5×#41C2211yescharacteristic
10C2×C5×#117C2211yescharacteristic
11C2×C5×#117C2211yescharacteristic
12C2×#4072C2211yescharacteristic
13C2×#4072C2211yescharacteristic
14C2×#4072C2211yescharacteristic
15C2×#4072C2211yescharacteristic
16#65853—22no—
17#65853—22no—
18#65851—22no—
19#65851—22no—
20#65834—22no—
21#65817C2214yes—
22#65804—22no—
23C5×#410—22no—
24C5×#410—22no—
25C2×C5×#99C2214yes—
26#65529—22no—
27#65529—22no—
28#65529—22no—
29#65529—22no—
30C2×#3966C2214yes—
31C2×#3966C2214yes—
32#64747C2214yes—
33#64747C2214yes—
34#64746C2214yes—
35#23474—51nomaximal, Sylow
36C2×C5×#41—21no—
37C2×C5×#41—21no—
38C2×C5×#41—21no—
39C2×C5×#41—21no—
40C2×C5×#41—21no—
41C2×C5×#41—21no—
42C2×C5×#41—21no—
43C2×C5×#41—21no—
44C2×C5×#41—21no—
45C2×C5×#41—21no—
46C2×C5×#41—21no—
47C2×C5×#41—21no—
48C22×C5×Q8Q811yescharacteristic, complemented
49C22×C5×D4Q811yescharacteristic, complemented
50C22×C5×D4C2312yes—
51C23×C4×C5Q812yescomplemented
52C23×C4×C5C2311yescharacteristic
53C5×#117—21no—
54C5×#117—21no—
55C5×#117—21no—
56C5×#117—21no—
57C5×#117—21no—
58C5×#117—21no—
59C5×#117D412yes—
60C5×#117D412yes—
61C2×C5×C4⋊C4—21no—
62C2×C5×C4⋊C4—21no—
63C2×C5×C22⋊C4C2312yes—
64C2×C5×C22⋊C4C2312yes—
65C2×C42×C5—21no—
66C2×C42×C5—21no—
67C2×C22⋊Dic5C2312yes—
68C2×C22⋊Dic5C2312yes—
69C2×C22⋊Dic5C2312yes—
70C2×C22⋊Dic5C2312yes—
71#4072—21no—
72#4072—21no—
73#4072—21no—
74#4072—21no—
75#4072—21no—
76#4072—21no—
77#4072—21no—
78#4072—21no—
79#4072—21no—
80#4072—21no—
81#4072—21no—
82#4072—21no—
83#4072D412yes—
84#4072D412yes—
85#4072D412yes—
86#4072D412yes—
87C2×C4⋊Dic5—21no—
88C2×C4⋊Dic5—21no—
89C2×C4⋊Dic5—21no—
90C2×C4⋊Dic5—21no—
91#4070—28no—
92C2×C4×Dic5—21no—
93C2×C4×Dic5—21no—
94C2×C4×Dic5—21no—
95C2×C4×Dic5—21no—
96#4036—24no—
97#4036—24no—
98#4034—24no—
99#4034—24no—
100#4024—28no—
101#4024—28no—
102#4016—24no—
103#4015—24no—
104C4×Dic10—28no—
105C5×#99—28no—
106#3966—28no—
107#3966—28no—
108#3458—51no—
109C2×#410—51no—
110C2×#410—51no—
111C2×#410D511yescharacteristic
112#2032—58no—
113#1740—51no—
114#1740—51no—
115#1740—51no—
116C24×C5C2×Q812yes—
117C24×C5C2411yescharacteristic
118C5×#41—41no—
119C5×#41—41no—
120C5×#41—41no—
121C5×#41—41no—
122C5×#41—41no—
123C5×#41—41no—
124C5×#41—41no—
125C5×#41—41no—
126C5×#41—41no—
127C5×#41—41no—
128C5×#41—41no—
129C5×#41—41no—
130C5×#41—41no—
131C5×#41—41no—
132C5×#41—41no—
133C5×#41—41no—
134C2×C5×Q8—21no—
135C2×C5×Q8—21no—
136C2×C5×Q8—21no—
137C2×C5×Q8—21no—
138C2×C5×Q8—21no—
139C2×C5×Q8—21no—
140C2×C5×D4—22no—
141C2×C5×D4—22no—
142C2×C5×D4—22no—
143C2×C5×D4—22no—
144C2×C5×D4—22no—
145C2×C5×D4—22no—
146C2×C5×D4—21no—
147C2×C5×D4—21no—
148C2×C5×D4—21no—
149C2×C5×D4—21no—
150C2×C5×D4—21no—
151C2×C5×D4—21no—
152C22×C4×C5—22no—
153C22×C4×C5—22no—
154C22×C4×C5—22no—
155C22×C4×C5—22no—
156C22×C4×C5—22no—
157C22×C4×C5—22no—
158C22×C4×C5—22no—
159C22×C4×C5—22no—
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×Q812yes—
167C22×C4×C5C2×Q811yescharacteristic
168C22×C4×C5C2×Q811yescharacteristic
169C22×C4×C5#4111yescharacteristic
170C22×C4×C5#4111yescharacteristic
171C22×C4×C5#4111yescharacteristic
172C22×C4×C5#4111yescharacteristic
173C22×C4×C5#4111yescharacteristic
174C22×C4×C5#4111yescharacteristic
175C22×Dic5—22no—
176C22×Dic5—22no—
177C22×Dic5—22no—
178C22×Dic5—22no—
179C2×Dic10—28no—
180C5×C4⋊C4—22no—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—41no—
183C5×C4⋊C4—41no—
184C5×C22⋊C4—22no—
185C5×C22⋊C4—22no—
186C5×C22⋊C4—22no—
187C5×C22⋊C4C2×D414yes—
188C42×C5—22no—
189C42×C5—22no—
190C42×C5—41no—
191C42×C5—41no—
192C22⋊Dic5—22no—
193C22⋊Dic5—22no—
194C22⋊Dic5—22no—
195C22⋊Dic5—22no—
196C22⋊Dic5—22no—
197C22⋊Dic5—22no—
198C22⋊Dic5C2×D414yes—
199C22⋊Dic5C2×D414yes—
200C4⋊Dic5—28no—
201C4⋊Dic5—22no—
202C4⋊Dic5—22no—
203C4⋊Dic5—22no—
204C4⋊Dic5—22no—
205C4⋊Dic5—41no—
206C4⋊Dic5—41no—
207C4⋊Dic5—41no—
208C4⋊Dic5—41no—
209#683—28no—
210#683—28no—
211C4×Dic5—22no—
212C4×Dic5—22no—
213C4×Dic5—22no—
214C4×Dic5—22no—
215C4×Dic5—41no—
216C4×Dic5—41no—
217C4×Dic5—41no—
218C4×Dic5—41no—
219C22×#41C2×D511yescharacteristic
220#544—102no—
221#544—102no—
222#544—102no—
223#543—54no—
224#533—102no—
225#533—102no—
226#533—102no—
227C2×#117—51no—
228C2×#117—51no—
229C2×#117—51no—
230C2×#117—51no—
231C2×#117C2×D511yescharacteristic
232C2×#117C2×D511yescharacteristic
233#410—22no—
234#410—22no—
235#410—102no—
236#410—102no—
237#410—102no—
238#410—102no—
239C2×#99—54no—
240C2×#99—54no—
241C2×#99C2×D514yes—
242#342—54no—
243#342—54no—
244#342—54no—
245C23×C5—22no—
246C23×C5—22no—
247C23×C5—22no—
248C23×C5—81no—
249C23×C5—82no—
250C23×C5C22×D411yescharacteristic
251C23×C5C22×D411yescharacteristic
252C23×C5C22×D411yescharacteristic
253C23×C5C22×Q811yescharacteristic
254C23×C5C2×#4111yescharacteristic
255C23×C5C2×#4111yescharacteristic
256C23×C5C2×#4111yescharacteristic
257C5×Q8—41no—
258C5×Q8—41no—
259C5×Q8—41no—
260C5×Q8—41no—
261C5×D4—42no—
262C5×D4—42no—
263C5×D4—42no—
264C5×D4—42no—
265C5×D4—41no—
266C5×D4—41no—
267C5×D4—41no—
268C5×D4—41no—
269C2×C4×C5—24no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5—42no—
277C2×C4×C5—42no—
278C2×C4×C5—82no—
279C2×C4×C5—82no—
280C2×C4×C5#12011yescharacteristic
281C2×C4×C5#12011yescharacteristic
282C2×C4×C5#12011yescharacteristic
283C2×C4×C5#12011yescharacteristic
284C2×C4×C5C22⋊Q811yescharacteristic
285C2×C4×C5C22⋊Q811yescharacteristic
286C2×C4×C5C22⋊Q811yescharacteristic
287C2×C4×C5C22⋊Q811yescharacteristic
288C2×C4×C5C22⋊Q811yescharacteristic
289C2×C4×C5C22⋊Q811yescharacteristic
290C2×C4×C5C22⋊Q811yescharacteristic
291C2×C4×C5C22⋊Q811yescharacteristic
292C2×C4×C5C22⋊Q811yescharacteristic
293C2×C4×C5C22⋊Q811yescharacteristic
294C2×C4×C5C22⋊Q811yescharacteristic
295C2×C4×C5C22⋊Q811yescharacteristic
296C2×C4×C5#12411yescharacteristic
297C2×C4×C5#12411yescharacteristic
298C2×C4×C5#12411yescharacteristic
299C2×C4×C5#12411yescharacteristic
300C2×C4×C5#12411yescharacteristic
301C2×C4×C5#12411yescharacteristic
302C2×C4×C5#12411yescharacteristic
303C2×C4×C5#12411yescharacteristic
304C2×C4×C5#12411yescharacteristic
305C2×C4×C5#12411yescharacteristic
306C2×C4×C5#12411yescharacteristic
307C2×C4×C5#12411yescharacteristic
308C2×Dic5—24no—
309C2×Dic5—24no—
310C2×Dic5—42no—
311C2×Dic5—42no—
312C2×Dic5—82no—
313C2×Dic5—82no—
314C2×Dic5—82no—
315C2×Dic5—82no—
316Dic10—88no—
317C2×#41—21no—
318C2×#41—21no—
319C2×#41—21no—
320C2×#41—21no—
321C2×#41—21no—
322C2×#41—21no—
323C2×#41—21no—
324C2×#41—21no—
325C2×#41—21no—
326C2×#41—21no—
327C2×#41—21no—
328C2×#41—21no—
329C22×Q8Dic1011yescharacteristic, complemented
330C22×D4Dic1011yescharacteristic, complemented
331C22×D4C22×D512yes—
332C23×C4Dic1012yescomplemented
333C23×C4C22×D511yescharacteristic
334C4⋊Q8—104no—
335C4⋊Q8—104no—
336C4⋊Q8—104no—
337#125—104no—
338#125—104no—
339#125—104no—
340C22⋊Q8—108no—
341C22⋊Q8—108no—
342C22⋊Q8—108no—
343C4×Q8—108no—
344#117—21no—
345#117—21no—
346#117—21no—
347#117—21no—
348#117—21no—
349#117—21no—
350#117—52no—
351#117—52no—
352#117—52no—
353#117—52no—
354#117—101no—
355#117—101no—
356#117—101no—
357#117—101no—
358#117—101no—
359#117—101no—
360#117—101no—
361#117—101no—
362#117—101no—
363#117—101no—
364#117—101no—
365#117—101no—
366#117C22⋊D512yes—
367#117C22⋊D512yes—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—101no—
371C2×C4⋊C4—101no—
372C2×C4⋊C4—101no—
373C2×C4⋊C4—101no—
374C2×C22⋊C4—52no—
375C2×C22⋊C4—52no—
376C2×C22⋊C4—52no—
377C2×C22⋊C4—52no—
378C2×C22⋊C4C22×D512yes—
379C2×C22⋊C4C22×D512yes—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—101no—
383C2×C42—101no—
384C2×C42—101no—
385C2×C42—101no—
386#99—28no—
387#99—108no—
388#99—108no—
389C22×C5—41no—
390C22×C5—41no—
391C22×C5—41no—
392C22×C5—41no—
393C22×C5—41no—
394C22×C5—41no—
395C22×C5—41no—
396C22×C5—42no—
397C22×C5—82no—
398C22×C5—82no—
399C22×C5—82no—
400C22×C5C2×#12011yescharacteristic
401C22×C5C2×C22⋊Q811yescharacteristic
402C22×C5C2×C22⋊Q811yescharacteristic
403C22×C5C2×C22⋊Q811yescharacteristic
404C22×C5C2×#12411yescharacteristic
405C22×C5C2×#12411yescharacteristic
406C22×C5C2×#12411yescharacteristic
407C4×C5—44no—
408C4×C5—42no—
409C4×C5—82no—
410C4×C5#39311yescharacteristic
411C4×C5#39311yescharacteristic
412C4×C5#39311yescharacteristic
413C4×C5#39311yescharacteristic
414C4×C5#39311yescharacteristic
415C4×C5#39311yescharacteristic
416C4×C5#39311yescharacteristic
417C4×C5#39311yescharacteristic
418Dic5—44no—
419Dic5—44no—
420Dic5—82no—
421Dic5—82no—
422C24C2×Dic1012yes—
423C24C23×D511yescharacteristic
424#41—41no—
425#41—41no—
426#41—41no—
427#41—41no—
428#41—41no—
429#41—41no—
430#41—41no—
431#41—41no—
432#41—41no—
433#41—41no—
434#41—41no—
435#41—41no—
436#41—41no—
437#41—41no—
438#41—41no—
439#41—41no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—108no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—102no—
468C22×C4—102no—
469C22×C4—102no—
470C22×C4—102no—
471C22×C4C2×Dic1012yes—
472C22×C4C2×Dic1011yescharacteristic
473C22×C4C2×Dic1011yescharacteristic
474C22×C4#70911yescharacteristic
475C22×C4#70911yescharacteristic
476C22×C4D4×D511yescharacteristic
477C22×C4D4×D511yescharacteristic
478C22×C4D4×D511yescharacteristic
479C22×C4D4×D511yescharacteristic
480C22×C4#71111yescharacteristic
481C22×C4#71111yescharacteristic
482C22×C4#71111yescharacteristic
483C22×C4#71111yescharacteristic
484C22×C4C2×C22⋊D511yescharacteristic
485C22×C4C2×C22⋊D511yescharacteristic
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—108no—
491C4⋊C4—108no—
492C4⋊C4—108no—
493C4⋊C4—102no—
494C4⋊C4—102no—
495C4⋊C4—102no—
496C4⋊C4—102no—
497C4⋊C4—201no—
498C4⋊C4—201no—
499C4⋊C4—201no—
500C4⋊C4—201no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—54no—
505C22⋊C4—54no—
506C22⋊C4—102no—
507C22⋊C4—102no—
508C22⋊C4—102no—
509C22⋊C4—102no—
510C22⋊C4—102no—
511C22⋊C4—102no—
512C22⋊C4C2×C22⋊D514yes—
513C42—22no—
514C42—22no—
515C42—41no—
516C42—41no—
517C42—102no—
518C42—102no—
519C42—102no—
520C42—102no—
521C42—201no—
522C42—201no—
523C42—201no—
524C42—201no—
525C2×C5—21no—
526C2×C5—21no—
527C2×C5—21no—
528C2×C5—21no—
529C2×C5—21no—
530C2×C5—21no—
531C2×C5—21no—
532C2×C5—82no—
533C2×C5C2×#39311yescharacteristic
534C23—22no—
535C23—22no—
536C23—22no—
537C23—81no—
538C23—82no—
539C23C22×Dic1011yescharacteristic
540C23C2×#70911yescharacteristic
541C23C2×D4×D511yescharacteristic
542C23C2×D4×D511yescharacteristic
543C23C2×#71111yescharacteristic
544C23C2×#71111yescharacteristic
545C23C22×C22⋊D511yescharacteristic
546Q8—41no—
547Q8—41no—
548Q8—41no—
549Q8—41no—
550Q8—408no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—41no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—104no—
571C2×C4—104no—
572C2×C4—202no—
573C2×C4—202no—
574C2×C4—402no—
575C2×C4—402no—
576C2×C4—402no—
577C2×C4—402no—
578C2×C4#402411yescharacteristic
579C2×C4#402411yescharacteristic
580C2×C4#402411yescharacteristic
581C2×C4#402411yescharacteristic
582C2×C4#402411yescharacteristic
583C2×C4#402411yescharacteristic
584C2×C4#402411yescharacteristic
585C2×C4#402411yescharacteristic
586C2×C4#403111yescharacteristic
587C2×C4#403111yescharacteristic
588C2×C4#403111yescharacteristic
589C2×C4#403111yescharacteristic
590C2×C4#403111yescharacteristic
591C2×C4#403111yescharacteristic
592C2×C4#403111yescharacteristic
593C2×C4#403111yescharacteristic
594C2×C4#407011yescharacteristic
595C2×C4#407011yescharacteristic
596C2×C4#407011yescharacteristic
597C2×C4#407011yescharacteristic
598C2×C4#408211yescharacteristic
599C2×C4#408211yescharacteristic
600C2×C4#408211yescharacteristic
601C2×C4#408211yescharacteristic
602C2×C4#408311yescharacteristic
603C2×C4#408311yescharacteristic
604C2×C4#408311yescharacteristic
605C2×C4#408311yescharacteristic
606C5#2347411yescharacteristic, complemented, Sylow
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22C2×#402411yescharacteristic
619C22C2×#402411yescharacteristic
620C22C2×#403111yescharacteristic
621C22C2×#403111yescharacteristic
622C22C2×#407011yescharacteristic
623C22C2×#408211yescharacteristic
624C22C2×#408311yescharacteristic
625C4—44no—
626C4—42no—
627C4—82no—
628C4—204no—
629C4—204no—
630C4—402no—
631C4—402no—
632C4#6523611yescharacteristic
633C4#6523611yescharacteristic
634C4#6523611yescharacteristic
635C4#6523611yescharacteristic
636C4#6523611yescharacteristic
637C4#6523611yescharacteristic
638C4#6523611yescharacteristic
639C4#6523611yescharacteristic
640C2—21no—
641C2—21no—
642C2—21no—
643C2—21no—
644C2—21no—
645C2—21no—
646C2—21no—
647C2—82no—
648C2C2×#6523611yescharacteristic
subgroupG/H
centerC4#65236
commutator, cosocleC24×C5C24
FrattiniC24C23×D5
FittingC2×C5×#410C2
socleC2×C5C2×#393
prime radicalC22×C5×#41C22
hypercenterC2×#410D5
hypocenterC5#23474
LCSG ▷ C24×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
1. #10617626C2semidirect
5. #10602386C2semidirect
6. #10602386C2semidirect
7. #10602385C2semidirect
8. #10600818C2semidirect
48. C22×C5×Q8Q8semidirect
49. C22×C5×D4Q8semidirect
51. C23×C4×C5Q8semidirect
329. C22×Q8Dic10semidirect
330. C22×D4Dic10semidirect
332. C23×C4Dic10semidirect
606. C5#23474semidirect

Maximal subgroups

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

subgroupindex
#106176262
C2×C5×#4102
C2×#655292
#106023862
#106023852
#106008182
#234745