Finite Group Databasebeta

← #12100034

Subgroups — #12100034

Order 1280.

1862 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D5×C4⋊C4C211yesmaximal, characteristic
2C2×#65150C212yesmaximal, complemented
3C2×#65015C211yesmaximal, characteristic
4#10605593C211yesmaximal, characteristic
5C5×#1853C211yesmaximal, characteristic, complemented
6#10602348C211yesmaximal, characteristic, complemented
7#10601372C212yesmaximal, complemented
8#10600720C211yesmaximal, characteristic, complemented
9#10600298C211yesmaximal, characteristic, complemented
10#10599667C212yesmaximal, complemented
11#10599660C212yesmaximal, complemented
12C23×C4×D5—21no—
13C23×C4×D5C2211yescharacteristic
14C22×C5×C4⋊C4C2211yescharacteristic
15C22×C4⋊Dic5C2211yescharacteristic
16C22×#683—21no—
17C22×#676C2212yes—
18C2×#4038C2211yescharacteristic
19C2×#4038C2211yescharacteristic
20C2×D5×C4⋊C4—28no—
21C2×D5×C4⋊C4—21no—
22C2×D5×C4⋊C4—21no—
23C2×D5×C4⋊C4C2212yes—
24C2×D5×C4⋊C4C2212yes—
25C2×C5×#106C2212yescomplemented
26C2×C5×C22⋊C8C2211yescharacteristic
27C5×#380C2211yescharacteristic
28#65501C2211yescharacteristic
29C2×C22⋊#168C2211yescharacteristic
30C2×#3952C2212yes—
31C2×#3950C2212yescomplemented
32#65276C2211yescharacteristic
33#65276C2211yescharacteristic
34C2×#3940C2212yescomplemented
35C2×#3939C2212yescomplemented
36#65244—21no—
37C22⋊C4×Dic5—21no—
38#65150—28no—
39#65015C418yescomplemented
40#64955—21no—
41#64953—21no—
42C5×#340C2212yescomplemented
43#64771C2212yescomplemented
44#64749C2212yescomplemented
45#64749C2212yescomplemented
46#20844—51nomaximal, Sylow
47C24×D5D411yescharacteristic
48C23×C4×C5—21no—
49C23×C4×C5C2311yescharacteristic
50C23×Dic5—21no—
51C23×Dic5D411yescharacteristic
52C22×C4×D5—24no—
53C22×C4×D5—24no—
54C22×C4×D5—28no—
55C22×C4×D5—22no—
56C22×C4×D5—22no—
57C22×C4×D5—44no—
58C22×C4×D5—44no—
59C22×C4×D5C2×C414yes—
60C22×C4×D5D411yescharacteristic
61C22×C4×D5D411yescharacteristic
62C22×C4×D5Q814yes—
63C22×C4×D5C2311yescharacteristic
64C22×C4×D5C2311yescharacteristic
65C22×C5×C8C2312yes—
66C2×C5×C4⋊C4—21no—
67C2×C5×C4⋊C4—24no—
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C4⋊C4C2312yes—
71C2×C5×C22⋊C4—21no—
72C2×C22⋊Dic5—21no—
73C2×#685—21no—
74C2×#685—21no—
75C2×#685—21no—
76C2×#685—21no—
77C2×C4⋊Dic5—21no—
78C2×C4⋊Dic5—24no—
79C2×C4⋊Dic5C2311yescharacteristic
80C2×C4⋊Dic5C2311yescharacteristic
81C2×C4⋊Dic5C2312yes—
82C2×#683—28no—
83C2×#683—22no—
84C2×#683—22no—
85C2×C4×Dic5—21no—
86C2×C4×Dic5—21no—
87C22×#168C2312yes—
88C2×#676—24no—
89C2×#676—24no—
90C2×#676C2×C418yescomplemented
91#4038—24no—
92#4038—24no—
93D5×C4⋊C4—28no—
94D5×C4⋊C4—28no—
95D5×C4⋊C4—24no—
96D5×C4⋊C4—24no—
97D5×C4⋊C4—24no—
98D5×C4⋊C4D418yes—
99C5×#106—24no—
100C5×C22⋊C8C2×C414yescomplemented
101C5×#95—21no—
102#3963—21no—
103#3963—21no—
104#3963—21no—
105C22⋊#168C2×C414yescomplemented
106#3952C2×C418yescomplemented
107#3950—24no—
108#3940—24no—
109#3939—24no—
110C23×C4⋊C4—51no—
111C2×#428—52no—
112C2×#407—51no—
113#2280—51no—
114#1853—52no—
115#1853—51no—
116#1853D511yescharacteristic, complemented
117#1852—51no—
118#1852—52no—
119#1852—51no—
120#1736—52no—
121C24×C5C2×D411yescharacteristic
122C23×D5—22no—
123C23×D5—22no—
124C23×D5—22no—
125C23×D5—22no—
126C23×D5—22no—
127C23×D5—22no—
128C23×D5C4⋊C414yes—
129C23×D5C2×D411yescharacteristic
130C23×D5C2×D411yescharacteristic
131C22×C4×C5—22no—
132C22×C4×C5—22no—
133C22×C4×C5—24no—
134C22×C4×C5—22no—
135C22×C4×C5—21no—
136C22×C4×C5—21no—
137C22×C4×C5—42no—
138C22×C4×C5—42no—
139C22×C4×C5C22×C412yes—
140C22×C4×C5C2×D411yescharacteristic
141C22×C4×C5C2×Q812yes—
142C22×C4×C5C2411yescharacteristic
143C22×Dic5—22no—
144C22×Dic5—22no—
145C22×Dic5—24no—
146C22×Dic5—22no—
147C22×Dic5—21no—
148C22×Dic5—21no—
149C22×Dic5—42no—
150C22×Dic5—42no—
151C22×Dic5C4⋊C414yes—
152C22×Dic5C2×D411yescharacteristic
153C22×Dic5C2×D411yescharacteristic
154C2×C4×D5—22no—
155C2×C4×D5—22no—
156C2×C4×D5—22no—
157C2×C4×D5—22no—
158C2×C4×D5—22no—
159C2×C4×D5—22no—
160C2×C4×D5—216no—
161C2×C4×D5—216no—
162C2×C4×D5—28no—
163C2×C4×D5—28no—
164C2×C4×D5—432no—
165C2×C4×D5—44no—
166C2×C4×D5—44no—
167C2×C4×D5—44no—
168C2×C4×D5—44no—
169C2×C4×D5C4⋊C414yes—
170C2×C4×D5C4⋊C414yes—
171C2×C4×D5C22×C412yes—
172C2×C4×D5C22×C412yes—
173C2×C4×D5C2×D414yes—
174C2×C4×D5C2×Q812yes—
175C2×C4×D5C2×Q812yes—
176C2×C4×D5#4114yes—
177C2×C5×C8—22no—
178C2×C5×C8—22no—
179C2×C5×C8C22×C414yescomplemented
180C5×C4⋊C4—24no—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—22no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4C2×D414yes—
185C5×C22⋊C4—42no—
186C22⋊Dic5—42no—
187#685—42no—
188#685—42no—
189#685—42no—
190#685—42no—
191C4⋊Dic5—24no—
192C4⋊Dic5—22no—
193C4⋊Dic5—22no—
194C4⋊Dic5—22no—
195C4⋊Dic5C2×D414yes—
196#683—28no—
197#683—28no—
198C4×Dic5—42no—
199C4×Dic5—42no—
200C2×#168—22no—
201C2×#168—22no—
202C2×#168C22×C414yescomplemented
203#676—216no—
204C24×C4—51no—
205C24×C4—101no—
206C22×M16—52no—
207C2×#117—51no—
208C2×#117—51no—
209C22×C4⋊C4—52no—
210C22×C4⋊C4—51no—
211C22×C4⋊C4—52no—
212C22×C4⋊C4—108no—
213C22×C4⋊C4—101no—
214C22×C4⋊C4—101no—
215C22×C4⋊C4—101no—
216C22×C4⋊C4C2×D511yescharacteristic
217#428—108no—
218C2×#107—52no—
219C2×#107—52no—
220C2×#106—52no—
221C2×#106C2×D512yescomplemented
222#407—58no—
223C2×C22⋊C8—51no—
224C2×C22⋊C8—52no—
225C2×C22⋊C8C2×D511yescharacteristic
226#381—101no—
227#380—51no—
228#380—51no—
229#380—51no—
230#380C2×D511yescharacteristic
231#379—101no—
232C4×C22⋊C4—101no—
233C4×C22⋊C4—101no—
234#340—52no—
235#340—52no—
236#340—52no—
237#340C2×D512yescomplemented
238C23×C5—21no—
239C23×C5—21no—
240C23×C5—21no—
241C23×C5—21no—
242C23×C5C2×C4⋊C412yes—
243C23×C5AGL(1,ℤ/8)12yes—
244C23×C5#13712yes—
245C23×C5C22×D411yescharacteristic
246C22×D5—28no—
247C22×D5—28no—
248C22×D5—28no—
249C22×D5—28no—
250C22×D5—21no—
251C22×D5—21no—
252C22×D5—21no—
253C22×D5—21no—
254C22×D5—21no—
255C22×D5—21no—
256C22×D5—21no—
257C22×D5—21no—
258C22×D5—28no—
259C22×D5—28no—
260C22×D5—28no—
261C22×D5C2×C4⋊C412yes—
262C22×D5C2×C4⋊C412yes—
263C22×D5#12012yes—
264C22×D5#12312yes—
265C22×D5AGL(1,ℤ/8)12yes—
266C22×D5#13712yes—
267C2×C4×C5—21no—
268C2×C4×C5—21no—
269C2×C4×C5—24no—
270C2×C4×C5—24no—
271C2×C4×C5—24no—
272C2×C4×C5—48no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—41no—
276C2×C4×C5—41no—
277C2×C4×C5—41no—
278C2×C4×C5—41no—
279C2×C4×C5—41no—
280C2×C4×C5—41no—
281C2×C4×C5C2×C4⋊C412yes—
282C2×C4×C5C23×C411yescharacteristic
283C2×C4×C5C22×D412yes—
284C2×C4×C5C22×Q811yescharacteristic
285C2×C4×C5C2×#4112yes—
286C2×Dic5—21no—
287C2×Dic5—21no—
288C2×Dic5—28no—
289C2×Dic5—24no—
290C2×Dic5—48no—
291C2×Dic5—42no—
292C2×Dic5—42no—
293C2×Dic5—41no—
294C2×Dic5—41no—
295C2×Dic5—41no—
296C2×Dic5—41no—
297C2×Dic5—41no—
298C2×Dic5—41no—
299C2×Dic5C2×C4⋊C412yes—
300C2×Dic5C2×C4⋊C412yes—
301C2×Dic5#12012yes—
302C2×Dic5#12312yes—
303C4×D5—28no—
304C4×D5—24no—
305C4×D5—24no—
306C4×D5—28no—
307C4×D5—416no—
308C4×D5—416no—
309C4×D5C4×D418yes—
310C4×D5C22⋊Q818yes—
311C5×C8—24no—
312#168—24no—
313C25—51no—
314C23×C4—21no—
315C23×C4—51no—
316C23×C4—51no—
317C23×C4—54no—
318C23×C4—51no—
319C23×C4—51no—
320C23×C4—54no—
321C23×C4—51no—
322C23×C4—104no—
323C23×C4—104no—
324C23×C4—102no—
325C23×C4—102no—
326C23×C4—108no—
327C23×C4—101no—
328C23×C4—204no—
329C23×C4—204no—
330C23×C4C22×D511yescharacteristic
331C2×M16—58no—
332C2×M16—104no—
333C2×M16—104no—
334C22×C8—52no—
335C22×C8C22×D512yes—
336#117—104no—
337#117—104no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—52no—
341C2×C4⋊C4—58no—
342C2×C4⋊C4—51no—
343C2×C4⋊C4—51no—
344C2×C4⋊C4—104no—
345C2×C4⋊C4—104no—
346C2×C4⋊C4—108no—
347C2×C4⋊C4—108no—
348C2×C4⋊C4—104no—
349C2×C4⋊C4—102no—
350C2×C4⋊C4—102no—
351C2×C4⋊C4—108no—
352C2×C4⋊C4—101no—
353C2×C4⋊C4—104no—
354C2×C4⋊C4C22×D511yescharacteristic
355C2×C4⋊C4C22×D511yescharacteristic
356C2×C4⋊C4C22×D512yes—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—101no—
359C2×C22⋊C4—101no—
360C2×C22⋊C4—101no—
361C2×C22⋊C4—101no—
362C2×C22⋊C4—101no—
363C2×C42—101no—
364C2×C42—101no—
365#107—104no—
366#107—104no—
367#106—24no—
368#106—104no—
369C22⋊C8—54no—
370C22⋊C8—58no—
371C22⋊C8C4×D514yescomplemented
372#95—21no—
373#95—101no—
374#95—101no—
375#95—101no—
376C22×C5—22no—
377C22×C5—22no—
378C22×C5—22no—
379C22×C5—22no—
380C22×C5—22no—
381C22×C5—22no—
382C22×C5#42814yescomplemented
383C22×C5C22×C4⋊C411yescharacteristic
384C22×C5C2×#12011yescharacteristic
385C22×C5C2×#12311yescharacteristic
386C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
387C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
388C22×C5C2×#13711yescharacteristic
389C22×C5C2×#13711yescharacteristic
390C2×D5—232no—
391C2×D5—22no—
392C2×D5—22no—
393C2×D5—22no—
394C2×D5—22no—
395C2×D5—22no—
396C2×D5—22no—
397C2×D5—22no—
398C2×D5—22no—
399C2×D5—22no—
400C2×D5—22no—
401C2×D5—22no—
402C2×D5—22no—
403C2×D5#38514yes—
404C2×D5#42814yescomplemented
405C4×C5—22no—
406C4×C5—44no—
407C4×C5—81no—
408C4×C5—81no—
409C4×C5C2×C4×D412yes—
410C4×C5C2×C22⋊Q812yes—
411Dic5—22no—
412Dic5—44no—
413Dic5—81no—
414Dic5—81no—
415Dic5#38514yes—
416C24—51no—
417C24—51no—
418C24—54no—
419C24—102no—
420C24—102no—
421C24—102no—
422C24—102no—
423C24—102no—
424C24—102no—
425C24D4×D511yescharacteristic
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—51no—
435C22×C4—52no—
436C22×C4—54no—
437C22×C4—52no—
438C22×C4—54no—
439C22×C4—51no—
440C22×C4—54no—
441C22×C4—52no—
442C22×C4—52no—
443C22×C4—54no—
444C22×C4—54no—
445C22×C4—102no—
446C22×C4—102no—
447C22×C4—1016no—
448C22×C4—1016no—
449C22×C4—102no—
450C22×C4—102no—
451C22×C4—108no—
452C22×C4—108no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4—104no—
458C22×C4—102no—
459C22×C4—101no—
460C22×C4—101no—
461C22×C4—204no—
462C22×C4—2032no—
463C22×C4—204no—
464C22×C4—204no—
465C22×C4—204no—
466C22×C4—202no—
467C22×C4—202no—
468C22×C4C2×C4×D512yes—
469C22×C4D4×D511yescharacteristic
470C22×C4Q8×D512yes—
471C22×C4C23×D511yescharacteristic
472M16—1016no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—54no—
476C2×C8—102no—
477C2×C8—102no—
478C2×C8C2×C4×D514yescomplemented
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—54no—
484C4⋊C4—108no—
485C4⋊C4—108no—
486C4⋊C4—104no—
487C4⋊C4—102no—
488C4⋊C4—102no—
489C4⋊C4—102no—
490C4⋊C4D4×D514yes—
491C22⋊C4—42no—
492C22⋊C4—202no—
493C22⋊C4—202no—
494C22⋊C4—202no—
495C22⋊C4—202no—
496C22⋊C4—202no—
497C42—202no—
498C42—202no—
499C2×C5—24no—
500C2×C5C2×#38511yescharacteristic
501C2×C5C2×#42811yescharacteristic
502C2×C5C2×#42811yescharacteristic
503C2×C5#300311yescharacteristic
504C2×C5#300411yescharacteristic
505C2×C5#309211yescharacteristic
506C2×C5#309211yescharacteristic
507D5—24no—
508D5—24no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—52no—
514C23—52no—
515C23—52no—
516C23—52no—
517C23—52no—
518C23—52no—
519C23—108no—
520C23—108no—
521C23—108no—
522C23—108no—
523C23—108no—
524C23—108no—
525C23—101no—
526C23—101no—
527C23—101no—
528C23—101no—
529C23—101no—
530C23—101no—
531C23—108no—
532C23—101no—
533C23—101no—
534C23D5×C4⋊C412yes—
535C23#406012yes—
536C23#406112yes—
537C23C2×D4×D511yescharacteristic
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—52no—
553C2×C4—52no—
554C2×C4—52no—
555C2×C4—58no—
556C2×C4—58no—
557C2×C4—52no—
558C2×C4—108no—
559C2×C4—108no—
560C2×C4—108no—
561C2×C4—101no—
562C2×C4—101no—
563C2×C4—104no—
564C2×C4—104no—
565C2×C4—104no—
566C2×C4—2016no—
567C2×C4—2016no—
568C2×C4—208no—
569C2×C4—202no—
570C2×C4—202no—
571C2×C4—201no—
572C2×C4—201no—
573C2×C4—201no—
574C2×C4—201no—
575C2×C4—201no—
576C2×C4—201no—
577C2×C4D5×C4⋊C412yes—
578C2×C4C22×C4×D511yescharacteristic
579C2×C4C2×D4×D512yes—
580C2×C4C2×Q8×D511yescharacteristic
581C2×C4D5×#4112yes—
582C8—24no—
583C8—104no—
584C5#2084411yescharacteristic, complemented, Sylow
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—54no—
592C22—54no—
593C22—1032no—
594C22—102no—
595C22—102no—
596C22—102no—
597C22—102no—
598C22—102no—
599C22—102no—
600C22—102no—
601C22—102no—
602C22—102no—
603C22—102no—
604C22—102no—
605C22—102no—
606C22#6515014yescomplemented
607C22C2×D5×C4⋊C411yescharacteristic
608C22D5×#12011yescharacteristic
609C22D5×#12311yescharacteristic
610C22C2×#406011yescharacteristic
611C22C2×#406111yescharacteristic
612C22D5×AGL(1,ℤ/8)11yescharacteristic
613C22D5×#13711yescharacteristic
614C4—22no—
615C4—44no—
616C4—54no—
617C4—81no—
618C4—81no—
619C4—102no—
620C4—204no—
621C4—401no—
622C4—401no—
623C4C4×D4×D512yes—
624C4D5×C22⋊Q812yes—
625C2—24no—
626C2—104no—
627C2—104no—
628C2D5×#38511yescharacteristic
629C2C2×#6515011yescharacteristic
630C2D5×#42811yescharacteristic
631C2#1061177011yescharacteristic
632C2#1061187711yescharacteristic
633C2#1061305611yescharacteristic
634C2#1061315811yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1853C2
socleC23×C5C22×D4
prime radicalC24×D5D4
cosocleC22×C4×C5C24
hypercenter#1853D5
hypocenterC5#20844
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. C2×#65150C2semidirect
5. C5×#1853C2semidirect
6. #10602348C2semidirect
7. #10601372C2semidirect
8. #10600720C2semidirect
9. #10600298C2semidirect
10. #10599667C2semidirect
11. #10599660C2semidirect
25. C2×C5×#106C22semidirect
31. C2×#3950C22semidirect
34. C2×#3940C22semidirect
35. C2×#3939C22semidirect
39. #65015C4semidirect
42. C5×#340C22semidirect
43. #64771C22semidirect
44. #64749C22semidirect
45. #64749C22semidirect
90. C2×#676C2×C4semidirect
100. C5×C22⋊C8C2×C4semidirect
105. C22⋊#168C2×C4semidirect
106. #3952C2×C4semidirect
116. #1853D5semidirect
179. C2×C5×C8C22×C4semidirect
202. C2×#168C22×C4semidirect
221. C2×#106C2×D5semidirect
237. #340C2×D5semidirect
371. C22⋊C8C4×D5semidirect
382. C22×C5#428semidirect
404. C2×D5#428semidirect
478. C2×C8C2×C4×D5semidirect
584. C5#20844semidirect
606. C22#65150semidirect

Maximal subgroups

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

subgroupindex
C22×D5×C4⋊C42
C2×#651502
C2×#650152
#106055932
C5×#18532
#106023482
#106013722
#106007202
#106002982
#105996672
#105996602
#208445