Finite Group Databasebeta

← #28589432

Subgroups — #28589432

Order 15104.

1630 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273685C211yesmaximal, characteristic
2C2×#16837443C212yesmaximal, complemented
3C2×#16837364C211yesmaximal, characteristic
4#380×D59C211yesmaximal, characteristic
5#21424296C212yesmaximal, complemented
6#21424292C212yesmaximal, complemented
7#21424291C211yesmaximal, characteristic, complemented
8#21424288C211yesmaximal, characteristic, complemented
9#21424284C211yesmaximal, characteristic, complemented
10#21423814C212yesmaximal, complemented
11C59×#1852C211yesmaximal, characteristic, complemented
12C23×C4×D59C2211yescharacteristic
13C22×#11745659C2212yes—
14C2×#14273685—21no—
15C2×#14273685—21no—
16C2×#14273685—28no—
17C2×#14273685C2212yes—
18C2×#14273685C2212yes—
19C22×C4⋊Dic59C2211yescharacteristic
20C2×C4⋊C4×D59C2211yescharacteristic
21C2×C4⋊C4×D59C2211yescharacteristic
22C22×(C2×D59)⋊C4—21no—
23C2×C22⋊C4×D59—21no—
24C22×C4×Dic59—21no—
25#16837443—28no—
26C2×#14273660C2212yescomplemented
27C2×#14273659C2212yescomplemented
28C2×#14273658C2212yescomplemented
29#16837364C418yescomplemented
30C2×C22⋊(C59⋊C8)C2211yescharacteristic
31C2×(C2×D59)⋊C8C2212yes—
32#16837262C2211yescharacteristic
33#16837261—21no—
34#16837260—21no—
35#16837259C2211yescharacteristic
36#16837259C2211yescharacteristic
37#95×D59—21no—
38#16837165C2212yescomplemented
39#16837164C2212yescomplemented
40#16837164C2212yescomplemented
41C22×C4⋊C4×C59C2211yescharacteristic
42C2×#107×C59C2212yescomplemented
43C2×C22⋊C8×C59C2211yescharacteristic
44C59×#380C2211yescharacteristic
45C59×#340C2212yescomplemented
46C24×D59D411yescharacteristic
47C23×Dic59—21no—
48C23×Dic59D411yescharacteristic
49C22×C4×D59—24no—
50C22×C4×D59—24no—
51C22×C4×D59—21no—
52C22×C4×D59—21no—
53C22×C4×D59C2×C414yes—
54C22×C4×D59D411yescharacteristic
55C22×C4×D59D411yescharacteristic
56C22×C4×D59Q814yes—
57C22×C4×D59C2311yescharacteristic
58C22×C4×D59C2311yescharacteristic
59C2×#11745659—24no—
60C2×#11745659—24no—
61C2×#11745659C2×C418yescomplemented
62C22×C59⋊C8C2312yes—
63#14273685—28no—
64#14273685—28no—
65#14273685—24no—
66#14273685—24no—
67#14273685—24no—
68#14273685D418yes—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—24no—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2312yes—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C4⋊C4×D59—24no—
77C4⋊C4×D59—24no—
78C2×C22⋊Dic59—21no—
79C2×(C2×D59)⋊C4—28no—
80C2×(C2×D59)⋊C4—22no—
81C2×(C2×D59)⋊C4—22no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C22⋊C4×D59—44no—
85C2×C4×Dic59—28no—
86C2×C4×Dic59—22no—
87C2×C4×Dic59—22no—
88#14273660—24no—
89#14273659—24no—
90#14273658—24no—
91C22⋊(C59⋊C8)C2×C414yescomplemented
92(C2×D59)⋊C8C2×C418yescomplemented
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C23×C4×C59—21no—
97C23×C4×C59C2311yescharacteristic
98C22×C8×C59C2312yes—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—24no—
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C4⋊C4×C59C2312yes—
104C2×C22⋊C4×C59—21no—
105#107×C59—24no—
106C22⋊C8×C59C2×C414yescomplemented
107#95×C59—21no—
108C24×C59C2×D411yescharacteristic
109C23×D59—22no—
110C23×D59—22no—
111C23×D59—22no—
112C23×D59—22no—
113C23×D59—41no—
114C23×D59—41no—
115C23×D59C4⋊C414yes—
116C23×D59C2×D411yescharacteristic
117C23×D59C2×D411yescharacteristic
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—24no—
121C22×C4×C59—22no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—42no—
125C22×C4×C59—42no—
126C22×C4×C59C22×C412yes—
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×Q812yes—
129C22×C4×C59C2411yescharacteristic
130C22×Dic59—22no—
131C22×Dic59—22no—
132C22×Dic59—24no—
133C22×Dic59—22no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—42no—
137C22×Dic59—42no—
138C22×Dic59C4⋊C414yes—
139C22×Dic59C2×D411yescharacteristic
140C22×Dic59C2×D411yescharacteristic
141C2×C4×D59—22no—
142C2×C4×D59—22no—
143C2×C4×D59—22no—
144C2×C4×D59—22no—
145C2×C4×D59—22no—
146C2×C4×D59—22no—
147C2×C4×D59—216no—
148C2×C4×D59—216no—
149C2×C4×D59—42no—
150C2×C4×D59—42no—
151C2×C4×D59—42no—
152C2×C4×D59—42no—
153C2×C4×D59—42no—
154C2×C4×D59—42no—
155C2×C4×D59C4⋊C414yes—
156C2×C4×D59C4⋊C414yes—
157C2×C4×D59C22×C412yes—
158C2×C4×D59C22×C412yes—
159C2×C4×D59C2×D414yes—
160C2×C4×D59C2×Q812yes—
161C2×C4×D59C2×Q812yes—
162C2×C4×D59#4114yes—
163C2×C8×C59—22no—
164C2×C8×C59—22no—
165C2×C8×C59C22×C414yes—
166C4⋊C4×C59—24no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—22no—
170C4⋊C4×C59C2×D414yes—
171C22⋊C4×C59—42no—
172C22⋊Dic59—42no—
173(C2×D59)⋊C4—28no—
174(C2×D59)⋊C4—28no—
175(C2×D59)⋊C4—42no—
176(C2×D59)⋊C4—42no—
177C4⋊Dic59—24no—
178C4⋊Dic59—22no—
179C4⋊Dic59—22no—
180C4⋊Dic59—22no—
181C4⋊Dic59C2×D414yes—
182Dic59⋊C4—42no—
183Dic59⋊C4—42no—
184C4×Dic59—28no—
185C4×Dic59—28no—
186C2×C59⋊C8—22no—
187C2×C59⋊C8—22no—
188C2×C59⋊C8C22×C414yes—
189#11745659—216no—
190C23×C59—21no—
191C23×C59—21no—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59C2×C4⋊C412yes—
195C23×C59AGL(1,ℤ/8)12yes—
196C23×C59#13712yes—
197C23×C59C22×D411yescharacteristic
198C22×D59—21no—
199C22×D59—21no—
200C22×D59—21no—
201C22×D59—21no—
202C22×D59—21no—
203C22×D59—21no—
204C22×D59—21no—
205C22×D59—21no—
206C22×D59—21no—
207C22×D59—21no—
208C22×D59—28no—
209C22×D59—28no—
210C22×D59—44no—
211C22×D59—44no—
212C22×D59—44no—
213C22×D59—44no—
214C22×D59—44no—
215C22×D59C2×C4⋊C412yes—
216C22×D59C2×C4⋊C412yes—
217C22×D59#12012yes—
218C22×D59#12312yes—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—24no—
222C2×C4×C59—24no—
223C2×C4×C59—24no—
224C2×C4×C59—48no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59C2×C4⋊C412yes—
234C2×C4×C59C23×C411yescharacteristic
235C2×C4×C59C22×D412yes—
236C2×C4×C59C22×Q811yescharacteristic
237C2×C4×C59C2×#4112yes—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—28no—
241C2×Dic59—24no—
242C2×Dic59—48no—
243C2×Dic59—42no—
244C2×Dic59—42no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59C2×C4⋊C412yes—
252C2×Dic59C2×C4⋊C412yes—
253C2×Dic59#12012yes—
254C2×Dic59#12312yes—
255C4×D59—28no—
256C4×D59—24no—
257C4×D59—24no—
258C4×D59—28no—
259C4×D59—84no—
260C4×D59—84no—
261C4×D59C4×D418yes—
262C4×D59C22⋊Q818yes—
263C8×C59—24no—
264C59⋊C8—24no—
265#20846—591nomaximal, Sylow
266C22×C59—22no—
267C22×C59—22no—
268C22×C59—22no—
269C22×C59—22no—
270C22×C59—22no—
271C22×C59—22no—
272C22×C59#42814yescomplemented
273C22×C59C22×C4⋊C411yescharacteristic
274C22×C59C2×#12011yescharacteristic
275C22×C59C2×#12311yescharacteristic
276C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C59C2×#13711yescharacteristic
278C22×C59#57511yescharacteristic
279C22×C59#57511yescharacteristic
280C2×D59—22no—
281C2×D59—22no—
282C2×D59—22no—
283C2×D59—22no—
284C2×D59—22no—
285C2×D59—22no—
286C2×D59—22no—
287C2×D59—22no—
288C2×D59—22no—
289C2×D59—416no—
290C2×D59—41no—
291C2×D59—41no—
292C2×D59—41no—
293C2×D59—41no—
294C2×D59#38514yes—
295C4×C59—22no—
296C4×C59—44no—
297C4×C59—81no—
298C4×C59—81no—
299C4×C59C2×C4×D412yes—
300C4×C59C2×C22⋊Q812yes—
301Dic59—22no—
302Dic59—44no—
303Dic59—81no—
304Dic59—81no—
305Dic59#38514yes—
306C22×#117—591no—
307C2×#428—592no—
308C2×#407—591no—
309C2×#380—591no—
310#1854—592no—
311#1854—592no—
312#1853—591no—
313#1853—591no—
314#1852—591no—
315#1852D5911yescharacteristic, complemented
316#1736—592no—
317C2×C59—24no—
318C2×C59C2×#38511yescharacteristic
319C2×C59C2×#42811yescharacteristic
320C2×C59#291111yescharacteristic
321C2×C59#300611yescharacteristic
322C2×C59#300711yescharacteristic
323C2×C59#309411yescharacteristic
324C2×C59#309511yescharacteristic
325D59—24no—
326D59—42no—
327C24×C4—591no—
328C22×M16—592no—
329C2×#117—592no—
330C2×#117—592no—
331C2×#117—1188no—
332C2×#117—1181no—
333C2×#117—1181no—
334C22×C4⋊C4—591no—
335C22×C4⋊C4—591no—
336C22×C4⋊C4—591no—
337C22×C4⋊C4C2×D5911yescharacteristic
338C22×C22⋊C4—1181no—
339C22×C22⋊C4—1181no—
340C22×C42—1181no—
341#428—1188no—
342C2×#107—592no—
343C2×#107C2×D5912yescomplemented
344C2×#106—592no—
345C2×#106—592no—
346#407—598no—
347C2×C22⋊C8—591no—
348C2×C22⋊C8—592no—
349C2×C22⋊C8C2×D5911yescharacteristic
350#380—591no—
351#380—591no—
352#380—591no—
353#380—1181no—
354#380—1181no—
355#380C2×D5911yescharacteristic
356C2×#95—1181no—
357#340—592no—
358#340—592no—
359#340—592no—
360#340C2×D5912yescomplemented
361C59#2084611yescharacteristic, complemented, Sylow
362C25—591no—
363C23×C4—21no—
364C23×C4—591no—
365C23×C4—591no—
366C23×C4—594no—
367C23×C4—591no—
368C23×C4—591no—
369C23×C4—594no—
370C23×C4—591no—
371C23×C4—1184no—
372C23×C4—1184no—
373C23×C4—1181no—
374C23×C4—1181no—
375C23×C4—1181no—
376C23×C4C22×D5911yescharacteristic
377C2×M16—598no—
378C2×M16—1184no—
379C2×M16—1184no—
380C22×C8—592no—
381C22×C8C22×D5912yes—
382#117—598no—
383#117—1184no—
384#117—1184no—
385#117—1188no—
386#117—1188no—
387#117—1184no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—592no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—1184no—
394C2×C4⋊C4—1184no—
395C2×C4⋊C4—1184no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C4⋊C4C22×D5912yes—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1182no—
404C2×C22⋊C4—1182no—
405C2×C22⋊C4—1188no—
406C2×C22⋊C4—1181no—
407C2×C22⋊C4—1181no—
408C2×C22⋊C4—1181no—
409C2×C22⋊C4—2364no—
410C2×C42—1182no—
411C2×C42—1182no—
412C2×C42—1188no—
413#107—24no—
414#107—1184no—
415#106—1184no—
416#106—1184no—
417C22⋊C8—594no—
418C22⋊C8—598no—
419C22⋊C8C4×D5914yescomplemented
420#95—21no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424C24—591no—
425C24—591no—
426C24—594no—
427C24—1182no—
428C24—1182no—
429C24—1182no—
430C24—1182no—
431C24—2361no—
432C24—2361no—
433C24D4×D5911yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—591no—
443C22×C4—592no—
444C22×C4—594no—
445C22×C4—592no—
446C22×C4—594no—
447C22×C4—594no—
448C22×C4—591no—
449C22×C4—594no—
450C22×C4—594no—
451C22×C4—592no—
452C22×C4—592no—
453C22×C4—1182no—
454C22×C4—1182no—
455C22×C4—11816no—
456C22×C4—11816no—
457C22×C4—1182no—
458C22×C4—1182no—
459C22×C4—1182no—
460C22×C4—1182no—
461C22×C4—1182no—
462C22×C4—1182no—
463C22×C4—1184no—
464C22×C4—1182no—
465C22×C4—1181no—
466C22×C4—1181no—
467C22×C4—2362no—
468C22×C4—2362no—
469C22×C4—2362no—
470C22×C4—2362no—
471C22×C4—2362no—
472C22×C4—2362no—
473C22×C4—2362no—
474C22×C4—2362no—
475C22×C4C2×C4×D5912yes—
476C22×C4D4×D5911yescharacteristic
477C22×C4Q8×D5912yes—
478C22×C4C23×D5911yescharacteristic
479M16—11816no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—594no—
483C2×C8—1182no—
484C2×C8—1182no—
485C2×C8C2×C4×D5914yes—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—594no—
491C4⋊C4—1184no—
492C4⋊C4—1182no—
493C4⋊C4—1182no—
494C4⋊C4—1182no—
495C4⋊C4—2362no—
496C4⋊C4—2362no—
497C4⋊C4D4×D5914yes—
498C22⋊C4—42no—
499C22⋊C4—1188no—
500C22⋊C4—1188no—
501C22⋊C4—2362no—
502C22⋊C4—2362no—
503C22⋊C4—2362no—
504C42—1188no—
505C42—1188no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—592no—
511C23—592no—
512C23—592no—
513C23—592no—
514C23—1188no—
515C23—1181no—
516C23—1181no—
517C23—1181no—
518C23—1181no—
519C23—1181no—
520C23—1181no—
521C23—1181no—
522C23—1181no—
523C23—1181no—
524C23—1188no—
525C23—1181no—
526C23—2364no—
527C23—2364no—
528C23—2364no—
529C23—2364no—
530C23—2364no—
531C23C4⋊C4×D5912yes—
532C23#1427375112yes—
533C23#1427375912yes—
534C23C2×D4×D5911yescharacteristic
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—592no—
550C2×C4—592no—
551C2×C4—592no—
552C2×C4—598no—
553C2×C4—598no—
554C2×C4—592no—
555C2×C4—1188no—
556C2×C4—1188no—
557C2×C4—1188no—
558C2×C4—1181no—
559C2×C4—1181no—
560C2×C4—1184no—
561C2×C4—1184no—
562C2×C4—1184no—
563C2×C4—4724no—
564C2×C4—4724no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2368no—
572C2×C4—2362no—
573C2×C4—2362no—
574C2×C4C4⋊C4×D5912yes—
575C2×C4C22×C4×D5911yescharacteristic
576C2×C4C2×D4×D5912yes—
577C2×C4C2×Q8×D5911yescharacteristic
578C2×C4#41×D5912yes—
579C8—24no—
580C8—1184no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—594no—
588C22—1182no—
589C22—1182no—
590C22—1182no—
591C22—1182no—
592C22—1182no—
593C22—1182no—
594C22—1182no—
595C22—1182no—
596C22—1182no—
597C22—23616no—
598C22—2361no—
599C22—2361no—
600C22—2361no—
601C22—2361no—
602C22#1683744314yescomplemented
603C22C2×C4⋊C4×D5911yescharacteristic
604C22#120×D5911yescharacteristic
605C22#123×D5911yescharacteristic
606C22C2×#1427375111yescharacteristic
607C22C2×#1427375911yescharacteristic
608C22#1683819811yescharacteristic
609C22#1683820111yescharacteristic
610C4—22no—
611C4—44no—
612C4—81no—
613C4—81no—
614C4—594no—
615C4—1182no—
616C4—4721no—
617C4—4721no—
618C4—2364no—
619C4C4×D4×D5912yes—
620C4C22⋊Q8×D5912yes—
621C2—24no—
622C2—1184no—
623C2—2362no—
624C2#385×D5911yescharacteristic
625C2C2×#1683744311yescharacteristic
626C2#2143268311yescharacteristic
627C2#2143334311yescharacteristic
628C2#2143335511yescharacteristic
629C2#2143403111yescharacteristic
630C2#2143404111yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1852C2
socleC23×C59C22×D4
prime radicalC24×D59D4
cosocleC22×C4×C59C24
hypercenter#1852D59
hypocenterC59#20846
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
2. C2×#16837443C2semidirect
5. #21424296C2semidirect
6. #21424292C2semidirect
7. #21424291C2semidirect
8. #21424288C2semidirect
9. #21424284C2semidirect
10. #21423814C2semidirect
11. C59×#1852C2semidirect
26. C2×#14273660C22semidirect
27. C2×#14273659C22semidirect
28. C2×#14273658C22semidirect
29. #16837364C4semidirect
38. #16837165C22semidirect
39. #16837164C22semidirect
40. #16837164C22semidirect
42. C2×#107×C59C22semidirect
45. C59×#340C22semidirect
61. C2×#11745659C2×C4semidirect
91. C22⋊(C59⋊C8)C2×C4semidirect
92. (C2×D59)⋊C8C2×C4semidirect
106. C22⋊C8×C59C2×C4semidirect
272. C22×C59#428semidirect
315. #1852D59semidirect
343. C2×#107C2×D59semidirect
360. #340C2×D59semidirect
361. C59#20846semidirect
419. C22⋊C8C4×D59semidirect
602. C22#16837443semidirect

Maximal subgroups

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

subgroupindex
C22×#142736852
C2×#168374432
C2×#168373642
#380×D592
#214242962
#214242922
#214242912
#214242882
#214242842
#214238142
C59×#18522
#2084659