Finite Group Databasebeta

← #28589423

Subgroups — #28589423

Order 15104.

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×C4⋊C4×D59C211yesmaximal, characteristic
2C2×#16837441C212yesmaximal, complemented
3C2×#16837364C211yesmaximal, characteristic
4#21426478C211yesmaximal, characteristic
5#21424288C211yesmaximal, characteristic, complemented
6#21424287C212yesmaximal, complemented
7#21424286C211yesmaximal, characteristic, complemented
8#21424285C212yesmaximal, complemented
9#21424284C211yesmaximal, characteristic, complemented
10#21423814C212yesmaximal, complemented
11C59×#1853C211yesmaximal, characteristic, complemented
12C23×C4×D59—21no—
13C23×C4×D59C2211yescharacteristic
14C22×#11745659C2212yes—
15C2×#14273685C2211yescharacteristic
16C2×#14273685C2211yescharacteristic
17C22×C4⋊Dic59C2211yescharacteristic
18C22×Dic59⋊C4—21no—
19C2×C4⋊C4×D59—21no—
20C2×C4⋊C4×D59—21no—
21C2×C4⋊C4×D59—28no—
22C2×C4⋊C4×D59C2212yes—
23C2×C4⋊C4×D59C2212yes—
24#16837441—28no—
25C2×#14273661C2212yescomplemented
26C2×#14273660C2212yescomplemented
27C2×#14273658C2212yescomplemented
28#16837364C418yescomplemented
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×(C2×D59)⋊C8C2212yes—
31#16837264—21no—
32#16837262C2211yescharacteristic
33#16837259C2211yescharacteristic
34#16837259C2211yescharacteristic
35#16837257—21no—
36C22⋊C4×Dic59—21no—
37#16837249—21no—
38#16837165C2212yescomplemented
39#16837164C2212yescomplemented
40#16837164C2212yescomplemented
41C22×C4⋊C4×C59C2211yescharacteristic
42C2×#106×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—28no—
52C22×C4×D59—22no—
53C22×C4×D59—22no—
54C22×C4×D59—44no—
55C22×C4×D59—44no—
56C22×C4×D59C2×C414yes—
57C22×C4×D59D411yescharacteristic
58C22×C4×D59D411yescharacteristic
59C22×C4×D59Q814yes—
60C22×C4×D59C2311yescharacteristic
61C22×C4×D59C2311yescharacteristic
62C2×#11745659—24no—
63C2×#11745659—24no—
64C2×#11745659C2×C418yescomplemented
65C22×C59⋊C8C2312yes—
66#14273685—24no—
67#14273685—24no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—24no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2312yes—
73C2×Dic59⋊C4—28no—
74C2×Dic59⋊C4—22no—
75C2×Dic59⋊C4—22no—
76C4⋊C4×D59—28no—
77C4⋊C4×D59—28no—
78C4⋊C4×D59—24no—
79C4⋊C4×D59—24no—
80C4⋊C4×D59—24no—
81C4⋊C4×D59D418yes—
82C2×C22⋊Dic59—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89#14273661—24no—
90#14273660—24no—
91#14273658—24no—
92C22⋊(C59⋊C8)C2×C414yescomplemented
93(C2×D59)⋊C8C2×C418yescomplemented
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C23×C4×C59—21no—
98C23×C4×C59C2311yescharacteristic
99C22×C8×C59C2312yes—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—24no—
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C4⋊C4×C59C2311yescharacteristic
104C2×C4⋊C4×C59C2312yes—
105C2×C22⋊C4×C59—21no—
106#106×C59—24no—
107C22⋊C8×C59C2×C414yescomplemented
108#95×C59—21no—
109C24×C59C2×D411yescharacteristic
110C23×D59—22no—
111C23×D59—22no—
112C23×D59—22no—
113C23×D59—22no—
114C23×D59—22no—
115C23×D59—22no—
116C23×D59C4⋊C414yes—
117C23×D59C2×D411yescharacteristic
118C23×D59C2×D411yescharacteristic
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—24no—
122C22×C4×C59—22no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—42no—
126C22×C4×C59—42no—
127C22×C4×C59C22×C412yes—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×Q812yes—
130C22×C4×C59C2411yescharacteristic
131C22×Dic59—22no—
132C22×Dic59—22no—
133C22×Dic59—24no—
134C22×Dic59—22no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—42no—
138C22×Dic59—42no—
139C22×Dic59C4⋊C414yes—
140C22×Dic59C2×D411yescharacteristic
141C22×Dic59C2×D411yescharacteristic
142C2×C4×D59—22no—
143C2×C4×D59—22no—
144C2×C4×D59—22no—
145C2×C4×D59—22no—
146C2×C4×D59—22no—
147C2×C4×D59—22no—
148C2×C4×D59—216no—
149C2×C4×D59—216no—
150C2×C4×D59—28no—
151C2×C4×D59—28no—
152C2×C4×D59—432no—
153C2×C4×D59—44no—
154C2×C4×D59—44no—
155C2×C4×D59—44no—
156C2×C4×D59—44no—
157C2×C4×D59C4⋊C414yes—
158C2×C4×D59C4⋊C414yes—
159C2×C4×D59C22×C412yes—
160C2×C4×D59C22×C412yes—
161C2×C4×D59C2×D414yes—
162C2×C4×D59C2×Q812yes—
163C2×C4×D59C2×Q812yes—
164C2×C4×D59#4114yes—
165C2×C8×C59—22no—
166C2×C8×C59—22no—
167C2×C8×C59C22×C414yescomplemented
168C4⋊C4×C59—24no—
169C4⋊C4×C59—22no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59C2×D414yes—
173C22⋊C4×C59—42no—
174C22⋊Dic59—42no—
175(C2×D59)⋊C4—42no—
176(C2×D59)⋊C4—42no—
177(C2×D59)⋊C4—42no—
178(C2×D59)⋊C4—42no—
179C4⋊Dic59—24no—
180C4⋊Dic59—22no—
181C4⋊Dic59—22no—
182C4⋊Dic59—22no—
183C4⋊Dic59C2×D414yes—
184Dic59⋊C4—28no—
185Dic59⋊C4—28no—
186C4×Dic59—42no—
187C4×Dic59—42no—
188C2×C59⋊C8—22no—
189C2×C59⋊C8—22no—
190C2×C59⋊C8C22×C414yescomplemented
191#11745659—216no—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59C2×C4⋊C412yes—
197C23×C59AGL(1,ℤ/8)12yes—
198C23×C59#13712yes—
199C23×C59C22×D411yescharacteristic
200C22×D59—28no—
201C22×D59—28no—
202C22×D59—28no—
203C22×D59—28no—
204C22×D59—21no—
205C22×D59—21no—
206C22×D59—21no—
207C22×D59—21no—
208C22×D59—21no—
209C22×D59—21no—
210C22×D59—21no—
211C22×D59—21no—
212C22×D59—28no—
213C22×D59—28no—
214C22×D59—28no—
215C22×D59C2×C4⋊C412yes—
216C22×D59C2×C4⋊C412yes—
217C22×D59#12012yes—
218C22×D59#12312yes—
219C22×D59AGL(1,ℤ/8)12yes—
220C22×D59#13712yes—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—24no—
224C2×C4×C59—24no—
225C2×C4×C59—24no—
226C2×C4×C59—48no—
227C2×C4×C59—42no—
228C2×C4×C59—42no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59C2×C4⋊C412yes—
236C2×C4×C59C23×C411yescharacteristic
237C2×C4×C59C22×D412yes—
238C2×C4×C59C22×Q811yescharacteristic
239C2×C4×C59C2×#4112yes—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—28no—
243C2×Dic59—24no—
244C2×Dic59—48no—
245C2×Dic59—42no—
246C2×Dic59—42no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59C2×C4⋊C412yes—
254C2×Dic59C2×C4⋊C412yes—
255C2×Dic59#12012yes—
256C2×Dic59#12312yes—
257C4×D59—28no—
258C4×D59—24no—
259C4×D59—24no—
260C4×D59—28no—
261C4×D59—416no—
262C4×D59—416no—
263C4×D59C4×D418yes—
264C4×D59C22⋊Q818yes—
265C8×C59—24no—
266C59⋊C8—24no—
267#20844—591nomaximal, Sylow
268C22×C59—22no—
269C22×C59—22no—
270C22×C59—22no—
271C22×C59—22no—
272C22×C59—22no—
273C22×C59—22no—
274C22×C59#42814yescomplemented
275C22×C59C22×C4⋊C411yescharacteristic
276C22×C59C2×#12011yescharacteristic
277C22×C59C2×#12311yescharacteristic
278C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C59C2×#13711yescharacteristic
281C22×C59C2×#13711yescharacteristic
282C2×D59—232no—
283C2×D59—22no—
284C2×D59—22no—
285C2×D59—22no—
286C2×D59—22no—
287C2×D59—22no—
288C2×D59—22no—
289C2×D59—22no—
290C2×D59—22no—
291C2×D59—22no—
292C2×D59—22no—
293C2×D59—22no—
294C2×D59—22no—
295C2×D59#38514yes—
296C2×D59#42814yescomplemented
297C4×C59—22no—
298C4×C59—44no—
299C4×C59—81no—
300C4×C59—81no—
301C4×C59C2×C4×D412yes—
302C4×C59C2×C22⋊Q812yes—
303Dic59—22no—
304Dic59—44no—
305Dic59—81no—
306Dic59—81no—
307Dic59#38514yes—
308C23×C4⋊C4—591no—
309C2×#428—592no—
310C2×#407—591no—
311#2280—591no—
312#1853—592no—
313#1853—591no—
314#1853D5911yescharacteristic, complemented
315#1852—591no—
316#1852—592no—
317#1852—591no—
318#1736—592no—
319C2×C59—24no—
320C2×C59C2×#38511yescharacteristic
321C2×C59C2×#42811yescharacteristic
322C2×C59C2×#42811yescharacteristic
323C2×C59#300311yescharacteristic
324C2×C59#300411yescharacteristic
325C2×C59#309211yescharacteristic
326C2×C59#309211yescharacteristic
327D59—24no—
328D59—24no—
329C24×C4—591no—
330C24×C4—1181no—
331C22×M16—592no—
332C2×#117—591no—
333C2×#117—591no—
334C22×C4⋊C4—592no—
335C22×C4⋊C4—591no—
336C22×C4⋊C4—592no—
337C22×C4⋊C4—1188no—
338C22×C4⋊C4—1181no—
339C22×C4⋊C4—1181no—
340C22×C4⋊C4—1181no—
341C22×C4⋊C4C2×D5911yescharacteristic
342#428—1188no—
343C2×#107—592no—
344C2×#107—592no—
345C2×#106—592no—
346C2×#106C2×D5912yescomplemented
347#407—598no—
348C2×C22⋊C8—591no—
349C2×C22⋊C8—592no—
350C2×C22⋊C8C2×D5911yescharacteristic
351#381—1181no—
352#380—591no—
353#380—591no—
354#380—591no—
355#380C2×D5911yescharacteristic
356#379—1181no—
357C4×C22⋊C4—1181no—
358C4×C22⋊C4—1181no—
359#340—592no—
360#340—592no—
361#340—592no—
362#340C2×D5912yescomplemented
363C59#2084411yescharacteristic, complemented, Sylow
364C25—591no—
365C23×C4—21no—
366C23×C4—591no—
367C23×C4—591no—
368C23×C4—594no—
369C23×C4—591no—
370C23×C4—591no—
371C23×C4—594no—
372C23×C4—591no—
373C23×C4—1184no—
374C23×C4—1184no—
375C23×C4—1182no—
376C23×C4—1182no—
377C23×C4—1188no—
378C23×C4—1181no—
379C23×C4—2364no—
380C23×C4—2364no—
381C23×C4C22×D5911yescharacteristic
382C2×M16—598no—
383C2×M16—1184no—
384C2×M16—1184no—
385C22×C8—592no—
386C22×C8C22×D5912yes—
387#117—1184no—
388#117—1184no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—592no—
392C2×C4⋊C4—598no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—1184no—
396C2×C4⋊C4—1184no—
397C2×C4⋊C4—1188no—
398C2×C4⋊C4—1188no—
399C2×C4⋊C4—1184no—
400C2×C4⋊C4—1182no—
401C2×C4⋊C4—1182no—
402C2×C4⋊C4—1188no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1184no—
405C2×C4⋊C4C22×D5911yescharacteristic
406C2×C4⋊C4C22×D5911yescharacteristic
407C2×C4⋊C4C22×D5912yes—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—1181no—
410C2×C22⋊C4—1181no—
411C2×C22⋊C4—1181no—
412C2×C22⋊C4—1181no—
413C2×C22⋊C4—1181no—
414C2×C42—1181no—
415C2×C42—1181no—
416#107—1184no—
417#107—1184no—
418#106—24no—
419#106—1184no—
420C22⋊C8—594no—
421C22⋊C8—598no—
422C22⋊C8C4×D5914yescomplemented
423#95—21no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427C24—591no—
428C24—591no—
429C24—594no—
430C24—1182no—
431C24—1182no—
432C24—1182no—
433C24—1182no—
434C24—1182no—
435C24—1182no—
436C24D4×D5911yescharacteristic
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—591no—
446C22×C4—592no—
447C22×C4—594no—
448C22×C4—592no—
449C22×C4—594no—
450C22×C4—594no—
451C22×C4—591no—
452C22×C4—594no—
453C22×C4—594no—
454C22×C4—592no—
455C22×C4—592no—
456C22×C4—1182no—
457C22×C4—1182no—
458C22×C4—11816no—
459C22×C4—11816no—
460C22×C4—1182no—
461C22×C4—1182no—
462C22×C4—1188no—
463C22×C4—1188no—
464C22×C4—1182no—
465C22×C4—1182no—
466C22×C4—1182no—
467C22×C4—1182no—
468C22×C4—1184no—
469C22×C4—1182no—
470C22×C4—1181no—
471C22×C4—1181no—
472C22×C4—2364no—
473C22×C4—23632no—
474C22×C4—2364no—
475C22×C4—2364no—
476C22×C4—2364no—
477C22×C4—2362no—
478C22×C4—2362no—
479C22×C4C2×C4×D5912yes—
480C22×C4D4×D5911yescharacteristic
481C22×C4Q8×D5912yes—
482C22×C4C23×D5911yescharacteristic
483M16—11816no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—594no—
487C2×C8—1182no—
488C2×C8—1182no—
489C2×C8C2×C4×D5914yescomplemented
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—594no—
495C4⋊C4—1188no—
496C4⋊C4—1188no—
497C4⋊C4—1184no—
498C4⋊C4—1182no—
499C4⋊C4—1182no—
500C4⋊C4—1182no—
501C4⋊C4D4×D5914yes—
502C22⋊C4—42no—
503C22⋊C4—2362no—
504C22⋊C4—2362no—
505C22⋊C4—2362no—
506C22⋊C4—2362no—
507C22⋊C4—2362no—
508C42—2362no—
509C42—2362no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—592no—
515C23—592no—
516C23—592no—
517C23—592no—
518C23—592no—
519C23—592no—
520C23—1188no—
521C23—1188no—
522C23—1188no—
523C23—1188no—
524C23—1188no—
525C23—1188no—
526C23—1181no—
527C23—1181no—
528C23—1181no—
529C23—1181no—
530C23—1181no—
531C23—1181no—
532C23—1181no—
533C23—1188no—
534C23—1181no—
535C23C4⋊C4×D5912yes—
536C23#1427375412yes—
537C23#1427375612yes—
538C23C2×D4×D5911yescharacteristic
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—592no—
554C2×C4—592no—
555C2×C4—592no—
556C2×C4—598no—
557C2×C4—598no—
558C2×C4—592no—
559C2×C4—1188no—
560C2×C4—1188no—
561C2×C4—1188no—
562C2×C4—1181no—
563C2×C4—1181no—
564C2×C4—1184no—
565C2×C4—1184no—
566C2×C4—1184no—
567C2×C4—23616no—
568C2×C4—23616no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2368no—
576C2×C4—2362no—
577C2×C4—2362no—
578C2×C4C4⋊C4×D5912yes—
579C2×C4C22×C4×D5911yescharacteristic
580C2×C4C2×D4×D5912yes—
581C2×C4C2×Q8×D5911yescharacteristic
582C2×C4#41×D5912yes—
583C8—24no—
584C8—1184no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—594no—
592C22—594no—
593C22—11832no—
594C22—1182no—
595C22—1182no—
596C22—1182no—
597C22—1182no—
598C22—1182no—
599C22—1182no—
600C22—1182no—
601C22—1182no—
602C22—1182no—
603C22—1182no—
604C22—1182no—
605C22—1182no—
606C22#1683744114yescomplemented
607C22C2×C4⋊C4×D5911yescharacteristic
608C22#120×D5911yescharacteristic
609C22#123×D5911yescharacteristic
610C22AGL(1,ℤ/8)×D5911yescharacteristic
611C22C2×#1427375411yescharacteristic
612C22#137×D5911yescharacteristic
613C22C2×#1427375611yescharacteristic
614C4—22no—
615C4—44no—
616C4—81no—
617C4—81no—
618C4—594no—
619C4—1182no—
620C4—4721no—
621C4—4721no—
622C4—2364no—
623C4C4×D4×D5912yes—
624C4C22⋊Q8×D5912yes—
625C2—24no—
626C2—1184no—
627C2—1184no—
628C2#385×D5911yescharacteristic
629C2#428×D5911yescharacteristic
630C2C2×#1683744111yescharacteristic
631C2#2143332411yescharacteristic
632C2#2143332711yescharacteristic
633C2#2143401811yescharacteristic
634C2#2143402011yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1853C2
socleC23×C59C22×D4
prime radicalC24×D59D4
cosocleC22×C4×C59C24
hypercenter#1853D59
hypocenterC59#20844
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. C2×#16837441C2semidirect
5. #21424288C2semidirect
6. #21424287C2semidirect
7. #21424286C2semidirect
8. #21424285C2semidirect
9. #21424284C2semidirect
10. #21423814C2semidirect
11. C59×#1853C2semidirect
25. C2×#14273661C22semidirect
26. C2×#14273660C22semidirect
27. C2×#14273658C22semidirect
28. #16837364C4semidirect
38. #16837165C22semidirect
39. #16837164C22semidirect
40. #16837164C22semidirect
42. C2×#106×C59C22semidirect
45. C59×#340C22semidirect
64. C2×#11745659C2×C4semidirect
92. C22⋊(C59⋊C8)C2×C4semidirect
93. (C2×D59)⋊C8C2×C4semidirect
107. C22⋊C8×C59C2×C4semidirect
167. C2×C8×C59C22×C4semidirect
190. C2×C59⋊C8C22×C4semidirect
274. C22×C59#428semidirect
296. C2×D59#428semidirect
314. #1853D59semidirect
346. C2×#106C2×D59semidirect
362. #340C2×D59semidirect
363. C59#20844semidirect
422. C22⋊C8C4×D59semidirect
489. C2×C8C2×C4×D59semidirect
606. C22#16837441semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D592
C2×#168374412
C2×#168373642
#214264782
#214242882
#214242872
#214242862
#214242852
#214242842
#214238142
C59×#18532
#2084459