Finite Group Databasebeta

← #28620947

Subgroups — #28620947

Order 15104.

1474 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837552C211yesmaximal, characteristic, complemented
2C22×#14273647C211yesmaximal, characteristic, complemented
3#21427452C211yesmaximal, characteristic
4#21425229C212yesmaximal, complemented
5#21425097C212yesmaximal, complemented
6#21424520C211yesmaximal, characteristic, complemented
7#21424451C212yesmaximal, complemented
8#21424413C212yesmaximal, complemented
9#21424389C211yesmaximal, characteristic, complemented
10#21424186C211yesmaximal, characteristic, complemented
11C2×C59×#554C211yesmaximal, characteristic, complemented
12C23×D236C2211yescharacteristic
13C22×#11745671C2212yes—
14C23×C59⋊C8C2211yescharacteristic
15C2×#14273701C2212yes—
16C2×#14273695—21no—
17#16837552—24no—
18C2×#14273651C2212yescomplemented
19C2×#14273647—24no—
20C2×#14273647C2212yes—
21C2×#14273647C2212yes—
22C2×#14273647C2212yes—
23C2×C22⋊(C59⋊C8)C2211yescharacteristic
24C2×C22⋊(C59⋊C8)C2212yescomplemented
25#16837319—21no—
26#16837311—21no—
27#16837304C2212yes—
28#16837262C2211yescharacteristic
29#16837257—21no—
30#16837164C2212yescomplemented
31#16837164C2212yescomplemented
32#16837164C2212yescomplemented
33#16837157C2212yescomplemented
34C59×#554—24no—
35C59×#554—24no—
36C2×C4⋊Q8×C59C2211yescharacteristic, complemented
37C2×#125×C59—21no—
38C2×C22⋊Q8×C59—21no—
39C2×C22⋊Q8×C59—21no—
40C2×C22⋊Q8×C59C2211yescharacteristic
41C2×C22⋊Q8×C59C2211yescharacteristic, complemented
42C2×C4×Q8×C59C2211yescharacteristic, complemented
43C2×C4×D4×C59—21no—
44C2×C4×D4×C59C2211yescharacteristic
45C22×C4⋊C4×C59C2211yescharacteristic
46C22×C4⋊C4×C59C2211yescharacteristic, complemented
47C24×D59—21no—
48C22×D236—22no—
49C22×D236—24no—
50C22×D236—22no—
51C22×D236—21no—
52C22×D236—42no—
53C22×D236C2312yes—
54C22×C22⋊D59—22no—
55C2×#11745671—24no—
56C2×#11745671—24no—
57C2×#11745671—42no—
58C22×C59⋊C8—22no—
59C22×C59⋊C8—42no—
60C22×C59⋊C8C2312yes—
61C22×C59⋊C8C2312yes—
62#14273701—24no—
63#14273701—42no—
64#14273695—44no—
65C2×C4⋊Dic59C2312yes—
66C2×C22⋊Dic59—21no—
67C2×(C2×D59)⋊C4—22no—
68C2×(C2×D59)⋊C4—22no—
69C2×(C2×D59)⋊C4—22no—
70#14273651—42no—
71#14273647—24no—
72#14273647—24no—
73#14273647—24no—
74C22⋊(C59⋊C8)—22no—
75C22⋊(C59⋊C8)—42no—
76C59⋊#95—21no—
77C22×Q8×C59—21no—
78C22×Q8×C59C2311yescharacteristic
79C22×D4×C59D411yescharacteristic
80C23×C4×C59—21no—
81C23×C4×C59—21no—
82C23×C4×C59C2311yescharacteristic
83C23×C4×C59C2311yescharacteristic
84C4⋊Q8×C59—22no—
85C4⋊Q8×C59—22no—
86#125×C59—28no—
87C22⋊Q8×C59—28no—
88C22⋊Q8×C59—28no—
89C22⋊Q8×C59—22no—
90C22⋊Q8×C59—22no—
91C22⋊Q8×C59—22no—
92C22⋊Q8×C59D414yescomplemented
93C4×Q8×C59—22no—
94C4×Q8×C59—22no—
95C4×D4×C59—28no—
96C4×D4×C59—22no—
97C4×D4×C59—22no—
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—22no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—22no—
106C2×C4⋊C4×C59—22no—
107C2×C4⋊C4×C59D411yescharacteristic
108C2×C4⋊C4×C59D414yescomplemented
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59D411yescharacteristic
115C2×C22⋊C4×C59D411yescharacteristic
116C2×C42×C59—21no—
117C2×C42×C59C2311yescharacteristic
118C24×C59C2×D411yescharacteristic
119C24×C59C2×D411yescharacteristic
120C23×D59—22no—
121C23×D59—44no—
122C23×D59—42no—
123C23×D59—42no—
124C23×D59—42no—
125C23×D59—82no—
126C2×Q8×C59—24no—
127C2×Q8×C59—21no—
128C2×Q8×C59—22no—
129C2×Q8×C59—22no—
130C2×Q8×C59—82no—
131C2×Q8×C59C2×D412yes—
132C2×D4×C59—22no—
133C2×D4×C59—22no—
134C2×D4×C59SD1614yescomplemented
135C22×C4×C59—22no—
136C22×C4×C59—22no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—24no—
142C22×C4×C59—24no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—22no—
146C22×C4×C59—22no—
147C22×C4×C59—82no—
148C22×C4×C59—82no—
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D412yes—
153C22×C4×C59C2×D412yes—
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2411yescharacteristic
156C2×C22⋊D59—44no—
157C2×C22⋊D59—44no—
158C2×C22⋊D59—82no—
159C22×Dic59—22no—
160C2×D236—28no—
161C2×D236—22no—
162C2×D236—22no—
163C2×D236—22no—
164C2×D236—22no—
165C2×D236—22no—
166C2×D236—48no—
167C2×D236—44no—
168C2×D236—44no—
169C2×D236—42no—
170C2×D236#4114yes—
171C4⋊C4×C59—24no—
172C4⋊C4×C59—24no—
173C4⋊C4×C59—24no—
174C4⋊C4×C59—24no—
175C4⋊C4×C59—24no—
176C4⋊C4×C59—24no—
177C4⋊C4×C59—22no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59C2×D412yes—
182C4⋊C4×C59C2×D412yes—
183C4⋊C4×C59C2×D412yes—
184C22⋊C4×C59—24no—
185C22⋊C4×C59—24no—
186C22⋊C4×C59—22no—
187C22⋊C4×C59—22no—
188C42×C59—24no—
189C42×C59—21no—
190C42×C59—21no—
191C22⋊Dic59—42no—
192#11745671—48no—
193(C2×D59)⋊C4—44no—
194(C2×D59)⋊C4—44no—
195(C2×D59)⋊C4—82no—
196C4⋊Dic59—22no—
197C4⋊Dic59—22no—
198C2×C59⋊C8—22no—
199C2×C59⋊C8—22no—
200C2×C59⋊C8—22no—
201C2×C59⋊C8—22no—
202C2×C59⋊C8—22no—
203C2×C59⋊C8—22no—
204C2×C59⋊C8—44no—
205C23×C59—22no—
206C23×C59—22no—
207C23×C59—21no—
208C23×C59—22no—
209C23×C59—21no—
210C23×C59—22no—
211C23×C59#12012yes—
212C23×C59C2×SD1612yes—
213C23×C59C2×SD1612yes—
214C23×C59#13712yes—
215C23×C59C22×D411yescharacteristic
216C22×D59—24no—
217C22×D59—48no—
218C22×D59—42no—
219C22×D59—42no—
220C22×D59—42no—
221C22×D59—42no—
222C22×D59—42no—
223C22×D59—42no—
224C22×D59—88no—
225C22×D59—88no—
226C22×D59—84no—
227C22×D59—84no—
228Q8×C59—24no—
229Q8×C59—41no—
230Q8×C59—41no—
231Q8×C59—84no—
232D4×C59—28no—
233C2×C4×C59—21no—
234C2×C4×C59—22no—
235C2×C4×C59—21no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—24no—
239C2×C4×C59—22no—
240C2×C4×C59—21no—
241C2×C4×C59—22no—
242C2×C4×C59—21no—
243C2×C4×C59—22no—
244C2×C4×C59—22no—
245C2×C4×C59—22no—
246C2×C4×C59—22no—
247C2×C4×C59—42no—
248C2×C4×C59—42no—
249C2×C4×C59—42no—
250C2×C4×C59—42no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—42no—
254C2×C4×C59—84no—
255C2×C4×C59—84no—
256C2×C4×C59#12012yes—
257C2×C4×C59C2×SD1612yes—
258C2×C4×C59C22×D411yescharacteristic
259C2×C4×C59C22×D411yescharacteristic
260C2×C4×C59C2×#4112yes—
261C2×C4×C59H(5,𝔽2)12yes—
262C22⋊D59—88no—
263C2×Dic59—42no—
264C2×Dic59—42no—
265C2×Dic59—42no—
266D236—24no—
267D236—24no—
268D236—48no—
269D236—42no—
270D236—42no—
271C59⋊C8—42no—
272C59⋊C8—42no—
273#23969—591nomaximal, Sylow
274C22×C59—22no—
275C22×C59—22no—
276C22×C59—24no—
277C22×C59—24no—
278C22×C59—22no—
279C22×C59—24no—
280C22×C59—41no—
281C22×C59—41no—
282C22×C59—42no—
283C22×C59#44814yes—
284C22×C59C2×#12011yescharacteristic
285C22×C59#54612yes—
286C22×C59C22×SD1611yescharacteristic
287C22×C59C2×#13711yescharacteristic
288C22×C59#57612yes—
289C2×D59—44no—
290C2×D59—44no—
291C2×D59—44no—
292C2×D59—816no—
293C2×D59—82no—
294C2×D59—82no—
295C2×D59—82no—
296C2×D59—82no—
297C4×C59—22no—
298C4×C59—22no—
299C4×C59—41no—
300C4×C59—42no—
301C4×C59—41no—
302C4×C59—42no—
303C4×C59—42no—
304C4×C59—42no—
305C4×C59#52511yescharacteristic
306C4×C59#53511yescharacteristic
307C4×C59#54612yes—
308Dic59—82no—
309C2×#554D5911yescharacteristic, complemented
310C2×#450—591no—
311C22×#102—591no—
312#2408—591no—
313#2023—592no—
314#2000—592no—
315#1892—591no—
316#1880—592no—
317#1874—592no—
318#1869—591no—
319#1824—591no—
320C2×C59—24no—
321C2×C59—42no—
322C2×C59#240411yescharacteristic
323C2×C59C2×#44811yescharacteristic
324C2×C59#300811yescharacteristic
325C2×C59#329512yes—
326C2×C59#331412yes—
327D59—84no—
328C23×D4—591no—
329C22×SD16—592no—
330C23×C8—591no—
331#554—24no—
332#554—24no—
333C2×C4⋊Q8C2×D5911yescharacteristic, complemented
334C2×#125—21no—
335C2×C22⋊Q8—21no—
336C2×C22⋊Q8—21no—
337C2×C22⋊Q8C2×D5911yescharacteristic
338C2×C22⋊Q8C2×D5911yescharacteristic, complemented
339C2×#121—592no—
340C2×#120—1181no—
341C2×C4×Q8C2×D5911yescharacteristic, complemented
342C2×C4×D4—21no—
343C2×C4×D4C2×D5911yescharacteristic
344C22×C4⋊C4C2×D5911yescharacteristic
345C22×C4⋊C4C2×D5911yescharacteristic, complemented
346#450—1184no—
347C2×#103—592no—
348C2×#102—592no—
349C2×#102—592no—
350C2×#102—592no—
351C2×#102—1184no—
352C2×C22⋊C8—591no—
353C2×C22⋊C8—592no—
354#394—1181no—
355#392—1181no—
356#390—592no—
357#380—591no—
358#379—1181no—
359#340—592no—
360#340—592no—
361#340—592no—
362#336—592no—
363C59#2396911yescharacteristic, complemented, Sylow
364C25—1181no—
365C22×Q8—21no—
366C22×Q8C22×D5911yescharacteristic
367C22×D4—592no—
368C22×D4—1184no—
369C22×D4—1182no—
370C22×D4—1182no—
371C22×D4—1182no—
372C22×D4—1181no—
373C22×D4—2362no—
374C22×D4C22⋊D5911yescharacteristic
375C23×C4—21no—
376C23×C4—21no—
377C23×C4C22×D5911yescharacteristic
378C23×C4C22×D5911yescharacteristic
379C2×SD16—1184no—
380C2×SD16—1184no—
381C2×SD16—2362no—
382C22×C8—592no—
383C22×C8—592no—
384C22×C8—1182no—
385C22×C8—2362no—
386C4⋊Q8—22no—
387C4⋊Q8—22no—
388#125—28no—
389C22⋊Q8—28no—
390C22⋊Q8—28no—
391C22⋊Q8—22no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8C22⋊D5914yescomplemented
395#121—1184no—
396#121—2362no—
397#120—2364no—
398C4×Q8—22no—
399C4×Q8—22no—
400C4×D4—28no—
401C4×D4—22no—
402C4×D4—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—592no—
413C2×C4⋊C4C22⋊D5911yescharacteristic
414C2×C4⋊C4C22⋊D5914yescomplemented
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—1182no—
421C2×C22⋊C4—1181no—
422C2×C22⋊C4—1182no—
423C2×C22⋊C4—1182no—
424C2×C22⋊C4C22⋊D5911yescharacteristic
425C2×C22⋊C4C22⋊D5911yescharacteristic
426C2×C42—21no—
427C2×C42C22×D5911yescharacteristic
428#103—2362no—
429#102—1184no—
430#102—1184no—
431#102—1184no—
432C22⋊C8—1182no—
433C22⋊C8—2362no—
434#95—1181no—
435C24—1182no—
436C24—4722no—
437C24—2364no—
438C24—2362no—
439C24—2362no—
440C24—2362no—
441C24C2×C22⋊D5911yescharacteristic
442C24C2×C22⋊D5911yescharacteristic
443C2×Q8—24no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—82no—
448C2×Q8C2×C22⋊D5912yes—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—594no—
452C2×D4—1188no—
453C2×D4—1182no—
454C2×D4—1182no—
455C2×D4—1182no—
456C2×D4—1182no—
457C2×D4—1182no—
458C2×D4—4722no—
459C2×D4—2368no—
460C2×D4—2364no—
461C2×D4—2364no—
462C2×D4—2364no—
463C2×D4—2364no—
464C2×D4—2362no—
465C2×D4#1174567114yescomplemented
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—1182no—
481C22×C4C2×C22⋊D5911yescharacteristic
482C22×C4C2×C22⋊D5911yescharacteristic
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C2×C22⋊D5912yes—
485C22×C4C2×C22⋊D5912yes—
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C23×D5911yescharacteristic
488SD16—2368no—
489C2×C8—1182no—
490C2×C8—1182no—
491C2×C8—1182no—
492C2×C8—1182no—
493C2×C8—1182no—
494C2×C8—1182no—
495C2×C8—2364no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—1182no—
507C4⋊C4—1182no—
508C4⋊C4C2×C22⋊D5912yes—
509C4⋊C4C2×C22⋊D5912yes—
510C4⋊C4C2×C22⋊D5912yes—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—4722no—
516C22⋊C4—2364no—
517C22⋊C4—2362no—
518C22⋊C4—2364no—
519C42—24no—
520C42—21no—
521C42—21no—
522C23—22no—
523C23—22no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—22no—
528C23—1184no—
529C23—4724no—
530C23—4728no—
531C23—4728no—
532C23—4724no—
533C23—2368no—
534C23—2362no—
535C23—2362no—
536C23—2362no—
537C23—2362no—
538C23—2362no—
539C23—2362no—
540C23#1427369612yes—
541C23C2×#1174567112yes—
542C23C2×#1174567112yes—
543C23#1427375812yes—
544C23C22×C22⋊D5911yescharacteristic
545Q8—24no—
546Q8—41no—
547Q8—41no—
548Q8—84no—
549D4—28no—
550D4—1184no—
551D4—1184no—
552D4—4728no—
553D4—2368no—
554D4—2362no—
555D4—2362no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—2362no—
580C2×C4—2362no—
581C2×C4—2362no—
582C2×C4#1427369612yes—
583C2×C4C2×#1174567112yes—
584C2×C4C22×C22⋊D5911yescharacteristic
585C2×C4C22×C22⋊D5911yescharacteristic
586C2×C4#41×D5912yes—
587C2×C4#1427377312yes—
588C8—2362no—
589C8—2362no—
590C22—24no—
591C22—24no—
592C22—22no—
593C22—22no—
594C22—24no—
595C22—22no—
596C22—42no—
597C22—41no—
598C22—41no—
599C22—47216no—
600C22—4722no—
601C22—4722no—
602C22—4722no—
603C22—4722no—
604C22—2364no—
605C22—2364no—
606C22—2364no—
607C22#1683754114yes—
608C22C2×#1427369611yescharacteristic
609C22#1683804012yes—
610C22C22×#1174567111yescharacteristic
611C22C2×#1427375811yescharacteristic
612C22#1683820512yes—
613C4—22no—
614C4—22no—
615C4—41no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—41no—
620C4—42no—
621C4—4722no—
622C4#1683790911yescharacteristic
623C4#1683796611yescharacteristic
624C4#1683804012yes—
625C2—24no—
626C2—42no—
627C2—4724no—
628C2#2142741111yescharacteristic
629C2C2×#1683754111yescharacteristic
630C2#2143336411yescharacteristic
631C2#2143591412yes—
632C2#2143617112yes—
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#554C2
socleC23×C59C22×D4
prime radical#21427452C2
hypercenterC2×#554D59
hypocenterC59#23969
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16837552C2semidirect
2. C22×#14273647C2semidirect
4. #21425229C2semidirect
5. #21425097C2semidirect
6. #21424520C2semidirect
7. #21424451C2semidirect
8. #21424413C2semidirect
9. #21424389C2semidirect
10. #21424186C2semidirect
11. C2×C59×#554C2semidirect
18. C2×#14273651C22semidirect
24. C2×C22⋊(C59⋊C8)C22semidirect
30. #16837164C22semidirect
31. #16837164C22semidirect
32. #16837164C22semidirect
33. #16837157C22semidirect
36. C2×C4⋊Q8×C59C22semidirect
41. C2×C22⋊Q8×C59C22semidirect
42. C2×C4×Q8×C59C22semidirect
46. C22×C4⋊C4×C59C22semidirect
92. C22⋊Q8×C59D4semidirect
108. C2×C4⋊C4×C59D4semidirect
134. C2×D4×C59SD16semidirect
309. C2×#554D59semidirect
333. C2×C4⋊Q8C2×D59semidirect
338. C2×C22⋊Q8C2×D59semidirect
341. C2×C4×Q8C2×D59semidirect
345. C22×C4⋊C4C2×D59semidirect
363. C59#23969semidirect
394. C22⋊Q8C22⋊D59semidirect
414. C2×C4⋊C4C22⋊D59semidirect
465. C2×D4#11745671semidirect

Maximal subgroups

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

subgroupindex
C2×#168375522
C22×#142736472
#214274522
#214252292
#214250972
#214245202
#214244512
#214244132
#214243892
#214241862
C2×C59×#5542
#2396959