Finite Group Databasebeta

← #28610245

Subgroups — #28610245

Order 15104.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837730C211yesmaximal, characteristic
2#21426971C211yesmaximal, characteristic
3#21424836C211yesmaximal, characteristic, complemented
4#21424775C211yesmaximal, characteristic, complemented
5#21424734C211yesmaximal, characteristic, complemented
6#21424681C211yesmaximal, characteristic, complemented
7#21424678C211yesmaximal, characteristic, complemented
8#21424530C211yesmaximal, characteristic
9#21424364C211yesmaximal, characteristic, complemented
10#21424295C211yesmaximal, characteristic
11C4×#14273658C211yesmaximal, characteristic, complemented
12#21423926C211yesmaximal, characteristic, complemented
13#21423880C211yesmaximal, characteristic
14#21423817C211yesmaximal, characteristic
15C59×#2362C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C22×C4⋊Dic59C2211yescharacteristic
18#16837730C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837285—21no—
33#16837285—21no—
34#16837262C2211yescharacteristic
35#16837260—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38C22⋊C4×Dic59—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C22⋊Q8×C59C2211yescharacteristic
50C2×C4×Q8×C59C2211yescharacteristic
51C2×#117×C59C2211yescharacteristic
52C59×#391C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#388—21no—
55C59×#388—21no—
56C59×#387—21no—
57C4×C4⋊C4×C59—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C22⋊C4×C59C2211yescharacteristic
60C23×Dic59—21no—
61C2×#11745664—22no—
62C2×#11745664—22no—
63C2×#11745664—41no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—22no—
70C2×C4⋊Dic59C2×C414yes—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80#14273660—41no—
81#14273658—22no—
82#14273658C2×C414yes—
83#14273658C2×C414yes—
84#14273651—22no—
85#14273651—41no—
86#14273651C2×C414yes—
87#14273651C2×C414yes—
88C22⋊(C59⋊C8)C2×C414yes—
89#14273638—41no—
90C4×C59⋊C8—22no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C22×Q8×C59C2311yescharacteristic
97C23×C4×C59C2311yescharacteristic
98C22⋊Q8×C59—22no—
99C22⋊Q8×C59C2×C414yes—
100C4×Q8×C59—22no—
101C4×Q8×C59—41no—
102#117×C59—22no—
103#117×C59—41no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59D411yescharacteristic
113C2×C22⋊C4×C59D411yescharacteristic
114C2×C42×C59—21no—
115C2×C42×C59—21no—
116C2×C42×C59—21no—
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119#95×C59—21no—
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
125C24×C59C2×D411yescharacteristic
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59C22×C412yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59C22×C412yes—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59#4112yes—
147C22×C4×C59C2411yescharacteristic
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—24no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—81no—
154C22×Dic59—81no—
155C4⋊C4×C59—24no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—42no—
161C4⋊C4×C59—42no—
162C4⋊C4×C59C22×C412yes—
163C4⋊C4×C59#4112yes—
164C22⋊C4×C59—24no—
165C22⋊C4×C59—22no—
166C22⋊C4×C59—41no—
167C22⋊C4×C59—42no—
168C42×C59—21no—
169C42×C59—21no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—42no—
173C42×C59—42no—
174C42×C59—42no—
175C22⋊Dic59—42no—
176C4⋊Dic59—24no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59C22×C412yes—
184C4⋊Dic59C22×C412yes—
185Dic59⋊C4—81no—
186Dic59⋊C4—81no—
187C4×Dic59—81no—
188C4×Dic59—81no—
189#11745664—44no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8C22×C412yes—
200C2×C59⋊C8C22×C412yes—
201C2×C59⋊C8#4112yes—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59C4×D412yes—
209C23×C59C22×D411yescharacteristic
210Q8×C59—22no—
211Q8×C59—22no—
212Q8×C59—41no—
213Q8×C59—41no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—22no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59C4×D412yes—
244C2×C4×C59#13712yes—
245C2×C4×C59C23×C411yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59H(5,𝔽2)11yescharacteristic
250C2×C4×C592−1+411yescharacteristic
251C2×Dic59—22no—
252C2×Dic59—22no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—84no—
264C59⋊C8—22no—
265C59⋊C8—22no—
266C59⋊C8—22no—
267C59⋊C8—41no—
268#22945—591nomaximal, Sylow
269C22×C59—22no—
270C22×C59—22no—
271C22×C59—22no—
272C22×C59—41no—
273C22×C59—41no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59C2×C4×D411yescharacteristic
277C22×C59#54611yescharacteristic
278C22×C59#54711yescharacteristic
279C22×C59C2×#13711yescharacteristic
280C22×C59#57511yescharacteristic
281C22×C59#57711yescharacteristic
282C22×C59#57711yescharacteristic
283C4×C59—22no—
284C4×C59—22no—
285C4×C59—41no—
286C4×C59—42no—
287C4×C59—41no—
288C4×C59—42no—
289C4×C59—42no—
290C4×C59—41no—
291C4×C59—81no—
292C4×C59C4×#4111yescharacteristic
293C4×C59#52011yescharacteristic
294C4×C59#55111yescharacteristic
295C4×C59#55211yescharacteristic
296Dic59—42no—
297Dic59—42no—
298Dic59—81no—
299Dic59—81no—
300C2×#483—591no—
301#2362D5911yescharacteristic
302#2345—591no—
303#1944—591no—
304#1933—591no—
305#1928—591no—
306#1920—591no—
307#1920—591no—
308#1894—591no—
309#1865—591no—
310#1854—591no—
311C4×#106—591no—
312#1763—591no—
313#1753—591no—
314#1736—591no—
315C2×C59—42no—
316C2×C59#235911yescharacteristic
317C2×C59C4×#13711yescharacteristic
318C2×C59#294911yescharacteristic
319C2×C59#332611yescharacteristic
320C2×C59#333111yescharacteristic
321C2×C59#334411yescharacteristic
322C2×C59#335011yescharacteristic
323C22×M16—591no—
324C2×C22⋊Q8C2×D5911yescharacteristic
325C2×C4×Q8C2×D5911yescharacteristic
326C2×#117C2×D5911yescharacteristic
327C22×C4⋊C4—591no—
328#483—598no—
329C2×#107—591no—
330C2×#106—591no—
331C2×#106—591no—
332C2×#106—591no—
333C2×#103—591no—
334C2×#103—591no—
335C2×#103—591no—
336C2×#103—591no—
337C2×C22⋊C8—591no—
338C2×#97—591no—
339C2×C4×C8—591no—
340#391C2×D5911yescharacteristic
341#389—591no—
342#389—591no—
343#389C2×D5911yescharacteristic
344#388—21no—
345#388—21no—
346#387—21no—
347#387—1181no—
348#387—1181no—
349#380—591no—
350#380—1181no—
351C4×C4⋊C4—21no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4C2×D5911yescharacteristic
355C4×C22⋊C4—1181no—
356C4×C22⋊C4C2×D5911yescharacteristic
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#336—591no—
366#336—591no—
367C59#2294511yescharacteristic, complemented, Sylow
368C22×Q8C22×D5911yescharacteristic
369C23×C4—1181no—
370C23×C4C22×D5911yescharacteristic
371C2×M16—1182no—
372C2×M16—1182no—
373C2×M16—2361no—
374C22×C8—591no—
375C22×C8—591no—
376C22×C8—591no—
377C22×C8—591no—
378C22⋊Q8—22no—
379C22⋊Q8C4×D5914yes—
380C4×Q8—22no—
381C4×Q8—41no—
382#117—22no—
383#117—41no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—594no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—1182no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4C22⋊D5911yescharacteristic
403C2×C22⋊C4C22⋊D5911yescharacteristic
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—1181no—
408C2×C42—1181no—
409C2×C42C22×D5911yescharacteristic
410C2×C42C22×D5911yescharacteristic
411#107—2361no—
412#106—594no—
413#106—594no—
414#106—1182no—
415#103—594no—
416#103—594no—
417#103—1182no—
418#103—2361no—
419C22⋊C8—594no—
420#97—2361no—
421C4×C8—1182no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95C22⋊D5911yescharacteristic
432#95C22⋊D5911yescharacteristic
433C24C2×C22⋊D5911yescharacteristic
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C2×C4×D5912yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—1184no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—4721no—
454C22×C4—4721no—
455C22×C4C2×C4×D5912yes—
456C22×C4#1174568512yes—
457C22×C4C2×C22⋊D5911yescharacteristic
458C22×C4C2×C22⋊D5911yescharacteristic
459C22×C4C2×C22⋊D5911yescharacteristic
460C22×C4C2×C22⋊D5911yescharacteristic
461C22×C4C2×C22⋊D5911yescharacteristic
462C22×C4C23×D5911yescharacteristic
463M16—2364no—
464C2×C8—592no—
465C2×C8—592no—
466C2×C8—592no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—592no—
484C4⋊C4—592no—
485C4⋊C4—1184no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—4721no—
493C4⋊C4—4721no—
494C4⋊C4C2×C4×D5912yes—
495C4⋊C4#1174568712yes—
496C22⋊C4—24no—
497C22⋊C4—22no—
498C22⋊C4—41no—
499C22⋊C4—42no—
500C22⋊C4—2362no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42—4721no—
509C42—4721no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23C4×C22⋊D5912yes—
517C23C22×C22⋊D5911yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—1182no—
552C2×C4—1182no—
553C2×C4—4724no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4C4×C22⋊D5912yes—
565C2×C4#1427375712yes—
566C2×C4C22×C4×D5911yescharacteristic
567C2×C4#41×D5911yescharacteristic
568C2×C4C2×#1174568711yescharacteristic
569C2×C4C2×#1174568511yescharacteristic
570C2×C4#1427377211yescharacteristic
571C2×C4#1427377411yescharacteristic
572C8—1182no—
573C8—1182no—
574C8—1182no—
575C8—2361no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×C4×C22⋊D5911yescharacteristic
584C22#1683804611yescharacteristic
585C22#1683805111yescharacteristic
586C22C2×#1427375711yescharacteristic
587C22#1683820011yescharacteristic
588C22#1683821011yescharacteristic
589C22#1683821011yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—42no—
598C4—81no—
599C4—4721no—
600C4—4721no—
601C4—2362no—
602C4—2362no—
603C4C4×#1174568711yescharacteristic
604C4#1683788111yescharacteristic
605C4#1683808111yescharacteristic
606C4#1683809211yescharacteristic
607C2—42no—
608C2#2142708211yescharacteristic
609C2C4×#1427375711yescharacteristic
610C2#2143295511yescharacteristic
611C2#2143634911yescharacteristic
612C2#2143641211yescharacteristic
613C2#2143655111yescharacteristic
614C2#2143664111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2362C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#2362D59
hypocenterC59#22945
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
3. #21424836C2semidirect
4. #21424775C2semidirect
5. #21424734C2semidirect
6. #21424681C2semidirect
7. #21424678C2semidirect
9. #21424364C2semidirect
11. C4×#14273658C2semidirect
12. #21423926C2semidirect
18. #16837730C4semidirect
367. C59#22945semidirect

Maximal subgroups

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

subgroupindex
C2×#168377302
#214269712
#214248362
#214247752
#214247342
#214246812
#214246782
#214245302
#214243642
#214242952
C4×#142736582
#214239262
#214238802
#214238172
C59×#23622
#2294559