Finite Group Databasebeta

← #28610568

Subgroups — #28610568

Order 15104.

886 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837690C211yesmaximal, characteristic
2#21427182C211yesmaximal, characteristic, complemented
3D4×Dic59⋊C4C211yesmaximal, characteristic
4#21425055C211yesmaximal, characteristic, complemented
5#21424851C211yesmaximal, characteristic, complemented
6#21424837C211yesmaximal, characteristic, complemented
7#21424836C211yesmaximal, characteristic, complemented
8#21424682C211yesmaximal, characteristic, complemented
9#21424679C211yesmaximal, characteristic, complemented
10#21424341C211yesmaximal, characteristic
11#21423962C211yesmaximal, characteristic, complemented
12#21423923C211yesmaximal, characteristic
13#21423921C211yesmaximal, characteristic
14#102×Dic59C211yesmaximal, characteristic
15C59×#1862C211yesmaximal, characteristic
16C2×#14273726C2211yescharacteristic
17C2×D4×Dic59C2211yescharacteristic
18C2×D4×Dic59C2211yescharacteristic
19C22×Dic59⋊C4—21no—
20#16837690C418yescomplemented
21C2×#14273660C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×Dic59⋊C8C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273638C2211yescharacteristic
28C2×#14273637C2211yescharacteristic
29C2×C8×Dic59C2211yescharacteristic
30#16837307—21no—
31#16837306—21no—
32#16837300C2211yescharacteristic
33#16837298C2211yescharacteristic
34#16837298C2211yescharacteristic
35#16837297—21no—
36#16837275—21no—
37#16837274—21no—
38#16837271C2211yescharacteristic
39#16837260—21no—
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4×Dic59⋊C4C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44#16837253—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C2×#107×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×#102×C59C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118—21no—
61C23×Dic59—21no—
62C23×Dic59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273726—22no—
66#14273726C2×C414yescomplemented
67D4×Dic59—22no—
68D4×Dic59—22no—
69D4×Dic59—41no—
70D4×Dic59C2×C414yes—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—44no—
82C2×Dic59⋊C4D411yescharacteristic
83C2×Dic59⋊C4D411yescharacteristic
84C2×C22⋊Dic59—21no—
85C2×C22⋊Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59C2311yescharacteristic
90C2×C4×Dic59C2311yescharacteristic
91#14273660—41no—
92#14273659C2×C414yescomplemented
93#14273658C2×C414yescomplemented
94Dic59⋊C8C2×C414yescomplemented
95#14273648C2×C414yes—
96#14273648Q814yescomplemented
97#14273638—41no—
98#14273637—41no—
99C8×Dic59—22no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95D411yescharacteristic
107C59⋊#95D411yescharacteristic
108C22×D4×C59C2311yescharacteristic
109C23×C4×C59—21no—
110C22×C8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C4×D4×C59—22no—
113C4×D4×C59—41no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C22⋊C4×C59—21no—
119C2×C42×C59C2311yescharacteristic
120#107×C59—41no—
121#102×C59—22no—
122#102×C59C2×C414yes—
123#95×C59—21no—
124C24×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59C22×C412yes—
129C2×D4×C59C2×Q812yes—
130C2×D4×C59#4112yes—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—42no—
135C22×C4×C59—42no—
136C22×C4×C59—81no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2411yescharacteristic
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—24no—
144C22×Dic59—21no—
145C22×Dic59—42no—
146C22×Dic59—42no—
147C22×Dic59—42no—
148C22×Dic59—42no—
149C22×Dic59—81no—
150C22×Dic59C2×D411yescharacteristic
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C2×Dic118—24no—
155C2×Dic118—81no—
156C2×Dic118—81no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59C22×C412yes—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—81no—
169C4⋊C4×C59C22×C412yes—
170C22⋊C4×C59—42no—
171C42×C59—21no—
172C42×C59—21no—
173C22⋊Dic59—24no—
174C22⋊Dic59—42no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59C22×C412yes—
181C4⋊Dic59C22×C412yes—
182C4⋊Dic59C2×Q812yes—
183Dic59⋊C4—24no—
184Dic59⋊C4—22no—
185Dic59⋊C4—22no—
186Dic59⋊C4—42no—
187Dic59⋊C4—42no—
188Dic59⋊C4—42no—
189Dic59⋊C4—81no—
190Dic59⋊C4—81no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193C4×Dic59—42no—
194C4×Dic59—81no—
195C4×Dic59—81no—
196C4×Dic59C22×C412yes—
197C4×Dic59#4112yes—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8C22×C412yes—
203C2×C59⋊C8C2×Q812yes—
204C23×C59—22no—
205C23×C59—22no—
206C23×C59—22no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59C22×D411yescharacteristic
212D4×C59—22no—
213D4×C59—22no—
214D4×C59—22no—
215D4×C59C4×Q814yes—
216C2×C4×C59—22no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—84no—
232C2×C4×C59C23×C411yescharacteristic
233C2×C4×C59C22×Q811yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59H(5,𝔽2)11yescharacteristic
237C2×C4×C592−1+411yescharacteristic
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—22no—
241C2×Dic59—22no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—22no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—48no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—84no—
260C2×Dic59C4×D412yes—
261C2×Dic59C4×D412yes—
262C2×Dic59AGL(1,ℤ/8)12yes—
263C2×Dic59#13712yes—
264Dic118—84no—
265C8×C59—22no—
266C8×C59—41no—
267C59⋊C8—22no—
268C59⋊C8—22no—
269#22969—591nomaximal, Sylow
270C22×C59—24no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59—42no—
275C22×C59—42no—
276C22×C59—42no—
277C22×C59C2×C4×D411yescharacteristic
278C22×C59#54611yescharacteristic
279C22×C59D4×Q811yescharacteristic
280C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C59C2×#13711yescharacteristic
282C22×C59#57611yescharacteristic
283C22×C59#57711yescharacteristic
284C4×C59—41no—
285C4×C59—42no—
286C4×C59—41no—
287C4×C59—81no—
288C4×C59—81no—
289C4×C59C2×C4×Q811yescharacteristic
290C4×C59#52011yescharacteristic
291C4×C59#55211yescharacteristic
292C4×C59#55411yescharacteristic
293Dic59—22no—
294Dic59—41no—
295Dic59—42no—
296Dic59—42no—
297Dic59—42no—
298Dic59—41no—
299Dic59—81no—
300Dic59#44014yes—
301C2×#476—591no—
302#2369—591no—
303D4×C4⋊C4—591no—
304#1990—591no—
305#1947—591no—
306#1944—591no—
307#1944—591no—
308#1920—591no—
309#1920—591no—
310#1862—591no—
311#1862D5911yescharacteristic
312#1770—591no—
313#1762—591no—
314#1762—591no—
315C4×#102—591no—
316C2×C59—44no—
317C2×C59#236011yescharacteristic
318C2×C59C2×#44011yescharacteristic
319C2×C59#294611yescharacteristic
320C2×C59#332511yescharacteristic
321C2×C59#333111yescharacteristic
322C2×C59#338411yescharacteristic
323C2×C59#338511yescharacteristic
324C2×C4⋊Q8—591no—
325C2×C4×D4—591no—
326C2×C4×D4—591no—
327C2×C4×D4C2×D5911yescharacteristic
328C22×C4⋊C4—1181no—
329#476—598no—
330C2×#107—591no—
331C2×#107C2×D5911yescharacteristic
332C2×#106—591no—
333C2×#106—591no—
334C2×C4⋊C8—591no—
335C2×#102—591no—
336C2×#102—591no—
337C2×#102C2×D5911yescharacteristic
338C2×#102C2×D5911yescharacteristic
339C2×#97—591no—
340C2×#97—591no—
341C2×C4×C8—591no—
342#391—1181no—
343#391—1181no—
344#389—591no—
345#389—591no—
346#389—591no—
347#389—1181no—
348#389C2×D5911yescharacteristic
349#385—1181no—
350#385—1181no—
351#384—591no—
352#380—1181no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4—591no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4—591no—
357C4×C4⋊C4—1181no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340C2×D5911yescharacteristic
366#336—591no—
367#336C2×D5911yescharacteristic
368C59#2296911yescharacteristic, complemented, Sylow
369C22×Q8—1181no—
370C22×D4C22×D5911yescharacteristic
371C23×C4—21no—
372C23×C4—1181no—
373C23×C4—1181no—
374C22×C8—591no—
375C22×C8—591no—
376C22×C8C22×D5911yescharacteristic
377C22×C8C22×D5911yescharacteristic
378C4⋊Q8—594no—
379C4⋊Q8—1182no—
380C4×D4—22no—
381C4×D4—41no—
382C4×D4—594no—
383C4×D4—1182no—
384C4×D4—1182no—
385C4×D4—2361no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—2364no—
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—1181no—
405C2×C22⋊C4—1181no—
406C2×C42—591no—
407C2×C42—591no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42—1181no—
411C2×C42C22×D5911yescharacteristic
412#107—41no—
413#107—2361no—
414#106—594no—
415#106—594no—
416C4⋊C8—594no—
417#102—22no—
418#102—594no—
419#102—594no—
420#102C4×D5914yes—
421#97—2361no—
422#97—2361no—
423C4×C8—1182no—
424#95—21no—
425#95—591no—
426#95—591no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433C24—21no—
434C2×Q8—1184no—
435C2×Q8—4721no—
436C2×Q8—4721no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4C2×C4×D5912yes—
441C2×D4#1174568712yes—
442C2×D4Q8×D5912yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4—591no—
450C22×C4—591no—
451C22×C4—591no—
452C22×C4—591no—
453C22×C4—1184no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—4721no—
459C22×C4—2362no—
460C22×C4—2362no—
461C22×C4—2362no—
462C22×C4—2362no—
463C22×C4D4×D5911yescharacteristic
464C22×C4D4×D5911yescharacteristic
465C22×C4C23×D5911yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—592no—
472C2×C8—592no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8C2×C4×D5912yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—81no—
484C4⋊C4—592no—
485C4⋊C4—592no—
486C4⋊C4—592no—
487C4⋊C4—1182no—
488C4⋊C4—1182no—
489C4⋊C4—1184no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—4721no—
496C4⋊C4—4721no—
497C4⋊C4—2362no—
498C4⋊C4—2362no—
499C4⋊C4—2362no—
500C4⋊C4C2×C4×D5912yes—
501C22⋊C4—42no—
502C22⋊C4—1184no—
503C22⋊C4—2362no—
504C42—21no—
505C42—21no—
506C42—592no—
507C42—592no—
508C42—1181no—
509C42—1181no—
510C42—4721no—
511C42—4721no—
512C42—2362no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C2×D4×D5911yescharacteristic
521Q8—4724no—
522D4—22no—
523D4—22no—
524D4—22no—
525D4#1427369214yes—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4—592no—
543C2×C4—592no—
544C2×C4—592no—
545C2×C4—592no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1182no—
555C2×C4—1182no—
556C2×C4—1182no—
557C2×C4—4724no—
558C2×C4—2368no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4C22×C4×D5911yescharacteristic
569C2×C4C2×Q8×D5911yescharacteristic
570C2×C4#41×D5911yescharacteristic
571C2×C4C2×#1174568711yescharacteristic
572C2×C4#1427377211yescharacteristic
573C2×C4#1427377411yescharacteristic
574C8—22no—
575C8—41no—
576C8—1182no—
577C8—1182no—
578C22—24no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22C4×D4×D5911yescharacteristic
586C22#1683803911yescharacteristic
587C22#1683806511yescharacteristic
588C22AGL(1,ℤ/8)×D5911yescharacteristic
589C22#137×D5911yescharacteristic
590C22#1683820311yescharacteristic
591C22#1683820911yescharacteristic
592C4—42no—
593C4—41no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—594no—
598C4—1182no—
599C4—4721no—
600C4—2362no—
601C4—2362no—
602C4—2362no—
603C4—2361no—
604C4—2361no—
605C4C2×#1427369211yescharacteristic
606C4#1683788111yescharacteristic
607C4#1683808911yescharacteristic
608C4#1683810611yescharacteristic
609C2—44no—
610C2#2142709611yescharacteristic
611C2#440×D5911yescharacteristic
612C2#2143292111yescharacteristic
613C2#2143633111yescharacteristic
614C2#2143640511yescharacteristic
615C2#2143699811yescharacteristic
616C2#2143700111yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1862C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1862D59
hypocenterC59#22969
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
2. #21427182C2semidirect
4. #21425055C2semidirect
5. #21424851C2semidirect
6. #21424837C2semidirect
7. #21424836C2semidirect
8. #21424682C2semidirect
9. #21424679C2semidirect
11. #21423962C2semidirect
20. #16837690C4semidirect
66. #14273726C2×C4semidirect
92. #14273659C2×C4semidirect
93. #14273658C2×C4semidirect
94. Dic59⋊C8C2×C4semidirect
96. #14273648Q8semidirect
368. C59#22969semidirect

Maximal subgroups

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

subgroupindex
C2×#168376902
#214271822
D4×Dic59⋊C42
#214250552
#214248512
#214248372
#214248362
#214246822
#214246792
#214243412
#214239622
#214239232
#214239212
#102×Dic592
C59×#18622
#2296959