Finite Group Databasebeta

← #28599243

Subgroups — #28599243

Order 15104.

858 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837458C211yesmaximal, characteristic
2#21426500C211yesmaximal, characteristic
3#382×D59C211yesmaximal, characteristic
4#21424866C211yesmaximal, characteristic, complemented
5#21424864C211yesmaximal, characteristic, complemented
6#21424862C211yesmaximal, characteristic, complemented
7#21424290C211yesmaximal, characteristic
8#21424287C211yesmaximal, characteristic
9#21424285C211yesmaximal, characteristic
10#21424282C211yesmaximal, characteristic
11#21424263C211yesmaximal, characteristic, complemented
12#21424261C211yesmaximal, characteristic, complemented
13#21424228C211yesmaximal, characteristic, complemented
14#21424228C211yesmaximal, characteristic, complemented
15C59×#1949C211yesmaximal, characteristic, complemented
16C2×#14273684—21no—
17C2×C4⋊C4×D59C2211yescharacteristic
18C2×C4⋊C4×D59C2211yescharacteristic
19C2×C42×D59C2211yescharacteristic
20#16837458C418yescomplemented
21C2×#14273661C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273659C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×(C2×D59)⋊C8C2211yescharacteristic
29C2×(C2×D59)⋊C8C2211yescharacteristic
30C2×#14273637C2211yescharacteristic
31C2×#14273637C2211yescharacteristic
32#16837271C2211yescharacteristic
33#16837271C2211yescharacteristic
34#16837268C2211yescharacteristic
35#16837267C2211yescharacteristic
36#16837267C2211yescharacteristic
37#16837266—21no—
38#16837266—21no—
39#16837266—21no—
40#16837261—21no—
41#16837259C2211yescharacteristic
42#16837259C2211yescharacteristic
43C4×Dic59⋊C4—21no—
44C4×C4⋊Dic59C2211yescharacteristic
45#95×D59—21no—
46#95×D59—21no—
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×C4⋊C8×C59C2211yescharacteristic
56C59×#382C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×C4×D59—21no—
61C22×C4×D59—21no—
62C22×C4×D59D411yescharacteristic
63C22×C4×D59D411yescharacteristic
64C22×C4×D59C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273684—44no—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C4⋊C4×D59—41no—
79C4⋊C4×D59—41no—
80C4⋊C4×D59—41no—
81C4⋊C4×D59—41no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59D411yescharacteristic
86C2×C4×Dic59D411yescharacteristic
87C2×C4×Dic59C2311yescharacteristic
88C42×D59C2×C414yes—
89C42×D59Q814yes—
90#14273661—41no—
91#14273660—41no—
92#14273660—41no—
93#14273659—41no—
94#14273658—41no—
95#14273658—41no—
96C4⋊(C59⋊C8)C2×C414yescomplemented
97(C2×D59)⋊C8C2×C414yes—
98(C2×D59)⋊C8C2×C414yes—
99#14273637C2×C414yescomplemented
100#14273637C2×C414yescomplemented
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C22×C8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C42×C59—21no—
117C2×C42×C59C2311yescharacteristic
118#107×C59—41no—
119#106×C59—41no—
120C4⋊C8×C59C2×C414yescomplemented
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124C23×D59C2×D411yescharacteristic
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2411yescharacteristic
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59C2×D411yescharacteristic
137C22×Dic59C2×D411yescharacteristic
138C22×Dic59C2×D411yescharacteristic
139C2×C4×D59—22no—
140C2×C4×D59—22no—
141C2×C4×D59—21no—
142C2×C4×D59—21no—
143C2×C4×D59—21no—
144C2×C4×D59—21no—
145C2×C4×D59—41no—
146C2×C4×D59—41no—
147C2×C4×D59—42no—
148C2×C4×D59—42no—
149C2×C4×D59—42no—
150C2×C4×D59—42no—
151C2×C4×D59—42no—
152C2×C4×D59—42no—
153C2×C4×D59C4⋊C414yes—
154C2×C4×D59C4⋊C414yes—
155C2×C4×D59C22×C412yes—
156C2×C4×D59C2×Q812yes—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59C22×C412yes—
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—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C42×C59—42no—
172C42×C59C22×C412yes—
173C42×C59C2×Q812yes—
174(C2×D59)⋊C4—42no—
175(C2×D59)⋊C4—42no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184Dic59⋊C4—42no—
185Dic59⋊C4—42no—
186Dic59⋊C4—42no—
187Dic59⋊C4—42no—
188Dic59⋊C4—42no—
189Dic59⋊C4—42no—
190C4×Dic59—42no—
191C4×Dic59C4⋊C414yes—
192C4×Dic59C4⋊C414yes—
193C4×Dic59C22×C412yes—
194C4×Dic59C2×Q812yes—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8C22×C412yes—
200C2×C59⋊C8C22×C412yes—
201C23×C59C22×D411yescharacteristic
202C22×D59—21no—
203C22×D59—21no—
204C22×D59—21no—
205C22×D59—21no—
206C22×D59C2×C4⋊C412yes—
207C22×D59AGL(1,ℤ/8)12yes—
208C22×D59#13712yes—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59C2×C4⋊C412yes—
226C2×C4×C59C2×C4⋊C412yes—
227C2×C4×C59C23×C411yescharacteristic
228C2×C4×C59C22×Q811yescharacteristic
229C2×C4×C59H(5,𝔽2)11yescharacteristic
230C2×C4×C59H(5,𝔽2)11yescharacteristic
231C2×C4×C592−1+411yescharacteristic
232C2×C4×C592−1+411yescharacteristic
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
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×Dic59C2×C4⋊C412yes—
254C4×D59—28no—
255C4×D59—22no—
256C4×D59—22no—
257C4×D59—22no—
258C4×D59—22no—
259C4×D59—44no—
260C4×D59—84no—
261C4×D59—84no—
262C8×C59—22no—
263C8×C59—22no—
264C59⋊C8—22no—
265C59⋊C8—22no—
266#21986—591nomaximal, Sylow
267C22×C59C22×C4⋊C411yescharacteristic
268C22×C59#53511yescharacteristic
269C22×C59#53611yescharacteristic
270C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C59C2×#13711yescharacteristic
272C22×C59#57511yescharacteristic
273C22×C59#57511yescharacteristic
274C2×D59—22no—
275C2×D59—22no—
276C2×D59—22no—
277C2×D59—22no—
278C2×D59—22no—
279C2×D59—22no—
280C2×D59#42814yes—
281C4×C59—22no—
282C4×C59—22no—
283C4×C59—81no—
284C4×C59—81no—
285C4×C59—81no—
286C4×C59—81no—
287C4×C59#52011yescharacteristic
288C4×C59#52011yescharacteristic
289C4×C59#54411yescharacteristic
290C4×C59#54411yescharacteristic
291Dic59—22no—
292Dic59—22no—
293Dic59—22no—
294Dic59—81no—
295Dic59—81no—
296Dic59—81no—
297Dic59—81no—
298C2×#432—591no—
299#2285—591no—
300C2×#382—591no—
301#1949—591no—
302#1949—591no—
303#1949—591no—
304#1949D5911yescharacteristic, complemented
305#1853—591no—
306#1853—591no—
307#1852—591no—
308#1852—591no—
309#1847—591no—
310#1847—591no—
311#1836—591no—
312#1836—591no—
313C2×C59#231711yescharacteristic
314C2×C59C2×#42811yescharacteristic
315C2×C59#291111yescharacteristic
316C2×C59#321311yescharacteristic
317C2×C59#321511yescharacteristic
318C2×C59#323411yescharacteristic
319C2×C59#323611yescharacteristic
320D59—24no—
321C2×#117—1181no—
322C22×C4⋊C4—591no—
323C22×C4⋊C4—591no—
324C22×C42—591no—
325#432—598no—
326C2×#107—591no—
327C2×#107—591no—
328C2×#107—591no—
329C2×#107C2×D5911yescharacteristic
330C2×#106—591no—
331C2×#106—591no—
332C2×#106—591no—
333C2×#106C2×D5911yescharacteristic
334C2×C4⋊C8—591no—
335C2×C4⋊C8C2×D5911yescharacteristic
336C2×C22⋊C8—591no—
337C2×C22⋊C8—591no—
338C2×#97—591no—
339C2×#97—591no—
340#384—591no—
341#384—591no—
342#382—591no—
343#382—591no—
344#382—591no—
345#382—1181no—
346#382—1181no—
347#382—1181no—
348#382C2×D5911yescharacteristic
349#380—591no—
350#380—591no—
351#380—1181no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—1181no—
354C4×C4⋊C4C2×D5911yescharacteristic
355C2×#95—1181no—
356C2×#95—1181no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340C2×D5911yescharacteristic
364#340C2×D5911yescharacteristic
365C59#2198611yescharacteristic, complemented, Sylow
366C23×C4—591no—
367C23×C4—591no—
368C23×C4—591no—
369C23×C4—1181no—
370C23×C4—1181no—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8C22×D5911yescharacteristic
374C22×C8C22×D5911yescharacteristic
375#117—2364no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—1181no—
382C2×C4⋊C4—1181no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4—1181no—
385C2×C4⋊C4—1181no—
386C2×C4⋊C4—2361no—
387C2×C4⋊C4—2361no—
388C2×C4⋊C4—2361no—
389C2×C4⋊C4—2361no—
390C2×C4⋊C4C22×D5911yescharacteristic
391C2×C4⋊C4C22×D5911yescharacteristic
392C2×C4⋊C4C22×D5911yescharacteristic
393C2×C4⋊C4C22×D5911yescharacteristic
394C2×C22⋊C4—1181no—
395C2×C22⋊C4—1181no—
396C2×C42—21no—
397C2×C42—591no—
398C2×C42—591no—
399C2×C42—591no—
400C2×C42—594no—
401C2×C42—594no—
402C2×C42—1181no—
403C2×C42C22×D5911yescharacteristic
404#107—41no—
405#107—2361no—
406#107—2361no—
407#107—2361no—
408#106—41no—
409#106—2361no—
410#106—2361no—
411#106—2361no—
412C4⋊C8—594no—
413C4⋊C8C4×D5914yescomplemented
414C22⋊C8—594no—
415C22⋊C8—594no—
416#97—594no—
417#97—594no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430C24—591no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—591no—
436C22×C4—591no—
437C22×C4—591no—
438C22×C4—594no—
439C22×C4—592no—
440C22×C4—592no—
441C22×C4—594no—
442C22×C4—1182no—
443C22×C4—1182no—
444C22×C4—1181no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—2361no—
453C22×C4—2361no—
454C22×C4—2362no—
455C22×C4—2362no—
456C22×C4—2362no—
457C22×C4—2362no—
458C22×C4—2362no—
459C22×C4—2362no—
460C22×C4D4×D5911yescharacteristic
461C22×C4D4×D5911yescharacteristic
462C22×C4C23×D5911yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—592no—
468C2×C8—592no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8C2×C4×D5912yes—
474C2×C8C2×C4×D5912yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—2362no—
492C4⋊C4—2362no—
493C4⋊C4—2362no—
494C4⋊C4—2362no—
495C4⋊C4—2362no—
496C4⋊C4—2362no—
497C22⋊C4—2362no—
498C22⋊C4—2362no—
499C42—42no—
500C42—594no—
501C42—594no—
502C42—592no—
503C42—592no—
504C42—2362no—
505C42C2×C4×D5912yes—
506C42Q8×D5912yes—
507C23—592no—
508C23—592no—
509C23—592no—
510C23—1181no—
511C23—1181no—
512C23—1181no—
513C23—1181no—
514C23C2×D4×D5911yescharacteristic
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—592no—
532C2×C4—592no—
533C2×C4—592no—
534C2×C4—1188no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1182no—
542C2×C4—1182no—
543C2×C4—1182no—
544C2×C4—1182no—
545C2×C4—4724no—
546C2×C4—4724no—
547C2×C4—2364no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4C4⋊C4×D5912yes—
561C2×C4C4⋊C4×D5912yes—
562C2×C4C22×C4×D5911yescharacteristic
563C2×C4C2×Q8×D5911yescharacteristic
564C2×C4#1427377211yescharacteristic
565C2×C4#1427377311yescharacteristic
566C2×C4#1427377411yescharacteristic
567C2×C4#1427377511yescharacteristic
568C8—22no—
569C8—22no—
570C8—1182no—
571C8—1182no—
572C22—594no—
573C22—1182no—
574C22—1182no—
575C22—1182no—
576C22—1182no—
577C22—1182no—
578C22—1182no—
579C22C2×C4⋊C4×D5911yescharacteristic
580C22#1683796811yescharacteristic
581C22#1683797511yescharacteristic
582C22AGL(1,ℤ/8)×D5911yescharacteristic
583C22#137×D5911yescharacteristic
584C22#1683819911yescharacteristic
585C22#1683819911yescharacteristic
586C4—22no—
587C4—22no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—1182no—
593C4—1182no—
594C4—1182no—
595C4—4721no—
596C4—4721no—
597C4—4721no—
598C4—4721no—
599C4#1683788111yescharacteristic
600C4#1683788411yescharacteristic
601C4#1683802911yescharacteristic
602C4#1683803211yescharacteristic
603C2—1184no—
604C2#2142671811yescharacteristic
605C2#428×D5911yescharacteristic
606C2#2143268411yescharacteristic
607C2#2143508111yescharacteristic
608C2#2143509711yescharacteristic
609C2#2143528411yescharacteristic
610C2#2143530011yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1949C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1949D59
hypocenterC59#21986
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #21424866C2semidirect
5. #21424864C2semidirect
6. #21424862C2semidirect
11. #21424263C2semidirect
12. #21424261C2semidirect
13. #21424228C2semidirect
14. #21424228C2semidirect
15. C59×#1949C2semidirect
20. #16837458C4semidirect
96. C4⋊(C59⋊C8)C2×C4semidirect
99. #14273637C2×C4semidirect
100. #14273637C2×C4semidirect
120. C4⋊C8×C59C2×C4semidirect
304. #1949D59semidirect
365. C59#21986semidirect
413. C4⋊C8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168374582
#214265002
#382×D592
#214248662
#214248642
#214248622
#214242902
#214242872
#214242852
#214242822
#214242632
#214242612
#214242282
C59×#19492
#2198659