Finite Group Databasebeta

← #28608081

Subgroups — #28608081

Order 15104.

894 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837500C211yesmaximal, characteristic
2C4⋊C4×D236C211yesmaximal, characteristic
3#21424789C211yesmaximal, characteristic, complemented
4#21424775C211yesmaximal, characteristic, complemented
5#21424688C211yesmaximal, characteristic, complemented
6#21424667C211yesmaximal, characteristic, complemented
7#21424433C211yesmaximal, characteristic
8#21424433C211yesmaximal, characteristic
9#21424229C211yesmaximal, characteristic, complemented
10#21424103C211yesmaximal, characteristic
11#21424101C211yesmaximal, characteristic, complemented
12#21423971C211yesmaximal, characteristic, complemented
13#21423924C211yesmaximal, characteristic
14C4×#14273647C211yesmaximal, characteristic
15C59×#2356C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19C2×C4×D236C2211yescharacteristic
20C2×C4⋊C4×D59—21no—
21#16837500C418yescomplemented
22C2×#14273660C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273647C2211yescharacteristic
30C2×#14273638C2211yescharacteristic
31C2×#14273638C2211yescharacteristic
32C2×C4×C59⋊C8C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837300C2211yescharacteristic
35#16837276—21no—
36#16837276—21no—
37#16837272C2211yescharacteristic
38#16837259—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C4×Q8×C59C2211yescharacteristic
50C59×#391C2211yescharacteristic
51C59×#391C2211yescharacteristic
52C59×#387—21no—
53C59×#387—21no—
54C59×#383—21no—
55C59×#382C2211yescharacteristic
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C59×#376—21no—
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C22×C4×D59—21no—
63C22×C4×D59—21no—
64C2×#11745671—22no—
65C2×#11745671—22no—
66C2×#11745671C2×C414yes—
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C4×D236—22no—
72C4×D236—22no—
73C4×D236—22no—
74C4×D236—41no—
75C4×D236—41no—
76C4×D236C2×C414yes—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59—21no—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C4⋊C4×D59—44no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87C2×(C2×D59)⋊C4—21no—
88#14273660C2×C414yescomplemented
89C4⋊(C59⋊C8)—22no—
90#14273651—41no—
91#14273651—41no—
92#14273651C2×C414yescomplemented
93#14273647—22no—
94#14273647—22no—
95#14273647C2×C414yes—
96#14273638—41no—
97#14273638C2×C414yescomplemented
98C4×C59⋊C8—41no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C22×Q8×C59C2311yescharacteristic
102C4×Q8×C59—22no—
103C4×Q8×C59C2×C414yescomplemented
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59D411yescharacteristic
106C2×C4⋊C4×C59D411yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C42×C59—21no—
111C2×C42×C59—21no—
112C2×C42×C59—21no—
113C2×C42×C59C2311yescharacteristic
114C2×C42×C59C2311yescharacteristic
115C2×C42×C59C2311yescharacteristic
116#95×C59—21no—
117#95×C59—21no—
118#95×C59—21no—
119#95×C59—21no—
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C23×D59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—21no—
132C2×Q8×C59C22×C412yes—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C2×D236—21no—
148C2×D236—21no—
149C2×D236—21no—
150C2×D236C22×C412yes—
151C2×D236#4112yes—
152C2×D236#4112yes—
153C2×C4×D59—24no—
154C2×C4×D59—42no—
155C2×C4×D59—42no—
156C2×C4×D59—42no—
157C2×C4×D59—42no—
158C2×C4×D59—42no—
159C2×C4×D59—42no—
160C2×C4×D59—81no—
161C2×C4×D59—81no—
162C4⋊C4×C59—24no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—41no—
170C4⋊C4×C59C22×C412yes—
171C42×C59—24no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—42no—
177C42×C59—42no—
178C42×C59C22×C412yes—
179C42×C59#4112yes—
180#11745671—28no—
181(C2×D59)⋊C4—24no—
182(C2×D59)⋊C4—42no—
183(C2×D59)⋊C4—42no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—42no—
190C4⋊Dic59—81no—
191C4⋊Dic59C22×C412yes—
192Dic59⋊C4—42no—
193Dic59⋊C4—42no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8C22×C412yes—
205C2×C59⋊C8C22×C412yes—
206C23×C59C22×D411yescharacteristic
207C22×D59—22no—
208C22×D59—22no—
209C22×D59—22no—
210C22×D59—41no—
211C22×D59—41no—
212C22×D59—41no—
213C22×D59—41no—
214Q8×C59—22no—
215Q8×C59—22no—
216Q8×C59—41no—
217Q8×C59—41no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—21no—
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—21no—
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×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59C4×D412yes—
246C2×C4×C59C4×D412yes—
247C2×C4×C59AGL(1,ℤ/8)12yes—
248C2×C4×C59C23×C411yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59H(5,𝔽2)11yescharacteristic
253C2×C4×C592−1+411yescharacteristic
254C2×Dic59—22no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263D236—22no—
264D236—22no—
265D236—22no—
266D236#11714yes—
267C4×D59—48no—
268C4×D59—84no—
269C4×D59—84no—
270C59⋊C8—22no—
271C59⋊C8—22no—
272C59⋊C8—41no—
273C59⋊C8—41no—
274#22778—591nomaximal, Sylow
275C22×C59C2×C4×D411yescharacteristic
276C22×C59#54611yescharacteristic
277C22×C59#54811yescharacteristic
278C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C59#57511yescharacteristic
280C22×C59#57711yescharacteristic
281C22×C59#57811yescharacteristic
282C2×D59—24no—
283C2×D59—42no—
284C2×D59—42no—
285C2×D59—42no—
286C2×D59—42no—
287C2×D59—42no—
288C2×D59—42no—
289C4×C59—22no—
290C4×C59—22no—
291C4×C59—22no—
292C4×C59—42no—
293C4×C59—42no—
294C4×C59—41no—
295C4×C59—41no—
296C4×C59—41no—
297C4×C59—81no—
298C4×C59—81no—
299C4×C59C2×#11711yescharacteristic
300C4×C59#52011yescharacteristic
301C4×C59#55211yescharacteristic
302C4×C59#55311yescharacteristic
303Dic59—42no—
304Dic59—81no—
305Dic59—81no—
306C2×#440—591no—
307#2356D5911yescharacteristic, complemented
308D4×C4⋊C4—591no—
309#1935—591no—
310#1933—591no—
311#1921—591no—
312#1918—591no—
313#1877—591no—
314#1877—591no—
315#1836—591no—
316#1804—591no—
317#1803—591no—
318#1771—591no—
319#1762—591no—
320C4×#102—591no—
321C2×C59#235211yescharacteristic
322C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
323C2×C59#294711yescharacteristic
324C2×C59#331011yescharacteristic
325C2×C59#331511yescharacteristic
326C2×C59#336711yescharacteristic
327C2×C59#337611yescharacteristic
328D59—44no—
329C22×SD16—591no—
330C2×C4×Q8C2×D5911yescharacteristic
331C2×C4×D4—591no—
332C2×C4×D4—591no—
333C2×C4×D4—591no—
334C22×C4⋊C4—1181no—
335#440—598no—
336C2×#107—591no—
337C2×C4⋊C8—591no—
338C2×#103—591no—
339C2×#103—591no—
340C2×#103—591no—
341C2×#102—591no—
342C2×#102—591no—
343C2×#102—591no—
344C2×#97—591no—
345C2×#97—591no—
346C2×C4×C8—591no—
347#391C2×D5911yescharacteristic
348#391C2×D5911yescharacteristic
349#389—591no—
350#389—591no—
351#387—21no—
352#387—21no—
353#385—1181no—
354#385—1181no—
355#384—591no—
356#383—21no—
357#382C2×D5911yescharacteristic
358#380—1181no—
359C4×C4⋊C4C2×D5911yescharacteristic
360C4×C4⋊C4C2×D5911yescharacteristic
361C4×C4⋊C4C2×D5911yescharacteristic
362#376—21no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#340—591no—
370#340—591no—
371#336—591no—
372#336—591no—
373C59#2277811yescharacteristic, complemented, Sylow
374C22×Q8C22×D5911yescharacteristic
375C22×D4—591no—
376C23×C4—1181no—
377C23×C4—1181no—
378C23×C4—1181no—
379C2×SD16—594no—
380C2×SD16—1182no—
381C2×SD16—1182no—
382C22×C8—591no—
383C22×C8—591no—
384C22×C8—591no—
385C22×C8—591no—
386C4×Q8—22no—
387C4×Q8C4×D5914yescomplemented
388C4×D4—594no—
389C4×D4—1182no—
390C4×D4—1182no—
391C4×D4—1182no—
392C4×D4—2361no—
393C4×D4—2361no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—2364no—
403C2×C4⋊C4C22⋊D5911yescharacteristic
404C2×C4⋊C4C22⋊D5911yescharacteristic
405C2×C4⋊C4C22×D5911yescharacteristic
406C2×C4⋊C4C22×D5911yescharacteristic
407C2×C4⋊C4C22×D5911yescharacteristic
408C2×C22⋊C4—1181no—
409C2×C22⋊C4—1181no—
410C2×C22⋊C4—1181no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42C22×D5911yescharacteristic
415C2×C42C22×D5911yescharacteristic
416C2×C42C22×D5911yescharacteristic
417#107—594no—
418C4⋊C8—1182no—
419#103—594no—
420#103—2361no—
421#103—2361no—
422#102—594no—
423#102—1182no—
424#102—1182no—
425#97—594no—
426#97—2361no—
427C4×C8—2361no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—1181no—
437#95—1181no—
438#95C22⋊D5911yescharacteristic
439#95C22⋊D5911yescharacteristic
440C24—1181no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8C2×C4×D5912yes—
447C2×D4—592no—
448C2×D4—592no—
449C2×D4—592no—
450C2×D4—1181no—
451C2×D4—1181no—
452C2×D4—1181no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—1181no—
458C22×C4—1184no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—4721no—
462C22×C4—4721no—
463C22×C4—2362no—
464C22×C4—2362no—
465C22×C4—2362no—
466C22×C4—2362no—
467C22×C4—2362no—
468C22×C4—2362no—
469C22×C4C2×C22⋊D5911yescharacteristic
470C22×C4C2×C22⋊D5911yescharacteristic
471C22×C4C2×C22⋊D5911yescharacteristic
472C22×C4C2×C22⋊D5911yescharacteristic
473C22×C4C2×C22⋊D5911yescharacteristic
474C22×C4C2×C22⋊D5911yescharacteristic
475C22×C4C23×D5911yescharacteristic
476SD16—1188no—
477C2×C8—592no—
478C2×C8—592no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
488C2×C8—1181no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—592no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—4721no—
504C4⋊C4—2362no—
505C4⋊C4—2362no—
506C4⋊C4—2362no—
507C4⋊C4C2×C4×D5912yes—
508C22⋊C4—1184no—
509C22⋊C4—2362no—
510C22⋊C4—2362no—
511C42—24no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—42no—
517C42—42no—
518C42C2×C4×D5912yes—
519C42#1174568512yes—
520C23—1182no—
521C23—1182no—
522C23—1182no—
523C23—2361no—
524C23—2361no—
525C23—2361no—
526C23—2361no—
527C23C22×C22⋊D5911yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532D4—594no—
533D4—1182no—
534D4—1182no—
535D4—1182no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—1182no—
564C2×C4—4724no—
565C2×C4—4724no—
566C2×C4—2368no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4C4×C22⋊D5912yes—
576C2×C4C4×C22⋊D5912yes—
577C2×C4#1427375212yes—
578C2×C4C22×C4×D5911yescharacteristic
579C2×C4#41×D5911yescharacteristic
580C2×C4#41×D5911yescharacteristic
581C2×C4C2×#1174568511yescharacteristic
582C2×C4#1427377311yescharacteristic
583C2×C4#1427377411yescharacteristic
584C8—1182no—
585C8—1182no—
586C8—2361no—
587C8—2361no—
588C22—1184no—
589C22—2362no—
590C22—2362no—
591C22—2362no—
592C22—2362no—
593C22—2362no—
594C22—2362no—
595C22C2×C4×C22⋊D5911yescharacteristic
596C22#1683804011yescharacteristic
597C22#1683805911yescharacteristic
598C22C2×#1427375211yescharacteristic
599C22#1683820011yescharacteristic
600C22#1683821011yescharacteristic
601C22#1683821711yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—81no—
612C4—4721no—
613C4—4721no—
614C4—2362no—
615C4#117×D5911yescharacteristic
616C4#1683788211yescharacteristic
617C4#1683809111yescharacteristic
618C4#1683809611yescharacteristic
619C2—2364no—
620C2#2142702611yescharacteristic
621C2C4×#1427375211yescharacteristic
622C2#2143293911yescharacteristic
623C2#2143611111yescharacteristic
624C2#2143618611yescharacteristic
625C2#2143681011yescharacteristic
626C2#2143691711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2356C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#2356D59
hypocenterC59#22778
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2356

Complements

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

NH ≅ G/Ntype
3. #21424789C2semidirect
4. #21424775C2semidirect
5. #21424688C2semidirect
6. #21424667C2semidirect
9. #21424229C2semidirect
11. #21424101C2semidirect
12. #21423971C2semidirect
15. C59×#2356C2semidirect
21. #16837500C4semidirect
88. #14273660C2×C4semidirect
92. #14273651C2×C4semidirect
97. #14273638C2×C4semidirect
103. C4×Q8×C59C2×C4semidirect
307. #2356D59semidirect
373. C59#22778semidirect
387. C4×Q8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168375002
C4⋊C4×D2362
#214247892
#214247752
#214246882
#214246672
#214244332
#214242292
#214241032
#214241012
#214239712
#214239242
C4×#142736472
C59×#23562
#2277859