Finite Group Databasebeta

← #28636863

Subgroups — #28636863

Order 15104.

806 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837740C211yesmaximal, characteristic
2#21428550C211yesmaximal, characteristic
3#21427127C211yesmaximal, characteristic
4#21425570C211yesmaximal, characteristic, complemented
5#21425517C211yesmaximal, characteristic, complemented
6#21425473C211yesmaximal, characteristic, complemented
7#21425416C211yesmaximal, characteristic, complemented
8#21424861C211yesmaximal, characteristic, complemented
9#21424740C211yesmaximal, characteristic, complemented
10#21424439C211yesmaximal, characteristic, complemented
11#21424409C211yesmaximal, characteristic
12#21424274C211yesmaximal, characteristic
13#21424099C211yesmaximal, characteristic, complemented
14#21423813C211yesmaximal, characteristic
15C2×C59×#467C211yesmaximal, characteristic
16C2×Q8×Dic59C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19#16837740—24no—
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×Dic59⋊C8C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29#16837330—21no—
30#16837330—21no—
31#16837309C2211yescharacteristic
32#16837309C2211yescharacteristic
33#16837309C2211yescharacteristic
34#16837301C2211yescharacteristic
35#16837295—21no—
36#16837295—21no—
37#16837288—21no—
38#16837283—21no—
39#16837283—21no—
40#16837271C2211yescharacteristic
41#16837267C2211yescharacteristic
42C4⋊C4×Dic59—21no—
43C4⋊C4×Dic59C2211yescharacteristic
44C4×C22⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C22×Q16×C59C2211yescharacteristic
54C23×C8×C59C2211yescharacteristic
55C2×C22⋊Q8×C59C2211yescharacteristic
56C2×C22⋊Q8×C59C2211yescharacteristic
57C59×#467—24no—
58C2×#107×C59C2211yescharacteristic
59C2×#103×C59C2211yescharacteristic
60C2×#103×C59C2211yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63Q8×Dic59—22no—
64Q8×Dic59—41no—
65#14273686—22no—
66#14273686—22no—
67#14273686—22no—
68#14273686—41no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
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×C22⋊Dic59—21no—
79C2×C22⋊Dic59D411yescharacteristic
80C2×C22⋊Dic59D411yescharacteristic
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59C2311yescharacteristic
86C2×C4×Dic59C2311yescharacteristic
87#14273660—22no—
88#14273658—22no—
89Dic59⋊C8—22no—
90Dic59⋊C8—41no—
91#14273651—22no—
92#14273651—22no—
93#14273651—22no—
94#14273651—41no—
95C22⋊(C59⋊C8)—22no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95D411yescharacteristic
107C59⋊#95D411yescharacteristic
108C22×Q8×C59C2311yescharacteristic
109C22×Q8×C59C2311yescharacteristic
110C23×C4×C59C2311yescharacteristic
111C2×Q16×C59—22no—
112C2×Q16×C59—22no—
113C2×Q16×C59—41no—
114C22×C8×C59—22no—
115C22×C8×C59—22no—
116C22×C8×C59—41no—
117C22×C8×C59C2311yescharacteristic
118C22×C8×C59C2311yescharacteristic
119C22⋊Q8×C59—22no—
120C22⋊Q8×C59—22no—
121C22⋊Q8×C59—22no—
122C22⋊Q8×C59D414yes—
123C2×C4⋊C4×C59—21no—
124C2×C4⋊C4×C59—21no—
125C2×C4⋊C4×C59C2311yescharacteristic
126C2×C4⋊C4×C59C2311yescharacteristic
127C2×C22⋊C4×C59—21no—
128C2×C22⋊C4×C59—21no—
129#107×C59—22no—
130#103×C59—22no—
131#103×C59—22no—
132C24×C59C2×D411yescharacteristic
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—21no—
142C2×Q8×C59—21no—
143C2×Q8×C59—21no—
144C2×Q8×C59C2×D412yes—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59—21no—
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D412yes—
155C22×C4×C59#4112yes—
156C22×C4×C59C2411yescharacteristic
157C22×Dic59—21no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59C2×D411yescharacteristic
162C22×Dic59C2×D411yescharacteristic
163C22×Dic59C2×D411yescharacteristic
164C22×Dic59C2×D411yescharacteristic
165Q16×C59—44no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59—44no—
172C2×C8×C59#4112yes—
173C4⋊C4×C59—24no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—42no—
178C4⋊C4×C59C2×D412yes—
179C22⋊C4×C59—24no—
180C22⋊C4×C59—42no—
181C22⋊Dic59—22no—
182C22⋊Dic59—41no—
183C22⋊Dic59—42no—
184C22⋊Dic59—42no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—42no—
193C4⋊Dic59—42no—
194C4⋊Dic59#4112yes—
195Dic59⋊C4—42no—
196Dic59⋊C4—42no—
197C4×Dic59—21no—
198C4×Dic59—21no—
199C4×Dic59—21no—
200C4×Dic59—21no—
201C4×Dic59—42no—
202C4×Dic59—42no—
203C4×Dic59—42no—
204C4×Dic59—42no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59#12112yes—
218C23×C59C22×D411yescharacteristic
219Q8×C59—22no—
220Q8×C59—22no—
221Q8×C59—41no—
222Q8×C59—41no—
223Q8×C59—41no—
224Q8×C59—41no—
225Q8×C59—41no—
226Q8×C59—41no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
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×C59#12112yes—
246C2×C4×C59C22×D411yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59H(5,𝔽2)11yescharacteristic
251C2×C4×C592−1+411yescharacteristic
252C2×Dic59—21no—
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—22no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—21no—
263C2×Dic59—21no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59#13512yes—
276C8×C59—22no—
277C8×C59—41no—
278C59⋊C8—41no—
279C59⋊C8—41no—
280#25201—591nomaximal, Sylow
281C22×C59—22no—
282C22×C59—22no—
283C22×C59—22no—
284C22×C59—41no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59C2×#12111yescharacteristic
289C22×C59#54611yescharacteristic
290C22×C59#54711yescharacteristic
291C22×C59C2×#13511yescharacteristic
292C22×C59#57511yescharacteristic
293C22×C59#57811yescharacteristic
294C22×C59#57811yescharacteristic
295C4×C59—22no—
296C4×C59—42no—
297C4×C59—42no—
298C4×C59—81no—
299C4×C59—81no—
300C4×C59#52911yescharacteristic
301C4×C59#53911yescharacteristic
302C4×C59#54611yescharacteristic
303C4×C59#54711yescharacteristic
304Dic59—22no—
305Dic59—41no—
306Dic59—41no—
307Dic59—41no—
308Dic59—42no—
309Dic59—81no—
310Dic59—81no—
311Dic59—81no—
312C2×#485—591no—
313C2×#467D5911yescharacteristic
314#2499—591no—
315#2362—591no—
316#2083—591no—
317#2074—591no—
318#2066—591no—
319#2056—591no—
320#1949—591no—
321#1928—591no—
322#1878—591no—
323#1873—591no—
324#1850—591no—
325#1803—591no—
326#1735—591no—
327C2×C59—42no—
328C2×C59#242511yescharacteristic
329C2×C59#303511yescharacteristic
330C2×C59#304411yescharacteristic
331C2×C59#330011yescharacteristic
332C2×C59#331511yescharacteristic
333C2×C59#334211yescharacteristic
334C2×C59#334911yescharacteristic
335C22×Q16C2×D5911yescharacteristic
336C23×C8C2×D5911yescharacteristic
337C2×C22⋊Q8C2×D5911yescharacteristic
338C2×C22⋊Q8C2×D5911yescharacteristic
339C2×C4×Q8—591no—
340C2×#117—591no—
341C2×#117—591no—
342#485—1184no—
343#467—24no—
344C2×#107—591no—
345C2×#107C2×D5911yescharacteristic
346C2×#106—591no—
347C2×C4⋊C8—591no—
348C2×C4⋊C8—591no—
349C2×#103—591no—
350C2×#103—591no—
351C2×#103—591no—
352C2×#103—591no—
353C2×#103C2×D5911yescharacteristic
354C2×#103C2×D5911yescharacteristic
355C2×C22⋊C8—591no—
356#397—1181no—
357#397—1181no—
358#391—591no—
359#391—591no—
360#391—591no—
361#389—591no—
362#388—1181no—
363#388—1181no—
364#387—1181no—
365#387—1181no—
366#387—1181no—
367#384—591no—
368#382—591no—
369C4×C4⋊C4—591no—
370C4×C4⋊C4—1181no—
371C4×C22⋊C4—591no—
372#340—591no—
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#340—591no—
378#336—591no—
379#336—591no—
380C59#2520111yescharacteristic, complemented, Sylow
381C22×Q8C22×D5911yescharacteristic
382C22×Q8C22×D5911yescharacteristic
383C23×C4C22×D5911yescharacteristic
384C2×Q16—22no—
385C2×Q16—22no—
386C2×Q16—41no—
387C22×C8—22no—
388C22×C8—22no—
389C22×C8—41no—
390C22×C8—591no—
391C22×C8—591no—
392C22×C8C22×D5911yescharacteristic
393C22×C8C22×D5911yescharacteristic
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8C22⋊D5914yes—
398C4×Q8—1182no—
399C4×Q8—2361no—
400#117—1182no—
401#117—1182no—
402#117—1182no—
403#117—2361no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4C22×D5911yescharacteristic
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—591no—
419C2×C22⋊C4—591no—
420C2×C22⋊C4—1181no—
421C2×C22⋊C4—1181no—
422C2×C42—591no—
423C2×C42—591no—
424C2×C42—1181no—
425C2×C42—1181no—
426C2×C42—1181no—
427C2×C42—1181no—
428#107—22no—
429#107—1182no—
430#106—1182no—
431C4⋊C8—1182no—
432C4⋊C8—2361no—
433#103—22no—
434#103—22no—
435#103—1182no—
436#103—1182no—
437#103—1182no—
438#103—2361no—
439C22⋊C8—1182no—
440#95—591no—
441#95—591no—
442#95—1181no—
443#95—1181no—
444#95—1181no—
445#95—1181no—
446#95—1181no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452C24D4×D5911yescharacteristic
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C2×C22⋊D5912yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—591no—
474C22×C4—591no—
475C22×C4—591no—
476C22×C4—591no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—1181no—
480C22×C4—1181no—
481C22×C4D4×D5911yescharacteristic
482C22×C4#1174568712yes—
483C22×C4C2×C22⋊D5912yes—
484C22×C4C23×D5911yescharacteristic
485Q16—44no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—44no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8#1174568712yes—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—592no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—1181no—
512C4⋊C4—2362no—
513C4⋊C4—2362no—
514C4⋊C4—2362no—
515C4⋊C4—2362no—
516C4⋊C4C2×C22⋊D5912yes—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—1182no—
520C22⋊C4—2361no—
521C22⋊C4—2362no—
522C22⋊C4—2362no—
523C42—1181no—
524C42—1181no—
525C42—1181no—
526C42—1181no—
527C42—2362no—
528C42—2362no—
529C42—2362no—
530C42—2362no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23#1427369912yes—
538C23C2×D4×D5911yescharacteristic
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
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—41no—
564C2×C4—41no—
565C2×C4—592no—
566C2×C4—1181no—
567C2×C4—1181no—
568C2×C4—1181no—
569C2×C4—1181no—
570C2×C4—1181no—
571C2×C4—1181no—
572C2×C4—1181no—
573C2×C4—1181no—
574C2×C4—1181no—
575C2×C4—1182no—
576C2×C4—1181no—
577C2×C4—1181no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4#1427369912yes—
590C2×C4C22×C22⋊D5911yescharacteristic
591C2×C4#41×D5911yescharacteristic
592C2×C4C2×#1174568711yescharacteristic
593C2×C4C2×#1174568711yescharacteristic
594C2×C4#1427377211yescharacteristic
595C2×C4#1427377411yescharacteristic
596C8—22no—
597C8—41no—
598C8—2361no—
599C8—2361no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22C2×#1427369911yescharacteristic
608C22#1683804111yescharacteristic
609C22#1683805211yescharacteristic
610C22#135×D5911yescharacteristic
611C22#1683819911yescharacteristic
612C22#1683821511yescharacteristic
613C22#1683821511yescharacteristic
614C4—22no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—81no—
619C4—1182no—
620C4—4721no—
621C4—4721no—
622C4—4721no—
623C4—2361no—
624C4—2362no—
625C4—2361no—
626C4—2361no—
627C4#1683793311yescharacteristic
628C4#1683799711yescharacteristic
629C4#1683804611yescharacteristic
630C4#1683805111yescharacteristic
631C2—42no—
632C2#2142765311yescharacteristic
633C2#2143354711yescharacteristic
634C2#2143362311yescharacteristic
635C2#2143597811yescharacteristic
636C2#2143619811yescharacteristic
637C2#2143652311yescharacteristic
638C2#2143663211yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#467C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenterC2×#467D59
hypocenterC59#25201
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
4. #21425570C2semidirect
5. #21425517C2semidirect
6. #21425473C2semidirect
7. #21425416C2semidirect
8. #21424861C2semidirect
9. #21424740C2semidirect
10. #21424439C2semidirect
13. #21424099C2semidirect
380. C59#25201semidirect

Maximal subgroups

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

subgroupindex
C2×#168377402
#214285502
#214271272
#214255702
#214255172
#214254732
#214254162
#214248612
#214247402
#214244392
#214244092
#214242742
#214240992
#214238132
C2×C59×#4672
#2520159