Finite Group Databasebeta

← #28596698

Subgroups — #28596698

Order 15104.

890 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837747C211yesmaximal, characteristic
2#21426840C211yesmaximal, characteristic, complemented
3#21426818C211yesmaximal, characteristic
4#21425067C211yesmaximal, characteristic, complemented
5#21424918C211yesmaximal, characteristic
6#21424794C211yesmaximal, characteristic, complemented
7#21424790C211yesmaximal, characteristic, complemented
8#21424781C211yesmaximal, characteristic, complemented
9#21424777C211yesmaximal, characteristic, complemented
10#21424225C211yesmaximal, characteristic, complemented
11#21423924C211yesmaximal, characteristic, complemented
12#21423921C211yesmaximal, characteristic, complemented
13C4×#14273649C211yesmaximal, characteristic, complemented
14#21423782C211yesmaximal, characteristic, complemented
15C4×#102×C59C211yesmaximal, characteristic, complemented
16C2×#14273726C2211yescharacteristic
17C2×#14273723C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19#16837747C418yescomplemented
20C2×#14273649C2211yescharacteristic
21C2×#14273649C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273647C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273638C2211yescharacteristic
27C2×#14273637C2211yescharacteristic
28C2×#14273637C2211yescharacteristic
29C2×C8×Dic59C2211yescharacteristic
30C2×C8×Dic59C2211yescharacteristic
31#16837308—21no—
32#16837306—21no—
33#16837305C2211yescharacteristic
34#16837303C2211yescharacteristic
35#16837301C2211yescharacteristic
36#16837298C2211yescharacteristic
37#16837289—21no—
38#16837286—21no—
39#16837281—21no—
40#16837281—21no—
41#16837278—21no—
42#16837278—21no—
43#16837267C2211yescharacteristic
44#16837267C2211yescharacteristic
45C4×C4⋊Dic59C2211yescharacteristic
46#16837248C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×C4×C8×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118—21no—
61C22×D236C2311yescharacteristic
62C22×C22⋊D59—21no—
63C22×C22⋊D59—21no—
64C22×C4×D59—21no—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273726—22no—
68#14273726C2×C414yescomplemented
69#14273723—22no—
70#14273723C2×C414yes—
71C4×D236—22no—
72C4×D236—22no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59C2311yescharacteristic
86C2×C4×Dic59C2311yescharacteristic
87#14273649—22no—
88#14273649C2×C414yescomplemented
89#14273648—22no—
90#14273648C2×C414yescomplemented
91#14273647—22no—
92#14273647C2×C414yescomplemented
93#14273638—22no—
94#14273637—22no—
95#14273637—22no—
96C8×Dic59—22no—
97C8×Dic59C2×C414yescomplemented
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95D411yescharacteristic
106C59⋊#95D411yescharacteristic
107C59⋊#95D411yescharacteristic
108C59⋊#95D411yescharacteristic
109C22×D4×C59C2311yescharacteristic
110C23×C4×C59—21no—
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C4×D4×C59—22no—
114C4×D4×C59—22no—
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C22⋊C4×C59—21no—
119C2×C42×C59—21no—
120C2×C42×C59C2311yescharacteristic
121#102×C59—22no—
122#102×C59C2×C414yescomplemented
123C4×C8×C59—22no—
124#95×C59—21no—
125C24×C59—21no—
126C23×D59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59C22×C412yes—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—42no—
137C22×C4×C59—81no—
138C22×C4×C59—81no—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2411yescharacteristic
142C2×C22⋊D59—24no—
143C2×C22⋊D59—24no—
144C2×C22⋊D59—81no—
145C2×C22⋊D59—81no—
146C2×C22⋊D59—81no—
147C2×C22⋊D59—81no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C2×D236—21no—
156C2×D236—21no—
157C2×D236—21no—
158C2×D236—21no—
159C2×D236—21no—
160C2×D236C22×C412yes—
161C2×C4×D59—42no—
162C2×C4×D59—81no—
163C2×C4×D59—81no—
164C2×Dic118—24no—
165C2×Dic118—81no—
166C2×Dic118—81no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59C22×C412yes—
172C2×C8×C59#4112yes—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59C22×C412yes—
179C22⋊C4×C59—42no—
180C42×C59—81no—
181C42×C59#4112yes—
182C42×C59#4112yes—
183C22⋊Dic59—81no—
184C22⋊Dic59—81no—
185(C2×D59)⋊C4—42no—
186(C2×D59)⋊C4—81no—
187(C2×D59)⋊C4—81no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—81no—
194C4⋊Dic59C22×C412yes—
195Dic59⋊C4—24no—
196Dic59⋊C4—81no—
197C4×Dic59—21no—
198C4×Dic59—21no—
199C4×Dic59—81no—
200C4×Dic59C22×C412yes—
201C4×Dic59#4112yes—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8C22×C412yes—
207C2×C59⋊C8#4112yes—
208C23×C59—22no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59C22×D411yescharacteristic
216C22×D59—22no—
217C22×D59—41no—
218C22×D59—41no—
219C22×D59—41no—
220C22×D59—41no—
221C22×D59—41no—
222C22×D59—41no—
223D4×C59—22no—
224D4×C59—22no—
225D4×C59—41no—
226D4×C59—41no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—22no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
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—84no—
241C2×C4×C59#12412yes—
242C2×C4×C59#12412yes—
243C2×C4×C59C23×C411yescharacteristic
244C2×C4×C59C2×#4111yescharacteristic
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C22⋊D59—84no—
250C22⋊D59—84no—
251C2×Dic59—21no—
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—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59C4×D412yes—
271C2×Dic59C4×D412yes—
272D236—22no—
273D236—22no—
274D236—41no—
275D236—41no—
276C4×D59—84no—
277Dic118—84no—
278C8×C59—22no—
279C8×C59—22no—
280C59⋊C8—22no—
281C59⋊C8—22no—
282#21644—591nomaximal, Sylow
283C22×C59—42no—
284C22×C59—42no—
285C22×C59—42no—
286C22×C59—81no—
287C22×C59—81no—
288C22×C59—81no—
289C22×C59—81no—
290C22×C59C2×C4×D411yescharacteristic
291C22×C59C2×#12411yescharacteristic
292C22×C59#53211yescharacteristic
293C22×C59#57611yescharacteristic
294C22×C59#57611yescharacteristic
295C22×C59#57711yescharacteristic
296C22×C59#57711yescharacteristic
297C2×D59—42no—
298C2×D59—42no—
299C2×D59—42no—
300C2×D59—81no—
301C2×D59—81no—
302C2×D59—81no—
303C2×D59—81no—
304C4×C59—22no—
305C4×C59—22no—
306C4×C59—42no—
307C4×C59—81no—
308C4×C59—81no—
309C4×C59C4×#4111yescharacteristic
310C4×C59C4×#4111yescharacteristic
311C4×C59#52911yescharacteristic
312C4×C59#52911yescharacteristic
313Dic59—22no—
314Dic59—22no—
315Dic59—41no—
316Dic59—41no—
317Dic59—42no—
318Dic59—81no—
319Dic59—81no—
320C2×#486—591no—
321#2331—591no—
322#2328—591no—
323#1993—591no—
324#1961—591no—
325#1936—591no—
326#1936—591no—
327#1934—591no—
328#1934—591no—
329#1835—591no—
330#1762—591no—
331#1762—591no—
332C4×#102—591no—
333C4×#102D5911yescharacteristic, complemented
334#1726—591no—
335C2×C59—82no—
336C2×C59C4×#12411yescharacteristic
337C2×C59#294811yescharacteristic
338C2×C59#294911yescharacteristic
339C2×C59#313611yescharacteristic
340C2×C59#313611yescharacteristic
341C2×C59#317911yescharacteristic
342C2×C59#317911yescharacteristic
343D59—82no—
344C2×C4⋊Q8—591no—
345C2×#127—591no—
346C2×C4×D4—591no—
347C2×C4×D4C2×D5911yescharacteristic
348#486—598no—
349C2×#102—591no—
350C2×#102—591no—
351C2×#102—591no—
352C2×#102—591no—
353C2×#102—591no—
354C2×#102—591no—
355C2×#102C2×D5911yescharacteristic
356C2×#102C2×D5911yescharacteristic
357C2×#97—591no—
358C2×#97—591no—
359C2×#97—591no—
360C2×C4×C8—591no—
361C2×C4×C8—591no—
362C2×C4×C8C2×D5911yescharacteristic
363#391—1181no—
364#391—1181no—
365#390—591no—
366#390—591no—
367#389—591no—
368#389—591no—
369#387—1181no—
370#387—1181no—
371#386—1181no—
372#386—1181no—
373#386—1181no—
374#386—1181no—
375#382—591no—
376#382—591no—
377C4×C4⋊C4—591no—
378C4×C4⋊C4C2×D5911yescharacteristic
379#376—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386#340C2×D5911yescharacteristic
387#340C2×D5911yescharacteristic
388C59#2164411yescharacteristic, complemented, Sylow
389C22×Q8—1181no—
390C22×D4—591no—
391C22×D4—1181no—
392C22×D4—1181no—
393C22×D4C22×D5911yescharacteristic
394C23×C4—21no—
395C23×C4—1181no—
396C22×C8—591no—
397C22×C8—591no—
398C22×C8C22×D5911yescharacteristic
399C22×C8C22×D5911yescharacteristic
400C4⋊Q8—594no—
401C4⋊Q8—1182no—
402#127—594no—
403#127—1182no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—1182no—
407C4×D4—1182no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4C22×D5911yescharacteristic
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4—1181no—
421C2×C22⋊C4—1181no—
422C2×C22⋊C4—1181no—
423C2×C42—21no—
424C2×C42—591no—
425C2×C42—591no—
426C2×C42—1181no—
427C2×C42C22×D5911yescharacteristic
428#102—22no—
429#102—594no—
430#102—594no—
431#102—594no—
432#102—1182no—
433#102—1182no—
434#102—1182no—
435#102C4×D5914yescomplemented
436#97—1182no—
437#97—1182no—
438#97—1182no—
439C4×C8—22no—
440C4×C8—594no—
441C4×C8—1182no—
442#95—21no—
443#95—591no—
444#95—591no—
445#95—591no—
446#95—591no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453#95—1181no—
454C24—21no—
455C24—1181no—
456C2×Q8—1184no—
457C2×Q8—4721no—
458C2×Q8—4721no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—592no—
465C2×D4—1184no—
466C2×D4—1184no—
467C2×D4—1181no—
468C2×D4—1181no—
469C2×D4—1181no—
470C2×D4—1181no—
471C2×D4—1181no—
472C2×D4—4721no—
473C2×D4—4721no—
474C2×D4—4721no—
475C2×D4—4721no—
476C2×D4C2×C4×D5912yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—81no—
482C22×C4—81no—
483C22×C4—591no—
484C22×C4—591no—
485C22×C4—591no—
486C22×C4—591no—
487C22×C4—1181no—
488C22×C4—1181no—
489C22×C4—1181no—
490C22×C4—4721no—
491C22×C4—4721no—
492C22×C4—2362no—
493C22×C4D4×D5911yescharacteristic
494C22×C4D4×D5911yescharacteristic
495C22×C4C23×D5911yescharacteristic
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—592no—
501C2×C8—592no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8C2×C4×D5912yes—
507C2×C8#1174568512yes—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—592no—
514C4⋊C4—1184no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—4721no—
521C4⋊C4—4721no—
522C4⋊C4C2×C4×D5912yes—
523C22⋊C4—42no—
524C22⋊C4—4721no—
525C22⋊C4—4721no—
526C22⋊C4—4721no—
527C22⋊C4—4721no—
528C22⋊C4—2362no—
529C42—81no—
530C42—592no—
531C42—592no—
532C42—1181no—
533C42—1181no—
534C42—4721no—
535C42#1174568512yes—
536C42#1174568712yes—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—1182no—
545C23—2361no—
546C23—2361no—
547C23—2361no—
548C23—2361no—
549C23—2361no—
550C23—2361no—
551C23C2×D4×D5911yescharacteristic
552Q8—4724no—
553D4—22no—
554D4—22no—
555D4—41no—
556D4—41no—
557D4—1182no—
558D4—1182no—
559D4—4724no—
560D4—4724no—
561D4—2361no—
562D4—2361no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—84no—
577C2×C4—592no—
578C2×C4—592no—
579C2×C4—1181no—
580C2×C4—1181no—
581C2×C4—1181no—
582C2×C4—1181no—
583C2×C4—1181no—
584C2×C4—1181no—
585C2×C4—1181no—
586C2×C4—1181no—
587C2×C4—1181no—
588C2×C4—1181no—
589C2×C4—1182no—
590C2×C4—4724no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4#1427371312yes—
600C2×C4#1427371312yes—
601C2×C4C22×C4×D5911yescharacteristic
602C2×C4#41×D5911yescharacteristic
603C2×C4#41×D5911yescharacteristic
604C2×C4C2×#1174568711yescharacteristic
605C2×C4C2×#1174568511yescharacteristic
606C2×C4C2×#1174568511yescharacteristic
607C8—22no—
608C8—22no—
609C8—1182no—
610C8—1182no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—4721no—
619C22—4721no—
620C22—4721no—
621C22—4721no—
622C22—2362no—
623C22—2362no—
624C22—2362no—
625C22C4×D4×D5911yescharacteristic
626C22C2×#1427371311yescharacteristic
627C22#1683794911yescharacteristic
628C22#1683820311yescharacteristic
629C22#1683820411yescharacteristic
630C22#1683820811yescharacteristic
631C22#1683820911yescharacteristic
632C4—22no—
633C4—22no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4—1182no—
638C4—1182no—
639C4—4721no—
640C4—4721no—
641C4—2362no—
642C4—2361no—
643C4—2361no—
644C4C4×#1174568711yescharacteristic
645C4C4×#1174568511yescharacteristic
646C4#1683793011yescharacteristic
647C4#1683793211yescharacteristic
648C2—82no—
649C2—4722no—
650C2C4×#1427371311yescharacteristic
651C2#2143294311yescharacteristic
652C2#2143295011yescharacteristic
653C2#2143444811yescharacteristic
654C2#2143445411yescharacteristic
655C2#2143477511yescharacteristic
656C2#2143478311yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC4×#102×C59C2
socleC23×C59C22×D4
prime radicalC2×#14273723C22
cosocleC22×C4×C59C24
hypercenterC4×#102D59
hypocenterC59#21644
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. #21426840C2semidirect
4. #21425067C2semidirect
6. #21424794C2semidirect
7. #21424790C2semidirect
8. #21424781C2semidirect
9. #21424777C2semidirect
10. #21424225C2semidirect
11. #21423924C2semidirect
12. #21423921C2semidirect
13. C4×#14273649C2semidirect
14. #21423782C2semidirect
15. C4×#102×C59C2semidirect
19. #16837747C4semidirect
68. #14273726C2×C4semidirect
88. #14273649C2×C4semidirect
90. #14273648C2×C4semidirect
92. #14273647C2×C4semidirect
97. C8×Dic59C2×C4semidirect
122. #102×C59C2×C4semidirect
333. C4×#102D59semidirect
388. C59#21644semidirect
435. #102C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377472
#214268402
#214268182
#214250672
#214249182
#214247942
#214247902
#214247812
#214247772
#214242252
#214239242
#214239212
C4×#142736492
#214237822
C4×#102×C592
#2164459