Finite Group Databasebeta

← #28558804

Subgroups — #28558804

Order 15104.

678 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434094C211yesmaximal, characteristic
2#21433173C211yesmaximal, characteristic, complemented
3#21433117C211yesmaximal, characteristic, complemented
4#21422962C211yesmaximal, characteristic, complemented
5#21422925C211yesmaximal, characteristic, complemented
6#21422923C211yesmaximal, characteristic, complemented
7#21422920C211yesmaximal, characteristic, complemented
8#21422919C211yesmaximal, characteristic, complemented
9#21422888C211yesmaximal, characteristic, complemented
10#21422888C211yesmaximal, characteristic, complemented
11#21422873C211yesmaximal, characteristic, complemented
12#21422537C211yesmaximal, characteristic, complemented
13#21422511C211yesmaximal, characteristic, complemented
14#21422394C211yesmaximal, characteristic, complemented
15C59×#1586C211yesmaximal, characteristic, complemented
16#16837949C2211yescharacteristic
17C4×D4×D59C2211yescharacteristic
18#16837717C411yescharacteristic
19#16837714C411yescharacteristic
20#16837681C411yescharacteristic
21#16837680C411yescharacteristic
22#16837596C411yescharacteristic
23#16837595C411yescharacteristic
24#16837537C411yescharacteristic
25#16837534C411yescharacteristic
26C4×D472C2211yescharacteristic
27C4×#11745669C2211yescharacteristic
28C4×#11745669C2211yescharacteristic
29#16837481C2211yescharacteristic, complemented
30#16837471—21no—
31#16837470—21no—
32C8×D236C2211yescharacteristic
33D4×C59⋊C8C2211yescharacteristic
34D4×C59⋊C8C2211yescharacteristic, complemented
35#16837467C2211yescharacteristic
36#16837467C2211yescharacteristic, complemented
37C8×C22⋊D59—21no—
38(C22⋊D59)⋊C8—21no—
39#16837458C2211yescharacteristic
40#16837454C2211yescharacteristic
41#16837453C2211yescharacteristic
42#16837453C2211yescharacteristic
43#16837370—21no—
44#16837367—21no—
45C2×(C2×D59)⋊C8—21no—
46C22⋊C8×D59—21no—
47C4×#11745659C2211yescharacteristic
48#16837154C2211yescharacteristic, complemented
49#16837154C2211yescharacteristic, complemented
50#16837153C2211yescharacteristic, complemented
51#16837131C2211yescharacteristic
52#16837131C2211yescharacteristic, complemented
53D4⋊(C59⋊C8)C2211yescharacteristic
54D4⋊(C59⋊C8)C2211yescharacteristic, complemented
55#16837129C2211yescharacteristic
56#16837129C2211yescharacteristic, complemented
57#16837123C2211yescharacteristic, complemented
58#16837123C2211yescharacteristic, complemented
59C8×C59⋊C8C2211yescharacteristic, complemented
60C4×D8×C59C2211yescharacteristic
61C59×#436C2211yescharacteristic, complemented
62C8×D4×C59C2211yescharacteristic
63C59×#334C2211yescharacteristic, complemented
64C59×D4⋊C8C2211yescharacteristic
65C59×D4⋊C8C2211yescharacteristic, complemented
66C59×#322C2211yescharacteristic, complemented
67C2×#11745685—21no—
68C2×D4×D59C2×C411yescharacteristic
69C22×C4×D59—21no—
70C2×D472C2×C411yescharacteristic
71C2×#11745669C2×C411yescharacteristic
72C2×#11745669C2×C411yescharacteristic
73C2×#11745659—21no—
74C2×#11745659—21no—
75C22×C59⋊C8—21no—
76C22×C59⋊C8—21no—
77C2×C8×D59—21no—
78C2×C8×D59—21no—
79C2×C8×D59—21no—
80#14273726C2×C411yescharacteristic
81#14273723C2×C411yescharacteristic
82#14273713—21no—
83#14273702C2×C411yescharacteristic
84#14273700—21no—
85#14273698C2×C411yescharacteristic
86C4×D236C2311yescharacteristic
87C4×D236C2311yescharacteristic
88D4×Dic59C2×C411yescharacteristic
89#14273689C2×C411yescharacteristic
90C4×C22⋊D59—21no—
91C4×C22⋊D59—21no—
92(C22⋊D59)⋊C4—21no—
93C4⋊C4×D59C2×C411yescharacteristic
94C22⋊C4×D59—21no—
95C42×D59C2311yescharacteristic
96#14273661C2×C411yescharacteristic
97#14273660C2×C411yescharacteristic
98#14273660C2×C411yescharacteristic
99C4⋊(C59⋊C8)C2311yescharacteristic
100C4⋊(C59⋊C8)C2311yescharacteristic
101Dic59⋊C8C2×C411yescharacteristic
102Dic59⋊C8C2×C411yescharacteristic
103Dic59⋊C8D411yescharacteristic
104Dic59⋊C8D411yescharacteristic
105#14273649C2×C411yescharacteristic
106#14273649C2×C411yescharacteristic
107#14273648C2×C411yescharacteristic
108#14273648C2×C411yescharacteristic
109#14273647C2×C411yescharacteristic
110#14273647C2×C411yescharacteristic
111C22⋊(C59⋊C8)—21no—
112C22⋊(C59⋊C8)—21no—
113(C2×D59)⋊C8—21no—
114(C2×D59)⋊C8—21no—
115(C2×D59)⋊C8—21no—
116(C2×D59)⋊C8—21no—
117(C2×D59)⋊C8—21no—
118(C2×D59)⋊C8—21no—
119(C2×D59)⋊C8C2×C411yescharacteristic
120(C2×D59)⋊C8C2×C411yescharacteristic
121(C2×D59)⋊C8D411yescharacteristic
122(C2×D59)⋊C8D411yescharacteristic
123#14273637—21no—
124#14273637—21no—
125C4×C59⋊C8C2311yescharacteristic
126C4×C59⋊C8C2311yescharacteristic
127C4×C59⋊C8C2311yescharacteristic
128C8×Dic59—21no—
129C2×D8×C59C2×C411yescharacteristic
130C2×M16×C59—21no—
131C22×C8×C59—21no—
132C4×D4×C59C2311yescharacteristic
133C4×D4×C59C2311yescharacteristic
134#107×C59C2×C411yescharacteristic
135C4⋊C8×C59C2311yescharacteristic
136C4⋊C8×C59C2311yescharacteristic
137#102×C59C2×C411yescharacteristic
138#102×C59C2×C411yescharacteristic
139C22⋊C8×C59—21no—
140C22⋊C8×C59—21no—
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C23×D59—21no—
145C2×D4×C59C22×C411yescharacteristic
146C2×D4×C59C22×C411yescharacteristic
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C2×C22⋊D59—21no—
150C2×C22⋊D59—21no—
151C2×C22⋊D59—21no—
152C2×C22⋊D59—21no—
153C22×Dic59—21no—
154D4×D59—22no—
155D4×D59—22no—
156#11745685—81no—
157#11745685—81no—
158C2×D236—21no—
159C2×D236C22×C411yescharacteristic
160C2×D236C22×C411yescharacteristic
161C2×C4×D59—21no—
162C2×C4×D59—21no—
163C2×C4×D59—21no—
164C2×C4×D59—21no—
165C2×C4×D59—81no—
166C2×C4×D59—81no—
167C2×C4×D59C22×C411yescharacteristic
168C2×C4×D59C2×D411yescharacteristic
169C2×C4×D59#4111yescharacteristic
170C2×Dic118—21no—
171D8×C59—22no—
172M16×C59—81no—
173C2×C8×C59—21no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59—81no—
177C2×C8×C59C22×C411yescharacteristic
178C2×C8×C59C22×C411yescharacteristic
179C2×C8×C59C2×D411yescharacteristic
180C2×C8×C59C2×D411yescharacteristic
181C4⋊C4×C59C22×C411yescharacteristic
182C4⋊C4×C59C22×C411yescharacteristic
183C22⋊C4×C59—21no—
184C22⋊C4×C59—21no—
185C42×C59C2411yescharacteristic
186C22⋊Dic59—21no—
187C22⋊Dic59—21no—
188#11745669—22no—
189#11745669—22no—
190(C2×D59)⋊C4—21no—
191(C2×D59)⋊C4—21no—
192(C2×D59)⋊C4—21no—
193(C2×D59)⋊C4—21no—
194C4⋊Dic59C22×C411yescharacteristic
195C4⋊Dic59C22×C411yescharacteristic
196Dic59⋊C4—21no—
197Dic59⋊C4—21no—
198C4×Dic59C22×C411yescharacteristic
199C4×Dic59C2×D411yescharacteristic
200C4×Dic59#4111yescharacteristic
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—81no—
205C2×C59⋊C8—81no—
206C2×C59⋊C8C22×C411yescharacteristic
207C2×C59⋊C8C22×C411yescharacteristic
208C2×C59⋊C8C2×D411yescharacteristic
209C2×C59⋊C8C2×D411yescharacteristic
210D472—22no—
211#11745659—42no—
212#11745659—81no—
213C8×D59—81no—
214C8×D59—81no—
215C8×D59—81no—
216C23×C59—21no—
217C23×C59—21no—
218C22×D59—21no—
219C22×D59—24no—
220C22×D59—21no—
221C22×D59—81no—
222C22×D59—81no—
223C22×D59C4×D411yescharacteristic
224D4×C59—21no—
225D4×C59—21no—
226D4×C59—21no—
227D4×C59#13112yes—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59C23×C411yescharacteristic
235C2×C4×C59C22×D411yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C22⋊D59—24no—
238C22⋊D59—81no—
239C22⋊D59—81no—
240C22⋊D59—81no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—81no—
244C2×Dic59C4×D411yescharacteristic
245C2×Dic59C4×D411yescharacteristic
246C2×Dic59C4×D411yescharacteristic
247D236—21no—
248D236—21no—
249D236—21no—
250D236—81no—
251D236#13112yes—
252C4×D59—21no—
253C4×D59—21no—
254C4×D59—21no—
255C4×D59—21no—
256C4×D59—41no—
257C4×D59—41no—
258C4×D59—41no—
259C4×D59—41no—
260C4×D59—41no—
261C4×D59—81no—
262C4×D59—81no—
263Dic118—81no—
264C8×C59—22no—
265C8×C59—21no—
266C8×C59—21no—
267C8×C59—21no—
268C8×C59—21no—
269C8×C59—41no—
270C8×C59—41no—
271C59⋊C8—21no—
272C59⋊C8—21no—
273C59⋊C8—21no—
274C59⋊C8—21no—
275C59⋊C8—41no—
276C59⋊C8—41no—
277C59⋊C8—41no—
278#17922—591nomaximal, Sylow
279C22×C59—22no—
280C22×C59—41no—
281C22×C59—41no—
282C22×C59—41no—
283C22×C59—41no—
284C22×C59—41no—
285C22×C59C2×C4×D411yescharacteristic
286C2×D59—21no—
287C2×D59—22no—
288C2×D59—21no—
289C2×D59—21no—
290C2×D59—41no—
291C2×D59—41no—
292C2×D59—41no—
293C2×D59—41no—
294C2×D59—41no—
295C2×D59—84no—
296C4×C59—21no—
297C4×C59—41no—
298C4×C59—41no—
299C4×C59C2×#13111yescharacteristic
300C4×C59C2×#13111yescharacteristic
301C4×C59#57611yescharacteristic
302C4×C59#57611yescharacteristic
303Dic59—21no—
304Dic59—21no—
305Dic59—21no—
306Dic59—41no—
307Dic59—41no—
308#3100—591no—
309#2981—591no—
310#2972—591no—
311#1586—591no—
312#1586D5911yescharacteristic, complemented
313#1581—591no—
314#1581—591no—
315#1580—591no—
316#1580—591no—
317C8×D8—591no—
318C8×D8—591no—
319#1572—591no—
320#1490—591no—
321#1484—591no—
322#1453—591no—
323C2×C59—42no—
324C2×C59—81no—
325C2×C59#292911yescharacteristic
326C2×C59#294811yescharacteristic
327C2×C59#295611yescharacteristic
328D59—41no—
329D59—42no—
330D59—81no—
331#532—591no—
332C2×C4×D4—591no—
333#480—591no—
334#480—591no—
335#474—591no—
336#474—591no—
337#459—591no—
338#459—591no—
339#447—591no—
340#447—591no—
341C4×D8—591no—
342C4×D8—591no—
343C4×D8—591no—
344C4×D8C2×D5911yescharacteristic
345#436—591no—
346#436C2×D5911yescharacteristic, complemented
347#435—1181no—
348#435—1181no—
349C8×D4—591no—
350C8×D4—591no—
351C8×D4—591no—
352C8×D4—591no—
353C8×D4—591no—
354C8×D4—1181no—
355C8×D4—1181no—
356C8×D4C2×D5911yescharacteristic
357#432—591no—
358#431—591no—
359#431—591no—
360#431—591no—
361#408—1181no—
362#408—1181no—
363C2×C22⋊C8—1181no—
364C2×C22⋊C8—1181no—
365C4×M16—591no—
366#335—591no—
367#335—591no—
368#334—591no—
369#334C2×D5911yescharacteristic, complemented
370D4⋊C8—591no—
371D4⋊C8—591no—
372D4⋊C8—591no—
373D4⋊C8—591no—
374D4⋊C8—591no—
375D4⋊C8—591no—
376D4⋊C8C2×D5911yescharacteristic
377D4⋊C8C2×D5911yescharacteristic, complemented
378#322—591no—
379#322—591no—
380#322C2×D5911yescharacteristic, complemented
381C82—591no—
382C59#1792211yescharacteristic, complemented, Sylow
383C2×#41—1181no—
384C22×D4—591no—
385C23×C4—1181no—
386C2×D8—591no—
387C2×D8—591no—
388C2×D8—591no—
389C2×D8C4×D5911yescharacteristic
390C2×M16—21no—
391C2×M16—1181no—
392C2×M16—1181no—
393C22×C8—21no—
394C22×C8—1181no—
395C22×C8—1181no—
396C22×C8—1181no—
397C22×C8—1181no—
398C22×C8—1181no—
399C4⋊Q8—591no—
400#127—591no—
401#124—1181no—
402#121—591no—
403#121—591no—
404#121—1181no—
405C4×D4—591no—
406C4×D4—591no—
407C4×D4—591no—
408C4×D4—591no—
409C4×D4—1181no—
410C4×D4—1181no—
411C4×D4—1181no—
412C4×D4C22×D5911yescharacteristic
413C4×D4C22×D5911yescharacteristic
414C2×C4⋊C4—591no—
415C2×C22⋊C4—1181no—
416C2×C42—591no—
417#107—591no—
418#107—591no—
419#107—591no—
420#107C4×D5911yescharacteristic
421C4⋊C8—591no—
422C4⋊C8—591no—
423C4⋊C8—591no—
424C4⋊C8—591no—
425C4⋊C8—591no—
426C4⋊C8—591no—
427C4⋊C8C22×D5911yescharacteristic
428C4⋊C8C22×D5911yescharacteristic
429#102—591no—
430#102—591no—
431#102—591no—
432#102—591no—
433#102—591no—
434#102—591no—
435#102C4×D5911yescharacteristic
436#102C4×D5911yescharacteristic
437C22⋊C8—21no—
438C22⋊C8—21no—
439C22⋊C8—591no—
440C22⋊C8—591no—
441C22⋊C8—591no—
442C22⋊C8—591no—
443C22⋊C8—1181no—
444C22⋊C8—1181no—
445C22⋊C8—1181no—
446C22⋊C8—1181no—
447C22⋊C8—1181no—
448C22⋊C8—1181no—
449C22⋊C8—1181no—
450C22⋊C8—1181no—
451#97—1181no—
452#97—1181no—
453C4×C8—591no—
454C4×C8—591no—
455C4×C8—591no—
456C4×C8—1181no—
457C4×C8C22×D5911yescharacteristic
458C4×C8C22×D5911yescharacteristic
459C4×C8C22×D5911yescharacteristic
460C24—1181no—
461#41—4721no—
462#41—4721no—
463C2×Q8—1181no—
464C2×D4—591no—
465C2×D4—591no—
466C2×D4—1181no—
467C2×D4—1181no—
468C2×D4—1181no—
469C2×D4—1181no—
470C2×D4—1182no—
471C2×D4—1182no—
472C2×D4—1181no—
473C2×D4C2×C4×D5911yescharacteristic
474C2×D4C2×C4×D5911yescharacteristic
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—591no—
478C22×C4—591no—
479C22×C4—591no—
480C22×C4—1181no—
481C22×C4—1181no—
482C22×C4—1181no—
483C22×C4—1181no—
484C22×C4—1181no—
485C22×C4—4721no—
486C22×C4—4721no—
487D8—22no—
488D8—1182no—
489D8—1182no—
490D8—1182no—
491M16—81no—
492M16—4721no—
493M16—2362no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—81no—
498C2×C8—591no—
499C2×C8—591no—
500C2×C8—591no—
501C2×C8—591no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—4721no—
506C2×C8—4721no—
507C2×C8—4721no—
508C2×C8—4721no—
509C2×C8—4721no—
510C2×C8C2×C4×D5911yescharacteristic
511C2×C8C2×C4×D5911yescharacteristic
512C2×C8D4×D5911yescharacteristic
513C2×C8D4×D5911yescharacteristic
514C4⋊C4—591no—
515C4⋊C4—591no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4C2×C4×D5911yescharacteristic
519C4⋊C4C2×C4×D5911yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—1181no—
523C22⋊C4—1181no—
524C22⋊C4—1181no—
525C22⋊C4—1181no—
526C22⋊C4—1181no—
527C22⋊C4—1181no—
528C42—591no—
529C42—591no—
530C42—591no—
531C42C23×D5911yescharacteristic
532C23—21no—
533C23—21no—
534C23—591no—
535C23—1184no—
536C23—1181no—
537C23—1181no—
538C23—4721no—
539C23—4721no—
540Q8—4721no—
541D4—21no—
542D4—21no—
543D4—21no—
544D4—592no—
545D4—1184no—
546D4—1181no—
547D4—1181no—
548D4—1181no—
549D4—4721no—
550D4—4721no—
551D4—4721no—
552D4—4721no—
553D4#1427373512yes—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—591no—
561C2×C4—591no—
562C2×C4—591no—
563C2×C4—1181no—
564C2×C4—1181no—
565C2×C4—1181no—
566C2×C4—1181no—
567C2×C4—1181no—
568C2×C4—1181no—
569C2×C4—4721no—
570C2×C4—4721no—
571C2×C4—4721no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4C22×C4×D5911yescharacteristic
578C2×C4C2×D4×D5911yescharacteristic
579C2×C4#41×D5911yescharacteristic
580C8—22no—
581C8—21no—
582C8—21no—
583C8—21no—
584C8—21no—
585C8—41no—
586C8—41no—
587C8—1181no—
588C8—1181no—
589C8—1181no—
590C8—1181no—
591C8—2361no—
592C8—2361no—
593C8—2361no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—1181no—
601C22—1181no—
602C22—1181no—
603C22—1182no—
604C22—4724no—
605C22—2361no—
606C22—2361no—
607C22—2361no—
608C22—2361no—
609C22—2361no—
610C22C4×D4×D5911yescharacteristic
611C4—21no—
612C4—41no—
613C4—41no—
614C4—1181no—
615C4—1181no—
616C4—1181no—
617C4—2361no—
618C4—2361no—
619C4#131×D5911yescharacteristic
620C4C2×#1427373511yescharacteristic
621C4#1683820611yescharacteristic
622C4#1683820611yescharacteristic
623C2—42no—
624C2—81no—
625C2—4721no—
626C2—2361no—
627C2—2362no—
628C2#2143278711yescharacteristic
629C2#2143294511yescharacteristic
630C2#2143300111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1586C2
socleC22×C59C2×C4×D4
prime radical#21434094C2
cosocleC42×C59C24
hypercenter#1586D59
hypocenterC59#17922
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1586

Complements

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

NH ≅ G/Ntype
2. #21433173C2semidirect
3. #21433117C2semidirect
4. #21422962C2semidirect
5. #21422925C2semidirect
6. #21422923C2semidirect
7. #21422920C2semidirect
8. #21422919C2semidirect
9. #21422888C2semidirect
10. #21422888C2semidirect
11. #21422873C2semidirect
12. #21422537C2semidirect
13. #21422511C2semidirect
14. #21422394C2semidirect
15. C59×#1586C2semidirect
29. #16837481C22semidirect
34. D4×C59⋊C8C22semidirect
36. #16837467C22semidirect
48. #16837154C22semidirect
49. #16837154C22semidirect
50. #16837153C22semidirect
52. #16837131C22semidirect
54. D4⋊(C59⋊C8)C22semidirect
56. #16837129C22semidirect
57. #16837123C22semidirect
58. #16837123C22semidirect
59. C8×C59⋊C8C22semidirect
61. C59×#436C22semidirect
63. C59×#334C22semidirect
65. C59×D4⋊C8C22semidirect
66. C59×#322C22semidirect
312. #1586D59semidirect
346. #436C2×D59semidirect
369. #334C2×D59semidirect
377. D4⋊C8C2×D59semidirect
380. #322C2×D59semidirect
382. C59#17922semidirect

Maximal subgroups

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

subgroupindex
#214340942
#214331732
#214331172
#214229622
#214229252
#214229232
#214229202
#214229192
#214228882
#214228732
#214225372
#214225112
#214223942
C59×#15862
#1792259