Finite Group Databasebeta

← #28596536

Subgroups — #28596536

Order 15104.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433886C211yesmaximal, characteristic
2#21433033C211yesmaximal, characteristic, complemented
3#21432525C211yesmaximal, characteristic
4#21425040C211yesmaximal, characteristic, complemented
5#21424694C211yesmaximal, characteristic, complemented
6#21424694C211yesmaximal, characteristic, complemented
7#21424368C211yesmaximal, characteristic
8#21424368C211yesmaximal, characteristic
9#21424113C211yesmaximal, characteristic
10#21423983C211yesmaximal, characteristic, complemented
11#21423978C211yesmaximal, characteristic, complemented
12#21423973C211yesmaximal, characteristic, complemented
13#21423939C211yesmaximal, characteristic
14#21423932C211yesmaximal, characteristic
15C59×#1950C211yesmaximal, characteristic, complemented
16C2×#14273735C2211yescharacteristic
17C2×#14273735C2211yescharacteristic
18#16837959C2211yescharacteristic
19C4×#11745685C2211yescharacteristic
20#16837710C411yescharacteristic, complemented
21#16837710C411yescharacteristic, complemented
22#16837703C411yescharacteristic, complemented
23#16837703C411yescharacteristic, complemented
24#16837698C411yescharacteristic, complemented
25#16837693C411yescharacteristic, complemented
26#16837688C411yescharacteristic, complemented
27#16837683C411yescharacteristic, complemented
28#16837531—21no—
29#16837531—21no—
30#16837531—21no—
31#16837531—21no—
32#16837440C2211yescharacteristic
33#16837440C2211yescharacteristic
34#16837430C2211yescharacteristic
35#16837430C2211yescharacteristic
36C2×#14273653C2211yescharacteristic
37C2×#14273653C2211yescharacteristic
38#16837399C2211yescharacteristic
39#16837399C2211yescharacteristic
40#16837360C2211yescharacteristic
41#16837360C2211yescharacteristic
42#16837360C2211yescharacteristic
43#16837360C2211yescharacteristic
44#16837356—21no—
45#16837356—21no—
46#16837354—21no—
47C2×#14273638C2211yescharacteristic
48C2×#14273637C2211yescharacteristic
49#97×D59—21no—
50C2×C8×Dic59C2211yescharacteristic
51C4×C4⋊Dic59C2211yescharacteristic
52#16837163C2211yescharacteristic
53#16837163C2211yescharacteristic
54#16837163C2211yescharacteristic
55#16837163C2211yescharacteristic
56#16837163C2211yescharacteristic
57#16837163C2211yescharacteristic
58#16837156C2211yescharacteristic
59#16837156C2211yescharacteristic
60C2×#108×C59C2211yescharacteristic
61C2×#108×C59C2211yescharacteristic
62C59×#423C2211yescharacteristic
63C2×C4⋊C8×C59C2211yescharacteristic
64C4×M16×C59C2211yescharacteristic
65C59×#341C2211yescharacteristic
66C59×#341C2211yescharacteristic
67C2×#11745685C2311yescharacteristic
68#14273735—22no—
69#14273735—22no—
70#14273735—22no—
71#14273735Q814yescomplemented
72C2×#11745659—21no—
73C2×#11745659—21no—
74C22×C59⋊C8C2311yescharacteristic
75C22×C59⋊C8C2311yescharacteristic
76C2×C8×D59—21no—
77C2×C8×D59—21no—
78#14273727C2×C411yescharacteristic
79#14273727C2×C411yescharacteristic
80#14273719C2×C411yescharacteristic
81#14273719C2×C411yescharacteristic
82#14273706—21no—
83#14273706—21no—
84C4×Dic118—21no—
85C4×Dic118—21no—
86C4×Dic118C2×C411yescharacteristic
87C4×Dic118C2×C411yescharacteristic
88C4×D236C2×C411yescharacteristic
89C4×D236C2×C411yescharacteristic
90C4×C22⋊D59—21no—
91C4×C22⋊D59—21no—
92#14273686C2311yescharacteristic
93#14273686C2311yescharacteristic
94#14273686C2311yescharacteristic
95#14273684—21no—
96C2×C4⋊Dic59—21no—
97C2×C4×Dic59C2311yescharacteristic
98C2×C4×Dic59C2311yescharacteristic
99C42×D59—21no—
100#14273661C2×C411yescharacteristic
101#14273661C2×C411yescharacteristic
102#14273661C2×C411yescharacteristic
103#14273661C2×C411yescharacteristic
104#14273659C2×C411yescharacteristic
105#14273659C2×C411yescharacteristic
106#14273659C2×C411yescharacteristic
107#14273659C2×C411yescharacteristic
108Dic59⋊C8—21no—
109Dic59⋊C8—21no—
110Dic59⋊C8—21no—
111Dic59⋊C8—21no—
112#14273653—22no—
113#14273653—22no—
114#14273652C2×C411yescharacteristic
115#14273652C2×C411yescharacteristic
116#14273652C2×C411yescharacteristic
117#14273652C2×C411yescharacteristic
118#14273649C2×C411yescharacteristic
119#14273649C2×C411yescharacteristic
120#14273649C2×C411yescharacteristic
121#14273649C2×C411yescharacteristic
122#14273638D411yescharacteristic
123#14273638D411yescharacteristic
124#14273638D411yescharacteristic
125#14273638D411yescharacteristic
126#14273637—21no—
127#14273637—21no—
128#14273637—21no—
129#14273637—21no—
130#14273637—22no—
131C8×Dic59—21no—
132C8×Dic59—21no—
133C8×Dic59—21no—
134C8×Dic59—21no—
135C8×Dic59—22no—
136C59⋊#95—21no—
137C2×M16×C59C2311yescharacteristic
138C2×M16×C59C2311yescharacteristic
139C2×M16×C59C2311yescharacteristic
140C2×M16×C59C2311yescharacteristic
141C22×C8×C59C2311yescharacteristic
142C22×C8×C59C2311yescharacteristic
143C2×C42×C59C2311yescharacteristic
144#108×C59—22no—
145#108×C59—22no—
146C4⋊C8×C59—21no—
147C4⋊C8×C59—21no—
148C4⋊C8×C59C2×C411yescharacteristic
149C4⋊C8×C59C2×C411yescharacteristic
150C4⋊C8×C59C2×C411yescharacteristic
151C4⋊C8×C59C2×C411yescharacteristic
152#97×C59—21no—
153C4×C8×C59—21no—
154C22×C4×C59#4111yescharacteristic
155C22×C4×C59#4111yescharacteristic
156C22×C4×C59C2411yescharacteristic
157C2×C22⋊D59—21no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160#11745685—21no—
161#11745685—21no—
162#11745685C2×Q812yes—
163#11745685#4112yes—
164C2×D236C22×C411yescharacteristic
165C2×C4×D59—21no—
166C2×C4×D59—21no—
167C2×C4×D59—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118C22×C411yescharacteristic
171M16×C59—21no—
172M16×C59—21no—
173M16×C59—21no—
174M16×C59—21no—
175M16×C59—21no—
176M16×C59—21no—
177M16×C59C2×Q812yes—
178M16×C59#4112yes—
179C2×C8×C59—21no—
180C2×C8×C59—21no—
181C2×C8×C59—21no—
182C2×C8×C59—21no—
183C2×C8×C59—21no—
184C2×C8×C59—21no—
185C2×C8×C59—21no—
186C2×C8×C59—21no—
187C2×C8×C59C22×C411yescharacteristic
188C2×C8×C59C22×C411yescharacteristic
189C2×C8×C59C22×C411yescharacteristic
190C2×C8×C59C22×C411yescharacteristic
191C42×C59C22×C411yescharacteristic
192C42×C59C22×C411yescharacteristic
193C42×C59C2×D411yescharacteristic
194C42×C59C2×D411yescharacteristic
195C22⋊Dic59—21no—
196C22⋊Dic59—21no—
197C22⋊Dic59—21no—
198(C2×D59)⋊C4—21no—
199(C2×D59)⋊C4—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—42no—
203C4⋊Dic59C22×C411yescharacteristic
204C4⋊Dic59C22×C411yescharacteristic
205C4⋊Dic59C22×C411yescharacteristic
206C4⋊Dic59C22×C411yescharacteristic
207C4⋊Dic59C22×C411yescharacteristic
208C4⋊Dic59C22×C411yescharacteristic
209Dic59⋊C4—21no—
210Dic59⋊C4—21no—
211Dic59⋊C4—21no—
212Dic59⋊C4—21no—
213Dic59⋊C4—21no—
214Dic59⋊C4—21no—
215C4×Dic59—21no—
216C4×Dic59—21no—
217C4×Dic59—21no—
218C4×Dic59—21no—
219C4×Dic59—21no—
220C4×Dic59—21no—
221C4×Dic59—21no—
222C4×Dic59—21no—
223C4×Dic59—21no—
224C2×C59⋊C8—21no—
225C2×C59⋊C8—21no—
226C2×C59⋊C8C2×D411yescharacteristic
227C2×C59⋊C8C2×D411yescharacteristic
228C2×C59⋊C8C2×D411yescharacteristic
229C2×C59⋊C8C2×D411yescharacteristic
230C2×C59⋊C8C2×Q812yes—
231C2×C59⋊C8#4112yes—
232#11745659—24no—
233#11745659—24no—
234C8×D59—24no—
235C8×D59—24no—
236C23×C59C2×#4111yescharacteristic
237C22×D59—21no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59C4×D411yescharacteristic
241C2×C4×C59C4×D411yescharacteristic
242C2×C4×C59C4×D411yescharacteristic
243C2×C4×C59C4×D411yescharacteristic
244C2×C4×C59C23×C411yescharacteristic
245C2×C4×C59C22×D411yescharacteristic
246C2×C4×C59C22×Q811yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C22⋊D59—22no—
251C22⋊D59—22no—
252C2×Dic59—21no—
253C2×Dic59—22no—
254C2×Dic59—21no—
255C2×Dic59—21no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265D236—21no—
266D236C4×Q812yes—
267C4×D59—22no—
268C4×D59—22no—
269C4×D59—81no—
270C4×D59—81no—
271Dic118—21no—
272Dic118—81no—
273Dic118—81no—
274Dic118C4×Q812yes—
275C8×C59—21no—
276C8×C59—21no—
277C8×C59—21no—
278C8×C59—21no—
279C8×C59—22no—
280C8×C59—22no—
281C8×C59—41no—
282C8×C59—41no—
283C59⋊C8—21no—
284C59⋊C8—21no—
285C59⋊C8C22⋊Q812yes—
286C59⋊C8C22⋊Q812yes—
287#21629—591nomaximal, Sylow
288C22×C59—21no—
289C22×C59—21no—
290C22×C59C2×C4×D411yescharacteristic
291C22×C59#52911yescharacteristic
292C22×C59#53311yescharacteristic
293C2×D59—22no—
294C2×D59—41no—
295C2×D59—41no—
296C4×C59—21no—
297C4×C59—21no—
298C4×C59—21no—
299C4×C59—21no—
300C4×C59C2×C4×Q811yescharacteristic
301C4×C59C4×#4111yescharacteristic
302C4×C59C2×C22⋊Q811yescharacteristic
303C4×C59#52511yescharacteristic
304Dic59—22no—
305Dic59—42no—
306Dic59—41no—
307Dic59—41no—
308Dic59—41no—
309Dic59—41no—
310Dic59—41no—
311Dic59—41no—
312Dic59—81no—
313#3076—591no—
314#2961—591no—
315#2875—591no—
316#1986—591no—
317#1950D5911yescharacteristic, complemented
318#1922—591no—
319#1922—591no—
320#1866—591no—
321#1866—591no—
322#1806—591no—
323#1773—591no—
324#1772—591no—
325#1771—591no—
326#1765—591no—
327#1764—591no—
328C2×C59—21no—
329C2×C59—21no—
330C2×C59C4×C22⋊Q811yescharacteristic
331C2×C59#295611yescharacteristic
332C2×C59#295711yescharacteristic
333D59—42no—
334C2×#131—591no—
335C2×#131—591no—
336#533—591no—
337C4×#41—591no—
338#479—591no—
339#479—591no—
340#478—591no—
341#478—591no—
342#477—591no—
343#476—591no—
344#475—591no—
345#474—591no—
346#446—1181no—
347#446—1181no—
348#446—1181no—
349#446—1181no—
350C2×#108C2×D5911yescharacteristic
351C2×#108C2×D5911yescharacteristic
352#427—591no—
353#427—591no—
354#424—591no—
355#424—591no—
356#423C2×D5911yescharacteristic
357C2×C4⋊C8C2×D5911yescharacteristic
358C2×#104—591no—
359C2×#104—591no—
360#416—591no—
361#416—591no—
362#405—591no—
363#405—591no—
364#405—591no—
365#405—591no—
366#405—1181no—
367#405—1181no—
368C4×M16—1181no—
369C4×M16C2×D5911yescharacteristic
370C2×#97—591no—
371C2×#97—591no—
372C2×#97—1181no—
373C2×C4×C8—591no—
374C4×C4⋊C4—591no—
375#341C2×D5911yescharacteristic
376#341C2×D5911yescharacteristic
377#339—591no—
378#339—591no—
379#339—591no—
380#339—591no—
381#339—591no—
382#339—591no—
383#336—591no—
384#336—591no—
385C59#2162911yescharacteristic, complemented, Sylow
386C2×#41—591no—
387#131—594no—
388#131—1182no—
389#131—1182no—
390#131—1182no—
391C2×M16—1181no—
392C2×M16—1181no—
393C2×M16C22×D5911yescharacteristic
394C2×M16C22×D5911yescharacteristic
395C2×M16C22×D5911yescharacteristic
396C2×M16C22×D5911yescharacteristic
397C22×C8—591no—
398C22×C8—591no—
399C22×C8—1181no—
400C22×C8—1181no—
401C22×C8C22×D5911yescharacteristic
402C22×C8C22×D5911yescharacteristic
403C4⋊Q8—591no—
404C4⋊Q8—591no—
405#125—591no—
406#125—591no—
407C22⋊Q8—1181no—
408C22⋊Q8—1181no—
409C4×Q8—591no—
410C4×Q8—591no—
411C4×Q8—1181no—
412C4×Q8—1181no—
413C4×D4—591no—
414C4×D4—591no—
415C4×D4—1181no—
416C4×D4—1181no—
417#117—591no—
418#117—591no—
419#117—591no—
420#117—1181no—
421C2×C4⋊C4—1181no—
422C2×C42—591no—
423C2×C42—591no—
424C2×C42—1181no—
425C2×C42C22×D5911yescharacteristic
426#108—22no—
427#108—22no—
428#107—591no—
429#107—591no—
430#107—591no—
431#107—591no—
432#106—591no—
433#106—591no—
434#106—591no—
435#106—591no—
436C4⋊C8—21no—
437C4⋊C8—21no—
438C4⋊C8—1181no—
439C4⋊C8—1181no—
440C4⋊C8—1181no—
441C4⋊C8—1181no—
442C4⋊C8C4×D5911yescharacteristic
443C4⋊C8C4×D5911yescharacteristic
444C4⋊C8C4×D5911yescharacteristic
445C4⋊C8C4×D5911yescharacteristic
446#104—1182no—
447#104—1182no—
448#103—591no—
449#103—591no—
450#103—591no—
451#103—591no—
452#102—591no—
453#102—591no—
454#102—591no—
455#102—591no—
456#97—21no—
457#97—591no—
458#97—591no—
459#97—591no—
460#97—591no—
461#97—1182no—
462#97—1181no—
463#97—1181no—
464#97—1181no—
465#97—1181no—
466C4×C8—21no—
467C4×C8—1182no—
468C4×C8—1181no—
469C4×C8—1181no—
470C4×C8—1181no—
471C4×C8—1181no—
472#95—1181no—
473#41—592no—
474#41—592no—
475#41—1181no—
476#41—1181no—
477C2×Q8—591no—
478C2×Q8—1181no—
479C2×Q8—1181no—
480C2×D4—591no—
481C2×D4—1181no—
482C22×C4—1181no—
483C22×C4—1181no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4#1174568911yescharacteristic
488C22×C4#1174568911yescharacteristic
489C22×C4C23×D5911yescharacteristic
490M16—21no—
491M16—21no—
492M16—21no—
493M16—21no—
494M16—21no—
495M16—21no—
496M16—1184no—
497M16—1184no—
498M16#1174568712yes—
499M16Q8×D5912yes—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—591no—
509C2×C8—591no—
510C2×C8—592no—
511C2×C8—592no—
512C2×C8—591no—
513C2×C8—591no—
514C2×C8—1184no—
515C2×C8—1184no—
516C2×C8—1181no—
517C2×C8—1181no—
518C2×C8C2×C4×D5911yescharacteristic
519C2×C8C2×C4×D5911yescharacteristic
520C2×C8C2×C4×D5911yescharacteristic
521C2×C8C2×C4×D5911yescharacteristic
522C4⋊C4—591no—
523C4⋊C4—591no—
524C4⋊C4—591no—
525C4⋊C4—591no—
526C4⋊C4—591no—
527C4⋊C4—591no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—1181no—
533C4⋊C4—1181no—
534C4⋊C4—1181no—
535C4⋊C4—1181no—
536C4⋊C4—2362no—
537C22⋊C4—1181no—
538C22⋊C4—1181no—
539C22⋊C4—1181no—
540C22⋊C4—1181no—
541C22⋊C4—1181no—
542C42—1181no—
543C42—1181no—
544C42—1181no—
545C42—1181no—
546C42—1181no—
547C42—1181no—
548C42—1181no—
549C42—1181no—
550C42—1181no—
551C42C2×C4×D5911yescharacteristic
552C42C2×C4×D5911yescharacteristic
553C42D4×D5911yescharacteristic
554C42D4×D5911yescharacteristic
555C23—1181no—
556C23C2×#1174568911yescharacteristic
557Q8—592no—
558Q8—1181no—
559Q8—4721no—
560Q8—4721no—
561D4—592no—
562D4—1182no—
563D4—1182no—
564D4—1181no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—1181no—
568C2×C4—1182no—
569C2×C4—1182no—
570C2×C4—1182no—
571C2×C4—1181no—
572C2×C4—1181no—
573C2×C4—1181no—
574C2×C4—1181no—
575C2×C4—1181no—
576C2×C4—1181no—
577C2×C4—4721no—
578C2×C4—4721no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4#1427368911yescharacteristic
585C2×C4#1427368911yescharacteristic
586C2×C4#1427368911yescharacteristic
587C2×C4#1427368911yescharacteristic
588C2×C4C22×C4×D5911yescharacteristic
589C2×C4C2×D4×D5911yescharacteristic
590C2×C4C2×Q8×D5911yescharacteristic
591C2×C4#41×D5911yescharacteristic
592C2×C4#41×D5911yescharacteristic
593C2×C4C2×#1174568711yescharacteristic
594C8—22no—
595C8—22no—
596C8—21no—
597C8—21no—
598C8—21no—
599C8—21no—
600C8—41no—
601C8—41no—
602C8—592no—
603C8—592no—
604C8—1181no—
605C8—1181no—
606C22—21no—
607C22—21no—
608C22—1182no—
609C22—2361no—
610C22—2361no—
611C22C2×#1427368911yescharacteristic
612C22#1683793711yescharacteristic
613C22#1683795711yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—1182no—
619C4—4721no—
620C4—2362no—
621C4—2361no—
622C4—2361no—
623C4—2361no—
624C4—2361no—
625C4—2361no—
626C4—2361no—
627C4C4×Q8×D5911yescharacteristic
628C4C4×#1174568711yescharacteristic
629C4C22⋊Q8×D5911yescharacteristic
630C4#1683791211yescharacteristic
631C2—21no—
632C2—21no—
633C2—2362no—
634C2#2142662611yescharacteristic
635C2#2143300511yescharacteristic
636C2#2143301211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1950C2
socleC22×C59C2×C4×D4
prime radicalC2×#11745685C23
cosocleC22×C4×C59C24
hypercenter#1950D59
hypocenterC59#21629
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1950

Complements

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

NH ≅ G/Ntype
2. #21433033C2semidirect
4. #21425040C2semidirect
5. #21424694C2semidirect
6. #21424694C2semidirect
10. #21423983C2semidirect
11. #21423978C2semidirect
12. #21423973C2semidirect
15. C59×#1950C2semidirect
20. #16837710C4semidirect
21. #16837710C4semidirect
22. #16837703C4semidirect
23. #16837703C4semidirect
24. #16837698C4semidirect
25. #16837693C4semidirect
26. #16837688C4semidirect
27. #16837683C4semidirect
71. #14273735Q8semidirect
317. #1950D59semidirect
385. C59#21629semidirect

Maximal subgroups

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

subgroupindex
#214338862
#214330332
#214325252
#214250402
#214246942
#214243682
#214241132
#214239832
#214239782
#214239732
#214239392
#214239322
C59×#19502
#2162959