Finite Group Databasebeta

← #28610780

Subgroups — #28610780

Order 15104.

692 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433884C211yesmaximal, characteristic
2#21433039C211yesmaximal, characteristic, complemented
3#21425076C211yesmaximal, characteristic, complemented
4#21424869C211yesmaximal, characteristic, complemented
5#21424846C211yesmaximal, characteristic, complemented
6#21424841C211yesmaximal, characteristic, complemented
7#21424692C211yesmaximal, characteristic, complemented
8#21424692C211yesmaximal, characteristic, complemented
9#21424371C211yesmaximal, characteristic
10#21424371C211yesmaximal, characteristic
11#21423955C211yesmaximal, characteristic, complemented
12#21423938C211yesmaximal, characteristic
13#21423931C211yesmaximal, characteristic
14#21423906C211yesmaximal, characteristic
15C59×#2906C211yesmaximal, characteristic
16#16837959C2211yescharacteristic
17#16837709C411yescharacteristic, complemented
18#16837709C411yescharacteristic, complemented
19#16837702C411yescharacteristic, complemented
20#16837702C411yescharacteristic, complemented
21#16837697C411yescharacteristic, complemented
22#16837692C411yescharacteristic, complemented
23#16837687C411yescharacteristic, complemented
24#16837682C411yescharacteristic, complemented
25#16837531—21no—
26#16837531—21no—
27#16837529—21no—
28#16837526—21no—
29C2×#14273662C2211yescharacteristic
30C2×#14273662C2211yescharacteristic
31#16837438C2211yescharacteristic
32#16837438C2211yescharacteristic
33#16837428C2211yescharacteristic
34#16837428C2211yescharacteristic
35C2×C4⋊(C59⋊C8)C2211yescharacteristic
36C2×#14273654C2211yescharacteristic
37C2×#14273654C2211yescharacteristic
38#16837398C2211yescharacteristic
39#16837398C2211yescharacteristic
40#16837357C2211yescharacteristic
41#16837357C2211yescharacteristic
42C4×#11745664C2211yescharacteristic
43C2×#14273637C2211yescharacteristic
44C2×C8×Dic59C2211yescharacteristic
45#16837248C2211yescharacteristic
46#16837166C2211yescharacteristic
47#16837166C2211yescharacteristic
48#16837163C2211yescharacteristic
49#16837163C2211yescharacteristic
50#16837163C2211yescharacteristic
51#16837163C2211yescharacteristic
52#16837161C2211yescharacteristic
53#16837161C2211yescharacteristic
54#16837156C2211yescharacteristic
55#16837156C2211yescharacteristic
56C2×#131×C59C2211yescharacteristic
57C2×#131×C59C2211yescharacteristic
58C4×#41×C59C2211yescharacteristic
59C59×#424—21no—
60C59×#424—21no—
61C59×#424C2211yescharacteristic
62C59×#424C2211yescharacteristic
63C59×#423—21no—
64C59×#423C2211yescharacteristic
65C2×C4⋊C8×C59—21no—
66C2×C4⋊C8×C59C2211yescharacteristic
67C2×#11745664C2311yescharacteristic
68C2×#11745664C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71#14273727C2×C411yescharacteristic
72#14273727C2×C411yescharacteristic
73#14273719C2×C411yescharacteristic
74#14273719C2×C411yescharacteristic
75#14273706—21no—
76#14273706—21no—
77C4×Dic118—21no—
78C4×Dic118—21no—
79#14273686C2311yescharacteristic
80#14273686C2311yescharacteristic
81C2×C4×Dic59C2311yescharacteristic
82C2×C4×Dic59C2311yescharacteristic
83#14273662—41no—
84#14273662—41no—
85#14273660C2×C411yescharacteristic
86#14273660C2×C411yescharacteristic
87#14273660C2×C411yescharacteristic
88#14273660C2×C411yescharacteristic
89#14273658C2×C411yescharacteristic
90#14273658C2×C411yescharacteristic
91#14273658C2×C411yescharacteristic
92#14273658C2×C411yescharacteristic
93C4⋊(C59⋊C8)C2×C411yescharacteristic
94C4⋊(C59⋊C8)C2×C411yescharacteristic
95C4⋊(C59⋊C8)C2×C411yescharacteristic
96C4⋊(C59⋊C8)C2×C411yescharacteristic
97Dic59⋊C8—21no—
98Dic59⋊C8—21no—
99Dic59⋊C8—21no—
100Dic59⋊C8—21no—
101#14273654—22no—
102#14273654Q814yescomplemented
103#14273651C2×C411yescharacteristic
104#14273651C2×C411yescharacteristic
105#14273651C2×C411yescharacteristic
106#14273651C2×C411yescharacteristic
107#14273648C2×C411yescharacteristic
108#14273648C2×C411yescharacteristic
109#14273648C2×C411yescharacteristic
110#14273648C2×C411yescharacteristic
111#14273638—21no—
112#14273637—21no—
113#14273637—21no—
114#14273637—41no—
115C4×C59⋊C8—21no—
116C8×Dic59—21no—
117C8×Dic59—21no—
118C8×Dic59—41no—
119C59⋊#95—21no—
120C59⋊#95—21no—
121C2×#41×C59C2311yescharacteristic
122#131×C59—22no—
123#131×C59—22no—
124#131×C59—41no—
125#131×C59—41no—
126C2×M16×C59—21no—
127C2×M16×C59—21no—
128C2×M16×C59C2311yescharacteristic
129C2×M16×C59C2311yescharacteristic
130C22×C8×C59—21no—
131C22×C8×C59—21no—
132C22×C8×C59C2311yescharacteristic
133C22×C8×C59C2311yescharacteristic
134C4×Q8×C59C2×C411yescharacteristic
135C4×Q8×C59C2×C411yescharacteristic
136C4×D4×C59—21no—
137C4×D4×C59—21no—
138C4×D4×C59C2×C411yescharacteristic
139C4×D4×C59C2×C411yescharacteristic
140#117×C59—21no—
141#117×C59C2311yescharacteristic
142C2×C42×C59—21no—
143C2×C42×C59C2311yescharacteristic
144C4⋊C8×C59—21no—
145C4⋊C8×C59—21no—
146C4⋊C8×C59—21no—
147C4⋊C8×C59—21no—
148C4⋊C8×C59—21no—
149C4⋊C8×C59—21no—
150C4⋊C8×C59D411yescharacteristic
151C4⋊C8×C59D411yescharacteristic
152C4⋊C8×C59D411yescharacteristic
153C4⋊C8×C59D411yescharacteristic
154#41×C59—21no—
155#41×C59—21no—
156#41×C59C2×Q812yes—
157#41×C59#4112yes—
158C2×Q8×C59C22×C411yescharacteristic
159C2×D4×C59—21no—
160C2×D4×C59C22×C411yescharacteristic
161C22×C4×C59—21no—
162C22×C4×C59—21no—
163C22×C4×C59—21no—
164C22×C4×C59#4111yescharacteristic
165C22×C4×C59#4111yescharacteristic
166C22×C4×C59C2411yescharacteristic
167C22×Dic59—21no—
168C22×Dic59—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171M16×C59—21no—
172M16×C59—21no—
173M16×C59—21no—
174M16×C59—21no—
175M16×C59—42no—
176M16×C59—42no—
177C2×C8×C59—21no—
178C2×C8×C59—21no—
179C2×C8×C59—21no—
180C2×C8×C59—21no—
181C2×C8×C59—21no—
182C2×C8×C59—21no—
183C2×C8×C59—42no—
184C2×C8×C59—42no—
185C2×C8×C59C2×D411yescharacteristic
186C2×C8×C59C2×D411yescharacteristic
187C2×C8×C59C2×D411yescharacteristic
188C2×C8×C59C2×D411yescharacteristic
189C4⋊C4×C59—21no—
190C4⋊C4×C59—21no—
191C4⋊C4×C59C22×C411yescharacteristic
192C4⋊C4×C59C22×C411yescharacteristic
193C22⋊C4×C59—21no—
194C22⋊C4×C59—21no—
195C22⋊C4×C59—21no—
196C42×C59—21no—
197C42×C59—21no—
198C42×C59—21no—
199C42×C59C22×C411yescharacteristic
200C42×C59C22×C411yescharacteristic
201C42×C59C2×D411yescharacteristic
202C42×C59C2×D411yescharacteristic
203C22⋊Dic59—21no—
204C22⋊Dic59—21no—
205C4⋊Dic59—21no—
206C4⋊Dic59—21no—
207C4⋊Dic59C22×C411yescharacteristic
208C4⋊Dic59C22×C411yescharacteristic
209C4⋊Dic59C22×C411yescharacteristic
210C4⋊Dic59C22×C411yescharacteristic
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×Dic59C2×Q812yes—
221#11745664—21no—
222#11745664—21no—
223#11745664—21no—
224#11745664C2×Q812yes—
225C2×C59⋊C8—21no—
226C2×C59⋊C8—21no—
227C2×C59⋊C8—21no—
228C2×C59⋊C8—21no—
229C2×C59⋊C8—21no—
230C2×C59⋊C8—21no—
231C2×C59⋊C8C22×C411yescharacteristic
232C2×C59⋊C8C22×C411yescharacteristic
233C2×C59⋊C8C22×C411yescharacteristic
234C2×C59⋊C8C22×C411yescharacteristic
235C23×C59—21no—
236C23×C59C2×#4111yescharacteristic
237Q8×C59—21no—
238Q8×C59C4×Q812yes—
239D4×C59—22no—
240D4×C59—22no—
241D4×C59—21no—
242D4×C59C4×Q812yes—
243C2×C4×C59—21no—
244C2×C4×C59—22no—
245C2×C4×C59—22no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—42no—
251C2×C4×C59—81no—
252C2×C4×C59C4×D411yescharacteristic
253C2×C4×C59C4×D411yescharacteristic
254C2×C4×C59C4×D411yescharacteristic
255C2×C4×C59C4×D411yescharacteristic
256C2×C4×C59C23×C411yescharacteristic
257C2×C4×C59C22×D411yescharacteristic
258C2×C4×C59C22×Q811yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59H(5,𝔽2)11yescharacteristic
261C2×C4×C592−1+411yescharacteristic
262C2×Dic59—21no—
263C2×Dic59—21no—
264C2×Dic59—21no—
265C2×Dic59—21no—
266C2×Dic59—22no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272Dic118—42no—
273Dic118—81no—
274C8×C59—21no—
275C8×C59—21no—
276C8×C59—21no—
277C8×C59—21no—
278C8×C59—41no—
279C8×C59—41no—
280C59⋊C8—21no—
281C59⋊C8—21no—
282C59⋊C8—21no—
283C59⋊C8—21no—
284C59⋊C8—22no—
285C59⋊C8—41no—
286#22984—591nomaximal, Sylow
287C22×C59—21no—
288C22×C59—21no—
289C22×C59—22no—
290C22×C59—41no—
291C22×C59—41no—
292C22×C59C2×C4×D411yescharacteristic
293C22×C59#55211yescharacteristic
294C22×C59#55411yescharacteristic
295C4×C59—22no—
296C4×C59—21no—
297C4×C59—21no—
298C4×C59—21no—
299C4×C59—21no—
300C4×C59—41no—
301C4×C59—41no—
302C4×C59C2×C4×Q811yescharacteristic
303C4×C59#52011yescharacteristic
304C4×C59#54611yescharacteristic
305C4×C59D4×Q811yescharacteristic
306Dic59—42no—
307Dic59—41no—
308Dic59—41no—
309Dic59—41no—
310Dic59—41no—
311Dic59—81no—
312#3076—591no—
313#2962—591no—
314#2906D5911yescharacteristic
315#1995—591no—
316#1950—591no—
317#1946—591no—
318#1945—591no—
319#1922—591no—
320#1922—591no—
321#1866—591no—
322#1866—591no—
323#1768—591no—
324#1765—591no—
325#1764—591no—
326#1759—591no—
327C2×C59—21no—
328C2×C59—21no—
329C2×C59—42no—
330C2×C59#236011yescharacteristic
331C2×C59#295511yescharacteristic
332C2×C59#295711yescharacteristic
333C2×#131C2×D5911yescharacteristic
334C2×#131C2×D5911yescharacteristic
335#533—591no—
336C4×#41C2×D5911yescharacteristic
337#479—591no—
338#479—591no—
339#478—591no—
340#478—591no—
341#477—591no—
342#476—591no—
343#475—591no—
344#474—591no—
345#446—1181no—
346#446—1181no—
347#446—1181no—
348C8×Q8—1181no—
349C2×#108—591no—
350C2×#108—591no—
351#427—591no—
352#427—591no—
353#424—21no—
354#424—21no—
355#424—591no—
356#424—591no—
357#424C2×D5911yescharacteristic
358#424C2×D5911yescharacteristic
359#423—21no—
360#423C2×D5911yescharacteristic
361C2×C4⋊C8—21no—
362C2×C4⋊C8—591no—
363C2×C4⋊C8C2×D5911yescharacteristic
364C2×#104—591no—
365C2×#104—591no—
366#416—591no—
367#416—591no—
368#405—591no—
369#405—591no—
370C4×M16—591no—
371C2×#97—591no—
372C2×C4×C8—591no—
373#376—591no—
374#341—591no—
375#341—591no—
376#339—591no—
377#339—591no—
378#339—591no—
379#339—591no—
380#339—591no—
381#339—591no—
382#336—591no—
383#336—591no—
384C59#2298411yescharacteristic, complemented, Sylow
385C2×#41C22×D5911yescharacteristic
386#131—22no—
387#131—22no—
388#131—41no—
389#131—41no—
390C2×M16—21no—
391C2×M16—21no—
392C2×M16—591no—
393C2×M16—591no—
394C2×M16C22×D5911yescharacteristic
395C2×M16C22×D5911yescharacteristic
396C22×C8—21no—
397C22×C8—21no—
398C22×C8—591no—
399C22×C8—591no—
400C22×C8C22×D5911yescharacteristic
401C22×C8C22×D5911yescharacteristic
402C4⋊Q8—591no—
403C4⋊Q8—591no—
404#125—591no—
405#125—591no—
406C22⋊Q8—1181no—
407C22⋊Q8—1181no—
408C4×Q8—1181no—
409C4×Q8—1181no—
410C4×Q8C4×D5911yescharacteristic
411C4×Q8C4×D5911yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4C4×D5911yescharacteristic
415C4×D4C4×D5911yescharacteristic
416#117—21no—
417#117—591no—
418#117—591no—
419#117C22×D5911yescharacteristic
420C2×C42—21no—
421C2×C42—591no—
422C2×C42—591no—
423C2×C42C22×D5911yescharacteristic
424#108—2361no—
425#108—2361no—
426#107—591no—
427#107—591no—
428#107—591no—
429#107—591no—
430#106—591no—
431#106—591no—
432#106—591no—
433#106—591no—
434C4⋊C8—21no—
435C4⋊C8—21no—
436C4⋊C8—21no—
437C4⋊C8—21no—
438C4⋊C8—21no—
439C4⋊C8—21no—
440C4⋊C8—591no—
441C4⋊C8—591no—
442C4⋊C8—591no—
443C4⋊C8—591no—
444C4⋊C8—1181no—
445C4⋊C8—1181no—
446C4⋊C8—1181no—
447C4⋊C8—1181no—
448C4⋊C8C22⋊D5911yescharacteristic
449C4⋊C8C22⋊D5911yescharacteristic
450C4⋊C8C22⋊D5911yescharacteristic
451C4⋊C8C22⋊D5911yescharacteristic
452#104—594no—
453#104—1182no—
454#103—591no—
455#103—591no—
456#103—591no—
457#103—591no—
458#102—591no—
459#102—591no—
460#102—591no—
461#102—591no—
462#97—1181no—
463#97—1181no—
464#97—1181no—
465#97—2361no—
466C4×C8—1181no—
467C4×C8—1181no—
468C4×C8—1181no—
469C4×C8—2361no—
470#95—1181no—
471#95—1181no—
472#41—21no—
473#41—21no—
474#41#1174568712yes—
475#41Q8×D5912yes—
476C2×Q8—1181no—
477C2×Q8—1181no—
478C2×Q8C2×C4×D5911yescharacteristic
479C2×D4—21no—
480C2×D4C2×C4×D5911yescharacteristic
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4#1174568511yescharacteristic
487C22×C4#1174568511yescharacteristic
488C22×C4C23×D5911yescharacteristic
489M16—21no—
490M16—21no—
491M16—21no—
492M16—21no—
493M16—42no—
494M16—42no—
495M16—592no—
496M16—1181no—
497M16—1181no—
498M16—1181no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—42no—
506C2×C8—42no—
507C2×C8—591no—
508C2×C8—591no—
509C2×C8—591no—
510C2×C8—591no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8—1181no—
514C2×C8—1181no—
515C2×C8—1181no—
516C2×C8—1181no—
517C2×C8C2×C22⋊D5911yescharacteristic
518C2×C8C2×C22⋊D5911yescharacteristic
519C2×C8C2×C22⋊D5911yescharacteristic
520C2×C8C2×C22⋊D5911yescharacteristic
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—591no—
524C4⋊C4—591no—
525C4⋊C4—591no—
526C4⋊C4—591no—
527C4⋊C4—1181no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—1181no—
533C4⋊C4C2×C4×D5911yescharacteristic
534C4⋊C4C2×C4×D5911yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—1181no—
539C22⋊C4—1181no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—592no—
544C42—1181no—
545C42—1181no—
546C42—1181no—
547C42—1181no—
548C42—1181no—
549C42C2×C4×D5911yescharacteristic
550C42C2×C4×D5911yescharacteristic
551C42C2×C22⋊D5911yescharacteristic
552C42C2×C22⋊D5911yescharacteristic
553C23—21no—
554C23C2×#1174568511yescharacteristic
555Q8—21no—
556Q8—4721no—
557Q8—2362no—
558Q8#1427369212yes—
559D4—22no—
560D4—22no—
561D4—21no—
562D4#1427369212yes—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—81no—
572C2×C4—1182no—
573C2×C4—1181no—
574C2×C4—1181no—
575C2×C4—1181no—
576C2×C4—1181no—
577C2×C4—2361no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4C4×C22⋊D5911yescharacteristic
583C2×C4C4×C22⋊D5911yescharacteristic
584C2×C4C4×C22⋊D5911yescharacteristic
585C2×C4C4×C22⋊D5911yescharacteristic
586C2×C4C22×C4×D5911yescharacteristic
587C2×C4C22×C22⋊D5911yescharacteristic
588C2×C4C2×Q8×D5911yescharacteristic
589C2×C4C2×#1174568711yescharacteristic
590C2×C4#1427377211yescharacteristic
591C2×C4#1427377511yescharacteristic
592C8—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—41no—
597C8—41no—
598C8—1182no—
599C8—1181no—
600C8—1181no—
601C8—1181no—
602C8—1181no—
603C8—2361no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22C2×C4×C22⋊D5911yescharacteristic
610C22#1683808711yescharacteristic
611C22#1683810711yescharacteristic
612C4—22no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—41no—
618C4—41no—
619C4—4721no—
620C4—2362no—
621C4—2361no—
622C4—2361no—
623C4—2361no—
624C4—2361no—
625C4C2×#1427369211yescharacteristic
626C4#1683788311yescharacteristic
627C4#1683804611yescharacteristic
628C4Q8×C22⋊D5911yescharacteristic
629C2—21no—
630C2—21no—
631C2—42no—
632C2#2142710111yescharacteristic
633C2#2143299711yescharacteristic
634C2#2143301111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2906C2
socleC22×C59C2×C4×D4
prime radicalC2×#41×C59C23
cosocleC22×C4×C59C24
hypercenter#2906D59
hypocenterC59#22984
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #21433039C2semidirect
3. #21425076C2semidirect
4. #21424869C2semidirect
5. #21424846C2semidirect
6. #21424841C2semidirect
7. #21424692C2semidirect
8. #21424692C2semidirect
11. #21423955C2semidirect
17. #16837709C4semidirect
18. #16837709C4semidirect
19. #16837702C4semidirect
20. #16837702C4semidirect
21. #16837697C4semidirect
22. #16837692C4semidirect
23. #16837687C4semidirect
24. #16837682C4semidirect
102. #14273654Q8semidirect
384. C59#22984semidirect

Maximal subgroups

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

subgroupindex
#214338842
#214330392
#214250762
#214248692
#214248462
#214248412
#214246922
#214243712
#214239552
#214239382
#214239312
#214239062
C59×#29062
#2298459