Finite Group Databasebeta

← #28591815

Subgroups — #28591815

Order 15104.

862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837505C211yesmaximal, characteristic
2#21426735C211yesmaximal, characteristic
3#21426461C211yesmaximal, characteristic, complemented
4#21424724C211yesmaximal, characteristic
5#21424715C211yesmaximal, characteristic
6#21424691C211yesmaximal, characteristic, complemented
7#21424664C211yesmaximal, characteristic, complemented
8#21424451C211yesmaximal, characteristic
9#21424449C211yesmaximal, characteristic
10#21424366C211yesmaximal, characteristic, complemented
11#21423995C211yesmaximal, characteristic, complemented
12#103×Dic59C211yesmaximal, characteristic, complemented
13#21423910C211yesmaximal, characteristic
14#21423846C211yesmaximal, characteristic, complemented
15C59×#1863C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×Q8×Dic59C2211yescharacteristic
18C2×Q8×Dic59C2211yescharacteristic
19C2×#14273689C2211yescharacteristic
20#16837505C418yescomplemented
21C2×#14273660C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273649C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273637C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29#16837304C2211yescharacteristic
30#16837303C2211yescharacteristic
31#16837301C2211yescharacteristic
32#16837294—21no—
33#16837292—21no—
34#16837264—21no—
35#16837259—21no—
36C4⋊C4×Dic59—21no—
37C4⋊C4×Dic59—21no—
38C4⋊C4×Dic59C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C4×C4⋊Dic59—21no—
42C4×C4⋊Dic59C2211yescharacteristic
43#16837248—21no—
44#16837248C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837157C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4×Q8×C59C2211yescharacteristic
54C2×#107×C59C2211yescharacteristic
55C2×#103×C59C2211yescharacteristic
56C2×#103×C59C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C2×#11745671—22no—
63C2×#11745671—22no—
64C2×#11745671C2×C414yes—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67Q8×Dic59—22no—
68Q8×Dic59—22no—
69Q8×Dic59—22no—
70Q8×Dic59C2×C414yes—
71#14273689—22no—
72#14273689C2×C414yes—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59C2311yescharacteristic
91C2×C4×Dic59C2311yescharacteristic
92#14273660C2×C414yes—
93#14273659—22no—
94#14273651—22no—
95#14273649—22no—
96#14273649C2×C414yes—
97#14273647—22no—
98#14273637C2×C414yes—
99C8×Dic59—22no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95D411yescharacteristic
106C59⋊#95D411yescharacteristic
107C59⋊#95D411yescharacteristic
108C59⋊#95D411yescharacteristic
109C22×Q8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C4×Q8×C59—22no—
113C4×Q8×C59—22no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C42×C59—21no—
120C2×C42×C59C2311yescharacteristic
121#107×C59—22no—
122#103×C59—22no—
123#103×C59C2×C414yes—
124#95×C59—21no—
125C23×D59—21no—
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59C22×C412yes—
130C2×Q8×C59#4112yes—
131C2×Q8×C59#4112yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2411yescharacteristic
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59C2×D411yescharacteristic
143C22×Dic59C2×D411yescharacteristic
144C22×Dic59C2×D411yescharacteristic
145C22×Dic59C2×D411yescharacteristic
146C2×D236—21no—
147C2×D236—21no—
148C2×D236—21no—
149C2×D236—21no—
150C2×D236—21no—
151C2×D236C22×C412yes—
152C2×C4×D59—24no—
153C2×C4×D59—81no—
154C2×C4×D59—81no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59C22×C412yes—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—81no—
167C4⋊C4×C59C22×C412yes—
168C4⋊C4×C59#4112yes—
169C42×C59—24no—
170C42×C59—21no—
171C42×C59—21no—
172#11745671—28no—
173(C2×D59)⋊C4—24no—
174(C2×D59)⋊C4—81no—
175(C2×D59)⋊C4—81no—
176(C2×D59)⋊C4—81no—
177(C2×D59)⋊C4—81no—
178C4⋊Dic59—24no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—24no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—81no—
185C4⋊Dic59C22×C412yes—
186C4⋊Dic59#4112yes—
187C4×Dic59—24no—
188C4×Dic59—21no—
189C4×Dic59—21no—
190C4×Dic59—24no—
191C4×Dic59—81no—
192C4×Dic59—81no—
193C4×Dic59—81no—
194C4×Dic59—81no—
195C4×Dic59C22×C412yes—
196C4×Dic59#4112yes—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8C22×C412yes—
203C23×C59C22×D411yescharacteristic
204C22×D59—22no—
205C22×D59—41no—
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—41no—
209C22×D59—41no—
210C22×D59—41no—
211Q8×C59—22no—
212Q8×C59—22no—
213Q8×C59—22no—
214Q8×C59#11714yes—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—22no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59C23×C411yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—22no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—22no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59C4×D412yes—
259C2×Dic59C4×D412yes—
260D236—22no—
261D236—22no—
262D236—41no—
263D236—41no—
264C4×D59—84no—
265C8×C59—22no—
266C8×C59—41no—
267C59⋊C8—22no—
268C59⋊C8—41no—
269#21128—591nomaximal, Sylow
270C22×C59C2×C4×D411yescharacteristic
271C22×C59#52511yescharacteristic
272C22×C59#52511yescharacteristic
273C22×C59#57611yescharacteristic
274C22×C59#57711yescharacteristic
275C22×C59#57711yescharacteristic
276C22×C59#57811yescharacteristic
277C2×D59—42no—
278C2×D59—42no—
279C2×D59—42no—
280C2×D59—81no—
281C2×D59—81no—
282C2×D59—81no—
283C2×D59—81no—
284C4×C59—24no—
285C4×C59—41no—
286C4×C59—42no—
287C4×C59—41no—
288C4×C59—42no—
289C4×C59—81no—
290C4×C59C2×#11711yescharacteristic
291C4×C59C4×#4111yescharacteristic
292C4×C59#52911yescharacteristic
293C4×C59#52911yescharacteristic
294Dic59—22no—
295Dic59—22no—
296Dic59—41no—
297Dic59—42no—
298Dic59—41no—
299Dic59—42no—
300Dic59—81no—
301C2×#440—591no—
302#2319—591no—
303#2276—591no—
304#1926—591no—
305#1925—591no—
306#1921—591no—
307#1918—591no—
308#1880—591no—
309#1880—591no—
310#1865—591no—
311#1863D5911yescharacteristic, complemented
312#1776—591no—
313C4×#103—591no—
314C4×#102—591no—
315#1746—591no—
316C2×C59#229011yescharacteristic
317C2×C59#294611yescharacteristic
318C2×C59#294711yescharacteristic
319C2×C59#311911yescharacteristic
320C2×C59#312111yescharacteristic
321C2×C59#312411yescharacteristic
322C2×C59#312611yescharacteristic
323D59—82no—
324C22×SD16—591no—
325C2×C4×Q8—591no—
326C2×C4×Q8—591no—
327C2×C4×Q8C2×D5911yescharacteristic
328C2×C4×D4—591no—
329#440—598no—
330C2×#107—591no—
331C2×#107C2×D5911yescharacteristic
332C2×#106—591no—
333C2×#103—591no—
334C2×#103C2×D5911yescharacteristic
335C2×#103C2×D5911yescharacteristic
336C2×#102—591no—
337C2×#102—591no—
338C2×#102—591no—
339C2×#97—591no—
340C2×C4×C8—591no—
341#390—591no—
342#390—591no—
343#389—591no—
344#389C2×D5911yescharacteristic
345#388—1181no—
346#388—1181no—
347#381—1181no—
348#380—1181no—
349C4×C4⋊C4—591no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—1181no—
354C4×C4⋊C4—1181no—
355C4×C4⋊C4—1181no—
356#376—591no—
357#376—1181no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340C2×D5911yescharacteristic
364#336—591no—
365#336—591no—
366#336—591no—
367#336C2×D5911yescharacteristic
368C59#2112811yescharacteristic, complemented, Sylow
369C22×Q8C22×D5911yescharacteristic
370C22×D4—591no—
371C23×C4—1181no—
372C2×SD16—594no—
373C2×SD16—1182no—
374C2×SD16—1182no—
375C22×C8—591no—
376C22×C8—591no—
377C22×C8C22×D5911yescharacteristic
378C22×C8C22×D5911yescharacteristic
379C4×Q8—22no—
380C4×Q8—22no—
381C4×Q8—594no—
382C4×Q8—1182no—
383C4×Q8—1182no—
384C4×Q8—1182no—
385C4×D4—594no—
386C4×D4—1182no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C22⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4—1181no—
403C2×C42—21no—
404C2×C42—591no—
405C2×C42—591no—
406C2×C42—1181no—
407C2×C42—1181no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42—1181no—
411C2×C42—1181no—
412C2×C42C22×D5911yescharacteristic
413#107—22no—
414#107—594no—
415#106—1182no—
416#103—22no—
417#103—1182no—
418#103C4×D5914yes—
419#102—594no—
420#102—1182no—
421#102—1182no—
422#97—594no—
423C4×C8—1182no—
424#95—21no—
425#95—591no—
426#95—591no—
427#95—591no—
428#95—591no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434C24—1181no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×C4×D5912yes—
439C2×Q8#1174568712yes—
440C2×Q8#1174568912yes—
441C2×D4—592no—
442C2×D4—1181no—
443C2×D4—1181no—
444C2×D4—1181no—
445C2×D4—1181no—
446C2×D4—1181no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—591no—
452C22×C4—591no—
453C22×C4—591no—
454C22×C4—591no—
455C22×C4—1184no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—4721no—
460C22×C4—4721no—
461C22×C4D4×D5911yescharacteristic
462C22×C4D4×D5911yescharacteristic
463C22×C4C23×D5911yescharacteristic
464SD16—1188no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—592no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8C2×C4×D5912yes—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—81no—
483C4⋊C4—592no—
484C4⋊C4—592no—
485C4⋊C4—1184no—
486C4⋊C4—1184no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—4721no—
492C4⋊C4C2×C4×D5912yes—
493C4⋊C4#1174568712yes—
494C22⋊C4—1184no—
495C22⋊C4—4721no—
496C22⋊C4—4721no—
497C22⋊C4—4721no—
498C22⋊C4—4721no—
499C42—24no—
500C42—21no—
501C42—21no—
502C42—592no—
503C42—592no—
504C42—1184no—
505C42—1184no—
506C42—1181no—
507C42—1181no—
508C42—4721no—
509C42—4721no—
510C42—4721no—
511C42—4721no—
512C23—1182no—
513C23—2361no—
514C23—2361no—
515C23—2361no—
516C23—2361no—
517C23—2361no—
518C23—2361no—
519C23C2×D4×D5911yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8#1427368514yes—
524D4—1182no—
525D4—1182no—
526D4—2361no—
527D4—2361no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—592no—
547C2×C4—592no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1182no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—1181no—
558C2×C4—1181no—
559C2×C4—1182no—
560C2×C4—4724no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4C22×C4×D5911yescharacteristic
569C2×C4#41×D5911yescharacteristic
570C2×C4#41×D5911yescharacteristic
571C2×C4C2×#1174568711yescharacteristic
572C2×C4C2×#1174568711yescharacteristic
573C2×C4C2×#1174568911yescharacteristic
574C8—22no—
575C8—41no—
576C8—1182no—
577C8—2361no—
578C22—4721no—
579C22—4721no—
580C22—4721no—
581C22—4721no—
582C22—2362no—
583C22—2362no—
584C22—2362no—
585C22C4×D4×D5911yescharacteristic
586C22#1683790811yescharacteristic
587C22#1683791511yescharacteristic
588C22#1683820611yescharacteristic
589C22#1683820911yescharacteristic
590C22#1683821111yescharacteristic
591C22#1683821511yescharacteristic
592C4—24no—
593C4—42no—
594C4—41no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—1182no—
599C4—1182no—
600C4—4721no—
601C4—2361no—
602C4—2362no—
603C4—2361no—
604C4—2362no—
605C4C2×#1427368511yescharacteristic
606C4C4×#1174568711yescharacteristic
607C4#1683792911yescharacteristic
608C4#1683793811yescharacteristic
609C2—4722no—
610C2#2142653211yescharacteristic
611C2#2143292711yescharacteristic
612C2#2143293411yescharacteristic
613C2#2143429611yescharacteristic
614C2#2143432611yescharacteristic
615C2#2143436511yescharacteristic
616C2#2143438911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1863C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1863D59
hypocenterC59#21128
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
3. #21426461C2semidirect
6. #21424691C2semidirect
7. #21424664C2semidirect
10. #21424366C2semidirect
11. #21423995C2semidirect
12. #103×Dic59C2semidirect
14. #21423846C2semidirect
15. C59×#1863C2semidirect
20. #16837505C4semidirect
311. #1863D59semidirect
368. C59#21128semidirect

Maximal subgroups

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

subgroupindex
C2×#168375052
#214267352
#214264612
#214247242
#214247152
#214246912
#214246642
#214244512
#214244492
#214243662
#214239952
#103×Dic592
#214239102
#214238462
C59×#18632
#2112859