Finite Group Databasebeta

← #28591761

Subgroups — #28591761

Order 15104.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837502C211yesmaximal, characteristic
2#21426742C211yesmaximal, characteristic, complemented
3#21426457C211yesmaximal, characteristic
4#21424735C211yesmaximal, characteristic, complemented
5#21424728C211yesmaximal, characteristic, complemented
6#21424686C211yesmaximal, characteristic, complemented
7#21424665C211yesmaximal, characteristic, complemented
8#21424428C211yesmaximal, characteristic
9#21424428C211yesmaximal, characteristic
10#21424355C211yesmaximal, characteristic
11#21424341C211yesmaximal, characteristic
12#21423994C211yesmaximal, characteristic, complemented
13C4×#14273652C211yesmaximal, characteristic, complemented
14#21423847C211yesmaximal, characteristic, complemented
15C4×#102×C59C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×D4×Dic59C2211yescharacteristic
19C2×D4×Dic59C2211yescharacteristic
20#16837502C418yescomplemented
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29#16837307C2211yescharacteristic
30#16837307C2211yescharacteristic
31#16837298C2211yescharacteristic
32#16837298C2211yescharacteristic
33#16837286—21no—
34#16837286—21no—
35#16837269—21no—
36#16837266—21no—
37C4⋊C4×Dic59C2211yescharacteristic
38C4⋊C4×Dic59C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40C22⋊C4×Dic59—21no—
41C22⋊C4×Dic59—21no—
42C4×C22⋊Dic59—21no—
43#16837248C2211yescharacteristic
44C2×C59⋊#95—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837156C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×C4×C8×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C23×Dic59—21no—
62C23×Dic59—21no—
63C2×#11745670—22no—
64C2×#11745670—22no—
65C2×#11745670C2×C414yes—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C4×Dic118—22no—
69C4×Dic118C2×C414yescomplemented
70D4×Dic59—22no—
71D4×Dic59—22no—
72D4×Dic59—22no—
73D4×Dic59—22no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59C2311yescharacteristic
87C2×C4×Dic59C2311yescharacteristic
88#14273660—22no—
89#14273660C2×C414yescomplemented
90#14273658—22no—
91#14273652—22no—
92#14273652—22no—
93#14273650C2×C414yescomplemented
94#14273648C2×C414yes—
95#14273638C2×C414yescomplemented
96C59⋊#95—21no—
97C59⋊#95—21no—
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×C59C2×C414yes—
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×C59—22no—
123C4×C8×C59—22no—
124#95×C59—21no—
125C24×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59C22×C412yes—
130C2×D4×C59#4112yes—
131C2×D4×C59#4112yes—
132C22×C4×C59—21no—
133C22×C4×C59—24no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—81no—
137C22×C4×C59—81no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2411yescharacteristic
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—24no—
146C22×Dic59—24no—
147C22×Dic59—81no—
148C22×Dic59—81no—
149C22×Dic59—81no—
150C22×Dic59—81no—
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118C22×C412yes—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59C22×C412yes—
173C22⋊C4×C59—24no—
174C42×C59—81no—
175C42×C59C22×C412yes—
176C42×C59#4112yes—
177C22⋊Dic59—24no—
178C22⋊Dic59—24no—
179#11745670—28no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59C22×C412yes—
184C4⋊Dic59#4112yes—
185C4⋊Dic59#4112yes—
186Dic59⋊C4—24no—
187Dic59⋊C4—81no—
188Dic59⋊C4—81no—
189C4×Dic59—24no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193C4×Dic59—21no—
194C4×Dic59—81no—
195C4×Dic59—81no—
196C4×Dic59—81no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8C22×C412yes—
202C2×C59⋊C8C22×C412yes—
203C23×C59—22no—
204C23×C59—22no—
205C23×C59—22no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59C22×D411yescharacteristic
211D4×C59—22no—
212D4×C59—22no—
213D4×C59—22no—
214D4×C59#11714yes—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—84no—
229C2×C4×C59C4×D412yes—
230C2×C4×C59C4×D412yes—
231C2×C4×C59C23×C411yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×Dic59—22no—
238C2×Dic59—22no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—22no—
244C2×Dic59—22no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—84no—
262C2×Dic59—84no—
263Dic118—22no—
264Dic118—22no—
265Dic118—41no—
266Dic118—41no—
267C8×C59—41no—
268C8×C59—41no—
269C59⋊C8—22no—
270C59⋊C8—22no—
271#21125—591nomaximal, Sylow
272C22×C59—24no—
273C22×C59—42no—
274C22×C59—42no—
275C22×C59—42no—
276C22×C59—42no—
277C22×C59—42no—
278C22×C59—42no—
279C22×C59C2×C4×D411yescharacteristic
280C22×C59#52511yescharacteristic
281C22×C59#52511yescharacteristic
282C22×C59#57611yescharacteristic
283C22×C59#57711yescharacteristic
284C22×C59#57711yescharacteristic
285C22×C59#57811yescharacteristic
286C4×C59—22no—
287C4×C59—22no—
288C4×C59—42no—
289C4×C59—81no—
290C4×C59—81no—
291C4×C59C2×#11711yescharacteristic
292C4×C59C4×#4111yescharacteristic
293C4×C59#52911yescharacteristic
294C4×C59#52911yescharacteristic
295Dic59—41no—
296Dic59—42no—
297Dic59—41no—
298Dic59—41no—
299Dic59—42no—
300Dic59—41no—
301Dic59—42no—
302Dic59—42no—
303C2×#440—591no—
304#2320—591no—
305#2275—591no—
306#1928—591no—
307#1927—591no—
308#1921—591no—
309#1918—591no—
310#1877—591no—
311#1877—591no—
312#1864—591no—
313#1862—591no—
314#1776—591no—
315C4×#103—591no—
316C4×#102D5911yescharacteristic
317#1746—591no—
318C2×C59—44no—
319C2×C59#229011yescharacteristic
320C2×C59#294611yescharacteristic
321C2×C59#294711yescharacteristic
322C2×C59#311811yescharacteristic
323C2×C59#312011yescharacteristic
324C2×C59#312311yescharacteristic
325C2×C59#312411yescharacteristic
326C22×SD16—591no—
327C2×C4×Q8—591no—
328C2×C4×D4—591no—
329C2×C4×D4—591no—
330C2×C4×D4C2×D5911yescharacteristic
331#440—598no—
332C2×#107—591no—
333C2×#107—591no—
334C2×#106—591no—
335C2×#103—591no—
336C2×#103—591no—
337C2×#103—591no—
338C2×#102—591no—
339C2×#102C2×D5911yescharacteristic
340C2×#102C2×D5911yescharacteristic
341C2×#97—591no—
342C2×C4×C8C2×D5911yescharacteristic
343#391—591no—
344#391—591no—
345#389—591no—
346#389—591no—
347#387—1181no—
348#387—1181no—
349#383—1181no—
350#382—1181no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4C2×D5911yescharacteristic
355C4×C22⋊C4—1181no—
356C4×C22⋊C4—1181no—
357C4×C22⋊C4—1181no—
358#376—591no—
359C2×#95—1181no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340C2×D5911yescharacteristic
365#340C2×D5911yescharacteristic
366#336—591no—
367#336—591no—
368#336—591no—
369#336—591no—
370C59#2112511yescharacteristic, complemented, Sylow
371C22×Q8—591no—
372C22×D4C22×D5911yescharacteristic
373C23×C4—21no—
374C23×C4—1181no—
375C23×C4—1181no—
376C2×SD16—594no—
377C2×SD16—1182no—
378C2×SD16—1182no—
379C22×C8—591no—
380C22×C8—591no—
381C22×C8C22×D5911yescharacteristic
382C22×C8C22×D5911yescharacteristic
383C4×Q8—594no—
384C4×Q8—1182no—
385C4×D4—22no—
386C4×D4—1182no—
387C4×D4—1182no—
388C4×D4—1182no—
389C4×D4—1182no—
390C4×D4C4×D5914yes—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C22⋊C4—21no—
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×C42C22×D5911yescharacteristic
411#107—594no—
412#107—1182no—
413#106—1182no—
414#103—594no—
415#103—1182no—
416#103—1182no—
417#102—22no—
418#102—22no—
419#102—594no—
420#97—594no—
421C4×C8—22no—
422#95—21no—
423#95—591no—
424#95—591no—
425#95—591no—
426#95—591no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436C24—21no—
437C2×Q8—592no—
438C2×Q8—1181no—
439C2×Q8—1181no—
440C2×Q8—1181no—
441C2×Q8—1181no—
442C2×Q8—1181no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4C2×C4×D5912yes—
447C2×D4#1174568512yes—
448C2×D4#1174568512yes—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—591no—
456C22×C4—591no—
457C22×C4—591no—
458C22×C4—591no—
459C22×C4—1184no—
460C22×C4—1184no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4—1181no—
464C22×C4—1181no—
465C22×C4—4721no—
466C22×C4—4721no—
467C22×C4—4721no—
468C22×C4—4721no—
469C22×C4D4×D5911yescharacteristic
470C22×C4D4×D5911yescharacteristic
471C22×C4C23×D5911yescharacteristic
472SD16—1188no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—592no—
480C2×C8—592no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—592no—
491C4⋊C4—592no—
492C4⋊C4—592no—
493C4⋊C4—1184no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—4721no—
498C4⋊C4—4721no—
499C4⋊C4C2×C4×D5912yes—
500C22⋊C4—24no—
501C22⋊C4—1184no—
502C22⋊C4—1184no—
503C42—81no—
504C42—1184no—
505C42—1181no—
506C42—1181no—
507C42—1181no—
508C42—1181no—
509C42—4721no—
510C42—4721no—
511C42—4721no—
512C42C2×C4×D5912yes—
513C42#1174568712yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D5911yescharacteristic
522Q8—1182no—
523Q8—1182no—
524Q8—2361no—
525Q8—2361no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4#1427368414yes—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—84no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1182no—
549C2×C4—1182no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—1181no—
558C2×C4—1182no—
559C2×C4—1182no—
560C2×C4—4724no—
561C2×C4—4724no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4(C22⋊D59)⋊C412yes—
571C2×C4(C22⋊D59)⋊C412yes—
572C2×C4C22×C4×D5911yescharacteristic
573C2×C4#41×D5911yescharacteristic
574C2×C4#41×D5911yescharacteristic
575C2×C4C2×#1174568711yescharacteristic
576C2×C4C2×#1174568511yescharacteristic
577C2×C4C2×#1174568511yescharacteristic
578C8—41no—
579C8—41no—
580C8—1182no—
581C8—1182no—
582C22—24no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22C2×(C22⋊D59)⋊C411yescharacteristic
590C22#1683791111yescharacteristic
591C22#1683791111yescharacteristic
592C22#1683820311yescharacteristic
593C22#1683820811yescharacteristic
594C22#1683820911yescharacteristic
595C22#1683821611yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—2362no—
602C4—2362no—
603C4—2361no—
604C4—2362no—
605C4—2361no—
606C4—2361no—
607C4—2362no—
608C4—2361no—
609C4C2×#1427368411yescharacteristic
610C4#1683787011yescharacteristic
611C4#1683793211yescharacteristic
612C4#1683793211yescharacteristic
613C2—44no—
614C2#2142653511yescharacteristic
615C2#2143292211yescharacteristic
616C2#2143294211yescharacteristic
617C2#2143427611yescharacteristic
618C2#2143430611yescharacteristic
619C2#2143434511yescharacteristic
620C2#2143435411yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC4×#102×C59C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenterC4×#102D59
hypocenterC59#21125
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. #21426742C2semidirect
4. #21424735C2semidirect
5. #21424728C2semidirect
6. #21424686C2semidirect
7. #21424665C2semidirect
12. #21423994C2semidirect
13. C4×#14273652C2semidirect
14. #21423847C2semidirect
20. #16837502C4semidirect
69. C4×Dic118C2×C4semidirect
89. #14273660C2×C4semidirect
93. #14273650C2×C4semidirect
95. #14273638C2×C4semidirect
370. C59#21125semidirect

Maximal subgroups

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

subgroupindex
C2×#168375022
#214267422
#214264572
#214247352
#214247282
#214246862
#214246652
#214244282
#214243552
#214243412
#214239942
C4×#142736522
#214238472
C4×#102×C592
#2112559