Finite Group Databasebeta

← #28596456

Subgroups — #28596456

Order 15104.

876 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837690C211yesmaximal, characteristic
2#21426843C211yesmaximal, characteristic, complemented
3#21426457C211yesmaximal, characteristic
4#21425046C211yesmaximal, characteristic, complemented
5#21424861C211yesmaximal, characteristic, complemented
6#21424678C211yesmaximal, characteristic, complemented
7#21424675C211yesmaximal, characteristic, complemented
8#21424357C211yesmaximal, characteristic
9#21424104C211yesmaximal, characteristic
10#21423995C211yesmaximal, characteristic, complemented
11#21423984C211yesmaximal, characteristic, complemented
12#21423952C211yesmaximal, characteristic, complemented
13#21423923C211yesmaximal, characteristic
14#102×Dic59C211yesmaximal, characteristic
15C59×#1862C211yesmaximal, characteristic
16C2×#14273726C2211yescharacteristic
17C2×D4×Dic59C2211yescharacteristic
18C2×D4×Dic59C2211yescharacteristic
19#16837690C418yescomplemented
20C2×#14273659C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×Dic59⋊C8C2211yescharacteristic
24C2×Dic59⋊C8C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273637C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29#16837307—21no—
30#16837306—21no—
31#16837300C2211yescharacteristic
32#16837284—21no—
33#16837283—21no—
34#16837267C2211yescharacteristic
35C4⋊C4×Dic59C2211yescharacteristic
36C4⋊C4×Dic59C2211yescharacteristic
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#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837157C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C2×#107×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×#102×C59C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118—21no—
61C23×Dic59—21no—
62C23×Dic59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273726—22no—
66#14273726C2×C414yescomplemented
67D4×Dic59—22no—
68D4×Dic59—22no—
69D4×Dic59C2×C414yes—
70D4×Dic59Q814yescomplemented
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59C2311yescharacteristic
85C2×C4×Dic59C2311yescharacteristic
86#14273659C2×C414yescomplemented
87#14273658—22no—
88#14273658C2×C414yescomplemented
89Dic59⋊C8—22no—
90Dic59⋊C8C2×C414yescomplemented
91#14273648—22no—
92#14273648C2×C414yes—
93#14273637—22no—
94C8×Dic59—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95D411yescharacteristic
104C59⋊#95D411yescharacteristic
105C59⋊#95D411yescharacteristic
106C59⋊#95D411yescharacteristic
107C22×D4×C59C2311yescharacteristic
108C23×C4×C59—21no—
109C22×C8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C4×D4×C59—22no—
112C4×D4×C59—22no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C42×C59C2311yescharacteristic
119#107×C59—22no—
120#102×C59—22no—
121#102×C59C2×C414yes—
122#95×C59—21no—
123C24×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59C22×C412yes—
128C2×D4×C59C2×Q812yes—
129C2×D4×C59#4112yes—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—24no—
133C22×C4×C59—21no—
134C22×C4×C59—81no—
135C22×C4×C59—81no—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2411yescharacteristic
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—24no—
143C22×Dic59—24no—
144C22×Dic59—21no—
145C22×Dic59—81no—
146C22×Dic59—81no—
147C22×Dic59—81no—
148C22×Dic59—81no—
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C2×Dic118—24no—
154C2×Dic118—81no—
155C2×Dic118—81no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59C22×C412yes—
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—
169C22⋊C4×C59—24no—
170C42×C59—21no—
171C42×C59—21no—
172C22⋊Dic59—24no—
173C22⋊Dic59—24no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59C22×C412yes—
180C4⋊Dic59C22×C412yes—
181C4⋊Dic59C2×Q812yes—
182Dic59⋊C4—24no—
183Dic59⋊C4—81no—
184C4×Dic59—81no—
185C4×Dic59—81no—
186C4×Dic59—81no—
187C4×Dic59—81no—
188C4×Dic59—81no—
189C4×Dic59C22×C412yes—
190C4×Dic59C2×Q812yes—
191C4×Dic59#4112yes—
192C4×Dic59#4112yes—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8C22×C412yes—
199C23×C59—22no—
200C23×C59—22no—
201C23×C59—22no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59C22×D411yescharacteristic
207D4×C59—22no—
208D4×C59—22no—
209D4×C59—22no—
210D4×C59C4×Q814yes—
211C2×C4×C59—22no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—22no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—41no—
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—84no—
227C2×C4×C59C23×C411yescharacteristic
228C2×C4×C59C22×Q811yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—22no—
242C2×Dic59—22no—
243C2×Dic59—22no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—84no—
254C2×Dic59—84no—
255C2×Dic59C4×D412yes—
256C2×Dic59C4×D412yes—
257C2×Dic59C22⋊Q812yes—
258C2×Dic59C22⋊Q812yes—
259Dic118—84no—
260C8×C59—22no—
261C8×C59—41no—
262C59⋊C8—22no—
263C59⋊C8—41no—
264#21623—591nomaximal, Sylow
265C22×C59—24no—
266C22×C59—42no—
267C22×C59—42no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59C2×C4×D411yescharacteristic
273C22×C59C2×C22⋊Q811yescharacteristic
274C22×C59#52511yescharacteristic
275C22×C59#57611yescharacteristic
276C22×C59#57711yescharacteristic
277C22×C59#57711yescharacteristic
278C22×C59#57711yescharacteristic
279C4×C59—41no—
280C4×C59—42no—
281C4×C59—41no—
282C4×C59—42no—
283C4×C59—81no—
284C4×C59C2×C4×Q811yescharacteristic
285C4×C59C4×#4111yescharacteristic
286C4×C59#52911yescharacteristic
287C4×C59#53311yescharacteristic
288Dic59—22no—
289Dic59—22no—
290Dic59—22no—
291Dic59—22no—
292Dic59—42no—
293Dic59—42no—
294Dic59—42no—
295Dic59—81no—
296C2×#476—591no—
297#2331—591no—
298#2275—591no—
299#1988—591no—
300#1949—591no—
301#1920—591no—
302#1919—591no—
303#1864—591no—
304#1862D5911yescharacteristic
305#1804—591no—
306#1776—591no—
307C4×#106—591no—
308#1768—591no—
309#1762—591no—
310C4×#102—591no—
311C2×C59—44no—
312C2×C59C4×C22⋊Q811yescharacteristic
313C2×C59#294611yescharacteristic
314C2×C59#294911yescharacteristic
315C2×C59#308111yescharacteristic
316C2×C59#308411yescharacteristic
317C2×C59#311811yescharacteristic
318C2×C59#312511yescharacteristic
319C2×C4⋊Q8—591no—
320C2×C4×D4—591no—
321C2×C4×D4—591no—
322C2×C4×D4C2×D5911yescharacteristic
323#476—598no—
324C2×#107C2×D5911yescharacteristic
325C2×#106—591no—
326C2×#106—591no—
327C2×#106—591no—
328C2×C4⋊C8—591no—
329C2×C4⋊C8—591no—
330C2×#102—591no—
331C2×#102—591no—
332C2×#102C2×D5911yescharacteristic
333C2×#102C2×D5911yescharacteristic
334C2×#97—591no—
335C2×C4×C8—591no—
336#391—1181no—
337#391—1181no—
338#389—591no—
339#389C2×D5911yescharacteristic
340#387—1181no—
341#387—1181no—
342#382—591no—
343C4×C4⋊C4—591no—
344C4×C4⋊C4—591no—
345C4×C4⋊C4—591no—
346C4×C4⋊C4—591no—
347C4×C4⋊C4—591no—
348C4×C22⋊C4—1181no—
349C4×C22⋊C4—1181no—
350C4×C22⋊C4—1181no—
351#376—591no—
352C2×#95—1181no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340C2×D5911yescharacteristic
361#336—591no—
362#336C2×D5911yescharacteristic
363C59#2162311yescharacteristic, complemented, Sylow
364C22×Q8—1181no—
365C22×D4C22×D5911yescharacteristic
366C23×C4—21no—
367C23×C4—1181no—
368C23×C4—1181no—
369C22×C8—591no—
370C22×C8—591no—
371C22×C8C22×D5911yescharacteristic
372C22×C8C22×D5911yescharacteristic
373C4⋊Q8—594no—
374C4⋊Q8—1182no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—594no—
378C4×D4—594no—
379C4×D4—1182no—
380C4×D4—1182no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4C22×D5911yescharacteristic
389C2×C4⋊C4C22×D5911yescharacteristic
390C2×C4⋊C4C22×D5911yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1181no—
393C2×C22⋊C4—1181no—
394C2×C42—591no—
395C2×C42—591no—
396C2×C42—1181no—
397C2×C42—1181no—
398C2×C42—1181no—
399C2×C42—1181no—
400C2×C42—1181no—
401C2×C42C22×D5911yescharacteristic
402#107—22no—
403#106—594no—
404#106—594no—
405#106—1182no—
406C4⋊C8—594no—
407C4⋊C8—1182no—
408#102—22no—
409#102—594no—
410#102—1182no—
411#102C4×D5914yes—
412#97—1182no—
413C4×C8—1182no—
414#95—21no—
415#95—591no—
416#95—591no—
417#95—591no—
418#95—591no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427C24—21no—
428C2×Q8—1184no—
429C2×Q8—4721no—
430C2×Q8—4721no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×C4×D5912yes—
435C2×D4#1174568712yes—
436C2×D4Q8×D5912yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—591no—
444C22×C4—591no—
445C22×C4—591no—
446C22×C4—591no—
447C22×C4—1184no—
448C22×C4—1184no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—4721no—
454C22×C4—4721no—
455C22×C4—4721no—
456C22×C4—4721no—
457C22×C4D4×D5911yescharacteristic
458C22×C4D4×D5911yescharacteristic
459C22×C4C23×D5911yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—592no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8C2×C4×D5912yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—81no—
477C4⋊C4—592no—
478C4⋊C4—592no—
479C4⋊C4—592no—
480C4⋊C4—1184no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—4721no—
487C4⋊C4C2×C4×D5912yes—
488C4⋊C4#1174568712yes—
489C22⋊C4—24no—
490C22⋊C4—1184no—
491C22⋊C4—1184no—
492C42—21no—
493C42—21no—
494C42—592no—
495C42—592no—
496C42—592no—
497C42—592no—
498C42—4721no—
499C42—4721no—
500C42—4721no—
501C42—4721no—
502C42—4721no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×D4×D5911yescharacteristic
511Q8—4724no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4#1427369214yes—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—592no—
533C2×C4—592no—
534C2×C4—592no—
535C2×C4—592no—
536C2×C4—1181no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1182no—
545C2×C4—1182no—
546C2×C4—1182no—
547C2×C4—4724no—
548C2×C4—4724no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4C22×C4×D5911yescharacteristic
559C2×C4C2×Q8×D5911yescharacteristic
560C2×C4#41×D5911yescharacteristic
561C2×C4#41×D5911yescharacteristic
562C2×C4C2×#1174568711yescharacteristic
563C2×C4C2×#1174568711yescharacteristic
564C8—22no—
565C8—41no—
566C8—1182no—
567C8—2361no—
568C22—24no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22C4×D4×D5911yescharacteristic
576C22C22⋊Q8×D5911yescharacteristic
577C22#1683790811yescharacteristic
578C22#1683820311yescharacteristic
579C22#1683820911yescharacteristic
580C22#1683821111yescharacteristic
581C22#1683821211yescharacteristic
582C4—42no—
583C4—41no—
584C4—42no—
585C4—41no—
586C4—81no—
587C4—1182no—
588C4—1182no—
589C4—1182no—
590C4—1182no—
591C4—4721no—
592C4—2362no—
593C4—2362no—
594C4—2362no—
595C4C2×#1427369211yescharacteristic
596C4C4×#1174568711yescharacteristic
597C4#1683792911yescharacteristic
598C4#1683795411yescharacteristic
599C2—44no—
600C2#2142662811yescharacteristic
601C2#2143292111yescharacteristic
602C2#2143295211yescharacteristic
603C2#2143392011yescharacteristic
604C2#2143395911yescharacteristic
605C2#2143427311yescharacteristic
606C2#2143436611yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1862C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1862D59
hypocenterC59#21623
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
2. #21426843C2semidirect
4. #21425046C2semidirect
5. #21424861C2semidirect
6. #21424678C2semidirect
7. #21424675C2semidirect
10. #21423995C2semidirect
11. #21423984C2semidirect
12. #21423952C2semidirect
19. #16837690C4semidirect
66. #14273726C2×C4semidirect
70. D4×Dic59Q8semidirect
86. #14273659C2×C4semidirect
88. #14273658C2×C4semidirect
90. Dic59⋊C8C2×C4semidirect
363. C59#21623semidirect

Maximal subgroups

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

subgroupindex
C2×#168376902
#214268432
#214264572
#214250462
#214248612
#214246782
#214246752
#214243572
#214241042
#214239952
#214239842
#214239522
#214239232
#102×Dic592
C59×#18622
#2162359