Finite Group Databasebeta

← #28591654

Subgroups — #28591654

Order 15104.

978 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837493C211yesmaximal, characteristic
2#21426567C211yesmaximal, characteristic, complemented
3C4×D4×Dic59C211yesmaximal, characteristic
4#21424731C211yesmaximal, characteristic, complemented
5#21424729C211yesmaximal, characteristic, complemented
6#21424684C211yesmaximal, characteristic, complemented
7#21424678C211yesmaximal, characteristic, complemented
8#21424430C211yesmaximal, characteristic
9#21424428C211yesmaximal, characteristic
10#21424359C211yesmaximal, characteristic
11#106×Dic59C211yesmaximal, characteristic, complemented
12#21423915C211yesmaximal, characteristic, complemented
13#102×Dic59C211yesmaximal, characteristic
14#21423840C211yesmaximal, characteristic, complemented
15C59×#1864C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×#14273692C2211yescharacteristic
18C2×D4×Dic59C2211yescharacteristic
19C2×D4×Dic59C2211yescharacteristic
20C22×C4×Dic59—21no—
21#16837493C418yescomplemented
22C2×#14273659C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29C2×C8×Dic59C2211yescharacteristic
30#16837308C2211yescharacteristic
31#16837307C2211yescharacteristic
32#16837301C2211yescharacteristic
33#16837289—21no—
34#16837283—21no—
35C4⋊C4×Dic59—21no—
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#16837247—21no—
44C42×Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837157C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×#102×C59C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C23×Dic59—21no—
62C23×Dic59—21no—
63C2×#11745670—22no—
64C2×#11745670—22no—
65C2×#11745670C2×C414yes—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273692—22no—
69#14273692C2×C414yescomplemented
70D4×Dic59—22no—
71D4×Dic59—22no—
72D4×Dic59—22no—
73D4×Dic59C2×C414yes—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—44no—
87C2×C4×Dic59D411yescharacteristic
88C2×C4×Dic59D411yescharacteristic
89C2×C4×Dic59D411yescharacteristic
90C2×C4×Dic59D411yescharacteristic
91C2×C4×Dic59C2311yescharacteristic
92C2×C4×Dic59C2311yescharacteristic
93#14273659—22no—
94#14273658C2×C414yescomplemented
95#14273652—22no—
96#14273652C2×C414yescomplemented
97#14273650—22no—
98#14273648—22no—
99C8×Dic59—22no—
100C8×Dic59C2×C414yescomplemented
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C22×D4×C59C2311yescharacteristic
111C23×C4×C59—21no—
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C4×D4×C59—22no—
115C4×D4×C59—22no—
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C22⋊C4×C59—21no—
121C2×C42×C59C2311yescharacteristic
122#106×C59—22no—
123#102×C59—22no—
124#102×C59C2×C414yes—
125#95×C59—21no—
126C24×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59C22×C412yes—
131C2×D4×C59#4112yes—
132C2×D4×C59#4112yes—
133C22×C4×C59—24no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—42no—
138C22×C4×C59—42no—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2411yescharacteristic
142C22×Dic59—24no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—24no—
147C22×Dic59—21no—
148C22×Dic59—42no—
149C22×Dic59—42no—
150C22×Dic59—42no—
151C22×Dic59—42no—
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C22×Dic59C2×D411yescharacteristic
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118C22×C412yes—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59C22×C412yes—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—42no—
173C4⋊C4×C59C22×C412yes—
174C4⋊C4×C59#4112yes—
175C22⋊C4×C59—24no—
176C42×C59—21no—
177C42×C59—21no—
178C22⋊Dic59—24no—
179C22⋊Dic59—24no—
180#11745670—28no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—42no—
186C4⋊Dic59C22×C412yes—
187C4⋊Dic59#4112yes—
188Dic59⋊C4—24no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—24no—
192C4×Dic59—22no—
193C4×Dic59—41no—
194C4×Dic59—41no—
195C4×Dic59—42no—
196C4×Dic59—42no—
197C4×Dic59—42no—
198C4×Dic59—42no—
199C4×Dic59SD1614yescomplemented
200C4×Dic59C22×C412yes—
201C4×Dic59#4112yes—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8C22×C412yes—
208C23×C59—22no—
209C23×C59—22no—
210C23×C59—22no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59C22×D411yescharacteristic
216D4×C59—22no—
217D4×C59—22no—
218D4×C59—22no—
219D4×C59#11714yes—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—22no—
223C2×C4×C59—21no—
224C2×C4×C59—22no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—48no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59C2×SD1612yes—
236C2×C4×C59C23×C411yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—22no—
247C2×Dic59—22no—
248C2×Dic59—21no—
249C2×Dic59—22no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—48no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—48no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59C4×D412yes—
264C2×Dic59C4×D412yes—
265C2×Dic59C2×SD1612yes—
266C2×Dic59C2×SD1612yes—
267C2×Dic59#13512yes—
268Dic118—22no—
269Dic118—22no—
270Dic118—41no—
271Dic118—41no—
272C8×C59—22no—
273C8×C59—41no—
274C59⋊C8—22no—
275C59⋊C8—41no—
276#21117—591nomaximal, Sylow
277C22×C59—24no—
278C22×C59—42no—
279C22×C59—42no—
280C22×C59—42no—
281C22×C59—42no—
282C22×C59—42no—
283C22×C59—42no—
284C22×C59C2×C4×D411yescharacteristic
285C22×C59#52511yescharacteristic
286C22×C59#52511yescharacteristic
287C22×C59C22×SD1611yescharacteristic
288C22×C59C2×#13511yescharacteristic
289C22×C59#57511yescharacteristic
290C22×C59#57511yescharacteristic
291C4×C59—22no—
292C4×C59—42no—
293C4×C59—41no—
294C4×C59—42no—
295C4×C59—81no—
296C4×C59C2×#11711yescharacteristic
297C4×C59C4×#4111yescharacteristic
298C4×C59#52911yescharacteristic
299C4×C59#52911yescharacteristic
300Dic59—22no—
301Dic59—22no—
302Dic59—42no—
303Dic59—41no—
304Dic59—42no—
305Dic59—42no—
306Dic59—81no—
307Dic59C4×SD1614yes—
308C2×C4×SD16—591no—
309#2295—591no—
310C42×D4—591no—
311#1927—591no—
312#1927—591no—
313#1920—591no—
314#1920—591no—
315#1877—591no—
316#1877—591no—
317#1864—591no—
318#1864D5911yescharacteristic
319C4×#106—591no—
320C4×#103—591no—
321C4×#102—591no—
322#1744—591no—
323C2×C59—44no—
324C2×C59#229011yescharacteristic
325C2×C59C2×C4×SD1611yescharacteristic
326C2×C59#294311yescharacteristic
327C2×C59#310111yescharacteristic
328C2×C59#310111yescharacteristic
329C2×C59#310211yescharacteristic
330C2×C59#310211yescharacteristic
331C22×SD16—591no—
332C2×C4×Q8—591no—
333C2×C4×D4—591no—
334C2×C4×D4—591no—
335C2×C4×D4C2×D5911yescharacteristic
336C22×C42—1181no—
337C4×SD16—598no—
338C2×#106—591no—
339C2×#106—591no—
340C2×#106C2×D5911yescharacteristic
341C2×#103—591no—
342C2×#103—591no—
343C2×#103—591no—
344C2×#102—591no—
345C2×#102C2×D5911yescharacteristic
346C2×#102C2×D5911yescharacteristic
347C2×C4×C8—591no—
348C2×C4×C8—591no—
349#391—591no—
350#391—591no—
351#389—591no—
352#389C2×D5911yescharacteristic
353#387—1181no—
354#387—1181no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4—591no—
357C4×C4⋊C4—591no—
358C4×C4⋊C4—591no—
359C4×C4⋊C4—1181no—
360C4×C22⋊C4—1181no—
361C4×C22⋊C4—1181no—
362C4×C22⋊C4—1181no—
363#376—1181no—
364C43—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#340—591no—
370#340C2×D5911yescharacteristic
371#336—591no—
372#336—591no—
373#336—591no—
374#336C2×D5911yescharacteristic
375C59#2111711yescharacteristic, complemented, Sylow
376C22×Q8—591no—
377C22×D4C22×D5911yescharacteristic
378C23×C4—21no—
379C23×C4—1181no—
380C23×C4—1181no—
381C2×SD16—594no—
382C2×SD16—1182no—
383C2×SD16—1182no—
384C22×C8—591no—
385C22×C8—591no—
386C22×C8C22×D5911yescharacteristic
387C22×C8C22×D5911yescharacteristic
388C4×Q8—594no—
389C4×Q8—1182no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—594no—
393C4×D4—1182no—
394C4×D4—1182no—
395C4×D4—1182no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4—591no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C4⋊C4C22×D5911yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1181no—
407C2×C22⋊C4—1181no—
408C2×C42—591no—
409C2×C42—591no—
410C2×C42—591no—
411C2×C42—591no—
412C2×C42—591no—
413C2×C42—591no—
414C2×C42—1181no—
415C2×C42—1181no—
416C2×C42—1181no—
417C2×C42—1181no—
418C2×C42—1181no—
419C2×C42—2364no—
420C2×C42C22×D5911yescharacteristic
421#106—22no—
422#106—594no—
423#106—1182no—
424#103—594no—
425#103—1182no—
426#103—1182no—
427#102—22no—
428#102—1182no—
429#102C4×D5914yes—
430C4×C8—594no—
431C4×C8—1182no—
432#95—21no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95—1181no—
440#95—1181no—
441#95—1181no—
442C24—21no—
443C2×Q8—592no—
444C2×Q8—1181no—
445C2×Q8—1181no—
446C2×Q8—1181no—
447C2×Q8—1181no—
448C2×Q8—1181no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D5912yes—
453C2×D4#1174568712yes—
454C2×D4#1174568912yes—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—591no—
462C22×C4—591no—
463C22×C4—591no—
464C22×C4—591no—
465C22×C4—1184no—
466C22×C4—1184no—
467C22×C4—1181no—
468C22×C4—1181no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4—2362no—
472C22×C4—2362no—
473C22×C4—2362no—
474C22×C4—2362no—
475C22×C4D4×D5911yescharacteristic
476C22×C4D4×D5911yescharacteristic
477C22×C4C23×D5911yescharacteristic
478SD16—1188no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—592no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
488C2×C8—1181no—
489C2×C8—1181no—
490C2×C8C2×C4×D5912yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—592no—
497C4⋊C4—592no—
498C4⋊C4—1184no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—2362no—
504C4⋊C4C2×C4×D5912yes—
505C4⋊C4#1174568712yes—
506C22⋊C4—24no—
507C22⋊C4—1184no—
508C22⋊C4—1184no—
509C42—21no—
510C42—21no—
511C42—594no—
512C42—592no—
513C42—592no—
514C42—1182no—
515C42—1184no—
516C42—1181no—
517C42—1181no—
518C42—2361no—
519C42—2361no—
520C42—2362no—
521C42—2362no—
522C42—2362no—
523C42—2362no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×D4×D5911yescharacteristic
532Q8—1182no—
533Q8—1182no—
534Q8—2361no—
535Q8—2361no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#1427368514yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—48no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—592no—
556C2×C4—592no—
557C2×C4—592no—
558C2×C4—592no—
559C2×C4—592no—
560C2×C4—1181no—
561C2×C4—1181no—
562C2×C4—1181no—
563C2×C4—1181no—
564C2×C4—1181no—
565C2×C4—1182no—
566C2×C4—1181no—
567C2×C4—1181no—
568C2×C4—1182no—
569C2×C4—1182no—
570C2×C4—2368no—
571C2×C4—2368no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2361no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4SD16×D5912yes—
582C2×C4C22×C4×D5911yescharacteristic
583C2×C4#41×D5911yescharacteristic
584C2×C4#41×D5911yescharacteristic
585C2×C4C2×#1174568711yescharacteristic
586C2×C4C2×#1174568711yescharacteristic
587C2×C4C2×#1174568911yescharacteristic
588C8—22no—
589C8—41no—
590C8—1182no—
591C8—2361no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C4×D4×D5911yescharacteristic
600C22#1683790811yescharacteristic
601C22#1683791511yescharacteristic
602C22C2×SD16×D5911yescharacteristic
603C22#135×D5911yescharacteristic
604C22#1683819811yescharacteristic
605C22#1683819811yescharacteristic
606C4—22no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—594no—
612C4—1182no—
613C4—1182no—
614C4—4721no—
615C4—2361no—
616C4—2362no—
617C4—2362no—
618C4—2362no—
619C4C2×#1427368511yescharacteristic
620C4C4×#1174568711yescharacteristic
621C4#1683792911yescharacteristic
622C4#1683793811yescharacteristic
623C2—44no—
624C2#2142653211yescharacteristic
625C2C4×SD16×D5911yescharacteristic
626C2#2143289311yescharacteristic
627C2#2143409911yescharacteristic
628C2#2143410611yescharacteristic
629C2#2143410711yescharacteristic
630C2#2143411911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1864C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1864D59
hypocenterC59#21117
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #21426567C2semidirect
4. #21424731C2semidirect
5. #21424729C2semidirect
6. #21424684C2semidirect
7. #21424678C2semidirect
11. #106×Dic59C2semidirect
12. #21423915C2semidirect
14. #21423840C2semidirect
21. #16837493C4semidirect
69. #14273692C2×C4semidirect
94. #14273658C2×C4semidirect
96. #14273652C2×C4semidirect
100. C8×Dic59C2×C4semidirect
199. C4×Dic59SD16semidirect
375. C59#21117semidirect

Maximal subgroups

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

subgroupindex
C2×#168374932
#214265672
C4×D4×Dic592
#214247312
#214247292
#214246842
#214246782
#214244302
#214244282
#214243592
#106×Dic592
#214239152
#102×Dic592
#214238402
C59×#18642
#2111759