Finite Group Databasebeta

← #28610107

Subgroups — #28610107

Order 15104.

858 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837723C211yesmaximal, characteristic
2#21427138C211yesmaximal, characteristic
3#21426983C211yesmaximal, characteristic, complemented
4#21424833C211yesmaximal, characteristic, complemented
5#21424790C211yesmaximal, characteristic, complemented
6#21424726C211yesmaximal, characteristic, complemented
7#21424677C211yesmaximal, characteristic, complemented
8#21424665C211yesmaximal, characteristic, complemented
9#21424341C211yesmaximal, characteristic, complemented
10#21424293C211yesmaximal, characteristic, complemented
11#21423992C211yesmaximal, characteristic, complemented
12#21423923C211yesmaximal, characteristic, complemented
13#21423872C211yesmaximal, characteristic, complemented
14#21423813C211yesmaximal, characteristic, complemented
15C59×#1893C211yesmaximal, characteristic
16C2×D4×Dic59C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19#16837723C418yescomplemented
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×C22⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273637C2211yescharacteristic
28C2×#14273637C2211yescharacteristic
29#16837305—21no—
30#16837305C2211yescharacteristic
31#16837301C2211yescharacteristic
32#16837298C2211yescharacteristic
33#16837298C2211yescharacteristic
34#16837296—21no—
35#16837296—21no—
36#16837295—21no—
37#16837282—21no—
38#16837266—21no—
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C22⋊C4×Dic59—21no—
43C4×C22⋊Dic59C2211yescharacteristic
44#16837247—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C23×C8×C59C2211yescharacteristic
54C2×#121×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×#102×C59C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C23×Dic59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63D4×Dic59—22no—
64D4×Dic59—41no—
65#14273686—22no—
66#14273686—22no—
67#14273686—41no—
68#14273686C2×C414yescomplemented
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×C22⋊Dic59—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59D411yescharacteristic
82C2×C22⋊Dic59D411yescharacteristic
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59C2311yescharacteristic
89C2×C4×Dic59C2311yescharacteristic
90#14273660—22no—
91#14273660C2×C414yescomplemented
92#14273658—41no—
93#14273658C2×C414yescomplemented
94#14273648C2×C414yes—
95#14273648C2×C414yes—
96C22⋊(C59⋊C8)C2×C414yescomplemented
97#14273637—22no—
98#14273637—41no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95D411yescharacteristic
106C59⋊#95D411yescharacteristic
107C22×D4×C59—21no—
108C22×D4×C59C2311yescharacteristic
109C23×C4×C59C2311yescharacteristic
110C22×C8×C59—22no—
111C22×C8×C59—22no—
112C22×C8×C59—41no—
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115#121×C59—22no—
116#121×C59C2×C414yes—
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122#102×C59—22no—
123#102×C59—41no—
124#95×C59—21no—
125C24×C59—21no—
126C24×C59C2×D411yescharacteristic
127C2×D4×C59—24no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—81no—
134C2×D4×C59—81no—
135C2×D4×C59C22×C412yes—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59C22×C412yes—
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59#4112yes—
146C22×C4×C59C2411yescharacteristic
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—42no—
152C22×Dic59—42no—
153C22×Dic59—81no—
154C22×Dic59C2×D411yescharacteristic
155C22×Dic59C2×D411yescharacteristic
156C22×Dic59C2×D411yescharacteristic
157C22×Dic59C2×D411yescharacteristic
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—44no—
164C2×C8×C59#4112yes—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59C22×C412yes—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—81no—
173C22⋊Dic59—24no—
174C22⋊Dic59—41no—
175C22⋊Dic59—41no—
176C22⋊Dic59—42no—
177C22⋊Dic59—42no—
178C22⋊Dic59—42no—
179C22⋊Dic59—81no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—81no—
186C4⋊Dic59C22×C412yes—
187C4⋊Dic59C22×C412yes—
188C4⋊Dic59#4112yes—
189Dic59⋊C4—81no—
190Dic59⋊C4—81no—
191C4×Dic59—24no—
192C4×Dic59—21no—
193C4×Dic59—21no—
194C4×Dic59—21no—
195C4×Dic59—21no—
196C4×Dic59—42no—
197C4×Dic59—42no—
198C4×Dic59—42no—
199C4×Dic59—81no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8C22×C412yes—
205C2×C59⋊C8C22×C412yes—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—22no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59C4×D412yes—
220C23×C59C22×D411yescharacteristic
221D4×C59—22no—
222D4×C59—22no—
223D4×C59—41no—
224D4×C59—41no—
225D4×C59—84no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—22no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59C4×D412yes—
242C2×C4×C59C23×C411yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×C4×C59C2×#4111yescharacteristic
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59H(5,𝔽2)11yescharacteristic
247C2×C4×C592−1+411yescharacteristic
248C2×Dic59—22no—
249C2×Dic59—22no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—22no—
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—21no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—21no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—84no—
273C8×C59—22no—
274C8×C59—41no—
275C59⋊C8—22no—
276C59⋊C8—22no—
277#22935—591nomaximal, Sylow
278C22×C59—22no—
279C22×C59—22no—
280C22×C59—22no—
281C22×C59—41no—
282C22×C59—41no—
283C22×C59—41no—
284C22×C59—41no—
285C22×C59—42no—
286C22×C59—42no—
287C22×C59—42no—
288C22×C59—81no—
289C22×C59—81no—
290C22×C59—81no—
291C22×C59—81no—
292C22×C59C2×C4×D411yescharacteristic
293C22×C59#54611yescharacteristic
294C22×C59#54711yescharacteristic
295C22×C59#57511yescharacteristic
296C22×C59#57511yescharacteristic
297C22×C59#57611yescharacteristic
298C22×C59#57711yescharacteristic
299C4×C59—22no—
300C4×C59—42no—
301C4×C59—81no—
302C4×C59—81no—
303C4×C59C4×#4111yescharacteristic
304C4×C59#52011yescharacteristic
305C4×C59#55111yescharacteristic
306C4×C59#55211yescharacteristic
307Dic59—41no—
308Dic59—41no—
309Dic59—42no—
310Dic59—42no—
311Dic59—41no—
312Dic59—41no—
313Dic59—42no—
314Dic59—81no—
315C2×#482—591no—
316#2363—591no—
317#2346—591no—
318#1943—591no—
319#1936—591no—
320#1926—591no—
321#1919—591no—
322#1918—591no—
323#1893D5911yescharacteristic
324#1862—591no—
325#1854—591no—
326#1776—591no—
327#1762—591no—
328#1752—591no—
329#1735—591no—
330C2×C59—42no—
331C2×C59—82no—
332C2×C59#235911yescharacteristic
333C2×C59#294311yescharacteristic
334C2×C59#294611yescharacteristic
335C2×C59#331711yescharacteristic
336C2×C59#331911yescharacteristic
337C2×C59#334711yescharacteristic
338C2×C59#334811yescharacteristic
339C23×C8C2×D5911yescharacteristic
340C2×#121C2×D5911yescharacteristic
341C2×C4×D4—591no—
342C2×#117—591no—
343C2×#117—591no—
344#482—598no—
345C2×#107—591no—
346C2×#107—591no—
347C2×#106—591no—
348C2×#106—591no—
349C2×#102—591no—
350C2×#102—591no—
351C2×#102C2×D5911yescharacteristic
352C2×#102C2×D5911yescharacteristic
353C2×C22⋊C8—591no—
354C2×#97—591no—
355C2×#97—591no—
356#390—591no—
357#390—1181no—
358#389—591no—
359#389—591no—
360#389—591no—
361#388—1181no—
362#388—1181no—
363#388—1181no—
364#386—1181no—
365#382—1181no—
366#380C2×D5911yescharacteristic
367C4×C4⋊C4—591no—
368C4×C4⋊C4—591no—
369C4×C4⋊C4—591no—
370C4×C22⋊C4—591no—
371C4×C22⋊C4—1181no—
372#376—1181no—
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#340—591no—
378#340—591no—
379#340C2×D5911yescharacteristic
380#340C2×D5911yescharacteristic
381#336—591no—
382#336—591no—
383C59#2293511yescharacteristic, complemented, Sylow
384C22×D4—21no—
385C22×D4C22×D5911yescharacteristic
386C23×C4—1181no—
387C23×C4C22×D5911yescharacteristic
388C22×C8—22no—
389C22×C8—22no—
390C22×C8—41no—
391C22×C8—591no—
392C22×C8—591no—
393C22×C8C22×D5911yescharacteristic
394C22×C8C22×D5911yescharacteristic
395#121—22no—
396#121C4×D5914yes—
397C4×D4—1182no—
398C4×D4—2361no—
399#117—594no—
400#117—1182no—
401#117—1182no—
402#117—2361no—
403C2×C4⋊C4—591no—
404C2×C4⋊C4—591no—
405C2×C4⋊C4—591no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4—1181no—
410C2×C4⋊C4C22×D5911yescharacteristic
411C2×C4⋊C4C22×D5911yescharacteristic
412C2×C4⋊C4C22×D5911yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—591no—
416C2×C22⋊C4—591no—
417C2×C22⋊C4—1181no—
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4—1181no—
421C2×C22⋊C4—1181no—
422C2×C42—591no—
423C2×C42—591no—
424C2×C42—1181no—
425C2×C42—1181no—
426C2×C42—1181no—
427C2×C42—1181no—
428C2×C42—1181no—
429#107—594no—
430#107—1182no—
431#106—594no—
432#106—2361no—
433#102—22no—
434#102—41no—
435#102—594no—
436#102—594no—
437C22⋊C8—594no—
438#97—1182no—
439#97—2361no—
440#95—21no—
441#95—591no—
442#95—591no—
443#95—1181no—
444#95—1181no—
445#95—1181no—
446#95—1181no—
447#95—1181no—
448#95—1181no—
449C24—21no—
450C24D4×D5911yescharacteristic
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—81no—
458C2×D4—81no—
459C2×D4C2×C4×D5912yes—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—591no—
468C22×C4—591no—
469C22×C4—591no—
470C22×C4—591no—
471C22×C4—1181no—
472C22×C4—1181no—
473C22×C4—1181no—
474C22×C4—1181no—
475C22×C4—4721no—
476C22×C4—2362no—
477C22×C4—2362no—
478C22×C4C2×C4×D5912yes—
479C22×C4D4×D5911yescharacteristic
480C22×C4#1174568712yes—
481C22×C4C23×D5911yescharacteristic
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—44no—
488C2×C8—592no—
489C2×C8—592no—
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8#1174568712yes—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—592no—
501C4⋊C4—592no—
502C4⋊C4—592no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—4721no—
509C4⋊C4—4721no—
510C4⋊C4—4721no—
511C4⋊C4C2×C4×D5912yes—
512C22⋊C4—24no—
513C22⋊C4—81no—
514C22⋊C4—1184no—
515C22⋊C4—4721no—
516C22⋊C4—2361no—
517C22⋊C4—2361no—
518C22⋊C4—2362no—
519C22⋊C4—2362no—
520C22⋊C4—2362no—
521C42—1184no—
522C42—1181no—
523C42—1181no—
524C42—1181no—
525C42—1181no—
526C42—4721no—
527C42—2362no—
528C42—2362no—
529C42—2362no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23(C22⋊D59)⋊C412yes—
544C23C2×D4×D5911yescharacteristic
545D4—22no—
546D4—22no—
547D4—41no—
548D4—41no—
549D4—84no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—1181no—
566C2×C4—1181no—
567C2×C4—1181no—
568C2×C4—1181no—
569C2×C4—1181no—
570C2×C4—1181no—
571C2×C4—1181no—
572C2×C4—1181no—
573C2×C4—1181no—
574C2×C4—1181no—
575C2×C4—1182no—
576C2×C4—1181no—
577C2×C4—1181no—
578C2×C4—1182no—
579C2×C4—1182no—
580C2×C4—4724no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4(C22⋊D59)⋊C412yes—
591C2×C4C22×C4×D5911yescharacteristic
592C2×C4#41×D5911yescharacteristic
593C2×C4C2×#1174568711yescharacteristic
594C2×C4C2×#1174568711yescharacteristic
595C2×C4#1427377211yescharacteristic
596C2×C4#1427377411yescharacteristic
597C8—22no—
598C8—41no—
599C8—1182no—
600C8—1182no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22C2×(C22⋊D59)⋊C411yescharacteristic
616C22#1683804111yescharacteristic
617C22#1683805211yescharacteristic
618C22#1683819911yescharacteristic
619C22#1683819911yescharacteristic
620C22#1683820311yescharacteristic
621C22#1683820811yescharacteristic
622C4—22no—
623C4—42no—
624C4—81no—
625C4—81no—
626C4—4721no—
627C4—2362no—
628C4—2361no—
629C4—2362no—
630C4—2361no—
631C4—2362no—
632C4—2361no—
633C4—2361no—
634C4C4×#1174568711yescharacteristic
635C4#1683788111yescharacteristic
636C4#1683808311yescharacteristic
637C4#1683808911yescharacteristic
638C2—42no—
639C2—82no—
640C2#2142709411yescharacteristic
641C2#2143289211yescharacteristic
642C2#2143292211yescharacteristic
643C2#2143622811yescharacteristic
644C2#2143625811yescharacteristic
645C2#2143660211yescharacteristic
646C2#2143661711yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1893C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
cosocleC22×C4×C59C24
hypercenter#1893D59
hypocenterC59#22935
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
3. #21426983C2semidirect
4. #21424833C2semidirect
5. #21424790C2semidirect
6. #21424726C2semidirect
7. #21424677C2semidirect
8. #21424665C2semidirect
9. #21424341C2semidirect
10. #21424293C2semidirect
11. #21423992C2semidirect
12. #21423923C2semidirect
13. #21423872C2semidirect
14. #21423813C2semidirect
19. #16837723C4semidirect
68. #14273686C2×C4semidirect
91. #14273660C2×C4semidirect
93. #14273658C2×C4semidirect
96. C22⋊(C59⋊C8)C2×C4semidirect
383. C59#22935semidirect

Maximal subgroups

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

subgroupindex
C2×#168377232
#214271382
#214269832
#214248332
#214247902
#214247262
#214246772
#214246652
#214243412
#214242932
#214239922
#214239232
#214238722
#214238132
C59×#18932
#2293559