Finite Group Databasebeta

← #28610095

Subgroups — #28610095

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#21426979C211yesmaximal, characteristic, complemented
3#21424831C211yesmaximal, characteristic, complemented
4#21424796C211yesmaximal, characteristic, complemented
5#21424720C211yesmaximal, characteristic, complemented
6#21424671C211yesmaximal, characteristic, complemented
7#21424667C211yesmaximal, characteristic, complemented
8#21424525C211yesmaximal, characteristic
9#21424343C211yesmaximal, characteristic, complemented
10#21424295C211yesmaximal, characteristic, complemented
11#21423994C211yesmaximal, characteristic, complemented
12#21423921C211yesmaximal, characteristic, complemented
13#21423874C211yesmaximal, characteristic, complemented
14#21423812C211yesmaximal, characteristic, complemented
15C59×#2363C211yesmaximal, characteristic
16C23×C59⋊C8C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18#16837723C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837290—21no—
33#16837290—21no—
34#16837266—21no—
35#16837262C2211yescharacteristic
36C4×C4⋊Dic59C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38#16837247—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×#121×C59C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C2×#117×C59C2211yescharacteristic
52C59×#390—21no—
53C59×#390C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#388—21no—
56C59×#386—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59—21no—
59C4×C22⋊C4×C59C2211yescharacteristic
60C22×C59⋊C8—22no—
61C22×C59⋊C8—22no—
62C22×C59⋊C8—41no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273686—22no—
68#14273686C2×C414yescomplemented
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×C22⋊Dic59—21no—
76C2×C22⋊Dic59—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80#14273660—22no—
81#14273660C2×C414yescomplemented
82#14273658—41no—
83#14273658C2×C414yescomplemented
84#14273648—22no—
85#14273648—41no—
86#14273648C2×C414yes—
87#14273648C2×C414yes—
88C22⋊(C59⋊C8)C2×C414yescomplemented
89#14273638—22no—
90#14273638—41no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C22×D4×C59—21no—
97C22×D4×C59C2311yescharacteristic
98C23×C4×C59—21no—
99C23×C4×C59C2311yescharacteristic
100#121×C59—22no—
101#121×C59C2×C414yes—
102C4×D4×C59—22no—
103C4×D4×C59—41no—
104#117×C59—22no—
105#117×C59—41no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59D411yescharacteristic
116C2×C22⋊C4×C59D411yescharacteristic
117C2×C42×C59—21no—
118C2×C42×C59—21no—
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121#95×C59—21no—
122#95×C59—21no—
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
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×C59—42no—
144C22×C4×C59—42no—
145C22×C4×C59—81no—
146C22×C4×C59C22×C412yes—
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59#4112yes—
153C22×C4×C59C2411yescharacteristic
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—81no—
163C4⋊C4×C59C22×C412yes—
164C4⋊C4×C59#4112yes—
165C22⋊C4×C59—24no—
166C22⋊C4×C59—41no—
167C22⋊C4×C59—41no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—42no—
170C22⋊C4×C59—42no—
171C22⋊C4×C59—81no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—42no—
177C42×C59—81no—
178C22⋊Dic59—24no—
179C22⋊Dic59—81no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59C22×C412yes—
187C4⋊Dic59C22×C412yes—
188Dic59⋊C4—81no—
189Dic59⋊C4—81no—
190C4×Dic59—24no—
191C4×Dic59—42no—
192C4×Dic59—42no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—44no—
203C2×C59⋊C8C22×C412yes—
204C2×C59⋊C8C22×C412yes—
205C2×C59⋊C8#4112yes—
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—21no—
234C2×C4×C59—21no—
235C2×C4×C59—22no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—84no—
254C2×C4×C59C4×D412yes—
255C2×C4×C59C23×C411yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59H(5,𝔽2)11yescharacteristic
260C2×C4×C592−1+411yescharacteristic
261C2×Dic59—22no—
262C2×Dic59—22no—
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—41no—
273C59⋊C8—22no—
274C59⋊C8—22no—
275C59⋊C8—22no—
276C59⋊C8—41no—
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—41no—
301C4×C59—41no—
302C4×C59—42no—
303C4×C59—41no—
304C4×C59—41no—
305C4×C59—42no—
306C4×C59—81no—
307C4×C59C4×#4111yescharacteristic
308C4×C59#52011yescharacteristic
309C4×C59#55111yescharacteristic
310C4×C59#55211yescharacteristic
311Dic59—42no—
312Dic59—42no—
313Dic59—81no—
314Dic59—81no—
315C2×#482—591no—
316#2363D5911yescharacteristic
317#2346—591no—
318#1943—591no—
319#1936—591no—
320#1926—591no—
321#1919—591no—
322#1918—591no—
323#1893—591no—
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×C8—591no—
340C2×#121C2×D5911yescharacteristic
341C2×C4×D4C2×D5911yescharacteristic
342C2×#117—591no—
343C2×#117C2×D5911yescharacteristic
344#482—598no—
345C2×#107—591no—
346C2×#107—591no—
347C2×#106—591no—
348C2×#106—591no—
349C2×#102—591no—
350C2×#102—591no—
351C2×#102—591no—
352C2×#102—591no—
353C2×C22⋊C8—591no—
354C2×#97—591no—
355C2×#97—591no—
356#390—21no—
357#390C2×D5911yescharacteristic
358#389—591no—
359#389—591no—
360#389C2×D5911yescharacteristic
361#388—21no—
362#388—1181no—
363#388—1181no—
364#386—21no—
365#382—1181no—
366#380—591no—
367C4×C4⋊C4—591no—
368C4×C4⋊C4—591no—
369C4×C4⋊C4C2×D5911yescharacteristic
370C4×C22⋊C4—21no—
371C4×C22⋊C4C2×D5911yescharacteristic
372#376—1181no—
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#336—591no—
382#336—591no—
383C59#2293511yescharacteristic, complemented, Sylow
384C22×D4—21no—
385C22×D4C22×D5911yescharacteristic
386C23×C4—21no—
387C23×C4C22×D5911yescharacteristic
388C22×C8—591no—
389C22×C8—591no—
390C22×C8—591no—
391C22×C8—591no—
392C22×C8—1182no—
393C22×C8—1182no—
394C22×C8—2361no—
395#121—22no—
396#121C4×D5914yes—
397C4×D4—22no—
398C4×D4—41no—
399#117—22no—
400#117—41no—
401#117—594no—
402#117—1182no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—591no—
405C2×C4⋊C4—591no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
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—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4C22⋊D5911yescharacteristic
421C2×C22⋊C4C22⋊D5911yescharacteristic
422C2×C42—21no—
423C2×C42—21no—
424C2×C42—1181no—
425C2×C42—1181no—
426C2×C42—1181no—
427C2×C42C22×D5911yescharacteristic
428C2×C42C22×D5911yescharacteristic
429#107—594no—
430#107—1182no—
431#106—594no—
432#106—2361no—
433#102—594no—
434#102—594no—
435#102—1182no—
436#102—2361no—
437C22⋊C8—594no—
438#97—1182no—
439#97—2361no—
440#95—21no—
441#95—21no—
442#95—1181no—
443#95—1181no—
444#95—1181no—
445#95—1181no—
446#95—1181no—
447#95C22⋊D5911yescharacteristic
448#95C22⋊D5911yescharacteristic
449C24—21no—
450C24C2×C22⋊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—42no—
468C22×C4—42no—
469C22×C4—81no—
470C22×C4—1181no—
471C22×C4—1181no—
472C22×C4—1181no—
473C22×C4—1181no—
474C22×C4C2×C4×D5912yes—
475C22×C4#1174568512yes—
476C22×C4C2×C22⋊D5911yescharacteristic
477C22×C4C2×C22⋊D5911yescharacteristic
478C22×C4C2×C22⋊D5911yescharacteristic
479C22×C4C2×C22⋊D5911yescharacteristic
480C22×C4C2×C22⋊D5911yescharacteristic
481C22×C4C23×D5911yescharacteristic
482C2×C8—592no—
483C2×C8—592no—
484C2×C8—592no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
488C2×C8—1181no—
489C2×C8—1181no—
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—2364no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—81no—
500C4⋊C4—592no—
501C4⋊C4—592no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—4721no—
509C4⋊C4—4721no—
510C4⋊C4C2×C4×D5912yes—
511C4⋊C4#1174568712yes—
512C22⋊C4—24no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—81no—
519C22⋊C4—1184no—
520C22⋊C4—4721no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—42no—
526C42—81no—
527C42—1184no—
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—
543C23C4×C22⋊D5912yes—
544C23C22×C22⋊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—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—84no—
578C2×C4—1182no—
579C2×C4—1182no—
580C2×C4—2361no—
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×C4C4×C22⋊D5912yes—
591C2×C4C22×C4×D5911yescharacteristic
592C2×C4#41×D5911yescharacteristic
593C2×C4C2×#1174568711yescharacteristic
594C2×C4C2×#1174568511yescharacteristic
595C2×C4#1427377211yescharacteristic
596C2×C4#1427377411yescharacteristic
597C8—1182no—
598C8—1182no—
599C8—1182no—
600C8—2361no—
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×C4×C22⋊D5911yescharacteristic
616C22#1683804611yescharacteristic
617C22#1683805111yescharacteristic
618C22#1683820011yescharacteristic
619C22#1683820011yescharacteristic
620C22#1683820511yescharacteristic
621C22#1683821311yescharacteristic
622C4—22no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—42no—
627C4—41no—
628C4—41no—
629C4—81no—
630C4—4721no—
631C4—4721no—
632C4—2362no—
633C4—2362no—
634C4C4×#1174568711yescharacteristic
635C4#1683788111yescharacteristic
636C4#1683808111yescharacteristic
637C4#1683809211yescharacteristic
638C2—42no—
639C2—82no—
640C2#2142708211yescharacteristic
641C2#2143289411yescharacteristic
642C2#2143292811yescharacteristic
643C2#2143622211yescharacteristic
644C2#2143625211yescharacteristic
645C2#2143659011yescharacteristic
646C2#2143661111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2363C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
cosocleC22×C4×C59C24
hypercenter#2363D59
hypocenterC59#22935
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
2. #21426979C2semidirect
3. #21424831C2semidirect
4. #21424796C2semidirect
5. #21424720C2semidirect
6. #21424671C2semidirect
7. #21424667C2semidirect
9. #21424343C2semidirect
10. #21424295C2semidirect
11. #21423994C2semidirect
12. #21423921C2semidirect
13. #21423874C2semidirect
14. #21423812C2semidirect
18. #16837723C4semidirect
68. #14273686C2×C4semidirect
81. #14273660C2×C4semidirect
83. #14273658C2×C4semidirect
88. 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
#214269792
#214248312
#214247962
#214247202
#214246712
#214246672
#214245252
#214243432
#214242952
#214239942
#214239212
#214238742
#214238122
C59×#23632
#2293559