Finite Group Databasebeta

← #28596036

Subgroups — #28596036

Order 15104.

900 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837726C211yesmaximal, characteristic
2#21426789C211yesmaximal, characteristic
3#21424716C211yesmaximal, characteristic, complemented
4#21424716C211yesmaximal, characteristic, complemented
5#21424688C211yesmaximal, characteristic, complemented
6#21424674C211yesmaximal, characteristic, complemented
7#21424413C211yesmaximal, characteristic
8#21424284C211yesmaximal, characteristic, complemented
9#21424191C211yesmaximal, characteristic, complemented
10#21423994C211yesmaximal, characteristic, complemented
11C4×#14273660C211yesmaximal, characteristic, complemented
12#21423924C211yesmaximal, characteristic, complemented
13C4×#14273647C211yesmaximal, characteristic, complemented
14#21423862C211yesmaximal, characteristic, complemented
15C59×#2274C211yesmaximal, characteristic, complemented
16C2×#14273701C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18#16837726C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273647C2211yescharacteristic
24C2×#14273647C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837304C2211yescharacteristic
32#16837302—21no—
33#16837300C2211yescharacteristic
34#16837292—21no—
35#16837278—21no—
36#16837273—21no—
37#16837262C2211yescharacteristic
38C4×C4⋊Dic59C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×#117×C59C2211yescharacteristic
50C2×#117×C59C2211yescharacteristic
51C22×C4⋊C4×C59C2211yescharacteristic
52C4×C4⋊C4×C59—21no—
53C4×C4⋊C4×C59—21no—
54C4×C4⋊C4×C59C2211yescharacteristic
55C4×C4⋊C4×C59C2211yescharacteristic
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59—21no—
59C4×C22⋊C4×C59—21no—
60C22×D236C2311yescharacteristic
61C22×C22⋊D59—21no—
62C22×C4×D59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273701—22no—
68#14273701C2×C414yes—
69C4×D236—22no—
70C4×D236—22no—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×C22⋊Dic59—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×C4×Dic59—21no—
81#14273660—22no—
82#14273660—22no—
83#14273660C2×C414yescomplemented
84#14273658C2×C414yescomplemented
85#14273647—22no—
86#14273647—22no—
87#14273647C2×C414yescomplemented
88#14273647C2×C414yescomplemented
89C22⋊(C59⋊C8)—22no—
90C22⋊(C59⋊C8)C2×C414yescomplemented
91#14273638—22no—
92C4×C59⋊C8—22no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C23×C4×C59—21no—
97C23×C4×C59C2311yescharacteristic
98#117×C59—22no—
99#117×C59—22no—
100#117×C59—22no—
101#117×C59C2×C414yescomplemented
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—22no—
104C2×C4⋊C4×C59—22no—
105C2×C4⋊C4×C59D411yescharacteristic
106C2×C4⋊C4×C59D411yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59D411yescharacteristic
113C2×C22⋊C4×C59D411yescharacteristic
114C2×C42×C59—21no—
115C2×C42×C59—21no—
116C2×C42×C59—21no—
117C2×C42×C59—21no—
118C2×C42×C59—21no—
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59—21no—
125C24×C59C2×D411yescharacteristic
126C23×D59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—24no—
136C22×C4×C59—81no—
137C22×C4×C59—81no—
138C22×C4×C59C22×C412yes—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59#4112yes—
145C22×C4×C59C2411yescharacteristic
146C2×C22⋊D59—24no—
147C2×C22⋊D59—81no—
148C2×C22⋊D59—81no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236—21no—
155C2×D236—21no—
156C2×D236—21no—
157C2×D236C22×C412yes—
158C2×C4×D59—42no—
159C2×C4×D59—81no—
160C2×C4×D59—81no—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59C22×C412yes—
169C4⋊C4×C59C22×C412yes—
170C4⋊C4×C59#4112yes—
171C4⋊C4×C59#4112yes—
172C22⋊C4×C59—24no—
173C22⋊C4×C59—22no—
174C22⋊C4×C59—22no—
175C42×C59—24no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—81no—
179C42×C59—81no—
180C42×C59—81no—
181C42×C59—81no—
182C42×C59#4112yes—
183C42×C59#4112yes—
184C22⋊Dic59—81no—
185(C2×D59)⋊C4—24no—
186(C2×D59)⋊C4—42no—
187(C2×D59)⋊C4—81no—
188(C2×D59)⋊C4—81no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59C22×C412yes—
195Dic59⋊C4—81no—
196C4×Dic59—81no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8C22×C412yes—
208C2×C59⋊C8C22×C412yes—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59C4×D412yes—
216C23×C59C22×D411yescharacteristic
217C22×D59—22no—
218C22×D59—41no—
219C22×D59—41no—
220C22×D59—41no—
221C22×D59—41no—
222C22×D59—41no—
223C22×D59—41no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—22no—
239C2×C4×C59—22no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
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—84no—
253C2×C4×C59C4×D412yes—
254C2×C4×C59#12312yes—
255C2×C4×C59#12312yes—
256C2×C4×C59C23×C411yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C22⋊D59—84no—
263C2×Dic59—22no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272D236—22no—
273D236—22no—
274D236—41no—
275D236—41no—
276C4×D59—84no—
277C59⋊C8—22no—
278C59⋊C8—22no—
279C59⋊C8—41no—
280C59⋊C8—41no—
281#21590—591nomaximal, Sylow
282C22×C59—22no—
283C22×C59—22no—
284C22×C59—22no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59—41no—
289C22×C59C2×C4×D411yescharacteristic
290C22×C59C2×#12311yescharacteristic
291C22×C59#52511yescharacteristic
292C22×C59#57611yescharacteristic
293C22×C59#57611yescharacteristic
294C22×C59#57711yescharacteristic
295C22×C59#57811yescharacteristic
296C2×D59—42no—
297C2×D59—42no—
298C2×D59—42no—
299C2×D59—81no—
300C2×D59—81no—
301C2×D59—81no—
302C2×D59—81no—
303C4×C59—22no—
304C4×C59—22no—
305C4×C59—22no—
306C4×C59—41no—
307C4×C59—41no—
308C4×C59—42no—
309C4×C59—42no—
310C4×C59—42no—
311C4×C59—42no—
312C4×C59C4×#4111yescharacteristic
313C4×C59C4×#4111yescharacteristic
314C4×C59#52911yescharacteristic
315C4×C59#52911yescharacteristic
316Dic59—42no—
317Dic59—81no—
318Dic59—81no—
319C2×#482—591no—
320#2324—591no—
321#2274D5911yescharacteristic, complemented
322#1925—591no—
323#1925—591no—
324#1921—591no—
325#1919—591no—
326#1874—591no—
327#1852—591no—
328#1825—591no—
329#1776—591no—
330C4×#107—591no—
331#1762—591no—
332C4×#102—591no—
333#1750—591no—
334C2×C59—42no—
335C2×C59C4×#12311yescharacteristic
336C2×C59#294611yescharacteristic
337C2×C59#295111yescharacteristic
338C2×C59#309611yescharacteristic
339C2×C59#309811yescharacteristic
340C2×C59#312111yescharacteristic
341C2×C59#312211yescharacteristic
342D59—82no—
343C2×#121—591no—
344C2×C4×D4—591no—
345C2×#117C2×D5911yescharacteristic
346C2×#117C2×D5911yescharacteristic
347C22×C4⋊C4C2×D5911yescharacteristic
348#482—598no—
349C2×#107—591no—
350C2×#107—591no—
351C2×#107—591no—
352C2×#106—591no—
353C2×#102—591no—
354C2×#102—591no—
355C2×#102—591no—
356C2×#102—591no—
357C2×C22⋊C8—591no—
358C2×C22⋊C8—591no—
359C2×#97—591no—
360C2×C4×C8—591no—
361#390—591no—
362#390—1181no—
363#389—591no—
364#388—1181no—
365#386—1181no—
366#385—1181no—
367#380—591no—
368C4×C4⋊C4—21no—
369C4×C4⋊C4—21no—
370C4×C4⋊C4—591no—
371C4×C4⋊C4C2×D5911yescharacteristic
372C4×C4⋊C4C2×D5911yescharacteristic
373C4×C4⋊C4C2×D5911yescharacteristic
374C4×C4⋊C4C2×D5911yescharacteristic
375C4×C22⋊C4—21no—
376C4×C22⋊C4—21no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#336—591no—
386#336—591no—
387C59#2159011yescharacteristic, complemented, Sylow
388C22×D4—591no—
389C22×D4—1181no—
390C23×C4—21no—
391C23×C4—1181no—
392C23×C4C22×D5911yescharacteristic
393C22×C8—591no—
394C22×C8—591no—
395C22×C8—591no—
396C22×C8—591no—
397#121—594no—
398#121—1182no—
399C4×D4—1182no—
400C4×D4—1182no—
401#117—22no—
402#117—22no—
403#117—22no—
404#117C4×D5914yescomplemented
405C2×C4⋊C4—21no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4C22⋊D5911yescharacteristic
413C2×C4⋊C4C22⋊D5911yescharacteristic
414C2×C4⋊C4C22×D5911yescharacteristic
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4—1181no—
421C2×C22⋊C4—1181no—
422C2×C22⋊C4—1181no—
423C2×C22⋊C4—1181no—
424C2×C22⋊C4C22⋊D5911yescharacteristic
425C2×C22⋊C4C22⋊D5911yescharacteristic
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—21no—
430C2×C42—21no—
431C2×C42—1181no—
432C2×C42C22×D5911yescharacteristic
433C2×C42C22×D5911yescharacteristic
434#107—594no—
435#107—1182no—
436#107—1182no—
437#106—594no—
438#102—594no—
439#102—594no—
440#102—1182no—
441#102—1182no—
442C22⋊C8—594no—
443C22⋊C8—1182no—
444#97—1182no—
445C4×C8—1182no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453C24—1181no—
454C24C2×C22⋊D5911yescharacteristic
455C2×D4—592no—
456C2×D4—1184no—
457C2×D4—1181no—
458C2×D4—1181no—
459C2×D4—1181no—
460C2×D4—1181no—
461C2×D4—1181no—
462C2×D4—4721no—
463C2×D4—4721no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—1181no—
476C22×C4—1181no—
477C22×C4—1181no—
478C22×C4—4721no—
479C22×C4—4721no—
480C22×C4—2362no—
481C22×C4C2×C4×D5912yes—
482C22×C4#1174568512yes—
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C2×C22⋊D5911yescharacteristic
485C22×C4C2×C22⋊D5911yescharacteristic
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C23×D5911yescharacteristic
489C2×C8—592no—
490C2×C8—592no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—592no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—1181no—
512C4⋊C4—1181no—
513C4⋊C4—1181no—
514C4⋊C4—4721no—
515C4⋊C4C2×C4×D5912yes—
516C4⋊C4C2×C4×D5912yes—
517C4⋊C4#1174568512yes—
518C4⋊C4#1174568512yes—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—1184no—
523C22⋊C4—4721no—
524C22⋊C4—4721no—
525C22⋊C4—4721no—
526C22⋊C4—2362no—
527C42—24no—
528C42—21no—
529C42—21no—
530C42—81no—
531C42—81no—
532C42—81no—
533C42—81no—
534C42—4721no—
535C42#1174568512yes—
536C42#1174568512yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—1182no—
544C23—2361no—
545C23—2361no—
546C23—2361no—
547C23—2361no—
548C23—2361no—
549C23—2361no—
550C23C4×C22⋊D5912yes—
551C23C22×C22⋊D5911yescharacteristic
552D4—1182no—
553D4—1182no—
554D4—4724no—
555D4—2361no—
556D4—2361no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—1182no—
587C2×C4—4724no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4C4×C22⋊D5912yes—
597C2×C4#1427371012yes—
598C2×C4#1427371012yes—
599C2×C4C22×C4×D5911yescharacteristic
600C2×C4C2×#1174568511yescharacteristic
601C2×C4C2×#1174568511yescharacteristic
602C2×C4C2×#1174568511yescharacteristic
603C2×C4C2×#1174568511yescharacteristic
604C2×C4C2×#1174568511yescharacteristic
605C8—1182no—
606C8—1182no—
607C8—2361no—
608C8—2361no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—4721no—
617C22—4721no—
618C22—4721no—
619C22—4721no—
620C22—2362no—
621C22—2362no—
622C22—2362no—
623C22C2×C4×C22⋊D5911yescharacteristic
624C22C2×#1427371011yescharacteristic
625C22#1683791311yescharacteristic
626C22#1683820511yescharacteristic
627C22#1683820511yescharacteristic
628C22#1683821011yescharacteristic
629C22#1683821711yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—42no—
638C4—42no—
639C4—4721no—
640C4—4721no—
641C4—2362no—
642C4C4×#1174568511yescharacteristic
643C4C4×#1174568511yescharacteristic
644C4#1683793611yescharacteristic
645C4#1683793611yescharacteristic
646C2—42no—
647C2—4722no—
648C2C4×#1427371011yescharacteristic
649C2#2143292411yescharacteristic
650C2#2143296911yescharacteristic
651C2#2143405411yescharacteristic
652C2#2143407411yescharacteristic
653C2#2143432511yescharacteristic
654C2#2143433911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2274C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
cosocleC22×C4×C59C24
hypercenter#2274D59
hypocenterC59#21590
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2274

Complements

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

NH ≅ G/Ntype
3. #21424716C2semidirect
4. #21424716C2semidirect
5. #21424688C2semidirect
6. #21424674C2semidirect
8. #21424284C2semidirect
9. #21424191C2semidirect
10. #21423994C2semidirect
11. C4×#14273660C2semidirect
12. #21423924C2semidirect
13. C4×#14273647C2semidirect
14. #21423862C2semidirect
15. C59×#2274C2semidirect
18. #16837726C4semidirect
83. #14273660C2×C4semidirect
84. #14273658C2×C4semidirect
87. #14273647C2×C4semidirect
88. #14273647C2×C4semidirect
90. C22⋊(C59⋊C8)C2×C4semidirect
101. #117×C59C2×C4semidirect
321. #2274D59semidirect
387. C59#21590semidirect
404. #117C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377262
#214267892
#214247162
#214246882
#214246742
#214244132
#214242842
#214241912
#214239942
C4×#142736602
#214239242
C4×#142736472
#214238622
C59×#22742
#2159059