Finite Group Databasebeta

← #28591780

Subgroups — #28591780

Order 15104.

938 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837512C211yesmaximal, characteristic
2#21426740C211yesmaximal, characteristic, complemented
3#21424723C211yesmaximal, characteristic, complemented
4#21424716C211yesmaximal, characteristic, complemented
5#21424681C211yesmaximal, characteristic, complemented
6#21424674C211yesmaximal, characteristic, complemented
7#21424444C211yesmaximal, characteristic
8#21424444C211yesmaximal, characteristic
9#21424357C211yesmaximal, characteristic, complemented
10#21424343C211yesmaximal, characteristic, complemented
11#21423994C211yesmaximal, characteristic, complemented
12C4×#14273647C211yesmaximal, characteristic, complemented
13C4×#14273648C211yesmaximal, characteristic, complemented
14#21423848C211yesmaximal, characteristic, complemented
15C59×#2275C211yesmaximal, characteristic, complemented
16C22×#11745669C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18#16837512C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837304C2211yescharacteristic
31#16837304C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837291—21no—
35#16837291—21no—
36#16837264—21no—
37#16837261—21no—
38C4×C4⋊Dic59C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C4×C4⋊C4×C59C2211yescharacteristic
53C4×C4⋊C4×C59C2211yescharacteristic
54C4×C4⋊C4×C59C2211yescharacteristic
55C4×C22⋊C4×C59—21no—
56C4×C22⋊C4×C59—21no—
57C4×C22⋊C4×C59—21no—
58C59×#376C2211yescharacteristic
59C2×#95×C59—21no—
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C2×#11745669—22no—
63C2×#11745669—22no—
64C2×#11745669C2×C414yes—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C4×D236—22no—
70C4×D236C2×C414yescomplemented
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×C4×Dic59—21no—
82#14273660—22no—
83#14273658—22no—
84#14273658C2×C414yescomplemented
85#14273648—22no—
86#14273648—22no—
87#14273648C2×C414yes—
88#14273647—22no—
89#14273647—22no—
90#14273647C2×C414yescomplemented
91#14273638C2×C414yescomplemented
92C4×C59⋊C8—22no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C22×D4×C59C2311yescharacteristic
97C23×C4×C59—21no—
98C23×C4×C59—21no—
99C23×C4×C59—21no—
100C4×D4×C59—22no—
101C4×D4×C59—22no—
102C4×D4×C59—22no—
103C4×D4×C59—22no—
104C4×D4×C59—22no—
105C4×D4×C59C2×C414yes—
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C42×C59—21no—
113C2×C42×C59—21no—
114C2×C42×C59—21no—
115C2×C42×C59C2311yescharacteristic
116C2×C42×C59C2311yescharacteristic
117C2×C42×C59C2311yescharacteristic
118#95×C59—21no—
119#95×C59—21no—
120#95×C59—21no—
121#95×C59D411yescharacteristic
122#95×C59D411yescharacteristic
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
125C24×C59—21no—
126C23×D59—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—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—24no—
137C22×C4×C59—24no—
138C22×C4×C59—24no—
139C22×C4×C59—81no—
140C22×C4×C59—81no—
141C22×C4×C59—81no—
142C22×C4×C59—81no—
143C22×C4×C59—81no—
144C22×C4×C59—81no—
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2411yescharacteristic
152C22×Dic59—21no—
153C22×Dic59—21no—
154C22×Dic59—21no—
155C2×D236—21no—
156C2×D236—21no—
157C2×D236—21no—
158C2×D236—21no—
159C2×D236—21no—
160C2×D236C22×C412yes—
161C2×C4×D59—24no—
162C2×C4×D59—81no—
163C2×C4×D59—81no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59C22×C412yes—
168C4⋊C4×C59#4112yes—
169C4⋊C4×C59#4112yes—
170C22⋊C4×C59—24no—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—24no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—81no—
178C42×C59—81no—
179C42×C59—81no—
180C42×C59C22×C412yes—
181C42×C59#4112yes—
182#11745669—28no—
183(C2×D59)⋊C4—24no—
184(C2×D59)⋊C4—81no—
185(C2×D59)⋊C4—81no—
186(C2×D59)⋊C4—81no—
187(C2×D59)⋊C4—81no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59C22×C412yes—
194Dic59⋊C4—81no—
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—22no—
210C23×C59—22no—
211C23×C59—22no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59C22×D411yescharacteristic
217C22×D59—22no—
218C22×D59—41no—
219C22×D59—41no—
220C22×D59—41no—
221C22×D59—41no—
222C22×D59—41no—
223C22×D59—41no—
224D4×C59—22no—
225D4×C59—22no—
226D4×C59—22no—
227D4×C59#11714yes—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—22no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
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—21no—
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×C59—84no—
255C2×C4×C59—84no—
256C2×C4×C59C4×D412yes—
257C2×C4×C59C4×D412yes—
258C2×C4×C59C23×C411yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×Dic59—22no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273D236—22no—
274D236—22no—
275D236—41no—
276D236—41no—
277C4×D59—84no—
278C59⋊C8—22no—
279C59⋊C8—22no—
280C59⋊C8—41no—
281C59⋊C8—41no—
282#21126—591nomaximal, Sylow
283C22×C59—24no—
284C22×C59—42no—
285C22×C59—42no—
286C22×C59—42no—
287C22×C59—42no—
288C22×C59—42no—
289C22×C59—42no—
290C22×C59C2×C4×D411yescharacteristic
291C22×C59#52511yescharacteristic
292C22×C59#52511yescharacteristic
293C22×C59#57611yescharacteristic
294C22×C59#57611yescharacteristic
295C22×C59#57711yescharacteristic
296C22×C59#57711yescharacteristic
297C2×D59—42no—
298C2×D59—42no—
299C2×D59—42no—
300C2×D59—81no—
301C2×D59—81no—
302C2×D59—81no—
303C2×D59—81no—
304C4×C59—22no—
305C4×C59—22no—
306C4×C59—41no—
307C4×C59—42no—
308C4×C59—41no—
309C4×C59—41no—
310C4×C59—42no—
311C4×C59—41no—
312C4×C59—42no—
313C4×C59C2×#11711yescharacteristic
314C4×C59C4×#4111yescharacteristic
315C4×C59#52911yescharacteristic
316C4×C59#52911yescharacteristic
317Dic59—42no—
318Dic59—81no—
319Dic59—81no—
320C2×#442—591no—
321#2319—591no—
322#2275D5911yescharacteristic, complemented
323#1926—591no—
324#1925—591no—
325#1920—591no—
326#1919—591no—
327#1879—591no—
328#1879—591no—
329#1864—591no—
330#1862—591no—
331#1776—591no—
332C4×#102—591no—
333C4×#102—591no—
334#1746—591no—
335C2×C59—44no—
336C2×C59#229011yescharacteristic
337C2×C59#294611yescharacteristic
338C2×C59#294611yescharacteristic
339C2×C59#311811yescharacteristic
340C2×C59#311911yescharacteristic
341C2×C59#312211yescharacteristic
342C2×C59#312511yescharacteristic
343D59—82no—
344C22×D8—591no—
345C2×C4×D4—591no—
346C2×C4×D4C2×D5911yescharacteristic
347C2×C4×D4C2×D5911yescharacteristic
348C2×C4×D4C2×D5911yescharacteristic
349#442—598no—
350C2×#107—591no—
351C2×#106—591no—
352C2×#106—591no—
353C2×#102—591no—
354C2×#102—591no—
355C2×#102—591no—
356C2×#102—591no—
357C2×#102—591no—
358C2×#102—591no—
359C2×#97—591no—
360C2×C4×C8—591no—
361#390—591no—
362#390—591no—
363#389—591no—
364#389—591no—
365#388—1181no—
366#388—1181no—
367#381—1181no—
368#380—1181no—
369C4×C4⋊C4—591no—
370C4×C4⋊C4C2×D5911yescharacteristic
371C4×C4⋊C4C2×D5911yescharacteristic
372C4×C4⋊C4C2×D5911yescharacteristic
373C4×C22⋊C4—21no—
374C4×C22⋊C4—21no—
375C4×C22⋊C4—21no—
376#376C2×D5911yescharacteristic
377C2×#95—21no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#336—591no—
385#336—591no—
386#336—591no—
387#336—591no—
388C59#2112611yescharacteristic, complemented, Sylow
389C22×D4—591no—
390C22×D4C22×D5911yescharacteristic
391C23×C4—21no—
392C23×C4—21no—
393C23×C4—21no—
394C23×C4—1181no—
395C2×D8—594no—
396C2×D8—1182no—
397C2×D8—1182no—
398C22×C8—591no—
399C22×C8—591no—
400C22×C8—591no—
401C22×C8—591no—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—594no—
408C4×D4—1182no—
409C4×D4C4×D5914yes—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1181no—
422C2×C22⋊C4—1181no—
423C2×C22⋊C4—1181no—
424C2×C22⋊C4—1181no—
425C2×C22⋊C4—1181no—
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—1181no—
430C2×C42C22×D5911yescharacteristic
431C2×C42C22×D5911yescharacteristic
432C2×C42C22×D5911yescharacteristic
433#107—1182no—
434#106—594no—
435#106—1182no—
436#102—594no—
437#102—594no—
438#102—1182no—
439#102—1182no—
440#102—1182no—
441#102—1182no—
442#97—594no—
443C4×C8—1182no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95C22⋊D5911yescharacteristic
451#95C22⋊D5911yescharacteristic
452#95C22⋊D5911yescharacteristic
453#95C22⋊D5911yescharacteristic
454C24—21no—
455C24—1181no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—592no—
460C2×D4—1181no—
461C2×D4—1181no—
462C2×D4—1181no—
463C2×D4—1181no—
464C2×D4—1181no—
465C2×D4C2×C4×D5912yes—
466C2×D4#1174568512yes—
467C2×D4#1174568512yes—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4—1181no—
481C22×C4—1184no—
482C22×C4—1181no—
483C22×C4—1181no—
484C22×C4—4721no—
485C22×C4—4721no—
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C2×C22⋊D5911yescharacteristic
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C23×D5911yescharacteristic
493D8—1188no—
494C2×C8—592no—
495C2×C8—592no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—592no—
510C4⋊C4—1181no—
511C4⋊C4—1181no—
512C4⋊C4—1181no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—4721no—
516C4⋊C4—4721no—
517C4⋊C4C2×C4×D5912yes—
518C4⋊C4#1174568512yes—
519C4⋊C4#1174568512yes—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—1184no—
524C22⋊C4—4721no—
525C22⋊C4—4721no—
526C22⋊C4—4721no—
527C22⋊C4—4721no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—81no—
533C42—81no—
534C42—81no—
535C42—4721no—
536C42C2×C4×D5912yes—
537C42#1174568512yes—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—1182no—
546C23—2361no—
547C23—2361no—
548C23—2361no—
549C23—2361no—
550C23—2361no—
551C23—2361no—
552C23C22×C22⋊D5911yescharacteristic
553D4—22no—
554D4—22no—
555D4—22no—
556D4—1182no—
557D4—1182no—
558D4—2361no—
559D4—2361no—
560D4#1427368414yes—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
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—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—1182no—
590C2×C4—4724no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4C4×C22⋊D5912yes—
600C2×C4C4×C22⋊D5912yes—
601C2×C4C22×C4×D5911yescharacteristic
602C2×C4C2×#1174568511yescharacteristic
603C2×C4C2×#1174568511yescharacteristic
604C2×C4C2×#1174568511yescharacteristic
605C2×C4C2×#1174568511yescharacteristic
606C2×C4C2×#1174568511yescharacteristic
607C8—1182no—
608C8—1182no—
609C8—2361no—
610C8—2361no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—4721no—
619C22—4721no—
620C22—4721no—
621C22—4721no—
622C22—2362no—
623C22—2362no—
624C22—2362no—
625C22C2×C4×C22⋊D5911yescharacteristic
626C22#1683791311yescharacteristic
627C22#1683791311yescharacteristic
628C22#1683820511yescharacteristic
629C22#1683820511yescharacteristic
630C22#1683821011yescharacteristic
631C22#1683821311yescharacteristic
632C4—22no—
633C4—22no—
634C4—41no—
635C4—42no—
636C4—41no—
637C4—42no—
638C4—41no—
639C4—41no—
640C4—42no—
641C4—4721no—
642C4—4721no—
643C4—2362no—
644C4C2×#1427368411yescharacteristic
645C4C4×#1174568511yescharacteristic
646C4#1683793611yescharacteristic
647C4#1683793611yescharacteristic
648C2—44no—
649C2—4722no—
650C2#2142653911yescharacteristic
651C2#2143292411yescharacteristic
652C2#2143292811yescharacteristic
653C2#2143428011yescharacteristic
654C2#2143429511yescharacteristic
655C2#2143433911yescharacteristic
656C2#2143437311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2275C2
socleC23×C59C22×D4
prime radicalC22×#11745669C22
cosocleC22×C4×C59C24
hypercenter#2275D59
hypocenterC59#21126
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ #376 ◁ #2275

Complements

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

NH ≅ G/Ntype
2. #21426740C2semidirect
3. #21424723C2semidirect
4. #21424716C2semidirect
5. #21424681C2semidirect
6. #21424674C2semidirect
9. #21424357C2semidirect
10. #21424343C2semidirect
11. #21423994C2semidirect
12. C4×#14273647C2semidirect
13. C4×#14273648C2semidirect
14. #21423848C2semidirect
15. C59×#2275C2semidirect
18. #16837512C4semidirect
70. C4×D236C2×C4semidirect
84. #14273658C2×C4semidirect
90. #14273647C2×C4semidirect
91. #14273638C2×C4semidirect
322. #2275D59semidirect
388. C59#21126semidirect

Maximal subgroups

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

subgroupindex
C2×#168375122
#214267402
#214247232
#214247162
#214246812
#214246742
#214244442
#214243572
#214243432
#214239942
C4×#142736472
C4×#142736482
#214238482
C59×#22752
#2112659