Finite Group Databasebeta

← #28595942

Subgroups — #28595942

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×#16837719C211yesmaximal, characteristic
2#21426783C211yesmaximal, characteristic
3#21426450C211yesmaximal, characteristic, complemented
4#21424725C211yesmaximal, characteristic, complemented
5#21424719C211yesmaximal, characteristic, complemented
6#21424679C211yesmaximal, characteristic, complemented
7#21424673C211yesmaximal, characteristic, complemented
8#21424415C211yesmaximal, characteristic
9#21424296C211yesmaximal, characteristic, complemented
10#21424181C211yesmaximal, characteristic, complemented
11#21423994C211yesmaximal, characteristic, complemented
12C4×#14273659C211yesmaximal, characteristic, complemented
13#21423921C211yesmaximal, characteristic, complemented
14#21423863C211yesmaximal, characteristic, complemented
15C4×#102×C59C211yesmaximal, characteristic, complemented
16C2×#14273698C2211yescharacteristic
17C2×#14273685C2211yescharacteristic
18C2×#14273684C2211yescharacteristic
19C2×C4⋊C4×D59C2211yescharacteristic
20#16837719C418yescomplemented
21C2×#14273660C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×(C2×D59)⋊C8C2211yescharacteristic
28C2×(C2×D59)⋊C8C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30#16837305C2211yescharacteristic
31#16837302—21no—
32#16837298C2211yescharacteristic
33#16837296—21no—
34#16837281—21no—
35#16837275—21no—
36#16837259C2211yescharacteristic
37C4⋊C4×Dic59C2211yescharacteristic
38C4⋊C4×Dic59C2211yescharacteristic
39C4×Dic59⋊C4—21no—
40C4×C4⋊Dic59C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42#16837253—21no—
43C4×(C2×D59)⋊C4—21no—
44#16837249—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×C4×C8×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×C22⋊D59—21no—
61C22×C4×D59—21no—
62C22×C4×D59C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273698—22no—
66#14273698C2×C414yes—
67#14273685—22no—
68#14273685—22no—
69#14273684—22no—
70#14273684—22no—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4D411yescharacteristic
78C2×Dic59⋊C4D411yescharacteristic
79C4⋊C4×D59—22no—
80C4⋊C4×D59C2×C414yescomplemented
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4D411yescharacteristic
87C2×(C2×D59)⋊C4D411yescharacteristic
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59—21no—
91C2×C4×Dic59—21no—
92C2×C4×Dic59—21no—
93C2×C4×Dic59C2311yescharacteristic
94#14273660—22no—
95#14273659—22no—
96#14273659C2×C414yescomplemented
97#14273658C2×C414yescomplemented
98#14273648—22no—
99#14273648C2×C414yes—
100(C2×D59)⋊C8—22no—
101(C2×D59)⋊C8C2×C414yescomplemented
102#14273638—22no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C22×D4×C59C2311yescharacteristic
110C23×C4×C59—21no—
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C4×D4×C59—22no—
114C4×D4×C59—22no—
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C22⋊C4×C59—21no—
119C2×C42×C59—21no—
120C2×C42×C59C2311yescharacteristic
121#102×C59—22no—
122#102×C59C2×C414yes—
123C4×C8×C59—22no—
124#95×C59—21no—
125C24×C59—21no—
126C23×D59C2×D411yescharacteristic
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59C22×C412yes—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—42no—
137C22×C4×C59—81no—
138C22×C4×C59—81no—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2411yescharacteristic
142C2×C22⋊D59—24no—
143C2×C22⋊D59—81no—
144C2×C22⋊D59—81no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151C22×Dic59C2×D411yescharacteristic
152C2×C4×D59—21no—
153C2×C4×D59—21no—
154C2×C4×D59—21no—
155C2×C4×D59—21no—
156C2×C4×D59—24no—
157C2×C4×D59—81no—
158C2×C4×D59—81no—
159C2×C4×D59C22×C412yes—
160C2×C4×D59#4112yes—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59C22×C412yes—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59C22×C412yes—
172C4⋊C4×C59#4112yes—
173C22⋊C4×C59—42no—
174C42×C59—81no—
175C42×C59#4112yes—
176C42×C59#4112yes—
177C22⋊Dic59—24no—
178C22⋊Dic59—81no—
179C22⋊Dic59—81no—
180(C2×D59)⋊C4—24no—
181(C2×D59)⋊C4—22no—
182(C2×D59)⋊C4—22no—
183(C2×D59)⋊C4—81no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59C22×C412yes—
190C4⋊Dic59C22×C412yes—
191C4⋊Dic59#4112yes—
192Dic59⋊C4—22no—
193Dic59⋊C4—22no—
194Dic59⋊C4—24no—
195Dic59⋊C4—81no—
196C4×Dic59—24no—
197C4×Dic59—21no—
198C4×Dic59—21no—
199C4×Dic59—81no—
200C4×Dic59—81no—
201C4×Dic59—81no—
202C4×Dic59—81no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8C22×C412yes—
209C23×C59—22no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59C22×D411yescharacteristic
217C22×D59—21no—
218C22×D59—21no—
219C22×D59—21no—
220C22×D59—21no—
221C22×D59—21no—
222C22×D59—21no—
223C22×D59C4×D412yes—
224D4×C59—22no—
225D4×C59—22no—
226D4×C59—41no—
227D4×C59—41no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
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×C59—84no—
242C2×C4×C59#12312yes—
243C2×C4×C59#12312yes—
244C2×C4×C59C23×C411yescharacteristic
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C22⋊D59—84no—
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—22no—
254C2×Dic59—22no—
255C2×Dic59—21no—
256C2×Dic59—21no—
257C2×Dic59—22no—
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—41no—
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×Dic59C4×D412yes—
272C4×D59—22no—
273C4×D59—22no—
274C4×D59—22no—
275C4×D59—22no—
276C4×D59—84no—
277C8×C59—22no—
278C8×C59—41no—
279C59⋊C8—22no—
280C59⋊C8—41no—
281#21584—591nomaximal, Sylow
282C22×C59—42no—
283C22×C59—42no—
284C22×C59—42no—
285C22×C59—81no—
286C22×C59—81no—
287C22×C59—81no—
288C22×C59—81no—
289C22×C59C2×C4×D411yescharacteristic
290C22×C59C2×#12311yescharacteristic
291C22×C59#52511yescharacteristic
292C22×C59#57611yescharacteristic
293C22×C59#57711yescharacteristic
294C22×C59#57711yescharacteristic
295C22×C59#57711yescharacteristic
296C2×D59—22no—
297C2×D59—22no—
298C2×D59—22no—
299C2×D59—41no—
300C2×D59—41no—
301C2×D59—41no—
302C2×D59—41no—
303C4×C59—22no—
304C4×C59—22no—
305C4×C59—42no—
306C4×C59—42no—
307C4×C59—81no—
308C4×C59C4×#4111yescharacteristic
309C4×C59C4×#4111yescharacteristic
310C4×C59#52911yescharacteristic
311C4×C59#52911yescharacteristic
312Dic59—22no—
313Dic59—41no—
314Dic59—41no—
315Dic59—42no—
316Dic59—42no—
317Dic59—42no—
318Dic59—81no—
319C2×#481—591no—
320#2324—591no—
321#2274—591no—
322#1926—591no—
323#1925—591no—
324#1920—591no—
325#1919—591no—
326#1875—591no—
327#1854—591no—
328#1823—591no—
329#1776—591no—
330C4×#106—591no—
331#1762—591no—
332C4×#102D5911yescharacteristic, complemented
333#1750—591no—
334C2×C59—82no—
335C2×C59C4×#12311yescharacteristic
336C2×C59#294611yescharacteristic
337C2×C59#294911yescharacteristic
338C2×C59#309611yescharacteristic
339C2×C59#309911yescharacteristic
340C2×C59#311911yescharacteristic
341C2×C59#312511yescharacteristic
342D59—42no—
343C2×#121—591no—
344C2×C4×D4C2×D5911yescharacteristic
345C2×#117—591no—
346C2×#117—591no—
347C22×C4⋊C4—591no—
348#481—598no—
349C2×#107—591no—
350C2×#106—591no—
351C2×#106—591no—
352C2×#106—591no—
353C2×#102—591no—
354C2×#102—591no—
355C2×#102C2×D5911yescharacteristic
356C2×#102C2×D5911yescharacteristic
357C2×C22⋊C8—591no—
358C2×C22⋊C8—591no—
359C2×#97—591no—
360C2×C4×C8C2×D5911yescharacteristic
361#390—591no—
362#390—1181no—
363#389—591no—
364#388—1181no—
365#386—1181no—
366#385—1181no—
367#380—591no—
368C4×C4⋊C4—591no—
369C4×C4⋊C4—591no—
370C4×C4⋊C4—591no—
371C4×C4⋊C4—591no—
372C4×C4⋊C4—1181no—
373C4×C4⋊C4—1181no—
374C4×C4⋊C4C2×D5911yescharacteristic
375C4×C22⋊C4—1181no—
376C4×C22⋊C4—1181no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340C2×D5911yescharacteristic
384#340C2×D5911yescharacteristic
385#336—591no—
386#336—591no—
387C59#2158411yescharacteristic, complemented, Sylow
388C22×D4—1181no—
389C22×D4C22×D5911yescharacteristic
390C23×C4—21no—
391C23×C4—591no—
392C23×C4—1181no—
393C22×C8—591no—
394C22×C8—591no—
395C22×C8C22×D5911yescharacteristic
396C22×C8C22×D5911yescharacteristic
397#121—594no—
398#121—1182no—
399C4×D4—22no—
400C4×D4—22no—
401#117—1182no—
402#117—1182no—
403#117—1182no—
404#117—1182no—
405C2×C4⋊C4—591no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—594no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—1182no—
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—591no—
420C2×C22⋊C4—591no—
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—591no—
428C2×C42—1181no—
429C2×C42—1181no—
430C2×C42—1181no—
431C2×C42—1181no—
432C2×C42—1181no—
433C2×C42C22×D5911yescharacteristic
434#107—1182no—
435#106—594no—
436#106—594no—
437#106—1182no—
438#102—22no—
439#102—594no—
440#102—1182no—
441#102C4×D5914yes—
442C22⋊C8—594no—
443C22⋊C8—1182no—
444#97—1182no—
445C4×C8—22no—
446#95—21no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453C24—21no—
454C24—591no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—1184no—
461C2×D4—4721no—
462C2×D4—4721no—
463C2×D4C2×C4×D5912yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—42no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—591no—
471C22×C4—592no—
472C22×C4—592no—
473C22×C4—591no—
474C22×C4—591no—
475C22×C4—1184no—
476C22×C4—1181no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—1181no—
480C22×C4—1181no—
481C22×C4—1181no—
482C22×C4—1181no—
483C22×C4—1181no—
484C22×C4—4721no—
485C22×C4—4721no—
486C22×C4D4×D5911yescharacteristic
487C22×C4D4×D5911yescharacteristic
488C22×C4C23×D5911yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—592no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8C2×C4×D5912yes—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—592no—
506C4⋊C4—592no—
507C4⋊C4—592no—
508C4⋊C4—1184no—
509C4⋊C4—1182no—
510C4⋊C4—1182no—
511C4⋊C4—1181no—
512C4⋊C4—1181no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—4721no—
517C4⋊C4C2×C4×D5912yes—
518C4⋊C4#1174568512yes—
519C22⋊C4—42no—
520C22⋊C4—1184no—
521C22⋊C4—1184no—
522C22⋊C4—1182no—
523C22⋊C4—1182no—
524C22⋊C4—4721no—
525C22⋊C4—4721no—
526C22⋊C4—4721no—
527C42—81no—
528C42—1184no—
529C42—1181no—
530C42—1181no—
531C42—4721no—
532C42—4721no—
533C42—4721no—
534C42—4721no—
535C42#1174568512yes—
536C42#1174568712yes—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—592no—
545C23—1181no—
546C23—1181no—
547C23—1181no—
548C23—1181no—
549C23—1181no—
550C23—1181no—
551C23C2×D4×D5911yescharacteristic
552D4—22no—
553D4—22no—
554D4—41no—
555D4—41no—
556D4—4724no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—84no—
571C2×C4—592no—
572C2×C4—1181no—
573C2×C4—1181no—
574C2×C4—1182no—
575C2×C4—1182no—
576C2×C4—1182no—
577C2×C4—1182no—
578C2×C4—1181no—
579C2×C4—1181no—
580C2×C4—1182no—
581C2×C4—1181no—
582C2×C4—1181no—
583C2×C4—1181no—
584C2×C4—1181no—
585C2×C4—1182no—
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×C4—2361no—
597C2×C4#1427370912yes—
598C2×C4#1427370912yes—
599C2×C4C22×C4×D5911yescharacteristic
600C2×C4#41×D5911yescharacteristic
601C2×C4#41×D5911yescharacteristic
602C2×C4C2×#1174568711yescharacteristic
603C2×C4C2×#1174568511yescharacteristic
604C2×C4C2×#1174568511yescharacteristic
605C8—22no—
606C8—41no—
607C8—1182no—
608C8—2361no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—1182no—
617C22—1182no—
618C22—1182no—
619C22—2361no—
620C22—2361no—
621C22—2361no—
622C22—2361no—
623C22C4×D4×D5911yescharacteristic
624C22C2×#1427370911yescharacteristic
625C22#1683791111yescharacteristic
626C22#1683820311yescharacteristic
627C22#1683820811yescharacteristic
628C22#1683820911yescharacteristic
629C22#1683820911yescharacteristic
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—81no—
635C4—1182no—
636C4—4721no—
637C4—2362no—
638C4—2362no—
639C4—2362no—
640C4—2361no—
641C4—2361no—
642C4C4×#1174568711yescharacteristic
643C4C4×#1174568511yescharacteristic
644C4#1683793011yescharacteristic
645C4#1683793411yescharacteristic
646C2—82no—
647C2—2362no—
648C2C4×#1427370911yescharacteristic
649C2#2143292111yescharacteristic
650C2#2143295011yescharacteristic
651C2#2143405011yescharacteristic
652C2#2143408311yescharacteristic
653C2#2143429111yescharacteristic
654C2#2143437911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC4×#102×C59C2
socleC23×C59C22×D4
prime radicalC2×#14273698C22
cosocleC22×C4×C59C24
hypercenterC4×#102D59
hypocenterC59#21584
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
3. #21426450C2semidirect
4. #21424725C2semidirect
5. #21424719C2semidirect
6. #21424679C2semidirect
7. #21424673C2semidirect
9. #21424296C2semidirect
10. #21424181C2semidirect
11. #21423994C2semidirect
12. C4×#14273659C2semidirect
13. #21423921C2semidirect
14. #21423863C2semidirect
15. C4×#102×C59C2semidirect
20. #16837719C4semidirect
80. C4⋊C4×D59C2×C4semidirect
96. #14273659C2×C4semidirect
97. #14273658C2×C4semidirect
101. (C2×D59)⋊C8C2×C4semidirect
332. C4×#102D59semidirect
387. C59#21584semidirect

Maximal subgroups

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

subgroupindex
C2×#168377192
#214267832
#214264502
#214247252
#214247192
#214246792
#214246732
#214244152
#214242962
#214241812
#214239942
C4×#142736592
#214239212
#214238632
C4×#102×C592
#2158459