Finite Group Databasebeta

← #28596048

Subgroups — #28596048

Order 15104.

844 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837745C211yesmaximal, characteristic
2#21426776C211yesmaximal, characteristic
3#21426450C211yesmaximal, characteristic
4#21424731C211yesmaximal, characteristic, complemented
5#21424730C211yesmaximal, characteristic, complemented
6#21424671C211yesmaximal, characteristic, complemented
7#21424402C211yesmaximal, characteristic
8#21424290C211yesmaximal, characteristic
9#21424190C211yesmaximal, characteristic
10#21423995C211yesmaximal, characteristic, complemented
11#21423984C211yesmaximal, characteristic, complemented
12#21423925C211yesmaximal, characteristic, complemented
13#21423915C211yesmaximal, characteristic, complemented
14#21423861C211yesmaximal, characteristic
15C59×#1921C211yesmaximal, characteristic, complemented
16C2×#14273707C2211yescharacteristic
17C2×#14273692C2211yescharacteristic
18C2×#14273685C2211yescharacteristic
19C2×#14273684C2211yescharacteristic
20C2×C4⋊C4×D59C2211yescharacteristic
21#16837745C418yescomplemented
22C2×#14273659C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×(C2×D59)⋊C8C2211yescharacteristic
30C2×(C2×D59)⋊C8C2211yescharacteristic
31C2×#14273637C2211yescharacteristic
32C2×C8×Dic59C2211yescharacteristic
33#16837308C2211yescharacteristic
34#16837298—21no—
35#16837292—21no—
36#16837291—21no—
37#16837285—21no—
38#16837259C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4×Dic59⋊C4—21no—
43C4×C4⋊Dic59C2211yescharacteristic
44#16837253—21no—
45C4×(C2×D59)⋊C4—21no—
46#16837249—21no—
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837157C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C4×C4⋊C4×C59C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C22×C4×D59—21no—
62C22×C4×D59C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273707—22no—
66#14273707C2×C414yes—
67#14273692—22no—
68#14273692—22no—
69#14273685—22no—
70#14273685C2×C414yes—
71#14273684—22no—
72#14273684—22no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4D411yescharacteristic
80C2×Dic59⋊C4D411yescharacteristic
81C4⋊C4×D59—22no—
82C4⋊C4×D59—22no—
83C2×(C2×D59)⋊C4—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×Dic59—21no—
94C2×C4×Dic59C2311yescharacteristic
95#14273659—22no—
96#14273658—22no—
97#14273658C2×C414yescomplemented
98#14273652—22no—
99#14273652C2×C414yescomplemented
100#14273650—22no—
101#14273650C2×C414yescomplemented
102(C2×D59)⋊C8—22no—
103(C2×D59)⋊C8C2×C414yes—
104#14273637—22no—
105C8×Dic59—22no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95—21no—
112C59⋊#95—21no—
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C42×C59—21no—
121C2×C42×C59C2311yescharacteristic
122#107×C59C2×C414yescomplemented
123#95×C59—21no—
124#95×C59—21no—
125C23×D59C2×D411yescharacteristic
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2411yescharacteristic
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59C2×D411yescharacteristic
138C22×Dic59C2×D411yescharacteristic
139C22×Dic59C2×D411yescharacteristic
140C2×C4×D59—21no—
141C2×C4×D59—21no—
142C2×C4×D59—21no—
143C2×C4×D59—21no—
144C2×C4×D59—24no—
145C2×C4×D59—81no—
146C2×C4×D59—81no—
147C2×C4×D59C22×C412yes—
148C2×C4×D59#4112yes—
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118C22×C412yes—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59C22×C412yes—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—81no—
167C4⋊C4×C59C22×C412yes—
168C4⋊C4×C59C22×C412yes—
169C4⋊C4×C59#4112yes—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—81no—
173(C2×D59)⋊C4—24no—
174(C2×D59)⋊C4—24no—
175(C2×D59)⋊C4—22no—
176(C2×D59)⋊C4—22no—
177(C2×D59)⋊C4—81no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—24no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59C22×C412yes—
184C4⋊Dic59#4112yes—
185Dic59⋊C4—24no—
186Dic59⋊C4—22no—
187Dic59⋊C4—22no—
188Dic59⋊C4—42no—
189C4×Dic59—24no—
190C4×Dic59—42no—
191C4×Dic59—81no—
192C4×Dic59—81no—
193C4×Dic59—81no—
194C4×Dic59—81no—
195C4×Dic59#4112yes—
196C4×Dic59#4112yes—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8C22×C412yes—
203C23×C59C22×D411yescharacteristic
204C22×D59—21no—
205C22×D59—21no—
206C22×D59—21no—
207C22×D59—21no—
208C22×D59—21no—
209C22×D59—21no—
210C22×D59C4×D412yes—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—22no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59C23×C411yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—22no—
242C2×Dic59—22no—
243C2×Dic59—22no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59C4×D412yes—
254C2×Dic59#12312yes—
255C2×Dic59#12312yes—
256C4×D59—22no—
257C4×D59—22no—
258C4×D59—22no—
259C4×D59—22no—
260C4×D59—84no—
261Dic118—22no—
262Dic118—22no—
263Dic118—41no—
264Dic118—41no—
265C8×C59—22no—
266C8×C59—41no—
267C59⋊C8—22no—
268C59⋊C8—41no—
269#21591—591nomaximal, Sylow
270C22×C59C2×C4×D411yescharacteristic
271C22×C59C2×#12311yescharacteristic
272C22×C59#52511yescharacteristic
273C22×C59#57711yescharacteristic
274C22×C59#57711yescharacteristic
275C22×C59#57711yescharacteristic
276C22×C59#57811yescharacteristic
277C2×D59—22no—
278C2×D59—22no—
279C2×D59—22no—
280C2×D59—41no—
281C2×D59—41no—
282C2×D59—41no—
283C2×D59—41no—
284C4×C59—41no—
285C4×C59—41no—
286C4×C59—42no—
287C4×C59—42no—
288C4×C59—42no—
289C4×C59—81no—
290C4×C59C4×#4111yescharacteristic
291C4×C59C4×#4111yescharacteristic
292C4×C59#52911yescharacteristic
293C4×C59#52911yescharacteristic
294Dic59—22no—
295Dic59—22no—
296Dic59—22no—
297Dic59—42no—
298Dic59—42no—
299Dic59—42no—
300Dic59—81no—
301C2×#485—591no—
302#2323—591no—
303#2274—591no—
304#1927—591no—
305#1927—591no—
306#1921D5911yescharacteristic, complemented
307#1919—591no—
308#1872—591no—
309#1853—591no—
310#1825—591no—
311#1776—591no—
312C4×#106—591no—
313#1763—591no—
314C4×#103—591no—
315#1750—591no—
316C2×C59C4×#12311yescharacteristic
317C2×C59#294711yescharacteristic
318C2×C59#295211yescharacteristic
319C2×C59#309711yescharacteristic
320C2×C59#309911yescharacteristic
321C2×C59#312111yescharacteristic
322C2×C59#312311yescharacteristic
323D59—42no—
324C2×C22⋊Q8—591no—
325C2×C4×Q8—591no—
326C2×#117—591no—
327C2×#117—591no—
328C22×C4⋊C4—591no—
329#485—598no—
330C2×#107C2×D5911yescharacteristic
331C2×#106—591no—
332C2×#106—591no—
333C2×#106—591no—
334C2×#103—591no—
335C2×#103—591no—
336C2×#103—591no—
337C2×#103—591no—
338C2×C22⋊C8—591no—
339C2×C22⋊C8—591no—
340C2×#97—591no—
341C2×C4×C8—591no—
342#391—591no—
343#389—1181no—
344#389C2×D5911yescharacteristic
345#388—1181no—
346#388—1181no—
347#387—1181no—
348#380—591no—
349C4×C4⋊C4—591no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—1181no—
354C4×C4⋊C4—1181no—
355C4×C4⋊C4C2×D5911yescharacteristic
356C4×C22⋊C4—1181no—
357C4×C22⋊C4—1181no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340C2×D5911yescharacteristic
364#340C2×D5911yescharacteristic
365#340C2×D5911yescharacteristic
366#336—591no—
367#336C2×D5911yescharacteristic
368C59#2159111yescharacteristic, complemented, Sylow
369C22×Q8—591no—
370C23×C4—591no—
371C23×C4—1181no—
372C22×C8—591no—
373C22×C8—591no—
374C22×C8C22×D5911yescharacteristic
375C22×C8C22×D5911yescharacteristic
376C22⋊Q8—594no—
377C22⋊Q8—1182no—
378C4×Q8—1182no—
379C4×Q8—1182no—
380#117—594no—
381#117—1182no—
382#117—1182no—
383#117—1182no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—1182no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1182no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C22⋊C4—591no—
400C2×C22⋊C4—591no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4—1181no—
403C2×C22⋊C4—1181no—
404C2×C42—21no—
405C2×C42—591no—
406C2×C42—1181no—
407C2×C42—1181no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42—1181no—
411C2×C42—1181no—
412C2×C42C22×D5911yescharacteristic
413#107C4×D5914yescomplemented
414#106—594no—
415#106—1182no—
416#106—1182no—
417#103—594no—
418#103—594no—
419#103—1182no—
420#103—1182no—
421C22⋊C8—594no—
422C22⋊C8—1182no—
423#97—1182no—
424C4×C8—1182no—
425#95—21no—
426#95—21no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434C24—591no—
435C2×Q8—592no—
436C2×Q8—1181no—
437C2×Q8—1181no—
438C2×Q8—1181no—
439C2×Q8—1181no—
440C2×Q8—1181no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—591no—
446C22×C4—591no—
447C22×C4—591no—
448C22×C4—592no—
449C22×C4—592no—
450C22×C4—1184no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—4721no—
460C22×C4—4721no—
461C22×C4D4×D5911yescharacteristic
462C22×C4D4×D5911yescharacteristic
463C22×C4C23×D5911yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—592no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8C2×C4×D5912yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—81no—
482C4⋊C4—592no—
483C4⋊C4—592no—
484C4⋊C4—1182no—
485C4⋊C4—1182no—
486C4⋊C4—1184no—
487C4⋊C4—1184no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—2362no—
493C4⋊C4C2×C4×D5912yes—
494C4⋊C4C2×C4×D5912yes—
495C4⋊C4#1174568912yes—
496C22⋊C4—1182no—
497C22⋊C4—1182no—
498C22⋊C4—1184no—
499C22⋊C4—1184no—
500C22⋊C4—4721no—
501C42—21no—
502C42—21no—
503C42—81no—
504C42—592no—
505C42—592no—
506C42—1184no—
507C42—4721no—
508C42—4721no—
509C42—4721no—
510C42—4721no—
511C42—2362no—
512C23—592no—
513C23—1181no—
514C23—1181no—
515C23—1181no—
516C23—1181no—
517C23—1181no—
518C23—1181no—
519C23C2×D4×D5911yescharacteristic
520Q8—1182no—
521Q8—1182no—
522Q8—2361no—
523Q8—2361no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—592no—
543C2×C4—592no—
544C2×C4—592no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1182no—
552C2×C4—1182no—
553C2×C4—1182no—
554C2×C4—1182no—
555C2×C4—1182no—
556C2×C4—1182no—
557C2×C4—1182no—
558C2×C4—4724no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4C22×C4×D5911yescharacteristic
569C2×C4#41×D5911yescharacteristic
570C2×C4#41×D5911yescharacteristic
571C2×C4#41×D5911yescharacteristic
572C2×C4#41×D5911yescharacteristic
573C2×C4C2×#1174568911yescharacteristic
574C8—22no—
575C8—41no—
576C8—1182no—
577C8—2361no—
578C22—1182no—
579C22—1182no—
580C22—1182no—
581C22—2361no—
582C22—2361no—
583C22—2361no—
584C22—2361no—
585C22C4×D4×D5911yescharacteristic
586C22#123×D5911yescharacteristic
587C22#1683791511yescharacteristic
588C22#1683820911yescharacteristic
589C22#1683821211yescharacteristic
590C22#1683821211yescharacteristic
591C22#1683821511yescharacteristic
592C4—42no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—1182no—
599C4—1182no—
600C4—1182no—
601C4—4721no—
602C4—2362no—
603C4—2362no—
604C4—2362no—
605C4#1683787011yescharacteristic
606C4C4×#1174568911yescharacteristic
607C4#1683793511yescharacteristic
608C4#1683793711yescharacteristic
609C2—2362no—
610C2#2142661711yescharacteristic
611C2#2143293411yescharacteristic
612C2#2143297811yescharacteristic
613C2#2143406411yescharacteristic
614C2#2143408811yescharacteristic
615C2#2143433011yescharacteristic
616C2#2143435011yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1921C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1921D59
hypocenterC59#21591
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
4. #21424731C2semidirect
5. #21424730C2semidirect
6. #21424671C2semidirect
10. #21423995C2semidirect
11. #21423984C2semidirect
12. #21423925C2semidirect
13. #21423915C2semidirect
15. C59×#1921C2semidirect
21. #16837745C4semidirect
97. #14273658C2×C4semidirect
99. #14273652C2×C4semidirect
101. #14273650C2×C4semidirect
122. #107×C59C2×C4semidirect
306. #1921D59semidirect
368. C59#21591semidirect
413. #107C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377452
#214267762
#214264502
#214247312
#214247302
#214246712
#214244022
#214242902
#214241902
#214239952
#214239842
#214239252
#214239152
#214238612
C59×#19212
#2159159