Finite Group Databasebeta

← #28630116

Subgroups — #28630116

Order 15104.

746 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837687C211yesmaximal, characteristic, complemented
2C2×#16837463C211yesmaximal, characteristic
3#21429917C211yesmaximal, characteristic
4#21425533C211yesmaximal, characteristic
5#21425532C211yesmaximal, characteristic
6#21425296C211yesmaximal, characteristic
7#21425289C211yesmaximal, characteristic
8#21425044C211yesmaximal, characteristic, complemented
9#21424864C211yesmaximal, characteristic, complemented
10#21424831C211yesmaximal, characteristic, complemented
11#21424821C211yesmaximal, characteristic, complemented
12#21424789C211yesmaximal, characteristic, complemented
13#21424775C211yesmaximal, characteristic, complemented
14#21424111C211yesmaximal, characteristic, complemented
15C59×#2323C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17#16837687—24no—
18#16837463—24no—
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×C22⋊(C59⋊C8)C2211yescharacteristic
31C2×#14273638C2211yescharacteristic
32C2×C4×C59⋊C8C2211yescharacteristic
33#16837341—21no—
34#16837338—21no—
35#16837322—21no—
36#16837314—21no—
37#16837300C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837268C2211yescharacteristic
40#16837262C2211yescharacteristic
41#16837262C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50C2×C22⋊Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C2×#117×C59C2211yescharacteristic
53C59×#391C2211yescharacteristic
54C59×#389—21no—
55C59×#389C2211yescharacteristic
56C59×#388—21no—
57C59×#388—21no—
58C59×#387—21no—
59C59×#380C2211yescharacteristic
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×Dic118—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66#14273727—22no—
67#14273727—22no—
68C2×C4⋊Dic59—21no—
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—
77#14273660—22no—
78#14273658—22no—
79#14273658—22no—
80#14273658—22no—
81C4⋊(C59⋊C8)—22no—
82C4⋊(C59⋊C8)D414yes—
83#14273651—22no—
84#14273651—22no—
85#14273651—22no—
86#14273651—22no—
87C22⋊(C59⋊C8)—22no—
88C22⋊(C59⋊C8)—22no—
89#14273638—22no—
90C4×C59⋊C8—22no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×Q8×C59C2311yescharacteristic
98C23×C4×C59C2311yescharacteristic
99C22⋊Q8×C59—22no—
100C22⋊Q8×C59—22no—
101C4×Q8×C59—22no—
102C4×Q8×C59—22no—
103#117×C59—22no—
104#117×C59—22no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59D411yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
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×C59—21no—
124#95×C59—21no—
125#95×C59—21no—
126C24×C59C2×D411yescharacteristic
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—21no—
132C2×Q8×C59#4112yes—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
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×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2411yescharacteristic
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C2×Dic118—42no—
154C2×Dic118—81no—
155C2×Dic118—81no—
156C4⋊C4×C59—24no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—22no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—42no—
165C4⋊C4×C59—81no—
166C4⋊C4×C59#4112yes—
167C22⋊C4×C59—24no—
168C22⋊C4×C59—22no—
169C22⋊C4×C59—22no—
170C22⋊C4×C59—81no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—42no—
174C42×C59—81no—
175C42×C59C2×D412yes—
176C42×C59#4112yes—
177C22⋊Dic59—81no—
178C22⋊Dic59—81no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—42no—
188Dic59⋊C4—81no—
189Dic59⋊C4—81no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
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⋊C8C2×D412yes—
201C2×C59⋊C8C2×D412yes—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59C22×D411yescharacteristic
210Q8×C59—22no—
211Q8×C59—22no—
212Q8×C59—41no—
213Q8×C59—41no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
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—41no—
242C2×C4×C59—41no—
243C2×C4×C59#12112yes—
244C2×C4×C59#12112yes—
245C2×C4×C59C22×D411yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C592−1+411yescharacteristic
250C2×C4×C592−1+411yescharacteristic
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263Dic118—84no—
264C59⋊C8—22no—
265C59⋊C8—22no—
266C59⋊C8—41no—
267C59⋊C8—41no—
268#24718—591nomaximal, Sylow
269C22×C59—41no—
270C22×C59—41no—
271C22×C59—41no—
272C22×C59—41no—
273C22×C59—41no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59C2×#12111yescharacteristic
277C22×C59#52511yescharacteristic
278C22×C59#53611yescharacteristic
279C22×C59#57711yescharacteristic
280C22×C59#57711yescharacteristic
281C22×C59#57711yescharacteristic
282C22×C59#57811yescharacteristic
283C4×C59—22no—
284C4×C59—22no—
285C4×C59—42no—
286C4×C59—41no—
287C4×C59—42no—
288C4×C59—41no—
289C4×C59—41no—
290C4×C59—81no—
291C4×C59—81no—
292C4×C59#54711yescharacteristic
293C4×C59#54811yescharacteristic
294C4×C59#55211yescharacteristic
295C4×C59#55611yescharacteristic
296Dic59—81no—
297Dic59—81no—
298Dic59—81no—
299Dic59—81no—
300C2×#475—591no—
301C2×#433—591no—
302#2628—591no—
303#2323D5911yescharacteristic
304#2077—591no—
305#2077—591no—
306#2036—591no—
307#2035—591no—
308#1987—591no—
309#1949—591no—
310#1943—591no—
311#1941—591no—
312#1935—591no—
313#1933—591no—
314#1805—591no—
315C2×C59—81no—
316C2×C59#241511yescharacteristic
317C2×C59#305711yescharacteristic
318C2×C59#306311yescharacteristic
319C2×C59#312111yescharacteristic
320C2×C59#312511yescharacteristic
321C2×C59#324211yescharacteristic
322C2×C59#324411yescharacteristic
323C2×C4⋊Q8—591no—
324C2×C22⋊Q8C2×D5911yescharacteristic
325C2×C4×Q8C2×D5911yescharacteristic
326C2×#117C2×D5911yescharacteristic
327#475—1184no—
328#433—1184no—
329C2×#107—591no—
330C2×#106—591no—
331C2×#106—591no—
332C2×#106—591no—
333C2×C4⋊C8—591no—
334C2×C4⋊C8—591no—
335C2×#103—591no—
336C2×#103—591no—
337C2×#103—591no—
338C2×#103—591no—
339C2×C22⋊C8—591no—
340C2×C22⋊C8—591no—
341C2×#97—591no—
342C2×C4×C8—591no—
343#400—1181no—
344#399—1181no—
345#395—1181no—
346#393—1181no—
347#391C2×D5911yescharacteristic
348#389—21no—
349#389—591no—
350#389—591no—
351#389C2×D5911yescharacteristic
352#388—21no—
353#388—21no—
354#387—21no—
355#382—591no—
356#380—591no—
357#380—591no—
358#380C2×D5911yescharacteristic
359C4×C4⋊C4C2×D5911yescharacteristic
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368C59#2471811yescharacteristic, complemented, Sylow
369C22×Q8—1181no—
370C22×Q8C22×D5911yescharacteristic
371C23×C4C22×D5911yescharacteristic
372C22×C8—591no—
373C22×C8—591no—
374C22×C8—591no—
375C22×C8—591no—
376C4⋊Q8—1182no—
377C4⋊Q8—1182no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C4×Q8—22no—
381C4×Q8—22no—
382#117—22no—
383#117—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4C22⋊D5911yescharacteristic
395C2×C4⋊C4C22⋊D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4—1181no—
403C2×C22⋊C4C22⋊D5911yescharacteristic
404C2×C22⋊C4C22⋊D5911yescharacteristic
405C2×C42—21no—
406C2×C42—21no—
407C2×C42C22×D5911yescharacteristic
408C2×C42C22×D5911yescharacteristic
409#107—1182no—
410#106—1182no—
411#106—1182no—
412#106—1182no—
413C4⋊C8—594no—
414C4⋊C8—1182no—
415#103—1182no—
416#103—1182no—
417#103—1182no—
418#103—1182no—
419C22⋊C8—1182no—
420C22⋊C8—1182no—
421#97—1182no—
422C4×C8—1182no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434C24C2×C22⋊D5911yescharacteristic
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—4721no—
441C2×Q8—4721no—
442C2×Q8—2362no—
443C2×Q8#1174568512yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4C2×C22⋊D5911yescharacteristic
459C22×C4C2×C22⋊D5911yescharacteristic
460C22×C4C2×C22⋊D5911yescharacteristic
461C22×C4C2×C22⋊D5911yescharacteristic
462C22×C4C2×C22⋊D5911yescharacteristic
463C22×C4C23×D5911yescharacteristic
464C2×C8—592no—
465C2×C8—592no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—81no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—4721no—
495C4⋊C4—4721no—
496C4⋊C4—2362no—
497C4⋊C4#1174568512yes—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—81no—
502C22⋊C4—4721no—
503C22⋊C4—4721no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—81no—
508C42D4×D5912yes—
509C42#1174568712yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23C22×C22⋊D5911yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522Q8—4724no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4#1427369912yes—
565C2×C4#1427369912yes—
566C2×C4C2×D4×D5911yescharacteristic
567C2×C4C2×#1174568711yescharacteristic
568C2×C4C2×#1174568511yescharacteristic
569C2×C4C2×#1174568511yescharacteristic
570C2×C4#1427377411yescharacteristic
571C2×C4#1427377411yescharacteristic
572C8—1182no—
573C8—1182no—
574C8—2361no—
575C8—2361no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#1427369911yescharacteristic
584C22#1683791311yescharacteristic
585C22#1683797311yescharacteristic
586C22#1683821011yescharacteristic
587C22#1683821011yescharacteristic
588C22#1683821311yescharacteristic
589C22#1683821711yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—4721no—
600C4—4721no—
601C4—4721no—
602C4—4721no—
603C4#1683805711yescharacteristic
604C4#1683806111yescharacteristic
605C4#1683809211yescharacteristic
606C4#1683812211yescharacteristic
607C2—81no—
608C2#2142752511yescharacteristic
609C2#2143373711yescharacteristic
610C2#2143378911yescharacteristic
611C2#2143432511yescharacteristic
612C2#2143437311yescharacteristic
613C2#2143535811yescharacteristic
614C2#2143538811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2323C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2323D59
hypocenterC59#24718
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
1. C2×#16837687C2semidirect
8. #21425044C2semidirect
9. #21424864C2semidirect
10. #21424831C2semidirect
11. #21424821C2semidirect
12. #21424789C2semidirect
13. #21424775C2semidirect
14. #21424111C2semidirect
368. C59#24718semidirect

Maximal subgroups

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

subgroupindex
C2×#168376872
C2×#168374632
#214299172
#214255332
#214255322
#214252962
#214252892
#214250442
#214248642
#214248312
#214248212
#214247892
#214247752
#214241112
C59×#23232
#2471859