Finite Group Databasebeta

← #28608126

Subgroups — #28608126

Order 15104.

866 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837502C211yesmaximal, characteristic
2#21427016C211yesmaximal, characteristic, complemented
3#21424796C211yesmaximal, characteristic
4#21424782C211yesmaximal, characteristic
5#21424688C211yesmaximal, characteristic, complemented
6#21424667C211yesmaximal, characteristic, complemented
7#21424454C211yesmaximal, characteristic
8#21424454C211yesmaximal, characteristic
9#21424231C211yesmaximal, characteristic, complemented
10#21424106C211yesmaximal, characteristic
11#21424098C211yesmaximal, characteristic, complemented
12#21423971C211yesmaximal, characteristic, complemented
13#21423929C211yesmaximal, characteristic, complemented
14C4×#14273650C211yesmaximal, characteristic, complemented
15C59×#2354C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20#16837502C418yescomplemented
21C2×#14273660C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273650C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273648C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×#14273638C2211yescharacteristic
31C2×C4×C59⋊C8C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837297—21no—
35#16837297—21no—
36#16837268C2211yescharacteristic
37#16837266—21no—
38#16837253—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C2×#117×C59—21no—
51C59×#390C2211yescharacteristic
52C59×#390C2211yescharacteristic
53C59×#388—21no—
54C59×#388—21no—
55C59×#384C2211yescharacteristic
56C59×#380—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C2×#11745670—22no—
62C2×#11745670—22no—
63C2×#11745670C2×C414yes—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C4×Dic118—22no—
69C4×Dic118—22no—
70C4×Dic118—22no—
71C4×Dic118—41no—
72C4×Dic118—41no—
73C4×Dic118C2×C414yes—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85#14273660C2×C414yes—
86C4⋊(C59⋊C8)—22no—
87#14273650—22no—
88#14273650—22no—
89#14273650C2×C414yes—
90#14273648—41no—
91#14273648—41no—
92#14273648C2×C414yes—
93#14273638—41no—
94#14273638C2×C414yes—
95C4×C59⋊C8—41no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C22×D4×C59C2311yescharacteristic
101C23×C4×C59—21no—
102C4×D4×C59—22no—
103C4×D4×C59C2×C414yes—
104#117×C59—44no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59D411yescharacteristic
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C42×C59—21no—
118C2×C42×C59—21no—
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C24×C59—21no—
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—24no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—42no—
138C22×C4×C59—42no—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2411yescharacteristic
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118C22×C412yes—
154C2×Dic118#4112yes—
155C2×Dic118#4112yes—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—41no—
163C4⋊C4×C59—42no—
164C4⋊C4×C59—81no—
165C4⋊C4×C59C22×C412yes—
166C22⋊C4×C59—24no—
167C22⋊C4×C59—42no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—81no—
170C22⋊C4×C59—81no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—42no—
176C42×C59—42no—
177C42×C59C22×C412yes—
178C42×C59#4112yes—
179#11745670—28no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59C22×C412yes—
186Dic59⋊C4—24no—
187Dic59⋊C4—42no—
188Dic59⋊C4—42no—
189Dic59⋊C4—42no—
190Dic59⋊C4—42no—
191C4×Dic59—24no—
192C4×Dic59—42no—
193C4×Dic59—42no—
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⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8C22×C412yes—
205C2×C59⋊C8C22×C412yes—
206C23×C59—22no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59C22×D411yescharacteristic
214D4×C59—22no—
215D4×C59—22no—
216D4×C59—41no—
217D4×C59—41no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—22no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—48no—
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×C59C4×D412yes—
244C2×C4×C59C4×D412yes—
245C2×C4×C59AGL(1,ℤ/8)12yes—
246C2×C4×C59C23×C411yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59H(5,𝔽2)11yescharacteristic
251C2×C4×C592−1+411yescharacteristic
252C2×Dic59—22no—
253C2×Dic59—22no—
254C2×Dic59—22no—
255C2×Dic59—22no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264Dic118—22no—
265Dic118—22no—
266Dic118—22no—
267Dic118#11714yescomplemented
268C59⋊C8—22no—
269C59⋊C8—22no—
270C59⋊C8—41no—
271C59⋊C8—41no—
272#22781—591nomaximal, Sylow
273C22×C59—42no—
274C22×C59—42no—
275C22×C59—42no—
276C22×C59—81no—
277C22×C59—81no—
278C22×C59—81no—
279C22×C59—81no—
280C22×C59C2×C4×D411yescharacteristic
281C22×C59#54611yescharacteristic
282C22×C59#54811yescharacteristic
283C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C59#57511yescharacteristic
285C22×C59#57711yescharacteristic
286C22×C59#57811yescharacteristic
287C4×C59—22no—
288C4×C59—22no—
289C4×C59—22no—
290C4×C59—42no—
291C4×C59—41no—
292C4×C59—41no—
293C4×C59—41no—
294C4×C59—81no—
295C4×C59—81no—
296C4×C59C2×#11711yescharacteristic
297C4×C59#52011yescharacteristic
298C4×C59#55211yescharacteristic
299C4×C59#55311yescharacteristic
300Dic59—24no—
301Dic59—42no—
302Dic59—81no—
303Dic59—81no—
304C2×#440—591no—
305#2354D5911yescharacteristic
306#2351—591no—
307#1936—591no—
308#1934—591no—
309#1921—591no—
310#1918—591no—
311#1880—591no—
312#1880—591no—
313#1836—591no—
314#1804—591no—
315#1803—591no—
316#1771—591no—
317#1763—591no—
318C4×#103—591no—
319C2×C59—82no—
320C2×C59#235211yescharacteristic
321C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
322C2×C59#294711yescharacteristic
323C2×C59#330811yescharacteristic
324C2×C59#332111yescharacteristic
325C2×C59#336811yescharacteristic
326C2×C59#337711yescharacteristic
327C22×SD16—591no—
328C2×C4×Q8—591no—
329C2×C4×Q8—591no—
330C2×C4×Q8—591no—
331C2×C4×D4C2×D5911yescharacteristic
332C2×#117—21no—
333#440—598no—
334C2×#107—591no—
335C2×C4⋊C8—591no—
336C2×#103—591no—
337C2×#103—591no—
338C2×#103—591no—
339C2×#102—591no—
340C2×#102—591no—
341C2×#102—591no—
342C2×#97—591no—
343C2×#97—591no—
344C2×C4×C8—591no—
345#390C2×D5911yescharacteristic
346#390C2×D5911yescharacteristic
347#389—591no—
348#389—591no—
349#389—1181no—
350#389—1181no—
351#388—21no—
352#388—21no—
353#384C2×D5911yescharacteristic
354#382—591no—
355#382—1181no—
356#380—21no—
357C4×C4⋊C4—1181no—
358C4×C4⋊C4C2×D5911yescharacteristic
359C4×C4⋊C4C2×D5911yescharacteristic
360C4×C4⋊C4C2×D5911yescharacteristic
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#336—591no—
370#336—591no—
371C59#2278111yescharacteristic, complemented, Sylow
372C22×Q8—591no—
373C22×D4C22×D5911yescharacteristic
374C23×C4—21no—
375C2×SD16—594no—
376C2×SD16—1182no—
377C2×SD16—1182no—
378C22×C8—591no—
379C22×C8—591no—
380C22×C8—591no—
381C22×C8—591no—
382C4×Q8—594no—
383C4×Q8—1182no—
384C4×Q8—1182no—
385C4×Q8—1182no—
386C4×Q8—2361no—
387C4×Q8—2361no—
388C4×D4—22no—
389C4×D4C4×D5914yes—
390#117—44no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4C22⋊D5911yescharacteristic
402C2×C4⋊C4C22⋊D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C4⋊C4C22×D5911yescharacteristic
405C2×C4⋊C4C22×D5911yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—1181no—
414C2×C42—1181no—
415C2×C42—1181no—
416C2×C42C22×D5911yescharacteristic
417C2×C42C22×D5911yescharacteristic
418C2×C42C22×D5911yescharacteristic
419#107—594no—
420C4⋊C8—1182no—
421#103—594no—
422#103—1182no—
423#103—1182no—
424#102—594no—
425#102—2361no—
426#102—2361no—
427#97—594no—
428#97—2361no—
429C4×C8—2361no—
430#95—21no—
431#95—21no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95C22⋊D5911yescharacteristic
437#95C22⋊D5911yescharacteristic
438C24—21no—
439C2×Q8—592no—
440C2×Q8—592no—
441C2×Q8—592no—
442C2×Q8—1181no—
443C2×Q8—1181no—
444C2×Q8—1181no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C2×C4×D5912yes—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4C2×C22⋊D5911yescharacteristic
462C22×C4C2×C22⋊D5911yescharacteristic
463C22×C4C2×C22⋊D5911yescharacteristic
464C22×C4C2×C22⋊D5911yescharacteristic
465C22×C4C2×C22⋊D5911yescharacteristic
466C22×C4C2×C22⋊D5911yescharacteristic
467C22×C4C23×D5911yescharacteristic
468SD16—1188no—
469C2×C8—592no—
470C2×C8—592no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—592no—
491C4⋊C4—1184no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—2362no—
498C4⋊C4—2362no—
499C4⋊C4—2362no—
500C4⋊C4—2362no—
501C4⋊C4C2×C4×D5912yes—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—42no—
512C42—42no—
513C42—1184no—
514C42—2362no—
515C42—2362no—
516C42C2×C4×D5912yes—
517C42#1174568512yes—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C22×C22⋊D5911yescharacteristic
526Q8—594no—
527Q8—1182no—
528Q8—1182no—
529Q8—1182no—
530D4—22no—
531D4—22no—
532D4—41no—
533D4—41no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—48no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—1182no—
560C2×C4—1182no—
561C2×C4—1182no—
562C2×C4—1182no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4C4×C22⋊D5912yes—
572C2×C4C4×C22⋊D5912yes—
573C2×C4#1427375312yes—
574C2×C4C22×C4×D5911yescharacteristic
575C2×C4#41×D5911yescharacteristic
576C2×C4#41×D5911yescharacteristic
577C2×C4C2×#1174568511yescharacteristic
578C2×C4#1427377311yescharacteristic
579C2×C4#1427377411yescharacteristic
580C8—1182no—
581C8—1182no—
582C8—2361no—
583C8—2361no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22C2×C4×C22⋊D5911yescharacteristic
592C22#1683804011yescharacteristic
593C22#1683805911yescharacteristic
594C22C2×#1427375311yescharacteristic
595C22#1683820011yescharacteristic
596C22#1683821311yescharacteristic
597C22#1683821711yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—42no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—1184no—
608C4—4721no—
609C4—4721no—
610C4—2362no—
611C4#117×D5911yescharacteristic
612C4#1683788211yescharacteristic
613C4#1683809111yescharacteristic
614C4#1683809611yescharacteristic
615C2—82no—
616C2#2142702611yescharacteristic
617C2C4×#1427375311yescharacteristic
618C2#2143293511yescharacteristic
619C2#2143608111yescharacteristic
620C2#2143627611yescharacteristic
621C2#2143682111yescharacteristic
622C2#2143693211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2354C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#2354D59
hypocenterC59#22781
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #21427016C2semidirect
5. #21424688C2semidirect
6. #21424667C2semidirect
9. #21424231C2semidirect
11. #21424098C2semidirect
12. #21423971C2semidirect
13. #21423929C2semidirect
14. C4×#14273650C2semidirect
20. #16837502C4semidirect
267. Dic118#117semidirect
371. C59#22781semidirect

Maximal subgroups

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

subgroupindex
C2×#168375022
#214270162
#214247962
#214247822
#214246882
#214246672
#214244542
#214242312
#214241062
#214240982
#214239712
#214239292
C4×#142736502
C59×#23542
#2278159