Finite Group Databasebeta

← #28661882

Subgroups — #28661882

Order 15104.

762 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427152C211yesmaximal, characteristic, complemented
2#21425640C211yesmaximal, characteristic, complemented
3#21425413C211yesmaximal, characteristic
4#21425342C211yesmaximal, characteristic, complemented
5#21425178C211yesmaximal, characteristic
6#21424949C211yesmaximal, characteristic
7#21424730C211yesmaximal, characteristic, complemented
8#21424720C211yesmaximal, characteristic
9#21424671C211yesmaximal, characteristic, complemented
10#21424655C211yesmaximal, characteristic, complemented
11#21424454C211yesmaximal, characteristic
12#21424454C211yesmaximal, characteristic
13#21424098C211yesmaximal, characteristic, complemented
14C4×#14273650C211yesmaximal, characteristic, complemented
15C59×#2512C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C22×#11745670C2211yescharacteristic
18C2×#14273727C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20C2×C4×Dic118C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×C4×C59⋊C8C2211yescharacteristic
33#16837308C2211yescharacteristic
34#16837308C2211yescharacteristic
35#16837298—21no—
36#16837285—21no—
37#16837285—21no—
38C4⋊C4×Dic59—21no—
39C4×C4⋊Dic59C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837157C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#126×C59—21no—
49C59×#397—21no—
50C59×#396—21no—
51C59×#390C2211yescharacteristic
52C59×#390C2211yescharacteristic
53C59×#390C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#388—21no—
56C59×#384C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C22×Dic118C2311yescharacteristic
60C22×Dic118C2311yescharacteristic
61C2×#11745670—22no—
62C2×#11745670—22no—
63C2×#11745670—22no—
64C2×#11745670—41no—
65C2×#11745670—41no—
66C2×#11745670—41no—
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71#14273727—22no—
72#14273727—22no—
73C4×Dic118—22no—
74C4×Dic118—22no—
75C4×Dic118—41no—
76C4×Dic118—41no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59—21no—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86#14273658—22no—
87#14273658—41no—
88C4⋊(C59⋊C8)—41no—
89#14273650—22no—
90#14273650—22no—
91#14273650—22no—
92#14273650—41no—
93#14273650—41no—
94#14273650—41no—
95#14273648—22no—
96#14273648—41no—
97C4×C59⋊C8—41no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×D4×C59C2311yescharacteristic
104#126×C59—44no—
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×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C42×C59—21no—
120C2×C42×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123#95×C59—21no—
124#95×C59—21no—
125#95×C59D411yescharacteristic
126#95×C59D411yescharacteristic
127C24×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C2×Dic118—21no—
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118#4112yes—
159C2×Dic118#4112yes—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—41no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59—81no—
170C4⋊C4×C59SD1614yescomplemented
171C22⋊C4×C59—42no—
172C22⋊C4×C59—42no—
173C22⋊C4×C59—42no—
174C22⋊C4×C59—42no—
175C22⋊C4×C59—81no—
176C22⋊C4×C59—81no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—42no—
184#11745670—44no—
185#11745670—44no—
186C4⋊Dic59—24no—
187C4⋊Dic59—24no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59#4112yes—
192Dic59⋊C4—42no—
193Dic59⋊C4—42no—
194C4×Dic59—42no—
195C4×Dic59—42no—
196C4×Dic59—42no—
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⋊C8—21no—
208C2×C59⋊C8#4112yes—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59C22×D411yescharacteristic
217D4×C59—41no—
218D4×C59—41no—
219D4×C59—41no—
220D4×C59—41no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
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—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
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—41no—
244C2×C4×C59—41no—
245C2×C4×C59C2×SD1612yes—
246C2×C4×C59C2×SD1612yes—
247C2×C4×C59C2×SD1612yes—
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59H(5,𝔽2)11yescharacteristic
253C2×C4×C59H(5,𝔽2)11yescharacteristic
254C2×Dic59—22no—
255C2×Dic59—22no—
256C2×Dic59—22no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266Dic118—22no—
267Dic118—22no—
268Dic118—22no—
269Dic118—22no—
270Dic118—41no—
271Dic118—41no—
272Dic118—41no—
273Dic118—41no—
274C59⋊C8—22no—
275C59⋊C8—41no—
276C59⋊C8—41no—
277C59⋊C8—41no—
278#27066—591nomaximal, Sylow
279C22×C59—81no—
280C22×C59—81no—
281C22×C59—81no—
282C22×C59—81no—
283C22×C59—81no—
284C22×C59—81no—
285C22×C59—81no—
286C22×C59#52511yescharacteristic
287C22×C59#54611yescharacteristic
288C22×C59#55011yescharacteristic
289C22×C59C22×SD1611yescharacteristic
290C22×C59#57511yescharacteristic
291C22×C59#57711yescharacteristic
292C22×C59#57811yescharacteristic
293C4×C59—22no—
294C4×C59—22no—
295C4×C59—22no—
296C4×C59—41no—
297C4×C59—41no—
298C4×C59—41no—
299C4×C59—81no—
300C4×C59—81no—
301C4×C59—81no—
302C4×C59#52911yescharacteristic
303C4×C59#53811yescharacteristic
304C4×C59#55111yescharacteristic
305C4×C59#55311yescharacteristic
306Dic59—42no—
307Dic59—42no—
308Dic59—42no—
309Dic59—81no—
310#2512D5911yescharacteristic
311#2365—591no—
312#2096—591no—
313#2055—591no—
314#2043—591no—
315#2013—591no—
316#1968—591no—
317#1927—591no—
318#1926—591no—
319#1919—591no—
320#1916—591no—
321#1880—591no—
322#1880—591no—
323#1803—591no—
324C4×#103—591no—
325C2×C59—161no—
326C2×C59#246711yescharacteristic
327C2×C59#310311yescharacteristic
328C2×C59#312111yescharacteristic
329C2×C59#329811yescharacteristic
330C2×C59#330711yescharacteristic
331C2×C59#339211yescharacteristic
332C2×C59#340211yescharacteristic
333C22×SD16—591no—
334C22×SD16—591no—
335C2×C4⋊Q8—591no—
336C2×#126—21no—
337C2×C4×Q8—591no—
338C2×C4×Q8—591no—
339C2×#106—591no—
340C2×#106—591no—
341C2×C4⋊C8—591no—
342C2×#103—591no—
343C2×#103—591no—
344C2×#103—591no—
345C2×#103—591no—
346C2×#103—591no—
347C2×#103—591no—
348C2×#102—591no—
349C2×#102—591no—
350C2×C4×C8—591no—
351#397—21no—
352#396—21no—
353#391—591no—
354#391—591no—
355#390C2×D5911yescharacteristic
356#390C2×D5911yescharacteristic
357#390C2×D5911yescharacteristic
358#389—1181no—
359#389C2×D5911yescharacteristic
360#388—21no—
361#387—1181no—
362#387—1181no—
363#384C2×D5911yescharacteristic
364C4×C4⋊C4—591no—
365C4×C4⋊C4—1181no—
366C4×C4⋊C4C2×D5911yescharacteristic
367C4×C4⋊C4C2×D5911yescharacteristic
368#340—591no—
369#340—591no—
370#340—591no—
371#340—591no—
372#336—591no—
373#336—591no—
374#336—591no—
375#336—591no—
376C59#2706611yescharacteristic, complemented, Sylow
377C22×Q8—591no—
378C22×Q8—591no—
379C22×D4C22×D5911yescharacteristic
380C2×SD16—1182no—
381C2×SD16—1182no—
382C2×SD16—1182no—
383C2×SD16—2361no—
384C2×SD16—2361no—
385C2×SD16—2361no—
386C22×C8—591no—
387C22×C8—591no—
388C22×C8—591no—
389C22×C8—591no—
390C4⋊Q8—1182no—
391C4⋊Q8—1182no—
392#126—44no—
393C4×Q8—1182no—
394C4×Q8—1182no—
395C4×Q8—2361no—
396C4×Q8—2361no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—591no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4—1181no—
406C2×C4⋊C4C22⋊D5911yescharacteristic
407C2×C4⋊C4C22⋊D5911yescharacteristic
408C2×C4⋊C4C22×D5911yescharacteristic
409C2×C4⋊C4C22×D5911yescharacteristic
410C2×C4⋊C4C22×D5911yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C42—21no—
418C2×C42—1181no—
419C2×C42—1181no—
420C2×C42—1181no—
421C2×C42C22×D5911yescharacteristic
422C2×C42C22×D5911yescharacteristic
423C2×C42C22×D5911yescharacteristic
424#106—1182no—
425#106—2361no—
426C4⋊C8—2361no—
427#103—1182no—
428#103—1182no—
429#103—1182no—
430#103—2361no—
431#103—2361no—
432#103—2361no—
433#102—1182no—
434#102—2361no—
435C4×C8—2361no—
436#95—21no—
437#95—21no—
438#95—1181no—
439#95—1181no—
440#95—1181no—
441#95—1181no—
442#95—1181no—
443#95C22⋊D5911yescharacteristic
444#95C22⋊D5911yescharacteristic
445C24—21no—
446C2×Q8—592no—
447C2×Q8—592no—
448C2×Q8—1181no—
449C2×Q8—1181no—
450C2×Q8—1181no—
451C2×Q8—1181no—
452C2×Q8—1181no—
453C2×Q8—1181no—
454C2×Q8—1181no—
455C2×Q8—1181no—
456C2×Q8—1181no—
457C2×Q8—1181no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—1181no—
468C22×C4—1181no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4C2×C22⋊D5911yescharacteristic
472C22×C4C2×C22⋊D5911yescharacteristic
473C22×C4C2×C22⋊D5911yescharacteristic
474C22×C4C2×C22⋊D5911yescharacteristic
475C22×C4C2×C22⋊D5911yescharacteristic
476C22×C4C2×C22⋊D5911yescharacteristic
477C22×C4C23×D5911yescharacteristic
478SD16—2364no—
479SD16—2364no—
480C2×C8—592no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
488C2×C8—1181no—
489C2×C8—1181no—
490C2×C8—1181no—
491C2×C8—1181no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—41no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—81no—
502C4⋊C4—592no—
503C4⋊C4—1184no—
504C4⋊C4—1184no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—2362no—
509C4⋊C4—2362no—
510C4⋊C4#1174567014yescomplemented
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—2362no—
525C42—2362no—
526C42—2362no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23C22×C22⋊D5911yescharacteristic
535Q8—1182no—
536Q8—1182no—
537Q8—1182no—
538Q8—1182no—
539Q8—2361no—
540Q8—2361no—
541Q8—2361no—
542Q8—2361no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—1182no—
572C2×C4—1182no—
573C2×C4—1182no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2361no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4C2×#1174567012yes—
584C2×C4C2×#1174567012yes—
585C2×C4C2×#1174567012yes—
586C2×C4#41×D5911yescharacteristic
587C2×C4#41×D5911yescharacteristic
588C2×C4#41×D5911yescharacteristic
589C2×C4#41×D5911yescharacteristic
590C2×C4#1427377311yescharacteristic
591C2×C4#1427377311yescharacteristic
592C8—1182no—
593C8—2361no—
594C8—2361no—
595C8—2361no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#1683790911yescharacteristic
604C22#1683804011yescharacteristic
605C22#1683807111yescharacteristic
606C22C22×#1174567011yescharacteristic
607C22#1683820011yescharacteristic
608C22#1683821311yescharacteristic
609C22#1683821711yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—4721no—
620C4—2362no—
621C4—2362no—
622C4—2362no—
623C4#1683793511yescharacteristic
624C4#1683799111yescharacteristic
625C4#1683807711yescharacteristic
626C4#1683810311yescharacteristic
627C2—161no—
628C2#2142814311yescharacteristic
629C2#2143412311yescharacteristic
630C2#2143431911yescharacteristic
631C2#2143595511yescharacteristic
632C2#2143606611yescharacteristic
633C2#2143707711yescharacteristic
634C2#2143717111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2512C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2512D59
hypocenterC59#27066
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2512

Complements

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

NH ≅ G/Ntype
1. #21427152C2semidirect
2. #21425640C2semidirect
4. #21425342C2semidirect
7. #21424730C2semidirect
9. #21424671C2semidirect
10. #21424655C2semidirect
13. #21424098C2semidirect
14. C4×#14273650C2semidirect
170. C4⋊C4×C59SD16semidirect
376. C59#27066semidirect
510. C4⋊C4#11745670semidirect

Maximal subgroups

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

subgroupindex
#214271522
#214256402
#214254132
#214253422
#214251782
#214249492
#214247302
#214247202
#214246712
#214246552
#214244542
#214240982
C4×#142736502
C59×#25122
#2706659