Finite Group Databasebeta

← #28620437

Subgroups — #28620437

Order 15104.

2252 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273706C211yesmaximal, characteristic, complemented
2C2×#16837540C212yesmaximal, complemented
3C2×#16837553C212yesmaximal, complemented
4#21425099C214yesmaximal, complemented
5#21424414C212yesmaximal, complemented
6#21424403C212yesmaximal, complemented
7#21424006C211yesmaximal, characteristic, complemented
8C22×#121×C59C211yesmaximal, characteristic
9C23×Dic118C2211yescharacteristic, complemented
10C22×#11745670C2214yescomplemented
11C2×#14273706—28no—
12C2×#14273706—21no—
13C2×#14273706—21no—
14C2×#14273706C2212yes—
15C2×#14273706C2212yescomplemented
16C22×C4⋊Dic59C2211yescharacteristic, complemented
17C22×Dic59⋊C4—21no—
18C22×C22⋊Dic59—21no—
19#16837553—28no—
20#16837540—28no—
21C2×#14273650C2214yescomplemented
22C2×#14273648C2214yescomplemented
23C2×C22⋊(C59⋊C8)C2212yes—
24C2×C22⋊(C59⋊C8)C2212yescomplemented
25C2×C22⋊(C59⋊C8)C2212yescomplemented
26#16837164C2214yescomplemented
27C23×D4×C59—21no—
28C23×D4×C59C2211yescharacteristic
29C24×C4×C59C2211yescharacteristic
30C2×#121×C59—21no—
31C2×#121×C59—21no—
32C2×#121×C59—28no—
33C2×#121×C59C2212yes—
34C2×#121×C59C2212yes—
35C22×C4⋊C4×C59C2211yescharacteristic
36C22×C22⋊C4×C59—21no—
37C22×Dic118—21no—
38C22×Dic118—24no—
39C22×Dic118—24no—
40C22×Dic118—42no—
41C22×Dic118C2312yes—
42C23×Dic59—21no—
43C23×Dic59—21no—
44C2×#11745670—28no—
45C2×#11745670—28no—
46C2×#11745670—44no—
47C22×C59⋊C8C2314yes—
48#14273706—28no—
49#14273706—28no—
50#14273706—24no—
51#14273706—24no—
52#14273706—24no—
53#14273706D418yescomplemented
54C2×C4⋊Dic59—21no—
55C2×C4⋊Dic59—24no—
56C2×C4⋊Dic59C2312yes—
57C2×Dic59⋊C4—28no—
58C2×Dic59⋊C4—22no—
59C2×Dic59⋊C4—22no—
60C2×C22⋊Dic59—28no—
61C2×C22⋊Dic59—22no—
62C2×C22⋊Dic59—22no—
63#14273650—28no—
64#14273648—28no—
65C22⋊(C59⋊C8)—24no—
66C22⋊(C59⋊C8)—24no—
67C22⋊(C59⋊C8)—42no—
68C25×C59—21no—
69C25×C59D411yescharacteristic
70C22×D4×C59—28no—
71C22×D4×C59—22no—
72C22×D4×C59—22no—
73C22×D4×C59—24no—
74C22×D4×C59—21no—
75C22×D4×C59—24no—
76C22×D4×C59—42no—
77C22×D4×C59—44no—
78C22×D4×C59—44no—
79C22×D4×C59C2312yes—
80C23×C4×C59—21no—
81C23×C4×C59—24no—
82C23×C4×C59—24no—
83C23×C4×C59—42no—
84C23×C4×C59D411yescharacteristic
85C23×C4×C59D411yescharacteristic
86C23×C4×C59D411yescharacteristic
87C23×C4×C59C2311yescharacteristic
88C23×C4×C59C2311yescharacteristic
89C23×C4×C59C2311yescharacteristic
90#121×C59—28no—
91#121×C59—28no—
92#121×C59—24no—
93#121×C59—24no—
94#121×C59—24no—
95#121×C59D418yes—
96C2×C4⋊C4×C59—21no—
97C2×C4⋊C4×C59—24no—
98C2×C4⋊C4×C59C2312yes—
99C2×C22⋊C4×C59—28no—
100C2×C22⋊C4×C59—22no—
101C2×C22⋊C4×C59—22no—
102C24×C59—22no—
103C24×C59—22no—
104C24×C59—22no—
105C24×C59—22no—
106C24×C59—24no—
107C24×C59—22no—
108C24×C59—41no—
109C24×C59—41no—
110C24×C59—42no—
111C24×C59—42no—
112C24×C59—42no—
113C24×C59—42no—
114C24×C59—42no—
115C24×C59—42no—
116C24×C59SD1614yescomplemented
117C24×C59C2×D411yescharacteristic
118C24×C59C2×D411yescharacteristic
119C24×C59C2×D411yescharacteristic
120C2×D4×C59—24no—
121C2×D4×C59—28no—
122C2×D4×C59—28no—
123C2×D4×C59—24no—
124C2×D4×C59—22no—
125C2×D4×C59—24no—
126C2×D4×C59—416no—
127C2×D4×C59—42no—
128C2×D4×C59—432no—
129C2×D4×C59—44no—
130C2×D4×C59—44no—
131C2×D4×C59—44no—
132C2×D4×C59—44no—
133C2×D4×C59—42no—
134C2×D4×C59C2×D414yes—
135C22×C4×C59—22no—
136C22×C4×C59—22no—
137C22×C4×C59—22no—
138C22×C4×C59—22no—
139C22×C4×C59—24no—
140C22×C4×C59—22no—
141C22×C4×C59—22no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—22no—
146C22×C4×C59—22no—
147C22×C4×C59—22no—
148C22×C4×C59—22no—
149C22×C4×C59—22no—
150C22×C4×C59—41no—
151C22×C4×C59—41no—
152C22×C4×C59—41no—
153C22×C4×C59—416no—
154C22×C4×C59—42no—
155C22×C4×C59—42no—
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D414yes—
160C22×C4×C59C2×D414yes—
161C22×C4×C59C2411yescharacteristic
162C22×Dic59—24no—
163C22×Dic59—24no—
164C22×Dic59—22no—
165C22×Dic59—22no—
166C22×Dic59—42no—
167C22×Dic59—42no—
168C22×Dic59—42no—
169C22×Dic59—42no—
170C2×Dic118—24no—
171C2×Dic118—24no—
172C2×Dic118—24no—
173C2×Dic118—22no—
174C2×Dic118—416no—
175C2×Dic118—42no—
176C2×Dic118—42no—
177C2×Dic118C2×D414yescomplemented
178C4⋊C4×C59—24no—
179C4⋊C4×C59—22no—
180C4⋊C4×C59C2×D414yes—
181C22⋊C4×C59—28no—
182C22⋊C4×C59—28no—
183C22⋊Dic59—28no—
184C22⋊Dic59—28no—
185#11745670—416no—
186C4⋊Dic59—24no—
187C4⋊Dic59—22no—
188C4⋊Dic59C2×D414yescomplemented
189Dic59⋊C4—28no—
190Dic59⋊C4—28no—
191C2×C59⋊C8—24no—
192C2×C59⋊C8—24no—
193C2×C59⋊C8—24no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—28no—
209C23×C59—28no—
210C23×C59—24no—
211C23×C59—44no—
212C23×C59—44no—
213C23×C59—44no—
214C23×C59—44no—
215C23×C59—44no—
216C23×C59—48no—
217C23×C59—48no—
218C23×C59—48no—
219C23×C59—42no—
220C23×C59—42no—
221C23×C59—42no—
222C23×C59—42no—
223C23×C59—42no—
224C23×C59—42no—
225C23×C59—84no—
226C23×C59—84no—
227C23×C59—84no—
228C23×C59—84no—
229C23×C59#12012yes—
230C23×C59#12012yes—
231C23×C59C2×SD1612yes—
232C23×C59C2×SD1612yes—
233C23×C59C2×SD1612yes—
234C23×C59AGL(1,ℤ/8)12yes—
235C23×C59#13712yes—
236C23×C59C22×D411yescharacteristic
237D4×C59—28no—
238D4×C59—48no—
239D4×C59—416no—
240D4×C59—416no—
241D4×C59—44no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—28no—
247C2×C4×C59—28no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—24no—
251C2×C4×C59—21no—
252C2×C4×C59—44no—
253C2×C4×C59—44no—
254C2×C4×C59—44no—
255C2×C4×C59—48no—
256C2×C4×C59—42no—
257C2×C4×C59—42no—
258C2×C4×C59—44no—
259C2×C4×C59#12012yes—
260C2×C4×C59#12012yes—
261C2×C4×C59C22×D412yes—
262C2×C4×C59C22×D412yes—
263C2×C4×C59H(5,𝔽2)12yes—
264C2×Dic59—24no—
265C2×Dic59—24no—
266C2×Dic59—48no—
267C2×Dic59—48no—
268C2×Dic59—42no—
269C2×Dic59—42no—
270C2×Dic59—42no—
271C2×Dic59—42no—
272Dic118—28no—
273Dic118—48no—
274Dic118—44no—
275C59⋊C8—44no—
276#23919—591nomaximal, Sylow
277C22×C59—22no—
278C22×C59—22no—
279C22×C59—22no—
280C22×C59—22no—
281C22×C59—22no—
282C22×C59—22no—
283C22×C59—22no—
284C22×C59—22no—
285C22×C59—22no—
286C22×C59—22no—
287C22×C59—22no—
288C22×C59—22no—
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—22no—
292C22×C59—416no—
293C22×C59—41no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59—41no—
297C22×C59—44no—
298C22×C59—44no—
299C22×C59—44no—
300C22×C59—816no—
301C22×C59—82no—
302C22×C59—82no—
303C22×C59—82no—
304C22×C59—82no—
305C22×C59#44814yescomplemented
306C22×C59#45014yescomplemented
307C22×C59C2×#12011yescharacteristic
308C22×C59C2×#12011yescharacteristic
309C22×C59#53411yescharacteristic
310C22×C59C22×SD1611yescharacteristic
311C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C59C2×#13711yescharacteristic
313C22×C59#57511yescharacteristic
314C4×C59—22no—
315C4×C59—22no—
316C4×C59—41no—
317C4×C59—44no—
318C4×C59D4214yes—
319Dic59—44no—
320Dic59—44no—
321C22×C22⋊Q8—591no—
322C22×#121D5911yescharacteristic
323C2×#448—592no—
324C2×#450—592no—
325#2000—594no—
326#1874—592no—
327#1872—592no—
328#1779—591no—
329C2×C59—24no—
330C2×C59—24no—
331C2×C59—42no—
332C2×C59—84no—
333C2×C59#240311yescharacteristic
334C2×C59C2×#45011yescharacteristic
335C2×C59C2×#44811yescharacteristic
336C2×C59#300611yescharacteristic
337C2×C59#300711yescharacteristic
338C2×C59#318711yescharacteristic
339C2×C59#319311yescharacteristic
340C23×Q8—591no—
341C23×D4—21no—
342C23×D4C2×D5911yescharacteristic
343C24×C4C2×D5911yescharacteristic
344C22×SD16—594no—
345C2×C22⋊Q8—592no—
346C2×C22⋊Q8—592no—
347C2×C22⋊Q8—1188no—
348C2×C22⋊Q8—1181no—
349C2×C22⋊Q8—1181no—
350C2×#121—21no—
351C2×#121—21no—
352C2×#121—28no—
353C2×#121C2×D5912yes—
354C2×#121C2×D5912yes—
355C22×C4⋊C4—591no—
356C22×C4⋊C4—1181no—
357C22×C4⋊C4C2×D5911yescharacteristic
358C22×C22⋊C4—21no—
359C22×C22⋊C4—1181no—
360#450—1188no—
361#448—1188no—
362C2×#103—594no—
363C2×#102—594no—
364C2×C22⋊C8—592no—
365C2×C22⋊C8—592no—
366C2×C22⋊C8—592no—
367#340—594no—
368C59#2391911yescharacteristic, complemented, Sylow
369C25—21no—
370C25C22⋊D5911yescharacteristic
371C22×Q8—592no—
372C22×Q8—1184no—
373C22×Q8—1181no—
374C22×Q8—1184no—
375C22×Q8—2362no—
376C22×D4—28no—
377C22×D4—22no—
378C22×D4—22no—
379C22×D4—24no—
380C22×D4—24no—
381C22×D4—21no—
382C22×D4—42no—
383C22×D4—44no—
384C22×D4—44no—
385C22×D4C22×D5912yes—
386C23×C4—21no—
387C23×C4—24no—
388C23×C4—24no—
389C23×C4—42no—
390C23×C4—1181no—
391C23×C4—1181no—
392C23×C4C22⋊D5911yescharacteristic
393C23×C4C22⋊D5911yescharacteristic
394C23×C4C22⋊D5911yescharacteristic
395C23×C4C22×D5911yescharacteristic
396C23×C4C22×D5911yescharacteristic
397C23×C4C22×D5911yescharacteristic
398C2×SD16—1188no—
399C2×SD16—1188no—
400C2×SD16—2364no—
401C22×C8—594no—
402C22⋊Q8—598no—
403C22⋊Q8—1184no—
404C22⋊Q8—1184no—
405C22⋊Q8—1188no—
406C22⋊Q8—1188no—
407C22⋊Q8—1184no—
408#121—28no—
409#121—28no—
410#121—24no—
411#121—24no—
412#121—24no—
413#121C22⋊D5918yes—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—24no—
416C2×C4⋊C4—592no—
417C2×C4⋊C4—1184no—
418C2×C4⋊C4—1188no—
419C2×C4⋊C4—1181no—
420C2×C4⋊C4—1182no—
421C2×C4⋊C4—1182no—
422C2×C4⋊C4C22×D5912yes—
423C2×C22⋊C4—28no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—1188no—
427C2×C22⋊C4—1182no—
428C2×C22⋊C4—1182no—
429#103—1188no—
430#102—1188no—
431C22⋊C8—1184no—
432C22⋊C8—1184no—
433C22⋊C8—2362no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—24no—
439C24—22no—
440C24—41no—
441C24—41no—
442C24—42no—
443C24—42no—
444C24—42no—
445C24—42no—
446C24—42no—
447C24—42no—
448C24#1174567014yescomplemented
449C24C2×C22⋊D5911yescharacteristic
450C24C2×C22⋊D5911yescharacteristic
451C24C2×C22⋊D5911yescharacteristic
452C2×Q8—594no—
453C2×Q8—1184no—
454C2×Q8—1182no—
455C2×Q8—1184no—
456C2×Q8—1184no—
457C2×Q8—23616no—
458C2×Q8—2362no—
459C2×Q8—2362no—
460C2×D4—24no—
461C2×D4—28no—
462C2×D4—28no—
463C2×D4—24no—
464C2×D4—22no—
465C2×D4—24no—
466C2×D4—416no—
467C2×D4—42no—
468C2×D4—432no—
469C2×D4—44no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—42no—
474C2×D4C2×C22⋊D5914yes—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—416no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—1184no—
497C22×C4—1184no—
498C22×C4—1182no—
499C22×C4—1182no—
500C22×C4—2362no—
501C22×C4—2362no—
502C22×C4—2362no—
503C22×C4—2362no—
504C22×C4D4×D5914yes—
505C22×C4C2×C22⋊D5911yescharacteristic
506C22×C4C2×C22⋊D5911yescharacteristic
507C22×C4C2×C22⋊D5911yescharacteristic
508C22×C4C2×C22⋊D5914yes—
509C22×C4C23×D5911yescharacteristic
510SD16—23616no—
511C2×C8—1184no—
512C2×C8—1184no—
513C2×C8—1184no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—594no—
517C4⋊C4—1188no—
518C4⋊C4—1184no—
519C4⋊C4—1188no—
520C4⋊C4—1182no—
521C4⋊C4C2×C22⋊D5914yes—
522C22⋊C4—28no—
523C22⋊C4—28no—
524C22⋊C4—1188no—
525C22⋊C4—1188no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—28no—
540C23—28no—
541C23—21no—
542C23—24no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—48no—
549C23—48no—
550C23—48no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23#1427369512yes—
562C23#1427369612yes—
563C23C2×#1174567012yes—
564C23C2×#1174567012yes—
565C23C2×#1174567012yes—
566C23#1427375312yes—
567C23#1427375712yes—
568C23C22×C22⋊D5911yescharacteristic
569Q8—1188no—
570Q8—2368no—
571Q8—2364no—
572D4—28no—
573D4—48no—
574D4—416no—
575D4—416no—
576D4—44no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—21no—
584C2×C4—24no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—1184no—
595C2×C4—1184no—
596C2×C4—2368no—
597C2×C4—2368no—
598C2×C4—2362no—
599C2×C4—2362no—
600C2×C4—2362no—
601C2×C4—2362no—
602C2×C4#1427369512yes—
603C2×C4#1427369612yes—
604C2×C4C2×D4×D5912yes—
605C2×C4C22×C22⋊D5912yes—
606C2×C4#1427377212yes—
607C8—2364no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—416no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—816no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22#1683754014yescomplemented
637C22#1683755314yescomplemented
638C22C2×#1427369511yescharacteristic
639C22C2×#1427369611yescharacteristic
640C22#1683796111yescharacteristic
641C22C22×#1174567011yescharacteristic
642C22C2×#1427375311yescharacteristic
643C22C2×#1427375711yescharacteristic
644C22#1683820011yescharacteristic
645C4—22no—
646C4—22no—
647C4—44no—
648C4—41no—
649C4—2364no—
650C4—2364no—
651C4D4×C22⋊D5914yes—
652C2—24no—
653C2—24no—
654C2—42no—
655C2—84no—
656C2#2142740611yescharacteristic
657C2C2×#1683755311yescharacteristic
658C2C2×#1683754011yescharacteristic
659C2#2143334611yescharacteristic
660C2#2143335611yescharacteristic
661C2#2143485311yescharacteristic
662C2#2143490211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
Fitting, prime radicalC22×#121×C59C2
socleC23×C59C22×D4
hypercenterC22×#121D59
hypocenterC59#23919
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. C22×#14273706C2semidirect
2. C2×#16837540C2semidirect
3. C2×#16837553C2semidirect
4. #21425099C2semidirect
5. #21424414C2semidirect
6. #21424403C2semidirect
7. #21424006C2semidirect
9. C23×Dic118C22semidirect
10. C22×#11745670C22semidirect
15. C2×#14273706C22semidirect
16. C22×C4⋊Dic59C22semidirect
21. C2×#14273650C22semidirect
22. C2×#14273648C22semidirect
24. C2×C22⋊(C59⋊C8)C22semidirect
25. C2×C22⋊(C59⋊C8)C22semidirect
26. #16837164C22semidirect
53. #14273706D4semidirect
116. C24×C59SD16semidirect
177. C2×Dic118C2×D4semidirect
188. C4⋊Dic59C2×D4semidirect
305. C22×C59#448semidirect
306. C22×C59#450semidirect
368. C59#23919semidirect
448. C24#11745670semidirect
636. C22#16837540semidirect
637. C22#16837553semidirect

Maximal subgroups

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

subgroupindex
C22×#142737062
C2×#168375402
C2×#168375532
#214250992
#214244142
#214244032
#214240062
C22×#121×C592
#2391959