Finite Group Databasebeta

← #28645260

Subgroups — #28645260

Order 15104.

1462 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837761C211yesmaximal, characteristic, complemented
2C2×#16837692C211yesmaximal, characteristic, complemented
3#21429088C211yesmaximal, characteristic, complemented
4#21425289C212yesmaximal, complemented
5#21425282C212yesmaximal, complemented
6#21425029C211yesmaximal, characteristic, complemented
7#21424895C211yesmaximal, characteristic, complemented
8#21424036C212yesmaximal, complemented
9#21424027C212yesmaximal, complemented
10#21423921C211yesmaximal, characteristic, complemented
11C2×C59×#546C211yesmaximal, characteristic
12C2×#14273727C2211yescharacteristic, complemented
13C2×#14273714—21no—
14#16837761—24no—
15#16837692—24no—
16C2×#14273658C2212yescomplemented
17C2×C4⋊(C59⋊C8)C2211yescharacteristic, complemented
18C2×#14273651C2212yescomplemented
19C2×#14273651C2212yescomplemented
20C2×#14273648C2212yes—
21C2×#14273648C2212yes—
22C2×#14273648C2212yes—
23C2×#14273648C2212yes—
24C2×C22⋊(C59⋊C8)C2212yes—
25C2×C22⋊(C59⋊C8)C2212yescomplemented
26C2×#14273638C2211yescharacteristic, complemented
27#16837327—21no—
28#16837327—21no—
29#16837307—21no—
30#16837262C2212yes—
31#16837262C2212yescomplemented
32C4×C4⋊Dic59C2211yescharacteristic, complemented
33#16837164C2212yescomplemented
34C22×#41×C59C2211yescharacteristic
35C23×D4×C59C2211yescharacteristic
36C59×#546—24no—
37C59×#546—24no—
38C2×#124×C59C2211yescharacteristic
39C2×#123×C59—21no—
40C2×C22⋊Q8×C59C2211yescharacteristic
41C2×#121×C59—21no—
42C2×#121×C59C2211yescharacteristic
43C2×#120×C59—21no—
44C2×C4×D4×C59C2211yescharacteristic
45C2×C4×D4×C59C2211yescharacteristic
46C22×C22⋊C4×C59—21no—
47C22×Dic118—21no—
48C22×C59⋊C8C2312yes—
49C22×C59⋊C8C2312yes—
50#14273727—22no—
51#14273727—22no—
52#14273714—44no—
53C2×C4⋊Dic59—21no—
54C2×C4⋊Dic59C2312yes—
55C2×C4⋊Dic59C2312yes—
56C2×C22⋊Dic59—22no—
57C2×C22⋊Dic59—22no—
58C2×C4×Dic59—21no—
59#14273658—24no—
60C4⋊(C59⋊C8)—22no—
61#14273651—24no—
62#14273651—42no—
63#14273648—24no—
64#14273648—24no—
65#14273648—24no—
66#14273648—42no—
67C22⋊(C59⋊C8)—42no—
68C22⋊(C59⋊C8)—42no—
69#14273638—22no—
70C59⋊#95—21no—
71C59⋊#95—22no—
72C59⋊#95—22no—
73C25×C59—21no—
74C2×#41×C59—22no—
75C2×#41×C59—22no—
76C2×#41×C59—24no—
77C2×#41×C59—42no—
78C22×Q8×C59C2311yescharacteristic
79C22×D4×C59—21no—
80C22×D4×C59—21no—
81C22×D4×C59—22no—
82C22×D4×C59—22no—
83C22×D4×C59—24no—
84C22×D4×C59—21no—
85C22×D4×C59—42no—
86C22×D4×C59D411yescharacteristic
87C22×D4×C59C2311yescharacteristic
88C22×D4×C59C2311yescharacteristic
89C23×C4×C59—21no—
90C23×C4×C59—21no—
91C23×C4×C59C2311yescharacteristic
92C23×C4×C59C2311yescharacteristic
93#124×C59—22no—
94#124×C59D414yes—
95#123×C59—28no—
96C22⋊Q8×C59—22no—
97C22⋊Q8×C59—22no—
98#121×C59—28no—
99#121×C59—22no—
100#121×C59—22no—
101#120×C59—28no—
102C4×D4×C59—22no—
103C4×D4×C59—22no—
104C4×D4×C59—22no—
105C4×D4×C59D414yes—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59D411yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C22⋊C4×C59—28no—
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59D411yescharacteristic
116C2×C22⋊C4×C59D411yescharacteristic
117C2×C42×C59C2311yescharacteristic
118C24×C59—21no—
119C24×C59—24no—
120C24×C59—21no—
121C24×C59—44no—
122C24×C59—42no—
123C24×C59—42no—
124C24×C59—82no—
125C24×C59C2×D411yescharacteristic
126C24×C59C2×D411yescharacteristic
127#41×C59—48no—
128#41×C59—44no—
129#41×C59—44no—
130C2×Q8×C59—21no—
131C2×Q8×C59—22no—
132C2×Q8×C59—22no—
133C2×Q8×C59C2×D412yes—
134C2×D4×C59—24no—
135C2×D4×C59—24no—
136C2×D4×C59—22no—
137C2×D4×C59—24no—
138C2×D4×C59—22no—
139C2×D4×C59—21no—
140C2×D4×C59—21no—
141C2×D4×C59—22no—
142C2×D4×C59—22no—
143C2×D4×C59—22no—
144C2×D4×C59—41no—
145C2×D4×C59—44no—
146C2×D4×C59—48no—
147C2×D4×C59—44no—
148C2×D4×C59—44no—
149C2×D4×C59—44no—
150C2×D4×C59—82no—
151C2×D4×C59C2×D412yes—
152C2×D4×C59C2×D412yes—
153C2×D4×C59#4114yes—
154C22×C4×C59—24no—
155C22×C4×C59—21no—
156C22×C4×C59—21no—
157C22×C4×C59—21no—
158C22×C4×C59—21no—
159C22×C4×C59—21no—
160C22×C4×C59—21no—
161C22×C4×C59—22no—
162C22×C4×C59—22no—
163C22×C4×C59—22no—
164C22×C4×C59—22no—
165C22×C4×C59—24no—
166C22×C4×C59—44no—
167C22×C4×C59—82no—
168C22×C4×C59C2×D411yescharacteristic
169C22×C4×C59C2×D411yescharacteristic
170C22×C4×C59C2×D411yescharacteristic
171C22×C4×C59C2×D411yescharacteristic
172C22×C4×C59C2411yescharacteristic
173C22×Dic59—22no—
174C22×Dic59—22no—
175C2×Dic118—42no—
176C2×Dic118—42no—
177C2×Dic118—42no—
178C4⋊C4×C59—24no—
179C4⋊C4×C59—22no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59C2×D412yes—
182C22⋊C4×C59—24no—
183C22⋊C4×C59—24no—
184C22⋊C4×C59—22no—
185C22⋊C4×C59—24no—
186C22⋊C4×C59—24no—
187C22⋊C4×C59—24no—
188C22⋊C4×C59—22no—
189C42×C59C2×D412yes—
190C42×C59C2×D412yes—
191C22⋊Dic59—44no—
192C22⋊Dic59—44no—
193C4⋊Dic59—22no—
194C4⋊Dic59—22no—
195C4⋊Dic59—22no—
196C4⋊Dic59—22no—
197C4⋊Dic59—42no—
198C4×Dic59—42no—
199C2×C59⋊C8—22no—
200C2×C59⋊C8—22no—
201C2×C59⋊C8—22no—
202C2×C59⋊C8—22no—
203C2×C59⋊C8—22no—
204C2×C59⋊C8—22no—
205C23×C59—21no—
206C23×C59—22no—
207C23×C59—22no—
208C23×C59—24no—
209C23×C59—21no—
210C23×C59—22no—
211C23×C59—22no—
212C23×C59—21no—
213C23×C59—22no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—22no—
217C23×C59—48no—
218C23×C59—48no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59—42no—
222C23×C59—42no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—42no—
226C23×C59—88no—
227C23×C59—84no—
228C23×C59—88no—
229C23×C59AGL(1,ℤ/8)12yes—
230C23×C59C22×D411yescharacteristic
231Q8×C59—24no—
232Q8×C59—41no—
233Q8×C59—41no—
234D4×C59—22no—
235D4×C59—24no—
236D4×C59—24no—
237D4×C59—44no—
238D4×C59—48no—
239D4×C59—48no—
240D4×C59—41no—
241D4×C59—41no—
242D4×C59—84no—
243D4×C59#12414yescomplemented
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—22no—
247C2×C4×C59—22no—
248C2×C4×C59—21no—
249C2×C4×C59—22no—
250C2×C4×C59—21no—
251C2×C4×C59—22no—
252C2×C4×C59—22no—
253C2×C4×C59—48no—
254C2×C4×C59—42no—
255C2×C4×C59—42no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—42no—
261C2×C4×C59—42no—
262C2×C4×C59—84no—
263C2×C4×C59#12012yes—
264C2×C4×C59#12012yes—
265C2×C4×C59C22×D411yescharacteristic
266C2×C4×C59C22×D411yescharacteristic
267C2×C4×C59C2×#4112yes—
268C2×C4×C59H(5,𝔽2)12yes—
269C2×Dic59—42no—
270C2×Dic59—42no—
271C2×Dic59—42no—
272C2×Dic59—42no—
273C2×Dic59—42no—
274C2×Dic59—42no—
275Dic118—84no—
276C59⋊C8—42no—
277C59⋊C8—42no—
278#25898—591nomaximal, Sylow
279C22×C59—24no—
280C22×C59—22no—
281C22×C59—24no—
282C22×C59—41no—
283C22×C59—42no—
284C22×C59—42no—
285C22×C59—41no—
286C22×C59—44no—
287C22×C59—42no—
288C22×C59—44no—
289C22×C59—41no—
290C22×C59—42no—
291C22×C59—41no—
292C22×C59—42no—
293C22×C59—41no—
294C22×C59—42no—
295C22×C59—42no—
296C22×C59—42no—
297C22×C59—816no—
298C22×C59—82no—
299C22×C59—82no—
300C22×C59C2×#12011yescharacteristic
301C22×C59#54612yes—
302C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C59#57511yescharacteristic
304C22×C59#57712yes—
305C4×C59—22no—
306C4×C59—22no—
307C4×C59—41no—
308C4×C59—41no—
309C4×C59—42no—
310C4×C59—42no—
311C4×C59C2×#12411yescharacteristic
312C4×C59#53511yescharacteristic
313C4×C59#54612yes—
314Dic59—82no—
315Dic59—82no—
316C2×#546D5911yescharacteristic
317C2×#489—591no—
318C2×#476—591no—
319#2547—591no—
320#2035—592no—
321#2034—592no—
322#1983—591no—
323#1955—591no—
324#1788—592no—
325#1786—592no—
326#1762—591no—
327C2×C59—42no—
328C2×C59—44no—
329C2×C59—81no—
330C2×C59—82no—
331C2×C59#243511yescharacteristic
332C2×C59#300611yescharacteristic
333C2×C59#300911yescharacteristic
334C2×C59#330612yes—
335C2×C59#331112yes—
336C22×#41C2×D5911yescharacteristic
337C23×D4C2×D5911yescharacteristic
338#546—24no—
339#546—24no—
340C2×C4⋊Q8—591no—
341C2×#124—1181no—
342C2×#124C2×D5911yescharacteristic
343C2×#123—21no—
344C2×C22⋊Q8C2×D5911yescharacteristic
345C2×#121—21no—
346C2×#121C2×D5911yescharacteristic
347C2×#120—21no—
348C2×C4×D4C2×D5911yescharacteristic
349C2×C4×D4C2×D5911yescharacteristic
350C22×C22⋊C4—21no—
351#489—1184no—
352#476—1184no—
353C2×#106—592no—
354C2×C4⋊C8—591no—
355C2×#103—592no—
356C2×#103—592no—
357C2×#102—592no—
358C2×#102—592no—
359C2×#102—592no—
360C2×#102—592no—
361C2×C22⋊C8—592no—
362C2×C22⋊C8—592no—
363C2×#97—591no—
364#397—1181no—
365#397—1181no—
366#391—1181no—
367#380—592no—
368#380—592no—
369C4×C4⋊C4—591no—
370#340—592no—
371C59#2589811yescharacteristic, complemented, Sylow
372C25—21no—
373C2×#41—22no—
374C2×#41—22no—
375C2×#41—24no—
376C2×#41—42no—
377C22×Q8—1181no—
378C22×Q8C22×D5911yescharacteristic
379C22×D4—21no—
380C22×D4—21no—
381C22×D4—22no—
382C22×D4—22no—
383C22×D4—24no—
384C22×D4—21no—
385C22×D4—42no—
386C22×D4C22⋊D5911yescharacteristic
387C22×D4C22×D5911yescharacteristic
388C22×D4C22×D5911yescharacteristic
389C23×C4—21no—
390C23×C4—21no—
391C23×C4C22×D5911yescharacteristic
392C23×C4C22×D5911yescharacteristic
393C22×C8—592no—
394C22×C8—592no—
395C4⋊Q8—1182no—
396C4⋊Q8—1182no—
397#124—22no—
398#124—2364no—
399#124C22⋊D5914yes—
400#123—28no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403#121—28no—
404#121—22no—
405#121—22no—
406#120—28no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4C22⋊D5914yes—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—592no—
413C2×C4⋊C4—592no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4C22⋊D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C22⋊C4—28no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1182no—
424C2×C22⋊C4—1182no—
425C2×C22⋊C4C22⋊D5911yescharacteristic
426C2×C22⋊C4C22⋊D5911yescharacteristic
427C2×C42—1181no—
428C2×C42C22×D5911yescharacteristic
429#106—1184no—
430C4⋊C8—1182no—
431#103—1184no—
432#103—2362no—
433#102—1184no—
434#102—1184no—
435#102—1184no—
436#102—2362no—
437C22⋊C8—2362no—
438C22⋊C8—2362no—
439#97—1182no—
440#95—1181no—
441#95—1182no—
442#95—1182no—
443C24—21no—
444C24—24no—
445C24—21no—
446C24—44no—
447C24—42no—
448C24—42no—
449C24—82no—
450C24C2×C22⋊D5911yescharacteristic
451C24C2×C22⋊D5911yescharacteristic
452#41—48no—
453#41—44no—
454#41—44no—
455C2×Q8—21no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—2362no—
459C2×Q8—2362no—
460C2×Q8—2362no—
461C2×Q8C2×C22⋊D5912yes—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—22no—
465C2×D4—24no—
466C2×D4—22no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—41no—
473C2×D4—44no—
474C2×D4—48no—
475C2×D4—44no—
476C2×D4—44no—
477C2×D4—44no—
478C2×D4—82no—
479C2×D4#1174568714yes—
480C2×D4C2×C22⋊D5912yes—
481C2×D4C2×C22⋊D5912yes—
482C22×C4—24no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—44no—
495C22×C4—82no—
496C22×C4—1182no—
497C22×C4—1182no—
498C22×C4C2×C22⋊D5911yescharacteristic
499C22×C4C2×C22⋊D5911yescharacteristic
500C22×C4C2×C22⋊D5911yescharacteristic
501C22×C4C2×C22⋊D5911yescharacteristic
502C22×C4C23×D5911yescharacteristic
503C2×C8—1182no—
504C2×C8—1182no—
505C2×C8—1182no—
506C2×C8—1182no—
507C2×C8—1182no—
508C2×C8—1182no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—1182no—
513C4⋊C4—1182no—
514C4⋊C4—1182no—
515C4⋊C4—1182no—
516C4⋊C4—2362no—
517C4⋊C4C2×C22⋊D5912yes—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—22no—
525C22⋊C4—2364no—
526C22⋊C4—2364no—
527C42—2362no—
528C42C2×C22⋊D5912yes—
529C42C2×C22⋊D5912yes—
530C23—21no—
531C23—22no—
532C23—21no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—22no—
542C23—48no—
543C23—48no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—88no—
552C23—84no—
553C23—88no—
554C23#1427375212yes—
555C23C22×C22⋊D5911yescharacteristic
556Q8—24no—
557Q8—41no—
558Q8—41no—
559Q8—4724no—
560D4—22no—
561D4—24no—
562D4—24no—
563D4—44no—
564D4—48no—
565D4—48no—
566D4—41no—
567D4—41no—
568D4—84no—
569D4#1427371414yescomplemented
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—2362no—
590C2×C4—2362no—
591C2×C4—2362no—
592C2×C4—2362no—
593C2×C4—2362no—
594C2×C4—2362no—
595C2×C4#1427369612yes—
596C2×C4#1427369612yes—
597C2×C4C22×C22⋊D5911yescharacteristic
598C2×C4C22×C22⋊D5911yescharacteristic
599C2×C4C2×#1174568712yes—
600C2×C4#1427377212yes—
601C8—2362no—
602C8—2362no—
603C22—22no—
604C22—24no—
605C22—24no—
606C22—42no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—44no—
611C22—42no—
612C22—44no—
613C22—41no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—816no—
622C22—82no—
623C22—82no—
624C22C2×#1427369611yescharacteristic
625C22#1683804612yes—
626C22C2×#1427375211yescharacteristic
627C22#1683820011yescharacteristic
628C22#1683821312yes—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—42no—
635C4—4722no—
636C4—4722no—
637C4C2×#1427371411yescharacteristic
638C4#1683796911yescharacteristic
639C4#1683804612yes—
640C2—42no—
641C2—44no—
642C2—81no—
643C2—82no—
644C2#2142774311yescharacteristic
645C2#2143334811yescharacteristic
646C2#2143337311yescharacteristic
647C2#2143605712yes—
648C2#2143613212yes—
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
Fitting, prime radicalC2×C59×#546C2
socleC23×C59C22×D4
hypercenterC2×#546D59
hypocenterC59#25898
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#546

Complements

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

NH ≅ G/Ntype
1. C2×#16837761C2semidirect
2. C2×#16837692C2semidirect
3. #21429088C2semidirect
4. #21425289C2semidirect
5. #21425282C2semidirect
6. #21425029C2semidirect
7. #21424895C2semidirect
8. #21424036C2semidirect
9. #21424027C2semidirect
10. #21423921C2semidirect
12. C2×#14273727C22semidirect
16. C2×#14273658C22semidirect
17. C2×C4⋊(C59⋊C8)C22semidirect
18. C2×#14273651C22semidirect
19. C2×#14273651C22semidirect
25. C2×C22⋊(C59⋊C8)C22semidirect
26. C2×#14273638C22semidirect
31. #16837262C22semidirect
32. C4×C4⋊Dic59C22semidirect
33. #16837164C22semidirect
243. D4×C59#124semidirect
371. C59#25898semidirect
569. D4#14273714semidirect

Maximal subgroups

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

subgroupindex
C2×#168377612
C2×#168376922
#214290882
#214252892
#214252822
#214250292
#214248952
#214240362
#214240272
#214239212
C2×C59×#5462
#2589859