Finite Group Databasebeta

← #17067333

Subgroups — #17067333

Order 4352.

1504 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763137C211yesmaximal, characteristic, complemented
2C22×#10587503C211yesmaximal, characteristic, complemented
3#14533092C211yesmaximal, characteristic, complemented
4#14530020C212yesmaximal, complemented
5#14529970C212yesmaximal, complemented
6#14529208C211yesmaximal, characteristic, complemented
7#14529077C211yesmaximal, characteristic, complemented
8#14529045C212yesmaximal, complemented
9#14528724C212yesmaximal, complemented
10#14528715C211yesmaximal, characteristic, complemented
11C2×C17×#546C211yesmaximal, characteristic
12C23×#3622C2211yescharacteristic
13C2×#10587621—21no—
14C22×C4⋊Dic17C2211yescharacteristic
15#11763137—24no—
16C2×#10587479C2212yescomplemented
17C2×#10587506C2212yescomplemented
18C2×#10587506C2212yescomplemented
19C2×#10587503—24no—
20C2×#10587503C2212yes—
21C2×#10587503C2212yes—
22C2×#10587503C2212yes—
23C2×#10587503C2212yes—
24C2×C22⋊#3622C2211yescharacteristic
25C2×C22⋊#3622C2212yescomplemented
26#11762739—21no—
27#11762734—21no—
28#11762707C2212yescomplemented
29#11762670—21no—
30#11762669C2211yescharacteristic
31#11762669C2212yescomplemented
32#11762571C2212yescomplemented
33#11762564C2212yescomplemented
34C22×#41×C17C2211yescharacteristic
35C23×D4×C17C2211yescharacteristic
36C17×#546—24no—
37C17×#546—24no—
38C2×C17×#124C2211yescharacteristic
39C2×C17×#123—21no—
40C2×C17×C22⋊Q8C2211yescharacteristic
41C2×C17×#121—21no—
42C2×C17×#121C2211yescharacteristic
43C2×C17×#120—21no—
44C2×C4×D4×C17C2211yescharacteristic
45C2×C4×D4×C17C2211yescharacteristic
46C22×C22⋊C4×C17—21no—
47C25×C17—21no—
48C2×#41×C17—22no—
49C2×#41×C17—22no—
50C2×#41×C17—24no—
51C2×#41×C17—42no—
52C22×Q8×C17C2311yescharacteristic
53C22×D4×C17—24no—
54C22×D4×C17—21no—
55C22×D4×C17—21no—
56C22×D4×C17—21no—
57C22×D4×C17—22no—
58C22×D4×C17—42no—
59C22×D4×C17D411yescharacteristic
60C22×D4×C17D414yes—
61C22×D4×C17C2311yescharacteristic
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C23×C4×C17—21no—
65C23×C4×C17C2311yescharacteristic
66C23×C4×C17C2311yescharacteristic
67C23×Dic17—21no—
68C17×#124—22no—
69C17×#124—22no—
70C17×#123—28no—
71C17×C22⋊Q8—22no—
72C17×C22⋊Q8D414yes—
73C17×#121—28no—
74C17×#121—22no—
75C17×#121—22no—
76C17×#120—28no—
77C4×D4×C17—22no—
78C4×D4×C17—22no—
79C4×D4×C17—22no—
80C4×D4×C17—22no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17D411yescharacteristic
83C2×C4⋊C4×C17C2311yescharacteristic
84C2×C22⋊C4×C17—28no—
85C2×C22⋊C4×C17—21no—
86C2×C22⋊C4×C17—21no—
87C2×C22⋊C4×C17—21no—
88C2×C22⋊C4×C17—21no—
89C2×C22⋊C4×C17—21no—
90C2×C22⋊C4×C17D411yescharacteristic
91C2×C22⋊C4×C17D411yescharacteristic
92C2×C42×C17C2311yescharacteristic
93C2×C22⋊Dic17—22no—
94C2×C22⋊Dic17—21no—
95#10587621—44no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17—22no—
98C2×C4⋊Dic17—22no—
99C2×C4⋊Dic17C2312yes—
100C2×C4⋊Dic17C2312yes—
101C2×#63225—22no—
102C22×#3622—22no—
103C22×#3622—42no—
104C22×#3622C2312yes—
105C22×#3622C2312yes—
106#10587506—24no—
107#10587506—42no—
108#10587503—24no—
109#10587503—24no—
110#10587503—24no—
111#10587503—42no—
112#10587502—22no—
113#10587502—21no—
114#10587502—22no—
115C22⋊#3622—22no—
116C22⋊#3622—42no—
117#10587479—24no—
118C24×C17—21no—
119C24×C17—21no—
120C24×C17—24no—
121C24×C17—44no—
122C24×C17—42no—
123C24×C17—42no—
124C24×C17—82no—
125C24×C17C2×D411yescharacteristic
126C24×C17C2×D411yescharacteristic
127#41×C17—48no—
128#41×C17—44no—
129#41×C17—44no—
130C2×Q8×C17—21no—
131C2×Q8×C17—22no—
132C2×Q8×C17—22no—
133C2×Q8×C17C2×D412yes—
134C2×D4×C17—28no—
135C2×D4×C17—24no—
136C2×D4×C17—22no—
137C2×D4×C17—22no—
138C2×D4×C17—22no—
139C2×D4×C17—24no—
140C2×D4×C17—24no—
141C2×D4×C17—21no—
142C2×D4×C17—21no—
143C2×D4×C17—22no—
144C2×D4×C17—22no—
145C2×D4×C17—22no—
146C2×D4×C17—44no—
147C2×D4×C17—48no—
148C2×D4×C17—44no—
149C2×D4×C17—44no—
150C2×D4×C17—82no—
151C2×D4×C17C2×D412yes—
152C2×D4×C17C2×D412yes—
153C2×D4×C17#4114yes—
154C22×C4×C17—22no—
155C22×C4×C17—22no—
156C22×C4×C17—24no—
157C22×C4×C17—21no—
158C22×C4×C17—21no—
159C22×C4×C17—24no—
160C22×C4×C17—21no—
161C22×C4×C17—21no—
162C22×C4×C17—22no—
163C22×C4×C17—22no—
164C22×C4×C17—44no—
165C22×C4×C17—82no—
166C22×C4×C17C2×D411yescharacteristic
167C22×C4×C17C2×D411yescharacteristic
168C22×C4×C17C2×D411yescharacteristic
169C22×C4×C17C2×D411yescharacteristic
170C22×C4×C17C2×D412yes—
171C22×C4×C17C2×D412yes—
172C22×C4×C17C2411yescharacteristic
173C22×Dic17—22no—
174C22×Dic17—22no—
175C22×Dic17—42no—
176C22×Dic17—42no—
177C22×Dic17—42no—
178C4⋊C4×C17—24no—
179C4⋊C4×C17—22no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17C2×D412yes—
182C22⋊C4×C17—24no—
183C22⋊C4×C17—24no—
184C22⋊C4×C17—24no—
185C22⋊C4×C17—24no—
186C22⋊C4×C17—24no—
187C22⋊C4×C17—22no—
188C22⋊C4×C17—22no—
189C42×C17—21no—
190C42×C17—21no—
191C22⋊Dic17—44no—
192C22⋊Dic17—42no—
193C4⋊Dic17—22no—
194C4⋊Dic17—22no—
195C4⋊Dic17—22no—
196C4⋊Dic17—22no—
197C4⋊Dic17—42no—
198#63225—44no—
199C2×#3622—22no—
200C2×#3622—22no—
201C2×#3622—22no—
202C2×#3622—22no—
203C2×#3622—22no—
204C2×#3622—22no—
205C2×#3622—44no—
206#24038—171nomaximal, Sylow
207C23×C17—22no—
208C23×C17—22no—
209C23×C17—22no—
210C23×C17—24no—
211C23×C17—22no—
212C23×C17—21no—
213C23×C17—22no—
214C23×C17—21no—
215C23×C17—21no—
216C23×C17—22no—
217C23×C17—48no—
218C23×C17—41no—
219C23×C17—41no—
220C23×C17—42no—
221C23×C17—42no—
222C23×C17—41no—
223C23×C17—41no—
224C23×C17—42no—
225C23×C17—48no—
226C23×C17—84no—
227C23×C17—88no—
228C23×C17—88no—
229C23×C17#12012yes—
230C23×C17#13512yes—
231C23×C17AGL(1,ℤ/8)12yes—
232C23×C17C22×D411yescharacteristic
233Q8×C17—24no—
234Q8×C17—41no—
235Q8×C17—41no—
236D4×C17—22no—
237D4×C17—22no—
238D4×C17—22no—
239D4×C17—24no—
240D4×C17—48no—
241D4×C17—48no—
242D4×C17—44no—
243D4×C17—42no—
244D4×C17—84no—
245D4×C17#12314yescomplemented
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—22no—
249C2×C4×C17—24no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—22no—
253C2×C4×C17—21no—
254C2×C4×C17—22no—
255C2×C4×C17—22no—
256C2×C4×C17—21no—
257C2×C4×C17—21no—
258C2×C4×C17—22no—
259C2×C4×C17—48no—
260C2×C4×C17—42no—
261C2×C4×C17—42no—
262C2×C4×C17—41no—
263C2×C4×C17—41no—
264C2×C4×C17—42no—
265C2×C4×C17—84no—
266C2×C4×C17#12012yes—
267C2×C4×C17C22×D411yescharacteristic
268C2×C4×C17C22×D411yescharacteristic
269C2×C4×C17C2×#4112yes—
270C2×C4×C17H(5,𝔽2)12yes—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C2×Dic17—42no—
275C2×Dic17—42no—
276C2×Dic17—42no—
277C2×Dic17—84no—
278#3622—42no—
279#3622—42no—
280C2×#546D1711yescharacteristic
281C2×#483—171no—
282C22×#102—171no—
283#2486—171no—
284#2042—172no—
285#2034—172no—
286#1892—171no—
287#1869—171no—
288#1864—172no—
289#1788—172no—
290#1786—171no—
291C22×C17—22no—
292C22×C17—22no—
293C22×C17—24no—
294C22×C17—24no—
295C22×C17—24no—
296C22×C17—42no—
297C22×C17—42no—
298C22×C17—44no—
299C22×C17—44no—
300C22×C17—41no—
301C22×C17—42no—
302C22×C17—41no—
303C22×C17—42no—
304C22×C17—41no—
305C22×C17—42no—
306C22×C17—42no—
307C22×C17—42no—
308C22×C17—816no—
309C22×C17—82no—
310C22×C17—82no—
311C22×C17#44914yes—
312C22×C17C2×#12011yescharacteristic
313C22×C17#54612yes—
314C22×C17C2×#13511yescharacteristic
315C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
316C22×C17#57712yes—
317C4×C17—22no—
318C4×C17—41no—
319C4×C17—42no—
320C4×C17—41no—
321C4×C17—41no—
322C4×C17—42no—
323C4×C17C2×#12311yescharacteristic
324C4×C17#53411yescharacteristic
325C4×C17#54612yes—
326Dic17—82no—
327Dic17—82no—
328C22×#41C2×D1711yescharacteristic
329C23×D4C2×D1711yescharacteristic
330C23×C8—171no—
331#546—24no—
332#546—24no—
333C2×#124C2×D1711yescharacteristic
334C2×#123—21no—
335C2×#123—341no—
336C2×C22⋊Q8C2×D1711yescharacteristic
337C2×#121—21no—
338C2×#121C2×D1711yescharacteristic
339C2×#120—21no—
340C2×C4×D4C2×D1711yescharacteristic
341C2×C4×D4C2×D1711yescharacteristic
342C22×C4⋊C4—171no—
343C22×C22⋊C4—21no—
344#483—344no—
345C2×#106—172no—
346C2×#103—172no—
347C2×#103—172no—
348C2×#102—172no—
349C2×#102—172no—
350C2×#102—172no—
351C2×#102—172no—
352C2×#102—344no—
353C2×C22⋊C8—171no—
354C2×C22⋊C8—172no—
355#398—341no—
356#397—341no—
357#389—172no—
358#381—341no—
359#380—171no—
360#380—172no—
361#340—172no—
362#336—172no—
363C2×C17—24no—
364C2×C17—44no—
365C2×C17—81no—
366C2×C17—82no—
367C2×C17#240511yescharacteristic
368C2×C17C2×#44911yescharacteristic
369C2×C17#300911yescharacteristic
370C2×C17#329612yes—
371C2×C17#330612yes—
372C25—21no—
373C2×#41—22no—
374C2×#41—22no—
375C2×#41—24no—
376C2×#41—42no—
377C22×Q8C22×D1711yescharacteristic
378C22×D4—24no—
379C22×D4—21no—
380C22×D4—21no—
381C22×D4—21no—
382C22×D4—22no—
383C22×D4—42no—
384C22×D4C22⋊D1711yescharacteristic
385C22×D4C22⋊D1714yes—
386C22×D4C22×D1711yescharacteristic
387C22×D4C22×D1711yescharacteristic
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—341no—
391C23×C4C22×D1711yescharacteristic
392C23×C4C22×D1711yescharacteristic
393C22×C8—172no—
394C22×C8—172no—
395C22×C8—342no—
396C22×C8—682no—
397#124—22no—
398#124—22no—
399#123—28no—
400#123—684no—
401C22⋊Q8—22no—
402C22⋊Q8C22⋊D1714yes—
403#121—28no—
404#121—22no—
405#121—22no—
406#120—28no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—172no—
413C2×C4⋊C4—172no—
414C2×C4⋊C4—342no—
415C2×C4⋊C4—342no—
416C2×C4⋊C4—341no—
417C2×C4⋊C4—342no—
418C2×C4⋊C4C22⋊D1711yescharacteristic
419C2×C4⋊C4C22×D1711yescharacteristic
420C2×C22⋊C4—28no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—341no—
427C2×C22⋊C4—342no—
428C2×C22⋊C4C22⋊D1711yescharacteristic
429C2×C22⋊C4C22⋊D1711yescharacteristic
430C2×C42C22×D1711yescharacteristic
431#106—344no—
432#103—344no—
433#103—682no—
434#102—344no—
435#102—344no—
436#102—344no—
437#102—682no—
438C22⋊C8—342no—
439C22⋊C8—682no—
440#95—342no—
441#95—341no—
442#95—342no—
443C17#2403811yescharacteristic, complemented, Sylow
444C24—21no—
445C24—21no—
446C24—24no—
447C24—44no—
448C24—42no—
449C24—42no—
450C24—82no—
451C24C2×C22⋊D1711yescharacteristic
452C24C2×C22⋊D1711yescharacteristic
453#41—48no—
454#41—44no—
455#41—44no—
456C2×Q8—21no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8C2×C22⋊D1712yes—
460C2×D4—28no—
461C2×D4—24no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—24no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—24no—
472C2×D4—44no—
473C2×D4—48no—
474C2×D4—44no—
475C2×D4—44no—
476C2×D4—82no—
477C2×D4#6325414yes—
478C2×D4C2×C22⋊D1712yes—
479C2×D4C2×C22⋊D1712yes—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—44no—
491C22×C4—82no—
492C22×C4—342no—
493C22×C4—342no—
494C22×C4—682no—
495C22×C4—682no—
496C22×C4—682no—
497C22×C4C2×C22⋊D1711yescharacteristic
498C22×C4C2×C22⋊D1711yescharacteristic
499C22×C4C2×C22⋊D1711yescharacteristic
500C22×C4C2×C22⋊D1712yes—
501C22×C4C2×C22⋊D1712yes—
502C22×C4C2×C22⋊D1711yescharacteristic
503C22×C4C23×D1711yescharacteristic
504C2×C8—342no—
505C2×C8—342no—
506C2×C8—342no—
507C2×C8—342no—
508C2×C8—342no—
509C2×C8—342no—
510C2×C8—684no—
511C4⋊C4—21no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—342no—
515C4⋊C4—342no—
516C4⋊C4—342no—
517C4⋊C4—342no—
518C4⋊C4—684no—
519C4⋊C4—682no—
520C4⋊C4C2×C22⋊D1712yes—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—22no—
526C22⋊C4—24no—
527C22⋊C4—22no—
528C22⋊C4—684no—
529C22⋊C4—682no—
530C42—21no—
531C42—21no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—22no—
542C23—48no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—48no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—84no—
552C23—88no—
553C23—88no—
554C23#1058763512yes—
555C23#1058763612yes—
556C23#1058763912yes—
557C23C22×C22⋊D1711yescharacteristic
558Q8—24no—
559Q8—41no—
560Q8—41no—
561D4—22no—
562D4—22no—
563D4—22no—
564D4—24no—
565D4—48no—
566D4—48no—
567D4—44no—
568D4—42no—
569D4—84no—
570D4#1058762114yescomplemented
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4—682no—
592C2×C4—682no—
593C2×C4—682no—
594C2×C4—682no—
595C2×C4—682no—
596C2×C4—682no—
597C2×C4—1364no—
598C2×C4#1058763912yes—
599C2×C4C2×#6325412yes—
600C2×C4#1058769112yes—
601C2×C4C22×C22⋊D1711yescharacteristic
602C2×C4C22×C22⋊D1711yescharacteristic
603C8—682no—
604C8—682no—
605C22—22no—
606C22—24no—
607C22—22no—
608C22—24no—
609C22—24no—
610C22—42no—
611C22—44no—
612C22—42no—
613C22—44no—
614C22—41no—
615C22—42no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—41no—
620C22—42no—
621C22—42no—
622C22—816no—
623C22—82no—
624C22—82no—
625C22#1176295414yes—
626C22C2×#1058763911yescharacteristic
627C22#1176345312yes—
628C22C2×#1058763611yescharacteristic
629C22C2×#1058763511yescharacteristic
630C22#1176362012yes—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—42no—
637C4—1362no—
638C4—1362no—
639C4C2×#1058762111yescharacteristic
640C4#1176336811yescharacteristic
641C4#1176345312yes—
642C2—24no—
643C2—44no—
644C2—81no—
645C2—82no—
646C2#1453210711yescharacteristic
647C2C2×#1176295411yescharacteristic
648C2#1453806111yescharacteristic
649C2#1454061912yes—
650C2#1454074512yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#546C2
socleC23×C17C22×D4
hypercenterC2×#546D17
hypocenterC17#24038
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#546

Complements

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

NH ≅ G/Ntype
1. C2×#11763137C2semidirect
2. C22×#10587503C2semidirect
3. #14533092C2semidirect
4. #14530020C2semidirect
5. #14529970C2semidirect
6. #14529208C2semidirect
7. #14529077C2semidirect
8. #14529045C2semidirect
9. #14528724C2semidirect
10. #14528715C2semidirect
16. C2×#10587479C22semidirect
17. C2×#10587506C22semidirect
18. C2×#10587506C22semidirect
25. C2×C22⋊#3622C22semidirect
28. #11762707C22semidirect
31. #11762669C22semidirect
32. #11762571C22semidirect
33. #11762564C22semidirect
245. D4×C17#123semidirect
443. C17#24038semidirect
570. D4#10587621semidirect

Maximal subgroups

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

subgroupindex
C2×#117631372
C22×#105875032
#145330922
#145300202
#145299702
#145292082
#145290772
#145290452
#145287242
#145287152
C2×C17×#5462
#2403817