Finite Group Databasebeta

← #17066745

Subgroups — #17066745

Order 4352.

1424 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762959C211yesmaximal, characteristic, complemented
2C2×#11762813C211yesmaximal, characteristic, complemented
3#14532140C211yesmaximal, characteristic
4#14529927C212yesmaximal, complemented
5#14529785C212yesmaximal, complemented
6#14529218C211yesmaximal, characteristic, complemented
7#14529139C212yesmaximal, complemented
8#14529101C212yesmaximal, complemented
9#14529082C211yesmaximal, characteristic, complemented
10#14528879C211yesmaximal, characteristic, complemented
11C2×C17×#552C211yesmaximal, characteristic, complemented
12C23×D68C2211yescharacteristic
13C22×#63230C2212yes—
14C22×#63223C2211yescharacteristic
15C2×#10587616C2212yes—
16C2×#10587623—21no—
17#11762959—24no—
18#11762813—24no—
19C2×#10587506C2212yescomplemented
20C2×#10587480C2212yes—
21C2×#10587480C2212yes—
22C2×#10587480C2212yes—
23C2×C22⋊#3622C2211yescharacteristic
24C2×C22⋊#3622C2212yescomplemented
25#11762726—21no—
26#11762718—21no—
27#11762711C2212yes—
28#11762669C2211yescharacteristic
29#11762664—21no—
30#11762571C2212yescomplemented
31#11762571C2212yescomplemented
32#11762571C2212yescomplemented
33#11762564C2212yescomplemented
34C17×#552—24no—
35C17×#552—24no—
36C2×C17×#126—21no—
37C2×C17×#125—21no—
38C2×C17×#125C2211yescharacteristic, complemented
39C2×C17×#123—21no—
40C2×C17×C22⋊Q8C2211yescharacteristic
41C2×C17×C22⋊Q8C2211yescharacteristic, complemented
42C2×C4×Q8×C17C2211yescharacteristic, complemented
43C2×C4×D4×C17C2211yescharacteristic
44C2×C17×#117—21no—
45C2×C17×#117C2211yescharacteristic
46C22×C4⋊C4×C17C2211yescharacteristic, complemented
47C24×D17—21no—
48C22×Q8×C17C2311yescharacteristic
49C22×D4×C17D411yescharacteristic
50C23×C4×C17—21no—
51C23×C4×C17C2311yescharacteristic
52C23×C4×C17C2311yescharacteristic
53C22×C22⋊D17—22no—
54C22×D68—22no—
55C22×D68—24no—
56C22×D68—22no—
57C22×D68—21no—
58C22×D68—42no—
59C22×D68C2312yes—
60C17×#126—28no—
61C17×#125—28no—
62C17×#125—22no—
63C17×#125—22no—
64C17×#123—28no—
65C17×C22⋊Q8—22no—
66C17×C22⋊Q8—22no—
67C17×C22⋊Q8—22no—
68C17×C22⋊Q8D414yescomplemented
69C4×Q8×C17—22no—
70C4×Q8×C17—22no—
71C4×D4×C17—22no—
72C4×D4×C17—22no—
73C17×#117—28no—
74C17×#117—22no—
75C17×#117D414yescomplemented
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17—21no—
79C2×C4⋊C4×C17—21no—
80C2×C4⋊C4×C17—21no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17—22no—
83C2×C4⋊C4×C17—22no—
84C2×C4⋊C4×C17D411yescharacteristic
85C2×C4⋊C4×C17C2311yescharacteristic
86C2×C4⋊C4×C17C2311yescharacteristic
87C2×C4⋊C4×C17C2311yescharacteristic
88C2×C22⋊C4×C17—21no—
89C2×C22⋊C4×C17—21no—
90C2×C22⋊C4×C17—21no—
91C2×C22⋊C4×C17D411yescharacteristic
92C2×C22⋊C4×C17D411yescharacteristic
93C2×C42×C17—21no—
94C2×C42×C17—21no—
95C2×C42×C17C2311yescharacteristic
96C2×C22⋊Dic17—21no—
97C2×#63230—24no—
98C2×#63230—24no—
99C2×#63230—42no—
100#10587623—44no—
101#10587616—24no—
102#10587616—42no—
103C2×#63227—22no—
104C2×#63227—22no—
105C2×#63227—22no—
106C2×C4⋊Dic17C2312yes—
107C2×#63223—22no—
108C2×#63223—42no—
109C22×#3622C2312yes—
110C22×#3622C2312yes—
111#10587506—42no—
112#10587502—21no—
113C22⋊#3622—22no—
114C22⋊#3622—42no—
115#10587480—24no—
116#10587480—24no—
117#10587480—24no—
118C24×C17C2×D411yescharacteristic
119C24×C17C2×D411yescharacteristic
120C23×D17—22no—
121C23×D17—44no—
122C23×D17—42no—
123C23×D17—42no—
124C23×D17—42no—
125C23×D17—82no—
126C2×Q8×C17—21no—
127C2×Q8×C17—22no—
128C2×Q8×C17—22no—
129C2×Q8×C17C2×D412yes—
130C2×D4×C17—22no—
131C2×D4×C17—22no—
132C2×D4×C17—41no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—24no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—22no—
143C22×C4×C17—22no—
144C22×C4×C17—42no—
145C22×C4×C17—42no—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D412yes—
151C22×C4×C17C2×D412yes—
152C22×C4×C17C2411yescharacteristic
153C2×C22⋊D17—44no—
154C2×C22⋊D17—44no—
155C2×C22⋊D17—82no—
156C22×Dic17—22no—
157C2×D68—28no—
158C2×D68—22no—
159C2×D68—22no—
160C2×D68—22no—
161C2×D68—22no—
162C2×D68—22no—
163C2×D68—48no—
164C2×D68—44no—
165C2×D68—44no—
166C2×D68—42no—
167C2×D68#4114yes—
168C4⋊C4×C17—24no—
169C4⋊C4×C17—24no—
170C4⋊C4×C17—24no—
171C4⋊C4×C17—24no—
172C4⋊C4×C17—24no—
173C4⋊C4×C17—24no—
174C4⋊C4×C17—22no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17C2×D412yes—
179C4⋊C4×C17C2×D412yes—
180C4⋊C4×C17C2×D412yes—
181C22⋊C4×C17—24no—
182C22⋊C4×C17—24no—
183C22⋊C4×C17—24no—
184C22⋊C4×C17—22no—
185C22⋊C4×C17—22no—
186C42×C17—24no—
187C42×C17—24no—
188C42×C17—21no—
189C42×C17—21no—
190C22⋊Dic17—42no—
191#63230—48no—
192#63227—44no—
193#63227—44no—
194#63227—82no—
195C4⋊Dic17—22no—
196C4⋊Dic17—22no—
197#63223—44no—
198C2×#3622—22no—
199C2×#3622—22no—
200C2×#3622—22no—
201C2×#3622—22no—
202C2×#3622—22no—
203C2×#3622—22no—
204#23982—171nomaximal, Sylow
205C23×C17—21no—
206C23×C17—22no—
207C23×C17—22no—
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—21no—
211C23×C17—22no—
212C23×C17—22no—
213C23×C17#12012yes—
214C23×C17AGL(1,ℤ/8)12yes—
215C23×C17C22×D411yescharacteristic
216C22×D17—24no—
217C22×D17—48no—
218C22×D17—42no—
219C22×D17—42no—
220C22×D17—42no—
221C22×D17—42no—
222C22×D17—42no—
223C22×D17—42no—
224C22×D17—88no—
225C22×D17—88no—
226C22×D17—84no—
227C22×D17—84no—
228Q8×C17—24no—
229Q8×C17—41no—
230Q8×C17—41no—
231D4×C17—44no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—24no—
236C2×C4×C17—21no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—22no—
247C2×C4×C17—48no—
248C2×C4×C17—42no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—42no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—42no—
257C2×C4×C17#12012yes—
258C2×C4×C17C22×D411yescharacteristic
259C2×C4×C17C22×D411yescharacteristic
260C2×C4×C17C2×#4112yes—
261C2×C4×C17H(5,𝔽2)12yes—
262C22⋊D17—88no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—42no—
266D68—24no—
267D68—24no—
268D68—48no—
269D68—42no—
270D68—42no—
271#3622—42no—
272#3622—42no—
273C2×#552D1711yescharacteristic, complemented
274C2×#450—171no—
275C2×#418—171no—
276#2408—171no—
277#2025—172no—
278#2000—172no—
279#1894—171no—
280#1880—172no—
281#1874—172no—
282#1870—171no—
283#1825—171no—
284C22×C17—22no—
285C22×C17—22no—
286C22×C17—22no—
287C22×C17—22no—
288C22×C17—24no—
289C22×C17—41no—
290C22×C17—41no—
291C22×C17—42no—
292C22×C17—42no—
293C22×C17—42no—
294C22×C17C2×#12011yescharacteristic
295C22×C17#54612yes—
296C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C17#57511yescharacteristic
298C22×C17#57712yes—
299C2×D17—44no—
300C2×D17—44no—
301C2×D17—44no—
302C2×D17—816no—
303C2×D17—82no—
304C2×D17—82no—
305C2×D17—82no—
306C2×D17—82no—
307C4×C17—22no—
308C4×C17—41no—
309C4×C17—42no—
310C4×C17—41no—
311C4×C17—42no—
312C4×C17—42no—
313C4×C17—41no—
314C4×C17—42no—
315C4×C17#52511yescharacteristic
316C4×C17#53511yescharacteristic
317C4×C17#54612yes—
318Dic17—82no—
319C23×D4—171no—
320C22×SD16—172no—
321C22×M16—171no—
322#552—24no—
323#552—24no—
324C2×#126—21no—
325C2×#125—21no—
326C2×#125C2×D1711yescharacteristic, complemented
327C2×#123—21no—
328C2×C22⋊Q8C2×D1711yescharacteristic
329C2×C22⋊Q8C2×D1711yescharacteristic, complemented
330C2×#121—172no—
331C2×#120—341no—
332C2×C4×Q8C2×D1711yescharacteristic, complemented
333C2×C4×D4C2×D1711yescharacteristic
334C2×#117—21no—
335C2×#117C2×D1711yescharacteristic
336C22×C4⋊C4C2×D1711yescharacteristic, complemented
337#450—344no—
338#418—344no—
339C2×#103—172no—
340C2×#102—172no—
341C2×#102—172no—
342C2×#102—172no—
343C2×C22⋊C8—171no—
344C2×C22⋊C8—172no—
345#394—341no—
346#392—341no—
347#390—172no—
348#380—171no—
349#379—341no—
350#340—172no—
351#340—172no—
352#340—172no—
353#336—172no—
354C2×C17—42no—
355C2×C17—42no—
356C2×C17#240411yescharacteristic
357C2×C17#300611yescharacteristic
358C2×C17#301011yescharacteristic
359C2×C17#331012yes—
360C2×C17#332312yes—
361D17—84no—
362C25—341no—
363C22×Q8C22×D1711yescharacteristic
364C22×D4—172no—
365C22×D4—344no—
366C22×D4—342no—
367C22×D4—342no—
368C22×D4—342no—
369C22×D4—341no—
370C22×D4—682no—
371C22×D4C22⋊D1711yescharacteristic
372C23×C4—21no—
373C23×C4C22×D1711yescharacteristic
374C23×C4C22×D1711yescharacteristic
375C2×SD16—344no—
376C2×SD16—344no—
377C2×SD16—682no—
378C2×M16—342no—
379C2×M16—682no—
380C22×C8—172no—
381C22×C8—172no—
382#126—28no—
383#125—28no—
384#125—22no—
385#125—22no—
386#123—28no—
387C22⋊Q8—22no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8C22⋊D1714yescomplemented
391#121—344no—
392#121—682no—
393#120—684no—
394C4×Q8—22no—
395C4×Q8—22no—
396C4×D4—22no—
397C4×D4—22no—
398#117—28no—
399#117—22no—
400#117C22⋊D1714yescomplemented
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—172no—
410C2×C4⋊C4C22⋊D1711yescharacteristic
411C2×C4⋊C4C22×D1711yescharacteristic
412C2×C4⋊C4C22×D1711yescharacteristic
413C2×C4⋊C4C22×D1711yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—341no—
418C2×C22⋊C4—342no—
419C2×C22⋊C4—342no—
420C2×C22⋊C4—342no—
421C2×C22⋊C4C22⋊D1711yescharacteristic
422C2×C22⋊C4C22⋊D1711yescharacteristic
423C2×C42—21no—
424C2×C42—21no—
425C2×C42C22×D1711yescharacteristic
426#103—682no—
427#102—344no—
428#102—344no—
429#102—344no—
430C22⋊C8—342no—
431C22⋊C8—682no—
432#95—341no—
433C17#2398211yescharacteristic, complemented, Sylow
434C24—342no—
435C24—684no—
436C24—682no—
437C24—682no—
438C24—682no—
439C24—1362no—
440C24C2×C22⋊D1711yescharacteristic
441C24C2×C22⋊D1711yescharacteristic
442C2×Q8—21no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8C2×C22⋊D1712yes—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—41no—
449C2×D4—174no—
450C2×D4—348no—
451C2×D4—342no—
452C2×D4—342no—
453C2×D4—342no—
454C2×D4—342no—
455C2×D4—342no—
456C2×D4—688no—
457C2×D4—684no—
458C2×D4—684no—
459C2×D4—684no—
460C2×D4—684no—
461C2×D4—682no—
462C2×D4—1362no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—342no—
477C22×C4C2×C22⋊D1711yescharacteristic
478C22×C4C2×C22⋊D1711yescharacteristic
479C22×C4C2×C22⋊D1711yescharacteristic
480C22×C4C2×C22⋊D1712yes—
481C22×C4C2×C22⋊D1712yes—
482C22×C4C2×C22⋊D1711yescharacteristic
483C22×C4C23×D1711yescharacteristic
484SD16—688no—
485M16—684no—
486C2×C8—342no—
487C2×C8—342no—
488C2×C8—342no—
489C2×C8—342no—
490C2×C8—342no—
491C2×C8—342no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—342no—
503C4⋊C4—342no—
504C4⋊C4C2×C22⋊D1712yes—
505C4⋊C4C2×C22⋊D1712yes—
506C4⋊C4C2×C22⋊D1712yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—684no—
513C22⋊C4—684no—
514C22⋊C4—682no—
515C22⋊C4—1362no—
516C42—24no—
517C42—21no—
518C42—21no—
519C42—24no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—344no—
529C23—688no—
530C23—682no—
531C23—682no—
532C23—682no—
533C23—682no—
534C23—682no—
535C23—682no—
536C23—1364no—
537C23—1364no—
538C23—1368no—
539C23—1368no—
540C23#1058763512yes—
541C23#1058763912yes—
542C23C22×C22⋊D1711yescharacteristic
543Q8—24no—
544Q8—41no—
545Q8—41no—
546D4—44no—
547D4—344no—
548D4—344no—
549D4—688no—
550D4—682no—
551D4—682no—
552D4—1368no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—682no—
579C2×C4—682no—
580C2×C4—682no—
581C2×C4#1058763912yes—
582C2×C4#41×D1712yes—
583C2×C4#1058769612yes—
584C2×C4C22×C22⋊D1711yescharacteristic
585C2×C4C22×C22⋊D1711yescharacteristic
586C8—682no—
587C8—682no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—24no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—684no—
599C22—684no—
600C22—684no—
601C22—13616no—
602C22—1362no—
603C22—1362no—
604C22—1362no—
605C22—1362no—
606C22C2×#1058763911yescharacteristic
607C22#1176344712yes—
608C22C2×#1058763511yescharacteristic
609C22#1176360711yescharacteristic
610C22#1176361712yes—
611C4—22no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—1362no—
620C4#1176331611yescharacteristic
621C4#1176337311yescharacteristic
622C4#1176344712yes—
623C2—42no—
624C2—42no—
625C2—1364no—
626C2#1453209911yescharacteristic
627C2#1453803611yescharacteristic
628C2#1453807011yescharacteristic
629C2#1454079912yes—
630C2#1454099412yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#552C2
socleC23×C17C22×D4
prime radical#14532140C2
hypercenterC2×#552D17
hypocenterC17#23982
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#11762959C2semidirect
2. C2×#11762813C2semidirect
4. #14529927C2semidirect
5. #14529785C2semidirect
6. #14529218C2semidirect
7. #14529139C2semidirect
8. #14529101C2semidirect
9. #14529082C2semidirect
10. #14528879C2semidirect
11. C2×C17×#552C2semidirect
19. C2×#10587506C22semidirect
24. C2×C22⋊#3622C22semidirect
30. #11762571C22semidirect
31. #11762571C22semidirect
32. #11762571C22semidirect
33. #11762564C22semidirect
38. C2×C17×#125C22semidirect
41. C2×C17×C22⋊Q8C22semidirect
42. C2×C4×Q8×C17C22semidirect
46. C22×C4⋊C4×C17C22semidirect
68. C17×C22⋊Q8D4semidirect
75. C17×#117D4semidirect
273. C2×#552D17semidirect
326. C2×#125C2×D17semidirect
329. C2×C22⋊Q8C2×D17semidirect
332. C2×C4×Q8C2×D17semidirect
336. C22×C4⋊C4C2×D17semidirect
390. C22⋊Q8C22⋊D17semidirect
400. #117C22⋊D17semidirect
433. C17#23982semidirect

Maximal subgroups

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

subgroupindex
C2×#117629592
C2×#117628132
#145321402
#145299272
#145297852
#145292182
#145291392
#145291012
#145290822
#145288792
C2×C17×#5522
#2398217