Finite Group Databasebeta

← #17066508

Subgroups — #17066508

Order 4352.

1500 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762942C211yesmaximal, characteristic, complemented
2C22×#10587480C211yesmaximal, characteristic, complemented
3#14532140C211yesmaximal, characteristic, complemented
4#14529927C212yesmaximal, complemented
5#14529780C212yesmaximal, complemented
6#14529213C211yesmaximal, characteristic, complemented
7#14529132C212yesmaximal, complemented
8#14529106C212yesmaximal, complemented
9#14529077C211yesmaximal, characteristic, complemented
10#14528874C211yesmaximal, characteristic, complemented
11C2×C17×#553C211yesmaximal, characteristic, complemented
12C23×D68C2211yescharacteristic
13C22×#63228C2212yes—
14C23×#3622C2211yescharacteristic
15C2×#10587616C2212yes—
16C2×#10587623—21no—
17#11762942—24no—
18C2×#10587503C2212yescomplemented
19C2×#10587480—24no—
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×#553—24no—
35C17×#553—24no—
36C2×C17×#126—21no—
37C2×C17×#125C2211yescharacteristic, complemented
38C2×C17×#123—21no—
39C2×C17×#121—21no—
40C2×C17×#121C2211yescharacteristic
41C2×C17×#121C2211yescharacteristic, complemented
42C2×C4×D4×C17—21no—
43C2×C4×D4×C17C2211yescharacteristic
44C2×C4×D4×C17C2211yescharacteristic, complemented
45C2×C17×#117C2211yescharacteristic, complemented
46C22×C4⋊C4×C17C2211yescharacteristic
47C24×D17—21no—
48C22×D4×C17—21no—
49C22×D4×C17—21no—
50C22×D4×C17D411yescharacteristic
51C22×D4×C17C2311yescharacteristic
52C23×C4×C17—21no—
53C23×C4×C17—21no—
54C23×C4×C17C2311yescharacteristic
55C23×C4×C17C2311yescharacteristic
56C22×C22⋊D17—22no—
57C22×D68—22no—
58C22×D68—24no—
59C22×D68—22no—
60C22×D68—21no—
61C22×D68—42no—
62C22×D68C2312yes—
63C17×#126—28no—
64C17×#125—22no—
65C17×#125—22no—
66C17×#123—28no—
67C17×#121—28no—
68C17×#121—22no—
69C17×#121—22no—
70C17×#121—22no—
71C17×#121D414yescomplemented
72C4×D4×C17—28no—
73C4×D4×C17—22no—
74C4×D4×C17—22no—
75C4×D4×C17—22no—
76C4×D4×C17—22no—
77C17×#117—22no—
78C17×#117—22no—
79C2×C4⋊C4×C17—21no—
80C2×C4⋊C4×C17—22no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17—21no—
83C2×C4⋊C4×C17D411yescharacteristic
84C2×C4⋊C4×C17D414yescomplemented
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×C17—21no—
92C2×C22⋊C4×C17D411yescharacteristic
93C2×C22⋊C4×C17D411yescharacteristic
94C2×C42×C17—21no—
95C2×C42×C17C2311yescharacteristic
96C2×C22⋊Dic17—21no—
97#10587623—44no—
98C2×#63228—24no—
99C2×#63228—24no—
100C2×#63228—42no—
101#10587616—24no—
102#10587616—42no—
103C2×#63227—22no—
104C2×#63227—22no—
105C2×#63227—22no—
106C2×C4⋊Dic17C2312yes—
107C22×#3622—22no—
108C22×#3622—42no—
109C22×#3622C2312yes—
110C22×#3622C2312yes—
111#10587503—42no—
112#10587502—21no—
113C22⋊#3622—22no—
114C22⋊#3622—42no—
115#10587480—24no—
116#10587480—24no—
117#10587480—24no—
118C24×C17—21no—
119C24×C17C2×D411yescharacteristic
120C24×C17C2×D411yescharacteristic
121C23×D17—22no—
122C23×D17—44no—
123C23×D17—42no—
124C23×D17—42no—
125C23×D17—42no—
126C23×D17—82no—
127C2×D4×C17—24no—
128C2×D4×C17—22no—
129C2×D4×C17—22no—
130C2×D4×C17—22no—
131C2×D4×C17—24no—
132C2×D4×C17—21no—
133C2×D4×C17—22no—
134C2×D4×C17—22no—
135C2×D4×C17—42no—
136C2×D4×C17—42no—
137C2×D4×C17—82no—
138C2×D4×C17C2×D412yes—
139C22×C4×C17—22no—
140C22×C4×C17—22no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—24no—
145C22×C4×C17—24no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—22no—
149C22×C4×C17—22no—
150C22×C4×C17—82no—
151C22×C4×C17—82no—
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D412yes—
157C22×C4×C17C2×D412yes—
158C22×C4×C17C2411yescharacteristic
159C2×C22⋊D17—44no—
160C2×C22⋊D17—44no—
161C2×C22⋊D17—82no—
162C22×Dic17—22no—
163C2×D68—28no—
164C2×D68—22no—
165C2×D68—22no—
166C2×D68—22no—
167C2×D68—22no—
168C2×D68—22no—
169C2×D68—48no—
170C2×D68—44no—
171C2×D68—44no—
172C2×D68—42no—
173C2×D68#4114yes—
174C4⋊C4×C17—24no—
175C4⋊C4×C17—24no—
176C4⋊C4×C17—24no—
177C4⋊C4×C17—22no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17C2×D412yes—
182C4⋊C4×C17C2×D412yes—
183C4⋊C4×C17C2×D412yes—
184C22⋊C4×C17—24no—
185C22⋊C4×C17—24no—
186C22⋊C4×C17—24no—
187C22⋊C4×C17—22no—
188C22⋊C4×C17—22no—
189C22⋊C4×C17—24no—
190C42×C17—24no—
191C42×C17—21no—
192C42×C17—21no—
193C22⋊Dic17—42no—
194#63228—48no—
195#63227—44no—
196#63227—44no—
197#63227—82no—
198C4⋊Dic17—22no—
199C4⋊Dic17—22no—
200C2×#3622—22no—
201C2×#3622—22no—
202C2×#3622—22no—
203C2×#3622—22no—
204C2×#3622—22no—
205C2×#3622—22no—
206C2×#3622—44no—
207#23963—171nomaximal, Sylow
208C23×C17—22no—
209C23×C17—22no—
210C23×C17—21no—
211C23×C17—22no—
212C23×C17—22no—
213C23×C17—21no—
214C23×C17—21no—
215C23×C17—22no—
216C23×C17—42no—
217C23×C17—41no—
218C23×C17—41no—
219C23×C17—42no—
220C23×C17#12012yes—
221C23×C17#13512yes—
222C23×C17AGL(1,ℤ/8)12yes—
223C23×C17C22×D411yescharacteristic
224C22×D17—24no—
225C22×D17—48no—
226C22×D17—42no—
227C22×D17—42no—
228C22×D17—42no—
229C22×D17—42no—
230C22×D17—42no—
231C22×D17—42no—
232C22×D17—88no—
233C22×D17—88no—
234C22×D17—84no—
235C22×D17—84no—
236D4×C17—24no—
237D4×C17—44no—
238D4×C17—48no—
239D4×C17—41no—
240D4×C17—41no—
241D4×C17—84no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—24no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—22no—
249C2×C4×C17—21no—
250C2×C4×C17—22no—
251C2×C4×C17—22no—
252C2×C4×C17—22no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—22no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—42no—
259C2×C4×C17—42no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—42no—
263C2×C4×C17—84no—
264C2×C4×C17—84no—
265C2×C4×C17#12012yes—
266C2×C4×C17C22×D411yescharacteristic
267C2×C4×C17C22×D411yescharacteristic
268C2×C4×C17C2×#4112yes—
269C2×C4×C17H(5,𝔽2)12yes—
270C22⋊D17—88no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274D68—24no—
275D68—24no—
276D68—48no—
277D68—42no—
278D68—42no—
279#3622—42no—
280#3622—42no—
281C2×#553D1711yescharacteristic, complemented
282C2×#447—171no—
283C22×#102—171no—
284#2408—171no—
285#2025—172no—
286#1999—172no—
287#1893—171no—
288#1879—172no—
289#1875—172no—
290#1869—171no—
291#1824—171no—
292C22×C17—22no—
293C22×C17—22no—
294C22×C17—24no—
295C22×C17—24no—
296C22×C17—41no—
297C22×C17—42no—
298C22×C17—41no—
299C22×C17—42no—
300C22×C17—42no—
301C22×C17—41no—
302C22×C17—42no—
303C22×C17—42no—
304C22×C17—82no—
305C22×C17—82no—
306C22×C17#44914yes—
307C22×C17C2×#12011yescharacteristic
308C22×C17#54612yes—
309C22×C17C2×#13511yescharacteristic
310C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
311C22×C17#57612yes—
312C2×D17—44no—
313C2×D17—44no—
314C2×D17—44no—
315C2×D17—816no—
316C2×D17—82no—
317C2×D17—82no—
318C2×D17—82no—
319C2×D17—82no—
320C4×C17—22no—
321C4×C17—41no—
322C4×C17—41no—
323C4×C17—42no—
324C4×C17—41no—
325C4×C17—42no—
326C4×C17—42no—
327C4×C17#52511yescharacteristic
328C4×C17#53511yescharacteristic
329C4×C17#54612yes—
330Dic17—82no—
331C23×D4—171no—
332C22×D8—172no—
333C23×C8—171no—
334#553—24no—
335#553—24no—
336C2×#126—21no—
337C2×#125C2×D1711yescharacteristic, complemented
338C2×#123—21no—
339C2×#121—21no—
340C2×#121—172no—
341C2×#121C2×D1711yescharacteristic
342C2×#121C2×D1711yescharacteristic, complemented
343C2×#120—341no—
344C2×C4×D4—21no—
345C2×C4×D4C2×D1711yescharacteristic
346C2×C4×D4C2×D1711yescharacteristic, complemented
347C2×#117C2×D1711yescharacteristic, complemented
348C22×C4⋊C4C2×D1711yescharacteristic
349#447—344no—
350C2×#102—172no—
351C2×#102—172no—
352C2×#102—172no—
353C2×#102—172no—
354C2×#102—344no—
355C2×C22⋊C8—171no—
356C2×C22⋊C8—172no—
357#394—341no—
358#392—341no—
359#390—172no—
360#380—171no—
361#379—341no—
362#340—172no—
363#340—172no—
364#340—172no—
365#336—172no—
366C2×C17—24no—
367C2×C17—81no—
368C2×C17—82no—
369C2×C17#240411yescharacteristic
370C2×C17C2×#44911yescharacteristic
371C2×C17#300811yescharacteristic
372C2×C17#329712yes—
373C2×C17#331812yes—
374D17—84no—
375C25—341no—
376C22×D4—21no—
377C22×D4—21no—
378C22×D4—172no—
379C22×D4—344no—
380C22×D4—342no—
381C22×D4—342no—
382C22×D4—342no—
383C22×D4—341no—
384C22×D4—682no—
385C22×D4C22⋊D1711yescharacteristic
386C22×D4C22×D1711yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C23×C4C22×D1711yescharacteristic
390C23×C4C22×D1711yescharacteristic
391C2×D8—344no—
392C2×D8—344no—
393C2×D8—682no—
394C22×C8—172no—
395C22×C8—172no—
396C22×C8—342no—
397C22×C8—682no—
398#126—28no—
399#125—22no—
400#125—22no—
401#123—28no—
402#121—28no—
403#121—22no—
404#121—22no—
405#121—22no—
406#121—344no—
407#121—682no—
408#121C22⋊D1714yescomplemented
409#120—684no—
410C4×D4—28no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415#117—22no—
416#117—22no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—22no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—172no—
422C2×C4⋊C4C22⋊D1711yescharacteristic
423C2×C4⋊C4C22⋊D1714yescomplemented
424C2×C4⋊C4C22×D1711yescharacteristic
425C2×C4⋊C4C22×D1711yescharacteristic
426C2×C4⋊C4C22×D1711yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—341no—
432C2×C22⋊C4—342no—
433C2×C22⋊C4—342no—
434C2×C22⋊C4—342no—
435C2×C22⋊C4C22⋊D1711yescharacteristic
436C2×C22⋊C4C22⋊D1711yescharacteristic
437C2×C42—21no—
438C2×C42C22×D1711yescharacteristic
439#102—344no—
440#102—344no—
441#102—344no—
442#102—682no—
443C22⋊C8—342no—
444C22⋊C8—682no—
445#95—341no—
446C17#2396311yescharacteristic, complemented, Sylow
447C24—21no—
448C24—342no—
449C24—684no—
450C24—682no—
451C24—682no—
452C24—682no—
453C24—1362no—
454C24C2×C22⋊D1711yescharacteristic
455C24C2×C22⋊D1711yescharacteristic
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—82no—
467C2×D4—174no—
468C2×D4—348no—
469C2×D4—342no—
470C2×D4—342no—
471C2×D4—342no—
472C2×D4—342no—
473C2×D4—342no—
474C2×D4—688no—
475C2×D4—684no—
476C2×D4—684no—
477C2×D4—684no—
478C2×D4—684no—
479C2×D4—682no—
480C2×D4—1362no—
481C2×D4C2×C22⋊D1712yes—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4—342no—
496C22×C4C2×C22⋊D1711yescharacteristic
497C22×C4C2×C22⋊D1711yescharacteristic
498C22×C4C2×C22⋊D1711yescharacteristic
499C22×C4C2×C22⋊D1712yes—
500C22×C4C2×C22⋊D1712yes—
501C22×C4C2×C22⋊D1711yescharacteristic
502C22×C4C23×D1711yescharacteristic
503D8—688no—
504C2×C8—342no—
505C2×C8—342no—
506C2×C8—342no—
507C2×C8—342no—
508C2×C8—342no—
509C2×C8—342no—
510C2×C8—684no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—342no—
519C4⋊C4—342no—
520C4⋊C4C2×C22⋊D1712yes—
521C4⋊C4C2×C22⋊D1712yes—
522C4⋊C4C2×C22⋊D1712yes—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—22no—
527C22⋊C4—24no—
528C22⋊C4—22no—
529C22⋊C4—684no—
530C22⋊C4—684no—
531C22⋊C4—682no—
532C22⋊C4—1362no—
533C42—24no—
534C42—21no—
535C42—21no—
536C23—22no—
537C23—22no—
538C23—21no—
539C23—22no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—344no—
549C23—688no—
550C23—682no—
551C23—682no—
552C23—682no—
553C23—682no—
554C23—682no—
555C23—682no—
556C23—1368no—
557C23—1368no—
558C23—1364no—
559C23—1364no—
560C23#1058763512yes—
561C23#1058763612yes—
562C23#1058763912yes—
563C23C22×C22⋊D1711yescharacteristic
564D4—24no—
565D4—44no—
566D4—48no—
567D4—41no—
568D4—41no—
569D4—84no—
570D4—344no—
571D4—344no—
572D4—688no—
573D4—682no—
574D4—682no—
575D4—1368no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—682no—
600C2×C4—682no—
601C2×C4—682no—
602C2×C4#1058763912yes—
603C2×C4#41×D1712yes—
604C2×C4#1058769612yes—
605C2×C4C22×C22⋊D1711yescharacteristic
606C2×C4C22×C22⋊D1711yescharacteristic
607C8—682no—
608C8—682no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—24no—
613C22—41no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—41no—
619C22—42no—
620C22—42no—
621C22—82no—
622C22—82no—
623C22—684no—
624C22—684no—
625C22—684no—
626C22—13616no—
627C22—1362no—
628C22—1362no—
629C22—1362no—
630C22—1362no—
631C22#1176295414yes—
632C22C2×#1058763911yescharacteristic
633C22#1176344712yes—
634C22C2×#1058763611yescharacteristic
635C22C2×#1058763511yescharacteristic
636C22#1176361212yes—
637C4—22no—
638C4—41no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—1362no—
645C4#1176331611yescharacteristic
646C4#1176337311yescharacteristic
647C4#1176344712yes—
648C2—24no—
649C2—81no—
650C2—82no—
651C2—1364no—
652C2#1453209911yescharacteristic
653C2C2×#1176295411yescharacteristic
654C2#1453805211yescharacteristic
655C2#1454062812yes—
656C2#1454091912yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#553C2
socleC23×C17C22×D4
hypercenterC2×#553D17
hypocenterC17#23963
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
1. C2×#11762942C2semidirect
2. C22×#10587480C2semidirect
3. #14532140C2semidirect
4. #14529927C2semidirect
5. #14529780C2semidirect
6. #14529213C2semidirect
7. #14529132C2semidirect
8. #14529106C2semidirect
9. #14529077C2semidirect
10. #14528874C2semidirect
11. C2×C17×#553C2semidirect
18. C2×#10587503C22semidirect
24. C2×C22⋊#3622C22semidirect
30. #11762571C22semidirect
31. #11762571C22semidirect
32. #11762571C22semidirect
33. #11762564C22semidirect
37. C2×C17×#125C22semidirect
41. C2×C17×#121C22semidirect
44. C2×C4×D4×C17C22semidirect
45. C2×C17×#117C22semidirect
71. C17×#121D4semidirect
84. C2×C4⋊C4×C17D4semidirect
281. C2×#553D17semidirect
337. C2×#125C2×D17semidirect
342. C2×#121C2×D17semidirect
346. C2×C4×D4C2×D17semidirect
347. C2×#117C2×D17semidirect
408. #121C22⋊D17semidirect
423. C2×C4⋊C4C22⋊D17semidirect
446. C17#23963semidirect

Maximal subgroups

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

subgroupindex
C2×#117629422
C22×#105874802
#145321402
#145299272
#145297802
#145292132
#145291322
#145291062
#145290772
#145288742
C2×C17×#5532
#2396317