Finite Group Databasebeta

← #17066519

Subgroups — #17066519

Order 4352.

1304 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762947C211yesmaximal, characteristic, complemented
2C22×#10587481C211yesmaximal, characteristic, complemented
3#14532149C211yesmaximal, characteristic, complemented
4#14529932C212yesmaximal, complemented
5#14529787C212yesmaximal, complemented
6#14529213C211yesmaximal, characteristic, complemented
7#14529118C212yesmaximal, complemented
8#14529096C212yesmaximal, complemented
9#14529077C211yesmaximal, characteristic, complemented
10#14528874C211yesmaximal, characteristic, complemented
11C2×C17×#553C211yesmaximal, characteristic
12C23×Dic34C2211yescharacteristic
13C22×#63229C2212yes—
14C23×#3622C2211yescharacteristic
15C2×#10587610C2212yes—
16#11762947—24no—
17C2×#10587481—24no—
18C2×#10587481C2212yes—
19C2×#10587481C2212yes—
20C2×#10587481C2212yes—
21C2×#10587503C2212yes—
22C2×C22⋊#3622C2211yescharacteristic
23C2×C22⋊#3622C2212yes—
24#11762729—21no—
25#11762721—21no—
26#11762715C2212yes—
27#11762689—21no—
28#11762681—21no—
29#11762669C2211yescharacteristic
30#11762571C2212yes—
31#11762571C2212yes—
32#11762571C2212yes—
33#11762564C2212yes—
34C17×#553—24no—
35C17×#553—24no—
36C2×C17×#126—21no—
37C2×C17×#125C2211yescharacteristic
38C2×C17×#123—21no—
39C2×C17×#121—21no—
40C2×C17×#121C2211yescharacteristic
41C2×C17×#121C2211yescharacteristic
42C2×C4×D4×C17—21no—
43C2×C4×D4×C17C2211yescharacteristic
44C2×C4×D4×C17C2211yescharacteristic
45C2×C17×#117C2211yescharacteristic
46C22×C4⋊C4×C17C2211yescharacteristic
47C22×D4×C17—21no—
48C22×D4×C17—21no—
49C22×D4×C17D411yescharacteristic
50C22×D4×C17C2311yescharacteristic
51C23×C4×C17—21no—
52C23×C4×C17—21no—
53C23×C4×C17C2311yescharacteristic
54C23×C4×C17C2311yescharacteristic
55C23×Dic17—21no—
56C22×Dic34—22no—
57C22×Dic34—24no—
58C22×Dic34—22no—
59C22×Dic34—21no—
60C22×Dic34—42no—
61C22×Dic34C2312yes—
62C17×#126—28no—
63C17×#125—22no—
64C17×#125—22no—
65C17×#123—28no—
66C17×#121—28no—
67C17×#121—22no—
68C17×#121—22no—
69C17×#121—22no—
70C17×#121D414yes—
71C4×D4×C17—28no—
72C4×D4×C17—22no—
73C4×D4×C17—22no—
74C4×D4×C17—22no—
75C4×D4×C17—22no—
76C17×#117—22no—
77C17×#117—22no—
78C2×C4⋊C4×C17—21no—
79C2×C4⋊C4×C17—22no—
80C2×C4⋊C4×C17—21no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17D411yescharacteristic
83C2×C4⋊C4×C17D414yes—
84C2×C4⋊C4×C17C2311yescharacteristic
85C2×C4⋊C4×C17C2311yescharacteristic
86C2×C4⋊C4×C17C2311yescharacteristic
87C2×C22⋊C4×C17—21no—
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×C17C2311yescharacteristic
95C2×C22⋊Dic17—22no—
96C2×C22⋊Dic17—21no—
97C2×#63229—24no—
98C2×#63229—24no—
99C2×#63229—42no—
100C2×C4⋊Dic17C2312yes—
101#10587610—24no—
102#10587610—42no—
103C2×#63225—22no—
104C22×#3622—22no—
105C22×#3622—42no—
106C22×#3622C2312yes—
107C22×#3622C2312yes—
108#10587503—42no—
109#10587502—22no—
110#10587502—22no—
111#10587502—21no—
112C22⋊#3622—22no—
113C22⋊#3622—42no—
114#10587481—24no—
115#10587481—24no—
116#10587481—24no—
117C24×C17—21no—
118C24×C17C2×D411yescharacteristic
119C24×C17C2×D411yescharacteristic
120C2×D4×C17—24no—
121C2×D4×C17—22no—
122C2×D4×C17—22no—
123C2×D4×C17—22no—
124C2×D4×C17—24no—
125C2×D4×C17—21no—
126C2×D4×C17—22no—
127C2×D4×C17—22no—
128C2×D4×C17—42no—
129C2×D4×C17—42no—
130C2×D4×C17—82no—
131C2×D4×C17C2×D412yes—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—24no—
138C22×C4×C17—24no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—22no—
142C22×C4×C17—22no—
143C22×C4×C17—82no—
144C22×C4×C17—82no—
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D412yes—
150C22×C4×C17C2×D412yes—
151C22×C4×C17C2411yescharacteristic
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—44no—
155C22×Dic17—42no—
156C2×Dic34—28no—
157C2×Dic34—22no—
158C2×Dic34—22no—
159C2×Dic34—22no—
160C2×Dic34—22no—
161C2×Dic34—22no—
162C2×Dic34—48no—
163C2×Dic34—44no—
164C2×Dic34—44no—
165C2×Dic34—42no—
166C2×Dic34#4114yes—
167C4⋊C4×C17—24no—
168C4⋊C4×C17—24no—
169C4⋊C4×C17—24no—
170C4⋊C4×C17—22no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17C2×D412yes—
175C4⋊C4×C17C2×D412yes—
176C4⋊C4×C17C2×D412yes—
177C22⋊C4×C17—24no—
178C22⋊C4×C17—24no—
179C22⋊C4×C17—24no—
180C22⋊C4×C17—22no—
181C22⋊C4×C17—22no—
182C22⋊C4×C17—24no—
183C42×C17—24no—
184C42×C17—21no—
185C42×C17—21no—
186C22⋊Dic17—44no—
187C22⋊Dic17—42no—
188#63229—48no—
189C4⋊Dic17—22no—
190C4⋊Dic17—22no—
191#63225—44no—
192C2×#3622—22no—
193C2×#3622—22no—
194C2×#3622—22no—
195C2×#3622—22no—
196C2×#3622—22no—
197C2×#3622—22no—
198C2×#3622—44no—
199#23964—171nomaximal, Sylow
200C23×C17—22no—
201C23×C17—22no—
202C23×C17—21no—
203C23×C17—22no—
204C23×C17—22no—
205C23×C17—21no—
206C23×C17—21no—
207C23×C17—22no—
208C23×C17—42no—
209C23×C17—41no—
210C23×C17—41no—
211C23×C17—42no—
212C23×C17#12012yes—
213C23×C17#13512yes—
214C23×C17#13712yes—
215C23×C17C22×D411yescharacteristic
216D4×C17—24no—
217D4×C17—44no—
218D4×C17—48no—
219D4×C17—41no—
220D4×C17—41no—
221D4×C17—84no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—24no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—22no—
229C2×C4×C17—21no—
230C2×C4×C17—22no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—22no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—42no—
243C2×C4×C17—84no—
244C2×C4×C17—84no—
245C2×C4×C17#12012yes—
246C2×C4×C17C22×D411yescharacteristic
247C2×C4×C17C22×D411yescharacteristic
248C2×C4×C17C2×#4112yes—
249C2×C4×C17H(5,𝔽2)12yes—
250C2×Dic17—24no—
251C2×Dic17—48no—
252C2×Dic17—42no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257Dic34—24no—
258Dic34—24no—
259Dic34—48no—
260Dic34—42no—
261Dic34—42no—
262#3622—42no—
263#3622—42no—
264C2×#553D1711yescharacteristic
265C2×#448—171no—
266C22×#103—171no—
267#2409—171no—
268#2026—172no—
269#2000—172no—
270#1893—171no—
271#1877—172no—
272#1873—172no—
273#1869—171no—
274#1824—171no—
275C22×C17—22no—
276C22×C17—22no—
277C22×C17—24no—
278C22×C17—24no—
279C22×C17—41no—
280C22×C17—42no—
281C22×C17—41no—
282C22×C17—42no—
283C22×C17—42no—
284C22×C17—41no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17—82no—
288C22×C17—82no—
289C22×C17#45214yes—
290C22×C17C2×#12011yescharacteristic
291C22×C17#54612yes—
292C22×C17C2×#13511yescharacteristic
293C22×C17C2×#13711yescharacteristic
294C22×C17#57712yes—
295C4×C17—22no—
296C4×C17—41no—
297C4×C17—41no—
298C4×C17—42no—
299C4×C17—41no—
300C4×C17—42no—
301C4×C17—42no—
302C4×C17#52511yescharacteristic
303C4×C17#53511yescharacteristic
304C4×C17#54612yes—
305Dic17—44no—
306Dic17—82no—
307C23×Q8—171no—
308C22×SD16—172no—
309C23×C8—171no—
310#553—24no—
311#553—24no—
312C2×#126—21no—
313C2×#125C2×D1711yescharacteristic
314C2×#123—21no—
315C2×C22⋊Q8—172no—
316C2×#121—21no—
317C2×#121C2×D1711yescharacteristic
318C2×#121C2×D1711yescharacteristic
319C2×C4×D4—21no—
320C2×C4×D4C2×D1711yescharacteristic
321C2×C4×D4C2×D1711yescharacteristic
322C2×#117C2×D1711yescharacteristic
323C22×C4⋊C4C2×D1711yescharacteristic
324#448—344no—
325C2×#103—172no—
326C2×#103—172no—
327C2×#103—172no—
328C2×#103—344no—
329C2×#102—172no—
330C2×C22⋊C8—171no—
331C2×C22⋊C8—172no—
332#395—341no—
333#393—341no—
334#391—172no—
335#386—341no—
336#385—341no—
337#380—171no—
338#340—172no—
339#340—172no—
340#340—172no—
341#336—172no—
342C2×C17—24no—
343C2×C17—81no—
344C2×C17—82no—
345C2×C17#240411yescharacteristic
346C2×C17C2×#45211yescharacteristic
347C2×C17#300911yescharacteristic
348C2×C17#330112yes—
349C2×C17#332012yes—
350C22×Q8—172no—
351C22×Q8—344no—
352C22×Q8—341no—
353C22×Q8—342no—
354C22×Q8—342no—
355C22×Q8—682no—
356C22×D4—21no—
357C22×D4—21no—
358C22×D4C22⋊D1711yescharacteristic
359C22×D4C22×D1711yescharacteristic
360C23×C4—21no—
361C23×C4—21no—
362C23×C4—341no—
363C23×C4C22×D1711yescharacteristic
364C23×C4C22×D1711yescharacteristic
365C2×SD16—344no—
366C2×SD16—344no—
367C2×SD16—682no—
368C22×C8—172no—
369C22×C8—172no—
370C22×C8—342no—
371C22×C8—682no—
372#126—28no—
373#125—22no—
374#125—22no—
375#123—28no—
376C22⋊Q8—344no—
377C22⋊Q8—682no—
378#121—28no—
379#121—22no—
380#121—22no—
381#121—22no—
382#121C22⋊D1714yes—
383C4×D4—28no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—22no—
388#117—22no—
389#117—22no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—172no—
395C2×C4⋊C4—342no—
396C2×C4⋊C4C22⋊D1711yescharacteristic
397C2×C4⋊C4C22⋊D1714yes—
398C2×C4⋊C4C22×D1711yescharacteristic
399C2×C4⋊C4C22×D1711yescharacteristic
400C2×C4⋊C4C22×D1711yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—342no—
406C2×C22⋊C4—341no—
407C2×C22⋊C4C22⋊D1711yescharacteristic
408C2×C22⋊C4C22⋊D1711yescharacteristic
409C2×C42—21no—
410C2×C42C22×D1711yescharacteristic
411#103—344no—
412#103—344no—
413#103—344no—
414#102—682no—
415C22⋊C8—342no—
416C22⋊C8—682no—
417#95—341no—
418#95—342no—
419#95—342no—
420C17#2396411yescharacteristic, complemented, Sylow
421C24—21no—
422C24C2×C22⋊D1711yescharacteristic
423C24C2×C22⋊D1711yescharacteristic
424C2×Q8—174no—
425C2×Q8—348no—
426C2×Q8—342no—
427C2×Q8—342no—
428C2×Q8—342no—
429C2×Q8—342no—
430C2×Q8—342no—
431C2×Q8—684no—
432C2×Q8—688no—
433C2×Q8—684no—
434C2×Q8—682no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—82no—
446C2×D4C2×C22⋊D1712yes—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—684no—
463C22×C4—682no—
464C22×C4C2×C22⋊D1711yescharacteristic
465C22×C4C2×C22⋊D1711yescharacteristic
466C22×C4C2×C22⋊D1711yescharacteristic
467C22×C4C2×C22⋊D1712yes—
468C22×C4C2×C22⋊D1712yes—
469C22×C4C2×C22⋊D1711yescharacteristic
470C22×C4C23×D1711yescharacteristic
471SD16—688no—
472C2×C8—342no—
473C2×C8—342no—
474C2×C8—342no—
475C2×C8—342no—
476C2×C8—342no—
477C2×C8—342no—
478C2×C8—684no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—342no—
487C4⋊C4—342no—
488C4⋊C4—684no—
489C4⋊C4C2×C22⋊D1712yes—
490C4⋊C4C2×C22⋊D1712yes—
491C4⋊C4C2×C22⋊D1712yes—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—22no—
496C22⋊C4—24no—
497C22⋊C4—22no—
498C22⋊C4—684no—
499C22⋊C4—682no—
500C42—24no—
501C42—21no—
502C42—21no—
503C23—22no—
504C23—22no—
505C23—21no—
506C23—22no—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23#1058763612yes—
516C23#1058763712yes—
517C23#1058763912yes—
518C23C22×C22⋊D1711yescharacteristic
519Q8—344no—
520Q8—344no—
521Q8—688no—
522Q8—682no—
523Q8—682no—
524D4—24no—
525D4—44no—
526D4—48no—
527D4—41no—
528D4—41no—
529D4—84no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—344no—
554C2×C4—688no—
555C2×C4—682no—
556C2×C4—682no—
557C2×C4—682no—
558C2×C4—682no—
559C2×C4—682no—
560C2×C4#1058763912yes—
561C2×C4#41×D1712yes—
562C2×C4#1058769612yes—
563C2×C4C22×C22⋊D1711yescharacteristic
564C2×C4C22×C22⋊D1711yescharacteristic
565C8—682no—
566C8—682no—
567C22—22no—
568C22—22no—
569C22—24no—
570C22—24no—
571C22—41no—
572C22—42no—
573C22—41no—
574C22—42no—
575C22—42no—
576C22—41no—
577C22—42no—
578C22—42no—
579C22—82no—
580C22—82no—
581C22#1176297114yes—
582C22C2×#1058763911yescharacteristic
583C22#1176344712yes—
584C22C2×#1058763611yescharacteristic
585C22C2×#1058763711yescharacteristic
586C22#1176362012yes—
587C4—22no—
588C4—41no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—684no—
595C4—1362no—
596C4#1176331611yescharacteristic
597C4#1176337311yescharacteristic
598C4#1176344712yes—
599C2—24no—
600C2—81no—
601C2—82no—
602C2#1453209911yescharacteristic
603C2C2×#1176297111yescharacteristic
604C2#1453806111yescharacteristic
605C2#1454068012yes—
606C2#1454094912yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#553C2
socleC23×C17C22×D4
hypercenterC2×#553D17
hypocenterC17#23964
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×#11762947C2semidirect
2. C22×#10587481C2semidirect
3. #14532149C2semidirect
4. #14529932C2semidirect
5. #14529787C2semidirect
6. #14529213C2semidirect
7. #14529118C2semidirect
8. #14529096C2semidirect
9. #14529077C2semidirect
10. #14528874C2semidirect
420. C17#23964semidirect

Maximal subgroups

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

subgroupindex
C2×#117629472
C22×#105874812
#145321492
#145299322
#145297872
#145292132
#145291182
#145290962
#145290772
#145288742
C2×C17×#5532
#2396417