Finite Group Databasebeta

← #17069994

Subgroups — #17069994

Order 4352.

1606 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#63229C211yesmaximal, characteristic, complemented
2C2×#11763036C212yesmaximal, complemented
3C2×D4×#3622C211yesmaximal, characteristic, complemented
4#14532149C211yesmaximal, characteristic, complemented
5#14530101C212yesmaximal, complemented
6#14529787C212yesmaximal, complemented
7#14529637C212yesmaximal, complemented
8#14528807C211yesmaximal, characteristic, complemented
9#14528718C211yesmaximal, characteristic, complemented
10#14528715C211yesmaximal, characteristic, complemented
11C17×#2408C211yesmaximal, characteristic
12C23×Dic34C2211yescharacteristic
13C22×#63229—28no—
14C22×#63229—24no—
15C22×#63229C2212yes—
16C22×#63229C2212yes—
17C23×#3622C2211yescharacteristic
18C23×#3622C2211yescharacteristic, complemented
19C2×#10587610C2212yescomplemented
20#11763036—28no—
21D4×#3622—24no—
22C2×#10587479C2212yescomplemented
23C2×C4⋊#3622C2211yescharacteristic, complemented
24C2×#10587481C2212yescomplemented
25C2×#10587481C2212yescomplemented
26C2×#10587503C2212yes—
27C2×#10587503C2212yes—
28C2×C22⋊#3622C2211yescharacteristic
29C2×C22⋊#3622C2211yescharacteristic, complemented
30C2×C4×#3622C2211yescharacteristic, complemented
31#11762729—21no—
32#11762721—21no—
33#11762715C2212yescomplemented
34#11762689—21no—
35#11762681—21no—
36#11762669C2211yescharacteristic
37#11762571C2212yescomplemented
38C23×D4×C17C2211yescharacteristic
39C2×C17×#121C2212yes—
40C2×C17×#120—21no—
41C2×C4×D4×C17C2211yescharacteristic
42C17×#394—21no—
43C17×#392—21no—
44C17×#390C2212yes—
45C17×#380C2211yescharacteristic
46C17×#379—21no—
47C25×C17—21no—
48C22×D4×C17—24no—
49C22×D4×C17—22no—
50C22×D4×C17—22no—
51C22×D4×C17—21no—
52C22×D4×C17—22no—
53C22×D4×C17—42no—
54C22×D4×C17D411yescharacteristic
55C22×D4×C17C2312yes—
56C23×C4×C17C2311yescharacteristic
57C23×C4×C17C2311yescharacteristic
58C23×Dic17—21no—
59C22×Dic34—22no—
60C22×Dic34—24no—
61C22×Dic34—22no—
62C22×Dic34—21no—
63C22×Dic34—42no—
64C22×Dic34C2312yes—
65C17×#121—24no—
66C17×#121—42no—
67C17×#120—44no—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C2×C4⋊C4×C17D411yescharacteristic
71C2×C4⋊C4×C17C2312yes—
72C2×C22⋊C4×C17—21no—
73C2×C22⋊C4×C17—22no—
74C2×C22⋊C4×C17—22no—
75C2×C22⋊C4×C17—22no—
76C2×C22⋊C4×C17D411yescharacteristic
77C2×C22⋊C4×C17D411yescharacteristic
78C2×C42×C17C2311yescharacteristic
79C2×C22⋊Dic17—22no—
80C2×C22⋊Dic17—21no—
81C2×#63229—232no—
82C2×#63229—24no—
83C2×#63229—24no—
84C2×#63229—24no—
85C2×#63229—24no—
86C2×#63229—24no—
87C2×#63229D418yescomplemented
88C2×C4⋊Dic17C2312yes—
89#10587610—24no—
90#10587610—42no—
91C2×#63225—22no—
92C22×#3622—24no—
93C22×#3622—24no—
94C22×#3622—22no—
95C22×#3622D414yescomplemented
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C17×#95—21no—
101#10587503—24no—
102#10587503—42no—
103#10587502—22no—
104#10587502—22no—
105#10587502—21no—
106C22⋊#3622—22no—
107C22⋊#3622—22no—
108#10587481—24no—
109#10587481—42no—
110#10587479—24no—
111C4⋊#3622—22no—
112C4×#3622—22no—
113C24×C17—22no—
114C24×C17—44no—
115C24×C17—42no—
116C24×C17—42no—
117C24×C17—42no—
118C24×C17—82no—
119C24×C17C2×D411yescharacteristic
120C24×C17C2×D411yescharacteristic
121C2×D4×C17—22no—
122C2×D4×C17—22no—
123C2×D4×C17—28no—
124C2×D4×C17—22no—
125C2×D4×C17—22no—
126C2×D4×C17—22no—
127C2×D4×C17—22no—
128C2×D4×C17—22no—
129C2×D4×C17—48no—
130C2×D4×C17—44no—
131C2×D4×C17—44no—
132C2×D4×C17—44no—
133C2×D4×C17—44no—
134C2×D4×C17—42no—
135C2×D4×C17—82no—
136C2×D4×C17SD1614yescomplemented
137C2×D4×C17C2×D414yes—
138C22×C4×C17—22no—
139C22×C4×C17—22no—
140C22×C4×C17—21no—
141C22×C4×C17—22no—
142C22×C4×C17—21no—
143C22×C4×C17—22no—
144C22×C4×C17—22no—
145C22×C4×C17C2×D412yes—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17#4112yes—
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×Dic34C2×D414yes—
167C4⋊C4×C17—22no—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—22no—
170C22⋊C4×C17—22no—
171C22⋊C4×C17—22no—
172C22⋊C4×C17—44no—
173C22⋊C4×C17—42no—
174C22⋊C4×C17—44no—
175C22⋊C4×C17—82no—
176C42×C17—21no—
177C42×C17—21no—
178C22⋊Dic17—44no—
179C22⋊Dic17—42no—
180#63229—216no—
181#63229—216no—
182C4⋊Dic17—22no—
183C4⋊Dic17—22no—
184#63225—44no—
185C2×#3622—22no—
186C2×#3622—28no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—22no—
190C2×#3622—22no—
191C2×#3622—44no—
192C2×#3622C2×D412yes—
193C2×#3622C2×D412yes—
194C2×#3622C2×D414yes—
195#24264—171nomaximal, Sylow
196C23×C17—22no—
197C23×C17—22no—
198C23×C17—21no—
199C23×C17—22no—
200C23×C17—21no—
201C23×C17—22no—
202C23×C17—24no—
203C23×C17—48no—
204C23×C17—42no—
205C23×C17—42no—
206C23×C17—42no—
207C23×C17—42no—
208C23×C17—42no—
209C23×C17—42no—
210C23×C17—88no—
211C23×C17—88no—
212C23×C17—84no—
213C23×C17—84no—
214C23×C17#12112yes—
215C23×C17C2×SD1612yes—
216C23×C17C2×SD1612yescomplemented
217C23×C17#13512yes—
218C23×C17C22×D411yescharacteristic
219D4×C17—28no—
220D4×C17—24no—
221D4×C17—24no—
222D4×C17—48no—
223D4×C17—42no—
224D4×C17—42no—
225D4×C17—88no—
226C2×C4×C17—21no—
227C2×C4×C17—22no—
228C2×C4×C17—21no—
229C2×C4×C17—22no—
230C2×C4×C17—22no—
231C2×C4×C17—24no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—42no—
241C2×C4×C17—42no—
242C2×C4×C17#12112yes—
243C2×C4×C17C2×SD1612yescomplemented
244C2×C4×C17C22×D411yescharacteristic
245C2×C4×C17C22×D412yes—
246C2×C4×C17C2×#4111yescharacteristic
247C2×C4×C17H(5,𝔽2)12yes—
248C2×Dic17—24no—
249C2×Dic17—48no—
250C2×Dic17—42no—
251C2×Dic17—42no—
252C2×Dic17—42no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255Dic34—24no—
256Dic34—24no—
257Dic34—48no—
258Dic34—42no—
259Dic34—42no—
260#3622—22no—
261#3622—41no—
262#3622—41no—
263#3622#12014yescomplemented
264C23×SD16—171no—
265C2×#465—172no—
266C2×C8×D4—171no—
267#2409—171no—
268#2408D1711yescharacteristic
269#2055—172no—
270#2000—172no—
271#1968—172no—
272#1807—171no—
273#1787—171no—
274#1786—171no—
275C22×C17—22no—
276C22×C17—24no—
277C22×C17—24no—
278C22×C17—22no—
279C22×C17—24no—
280C22×C17—22no—
281C22×C17—41no—
282C22×C17—42no—
283C22×C17—41no—
284C22×C17—44no—
285C22×C17—44no—
286C22×C17—44no—
287C22×C17—816no—
288C22×C17—82no—
289C22×C17—82no—
290C22×C17—82no—
291C22×C17—82no—
292C22×C17#46514yescomplemented
293C22×C17C2×#12111yescharacteristic
294C22×C17D4212yes—
295C22×C17C22×SD1611yescharacteristic
296C22×C17C2×#13511yescharacteristic
297C22×C17#57712yes—
298C4×C17—22no—
299C4×C17—22no—
300C4×C17—41no—
301C4×C17—41no—
302C4×C17—82no—
303C4×C17C2×#12011yescharacteristic
304C4×C17#53511yescharacteristic
305C4×C17#54612yes—
306Dic17—44no—
307Dic17—82no—
308C23×Q8—171no—
309C23×D4C2×D1711yescharacteristic
310C22×SD16—172no—
311C22×SD16—172no—
312C22×SD16—344no—
313C22×SD16—348no—
314C23×C8—171no—
315C23×C8—171no—
316C2×C22⋊Q8—172no—
317C2×#121C2×D1712yes—
318C2×#120—21no—
319C2×C4×D4C2×D1711yescharacteristic
320#465—348no—
321C8×D4—344no—
322C2×#106—172no—
323C2×C4⋊C8—171no—
324C2×#103—172no—
325C2×#103—172no—
326C2×#102—172no—
327C2×#102—172no—
328C2×C22⋊C8—171no—
329C2×C22⋊C8—171no—
330C2×C4×C8—171no—
331#395—341no—
332#394—21no—
333#393—341no—
334#392—21no—
335#391—172no—
336#390C2×D1712yes—
337#386—341no—
338#385—341no—
339#380—171no—
340#380C2×D1711yescharacteristic
341#379—21no—
342#340—172no—
343C2×C17—24no—
344C2×C17—42no—
345C2×C17—84no—
346C2×C17#240811yescharacteristic
347C2×C17C2×#46511yescharacteristic
348C2×C17#305711yescharacteristic
349C2×C17D4×SD1612yes—
350C2×C17#328212yes—
351C25—21no—
352C22×Q8—172no—
353C22×Q8—344no—
354C22×Q8—341no—
355C22×Q8—342no—
356C22×Q8—342no—
357C22×Q8—682no—
358C22×D4—24no—
359C22×D4—22no—
360C22×D4—22no—
361C22×D4—22no—
362C22×D4—21no—
363C22×D4—42no—
364C22×D4C22⋊D1711yescharacteristic
365C22×D4C22×D1712yes—
366C23×C4—341no—
367C23×C4C22×D1711yescharacteristic
368C23×C4C22×D1711yescharacteristic
369C2×SD16—178no—
370C2×SD16—344no—
371C2×SD16—344no—
372C2×SD16—344no—
373C2×SD16—344no—
374C2×SD16—3432no—
375C2×SD16—344no—
376C22×C8—171no—
377C22×C8—174no—
378C22×C8—171no—
379C22×C8—171no—
380C22×C8—171no—
381C22×C8—344no—
382C22×C8—342no—
383C22×C8—344no—
384C22⋊Q8—344no—
385C22⋊Q8—682no—
386#121—24no—
387#121—42no—
388#120—44no—
389C4×D4—22no—
390C4×D4—22no—
391C2×C4⋊C4—172no—
392C2×C4⋊C4—342no—
393C2×C4⋊C4C22⋊D1711yescharacteristic
394C2×C4⋊C4C22×D1712yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—342no—
400C2×C22⋊C4—341no—
401C2×C22⋊C4C22⋊D1711yescharacteristic
402C2×C22⋊C4C22⋊D1711yescharacteristic
403C2×C42C22×D1711yescharacteristic
404#106—344no—
405C4⋊C8—342no—
406#103—344no—
407#103—682no—
408#102—344no—
409#102—682no—
410C22⋊C8—342no—
411C22⋊C8—342no—
412C4×C8—342no—
413#95—21no—
414#95—341no—
415#95—342no—
416#95—342no—
417C17#2426411yescharacteristic, complemented, Sylow
418C24—22no—
419C24—44no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—82no—
424C24C2×C22⋊D1711yescharacteristic
425C24C2×C22⋊D1711yescharacteristic
426C2×Q8—174no—
427C2×Q8—348no—
428C2×Q8—342no—
429C2×Q8—342no—
430C2×Q8—342no—
431C2×Q8—342no—
432C2×Q8—342no—
433C2×Q8—684no—
434C2×Q8—688no—
435C2×Q8—684no—
436C2×Q8—682no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—28no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—48no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—42no—
451C2×D4—82no—
452C2×D4#6322914yescomplemented
453C2×D4D4×D1714yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—684no—
464C22×C4—682no—
465C22×C4D4×D1712yes—
466C22×C4#6325412yes—
467C22×C4C2×C22⋊D1711yescharacteristic
468C22×C4C2×C22⋊D1711yescharacteristic
469C22×C4C2×C22⋊D1711yescharacteristic
470C22×C4C2×C22⋊D1711yescharacteristic
471C22×C4C23×D1711yescharacteristic
472SD16—3416no—
473SD16—3416no—
474C2×C8—172no—
475C2×C8—174no—
476C2×C8—172no—
477C2×C8—348no—
478C2×C8—342no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—342no—
482C2×C8—342no—
483C2×C8—684no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—342no—
488C4⋊C4—342no—
489C4⋊C4—684no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—44no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C22⋊C4—82no—
496C22⋊C4—684no—
497C22⋊C4—682no—
498C42—21no—
499C42—21no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—48no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—88no—
515C23—88no—
516C23—84no—
517C23—84no—
518C23C2×#6322912yes—
519C23C2×#6322912yescomplemented
520C23#1058762512yes—
521C23#1058763612yes—
522C23C22×C22⋊D1711yescharacteristic
523Q8—344no—
524Q8—344no—
525Q8—688no—
526Q8—682no—
527Q8—682no—
528D4—28no—
529D4—24no—
530D4—24no—
531D4—48no—
532D4—42no—
533D4—42no—
534D4—88no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—344no—
552C2×C4—688no—
553C2×C4—682no—
554C2×C4—682no—
555C2×C4—682no—
556C2×C4—682no—
557C2×C4—682no—
558C2×C4C2×#6322912yescomplemented
559C2×C4#1058762512yes—
560C2×C4C2×D4×D1711yescharacteristic
561C2×C4C2×D4×D1712yes—
562C2×C4C2×#6325411yescharacteristic
563C2×C4#1058769112yes—
564C8—174no—
565C8—342no—
566C8—681no—
567C8—681no—
568C22—22no—
569C22—22no—
570C22—24no—
571C22—24no—
572C22—22no—
573C22—24no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—816no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22#1176303614yescomplemented
586C22C2×#1058762511yescharacteristic
587C22D4×C22⋊D1712yes—
588C22C22×#6322911yescharacteristic
589C22C2×#1058763611yescharacteristic
590C22#1176362012yes—
591C4—22no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—82no—
596C4—684no—
597C4—1362no—
598C4#120×D1711yescharacteristic
599C4#1176337411yescharacteristic
600C4#1176344612yes—
601C2—24no—
602C2—42no—
603C2—84no—
604C2#1453213711yescharacteristic
605C2C2×#1176303611yescharacteristic
606C2#1453842011yescharacteristic
607C2D4×#6322912yes—
608C2#1454048712yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC17×#2408C2
socleC23×C17C22×D4
hypercenter#2408D17
hypocenterC17#24264
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. C23×#63229C2semidirect
2. C2×#11763036C2semidirect
3. C2×D4×#3622C2semidirect
4. #14532149C2semidirect
5. #14530101C2semidirect
6. #14529787C2semidirect
7. #14529637C2semidirect
8. #14528807C2semidirect
9. #14528718C2semidirect
10. #14528715C2semidirect
18. C23×#3622C22semidirect
19. C2×#10587610C22semidirect
22. C2×#10587479C22semidirect
23. C2×C4⋊#3622C22semidirect
24. C2×#10587481C22semidirect
25. C2×#10587481C22semidirect
29. C2×C22⋊#3622C22semidirect
30. C2×C4×#3622C22semidirect
33. #11762715C22semidirect
37. #11762571C22semidirect
87. C2×#63229D4semidirect
95. C22×#3622D4semidirect
136. C2×D4×C17SD16semidirect
216. C23×C17C2×SD16semidirect
243. C2×C4×C17C2×SD16semidirect
263. #3622#120semidirect
292. C22×C17#465semidirect
417. C17#24264semidirect
452. C2×D4#63229semidirect
519. C23C2×#63229semidirect
558. C2×C4C2×#63229semidirect
585. C22#11763036semidirect

Maximal subgroups

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

subgroupindex
C23×#632292
C2×#117630362
C2×D4×#36222
#145321492
#145301012
#145297872
#145296372
#145288072
#145287182
#145287152
C17×#24082
#2426417