Finite Group Databasebeta

← #17067377

Subgroups — #17067377

Order 4352.

1338 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic34C211yesmaximal, characteristic
2C2×#11763139C211yesmaximal, characteristic, complemented
3C22×#10587481C211yesmaximal, characteristic, complemented
4#14530030C212yesmaximal, complemented
5#14529980C212yesmaximal, complemented
6#14529205C211yesmaximal, characteristic, complemented
7#14529077C211yesmaximal, characteristic
8#14529051C212yesmaximal, complemented
9#14528718C211yesmaximal, characteristic, complemented
10#14528718C212yesmaximal, complemented
11C17×#2486C211yesmaximal, characteristic
12C23×Dic34C2211yescharacteristic
13C23×Dic34C2211yescharacteristic
14C23×#3622C2211yescharacteristic
15D4×Dic34—24no—
16D4×Dic34—24no—
17C2×#10587555C2211yescharacteristic
18C2×#10587574—21no—
19C2×#10587610C2211yescharacteristic
20C2×#10587610C2211yescharacteristic
21C2×#10587564—21no—
22C2×#10587564—21no—
23C2×C4×Dic34C2211yescharacteristic
24C2×D4×Dic17—21no—
25#11763139—24no—
26C2×#10587479C2212yes—
27C2×#10587481—24no—
28C2×#10587481C2212yes—
29C2×#10587481C2212yes—
30C2×#10587481C2212yes—
31C2×#10587481C2212yes—
32C2×#10587481C2212yes—
33C2×#10587481C2212yes—
34C2×C22⋊#3622C2211yescharacteristic
35C2×C22⋊#3622C2212yes—
36#11762571C2212yes—
37#11762564C2212yes—
38C2×C17×#123—21no—
39C2×C4×D4×C17C2211yescharacteristic
40C22×C4⋊C4×C17C2211yescharacteristic
41C17×#398—21no—
42C17×#397—21no—
43C17×#389C2212yes—
44C17×#381—21no—
45C17×#380C2211yescharacteristic
46C17×#380C2212yes—
47C22×D4×C17D411yescharacteristic
48C23×C4×C17—21no—
49C23×C4×C17C2311yescharacteristic
50C23×C4×C17C2311yescharacteristic
51C23×Dic17—21no—
52C23×Dic17—21no—
53C22×Dic34—24no—
54C22×Dic34—22no—
55C22×Dic34—21no—
56C22×Dic34—21no—
57C22×Dic34—22no—
58C22×Dic34—22no—
59C22×Dic34—24no—
60C22×Dic34—42no—
61C22×Dic34—42no—
62C22×Dic34D414yes—
63C22×Dic34C2311yescharacteristic
64C22×Dic34C2311yescharacteristic
65C22×Dic34C2311yescharacteristic
66C17×#123—44no—
67C4×D4×C17—22no—
68C4×D4×C17—22no—
69C2×C4⋊C4×C17—22no—
70C2×C4⋊C4×C17—22no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—22no—
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17C2312yes—
75C2×C4⋊C4×C17C2312yes—
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—22no—
78C2×C22⋊C4×C17D411yescharacteristic
79C2×C22⋊C4×C17D411yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C22⋊Dic17—21no—
82C2×C22⋊Dic17—21no—
83D4×Dic17—28no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17C2311yescharacteristic
87#10587610—22no—
88#10587610—22no—
89#10587610—22no—
90#10587610D414yes—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×C4×Dic17—21no—
95C22×#3622—22no—
96C22×#3622—42no—
97C22×#3622C2312yes—
98C22×#3622C2312yes—
99#10587574—28no—
100#10587564—28no—
101#10587564—28no—
102#10587555—22no—
103#10587555—22no—
104C4×Dic34—22no—
105C4×Dic34—22no—
106C17×#95—22no—
107C17×#95—21no—
108C17×#95—22no—
109C22⋊#3622—22no—
110C22⋊#3622—42no—
111#10587481—24no—
112#10587481—24no—
113#10587481—24no—
114#10587481—24no—
115#10587481—42no—
116#10587481—42no—
117#10587479—24no—
118C24×C17C2×D411yescharacteristic
119C24×C17C2×D411yescharacteristic
120C2×D4×C17—22no—
121C2×D4×C17—22no—
122C2×D4×C17SD1614yes—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17—42no—
132C22×C4×C17—42no—
133C22×C4×C17—42no—
134C22×C4×C17C2×D412yes—
135C22×C4×C17C2×D412yes—
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2411yescharacteristic
141C22×Dic17—21no—
142C22×Dic17—24no—
143C22×Dic17—24no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—44no—
148C22×Dic17—44no—
149C2×Dic34—21no—
150C2×Dic34—22no—
151C2×Dic34—22no—
152C2×Dic34—22no—
153C2×Dic34—22no—
154C2×Dic34—22no—
155C2×Dic34—28no—
156C2×Dic34—24no—
157C2×Dic34—24no—
158C2×Dic34—21no—
159C2×Dic34—21no—
160C2×Dic34—44no—
161C2×Dic34—48no—
162C2×Dic34—48no—
163C2×Dic34—44no—
164C2×Dic34—44no—
165C2×Dic34—44no—
166C2×Dic34—44no—
167C2×Dic34C2×D412yes—
168C2×Dic34C2×D412yes—
169C2×Dic34C2×D412yes—
170C2×Dic34#4114yes—
171C4⋊C4×C17—22no—
172C4⋊C4×C17—22no—
173C4⋊C4×C17—22no—
174C4⋊C4×C17—22no—
175C4⋊C4×C17—22no—
176C4⋊C4×C17—42no—
177C4⋊C4×C17—44no—
178C22⋊C4×C17—22no—
179C22⋊C4×C17—22no—
180C22⋊C4×C17—42no—
181C22⋊C4×C17—44no—
182C42×C17—21no—
183C42×C17—21no—
184C22⋊Dic17—24no—
185C22⋊Dic17—24no—
186C4⋊Dic17—24no—
187C4⋊Dic17—24no—
188C4⋊Dic17—21no—
189C4⋊Dic17C2×D412yes—
190#63225—24no—
191#63225—24no—
192#63225—24no—
193C4×Dic17—24no—
194C2×#3622—22no—
195C2×#3622—22no—
196C2×#3622—22no—
197C2×#3622—22no—
198C2×#3622—22no—
199C2×#3622—22no—
200C2×#3622—44no—
201#24041—171nomaximal, Sylow
202C23×C17—22no—
203C23×C17—22no—
204C23×C17—21no—
205C23×C17—22no—
206C23×C17—21no—
207C23×C17—22no—
208C23×C17#12012yes—
209C23×C17C2×SD1612yes—
210C23×C17C2×SD1612yes—
211C23×C17#13712yes—
212C23×C17C22×D411yescharacteristic
213D4×C17—28no—
214C2×C4×C17—21no—
215C2×C4×C17—22no—
216C2×C4×C17—21no—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—24no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—42no—
227C2×C4×C17—42no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—42no—
233C2×C4×C17—84no—
234C2×C4×C17#12012yes—
235C2×C4×C17C2×SD1612yes—
236C2×C4×C17C22×D411yescharacteristic
237C2×C4×C17C22×D411yescharacteristic
238C2×C4×C17C2×#4112yes—
239C2×C4×C17H(5,𝔽2)12yes—
240C2×Dic17—22no—
241C2×Dic17—24no—
242C2×Dic17—22no—
243C2×Dic17—22no—
244C2×Dic17—22no—
245C2×Dic17—48no—
246C2×Dic17—48no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17—42no—
250Dic34—22no—
251Dic34—24no—
252Dic34—22no—
253Dic34—22no—
254Dic34—24no—
255Dic34—48no—
256Dic34—48no—
257Dic34—41no—
258Dic34—41no—
259Dic34—42no—
260Dic34#12314yescomplemented
261#3622—42no—
262#3622—42no—
263C2×D4×Q8—171no—
264C2×#483—171no—
265C22×#103—171no—
266#2486D1711yescharacteristic
267#2043—172no—
268#2035—172no—
269#1892—171no—
270#1869—171no—
271#1865—172no—
272#1787—171no—
273#1787—172no—
274C22×C17—22no—
275C22×C17—22no—
276C22×C17—24no—
277C22×C17—24no—
278C22×C17—22no—
279C22×C17—24no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17—42no—
283C22×C17#44814yes—
284C22×C17C2×#12011yescharacteristic
285C22×C17#54612yes—
286C22×C17C22×SD1611yescharacteristic
287C22×C17C2×#13711yescharacteristic
288C22×C17#57812yes—
289C4×C17—22no—
290C4×C17—22no—
291C4×C17—41no—
292C4×C17—41no—
293C4×C17—82no—
294C4×C17—82no—
295C4×C17C2×#12311yescharacteristic
296C4×C17#53411yescharacteristic
297C4×C17#54612yes—
298Dic17—24no—
299Dic17—42no—
300Dic17—42no—
301Dic17—42no—
302C23×Q8—171no—
303C23×Q8—171no—
304C23×C8—171no—
305D4×Q8—344no—
306D4×Q8—344no—
307C2×C4⋊Q8—171no—
308C2×C4⋊Q8—341no—
309C2×#123—21no—
310C2×C22⋊Q8—171no—
311C2×C22⋊Q8—171no—
312C2×C22⋊Q8—341no—
313C2×C22⋊Q8—341no—
314C2×C4×Q8—171no—
315C2×C4×D4—341no—
316C2×C4×D4C2×D1711yescharacteristic
317C22×C4⋊C4C2×D1711yescharacteristic
318#483—344no—
319C2×#106—172no—
320C2×#103—172no—
321C2×#103—172no—
322C2×#103—172no—
323C2×#103—172no—
324C2×#103—172no—
325C2×#103—172no—
326C2×#103—344no—
327C2×C22⋊C8—171no—
328C2×C22⋊C8—172no—
329#398—21no—
330#397—21no—
331#389C2×D1712yes—
332#381—21no—
333#380C2×D1711yescharacteristic
334#380C2×D1712yes—
335#340—172no—
336#336—172no—
337C2×C17—24no—
338C2×C17—42no—
339C2×C17#240511yescharacteristic
340C2×C17C2×#44811yescharacteristic
341C2×C17#301111yescharacteristic
342C2×C17#329812yes—
343C2×C17#331212yes—
344C22×Q8—174no—
345C22×Q8—171no—
346C22×Q8—171no—
347C22×Q8—171no—
348C22×Q8—342no—
349C22×Q8—342no—
350C22×Q8—344no—
351C22×Q8—344no—
352C22×Q8—342no—
353C22×Q8—341no—
354C22×Q8—341no—
355C22×Q8—682no—
356C22×Q8—682no—
357C22×D4C22⋊D1711yescharacteristic
358C23×C4—21no—
359C23×C4—341no—
360C23×C4—341no—
361C23×C4C22×D1711yescharacteristic
362C23×C4C22×D1711yescharacteristic
363C22×C8—172no—
364C22×C8—172no—
365C22×C8—342no—
366C22×C8—682no—
367C4⋊Q8—342no—
368C4⋊Q8—342no—
369C4⋊Q8—348no—
370#123—44no—
371C22⋊Q8—174no—
372C22⋊Q8—348no—
373C22⋊Q8—348no—
374C22⋊Q8—342no—
375C22⋊Q8—342no—
376C22⋊Q8—342no—
377C4×Q8—342no—
378C4×Q8—342no—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—348no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4C22⋊D1711yescharacteristic
393C2×C4⋊C4C22×D1712yes—
394C2×C4⋊C4C22×D1712yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—341no—
398C2×C22⋊C4—341no—
399C2×C22⋊C4C22⋊D1711yescharacteristic
400C2×C22⋊C4C22⋊D1711yescharacteristic
401C2×C42—341no—
402C2×C42C22×D1711yescharacteristic
403#106—344no—
404#103—344no—
405#103—344no—
406#103—344no—
407#103—344no—
408#103—682no—
409#103—682no—
410C22⋊C8—342no—
411C22⋊C8—682no—
412#95—22no—
413#95—21no—
414#95—22no—
415C17#2404111yescharacteristic, complemented, Sylow
416C24C2×C22⋊D1711yescharacteristic
417C24C2×C22⋊D1711yescharacteristic
418C2×Q8—172no—
419C2×Q8—172no—
420C2×Q8—172no—
421C2×Q8—174no—
422C2×Q8—344no—
423C2×Q8—348no—
424C2×Q8—341no—
425C2×Q8—341no—
426C2×Q8—341no—
427C2×Q8—342no—
428C2×Q8—344no—
429C2×Q8—342no—
430C2×Q8—342no—
431C2×Q8—342no—
432C2×Q8—342no—
433C2×Q8—684no—
434C2×Q8—684no—
435C2×Q8—684no—
436C2×Q8—688no—
437C2×Q8—684no—
438C2×Q8—688no—
439C2×Q8—684no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4#6322914yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—341no—
455C22×C4—344no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—344no—
460C22×C4—684no—
461C22×C4—684no—
462C22×C4D4×D1712yes—
463C22×C4D4×D1712yes—
464C22×C4C2×C22⋊D1711yescharacteristic
465C22×C4C2×C22⋊D1711yescharacteristic
466C22×C4C2×C22⋊D1711yescharacteristic
467C22×C4C2×C22⋊D1711yescharacteristic
468C22×C4C23×D1711yescharacteristic
469C2×C8—342no—
470C2×C8—342no—
471C2×C8—342no—
472C2×C8—342no—
473C2×C8—342no—
474C2×C8—342no—
475C2×C8—684no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—172no—
484C4⋊C4—344no—
485C4⋊C4—344no—
486C4⋊C4—344no—
487C4⋊C4—344no—
488C4⋊C4—341no—
489C4⋊C4—344no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—344no—
495C22⋊C4—344no—
496C42—21no—
497C42—21no—
498C42—344no—
499C23—22no—
500C23—22no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—22no—
505C23C2×#6322912yes—
506C23C2×#6322912yes—
507C23#1058762312yes—
508C23#1058763712yes—
509C23C22×C22⋊D1711yescharacteristic
510Q8—174no—
511Q8—342no—
512Q8—344no—
513Q8—342no—
514Q8—342no—
515Q8—344no—
516Q8—688no—
517Q8—688no—
518Q8—681no—
519Q8—681no—
520Q8—682no—
521D4—28no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—84no—
542C2×C4—342no—
543C2×C4—344no—
544C2×C4—342no—
545C2×C4—342no—
546C2×C4—342no—
547C2×C4—688no—
548C2×C4—688no—
549C2×C4—682no—
550C2×C4—682no—
551C2×C4—682no—
552C2×C4C2×#6322912yes—
553C2×C4#1058762312yes—
554C2×C4C2×D4×D1711yescharacteristic
555C2×C4C2×D4×D1711yescharacteristic
556C2×C4#41×D1712yes—
557C2×C4#1058769612yes—
558C8—682no—
559C8—682no—
560C22—22no—
561C22—24no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—22no—
566C22—41no—
567C22—42no—
568C22—41no—
569C22#1176294714yes—
570C22C2×#1058762311yescharacteristic
571C22#1176344712yes—
572C22C22×#6322911yescharacteristic
573C22C2×#1058763711yescharacteristic
574C22#1176362412yes—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—82no—
580C4—82no—
581C4—344no—
582C4—682no—
583C4—682no—
584C4—682no—
585C4#123×D1711yescharacteristic
586C4#1176337011yescharacteristic
587C4#1176345112yes—
588C2—24no—
589C2—42no—
590C2#1453210811yescharacteristic
591C2C2×#1176294711yescharacteristic
592C2#1453807711yescharacteristic
593C2#1454064312yes—
594C2#1454082912yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2486C2
socleC23×C17C22×D4
prime radicalC22×D4×C17D4
hypercenter#2486D17
hypocenterC17#24041
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. C2×#11763139C2semidirect
3. C22×#10587481C2semidirect
4. #14530030C2semidirect
5. #14529980C2semidirect
6. #14529205C2semidirect
8. #14529051C2semidirect
9. #14528718C2semidirect
10. #14528718C2semidirect
260. Dic34#123semidirect
415. C17#24041semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic342
C2×#117631392
C22×#105874812
#145300302
#145299802
#145292052
#145290772
#145290512
#145287182
C17×#24862
#2404117