Finite Group Databasebeta

← #17067388

Subgroups — #17067388

Order 4352.

1318 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic34C211yesmaximal, characteristic
2C2×#11763144C211yesmaximal, characteristic, complemented
3C22×#10587481C211yesmaximal, characteristic, complemented
4#14530030C212yesmaximal, complemented
5#14529987C212yesmaximal, complemented
6#14529205C211yesmaximal, characteristic, complemented
7#14529077C211yesmaximal, characteristic
8#14529037C212yesmaximal, complemented
9#14528718C211yesmaximal, characteristic, complemented
10#14528718C212yesmaximal, complemented
11C17×#2487C211yesmaximal, 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#11763144—24no—
26C2×#10587478C2212yes—
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×C4×D4×C17C2211yescharacteristic
39C22×C4⋊C4×C17C2211yescharacteristic
40C17×#400—21no—
41C17×#399—21no—
42C17×#389C2212yes—
43C17×#386—21no—
44C17×#385—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
66C4×D4×C17—22no—
67C4×D4×C17—22no—
68C2×C4⋊C4×C17—22no—
69C2×C4⋊C4×C17—22no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—22no—
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C4⋊C4×C17C2312yes—
74C2×C4⋊C4×C17C2312yes—
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17—22no—
77C2×C22⋊C4×C17D411yescharacteristic
78C2×C22⋊C4×C17D411yescharacteristic
79C2×C42×C17C2311yescharacteristic
80C2×C22⋊Dic17—21no—
81C2×C22⋊Dic17—21no—
82D4×Dic17—28no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17C2311yescharacteristic
86#10587610—22no—
87#10587610—22no—
88#10587610—22no—
89#10587610D414yes—
90C2×#63225—21no—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×C4×Dic17—21no—
94C22×#3622—22no—
95C22×#3622—42no—
96C22×#3622C2312yes—
97C22×#3622C2312yes—
98#10587574—28no—
99#10587564—28no—
100#10587564—28no—
101#10587555—22no—
102#10587555—22no—
103C4×Dic34—22no—
104C4×Dic34—22no—
105C17×#95—22no—
106C17×#95—21no—
107C17×#95—22no—
108C22⋊#3622—22no—
109C22⋊#3622—42no—
110#10587481—24no—
111#10587481—24no—
112#10587481—24no—
113#10587481—24no—
114#10587481—42no—
115#10587481—42no—
116#10587478—24no—
117C24×C17C2×D411yescharacteristic
118C24×C17C2×D411yescharacteristic
119C2×D4×C17—22no—
120C2×D4×C17—22no—
121C2×D4×C17Q1614yes—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—42no—
131C22×C4×C17—82no—
132C22×C4×C17C2×D412yes—
133C22×C4×C17C2×D412yes—
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2411yescharacteristic
139C22×Dic17—21no—
140C22×Dic17—24no—
141C22×Dic17—24no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—44no—
146C22×Dic17—44no—
147C2×Dic34—21no—
148C2×Dic34—22no—
149C2×Dic34—22no—
150C2×Dic34—22no—
151C2×Dic34—22no—
152C2×Dic34—22no—
153C2×Dic34—28no—
154C2×Dic34—24no—
155C2×Dic34—24no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—44no—
159C2×Dic34—48no—
160C2×Dic34—48no—
161C2×Dic34—44no—
162C2×Dic34—44no—
163C2×Dic34—44no—
164C2×Dic34—44no—
165C2×Dic34C2×D412yes—
166C2×Dic34C2×D412yes—
167C2×Dic34C2×D412yes—
168C2×Dic34#4114yes—
169C4⋊C4×C17—22no—
170C4⋊C4×C17—22no—
171C4⋊C4×C17—22no—
172C4⋊C4×C17—22no—
173C4⋊C4×C17—22no—
174C4⋊C4×C17—42no—
175C4⋊C4×C17—82no—
176C22⋊C4×C17—22no—
177C22⋊C4×C17—22no—
178C22⋊C4×C17—42no—
179C22⋊C4×C17—82no—
180C42×C17—21no—
181C42×C17—21no—
182C22⋊Dic17—24no—
183C22⋊Dic17—24no—
184C4⋊Dic17—24no—
185C4⋊Dic17—24no—
186C4⋊Dic17—21no—
187C4⋊Dic17C2×D412yes—
188#63225—24no—
189#63225—24no—
190#63225—24no—
191C4×Dic17—24no—
192C2×#3622—22no—
193C2×#3622—22no—
194C2×#3622—22no—
195C2×#3622—22no—
196C2×#3622—22no—
197C2×#3622—22no—
198C2×#3622—44no—
199#24042—171nomaximal, Sylow
200C23×C17—22no—
201C23×C17—22no—
202C23×C17—21no—
203C23×C17—22no—
204C23×C17—21no—
205C23×C17—22no—
206C23×C17#12012yes—
207C23×C17C2×Q1612yes—
208C23×C17C2×Q1612yes—
209C23×C17#13712yes—
210C23×C17C22×D411yescharacteristic
211D4×C17—28no—
212C2×C4×C17—21no—
213C2×C4×C17—22no—
214C2×C4×C17—21no—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
217C2×C4×C17—24no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—42no—
225C2×C4×C17—42no—
226C2×C4×C17—42no—
227C2×C4×C17—42no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—84no—
232C2×C4×C17#12012yes—
233C2×C4×C17C2×Q1612yes—
234C2×C4×C17C22×D411yescharacteristic
235C2×C4×C17C22×D411yescharacteristic
236C2×C4×C17C2×#4112yes—
237C2×C4×C17H(5,𝔽2)12yes—
238C2×Dic17—22no—
239C2×Dic17—24no—
240C2×Dic17—22no—
241C2×Dic17—22no—
242C2×Dic17—22no—
243C2×Dic17—48no—
244C2×Dic17—48no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248Dic34—22no—
249Dic34—24no—
250Dic34—22no—
251Dic34—22no—
252Dic34—24no—
253Dic34—48no—
254Dic34—48no—
255Dic34—41no—
256Dic34—41no—
257Dic34—42no—
258Dic34#12314yes—
259#3622—42no—
260#3622—42no—
261C2×D4×Q8—171no—
262C2×#484—171no—
263C22×#103—171no—
264#2487D1711yescharacteristic
265#2043—172no—
266#2036—172no—
267#1892—171no—
268#1869—171no—
269#1863—172no—
270#1787—171no—
271#1787—172no—
272C22×C17—22no—
273C22×C17—22no—
274C22×C17—24no—
275C22×C17—24no—
276C22×C17—22no—
277C22×C17—24no—
278C22×C17—41no—
279C22×C17—41no—
280C22×C17—42no—
281C22×C17#45114yes—
282C22×C17C2×#12011yescharacteristic
283C22×C17#54612yes—
284C22×C17C22×Q1611yescharacteristic
285C22×C17C2×#13711yescharacteristic
286C22×C17#57712yes—
287C4×C17—22no—
288C4×C17—22no—
289C4×C17—41no—
290C4×C17—41no—
291C4×C17—82no—
292C4×C17—82no—
293C4×C17C2×#12311yescharacteristic
294C4×C17#53411yescharacteristic
295C4×C17#54612yes—
296Dic17—24no—
297Dic17—42no—
298Dic17—42no—
299Dic17—42no—
300C23×Q8—171no—
301C23×Q8—171no—
302C23×C8—171no—
303D4×Q8—344no—
304D4×Q8—344no—
305C2×C4⋊Q8—171no—
306C2×C4⋊Q8—341no—
307C2×C22⋊Q8—171no—
308C2×C22⋊Q8—171no—
309C2×C22⋊Q8—341no—
310C2×C22⋊Q8—341no—
311C2×C4×Q8—171no—
312C2×C4×D4—341no—
313C2×C4×D4C2×D1711yescharacteristic
314C22×C4⋊C4C2×D1711yescharacteristic
315#484—344no—
316C2×#107—172no—
317C2×#103—172no—
318C2×#103—172no—
319C2×#103—172no—
320C2×#103—172no—
321C2×#103—172no—
322C2×#103—172no—
323C2×#103—344no—
324C2×C22⋊C8—171no—
325C2×C22⋊C8—172no—
326#400—21no—
327#399—21no—
328#389C2×D1712yes—
329#386—21no—
330#385—21no—
331#380C2×D1711yescharacteristic
332#380C2×D1712yes—
333#340—172no—
334#336—172no—
335C2×C17—24no—
336C2×C17—42no—
337C2×C17#240511yescharacteristic
338C2×C17C2×#45111yescharacteristic
339C2×C17#301011yescharacteristic
340C2×C17#329912yes—
341C2×C17#332012yes—
342C22×Q8—174no—
343C22×Q8—171no—
344C22×Q8—171no—
345C22×Q8—171no—
346C22×Q8—342no—
347C22×Q8—342no—
348C22×Q8—344no—
349C22×Q8—344no—
350C22×Q8—342no—
351C22×Q8—341no—
352C22×Q8—341no—
353C22×Q8—682no—
354C22×Q8—682no—
355C22×D4C22⋊D1711yescharacteristic
356C23×C4—21no—
357C23×C4—341no—
358C23×C4—341no—
359C23×C4C22×D1711yescharacteristic
360C23×C4C22×D1711yescharacteristic
361C22×C8—172no—
362C22×C8—172no—
363C22×C8—342no—
364C22×C8—682no—
365C4⋊Q8—342no—
366C4⋊Q8—342no—
367C4⋊Q8—348no—
368C22⋊Q8—174no—
369C22⋊Q8—348no—
370C22⋊Q8—348no—
371C22⋊Q8—342no—
372C22⋊Q8—342no—
373C22⋊Q8—342no—
374C4×Q8—342no—
375C4×Q8—342no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—348no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4C22⋊D1711yescharacteristic
390C2×C4⋊C4C22×D1712yes—
391C2×C4⋊C4C22×D1712yes—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—341no—
395C2×C22⋊C4—341no—
396C2×C22⋊C4C22⋊D1711yescharacteristic
397C2×C22⋊C4C22⋊D1711yescharacteristic
398C2×C42—341no—
399C2×C42C22×D1711yescharacteristic
400#107—344no—
401#103—344no—
402#103—344no—
403#103—344no—
404#103—344no—
405#103—682no—
406#103—682no—
407C22⋊C8—342no—
408C22⋊C8—682no—
409#95—22no—
410#95—21no—
411#95—22no—
412C17#2404211yescharacteristic, complemented, Sylow
413C24C2×C22⋊D1711yescharacteristic
414C24C2×C22⋊D1711yescharacteristic
415C2×Q8—172no—
416C2×Q8—172no—
417C2×Q8—172no—
418C2×Q8—174no—
419C2×Q8—344no—
420C2×Q8—348no—
421C2×Q8—341no—
422C2×Q8—341no—
423C2×Q8—341no—
424C2×Q8—342no—
425C2×Q8—344no—
426C2×Q8—342no—
427C2×Q8—342no—
428C2×Q8—342no—
429C2×Q8—342no—
430C2×Q8—684no—
431C2×Q8—684no—
432C2×Q8—684no—
433C2×Q8—688no—
434C2×Q8—684no—
435C2×Q8—688no—
436C2×Q8—684no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4#6323114yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—82no—
450C22×C4—341no—
451C22×C4—344no—
452C22×C4—341no—
453C22×C4—341no—
454C22×C4—341no—
455C22×C4—344no—
456C22×C4—684no—
457C22×C4—684no—
458C22×C4D4×D1712yes—
459C22×C4D4×D1712yes—
460C22×C4C2×C22⋊D1711yescharacteristic
461C22×C4C2×C22⋊D1711yescharacteristic
462C22×C4C2×C22⋊D1711yescharacteristic
463C22×C4C2×C22⋊D1711yescharacteristic
464C22×C4C23×D1711yescharacteristic
465C2×C8—342no—
466C2×C8—342no—
467C2×C8—342no—
468C2×C8—342no—
469C2×C8—342no—
470C2×C8—342no—
471C2×C8—684no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—82no—
479C4⋊C4—172no—
480C4⋊C4—344no—
481C4⋊C4—344no—
482C4⋊C4—344no—
483C4⋊C4—344no—
484C4⋊C4—341no—
485C4⋊C4—344no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—42no—
489C22⋊C4—82no—
490C22⋊C4—344no—
491C22⋊C4—344no—
492C42—21no—
493C42—21no—
494C42—344no—
495C23—22no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23#1058762312yes—
502C23C2×#6323112yes—
503C23C2×#6323112yes—
504C23#1058763712yes—
505C23C22×C22⋊D1711yescharacteristic
506Q8—174no—
507Q8—342no—
508Q8—344no—
509Q8—342no—
510Q8—342no—
511Q8—344no—
512Q8—688no—
513Q8—688no—
514Q8—681no—
515Q8—681no—
516Q8—682no—
517D4—28no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—84no—
538C2×C4—342no—
539C2×C4—344no—
540C2×C4—342no—
541C2×C4—342no—
542C2×C4—342no—
543C2×C4—688no—
544C2×C4—688no—
545C2×C4—682no—
546C2×C4—682no—
547C2×C4—682no—
548C2×C4#1058762312yes—
549C2×C4C2×#6323112yes—
550C2×C4C2×D4×D1711yescharacteristic
551C2×C4C2×D4×D1711yescharacteristic
552C2×C4#41×D1712yes—
553C2×C4#1058769612yes—
554C8—682no—
555C8—682no—
556C22—22no—
557C22—22no—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—42no—
565C22#1176296414yes—
566C22C2×#1058762311yescharacteristic
567C22#1176344712yes—
568C22C22×#6323111yescharacteristic
569C22C2×#1058763711yescharacteristic
570C22#1176362012yes—
571C4—22no—
572C4—22no—
573C4—41no—
574C4—41no—
575C4—82no—
576C4—82no—
577C4—344no—
578C4—682no—
579C4—682no—
580C4—682no—
581C4#123×D1711yescharacteristic
582C4#1176337011yescharacteristic
583C4#1176345112yes—
584C2—24no—
585C2—42no—
586C2#1453210811yescharacteristic
587C2C2×#1176296411yescharacteristic
588C2#1453806811yescharacteristic
589C2#1454065412yes—
590C2#1454094912yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2487C2
socleC23×C17C22×D4
prime radicalC22×D4×C17D4
hypercenter#2487D17
hypocenterC17#24042
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. C2×#11763144C2semidirect
3. C22×#10587481C2semidirect
4. #14530030C2semidirect
5. #14529987C2semidirect
6. #14529205C2semidirect
8. #14529037C2semidirect
9. #14528718C2semidirect
10. #14528718C2semidirect
412. C17#24042semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic342
C2×#117631442
C22×#105874812
#145300302
#145299872
#145292052
#145290772
#145290372
#145287182
C17×#24872
#2404217