Finite Group Databasebeta

← #17067381

Subgroups — #17067381

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×#11763142C211yesmaximal, characteristic, complemented
2C22×#10587506C211yesmaximal, characteristic, complemented
3#14533101C211yesmaximal, characteristic
4#14530027C212yesmaximal, complemented
5#14529984C212yesmaximal, complemented
6#14529208C211yesmaximal, characteristic, complemented
7#14529077C211yesmaximal, characteristic
8#14529038C212yesmaximal, complemented
9#14528719C211yesmaximal, characteristic, complemented
10#14528719C212yesmaximal, complemented
11C2×D4×Q8×C17C211yesmaximal, characteristic
12C23×#3622C2211yescharacteristic
13C22×C4⋊Dic17C2211yescharacteristic
14#11763142—24no—
15C2×#10587478C2212yes—
16C2×#10587506—24no—
17C2×#10587506C2212yes—
18C2×#10587506C2212yes—
19C2×#10587506C2212yes—
20C2×#10587506C2212yes—
21C2×#10587506C2212yes—
22C2×#10587506C2212yes—
23C2×C22⋊#3622C2211yescharacteristic
24C2×C22⋊#3622C2212yes—
25#11762748—21no—
26#11762745—21no—
27#11762707C2212yes—
28#11762689—21no—
29#11762681—21no—
30#11762669C2211yescharacteristic
31#11762669C2212yes—
32#11762571C2212yes—
33#11762564C2212yes—
34C23×Q8×C17C2211yescharacteristic
35C23×Q8×C17C2211yescharacteristic
36D4×Q8×C17—24no—
37D4×Q8×C17—24no—
38C2×C17×C4⋊Q8—21no—
39C2×C17×C4⋊Q8C2211yescharacteristic
40C2×C17×C22⋊Q8—21no—
41C2×C17×C22⋊Q8—21no—
42C2×C17×C22⋊Q8C2211yescharacteristic
43C2×C17×C22⋊Q8C2211yescharacteristic
44C2×C4×Q8×C17C2211yescharacteristic
45C2×C4×D4×C17—21no—
46C2×C4×D4×C17C2211yescharacteristic
47C22×Q8×C17—22no—
48C22×Q8×C17—22no—
49C22×Q8×C17—21no—
50C22×Q8×C17—21no—
51C22×Q8×C17—22no—
52C22×Q8×C17—24no—
53C22×Q8×C17—24no—
54C22×Q8×C17—42no—
55C22×Q8×C17—42no—
56C22×Q8×C17D414yes—
57C22×Q8×C17C2311yescharacteristic
58C22×Q8×C17C2311yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C22×D4×C17D411yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17—21no—
63C23×C4×C17C2311yescharacteristic
64C23×C4×C17C2311yescharacteristic
65C23×Dic17—21no—
66C17×C4⋊Q8—28no—
67C17×C4⋊Q8—22no—
68C17×C4⋊Q8—22no—
69C17×C22⋊Q8—28no—
70C17×C22⋊Q8—28no—
71C17×C22⋊Q8—22no—
72C17×C22⋊Q8—22no—
73C17×C22⋊Q8—22no—
74C17×C22⋊Q8D414yes—
75C4×Q8×C17—22no—
76C4×Q8×C17—22no—
77C4×D4×C17—28no—
78C4×D4×C17—22no—
79C4×D4×C17—22no—
80C2×C4⋊C4×C17—21no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17—21no—
83C2×C4⋊C4×C17—21no—
84C2×C4⋊C4×C17—21no—
85C2×C4⋊C4×C17D411yescharacteristic
86C2×C4⋊C4×C17C2311yescharacteristic
87C2×C22⋊C4×C17—21no—
88C2×C22⋊C4×C17—21no—
89C2×C22⋊C4×C17D411yescharacteristic
90C2×C22⋊C4×C17D411yescharacteristic
91C2×C42×C17—21no—
92C2×C42×C17C2311yescharacteristic
93C2×C22⋊Dic17—21no—
94C2×C22⋊Dic17—22no—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17—22no—
97C2×C4⋊Dic17—22no—
98C2×C4⋊Dic17C2312yes—
99C2×C4⋊Dic17C2312yes—
100C2×#63225—22no—
101C22×#3622—22no—
102C22×#3622—42no—
103C22×#3622C2312yes—
104C22×#3622C2312yes—
105#10587506—24no—
106#10587506—24no—
107#10587506—24no—
108#10587506—24no—
109#10587506—42no—
110#10587506—42no—
111#10587502—22no—
112#10587502—21no—
113#10587502—22no—
114C22⋊#3622—22no—
115C22⋊#3622—42no—
116#10587478—24no—
117C24×C17C2×D411yescharacteristic
118C24×C17C2×D411yescharacteristic
119C2×Q8×C17—24no—
120C2×Q8×C17—24no—
121C2×Q8×C17—21no—
122C2×Q8×C17—21no—
123C2×Q8×C17—21no—
124C2×Q8×C17—22no—
125C2×Q8×C17—28no—
126C2×Q8×C17—22no—
127C2×Q8×C17—22no—
128C2×Q8×C17—22no—
129C2×Q8×C17—22no—
130C2×Q8×C17—44no—
131C2×Q8×C17—44no—
132C2×Q8×C17—44no—
133C2×Q8×C17—48no—
134C2×Q8×C17—48no—
135C2×Q8×C17—44no—
136C2×Q8×C17—44no—
137C2×Q8×C17C2×D412yes—
138C2×Q8×C17C2×D412yes—
139C2×Q8×C17C2×D412yes—
140C2×Q8×C17#4114yes—
141C2×D4×C17—22no—
142C2×D4×C17—22no—
143C2×D4×C17Q1614yes—
144C22×C4×C17—21no—
145C22×C4×C17—22no—
146C22×C4×C17—22no—
147C22×C4×C17—24no—
148C22×C4×C17—24no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17—22no—
155C22×C4×C17—22no—
156C22×C4×C17—44no—
157C22×C4×C17—44no—
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2×D412yes—
162C22×C4×C17C2×D412yes—
163C22×C4×C17C2×D411yescharacteristic
164C22×C4×C17C2411yescharacteristic
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17—42no—
168C22×Dic17—82no—
169C4⋊C4×C17—24no—
170C4⋊C4×C17—24no—
171C4⋊C4×C17—24no—
172C4⋊C4×C17—24no—
173C4⋊C4×C17—24no—
174C4⋊C4×C17—22no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17C2×D412yes—
177C22⋊C4×C17—24no—
178C22⋊C4×C17—22no—
179C22⋊C4×C17—24no—
180C22⋊C4×C17—22no—
181C42×C17—24no—
182C42×C17—21no—
183C42×C17—21no—
184C22⋊Dic17—42no—
185C22⋊Dic17—82no—
186C4⋊Dic17—22no—
187C4⋊Dic17—22no—
188C4⋊Dic17—22no—
189C4⋊Dic17—22no—
190C4⋊Dic17—42no—
191#63225—82no—
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
211Q8×C17—22no—
212Q8×C17—24no—
213Q8×C17—22no—
214Q8×C17—22no—
215Q8×C17—24no—
216Q8×C17—48no—
217Q8×C17—48no—
218Q8×C17—41no—
219Q8×C17—41no—
220Q8×C17—42no—
221Q8×C17#12314yes—
222D4×C17—28no—
223C2×C4×C17—21no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—24no—
227C2×C4×C17—21no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—24no—
231C2×C4×C17—21no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—21no—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—48no—
239C2×C4×C17—48no—
240C2×C4×C17—42no—
241C2×C4×C17—42no—
242C2×C4×C17—42no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—42no—
246C2×C4×C17#12012yes—
247C2×C4×C17C2×Q1612yes—
248C2×C4×C17C22×D411yescharacteristic
249C2×C4×C17C22×D411yescharacteristic
250C2×C4×C17C2×#4112yes—
251C2×C4×C17H(5,𝔽2)12yes—
252C2×Dic17—42no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—84no—
259#3622—42no—
260#3622—42no—
261C2×D4×Q8D1711yescharacteristic
262C2×#484—171no—
263C22×#103—171no—
264#2487—171no—
265#2043—172no—
266#2036—172no—
267#1892—171no—
268#1869—171no—
269#1863—172no—
270#1787—172no—
271#1787—171no—
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—24no—
289C4×C17—22no—
290C4×C17—41no—
291C4×C17—42no—
292C4×C17—42no—
293C4×C17—41no—
294C4×C17—42no—
295C4×C17C2×#12311yescharacteristic
296C4×C17#53411yescharacteristic
297C4×C17#54612yes—
298Dic17—82no—
299Dic17—82no—
300C23×Q8C2×D1711yescharacteristic
301C23×Q8C2×D1711yescharacteristic
302C23×C8—171no—
303D4×Q8—24no—
304D4×Q8—24no—
305C2×C4⋊Q8—21no—
306C2×C4⋊Q8C2×D1711yescharacteristic
307C2×C22⋊Q8—21no—
308C2×C22⋊Q8—21no—
309C2×C22⋊Q8C2×D1711yescharacteristic
310C2×C22⋊Q8C2×D1711yescharacteristic
311C2×C4×Q8C2×D1711yescharacteristic
312C2×C4×D4—21no—
313C2×C4×D4C2×D1711yescharacteristic
314C22×C4⋊C4—171no—
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—341no—
327#399—341no—
328#389—172no—
329#386—341no—
330#385—341no—
331#380—171no—
332#380—172no—
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—22no—
343C22×Q8—22no—
344C22×Q8—21no—
345C22×Q8—21no—
346C22×Q8—22no—
347C22×Q8—24no—
348C22×Q8—24no—
349C22×Q8—42no—
350C22×Q8—42no—
351C22×Q8C22⋊D1714yes—
352C22×Q8C22×D1711yescharacteristic
353C22×Q8C22×D1711yescharacteristic
354C22×Q8C22×D1711yescharacteristic
355C22×D4C22⋊D1711yescharacteristic
356C23×C4—21no—
357C23×C4—21no—
358C23×C4—341no—
359C23×C4C22×D1711yescharacteristic
360C23×C4C22×D1711yescharacteristic
361C22×C8—172no—
362C22×C8—172no—
363C22×C8—342no—
364C22×C8—682no—
365C4⋊Q8—28no—
366C4⋊Q8—22no—
367C4⋊Q8—22no—
368C22⋊Q8—28no—
369C22⋊Q8—28no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8C22⋊D1714yes—
374C4×Q8—22no—
375C4×Q8—22no—
376C4×D4—28no—
377C4×D4—22no—
378C4×D4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—172no—
385C2×C4⋊C4—172no—
386C2×C4⋊C4—342no—
387C2×C4⋊C4—342no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—342no—
390C2×C4⋊C4C22⋊D1711yescharacteristic
391C2×C4⋊C4C22×D1711yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—341no—
395C2×C22⋊C4—342no—
396C2×C22⋊C4C22⋊D1711yescharacteristic
397C2×C22⋊C4C22⋊D1711yescharacteristic
398C2×C42—21no—
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—342no—
410#95—341no—
411#95—342no—
412C17#2404211yescharacteristic, complemented, Sylow
413C24C2×C22⋊D1711yescharacteristic
414C24C2×C22⋊D1711yescharacteristic
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—24no—
421C2×Q8—28no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—48no—
430C2×Q8—48no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8#6325414yes—
434C2×Q8C2×C22⋊D1712yes—
435C2×Q8C2×C22⋊D1712yes—
436C2×Q8C2×C22⋊D1712yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4#6323114yes—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—342no—
455C22×C4—342no—
456C22×C4—682no—
457C22×C4—1362no—
458C22×C4C2×C22⋊D1711yescharacteristic
459C22×C4C2×C22⋊D1711yescharacteristic
460C22×C4C2×C22⋊D1712yes—
461C22×C4C2×C22⋊D1712yes—
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—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—342no—
480C4⋊C4—342no—
481C4⋊C4—342no—
482C4⋊C4—342no—
483C4⋊C4—682no—
484C4⋊C4—1362no—
485C4⋊C4C2×C22⋊D1712yes—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—682no—
491C22⋊C4—1362no—
492C42—24no—
493C42—21no—
494C42—21no—
495C23—22no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23#1058762812yes—
502C23C2×#6323112yes—
503C23C2×#6323112yes—
504C23#1058763912yes—
505C23C22×C22⋊D1711yescharacteristic
506Q8—22no—
507Q8—24no—
508Q8—22no—
509Q8—22no—
510Q8—24no—
511Q8—48no—
512Q8—48no—
513Q8—41no—
514Q8—41no—
515Q8—42no—
516Q8#1058762114yes—
517D4—28no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—682no—
542C2×C4—682no—
543C2×C4—682no—
544C2×C4—682no—
545C2×C4—682no—
546C2×C4—682no—
547C2×C4—1364no—
548C2×C4C2×#6323112yes—
549C2×C4#1058763912yes—
550C2×C4C2×#6325412yes—
551C2×C4#1058769112yes—
552C2×C4C22×C22⋊D1711yescharacteristic
553C2×C4C22×C22⋊D1711yescharacteristic
554C8—682no—
555C8—682no—
556C22—22no—
557C22—22no—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—42no—
565C22#1176296514yes—
566C22C2×#1058763911yescharacteristic
567C22#1176345312yes—
568C22C22×#6323111yescharacteristic
569C22C2×#1058762811yescharacteristic
570C22#1176361712yes—
571C4—22no—
572C4—24no—
573C4—22no—
574C4—41no—
575C4—42no—
576C4—42no—
577C4—41no—
578C4—42no—
579C4—1362no—
580C4—1362no—
581C4C2×#1058762111yescharacteristic
582C4#1176336811yescharacteristic
583C4#1176345312yes—
584C2—24no—
585C2—42no—
586C2#1453210711yescharacteristic
587C2C2×#1176296511yescharacteristic
588C2#1453807011yescharacteristic
589C2#1454066012yes—
590C2#1454095512yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×D4×Q8×C17C2
socleC23×C17C22×D4
prime radicalC22×D4×C17D4
hypercenterC2×D4×Q8D17
hypocenterC17#24042
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#11763142C2semidirect
2. C22×#10587506C2semidirect
4. #14530027C2semidirect
5. #14529984C2semidirect
6. #14529208C2semidirect
8. #14529038C2semidirect
9. #14528719C2semidirect
10. #14528719C2semidirect
412. C17#24042semidirect

Maximal subgroups

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

subgroupindex
C2×#117631422
C22×#105875062
#145331012
#145300272
#145299842
#145292082
#145290772
#145290382
#145287192
C2×D4×Q8×C172
#2404217