Finite Group Databasebeta

← #17068222

Subgroups — #17068222

Order 4352.

1238 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
1C2×#11763096C211yesmaximal, characteristic, complemented
2C2×C4×#63228C211yesmaximal, characteristic, complemented
3#14533697C211yesmaximal, characteristic, complemented
4#14529970C212yesmaximal, complemented
5#14529780C212yesmaximal, complemented
6#14529746C211yesmaximal, characteristic, complemented
7#14529579C211yesmaximal, characteristic, complemented
8#14529106C212yesmaximal, complemented
9#14528983C212yesmaximal, complemented
10#14528784C211yesmaximal, characteristic, complemented
11C2×C17×#555C211yesmaximal, characteristic, complemented
12C22×#63228C2211yescharacteristic
13C2×#10587616C2212yes—
14C2×#10587624—21no—
15C2×C4×D68C2211yescharacteristic
16#11763096—24no—
17C4×#63228—24no—
18C2×#10587478C2211yescharacteristic
19C2×#10587479C2212yes—
20C2×C4⋊#3622C2211yescharacteristic
21C2×#10587503C2211yescharacteristic
22C2×#10587503C2212yes—
23C2×#10587480C2212yes—
24C2×#10587480C2211yescharacteristic
25C2×C22⋊#3622C2212yes—
26C2×C22⋊#3622C2212yes—
27C2×C4×#3622C2211yescharacteristic
28#11762725—22no—
29#11762679C2211yescharacteristic
30#11762669C2212yes—
31#11762666—21no—
32#11762571C2212yes—
33#11762571C2212yes—
34#11762571C2212yes—
35C17×#555—24no—
36C17×#555—24no—
37C2×C17×C4⋊Q8C2211yescharacteristic
38C2×C17×#124—22no—
39C2×C17×#121—21no—
40C2×C17×#121C2212yes—
41C2×C4×D4×C17C2211yescharacteristic
42C2×C4×D4×C17C2211yescharacteristic
43C2×C17×#117—21no—
44C2×C17×#117C2212yes—
45C22×Q8×C17—21no—
46C22×D4×C17—22no—
47C22×D4×C17D411yescharacteristic
48C22×D4×C17C2311yescharacteristic
49C23×C4×C17—21no—
50C23×C4×C17C2312yes—
51C22×C22⋊D17—22no—
52C22×D68C2311yescharacteristic
53C22×C4×D17—21no—
54C17×C4⋊Q8—22no—
55C17×C4⋊Q8D414yescomplemented
56C17×#124—216no—
57C17×#121—28no—
58C17×#121—24no—
59C17×#121—24no—
60C4×D4×C17—22no—
61C4×D4×C17—22no—
62C4×D4×C17—22no—
63C4×D4×C17D414yescomplemented
64C17×#117—28no—
65C17×#117—24no—
66C17×#117—24no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17D411yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2312yes—
71C2×C22⋊C4×C17—22no—
72C2×C22⋊C4×C17—22no—
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17D412yes—
75C2×C42×C17—22no—
76C2×C42×C17C2311yescharacteristic
77C2×C22⋊Dic17—22no—
78#10587624—44no—
79C2×#63228—22no—
80C2×#63228—22no—
81C2×#63228—22no—
82#10587616—24no—
83#10587616—42no—
84C2×#63227—21no—
85C2×#63227—22no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2312yes—
90C22×#3622C2311yescharacteristic
91C22×#3622C2312yes—
92C22×#3622C2311yescharacteristic
93C4×D68—22no—
94C4×D68—22no—
95#10587503—22no—
96#10587503—42no—
97#10587502—22no—
98C22⋊#3622—24no—
99C22⋊#3622—42no—
100#10587480—24no—
101#10587480—22no—
102#10587479—24no—
103#10587478—22no—
104C4⋊#3622—22no—
105C4×#3622—22no—
106C24×C17—21no—
107C24×C17C2×D412yes—
108C23×D17—21no—
109C2×Q8×C17—24no—
110C2×Q8×C17—81no—
111C2×Q8×C17—81no—
112C2×D4×C17—28no—
113C2×D4×C17—22no—
114C2×D4×C17—22no—
115C2×D4×C17—22no—
116C2×D4×C17—21no—
117C2×D4×C17—21no—
118C2×D4×C17—21no—
119C2×D4×C17—21no—
120C2×D4×C17—21no—
121C2×D4×C17—82no—
122C2×D4×C17—82no—
123C2×D4×C17C2×D412yes—
124C22×C4×C17—21no—
125C22×C4×C17—22no—
126C22×C4×C17—24no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—42no—
134C22×C4×C17—42no—
135C22×C4×C17C2×D412yes—
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2411yescharacteristic
139C2×C22⋊D17—44no—
140C2×C22⋊D17—44no—
141C2×C22⋊D17—82no—
142C22×Dic17—21no—
143C22×Dic17—22no—
144C2×D68—21no—
145C2×D68—21no—
146C2×D68—21no—
147C2×D68—21no—
148C2×D68—21no—
149C2×D68#4112yes—
150C2×C4×D17—42no—
151C2×C4×D17—42no—
152C2×C4×D17—42no—
153C4⋊C4×C17—24no—
154C4⋊C4×C17—22no—
155C4⋊C4×C17—22no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17C2×D414yes—
158C4⋊C4×C17C2×D412yes—
159C22⋊C4×C17—28no—
160C22⋊C4×C17—28no—
161C22⋊C4×C17—24no—
162C22⋊C4×C17—24no—
163C42×C17—28no—
164C42×C17C2×D412yes—
165C42×C17C2×D412yes—
166C22⋊Dic17—44no—
167#63228—28no—
168#63227—44no—
169#63227—42no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—22no—
173C4⋊Dic17—22no—
174C4⋊Dic17—42no—
175C4⋊Dic17—81no—
176C2×#3622—21no—
177C2×#3622—21no—
178C2×#3622—21no—
179C2×#3622—22no—
180C2×#3622—22no—
181C2×#3622—21no—
182C2×#3622—22no—
183C2×#3622—21no—
184C2×#3622#4112yes—
185#24117—171nomaximal, Sylow
186C23×C17—22no—
187C23×C17—22no—
188C23×C17—22no—
189C23×C17—22no—
190C23×C17—22no—
191C23×C17—22no—
192C23×C17—22no—
193C23×C17—22no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17C22×D411yescharacteristic
201C22×D17—22no—
202C22×D17—41no—
203C22×D17—41no—
204C22×D17—41no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17—41no—
208Q8×C17—84no—
209D4×C17—22no—
210D4×C17—22no—
211D4×C17—41no—
212D4×C17—41no—
213D4×C17—44no—
214D4×C17—88no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—24no—
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—48no—
231C2×C4×C17#12012yes—
232C2×C4×C17#12012yes—
233C2×C4×C17#13512yes—
234C2×C4×C17AGL(1,ℤ/8)12yes—
235C2×C4×C17C22×D411yescharacteristic
236C2×C4×C17C22×D411yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17H(5,𝔽2)11yescharacteristic
240C2×C4×C172−1+411yescharacteristic
241C22⋊D17—88no—
242C2×Dic17—41no—
243C2×Dic17—42no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248D68—22no—
249D68—22no—
250D68—41no—
251D68—41no—
252C4×D17—84no—
253#3622—22no—
254#3622—41no—
255#3622—42no—
256C2×#555D1711yescharacteristic, complemented
257C2×#476—171no—
258C2×C4×D8—171no—
259#2537—171no—
260#2034—172no—
261#1999—172no—
262#1990—171no—
263#1954—171no—
264#1875—172no—
265#1854—172no—
266#1802—171no—
267C22×C17—42no—
268C22×C17—42no—
269C22×C17—42no—
270C22×C17—42no—
271C22×C17—42no—
272C22×C17—42no—
273C22×C17—42no—
274C22×C17—42no—
275C22×C17—42no—
276C22×C17—42no—
277C22×C17—81no—
278C22×C17—81no—
279C22×C17—81no—
280C22×C17—81no—
281C22×C17C2×#12011yescharacteristic
282C22×C17#54611yescharacteristic
283C22×C17#54711yescharacteristic
284C22×C17C2×#13511yescharacteristic
285C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C17#57611yescharacteristic
287C22×C17#57711yescharacteristic
288C2×D17—42no—
289C2×D17—42no—
290C2×D17—42no—
291C2×D17—81no—
292C2×D17—81no—
293C2×D17—81no—
294C2×D17—81no—
295C4×C17—22no—
296C4×C17—42no—
297C4×C17—42no—
298C4×C17—44no—
299C4×C17#44914yes—
300C4×C17#52511yescharacteristic
301C4×C17#53711yescharacteristic
302C4×C17#54811yescharacteristic
303C4×C17#55011yescharacteristic
304Dic17—81no—
305Dic17—82no—
306C22×D8—171no—
307#555—24no—
308#555—24no—
309C2×C4⋊Q8C2×D1711yescharacteristic
310C2×#124—22no—
311C2×#121—21no—
312C2×#121—172no—
313C2×#121—341no—
314C2×#121C2×D1712yes—
315C2×C4×D4—171no—
316C2×C4×D4C2×D1711yescharacteristic
317C2×C4×D4C2×D1711yescharacteristic
318C2×#117—21no—
319C2×#117C2×D1712yes—
320#476—344no—
321C4×D8—344no—
322C2×#107—171no—
323C2×#106—172no—
324C2×C4⋊C8—171no—
325C2×#102—171no—
326C2×#102—171no—
327C2×#102—172no—
328C2×#102—172no—
329C2×C22⋊C8—172no—
330C2×C22⋊C8—172no—
331C2×C4×C8—171no—
332#394—342no—
333#384—171no—
334#380—172no—
335#380—341no—
336#340—172no—
337#340—172no—
338#340—172no—
339C2×C17—82no—
340C2×C17—82no—
341C2×C17#240711yescharacteristic
342C2×C17C2×#44911yescharacteristic
343C2×C17#301211yescharacteristic
344C2×C17#330311yescharacteristic
345C2×C17#332511yescharacteristic
346C2×C17#334111yescharacteristic
347C2×C17#335411yescharacteristic
348D17—82no—
349C22×Q8—21no—
350C22×D4—22no—
351C22×D4—171no—
352C22×D4—342no—
353C22×D4C22⋊D1711yescharacteristic
354C22×D4C22×D1711yescharacteristic
355C23×C4—21no—
356C23×C4—341no—
357C23×C4C22×D1712yes—
358C2×D8—342no—
359C2×D8—342no—
360C2×D8—342no—
361C22×C8—171no—
362C22×C8—171no—
363C22×C8—172no—
364C4⋊Q8—22no—
365C4⋊Q8C22⋊D1714yescomplemented
366#124—216no—
367#121—28no—
368#121—24no—
369#121—24no—
370#121—344no—
371#121—682no—
372#121—684no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—342no—
377C4×D4—342no—
378C4×D4C22⋊D1714yescomplemented
379#117—28no—
380#117—24no—
381#117—24no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—172no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4C22⋊D1711yescharacteristic
388C2×C4⋊C4C22×D1711yescharacteristic
389C2×C4⋊C4C22×D1712yes—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—342no—
394C2×C22⋊C4—341no—
395C2×C22⋊C4—342no—
396C2×C22⋊C4C22⋊D1712yes—
397C2×C42—22no—
398C2×C42C22×D1711yescharacteristic
399#107—342no—
400#106—344no—
401C4⋊C8—342no—
402#102—342no—
403#102—342no—
404#102—344no—
405#102—682no—
406C22⋊C8—344no—
407C22⋊C8—682no—
408C4×C8—342no—
409#95—342no—
410C17#2411711yescharacteristic, complemented, Sylow
411C24—21no—
412C24—341no—
413C24C2×C22⋊D1712yes—
414C2×Q8—24no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×D4—28no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4—172no—
429C2×D4—341no—
430C2×D4—341no—
431C2×D4—341no—
432C2×D4—341no—
433C2×D4—341no—
434C2×D4—684no—
435C2×D4—684no—
436C2×D4—1362no—
437C2×D4C2×C22⋊D1712yes—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—341no—
450C22×C4—342no—
451C22×C4—682no—
452C22×C4—682no—
453C22×C4—682no—
454C22×C4C2×C22⋊D1712yes—
455C22×C4C2×C22⋊D1711yescharacteristic
456C22×C4C2×C22⋊D1711yescharacteristic
457C22×C4C23×D1711yescharacteristic
458D8—348no—
459C2×C8—172no—
460C2×C8—341no—
461C2×C8—341no—
462C2×C8—342no—
463C2×C8—341no—
464C2×C8—342no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—342no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—342no—
473C4⋊C4—342no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—682no—
477C4⋊C4—1361no—
478C4⋊C4C2×C22⋊D1714yes—
479C4⋊C4C2×C22⋊D1712yes—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—684no—
485C22⋊C4—684no—
486C22⋊C4—682no—
487C42—28no—
488C42C2×C22⋊D1712yes—
489C42C2×C22⋊D1712yes—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—342no—
505C23—681no—
506C23—681no—
507C23—681no—
508C23—681no—
509C23—681no—
510C23—681no—
511C23C22×C22⋊D1711yescharacteristic
512Q8—84no—
513D4—22no—
514D4—22no—
515D4—44no—
516D4—41no—
517D4—41no—
518D4—88no—
519D4—342no—
520D4—342no—
521D4—681no—
522D4—681no—
523D4—1368no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—681no—
541C2×C4—681no—
542C2×C4—681no—
543C2×C4—682no—
544C2×C4—682no—
545C2×C4—682no—
546C2×C4—1364no—
547C2×C4#1058763512yes—
548C2×C4#1058763612yes—
549C2×C4#1058763912yes—
550C2×C4#1058763912yes—
551C2×C4#41×D1711yescharacteristic
552C2×C4#41×D1711yescharacteristic
553C2×C4#1058769611yescharacteristic
554C2×C4#1058769711yescharacteristic
555C2×C4C22×C22⋊D1711yescharacteristic
556C2×C4C22×C22⋊D1711yescharacteristic
557C8—342no—
558C8—682no—
559C8—681no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—682no—
575C22—682no—
576C22—682no—
577C22—1361no—
578C22—1361no—
579C22—1361no—
580C22—1361no—
581C22C2×#1058763911yescharacteristic
582C22#1176344711yescharacteristic
583C22#1176345811yescharacteristic
584C22C2×#1058763611yescharacteristic
585C22C2×#1058763511yescharacteristic
586C22#1176361211yescharacteristic
587C22#1176362011yescharacteristic
588C4—22no—
589C4—42no—
590C4—42no—
591C4—44no—
592C4—1361no—
593C4—1362no—
594C4#1176295414yes—
595C4#1176331611yescharacteristic
596C4#1176338711yescharacteristic
597C4#1176346611yescharacteristic
598C4#1176347811yescharacteristic
599C2—82no—
600C2—82no—
601C2—1362no—
602C2#1453212311yescharacteristic
603C2C2×#1176295411yescharacteristic
604C2#1453808811yescharacteristic
605C2#1454070611yescharacteristic
606C2#1454102011yescharacteristic
607C2#1454120211yescharacteristic
608C2#1454137311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#555C2
socleC23×C17C22×D4
hypercenterC2×#555D17
hypocenterC17#24117
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. C2×#11763096C2semidirect
2. C2×C4×#63228C2semidirect
3. #14533697C2semidirect
4. #14529970C2semidirect
5. #14529780C2semidirect
6. #14529746C2semidirect
7. #14529579C2semidirect
8. #14529106C2semidirect
9. #14528983C2semidirect
10. #14528784C2semidirect
11. C2×C17×#555C2semidirect
55. C17×C4⋊Q8D4semidirect
63. C4×D4×C17D4semidirect
256. C2×#555D17semidirect
365. C4⋊Q8C22⋊D17semidirect
378. C4×D4C22⋊D17semidirect
410. C17#24117semidirect

Maximal subgroups

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

subgroupindex
C2×#117630962
C2×C4×#632282
#145336972
#145299702
#145297802
#145297462
#145295792
#145291062
#145289832
#145287842
C2×C17×#5552
#2411717