Finite Group Databasebeta

← #17035205

Subgroups — #17035205

Order 4352.

1422 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763289C211yesmaximal, characteristic
2#14537373C211yesmaximal, characteristic, complemented
3C2×#11762827C211yesmaximal, characteristic
4#14537304C211yesmaximal, characteristic
5#14531407C211yesmaximal, characteristic, complemented
6#14529018C214yesmaximal, complemented
7#14529011C214yesmaximal, complemented
8C2×#11762577C211yesmaximal, characteristic, complemented
9C17×#1733C211yesmaximal, characteristic, complemented
10C22×#63252C2211yescharacteristic
11C22×#63223C2211yescharacteristic
12#11763289—22no—
13#11763289—22no—
14#11763289—22no—
15#11763289C2214yes—
16C2×C4×Dic34—21no—
17C2×C4×D68—21no—
18C2×#10587567—22no—
19C2×#10587612C2211yescharacteristic
20C2×#10587578—21no—
21C22×C4⋊Dic17C2211yescharacteristic
22C2×C4⋊C4×D17—21no—
23#11762852C2212yescomplemented
24#11762852C2212yescomplemented
25#11762849—21no—
26#11762849—21no—
27#11762845—22no—
28#11762827C418yescomplemented
29C2×#10587476C2214yes—
30#11762819—21no—
31#11762815—21no—
32#11762811C2212yes—
33#11762811C2212yes—
34#11762806—22no—
35#11762708—22no—
36#11762684—22no—
37#11762679C2212yescomplemented
38#11762675C2212yescomplemented
39#11762577C2214yescomplemented
40C22×M16×C17C2211yescharacteristic
41C2×C17×#117C2211yescharacteristic
42C2×C17×#117C2211yescharacteristic
43C17×#339C2214yescomplemented
44C23×C4×C17C2311yescharacteristic
45C22×C22⋊D17—21no—
46C23×Dic17D412yes—
47C2×#63252—21no—
48C2×#63252—21no—
49C2×#63252—22no—
50C2×#63252—22no—
51C2×#63252C2×C414yes—
52C2×#63252D412yes—
53C2×#63252Q814yes—
54C2×#63252C2312yes—
55C22×D68D411yescharacteristic
56C22×C4×D17—21no—
57C22×C4×D17—22no—
58C22×Dic34D411yescharacteristic
59C2×M16×C17—21no—
60C2×M16×C17—22no—
61C2×M16×C17—22no—
62C2×M16×C17C2312yes—
63C22×C8×C17—21no—
64C17×#117—21no—
65C17×#117—21no—
66C17×#117—22no—
67C17×#117C2×C414yescomplemented
68C17×#117D412yes—
69C17×#117C2312yes—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17D411yescharacteristic
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17—21no—
75C2×C42×C17—21no—
76C2×C42×C17C2312yes—
77C2×C22⋊Dic17—21no—
78C2×#63227—22no—
79#10587612—21no—
80#10587612—21no—
81#10587612—21no—
82#10587612—21no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—24no—
88C2×C4⋊Dic17D412yes—
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2312yes—
92C2×#63225—22no—
93C2×C4×Dic17—22no—
94C2×C4×Dic17—21no—
95C2×#63223—21no—
96C2×#63223—22no—
97C2×#63223C2×C414yescomplemented
98C2×#63223C2312yes—
99C22×#3622—21no—
100#10587578—24no—
101#10587578—24no—
102C4⋊C4×D17—24no—
103C4⋊C4×D17—24no—
104#10587567—28no—
105#10587567—28no—
106C4×D68—24no—
107C4×D68—24no—
108C4×Dic34—24no—
109C4×Dic34—24no—
110#10587502—24no—
111#10587492—22no—
112#10587492—22no—
113#10587487—22no—
114#10587487—22no—
115#10587479—22no—
116#10587479—22no—
117#10587478—22no—
118#10587478—22no—
119#10587476C2×C4116yescomplemented
120C24×C17C2×D411yescharacteristic
121C23×D17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—22no—
125C22×C4×C17C22×C412yes—
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×Q812yes—
132C22×C4×C17#4112yes—
133C22×C4×C17#4112yes—
134C22×C4×C17C2411yescharacteristic
135C2×C22⋊D17—24no—
136C2×C22⋊D17—22no—
137C2×C22⋊D17—24no—
138C2×C22⋊D17—22no—
139C22×Dic17—21no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—82no—
145C22×Dic17C2×D414yes—
146#63252—24no—
147#63252—24no—
148#63252—24no—
149#63252—24no—
150#63252—24no—
151#63252—22no—
152#63252—22no—
153#63252C4⋊C418yescomplemented
154#63252C22×C414yes—
155#63252C2×Q814yes—
156C2×D68—21no—
157C2×D68—22no—
158C2×D68C4⋊C414yescomplemented
159C2×D68C2×D412yes—
160C2×C4×D17—24no—
161C2×C4×D17—22no—
162C2×C4×D17—24no—
163C2×C4×D17—22no—
164C2×C4×D17—28no—
165C2×C4×D17—84no—
166C2×Dic34—21no—
167C2×Dic34—22no—
168C2×Dic34C4⋊C414yescomplemented
169C2×Dic34C2×D412yes—
170M16×C17—22no—
171M16×C17—22no—
172M16×C17—42no—
173C2×C8×C17—22no—
174C2×C8×C17—42no—
175C4⋊C4×C17—24no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17C2×D412yes—
179C4⋊C4×C17C2×D412yes—
180C22⋊C4×C17—24no—
181C22⋊C4×C17—22no—
182C22⋊C4×C17—22no—
183C42×C17—22no—
184C42×C17—22no—
185C42×C17—22no—
186C42×C17C22×C414yescomplemented
187C22⋊Dic17—41no—
188C22⋊Dic17—41no—
189#63227—28no—
190C4⋊Dic17—22no—
191C4⋊Dic17—22no—
192C4⋊Dic17—22no—
193C4⋊Dic17—22no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—84no—
199#63225—28no—
200C4×Dic17—28no—
201C4×Dic17—41no—
202C4×Dic17—41no—
203#63223—24no—
204#63223—22no—
205#63223C22×C414yescomplemented
206C2×#3622—24no—
207C2×#3622—22no—
208#20864—171nomaximal, Sylow
209C23×C17—41no—
210C23×C17—41no—
211C23×C17C4×D412yes—
212C23×C17#12011yescharacteristic
213C23×C17#12011yescharacteristic
214C23×C17C22⋊Q812yes—
215C23×C17C22×D411yescharacteristic
216C22×D17—22no—
217C22×D17—24no—
218C22×D17—42no—
219C22×D17—42no—
220C2×C4×C17—24no—
221C2×C4×C17—22no—
222C2×C4×C17—22no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17—42no—
226C2×C4×C17—42no—
227C2×C4×C17C2×C4⋊C412yes—
228C2×C4×C17C2×C4⋊C412yes—
229C2×C4×C17C4×D412yes—
230C2×C4×C17C4×D412yes—
231C2×C4×C17C4×D412yes—
232C2×C4×C17#12011yescharacteristic
233C2×C4×C17#12011yescharacteristic
234C2×C4×C17C22⋊Q812yes—
235C2×C4×C17C22⋊Q812yes—
236C2×C4×C17C22⋊Q812yes—
237C2×C4×C17#12312yes—
238C2×C4×C17#12312yes—
239C2×C4×C17C23×C411yescharacteristic
240C2×C4×C17C22×D411yescharacteristic
241C2×C4×C17C22×D411yescharacteristic
242C2×C4×C17C22×Q811yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C22⋊D17—24no—
246C22⋊D17—24no—
247C22⋊D17—28no—
248C2×Dic17—22no—
249C2×Dic17—24no—
250C2×Dic17—24no—
251C2×Dic17—44no—
252C2×Dic17—44no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—82no—
258D68—22no—
259D68—24no—
260D68C2×C4⋊C414yes—
261C4×D17—24no—
262C4×D17—24no—
263C4×D17—28no—
264C4×D17—88no—
265Dic34—22no—
266Dic34—24no—
267Dic34C2×C4⋊C414yes—
268C8×C17—42no—
269#3622—24no—
270C2×#520—171no—
271#2911—171no—
272C2×#421—171no—
273#2896—171no—
274#2317—171no—
275#1860—174no—
276#1859—174no—
277C2×#343—171no—
278#1733D1711yescharacteristic, complemented
279C22×C17—21no—
280C22×C17—21no—
281C22×C17—21no—
282C22×C17—21no—
283C22×C17—21no—
284C22×C17—21no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17#38512yes—
288C22×C17#38512yes—
289C22×C17C2×C4×D411yescharacteristic
290C22×C17C2×#12011yescharacteristic
291C22×C17C2×C22⋊Q811yescharacteristic
292C2×D17—24no—
293C2×D17—48no—
294C2×D17—42no—
295C2×D17—42no—
296C4×C17—44no—
297C4×C17—42no—
298C4×C17—42no—
299C4×C17#38512yes—
300C4×C17#38512yes—
301C4×C17C22×C4⋊C411yescharacteristic
302C4×C17C2×C4×D411yescharacteristic
303C4×C17C2×C22⋊Q811yescharacteristic
304C4×C17C2×#12311yescharacteristic
305Dic17—24no—
306Dic17—44no—
307Dic17—81no—
308Dic17—81no—
309C22×#41—171no—
310C22×M16—171no—
311C22×M16C2×D1711yescharacteristic
312#520—174no—
313#520—342no—
314#520—342no—
315#520—342no—
316C2×C4×Q8—341no—
317C2×C4×D4—341no—
318C2×C4×D4—342no—
319C2×#117—171no—
320C2×#117—341no—
321C2×#117C2×D1711yescharacteristic
322C2×#117C2×D1711yescharacteristic
323C22×C4⋊C4—171no—
324C22×C4⋊C4—341no—
325#428—172no—
326#428—172no—
327#428—341no—
328#428—341no—
329#427—342no—
330#421—178no—
331C2×#104—174no—
332#419—341no—
333#418—341no—
334#417—172no—
335#417—172no—
336#416—342no—
337#389—342no—
338#385—342no—
339#384—172no—
340#382—172no—
341#343—174no—
342#339C2×D1714yescomplemented
343C2×C17—21no—
344C2×C17—22no—
345C2×C17—21no—
346C2×C17—22no—
347C2×C17C2×#38511yescharacteristic
348C2×C17#301911yescharacteristic
349C2×C17#302011yescharacteristic
350D17—44no—
351C2×#41—174no—
352C2×#41—172no—
353C2×#41—174no—
354C2×#41—172no—
355C2×#41—341no—
356C2×#41—342no—
357C2×#41—341no—
358C2×#41—342no—
359C22×Q8—171no—
360C22×D4—171no—
361C22×D4—341no—
362C23×C4—172no—
363C23×C4—342no—
364C23×C4—341no—
365C23×C4C22×D1711yescharacteristic
366C2×M16—21no—
367C2×M16—22no—
368C2×M16—22no—
369C2×M16—174no—
370C2×M16—172no—
371C2×M16—341no—
372C2×M16—342no—
373C2×M16C22×D1712yes—
374C22×C8—21no—
375C22×C8—341no—
376C4×Q8—344no—
377C4×Q8—344no—
378C4×D4—344no—
379C4×D4—344no—
380C4×D4—348no—
381C4×D4—348no—
382#117—21no—
383#117—21no—
384#117—22no—
385#117—344no—
386#117—344no—
387#117—341no—
388#117—341no—
389#117—341no—
390#117—341no—
391#117C4×D1714yescomplemented
392#117D6812yes—
393#117C22×D1712yes—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—172no—
396C2×C4⋊C4—172no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—171no—
399C2×C4⋊C4—344no—
400C2×C4⋊C4—342no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4—341no—
403C2×C4⋊C4—341no—
404C2×C4⋊C4—341no—
405C2×C4⋊C4—344no—
406C2×C4⋊C4—344no—
407C2×C4⋊C4D6811yescharacteristic
408C2×C4⋊C4D6811yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—342no—
412C2×C22⋊C4—341no—
413C2×C42—21no—
414C2×C42—342no—
415C2×C42—341no—
416C2×C42C22×D1712yes—
417#107—342no—
418#107—342no—
419#106—342no—
420#106—342no—
421#104—1716no—
422#103—342no—
423#103—342no—
424#102—342no—
425#102—342no—
426#95—344no—
427C17#2086411yescharacteristic, complemented, Sylow
428C24—341no—
429C24D4×D1711yescharacteristic
430#41—174no—
431#41—174no—
432#41—178no—
433#41—342no—
434#41—342no—
435#41—344no—
436#41—344no—
437#41—344no—
438#41—344no—
439#41—344no—
440C2×Q8—172no—
441C2×Q8—174no—
442C2×Q8—341no—
443C2×Q8—342no—
444C2×D4—172no—
445C2×D4—174no—
446C2×D4—342no—
447C2×D4—344no—
448C2×D4—342no—
449C2×D4—344no—
450C2×D4—341no—
451C2×D4—342no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—174no—
456C22×C4—348no—
457C22×C4—341no—
458C22×C4—342no—
459C22×C4—344no—
460C22×C4—342no—
461C22×C4—344no—
462C22×C4—342no—
463C22×C4—342no—
464C22×C4—341no—
465C22×C4—341no—
466C22×C4—1364no—
467C22×C4—1362no—
468C22×C4C2×C4×D1712yes—
469C22×C4C2×D6811yescharacteristic
470C22×C4C2×D6811yescharacteristic
471C22×C4#6325212yes—
472C22×C4D4×D1711yescharacteristic
473C22×C4D4×D1711yescharacteristic
474C22×C4D4×D1711yescharacteristic
475C22×C4#6325412yes—
476C22×C4Q8×D1712yes—
477C22×C4C23×D1711yescharacteristic
478M16—22no—
479M16—22no—
480M16—42no—
481M16—174no—
482M16—342no—
483M16—344no—
484C2×C8—22no—
485C2×C8—42no—
486C2×C8—342no—
487C2×C8—344no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—348no—
492C4⋊C4—342no—
493C4⋊C4—342no—
494C4⋊C4—342no—
495C4⋊C4—342no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—1364no—
501C4⋊C4C2×D6812yes—
502C4⋊C4C2×D6812yes—
503C22⋊C4—22no—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—348no—
507C22⋊C4—681no—
508C22⋊C4—681no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—348no—
513C42—681no—
514C42—681no—
515C42C2×C4×D1714yescomplemented
516C23—41no—
517C23—41no—
518C23—342no—
519C23—344no—
520C23—682no—
521C23—682no—
522C23#1058756712yes—
523C23#1058756811yescharacteristic
524C23#1058756811yescharacteristic
525C23#1058758212yes—
526C23C2×D4×D1711yescharacteristic
527Q8—174no—
528Q8—342no—
529Q8—344no—
530D4—174no—
531D4—344no—
532D4—344no—
533D4—348no—
534D4—342no—
535D4—344no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—342no—
544C2×C4—344no—
545C2×C4—344no—
546C2×C4—344no—
547C2×C4—348no—
548C2×C4—344no—
549C2×C4—684no—
550C2×C4—684no—
551C2×C4—682no—
552C2×C4—682no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—1368no—
556C2×C4—1362no—
557C2×C4C4×D6812yes—
558C2×C4C4×D6812yes—
559C2×C4#1058756712yes—
560C2×C4#1058756811yescharacteristic
561C2×C4#1058756811yescharacteristic
562C2×C4#1058756912yes—
563C2×C4#1058756912yes—
564C2×C4C4⋊C4×D1712yes—
565C2×C4C4⋊C4×D1712yes—
566C2×C4#1058758212yes—
567C2×C4#1058758312yes—
568C2×C4#1058758312yes—
569C2×C4C22×C4×D1711yescharacteristic
570C2×C4C22×D6811yescharacteristic
571C2×C4C2×#6325211yescharacteristic
572C2×C4C2×D4×D1711yescharacteristic
573C2×C4C2×#6325411yescharacteristic
574C2×C4C2×Q8×D1711yescharacteristic
575C8—42no—
576C8—344no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—42no—
584C22—42no—
585C22—344no—
586C22—688no—
587C22—682no—
588C22—682no—
589C22#1176268312yes—
590C22#1176268312yes—
591C22C2×#1058756711yescharacteristic
592C22C2×#1058756811yescharacteristic
593C22C2×#1058758211yescharacteristic
594C4—44no—
595C4—42no—
596C4—42no—
597C4—344no—
598C4—684no—
599C4—1361no—
600C4—1361no—
601C4#1176268312yes—
602C4#1176268312yes—
603C4C2×C4⋊C4×D1711yescharacteristic
604C4C2×C4×D6811yescharacteristic
605C4C2×#1058758311yescharacteristic
606C4C2×#1058756911yescharacteristic
607C2—21no—
608C2—22no—
609C2—21no—
610C2—22no—
611C2—684no—
612C2C2×#1176268311yescharacteristic
613C2#1453814311yescharacteristic
614C2#1453815011yescharacteristic
subgroupG/H
centerC22C2×#10587568
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1733C2
socleC22×C17C2×#120
prime radicalC22×#63252C22
cosocleC22×C4×C17C24
hypercenter#1733D17
hypocenterC17#20864
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
2. #14537373C2semidirect
5. #14531407C2semidirect
6. #14529018C2semidirect
7. #14529011C2semidirect
8. C2×#11762577C2semidirect
9. C17×#1733C2semidirect
23. #11762852C22semidirect
24. #11762852C22semidirect
28. #11762827C4semidirect
37. #11762679C22semidirect
38. #11762675C22semidirect
39. #11762577C22semidirect
43. C17×#339C22semidirect
67. C17×#117C2×C4semidirect
97. C2×#63223C2×C4semidirect
119. #10587476C2×C4semidirect
153. #63252C4⋊C4semidirect
158. C2×D68C4⋊C4semidirect
168. C2×Dic34C4⋊C4semidirect
186. C42×C17C22×C4semidirect
205. #63223C22×C4semidirect
278. #1733D17semidirect
342. #339C2×D17semidirect
391. #117C4×D17semidirect
427. C17#20864semidirect
515. C42C2×C4×D17semidirect

Maximal subgroups

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

subgroupindex
C2×#117632892
#145373732
C2×#117628272
#145373042
#145314072
#145290182
#145290112
C2×#117625772
C17×#17332
#2086417