Finite Group Databasebeta

← #17004385

Subgroups — #17004385

Order 4352.

648 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538835C211yesmaximal, characteristic
2#14537932C211yesmaximal, characteristic, complemented
3#14537805C211yesmaximal, characteristic, complemented
4#14527645C211yesmaximal, characteristic, complemented
5#14527644C211yesmaximal, characteristic, complemented
6#14527621C211yesmaximal, characteristic, complemented
7#14527612C211yesmaximal, characteristic, complemented
8#14527609C211yesmaximal, characteristic, complemented
9C8×#63230C211yesmaximal, characteristic, complemented
10C8×#63228C211yesmaximal, characteristic, complemented
11#14527574C211yesmaximal, characteristic, complemented
12#14527207C211yesmaximal, characteristic, complemented
13#14527191C211yesmaximal, characteristic, complemented
14#14527085C211yesmaximal, characteristic, complemented
15C17×SD16⋊C8C211yesmaximal, characteristic, complemented
16#11763346C2211yescharacteristic
17C4×D4×D17C2211yescharacteristic
18#11763143C411yescharacteristic
19#11763129C411yescharacteristic
20#11763108C411yescharacteristic
21#11763107C411yescharacteristic
22#11763027C411yescharacteristic
23#11763026C411yescharacteristic
24#11762958C411yescharacteristic
25#11762944C411yescharacteristic
26#11762940C2211yescharacteristic, complemented
27C4×#63230C2211yescharacteristic
28C4×D136C2211yescharacteristic
29C4×#63228C2211yescharacteristic
30#11762888—21no—
31#11762886C2211yescharacteristic, complemented
32#11762879—21no—
33C8×D68C2211yescharacteristic
34C8×D68C2211yescharacteristic, complemented
35D4×#3622C2211yescharacteristic
36#11762874C2211yescharacteristic
37C8×C22⋊D17—21no—
38#11762872—21no—
39#11762868—21no—
40#11762861C2211yescharacteristic
41#11762860C2211yescharacteristic
42#11762860C2211yescharacteristic
43#11762860C2211yescharacteristic
44#11762834—21no—
45#11762831C2211yescharacteristic
46C2×#10587491—21no—
47C22⋊C8×D17—21no—
48#11762561C2211yescharacteristic, complemented
49#11762560C2211yescharacteristic, complemented
50#11762559C2211yescharacteristic, complemented
51Q8⋊#3622C2211yescharacteristic, complemented
52#11762538C2211yescharacteristic
53#11762538C2211yescharacteristic, complemented
54D4⋊#3622C2211yescharacteristic
55#11762536C2211yescharacteristic
56#11762536C2211yescharacteristic, complemented
57#11762531C2211yescharacteristic, complemented
58#11762530C2211yescharacteristic, complemented
59C8×#3622C2211yescharacteristic, complemented
60C8×Q8×C17C2211yescharacteristic, complemented
61C4×SD16×C17C2211yescharacteristic
62C8×D4×C17C2211yescharacteristic
63C17×#335C2211yescharacteristic, complemented
64C17×Q8⋊C8C2211yescharacteristic, complemented
65C17×D4⋊C8C2211yescharacteristic
66C17×#322C2211yescharacteristic, complemented
67C2×D4×D17C2×C411yescharacteristic
68C22×C4×D17—21no—
69C2×SD16×C17C2×C411yescharacteristic
70C22×C8×C17—21no—
71C4×Q8×C17C2311yescharacteristic
72C4×D4×C17C2311yescharacteristic
73#10587633C2×C411yescharacteristic
74#10587632C2×C411yescharacteristic
75C2×#63230C2×C411yescharacteristic
76D4×Dic17C2×C411yescharacteristic
77C2×#63228C2×C411yescharacteristic
78C4×C22⋊D17—21no—
79C22×#3622—21no—
80C2×D136C2×C411yescharacteristic
81C2×#63218—21no—
82C2×#63218—21no—
83C2×C8×D17—21no—
84C2×C8×D17—21no—
85C2×C8×D17—21no—
86#10587584—21no—
87#10587581C2×C411yescharacteristic
88#10587580—21no—
89#10587579C2×C411yescharacteristic
90C4⋊C4×D17C2×C411yescharacteristic
91#10587576C2×C411yescharacteristic
92#10587567—21no—
93C22⋊C4×D17—21no—
94C4×D68—21no—
95C4×D68C2311yescharacteristic
96C4×D68C2311yescharacteristic
97#10587558—21no—
98C42×D17C2311yescharacteristic
99C17×#106C2×C411yescharacteristic
100C17×C4⋊C8—21no—
101C17×C4⋊C8C2311yescharacteristic
102C17×C4⋊C8C2311yescharacteristic
103C17×#103C2×C411yescharacteristic
104C17×#102C2×C411yescharacteristic
105C17×C22⋊C8—21no—
106C4×C8×C17—21no—
107C4×C8×C17C2311yescharacteristic
108C4×C8×C17C2311yescharacteristic
109C4×C8×C17C2311yescharacteristic
110#10587506C2×C411yescharacteristic
111#10587503C2×C411yescharacteristic
112C22⋊#3622—21no—
113#10587492C2×C411yescharacteristic
114#10587492C2×C411yescharacteristic
115#10587491—21no—
116#10587491—21no—
117#10587491—21no—
118#10587491—21no—
119#10587491—21no—
120#10587491—21no—
121#10587491C2×C411yescharacteristic
122#10587491C2×C411yescharacteristic
123#10587491D411yescharacteristic
124#10587491D411yescharacteristic
125#10587489C2×C411yescharacteristic
126#10587485—21no—
127#10587485—21no—
128#10587485C2×C411yescharacteristic
129#10587485C2×C411yescharacteristic
130#10587485D411yescharacteristic
131#10587485D411yescharacteristic
132C8×Dic17—21no—
133#10587480C2×C411yescharacteristic
134#10587480C2×C411yescharacteristic
135#10587479C2×C411yescharacteristic
136#10587478C2×C411yescharacteristic
137C4⋊#3622—21no—
138C4⋊#3622C2311yescharacteristic
139C4⋊#3622C2311yescharacteristic
140#10587474—21no—
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C23×D17—21no—
145C2×Q8×C17C22×C411yescharacteristic
146C2×D4×C17C22×C411yescharacteristic
147C22×C4×C17—21no—
148C2×C22⋊D17—21no—
149C22×Dic17—21no—
150D4×D17—22no—
151D4×D17—22no—
152C2×D68—21no—
153C2×D68C22×C411yescharacteristic
154C2×D68C22×C411yescharacteristic
155C2×C4×D17—21no—
156C2×C4×D17—21no—
157C2×C4×D17—21no—
158C2×C4×D17—21no—
159C2×C4×D17—81no—
160C2×C4×D17—81no—
161C2×C4×D17C22×C411yescharacteristic
162C2×C4×D17C2×D411yescharacteristic
163C2×C4×D17#4111yescharacteristic
164SD16×C17—22no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—81no—
169C2×C8×C17C22×C411yescharacteristic
170C2×C8×C17C22×C411yescharacteristic
171C2×C8×C17C2×D411yescharacteristic
172C2×C8×C17C2×D411yescharacteristic
173C4⋊C4×C17—21no—
174C4⋊C4×C17C22×C411yescharacteristic
175C4⋊C4×C17C22×C411yescharacteristic
176C22⋊C4×C17—21no—
177C42×C17—21no—
178C42×C17C2411yescharacteristic
179C22⋊Dic17—21no—
180#63230—22no—
181#63228—22no—
182#63227—21no—
183#63227—21no—
184#63227—21no—
185#63227—21no—
186#63227—21no—
187#63227—21no—
188#63227—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17C22×C411yescharacteristic
191C4⋊Dic17C22×C411yescharacteristic
192#63225—21no—
193#63225—21no—
194#63225—21no—
195C4×Dic17C22×C411yescharacteristic
196C4×Dic17C2×D411yescharacteristic
197C4×Dic17#4111yescharacteristic
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—81no—
202C2×#3622C22×C411yescharacteristic
203C2×#3622C22×C411yescharacteristic
204C2×#3622C2×D411yescharacteristic
205C2×#3622C2×D411yescharacteristic
206D136—22no—
207#63218—81no—
208#63218—81no—
209C8×D17—81no—
210C8×D17—81no—
211C8×D17—81no—
212#17918—171nomaximal, Sylow
213C23×C17—21no—
214C22×D17—21no—
215C22×D17—24no—
216C22×D17—21no—
217C22×D17—81no—
218C22×D17—81no—
219C22×D17C4×D411yescharacteristic
220Q8×C17—21no—
221Q8×C17—21no—
222D4×C17—21no—
223D4×C17#13112yes—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17C23×C411yescharacteristic
230C2×C4×C17C22×D411yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C22⋊D17—24no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—81no—
236C2×Dic17C4×D411yescharacteristic
237C2×Dic17C4×D411yescharacteristic
238C2×Dic17C4×D411yescharacteristic
239D68—21no—
240D68—21no—
241D68—21no—
242D68—81no—
243D68#13112yes—
244C4×D17—21no—
245C4×D17—21no—
246C4×D17—21no—
247C4×D17—21no—
248C4×D17—41no—
249C4×D17—41no—
250C4×D17—41no—
251C4×D17—41no—
252C4×D17—41no—
253C4×D17—81no—
254C4×D17—81no—
255C8×C17—21no—
256C8×C17—21no—
257C8×C17—21no—
258C8×C17—21no—
259C8×C17—41no—
260C8×C17—41no—
261C8×C17—41no—
262#3622—21no—
263#3622—21no—
264#3622—21no—
265#3622—21no—
266#3622—41no—
267#3622—41no—
268#3622—41no—
269#3105—171no—
270#2993—171no—
271#2972—171no—
272#1586—171no—
273#1585—171no—
274#1582—171no—
275#1581—171no—
276#1580—171no—
277SD16⋊C8D1711yescharacteristic, complemented
278C8×SD16—171no—
279C8×D8—171no—
280#1574—171no—
281#1486—171no—
282#1482—171no—
283#1454—171no—
284C22×C17—22no—
285C22×C17—41no—
286C22×C17—41no—
287C22×C17C2×C4×D411yescharacteristic
288C2×D17—21no—
289C2×D17—22no—
290C2×D17—21no—
291C2×D17—21no—
292C2×D17—41no—
293C2×D17—41no—
294C2×D17—41no—
295C2×D17—41no—
296C2×D17—41no—
297C2×D17—84no—
298C4×C17—21no—
299C4×C17—41no—
300C4×C17—41no—
301C4×C17—41no—
302C4×C17C2×#13111yescharacteristic
303C4×C17C2×#13111yescharacteristic
304C4×C17#57611yescharacteristic
305C4×C17#57711yescharacteristic
306Dic17—21no—
307Dic17—21no—
308Dic17—21no—
309Dic17—41no—
310Dic17—41no—
311#529—171no—
312C2×C4×D4—171no—
313#484—171no—
314#481—171no—
315#478—171no—
316#478—171no—
317#463—171no—
318#463—171no—
319#450—171no—
320#447—171no—
321#446—171no—
322C8×Q8C2×D1711yescharacteristic, complemented
323C4×SD16—171no—
324C4×SD16C2×D1711yescharacteristic
325C4×D8—171no—
326C4×D8—171no—
327#436—171no—
328#436—341no—
329#435—341no—
330C8×D4—171no—
331C8×D4—171no—
332C8×D4—171no—
333C8×D4—171no—
334C8×D4—341no—
335C8×D4—341no—
336C8×D4C2×D1711yescharacteristic
337#433—341no—
338#431—171no—
339#431—171no—
340#431—171no—
341#431—171no—
342#424—341no—
343#423—171no—
344C2×C22⋊C8—341no—
345C2×C22⋊C8—341no—
346#335—171no—
347#335C2×D1711yescharacteristic, complemented
348#334—171no—
349#334—171no—
350Q8⋊C8—171no—
351Q8⋊C8C2×D1711yescharacteristic, complemented
352D4⋊C8—171no—
353D4⋊C8—171no—
354D4⋊C8—171no—
355D4⋊C8—171no—
356D4⋊C8—171no—
357D4⋊C8C2×D1711yescharacteristic
358#322—171no—
359#322—171no—
360#322C2×D1711yescharacteristic, complemented
361C82—171no—
362C2×C17—42no—
363C2×C17#292911yescharacteristic
364C2×C17#294611yescharacteristic
365C2×C17#295711yescharacteristic
366D17—41no—
367D17—42no—
368D17—81no—
369C22×D4—171no—
370C23×C4—341no—
371C2×SD16—171no—
372C2×SD16C4×D1711yescharacteristic
373C2×D8—171no—
374C2×D8—171no—
375C2×M16—341no—
376C2×M16—341no—
377C22×C8—21no—
378C22×C8—341no—
379C22×C8—341no—
380C22×C8—341no—
381C22×C8—341no—
382#126—341no—
383#125—171no—
384#124—171no—
385#123—341no—
386C22⋊Q8—171no—
387#121—171no—
388C4×Q8C22×D1711yescharacteristic
389C4×D4—171no—
390C4×D4—171no—
391C4×D4—171no—
392C4×D4—171no—
393C4×D4—341no—
394C4×D4—341no—
395C4×D4—341no—
396C4×D4C22×D1711yescharacteristic
397#117—341no—
398C2×C4⋊C4—171no—
399C2×C22⋊C4—341no—
400C2×C42—171no—
401#107—171no—
402#107—171no—
403#106—171no—
404#106C4×D1711yescharacteristic
405C4⋊C8—21no—
406C4⋊C8—171no—
407C4⋊C8—171no—
408C4⋊C8—171no—
409C4⋊C8—171no—
410C4⋊C8—171no—
411C4⋊C8—171no—
412C4⋊C8—341no—
413C4⋊C8—341no—
414C4⋊C8—341no—
415C4⋊C8C22×D1711yescharacteristic
416C4⋊C8C22×D1711yescharacteristic
417#103—171no—
418#103C4×D1711yescharacteristic
419#102—171no—
420#102—171no—
421#102—171no—
422#102—171no—
423#102—171no—
424#102C4×D1711yescharacteristic
425C22⋊C8—21no—
426C22⋊C8—171no—
427C22⋊C8—171no—
428C22⋊C8—171no—
429C22⋊C8—171no—
430C22⋊C8—341no—
431C22⋊C8—341no—
432C22⋊C8—341no—
433C22⋊C8—341no—
434C22⋊C8—341no—
435C22⋊C8—341no—
436C22⋊C8—341no—
437#97—341no—
438C4×C8—21no—
439C4×C8—171no—
440C4×C8—171no—
441C4×C8—171no—
442C4×C8—341no—
443C4×C8C22×D1711yescharacteristic
444C4×C8C22×D1711yescharacteristic
445C4×C8C22×D1711yescharacteristic
446C17#1791811yescharacteristic, complemented, Sylow
447C24—341no—
448C2×Q8C2×C4×D1711yescharacteristic
449C2×D4—171no—
450C2×D4—171no—
451C2×D4—341no—
452C2×D4—341no—
453C2×D4—342no—
454C2×D4—342no—
455C2×D4C2×C4×D1711yescharacteristic
456C22×C4—21no—
457C22×C4—171no—
458C22×C4—171no—
459C22×C4—171no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—341no—
464C22×C4—341no—
465C22×C4—1361no—
466C22×C4—1361no—
467SD16—22no—
468SD16—342no—
469D8—342no—
470D8—342no—
471M16—1361no—
472M16—1361no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—81no—
477C2×C8—171no—
478C2×C8—171no—
479C2×C8—171no—
480C2×C8—171no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—1361no—
485C2×C8—1361no—
486C2×C8—1361no—
487C2×C8—1361no—
488C2×C8C2×C4×D1711yescharacteristic
489C2×C8C2×C4×D1711yescharacteristic
490C2×C8D4×D1711yescharacteristic
491C2×C8D4×D1711yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—171no—
494C4⋊C4—171no—
495C4⋊C4—341no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4C2×C4×D1711yescharacteristic
500C4⋊C4C2×C4×D1711yescharacteristic
501C22⋊C4—21no—
502C22⋊C4—341no—
503C22⋊C4—341no—
504C22⋊C4—341no—
505C22⋊C4—341no—
506C22⋊C4—341no—
507C22⋊C4—341no—
508C22⋊C4—341no—
509C22⋊C4—341no—
510C42—21no—
511C42—171no—
512C42—171no—
513C42—171no—
514C42C23×D1711yescharacteristic
515C23—21no—
516C23—171no—
517C23—344no—
518C23—341no—
519C23—341no—
520C23—1361no—
521C23—1361no—
522Q8—21no—
523Q8—21no—
524D4—21no—
525D4—172no—
526D4—344no—
527D4—341no—
528D4—341no—
529D4—341no—
530D4—1361no—
531D4#1058758712yes—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—171no—
538C2×C4—171no—
539C2×C4—171no—
540C2×C4—341no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—1361no—
552C2×C4—1361no—
553C2×C4—1361no—
554C2×C4C22×C4×D1711yescharacteristic
555C2×C4C2×D4×D1711yescharacteristic
556C2×C4#41×D1711yescharacteristic
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—341no—
565C8—341no—
566C8—341no—
567C8—341no—
568C8—681no—
569C8—681no—
570C8—681no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—341no—
575C22—342no—
576C22—341no—
577C22—341no—
578C22—681no—
579C22—681no—
580C22—681no—
581C22—681no—
582C22—681no—
583C22—1364no—
584C22C4×D4×D1711yescharacteristic
585C4—21no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—341no—
590C4—341no—
591C4—341no—
592C4—681no—
593C4—681no—
594C4#131×D1711yescharacteristic
595C4C2×#1058758711yescharacteristic
596C4#1176361311yescharacteristic
597C4#1176361811yescharacteristic
598C2—42no—
599C2—681no—
600C2—682no—
601C2—1361no—
602C2#1453747311yescharacteristic
603C2#1453761511yescharacteristic
604C2#1453769811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×SD16⋊C8C2
socleC22×C17C2×C4×D4
prime radical#11762944C4
cosocleC42×C17C24
hypercenterSD16⋊C8D17
hypocenterC17#17918
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
2. #14537932C2semidirect
3. #14537805C2semidirect
4. #14527645C2semidirect
5. #14527644C2semidirect
6. #14527621C2semidirect
7. #14527612C2semidirect
8. #14527609C2semidirect
9. C8×#63230C2semidirect
10. C8×#63228C2semidirect
11. #14527574C2semidirect
12. #14527207C2semidirect
13. #14527191C2semidirect
14. #14527085C2semidirect
15. C17×SD16⋊C8C2semidirect
26. #11762940C22semidirect
31. #11762886C22semidirect
34. C8×D68C22semidirect
48. #11762561C22semidirect
49. #11762560C22semidirect
50. #11762559C22semidirect
51. Q8⋊#3622C22semidirect
53. #11762538C22semidirect
56. #11762536C22semidirect
57. #11762531C22semidirect
58. #11762530C22semidirect
59. C8×#3622C22semidirect
60. C8×Q8×C17C22semidirect
63. C17×#335C22semidirect
64. C17×Q8⋊C8C22semidirect
66. C17×#322C22semidirect
277. SD16⋊C8D17semidirect
322. C8×Q8C2×D17semidirect
347. #335C2×D17semidirect
351. Q8⋊C8C2×D17semidirect
360. #322C2×D17semidirect
446. C17#17918semidirect

Maximal subgroups

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

subgroupindex
#145388352
#145379322
#145378052
#145276452
#145276442
#145276212
#145276122
#145276092
C8×#632302
C8×#632282
#145275742
#145272072
#145271912
#145270852
C17×SD16⋊C82
#1791817