Finite Group Databasebeta

← #17190909

Subgroups — #17190909

Order 4352.

712 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763245C211yesmaximal, characteristic, complemented
2C2×#11763244C211yesmaximal, characteristic, complemented
3#14540471C211yesmaximal, characteristic, complemented
4#14531043C211yesmaximal, characteristic, complemented
5#14530899C211yesmaximal, characteristic, complemented
6#14530895C211yesmaximal, characteristic, complemented
7#14530839C211yesmaximal, characteristic, complemented
8#14530836C211yesmaximal, characteristic, complemented
9#14530724C211yesmaximal, characteristic, complemented
10#14530722C211yesmaximal, characteristic, complemented
11#14530696C211yesmaximal, characteristic, complemented
12#14530695C211yesmaximal, characteristic, complemented
13C4×#10587497C211yesmaximal, characteristic, complemented
14#14530616C211yesmaximal, characteristic, complemented
15C17×#3280C211yesmaximal, characteristic, complemented
16C2×#10587590C2211yescharacteristic
17C22×D136C2211yescharacteristic, complemented
18#11763455—21no—
19#11763245—24no—
20#11763244—24no—
21C2×#10587500C2211yescharacteristic, complemented
22C2×#10587498C2211yescharacteristic, complemented
23C2×#10587498C2211yescharacteristic, complemented
24C2×#10587499C2211yescharacteristic, complemented
25C2×#10587499C2211yescharacteristic, complemented
26C2×#10587497C2211yescharacteristic
27C2×#10587497C2211yescharacteristic
28C2×#10587497C2211yescharacteristic
29C2×#10587497C2211yescharacteristic
30C2×#10587483C2211yescharacteristic, complemented
31C2×#10587483C2211yescharacteristic, complemented
32#11763193C2211yescharacteristic, complemented
33#11763050C2211yescharacteristic, complemented
34#11763045C2211yescharacteristic
35#11763025—21no—
36#11762962—21no—
37#11762956—21no—
38C4×D136C2211yescharacteristic, complemented
39#11762634C2211yescharacteristic, complemented
40C4⋊#63214C2211yescharacteristic, complemented
41#11762613C2211yescharacteristic, complemented
42#11762612C2211yescharacteristic, complemented
43#11762610C2211yescharacteristic, complemented
44#11762610C2211yescharacteristic, complemented
45#11762609C2211yescharacteristic, complemented
46#11762609C2211yescharacteristic, complemented
47C22⋊#63214C2211yescharacteristic, complemented
48C22⋊#63214C2211yescharacteristic, complemented
49C4×#63214C2211yescharacteristic, complemented
50C2×C17×#135C2211yescharacteristic
51C22×D8×C17C2211yescharacteristic, complemented
52C17×#546—21no—
53C17×#495C2211yescharacteristic, complemented
54C17×#467C2211yescharacteristic, complemented
55C17×#466C2211yescharacteristic
56C17×#463—21no—
57C17×#450—21no—
58C17×#449—21no—
59C4×D8×C17C2211yescharacteristic, complemented
60C8×D4×C17C2211yescharacteristic
61C2×#41×C17—21no—
62C22×D4×C17—21no—
63C2×#63252—21no—
64C22×D68—21no—
65C17×#135—22no—
66C17×#135—41no—
67C2×Q16×C17C2311yescharacteristic
68C2×SD16×C17—21no—
69C2×SD16×C17—21no—
70C2×D8×C17—22no—
71C2×D8×C17—21no—
72C2×D8×C17—41no—
73C2×D8×C17C2311yescharacteristic
74C2×D8×C17C2311yescharacteristic
75C22×C8×C17C2311yescharacteristic
76C22×C8×C17C2311yescharacteristic
77C17×#124—21no—
78C17×#123—41no—
79C17×C22⋊Q8—21no—
80C17×#121—21no—
81C17×#121—41no—
82C17×#120—41no—
83C4×D4×C17—21no—
84C4×D4×C17D411yescharacteristic
85C2×C22⋊C4×C17—41no—
86#10587623—41no—
87#10587616—21no—
88#10587610—21no—
89C2×Dic68C2311yescharacteristic
90#10587590—22no—
91#10587590—41no—
92C2×D136—22no—
93C2×D136—21no—
94C2×D136—41no—
95C2×D136C2311yescharacteristic
96C2×D136C2311yescharacteristic
97C2×#63219—21no—
98C2×#63219—21no—
99#10587580—41no—
100#10587570—41no—
101C22⋊C4×D17—41no—
102#10587561—21no—
103C4×D68—21no—
104C17×#107C2311yescharacteristic
105C17×C4⋊C8D411yescharacteristic
106C17×#103—21no—
107C17×#102—21no—
108C17×#102—21no—
109C17×C22⋊C8D411yescharacteristic
110C17×C22⋊C8D411yescharacteristic
111C4×C8×C17C2311yescharacteristic
112#10587500—41no—
113#10587499—22no—
114#10587499—22no—
115#10587498—22no—
116#10587498—22no—
117#10587497—22no—
118#10587497—22no—
119#10587497—41no—
120#10587497D414yescomplemented
121#10587492—21no—
122#10587492—21no—
123#10587489C2311yescharacteristic
124#10587487—21no—
125#10587483—22no—
126#10587483—22no—
127C2×#63214C2311yescharacteristic
128C2×#63214C2311yescharacteristic
129C2×#63214C2311yescharacteristic
130C2×#63214C2311yescharacteristic
131C24×C17—41no—
132C23×D17—41no—
133#41×C17—42no—
134#41×C17—81no—
135C2×Q8×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17—42no—
139C2×D4×C17—41no—
140C2×D4×C17—41no—
141C2×D4×C17—41no—
142C2×D4×C17—81no—
143C2×D4×C17D811yescharacteristic
144C22×C4×C17—41no—
145C22×C4×C17—41no—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C2×C22⋊D17—41no—
149C2×C22⋊D17—41no—
150#63252—42no—
151#63252—81no—
152C2×D68—21no—
153C2×D68—21no—
154C2×D68—42no—
155C2×D68—41no—
156C2×D68—81no—
157C2×C4×D17—41no—
158C2×C4×D17—41no—
159C2×Dic34—21no—
160Q16×C17—21no—
161Q16×C17—21no—
162SD16×C17—42no—
163SD16×C17—81no—
164D8×C17—21no—
165D8×C17—21no—
166D8×C17—21no—
167D8×C17—42no—
168D8×C17C2×D412yes—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17C2×D411yescharacteristic
174C2×C8×C17C2×D411yescharacteristic
175C2×C8×C17C2×D411yescharacteristic
176C2×C8×C17C2411yescharacteristic
177C4⋊C4×C17—21no—
178C4⋊C4×C17—41no—
179C4⋊C4×C17D811yescharacteristic
180C22⋊C4×C17—41no—
181C22⋊C4×C17—41no—
182C22⋊C4×C17—41no—
183C22⋊C4×C17—41no—
184C22⋊C4×C17—41no—
185C22⋊C4×C17D811yescharacteristic
186C22⋊C4×C17D811yescharacteristic
187C42×C17C2×D411yescharacteristic
188C22⋊Dic17—41no—
189#63227—41no—
190#63227—41no—
191#63227—41no—
192#63227—41no—
193C4⋊Dic17—21no—
194#63225—41no—
195Dic68—21no—
196Dic68—21no—
197D136—21no—
198D136—21no—
199D136—21no—
200D136—42no—
201D136C2×D412yes—
202#63219—42no—
203#63219—81no—
204#63214—21no—
205#63214—21no—
206#63214—21no—
207#63214C2×D412yes—
208#33839—171nomaximal, Sylow
209C23×C17—41no—
210C23×C17—41no—
211C23×C17—82no—
212C23×C17—82no—
213C23×C17—161no—
214C23×C17C2×D811yescharacteristic
215C23×C17C2×D811yescharacteristic
216C22×D17—41no—
217C22×D17—41no—
218C22×D17—82no—
219C22×D17—82no—
220C22×D17—161no—
221Q8×C17—41no—
222Q8×C17—41no—
223D4×C17—22no—
224D4×C17—41no—
225D4×C17—41no—
226D4×C17—41no—
227D4×C17—41no—
228D4×C17—82no—
229D4×C17—82no—
230D4×C17—161no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—82no—
238C2×C4×C17—161no—
239C2×C4×C17C2×D811yescharacteristic
240C2×C4×C17C2×D811yescharacteristic
241C2×C4×C17C2×D811yescharacteristic
242C2×C4×C17C2×D811yescharacteristic
243C2×C4×C17C22×D411yescharacteristic
244C22⋊D17—82no—
245C22⋊D17—161no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248D68—22no—
249D68—41no—
250D68—41no—
251D68—41no—
252D68—82no—
253C4×D17—82no—
254C4×D17—161no—
255Dic34—41no—
256Dic34—41no—
257C8×C17—21no—
258C8×C17—21no—
259C8×C17—21no—
260C8×C17C22×D411yescharacteristic
261C8×C17H(5,𝔽2)11yescharacteristic
262C2×#509—171no—
263C2×#509—171no—
264#3280—171no—
265#3280D1711yescharacteristic, complemented
266#2249—171no—
267#2220—171no—
268#2220—171no—
269#2210—171no—
270#2210—171no—
271#2189—171no—
272#2189—171no—
273#2184—171no—
274#2184—171no—
275C4×D16—171no—
276#2169—171no—
277C22×C17—21no—
278C22×C17—21no—
279C22×C17—21no—
280C22×C17—21no—
281C22×C17—21no—
282C22×C17—21no—
283C22×C17—42no—
284C22×C17—81no—
285C22×C17—81no—
286C22×C17—81no—
287C22×C17—81no—
288C22×C17—81no—
289C22×C17—164no—
290C22×C17C22×D811yescharacteristic
291C2×D17—42no—
292C2×D17—81no—
293C2×D17—81no—
294C2×D17—81no—
295C2×D17—81no—
296C2×D17—81no—
297C2×D17—164no—
298C4×C17—21no—
299C4×C17—21no—
300C4×C17—21no—
301C4×C17—21no—
302C4×C17—81no—
303C4×C17—81no—
304C4×C17D4211yescharacteristic
305C4×C17#57611yescharacteristic
306Dic17—81no—
307Dic17—81no—
308C2×#135—171no—
309C2×#135C2×D1711yescharacteristic
310C22×D8—171no—
311C22×D8C2×D1711yescharacteristic, complemented
312#546—21no—
313#546—341no—
314#509—344no—
315#509—344no—
316C2×Q32—171no—
317C2×SD32—171no—
318C2×SD32—171no—
319C2×SD32—171no—
320C2×SD32—171no—
321C2×D16—171no—
322C2×D16—171no—
323C2×D16—171no—
324C2×D16—171no—
325C2×M32—171no—
326C2×M32—171no—
327#495—171no—
328#495C2×D1711yescharacteristic, complemented
329#467—171no—
330#467C2×D1711yescharacteristic, complemented
331#466—171no—
332#466C2×D1711yescharacteristic
333#463—21no—
334#463—341no—
335#450—21no—
336#450—341no—
337#449—21no—
338#449—341no—
339C4×D8—171no—
340C4×D8C2×D1711yescharacteristic, complemented
341C8×D4C2×D1711yescharacteristic
342#366—171no—
343C4⋊C16—171no—
344#358—171no—
345#358—171no—
346#357—171no—
347#357—171no—
348#357—171no—
349#357—171no—
350C22⋊C16—171no—
351C22⋊C16—171no—
352C4×C16—171no—
353C2×C17—41no—
354C2×C17—41no—
355C2×C17—82no—
356C2×C17—161no—
357C2×C17D4×D811yescharacteristic
358C2×C17#341411yescharacteristic
359C2×C17#341511yescharacteristic
360D17—82no—
361D17—161no—
362C2×#41—21no—
363C2×#41—341no—
364C22×D4—21no—
365C22×D4—341no—
366#135—22no—
367#135—41no—
368#135—342no—
369#135—681no—
370C2×Q16—171no—
371C2×Q16C22×D1711yescharacteristic
372C2×SD16—21no—
373C2×SD16—21no—
374C2×SD16—341no—
375C2×SD16—341no—
376C2×D8—22no—
377C2×D8—21no—
378C2×D8—41no—
379C2×D8—171no—
380C2×D8—171no—
381C2×D8—342no—
382C2×D8—341no—
383C2×D8—681no—
384C2×D8C22×D1711yescharacteristic
385C2×D8C22×D1711yescharacteristic
386C22×C8C22×D1711yescharacteristic
387C22×C8C22×D1711yescharacteristic
388#124—21no—
389#124—341no—
390#123—41no—
391#123—681no—
392C22⋊Q8—21no—
393C22⋊Q8—341no—
394#121—21no—
395#121—41no—
396#121—341no—
397#121—681no—
398#120—41no—
399#120—681no—
400C4×D4—21no—
401C4×D4—341no—
402C4×D4C22⋊D1711yescharacteristic
403C2×C22⋊C4—41no—
404C2×C22⋊C4—681no—
405Q32—681no—
406SD32—342no—
407SD32—342no—
408SD32—342no—
409SD32—342no—
410D16—174no—
411D16—342no—
412D16—342no—
413D16—681no—
414M32—342no—
415M32—342no—
416C2×C16—171no—
417C2×C16—171no—
418C2×C16—171no—
419C2×C16—171no—
420#107—171no—
421#107C22×D1711yescharacteristic
422C4⋊C8C22⋊D1711yescharacteristic
423#103—21no—
424#103—341no—
425#102—21no—
426#102—21no—
427#102—341no—
428#102—341no—
429C22⋊C8C22⋊D1711yescharacteristic
430C22⋊C8C22⋊D1711yescharacteristic
431C4×C8C22×D1711yescharacteristic
432C17#3383911yescharacteristic, complemented, Sylow
433C24—41no—
434C24—681no—
435#41—42no—
436#41—81no—
437#41—682no—
438#41—1361no—
439C2×Q8—21no—
440C2×Q8—341no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—42no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—41no—
447C2×D4—81no—
448C2×D4—341no—
449C2×D4—341no—
450C2×D4—682no—
451C2×D4—681no—
452C2×D4—681no—
453C2×D4—681no—
454C2×D4—1361no—
455C2×D4#6322811yescharacteristic
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—681no—
459C22×C4—681no—
460C22×C4C2×C22⋊D1711yescharacteristic
461C22×C4C2×C22⋊D1711yescharacteristic
462Q16—21no—
463Q16—21no—
464Q16—341no—
465Q16—341no—
466SD16—42no—
467SD16—81no—
468SD16—682no—
469SD16—1361no—
470D8—21no—
471D8—21no—
472D8—21no—
473D8—42no—
474D8—172no—
475D8—341no—
476D8—341no—
477D8—341no—
478D8—682no—
479D8D4×D1712yes—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8C2×C22⋊D1711yescharacteristic
485C2×C8C2×C22⋊D1711yescharacteristic
486C2×C8C2×C22⋊D1711yescharacteristic
487C2×C8C23×D1711yescharacteristic
488C4⋊C4—21no—
489C4⋊C4—41no—
490C4⋊C4—341no—
491C4⋊C4—681no—
492C4⋊C4#6322811yescharacteristic
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—681no—
499C22⋊C4—681no—
500C22⋊C4—681no—
501C22⋊C4—681no—
502C22⋊C4—681no—
503C22⋊C4#6322811yescharacteristic
504C22⋊C4#6322811yescharacteristic
505C42C2×C22⋊D1711yescharacteristic
506C16—172no—
507C16—341no—
508C16—341no—
509C16—341no—
510C23—41no—
511C23—41no—
512C23—82no—
513C23—82no—
514C23—161no—
515C23—2721no—
516C23—681no—
517C23—681no—
518C23—1362no—
519C23—1362no—
520C23C2×#6322811yescharacteristic
521C23C2×#6322811yescharacteristic
522Q8—41no—
523Q8—41no—
524Q8—681no—
525Q8—681no—
526D4—22no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—82no—
532D4—82no—
533D4—161no—
534D4—2721no—
535D4—342no—
536D4—681no—
537D4—681no—
538D4—681no—
539D4—1362no—
540D4—1362no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—82no—
548C2×C4—161no—
549C2×C4—2721no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—1362no—
553C2×C4C2×#6322811yescharacteristic
554C2×C4C2×#6322811yescharacteristic
555C2×C4C2×#6322811yescharacteristic
556C2×C4C2×#6322811yescharacteristic
557C2×C4C22×C22⋊D1711yescharacteristic
558C8—21no—
559C8—21no—
560C8—21no—
561C8C2×D4×D1711yescharacteristic
562C8#1058769111yescharacteristic
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—42no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—164no—
576C22—2724no—
577C22—682no—
578C22—1361no—
579C22—1361no—
580C22—1361no—
581C22—1361no—
582C22—1361no—
583C22C22×#6322811yescharacteristic
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—81no—
589C4—81no—
590C4—1361no—
591C4—1361no—
592C4D4×C22⋊D1711yescharacteristic
593C4#1176361211yescharacteristic
594C2—41no—
595C2—41no—
596C2—82no—
597C2—161no—
598C2—2721no—
599C2—1362no—
600C2D4×#6322811yescharacteristic
601C2#1454191811yescharacteristic
602C2#1454192311yescharacteristic
subgroupG/H
centerC22C22×#63228
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#3280C2
socleC22×C17C22×D8
hypercenter#3280D17
hypocenterC17#33839
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. C2×#11763245C2semidirect
2. C2×#11763244C2semidirect
3. #14540471C2semidirect
4. #14531043C2semidirect
5. #14530899C2semidirect
6. #14530895C2semidirect
7. #14530839C2semidirect
8. #14530836C2semidirect
9. #14530724C2semidirect
10. #14530722C2semidirect
11. #14530696C2semidirect
12. #14530695C2semidirect
13. C4×#10587497C2semidirect
14. #14530616C2semidirect
15. C17×#3280C2semidirect
17. C22×D136C22semidirect
21. C2×#10587500C22semidirect
22. C2×#10587498C22semidirect
23. C2×#10587498C22semidirect
24. C2×#10587499C22semidirect
25. C2×#10587499C22semidirect
30. C2×#10587483C22semidirect
31. C2×#10587483C22semidirect
32. #11763193C22semidirect
33. #11763050C22semidirect
38. C4×D136C22semidirect
39. #11762634C22semidirect
40. C4⋊#63214C22semidirect
41. #11762613C22semidirect
42. #11762612C22semidirect
43. #11762610C22semidirect
44. #11762610C22semidirect
45. #11762609C22semidirect
46. #11762609C22semidirect
47. C22⋊#63214C22semidirect
48. C22⋊#63214C22semidirect
49. C4×#63214C22semidirect
51. C22×D8×C17C22semidirect
53. C17×#495C22semidirect
54. C17×#467C22semidirect
59. C4×D8×C17C22semidirect
120. #10587497D4semidirect
265. #3280D17semidirect
311. C22×D8C2×D17semidirect
328. #495C2×D17semidirect
330. #467C2×D17semidirect
340. C4×D8C2×D17semidirect
432. C17#33839semidirect

Maximal subgroups

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

subgroupindex
C2×#117632452
C2×#117632442
#145404712
#145310432
#145308992
#145308952
#145308392
#145308362
#145307242
#145307222
#145306962
#145306952
C4×#105874972
#145306162
C17×#32802
#3383917