Finite Group Databasebeta

← #17940856

Subgroups — #17940856

Order 4352.

2038 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14543553C212yesmaximal, complemented
2#14541936C214yesmaximal, complemented
3#14541930C214yesmaximal, complemented
4#14541929C214yesmaximal, complemented
5#14541924C214yesmaximal, complemented
6#14541920C211yesmaximal, characteristic, complemented
7#14541917C214yesmaximal, complemented
8C2×#11763239C212yesmaximal, complemented
9D17×#508C214yesmaximal, complemented
10C2×#11763227C211yesmaximal, characteristic, complemented
11C17×#3414C211yesmaximal, characteristic, complemented
12#11763645—22no—
13#11763623C2214yescomplemented
14#11763622C2214yescomplemented
15#11763616—24no—
16#11763615—24no—
17#11763611C2214yescomplemented
18#11763610C2214yescomplemented
19#11763607—22no—
20#11763606—24no—
21C2×#10587590C2212yescomplemented
22#135×D17C2218yescomplemented
23C2×#10587587C2211yescharacteristic
24#11763252C2214yescomplemented
25#11763251C2218yescomplemented
26#11763248C2218yescomplemented
27#11763247C2214yescomplemented
28#11763246C2218yescomplemented
29#11763243C2218yescomplemented
30#11763242C2214yescomplemented
31#11763241C2214yescomplemented
32#11763240C2218yescomplemented
33#11763239C2214yescomplemented
34#11763239C2214yescomplemented
35#11763239C2214yescomplemented
36#11763239C2214yescomplemented
37#11763238C2214yescomplemented
38C2×#10587500C2212yescomplemented
39Q32×D17C2214yescomplemented
40C2×#10587498C2212yescomplemented
41C2×#10587499C2212yescomplemented
42SD32×D17C2218yescomplemented
43C2×#10587497C2212yescomplemented
44D16×D17C2214yescomplemented
45#11763227C2214yescomplemented
46#11763227C2214yescomplemented
47C2×#10587469C2212yescomplemented
48C22×#63214C2211yescharacteristic, complemented
49C2×C16×D17C2212yescomplemented
50C2×C17×#135C2212yescomplemented
51C17×#510C2214yescomplemented
52C17×#509C2214yescomplemented
53C17×#508C2214yescomplemented
54C2×C17×M32C2211yescharacteristic, complemented
55C2×#41×C17—22no—
56#10587697—24no—
57#10587697—42no—
58#10587696—24no—
59#10587696—42no—
60#41×D17—28no—
61#41×D17—44no—
62#10587694—24no—
63#10587691—24no—
64#10587691—44no—
65C2×#63252—22no—
66C2×#63252—44no—
67C2×#63252D411yescharacteristic
68C17×#135—22no—
69C17×#135—22no—
70C17×#135C2314yes—
71C17×#135C2314yes—
72C2×Q16×C17C2312yes—
73C2×SD16×C17—22no—
74C2×D8×C17C2312yes—
75C22×C8×C17C2311yescharacteristic
76#10587637—24no—
77#10587636—28no—
78#10587635—24no—
79#10587628—24no—
80#10587618—24no—
81C2×#63223D411yescharacteristic
82#10587605C2318yes—
83#10587604—28no—
84Q16×D17C2318yes—
85#10587602—28no—
86#10587601—28no—
87#10587600—28no—
88SD16×D17—28no—
89#10587598C2318yes—
90#10587597—28no—
91D8×D17C2318yes—
92C2×Dic68C2312yes—
93#10587590—22no—
94#10587590—22no—
95#10587590C2314yes—
96#10587590C2314yes—
97C2×D136C2312yes—
98C2×#63219—22no—
99#10587587D414yes—
100#10587587C2314yes—
101C2×#63218D412yes—
102C2×C8×D17C2312yes—
103C17×Q32C2314yes—
104C17×SD32C2318yes—
105C17×D16C2314yes—
106C17×M32C2314yes—
107C2×C16×C17C2312yes—
108#10587500C2314yes—
109#10587500C2314yes—
110#10587499C2314yes—
111#10587499C2314yes—
112#10587498C2314yes—
113#10587498C2314yes—
114#10587497C2314yes—
115#10587497C2314yes—
116C2×#63214C2311yescharacteristic
117C2×#63214C2311yescharacteristic
118C2×#63214C2312yes—
119C2×#63214C2312yes—
120Dic136C2314yes—
121#10587471C2318yes—
122D272C2314yes—
123#10587469C2318yes—
124C16×D17C2314yes—
125C16×D17C2314yes—
126#41×C17—24no—
127#41×C17—24no—
128#41×C17—42no—
129#41×C17—42no—
130C2×Q8×C17—22no—
131C2×D4×C17—22no—
132C2×D4×C17—42no—
133C22×C4×C17—42no—
134C22×C4×C17C2×D411yescharacteristic
135C2×C22⋊D17—42no—
136C2×C22⋊D17—48no—
137C2×C22⋊D17D812yes—
138#63256—28no—
139#63256—44no—
140Q8×D17—28no—
141Q8×D17—44no—
142#63254—28no—
143#63254—48no—
144#63254—48no—
145#63254—44no—
146D4×D17—28no—
147D4×D17—48no—
148D4×D17—48no—
149D4×D17—44no—
150#63252—24no—
151#63252—24no—
152#63252—42no—
153#63252—42no—
154#63252—48no—
155#63252—44no—
156#63252—48no—
157#63252—44no—
158#63252D814yes—
159#63252C2×D414yes—
160C2×D68—22no—
161C2×D68—44no—
162C2×D68D811yescharacteristic
163C2×C4×D17—42no—
164C2×C4×D17—48no—
165C2×C4×D17C2×D412yes—
166C2×Dic34—22no—
167C2×Dic34—44no—
168C2×Dic34D811yescharacteristic
169Q16×C17—22no—
170Q16×C17C2414yes—
171SD16×C17—24no—
172SD16×C17—24no—
173D8×C17—22no—
174D8×C17C2414yes—
175C2×C8×C17C2×D412yes—
176C2×C8×C17C2×D411yescharacteristic
177C2×C8×C17C2412yes—
178C2×C8×C17C2411yescharacteristic
179#63231—28no—
180#63230—28no—
181#63229—28no—
182#63228—28no—
183#63223C2×D414yes—
184C2×#3622C2×D412yes—
185Dic68—22no—
186Dic68C2414yes—
187D136—22no—
188D136C2414yes—
189#63219—24no—
190#63219—24no—
191#63218C2×D418yes—
192C8×D17C2×D414yes—
193C8×D17C2414yes—
194C16×C17C2414yes—
195#63214C2411yescharacteristic
196#63214C2411yescharacteristic
197#63214C2412yes—
198#62635—171nomaximal, Sylow
199C23×C17—42no—
200C23×C17C2×D811yescharacteristic
201C22×D17—42no—
202C22×D17—48no—
203C22×D17C2×D812yes—
204Q8×C17—24no—
205Q8×C17—42no—
206D4×C17—24no—
207D4×C17—44no—
208D4×C17—44no—
209D4×C17—42no—
210C2×C4×C17—42no—
211C2×C4×C17—44no—
212C2×C4×C17—44no—
213C2×C4×C17C2×D811yescharacteristic
214C2×C4×C17C2×D812yes—
215C2×C4×C17C22×D412yes—
216C2×C4×C17C22×D411yescharacteristic
217C22⋊D17—44no—
218C22⋊D17—44no—
219C22⋊D17—48no—
220C22⋊D17—48no—
221C22⋊D17C2×D818yes—
222C2×Dic17—42no—
223C2×Dic17—48no—
224C2×Dic17C2×D812yes—
225D68—24no—
226D68—48no—
227D68—42no—
228D68C2×D814yes—
229C4×D17—48no—
230C4×D17—44no—
231C4×D17—44no—
232C4×D17—48no—
233C4×D17C2×D814yes—
234C4×D17C22×D414yes—
235Dic34—24no—
236Dic34—42no—
237Dic34—48no—
238Dic34C2×D814yes—
239C8×C17C22×D412yes—
240C8×C17C22×D411yescharacteristic
241C8×C17C2511yescharacteristic
242#3622C22×D414yes—
243#3584—172no—
244#3417—174no—
245#3416—174no—
246#3416—174no—
247#3415—174no—
248#3414—174no—
249#3414—171no—
250#3414D1711yescharacteristic, complemented
251C2×#508—174no—
252C2×#508—172no—
253C2×#504—171no—
254C22×C17—41no—
255C22×C17—44no—
256C22×C17—162no—
257C22×C17C22×D812yes—
258C22×C17C22×D811yescharacteristic
259C2×D17—42no—
260C2×D17—44no—
261C2×D17—168no—
262C2×D17—162no—
263C2×D17C22×D814yes—
264C4×C17—44no—
265C4×C17C22×D812yes—
266C4×C17C22×D811yescharacteristic
267C4×C17C23×D411yescharacteristic
268Dic17—44no—
269Dic17C22×D814yes—
270#585—342no—
271#578—174no—
272#578—174no—
273#577—344no—
274#577—344no—
275#576—174no—
276#576—174no—
277#575—344no—
278#575—342no—
279C2×#135—178no—
280C2×#135—172no—
281C2×#135C2×D1712yescomplemented
282C2×#131—171no—
283#510—178no—
284#510—178no—
285#510—174no—
286#510C2×D1714yescomplemented
287#509—178no—
288#509—178no—
289#509—174no—
290#509C2×D1714yescomplemented
291#508—174no—
292#508—174no—
293#508—174no—
294#508—174no—
295#508—174no—
296#508—178no—
297#508—174no—
298#508—174no—
299#508C2×D1714yescomplemented
300C2×Q32—172no—
301C2×Q32—174no—
302C2×SD32—172no—
303C2×SD32—172no—
304C2×SD32—178no—
305C2×D16—172no—
306C2×D16—174no—
307#504—174no—
308#504—174no—
309C2×M32—172no—
310C2×M32C2×D1711yescharacteristic, complemented
311C22×C16—171no—
312C22×C16—172no—
313C2×C17—21no—
314C2×C17—22no—
315C2×C17—84no—
316C2×C17C23×D811yescharacteristic
317D17—24no—
318D17—84no—
3192−1+4—344no—
3202−1+4—344no—
3212−1+4—682no—
322H(5,𝔽2)—344no—
323H(5,𝔽2)—344no—
324H(5,𝔽2)—684no—
325H(5,𝔽2)—682no—
326C2×#41—22no—
327C2×#41—171no—
328C2×#41—348no—
329C2×#41—342no—
330C2×#41—684no—
331C2×#41—684no—
332#137—348no—
333#137—348no—
334#137—344no—
335#137—344no—
336AGL(1,ℤ/8)—348no—
337AGL(1,ℤ/8)—348no—
338AGL(1,ℤ/8)—344no—
339AGL(1,ℤ/8)—344no—
340#135—22no—
341#135—22no—
342#135—178no—
343#135—178no—
344#135—174no—
345#135—174no—
346#135—348no—
347#135—348no—
348#135—342no—
349#135—342no—
350#135C22×D1714yes—
351#135C22×D1714yes—
352C2×Q16—178no—
353C2×Q16—172no—
354C2×Q16C22×D1712yes—
355C2×SD16—22no—
356C2×SD16—348no—
357C2×SD16—342no—
358C2×D8—178no—
359C2×D8—172no—
360C2×D8C22×D1712yes—
361#131—174no—
362#131—174no—
363C2×M16—171no—
364C2×M16—172no—
365C22×C8—172no—
366C22×C8C22×D1711yescharacteristic
367Q32—174no—
368Q32—174no—
369Q32—174no—
370Q32C22×D1714yes—
371SD32—174no—
372SD32—174no—
373SD32—174no—
374SD32—174no—
375SD32—178no—
376SD32C22×D1718yes—
377D16—174no—
378D16—174no—
379D16—174no—
380D16C22×D1714yes—
381M32—178no—
382M32C22×D1714yes—
383C2×C16—171no—
384C2×C16—171no—
385C2×C16—172no—
386C2×C16—172no—
387C2×C16—174no—
388C2×C16—174no—
389C2×C16C22×D1712yes—
390C17#6263511yescharacteristic, complemented, Sylow
391#41—24no—
392#41—24no—
393#41—42no—
394#41—42no—
395#41—174no—
396#41—174no—
397#41—348no—
398#41—348no—
399#41—344no—
400#41—344no—
401#41—688no—
402#41—688no—
403#41—684no—
404#41—684no—
405#41—688no—
406#41—688no—
407#41—684no—
408#41—684no—
409#41—682no—
410#41—682no—
411C2×Q8—22no—
412C2×Q8—171no—
413C2×Q8—348no—
414C2×Q8—342no—
415C2×Q8—684no—
416C2×Q8—684no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4—172no—
420C2×D4—171no—
421C2×D4—348no—
422C2×D4—342no—
423C2×D4—684no—
424C2×D4—688no—
425C2×D4—684no—
426C2×D4—688no—
427C2×D4—688no—
428C2×D4—682no—
429C22×C4—42no—
430C22×C4—172no—
431C22×C4—688no—
432C22×C4—682no—
433C22×C4D4×D1711yescharacteristic
434Q16—22no—
435Q16—174no—
436Q16—348no—
437Q16—342no—
438Q16C23×D1714yes—
439SD16—24no—
440SD16—24no—
441SD16—348no—
442SD16—348no—
443SD16—344no—
444SD16—344no—
445D8—22no—
446D8—174no—
447D8—348no—
448D8—342no—
449D8C23×D1714yes—
450M16—174no—
451M16—178no—
452C2×C8—172no—
453C2×C8—174no—
454C2×C8—174no—
455C2×C8D4×D1711yescharacteristic
456C2×C8D4×D1712yes—
457C2×C8C23×D1711yescharacteristic
458C2×C8C23×D1712yes—
459C16—171no—
460C16—171no—
461C16—172no—
462C16C23×D1714yes—
463C23—42no—
464C23—172no—
465C23—688no—
466C23—682no—
467C23D8×D1711yescharacteristic
468Q8—24no—
469Q8—42no—
470Q8—174no—
471Q8—344no—
472Q8—688no—
473Q8—682no—
474D4—24no—
475D4—44no—
476D4—44no—
477D4—42no—
478D4—178no—
479D4—174no—
480D4—344no—
481D4—688no—
482D4—688no—
483D4—688no—
484D4—684no—
485D4—684no—
486D4—682no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—172no—
491C2×C4—174no—
492C2×C4—174no—
493C2×C4—688no—
494C2×C4—688no—
495C2×C4—688no—
496C2×C4—682no—
497C2×C4—684no—
498C2×C4—684no—
499C2×C4D8×D1711yescharacteristic
500C2×C4D8×D1712yes—
501C2×C4C2×D4×D1712yes—
502C2×C4C2×D4×D1711yescharacteristic
503C8—174no—
504C8C2×D4×D1711yescharacteristic
505C8C2×D4×D1712yes—
506C8C24×D1711yescharacteristic
507C22—41no—
508C22—44no—
509C22—162no—
510C22—2722no—
511C22—2728no—
512C22—174no—
513C22—682no—
514C22—684no—
515C22C2×D8×D1712yes—
516C22C2×D8×D1711yescharacteristic
517C4—44no—
518C4—174no—
519C4—684no—
520C4C2×D8×D1712yes—
521C4C2×D8×D1711yescharacteristic
522C4C22×D4×D1711yescharacteristic
523C2—22no—
524C2—21no—
525C2—84no—
526C2—344no—
527C2—1364no—
528C2C22×D8×D1711yescharacteristic
subgroupG/H
centerC4C2×D8×D17
commutator, cosocleC8×C17C25
FrattiniC8C24×D17
FittingC17×#3414C2
socleC2×C17C23×D8
hypercenter#3414D17
hypocenterC17#62635
LCSG ▷ C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. #14543553C2semidirect
2. #14541936C2semidirect
3. #14541930C2semidirect
4. #14541929C2semidirect
5. #14541924C2semidirect
6. #14541920C2semidirect
7. #14541917C2semidirect
8. C2×#11763239C2semidirect
9. D17×#508C2semidirect
10. C2×#11763227C2semidirect
11. C17×#3414C2semidirect
13. #11763623C22semidirect
14. #11763622C22semidirect
17. #11763611C22semidirect
18. #11763610C22semidirect
21. C2×#10587590C22semidirect
22. #135×D17C22semidirect
24. #11763252C22semidirect
25. #11763251C22semidirect
26. #11763248C22semidirect
27. #11763247C22semidirect
28. #11763246C22semidirect
29. #11763243C22semidirect
30. #11763242C22semidirect
31. #11763241C22semidirect
32. #11763240C22semidirect
33. #11763239C22semidirect
34. #11763239C22semidirect
35. #11763239C22semidirect
36. #11763239C22semidirect
37. #11763238C22semidirect
38. C2×#10587500C22semidirect
39. Q32×D17C22semidirect
40. C2×#10587498C22semidirect
41. C2×#10587499C22semidirect
42. SD32×D17C22semidirect
43. C2×#10587497C22semidirect
44. D16×D17C22semidirect
45. #11763227C22semidirect
46. #11763227C22semidirect
47. C2×#10587469C22semidirect
48. C22×#63214C22semidirect
49. C2×C16×D17C22semidirect
50. C2×C17×#135C22semidirect
51. C17×#510C22semidirect
52. C17×#509C22semidirect
53. C17×#508C22semidirect
54. C2×C17×M32C22semidirect
250. #3414D17semidirect
281. C2×#135C2×D17semidirect
286. #510C2×D17semidirect
290. #509C2×D17semidirect
299. #508C2×D17semidirect
310. C2×M32C2×D17semidirect
390. C17#62635semidirect

Maximal subgroups

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

subgroupindex
#145435532
#145419362
#145419302
#145419292
#145419242
#145419202
#145419172
C2×#117632392
D17×#5082
C2×#117632272
C17×#34142
#6263517