Finite Group Databasebeta

← #17189157

Subgroups — #17189157

Order 4352.

580 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D17×#467C211yesmaximal, characteristic
2D17×#427C211yesmaximal, characteristic
3#14530993C211yesmaximal, characteristic, complemented
4#14530992C211yesmaximal, characteristic, complemented
5#14530989C211yesmaximal, characteristic, complemented
6#14530988C211yesmaximal, characteristic, complemented
7#14530985C211yesmaximal, characteristic, complemented
8#14530983C211yesmaximal, characteristic, complemented
9#14530980C211yesmaximal, characteristic, complemented
10#14530571C211yesmaximal, characteristic, complemented
11#14530569C211yesmaximal, characteristic, complemented
12#14530502C211yesmaximal, characteristic, complemented
13#14530502C211yesmaximal, characteristic, complemented
14#14530393C211yesmaximal, characteristic
15C17×#2237C211yesmaximal, characteristic, complemented
16C2×Q16×D17C2211yescharacteristic
17C22×C8×D17C2211yescharacteristic
18C22⋊Q8×D17—21no—
19#117×D17—21no—
20C2×#10587469C2211yescharacteristic
21C2×#10587469C2211yescharacteristic
22#11763051C2211yescharacteristic
23#11763050C2211yescharacteristic
24#11763049—21no—
25#11763048—21no—
26#11763047C2211yescharacteristic
27#11762847C2211yescharacteristic
28#11762846C2211yescharacteristic
29#11762846C2211yescharacteristic
30#11762845—21no—
31#11762844—21no—
32#107×D17C2211yescharacteristic
33#107×D17C2211yescharacteristic
34#107×D17C2211yescharacteristic
35#106×D17—21no—
36#103×D17—21no—
37#11762637C2211yescharacteristic, complemented
38#11762636C2211yescharacteristic, complemented
39#11762636C2211yescharacteristic, complemented
40#11762635C2211yescharacteristic, complemented
41#11762634C2211yescharacteristic, complemented
42#11762634C2211yescharacteristic, complemented
43#11762614C2211yescharacteristic, complemented
44#11762614C2211yescharacteristic, complemented
45#11762613C2211yescharacteristic, complemented
46#11762613C2211yescharacteristic, complemented
47#11762612C2211yescharacteristic, complemented
48#11762612C2211yescharacteristic, complemented
49C22⋊#63214C2211yescharacteristic
50#11762588C2211yescharacteristic
51#11762588C2211yescharacteristic
52C17×#467C2211yescharacteristic
53C17×#427C2211yescharacteristic
54C17×#367C2211yescharacteristic, complemented
55C17×#366C2211yescharacteristic, complemented
56C17×#358C2211yescharacteristic, complemented
57C17×#358C2211yescharacteristic, complemented
58C17×C22⋊C16C2211yescharacteristic
59C2×Q8×D17—21no—
60C22×C4×D17D411yescharacteristic
61C2×Q16×C17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C17×C22⋊Q8—21no—
64C17×#117—21no—
65#10587632—21no—
66C2×#63231—21no—
67#10587612—21no—
68#10587610—21no—
69C22×#3622D411yescharacteristic
70Q16×D17—22no—
71Q16×D17—22no—
72C2×Dic68C2311yescharacteristic
73C2×C8×D17—22no—
74C2×C8×D17—22no—
75C2×C8×D17D411yescharacteristic
76C2×C8×D17D411yescharacteristic
77C2×C8×D17C2311yescharacteristic
78C2×C8×D17C2311yescharacteristic
79#10587583—21no—
80#10587582—41no—
81#10587578—21no—
82#10587578—21no—
83C4⋊C4×D17—21no—
84C4⋊C4×D17—21no—
85C4⋊C4×D17—21no—
86C4⋊C4×D17—41no—
87C22⋊C4×D17—41no—
88C22⋊C4×D17—41no—
89#10587564—41no—
90#10587563—41no—
91#10587558—41no—
92C42×D17—41no—
93C2×C16×C17C2311yescharacteristic
94C2×C16×C17C2311yescharacteristic
95C17×#107C2311yescharacteristic
96C17×#107C2311yescharacteristic
97C17×#107C2311yescharacteristic
98C17×#106—21no—
99C17×#103—21no—
100#10587506—21no—
101#10587489C2311yescharacteristic
102#10587489C2311yescharacteristic
103#10587489C2311yescharacteristic
104#10587488—21no—
105#10587487—21no—
106C2×#63214C2311yescharacteristic
107C2×#63214C2311yescharacteristic
108#10587481—21no—
109#10587479—21no—
110#10587479—21no—
111#10587478—21no—
112#10587478—21no—
113#10587478—21no—
114#10587469—22no—
115#10587469—22no—
116C23×D17D811yescharacteristic
117C2×Q8×C17—21no—
118C22×C4×C17C2×D411yescharacteristic
119C22×Dic17D811yescharacteristic
120Q8×D17—42no—
121Q8×D17—42no—
122C2×C4×D17—22no—
123C2×C4×D17—22no—
124C2×C4×D17—41no—
125C2×C4×D17—41no—
126C2×C4×D17—41no—
127C2×C4×D17—41no—
128C2×C4×D17D811yescharacteristic
129C2×C4×D17D811yescharacteristic
130C2×C4×D17C2×D411yescharacteristic
131C2×C4×D17C2×D411yescharacteristic
132C2×Dic34—21no—
133C2×Dic34—41no—
134Q16×C17—21no—
135Q16×C17—21no—
136C2×C8×C17—21no—
137C2×C8×C17—21no—
138C2×C8×C17C2×D411yescharacteristic
139C2×C8×C17C2411yescharacteristic
140C4⋊C4×C17—21no—
141C4⋊C4×C17—21no—
142C4⋊C4×C17—21no—
143C4⋊C4×C17—41no—
144C22⋊C4×C17—41no—
145C22⋊C4×C17—41no—
146C42×C17—41no—
147C22⋊Dic17—41no—
148C22⋊Dic17—41no—
149#63231—42no—
150#63227—41no—
151#63227—41no—
152#63227—41no—
153#63227—41no—
154C4⋊Dic17—21no—
155C4⋊Dic17—21no—
156C4⋊Dic17—21no—
157C4⋊Dic17—41no—
158#63225—41no—
159#63225—41no—
160#63225—41no—
161#63225—41no—
162#63225—41no—
163C4×Dic17—41no—
164C4×Dic17—41no—
165C4×Dic17—41no—
166C2×#3622—22no—
167C2×#3622C2×D411yescharacteristic
168C2×#3622C2×D411yescharacteristic
169Dic68—21no—
170Dic68—21no—
171C8×D17—22no—
172C8×D17—21no—
173C8×D17—21no—
174C8×D17—22no—
175C8×D17#4112yes—
176C8×D17#4112yes—
177C16×C17—21no—
178C16×C17—21no—
179#63214—21no—
180#63214—21no—
181#33649—171nomaximal, Sylow
182C23×C17C2×D811yescharacteristic
183C22×D17—22no—
184C22×D17—22no—
185C22×D17—41no—
186C22×D17C2×D811yescharacteristic
187C22×D17C2×D811yescharacteristic
188Q8×C17—41no—
189Q8×C17—41no—
190C2×C4×C17—21no—
191C2×C4×C17—21no—
192C2×C4×C17—41no—
193C2×C4×C17—41no—
194C2×C4×C17—41no—
195C2×C4×C17—41no—
196C2×C4×C17C2×D811yescharacteristic
197C2×C4×C17C22×D411yescharacteristic
198C2×Dic17—22no—
199C2×Dic17—41no—
200C2×Dic17—41no—
201C2×Dic17—41no—
202C2×Dic17—41no—
203C2×Dic17C2×D811yescharacteristic
204C2×Dic17C2×D811yescharacteristic
205C4×D17—22no—
206C4×D17—21no—
207C4×D17—21no—
208C4×D17—22no—
209C4×D17—82no—
210C4×D17—82no—
211C4×D17—82no—
212C4×D17—82no—
213C4×D17#12312yes—
214C4×D17#13712yes—
215Dic34—41no—
216Dic34—41no—
217Dic34—82no—
218C8×C17—21no—
219C8×C17C2×#4111yescharacteristic
220C8×C17C2×#4111yescharacteristic
221#3622—21no—
222#3622#12312yes—
223C2×#467—171no—
224C2×#427—171no—
225#2237—171no—
226#2237—171no—
227#2237—171no—
228#2237D1711yescharacteristic, complemented
229#2236—171no—
230#2236—171no—
231#2235—171no—
232#2235—171no—
233#2159—171no—
234#2159—171no—
235#2141—171no—
236#2141—171no—
237#2112—171no—
238C22×C17—21no—
239C22×C17—21no—
240C22×C17—21no—
241C22×C17C22×D811yescharacteristic
242C2×D17—21no—
243C2×D17—21no—
244C2×D17—21no—
245C2×D17—21no—
246C2×D17—44no—
247C2×D17#48012yes—
248C2×D17#51012yes—
249C4×C17—21no—
250C4×C17—81no—
251C4×C17—81no—
252C4×C17—81no—
253C4×C17—81no—
254C4×C17C2×#12311yescharacteristic
255C4×C17C2×#13711yescharacteristic
256Dic17—21no—
257Dic17—81no—
258Dic17—81no—
259Dic17—81no—
260Dic17—81no—
261Dic17#48012yes—
262C22×Q16—171no—
263C23×C8—171no—
264C2×C22⋊Q8—341no—
265C2×#117—341no—
266C2×M32—171no—
267C2×M32—171no—
268#467—171no—
269#467—171no—
270#467—171no—
271#467—341no—
272#467—341no—
273#467C2×D1711yescharacteristic
274#427—171no—
275#427—171no—
276#427—171no—
277#427—341no—
278#427—341no—
279#427C2×D1711yescharacteristic
280C2×#107—171no—
281C2×#107—171no—
282C2×#107—171no—
283C2×#106—341no—
284C2×#103—341no—
285#367—171no—
286#367—171no—
287#367—171no—
288#367C2×D1711yescharacteristic, complemented
289#366—171no—
290#366—171no—
291#366—171no—
292#366C2×D1711yescharacteristic, complemented
293#358—171no—
294#358—171no—
295#358—171no—
296#358—171no—
297#358—171no—
298#358—171no—
299#358C2×D1711yescharacteristic, complemented
300#358C2×D1711yescharacteristic, complemented
301C22⋊C16—171no—
302C22⋊C16—171no—
303C22⋊C16—171no—
304C22⋊C16C2×D1711yescharacteristic
305C2×C17—41no—
306C2×C17C2×#48011yescharacteristic
307C2×C17C2×#51011yescharacteristic
308C2×C17#341411yescharacteristic
309D17—22no—
310D17—22no—
311C22×Q8—341no—
312C23×C4—171no—
313C2×Q16—171no—
314C2×Q16—342no—
315C2×Q16—342no—
316C2×Q16—341no—
317C2×Q16C22×D1711yescharacteristic
318C22×C8—171no—
319C22×C8—171no—
320C22×C8—171no—
321C22×C8—171no—
322C22×C8—171no—
323C22×C8—342no—
324C22×C8—342no—
325C22×C8C22×D1711yescharacteristic
326C22⋊Q8—21no—
327C22⋊Q8—341no—
328C22⋊Q8—341no—
329C22⋊Q8—341no—
330C22⋊Q8—681no—
331C22⋊Q8—681no—
332#117—21no—
333#117—341no—
334#117—341no—
335#117—341no—
336#117—681no—
337#117—681no—
338C2×C4⋊C4—341no—
339C2×C4⋊C4—341no—
340C2×C4⋊C4—341no—
341C2×C4⋊C4—681no—
342C2×C22⋊C4—681no—
343C2×C22⋊C4—681no—
344C2×C42—681no—
345M32—342no—
346M32—342no—
347C2×C16—171no—
348C2×C16—171no—
349C2×C16C22×D1711yescharacteristic
350C2×C16C22×D1711yescharacteristic
351#107—171no—
352#107—171no—
353#107—171no—
354#107—341no—
355#107—341no—
356#107—341no—
357#107C22×D1711yescharacteristic
358#107C22×D1711yescharacteristic
359#107C22×D1711yescharacteristic
360#106—21no—
361#106—341no—
362#106—341no—
363#106—341no—
364#103—21no—
365#103—341no—
366#103—341no—
367#103—341no—
368C17#3364911yescharacteristic, complemented, Sylow
369C24—171no—
370C2×Q8—21no—
371C2×Q8—341no—
372C2×Q8—681no—
373C2×Q8—682no—
374C2×Q8—682no—
375C22×C4—171no—
376C22×C4—171no—
377C22×C4—171no—
378C22×C4—171no—
379C22×C4—171no—
380C22×C4—342no—
381C22×C4—342no—
382C22×C4—681no—
383C22×C4—681no—
384C22×C4—681no—
385C22×C4—681no—
386C22×C4D4×D1711yescharacteristic
387Q16—21no—
388Q16—21no—
389Q16—341no—
390Q16—341no—
391Q16—682no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—171no—
395C2×C8—172no—
396C2×C8—172no—
397C2×C8—171no—
398C2×C8—342no—
399C2×C8—342no—
400C2×C8—342no—
401C2×C8—341no—
402C2×C8—341no—
403C2×C8D4×D1711yescharacteristic
404C2×C8C23×D1711yescharacteristic
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—41no—
409C4⋊C4—341no—
410C4⋊C4—341no—
411C4⋊C4—341no—
412C4⋊C4—681no—
413C4⋊C4—681no—
414C4⋊C4—681no—
415C4⋊C4—681no—
416C4⋊C4—681no—
417C4⋊C4—681no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—681no—
421C22⋊C4—681no—
422C22⋊C4—681no—
423C22⋊C4—681no—
424C22⋊C4—681no—
425C22⋊C4—681no—
426C42—41no—
427C42—681no—
428C42—681no—
429C42—681no—
430C16—21no—
431C16—21no—
432C16—341no—
433C16—341no—
434C23—171no—
435C23—171no—
436C23—342no—
437C23—342no—
438C23—681no—
439C23D8×D1711yescharacteristic
440Q8—41no—
441Q8—41no—
442Q8—681no—
443Q8—681no—
444Q8—1362no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—171no—
452C2×C4—172no—
453C2×C4—172no—
454C2×C4—171no—
455C2×C4—342no—
456C2×C4—342no—
457C2×C4—342no—
458C2×C4—341no—
459C2×C4—341no—
460C2×C4—681no—
461C2×C4—681no—
462C2×C4—681no—
463C2×C4—681no—
464C2×C4—1362no—
465C2×C4—1362no—
466C2×C4—1362no—
467C2×C4—1362no—
468C2×C4D8×D1711yescharacteristic
469C2×C4C2×D4×D1711yescharacteristic
470C8—21no—
471C8—172no—
472C8—341no—
473C8#41×D1711yescharacteristic
474C8#41×D1711yescharacteristic
475C22—21no—
476C22—21no—
477C22—21no—
478C22—172no—
479C22—172no—
480C22—341no—
481C22—341no—
482C22—341no—
483C22—341no—
484C22—684no—
485C22C2×D8×D1711yescharacteristic
486C4—21no—
487C4—81no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—172no—
492C4—341no—
493C4—1361no—
494C4—1361no—
495C4—1361no—
496C4—1361no—
497C4#123×D1711yescharacteristic
498C4#137×D1711yescharacteristic
499C2—41no—
500C2—342no—
501C2—342no—
502C2D17×#48011yescharacteristic
503C2D17×#51011yescharacteristic
504C2#1454191711yescharacteristic
subgroupG/H
centerC22C2×D8×D17
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#2237C2
socleC22×C17C22×D8
prime radicalC23×D17D8
hypercenter#2237D17
hypocenterC17#33649
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2237

Complements

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

NH ≅ G/Ntype
3. #14530993C2semidirect
4. #14530992C2semidirect
5. #14530989C2semidirect
6. #14530988C2semidirect
7. #14530985C2semidirect
8. #14530983C2semidirect
9. #14530980C2semidirect
10. #14530571C2semidirect
11. #14530569C2semidirect
12. #14530502C2semidirect
13. #14530502C2semidirect
15. C17×#2237C2semidirect
37. #11762637C22semidirect
38. #11762636C22semidirect
39. #11762636C22semidirect
40. #11762635C22semidirect
41. #11762634C22semidirect
42. #11762634C22semidirect
43. #11762614C22semidirect
44. #11762614C22semidirect
45. #11762613C22semidirect
46. #11762613C22semidirect
47. #11762612C22semidirect
48. #11762612C22semidirect
54. C17×#367C22semidirect
55. C17×#366C22semidirect
56. C17×#358C22semidirect
57. C17×#358C22semidirect
228. #2237D17semidirect
288. #367C2×D17semidirect
292. #366C2×D17semidirect
299. #358C2×D17semidirect
300. #358C2×D17semidirect
368. C17#33649semidirect

Maximal subgroups

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

subgroupindex
D17×#4672
D17×#4272
#145309932
#145309922
#145309892
#145309882
#145309852
#145309832
#145309802
#145305712
#145305692
#145305022
#145303932
C17×#22372
#3364917