Finite Group Databasebeta

← #17189008

Subgroups — #17189008

Order 4352.

556 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539317C211yesmaximal, characteristic, complemented
2#14537631C211yesmaximal, characteristic
3#14530943C211yesmaximal, characteristic, complemented
4#14530942C211yesmaximal, characteristic, complemented
5#14530941C211yesmaximal, characteristic, complemented
6#14530940C211yesmaximal, characteristic, complemented
7#14530930C211yesmaximal, characteristic, complemented
8#14530919C211yesmaximal, characteristic, complemented
9#14530918C211yesmaximal, characteristic, complemented
10#14530571C211yesmaximal, characteristic, complemented
11#14530567C211yesmaximal, characteristic, complemented
12#14530541C211yesmaximal, characteristic, complemented
13#14530498C211yesmaximal, characteristic
14D17×#357C211yesmaximal, characteristic
15C17×#2226C211yesmaximal, characteristic, complemented
16C2×D8×D17C2211yescharacteristic
17#11763436—21no—
18#11763281—21no—
19C2×#10587469C2211yescharacteristic
20C2×C16×D17C2211yescharacteristic
21#11763217—21no—
22#11763215—21no—
23#11763213C2211yescharacteristic, complemented
24#11763211C2211yescharacteristic, complemented
25#11763210C2211yescharacteristic, complemented
26#11762919—21no—
27#11762918—21no—
28C4×D136C2211yescharacteristic
29D8×Dic17C2211yescharacteristic
30#11762891C2211yescharacteristic
31#11762850—21no—
32#11762846C2211yescharacteristic
33#11762846C2211yescharacteristic
34#107×D17C2211yescharacteristic
35#11762812—21no—
36#11762763C2211yescharacteristic
37#11762637C2211yescharacteristic, complemented
38#11762636C2211yescharacteristic, complemented
39#11762635C2211yescharacteristic, complemented
40#11762634C2211yescharacteristic, complemented
41#11762634C2211yescharacteristic, complemented
42#11762634C2211yescharacteristic, complemented
43C4⋊#63214C2211yescharacteristic, complemented
44#11762627C2211yescharacteristic, complemented
45#11762627C2211yescharacteristic, complemented
46#11762611C2211yescharacteristic
47#11762611C2211yescharacteristic
48#11762610C2211yescharacteristic
49#11762610C2211yescharacteristic
50#11762609C2211yescharacteristic
51#11762609C2211yescharacteristic
52C17×#500C2211yescharacteristic, complemented
53C4×D8×C17C2211yescharacteristic
54C17×#367C2211yescharacteristic, complemented
55C17×#367C2211yescharacteristic, complemented
56C17×C4⋊C16C2211yescharacteristic, complemented
57C17×#357C2211yescharacteristic
58C17×#357C2211yescharacteristic
59C2×D4×D17—21no—
60C2×D8×C17C2311yescharacteristic
61C17×C4⋊Q8—21no—
62C4×D4×C17—21no—
63#10587632—41no—
64#10587630—41no—
65D4×Dic17—21no—
66C2×#63228—21no—
67C4×C22⋊D17—41no—
68C2×#63227—41no—
69D8×D17—22no—
70D8×D17Q814yescomplemented
71C2×D136C2311yescharacteristic
72C2×#63218D411yescharacteristic
73C2×C8×D17C2311yescharacteristic
74#10587582—41no—
75#10587579—21no—
76#10587578—21no—
77#10587578—21no—
78C4⋊C4×D17—21no—
79#10587576—21no—
80#10587575—41no—
81#10587574—21no—
82#10587567—41no—
83C22⋊C4×D17—41no—
84C4×D68—21no—
85#10587558D411yescharacteristic
86#10587556—21no—
87C2×C16×C17C2311yescharacteristic
88C2×C16×C17C2311yescharacteristic
89C17×#107—21no—
90C17×#107C2311yescharacteristic
91C17×#107C2311yescharacteristic
92C17×#107C2311yescharacteristic
93C17×#102—21no—
94C4×C8×C17C2311yescharacteristic
95#10587503—21no—
96#10587492—21no—
97#10587489C2311yescharacteristic
98#10587489C2311yescharacteristic
99#10587489C2311yescharacteristic
100#10587488—21no—
101#10587486D411yescharacteristic
102C8×Dic17C2311yescharacteristic
103C2×#63214C2311yescharacteristic
104C2×#63214C2311yescharacteristic
105#10587480—21no—
106#10587479—21no—
107#10587479—21no—
108#10587479—21no—
109#10587478—21no—
110#10587478—21no—
111#10587474D411yescharacteristic
112#10587469—22no—
113C16×D17—22no—
114C23×D17—41no—
115C2×Q8×C17—41no—
116C2×D4×C17—21no—
117C22×C4×C17—41no—
118C2×C22⋊D17—41no—
119C22×Dic17—41no—
120D4×D17—24no—
121D4×D17—42no—
122C2×D68—21no—
123C2×C4×D17—41no—
124C2×C4×D17—41no—
125C2×C4×D17—41no—
126C2×C4×D17C2×D411yescharacteristic
127C2×Dic34—41no—
128D8×C17—21no—
129D8×C17C2×Q812yes—
130C2×C8×C17C2×D411yescharacteristic
131C2×C8×C17C2411yescharacteristic
132C4⋊C4×C17—21no—
133C4⋊C4×C17—21no—
134C4⋊C4×C17—21no—
135C4⋊C4×C17—41no—
136C22⋊C4×C17—41no—
137C42×C17C2×D411yescharacteristic
138C22⋊Dic17—41no—
139#63228—24no—
140#63227—41no—
141#63227—41no—
142#63227—41no—
143#63227—41no—
144#63227D811yescharacteristic
145#63227D811yescharacteristic
146C4⋊Dic17—21no—
147C4⋊Dic17—21no—
148C4⋊Dic17—21no—
149C4⋊Dic17—41no—
150#63225—41no—
151#63225—41no—
152#63225—41no—
153#63225—41no—
154#63225—41no—
155#63225D811yescharacteristic
156#63225D811yescharacteristic
157C4×Dic17—41no—
158C4×Dic17—41no—
159C4×Dic17C2×D411yescharacteristic
160C2×#3622C2×D411yescharacteristic
161C2×#3622C2×D411yescharacteristic
162D136—21no—
163D136C2×Q812yes—
164#63218—22no—
165C8×D17C2×Q812yes—
166C8×D17#4112yes—
167C16×C17—21no—
168C16×C17—21no—
169#63214—21no—
170#63214—21no—
171#33631—171nomaximal, Sylow
172C23×C17—41no—
173C22×D17—41no—
174C22×D17—44no—
175C22×D17—161no—
176C22×D17—161no—
177C22×D17C2×D811yescharacteristic
178Q8×C17—161no—
179D4×C17—22no—
180D4×C17—41no—
181C2×C4×C17—41no—
182C2×C4×C17—41no—
183C2×C4×C17—41no—
184C2×C4×C17—161no—
185C2×C4×C17C2×D811yescharacteristic
186C2×C4×C17C2×D811yescharacteristic
187C2×C4×C17C22×D411yescharacteristic
188C22⋊D17—44no—
189C2×Dic17—41no—
190C2×Dic17—41no—
191C2×Dic17—41no—
192C2×Dic17—161no—
193C2×Dic17C2×D811yescharacteristic
194C2×Dic17C2×D811yescharacteristic
195C2×Dic17C2×D811yescharacteristic
196D68—22no—
197D68—41no—
198C4×D17—161no—
199C4×D17—161no—
200C4×D17—161no—
201C4×D17C22⋊Q812yes—
202C4×D17#13712yes—
203Dic34—161no—
204C8×C17—21no—
205C8×C17C22×Q811yescharacteristic
206C8×C17C2×#4111yescharacteristic
207#3622—21no—
208#3622C22⋊Q812yes—
209#3164—171no—
210#2948—171no—
211#2228—171no—
212#2228—171no—
213#2228—171no—
214#2228—171no—
215#2226—171no—
216#2226D1711yescharacteristic, complemented
217#2224—171no—
218#2224—171no—
219#2159—171no—
220#2158—171no—
221#2151—171no—
222#2140—171no—
223C2×#357—171no—
224C22×C17—42no—
225C22×C17—81no—
226C22×C17—81no—
227C22×C17C22×D811yescharacteristic
228C2×D17—21no—
229C2×D17—21no—
230C2×D17—42no—
231C2×D17—81no—
232C2×D17—81no—
233C2×D17—164no—
234C2×D17#47412yes—
235C4×C17—21no—
236C4×C17—21no—
237C4×C17—81no—
238C4×C17—81no—
239C4×C17—81no—
240C4×C17C2×C22⋊Q811yescharacteristic
241C4×C17C2×#13711yescharacteristic
242Dic17—21no—
243Dic17—21no—
244Dic17—81no—
245Dic17—81no—
246Dic17—81no—
247Dic17#47412yes—
248C22×D8—171no—
249#544—341no—
250#518—341no—
251C2×M32—171no—
252C22×C16—171no—
253#501—341no—
254#501—341no—
255#500—171no—
256#500C2×D1711yescharacteristic, complemented
257#499—171no—
258#499—171no—
259#442—341no—
260#442—341no—
261C4×D8—171no—
262C4×D8—171no—
263C4×D8—171no—
264C4×D8C2×D1711yescharacteristic
265#428—341no—
266#427—171no—
267#427—171no—
268C2×#107—171no—
269#418—341no—
270#405—171no—
271#367—171no—
272#367—171no—
273#367C2×D1711yescharacteristic, complemented
274#367C2×D1711yescharacteristic, complemented
275#366—171no—
276#366—171no—
277#366—171no—
278#366—171no—
279C4⋊C16—171no—
280C4⋊C16—171no—
281C4⋊C16—171no—
282C4⋊C16C2×D1711yescharacteristic, complemented
283#357—171no—
284#357—171no—
285#357—171no—
286#357—171no—
287#357—171no—
288#357—171no—
289#357C2×D1711yescharacteristic
290#357C2×D1711yescharacteristic
291C2×C17—82no—
292C2×C17C2×#47411yescharacteristic
293C2×C17#341511yescharacteristic
294C2×C17#341611yescharacteristic
295D17—22no—
296D17—82no—
297C22×D4—341no—
298C2×D8—174no—
299C2×D8—171no—
300C2×D8—342no—
301C2×D8—341no—
302C2×D8C22×D1711yescharacteristic
303C2×M16—171no—
304C22×C8—171no—
305C4⋊Q8—21no—
306C4⋊Q8—341no—
307C4⋊Q8—681no—
308#125—341no—
309#125—341no—
310#125—681no—
311C22⋊Q8—681no—
312C22⋊Q8—681no—
313C4×D4—21no—
314C4×D4—341no—
315C4×D4—341no—
316C4×D4—341no—
317C4×D4—681no—
318C4×D4—681no—
319#117—171no—
320#117—341no—
321#117—341no—
322C2×C4⋊C4—341no—
323C2×C22⋊C4—681no—
324C2×C22⋊C4—681no—
325M32—342no—
326C2×C16—171no—
327C2×C16—171no—
328C2×C16—342no—
329C2×C16C22×D1711yescharacteristic
330C2×C16C22×D1711yescharacteristic
331#107—21no—
332#107—171no—
333#107—171no—
334#107—171no—
335#107—341no—
336#107—341no—
337#107C22×D1711yescharacteristic
338#107C22×D1711yescharacteristic
339#107C22×D1711yescharacteristic
340#106—341no—
341#106—341no—
342#106—341no—
343#106—341no—
344#102—21no—
345#102—341no—
346#102—341no—
347#102—341no—
348#97—171no—
349#97—171no—
350C4×C8—171no—
351C4×C8C22×D1711yescharacteristic
352C17#3363111yescharacteristic, complemented, Sylow
353C24—681no—
354C2×Q8—41no—
355C2×Q8—681no—
356C2×D4—21no—
357C2×D4—344no—
358C2×D4—341no—
359C2×D4—682no—
360C2×D4—681no—
361C22×C4—41no—
362C22×C4—171no—
363C22×C4—681no—
364C22×C4—681no—
365C22×C4—681no—
366C22×C4—681no—
367D8—21no—
368D8—172no—
369D8—344no—
370D8—341no—
371D8Q8×D1712yes—
372M16—342no—
373C2×C8—171no—
374C2×C8—172no—
375C2×C8—172no—
376C2×C8—171no—
377C2×C8D4×D1711yescharacteristic
378C2×C8C23×D1711yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—171no—
384C4⋊C4—171no—
385C4⋊C4—341no—
386C4⋊C4—341no—
387C4⋊C4—341no—
388C4⋊C4—681no—
389C4⋊C4—681no—
390C4⋊C4—681no—
391C4⋊C4—681no—
392C4⋊C4—681no—
393C4⋊C4—681no—
394C22⋊C4—41no—
395C22⋊C4—171no—
396C22⋊C4—171no—
397C22⋊C4—681no—
398C22⋊C4—681no—
399C22⋊C4—681no—
400C22⋊C4—681no—
401C22⋊C4—681no—
402C42—171no—
403C42—681no—
404C42—681no—
405C42D4×D1711yescharacteristic
406C16—21no—
407C16—21no—
408C16—341no—
409C16—341no—
410C23—41no—
411C23—2721no—
412C23—2721no—
413C23—171no—
414C23—684no—
415C23—681no—
416Q8—161no—
417Q8—2721no—
418D4—22no—
419D4—41no—
420D4—342no—
421D4—684no—
422D4—681no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—161no—
427C2×C4—2721no—
428C2×C4—2721no—
429C2×C4—2721no—
430C2×C4—2721no—
431C2×C4—171no—
432C2×C4—172no—
433C2×C4—172no—
434C2×C4—171no—
435C2×C4—171no—
436C2×C4—681no—
437C2×C4—681no—
438C2×C4—681no—
439C2×C4D8×D1711yescharacteristic
440C2×C4D8×D1711yescharacteristic
441C2×C4C2×D4×D1711yescharacteristic
442C8—21no—
443C8—172no—
444C8—341no—
445C8C2×Q8×D1711yescharacteristic
446C8#41×D1711yescharacteristic
447C22—42no—
448C22—81no—
449C22—81no—
450C22—2724no—
451C22—172no—
452C22—341no—
453C22—341no—
454C22—682no—
455C22—1361no—
456C22—1361no—
457C22C2×D8×D1711yescharacteristic
458C4—21no—
459C4—21no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—172no—
464C4—341no—
465C4—341no—
466C4—1361no—
467C4—1361no—
468C4—1361no—
469C4C22⋊Q8×D1711yescharacteristic
470C4#137×D1711yescharacteristic
471C2—82no—
472C2—342no—
473C2—1362no—
474C2D17×#47411yescharacteristic
475C2#1454192411yescharacteristic
476C2#1454192611yescharacteristic
subgroupG/H
centerC22C2×D8×D17
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#2226C2
socleC22×C17C22×D8
prime radicalC2×D8×D17C22
hypercenter#2226D17
hypocenterC17#33631
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2226

Complements

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

NH ≅ G/Ntype
1. #14539317C2semidirect
3. #14530943C2semidirect
4. #14530942C2semidirect
5. #14530941C2semidirect
6. #14530940C2semidirect
7. #14530930C2semidirect
8. #14530919C2semidirect
9. #14530918C2semidirect
10. #14530571C2semidirect
11. #14530567C2semidirect
12. #14530541C2semidirect
15. C17×#2226C2semidirect
23. #11763213C22semidirect
24. #11763211C22semidirect
25. #11763210C22semidirect
37. #11762637C22semidirect
38. #11762636C22semidirect
39. #11762635C22semidirect
40. #11762634C22semidirect
41. #11762634C22semidirect
42. #11762634C22semidirect
43. C4⋊#63214C22semidirect
44. #11762627C22semidirect
45. #11762627C22semidirect
52. C17×#500C22semidirect
54. C17×#367C22semidirect
55. C17×#367C22semidirect
56. C17×C4⋊C16C22semidirect
70. D8×D17Q8semidirect
216. #2226D17semidirect
256. #500C2×D17semidirect
273. #367C2×D17semidirect
274. #367C2×D17semidirect
282. C4⋊C16C2×D17semidirect
352. C17#33631semidirect

Maximal subgroups

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

subgroupindex
#145393172
#145376312
#145309432
#145309422
#145309412
#145309402
#145309302
#145309192
#145309182
#145305712
#145305672
#145305412
#145304982
D17×#3572
C17×#22262
#3363117