Finite Group Databasebeta

← #17191156

Subgroups — #17191156

Order 4352.

622 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14541174C211yesmaximal, characteristic, complemented
2#14540458C211yesmaximal, characteristic
3#14531006C211yesmaximal, characteristic, complemented
4#14530971C211yesmaximal, characteristic, complemented
5#14530930C211yesmaximal, characteristic, complemented
6#14530881C211yesmaximal, characteristic, complemented
7#14530878C211yesmaximal, characteristic, complemented
8#14530823C211yesmaximal, characteristic, complemented
9#14530715C211yesmaximal, characteristic, complemented
10#14530714C211yesmaximal, characteristic, complemented
11#14530635C211yesmaximal, characteristic, complemented
12#14530601C211yesmaximal, characteristic, complemented
13C2×#11762609C211yesmaximal, characteristic, complemented
14D17×#357C211yesmaximal, characteristic, complemented
15C17×#2233C211yesmaximal, characteristic, complemented
16C22×D136C2211yescharacteristic
17C2×D8×D17C2211yescharacteristic
18#11763514—21no—
19#11763443—21no—
20C2×#10587471C2211yescharacteristic
21C2×#10587471C2211yescharacteristic, complemented
22C2×#10587498C2211yescharacteristic, complemented
23C22×#63214C2211yescharacteristic, complemented
24C2×C16×D17C2211yescharacteristic, complemented
25#11763213C2211yescharacteristic, complemented
26#11763184C2211yescharacteristic
27#11763146—21no—
28#11763144—21no—
29#11763103—21no—
30#11763051C2211yescharacteristic, complemented
31#11763050C2211yescharacteristic, complemented
32#11763046C2211yescharacteristic
33#11763003—21no—
34#11762944—21no—
35#11762942—21no—
36#11762903C2211yescharacteristic, complemented
37#11762891C2211yescharacteristic
38C8×C22⋊D17C2211yescharacteristic
39#107×D17C2211yescharacteristic, complemented
40#11762636C2211yescharacteristic, complemented
41#11762627C2211yescharacteristic, complemented
42#11762612C2211yescharacteristic, complemented
43#11762611C2211yescharacteristic
44#11762611C2211yescharacteristic, complemented
45#11762610C2211yescharacteristic, complemented
46#11762609C2211yescharacteristic
47#11762609C2211yescharacteristic
48#11762609C2211yescharacteristic, complemented
49#11762609C2211yescharacteristic, complemented
50#11762588C2211yescharacteristic, complemented
51C16×Dic17C2211yescharacteristic, complemented
52C17×#466C2211yescharacteristic
53C2×C17×#107C2211yescharacteristic, complemented
54C17×#367C2211yescharacteristic, complemented
55C17×#367C2211yescharacteristic, complemented
56C17×#357C2211yescharacteristic, complemented
57C17×#357C2211yescharacteristic, complemented
58C17×C22⋊C16C2211yescharacteristic, complemented
59C2×D4×D17—21no—
60C2×D4×D17—41no—
61C22×D68—21no—
62C2×D8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C17×#121—21no—
65C2×C4⋊C4×C17—21no—
66#10587626—21no—
67#10587623—41no—
68C2×#63228—21no—
69C2×#63228—21no—
70C4×C22⋊D17—41no—
71C4×C22⋊D17D411yescharacteristic
72#10587610—21no—
73#10587610—41no—
74D8×D17—22no—
75D8×D17—41no—
76C2×Dic68C2311yescharacteristic
77C2×D136—22no—
78C2×D136—21no—
79C2×D136—41no—
80C2×D136C2311yescharacteristic
81C2×D136C2311yescharacteristic
82C2×C8×D17C2311yescharacteristic
83#10587583—21no—
84#10587582—41no—
85#10587581—21no—
86#10587579—21no—
87C4⋊C4×D17—21no—
88#10587575—41no—
89#10587574—21no—
90#10587573—21no—
91#10587570—41no—
92#10587568—41no—
93C2×C16×C17C2311yescharacteristic
94C2×C16×C17C2311yescharacteristic
95C17×#107—21no—
96C17×#107C2311yescharacteristic
97C17×#107C2311yescharacteristic
98C17×#107C2311yescharacteristic
99C17×#102—21no—
100C22⋊#3622D411yescharacteristic
101#10587498—41no—
102#10587491D411yescharacteristic
103#10587489C2311yescharacteristic
104#10587487—21no—
105#10587485D411yescharacteristic
106C8×Dic17C2311yescharacteristic
107C2×#63214—41no—
108C2×#63214C2311yescharacteristic
109C2×#63214C2311yescharacteristic
110#10587481—21no—
111#10587481—21no—
112#10587480—21no—
113#10587480—21no—
114#10587478—21no—
115#10587478—21no—
116#10587471—22no—
117#10587471—41no—
118C16×D17—41no—
119C23×D17—41no—
120C23×D17—41no—
121C2×D4×C17—21no—
122C2×D4×C17—41no—
123C22×C4×C17—41no—
124C22×C4×C17C2×D411yescharacteristic
125C2×C22⋊D17—41no—
126C2×C22⋊D17—41no—
127C2×C22⋊D17—41no—
128C2×C22⋊D17D811yescharacteristic
129D4×D17—42no—
130D4×D17—81no—
131D4×D17—84no—
132C2×D68—21no—
133C2×D68—21no—
134C2×D68—42no—
135C2×D68—41no—
136C2×D68—41no—
137C2×D68—81no—
138C2×C4×D17—41no—
139C2×C4×D17—41no—
140C2×C4×D17C2×D411yescharacteristic
141C2×Dic34—21no—
142C2×Dic34—41no—
143D8×C17—21no—
144D8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17C2×D411yescharacteristic
148C2×C8×C17C2411yescharacteristic
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—41no—
153C22⋊C4×C17—41no—
154C22⋊Dic17—41no—
155C22⋊Dic17D811yescharacteristic
156#63228—42no—
157#63228—81no—
158#63227—41no—
159#63227—41no—
160#63227—41no—
161#63227D811yescharacteristic
162C4⋊Dic17—21no—
163C4⋊Dic17—41no—
164#63225—41no—
165#63225—41no—
166#63225—41no—
167#63225—41no—
168#63225D811yescharacteristic
169C4×Dic17—41no—
170C4×Dic17C2×D411yescharacteristic
171C2×#3622C2×D411yescharacteristic
172C2×#3622C2×D411yescharacteristic
173Dic68—21no—
174Dic68—21no—
175D136—21no—
176D136—21no—
177D136—21no—
178D136—42no—
179D136#4112yes—
180C8×D17—21no—
181C8×D17—21no—
182C16×C17—21no—
183C16×C17—21no—
184#63214—21no—
185#63214—21no—
186#33862—171nomaximal, Sylow
187C23×C17—41no—
188C23×C17C2×D811yescharacteristic
189C22×D17—41no—
190C22×D17—41no—
191C22×D17—82no—
192C22×D17—82no—
193C22×D17—82no—
194C22×D17—82no—
195C22×D17—161no—
196C22×D17—161no—
197C22×D17C2×D811yescharacteristic
198D4×C17—41no—
199D4×C17—41no—
200D4×C17—82no—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—41no—
204C2×C4×C17—41no—
205C2×C4×C17—41no—
206C2×C4×C17—161no—
207C2×C4×C17C2×D811yescharacteristic
208C2×C4×C17C22×D411yescharacteristic
209C22⋊D17—82no—
210C22⋊D17—82no—
211C22⋊D17—161no—
212C22⋊D17SD3214yescomplemented
213C2×Dic17—41no—
214C2×Dic17—41no—
215C2×Dic17C2×D811yescharacteristic
216C2×Dic17C2×D811yescharacteristic
217C2×Dic17C2×D811yescharacteristic
218D68—22no—
219D68—41no—
220D68—41no—
221D68—41no—
222D68—82no—
223D68—82no—
224C4×D17—21no—
225C4×D17—21no—
226C4×D17—82no—
227C4×D17—161no—
228Dic34—41no—
229Dic34—41no—
230Dic34—161no—
231C8×C17—21no—
232C8×C17C2×#4111yescharacteristic
233C8×C17H(5,𝔽2)11yescharacteristic
234#3622—21no—
235#3622—21no—
236#3338—171no—
237D4×D8—171no—
238#2240—171no—
239#2233—171no—
240#2233D1711yescharacteristic, complemented
241#2226—171no—
242#2217—171no—
243#2217—171no—
244#2207—171no—
245#2187—171no—
246#2187—171no—
247C4×SD32—171no—
248D4×C16—171no—
249C2×#357—171no—
250C2×#357—171no—
251C22×C17—21no—
252C22×C17—21no—
253C22×C17—81no—
254C22×C17—81no—
255C22×C17—81no—
256C22×C17C2×SD3212yes—
257C22×C17C22×D811yescharacteristic
258C2×D17—21no—
259C2×D17—21no—
260C2×D17—42no—
261C2×D17—81no—
262C2×D17—81no—
263C2×D17—81no—
264C2×D17—81no—
265C2×D17—81no—
266C2×D17—164no—
267C2×D17—164no—
268C2×D17C2×SD3212yes—
269C4×C17—21no—
270C4×C17—81no—
271C4×C17—81no—
272C4×C17—81no—
273C4×C17#54611yescharacteristic
274C4×C17#57711yescharacteristic
275Dic17—21no—
276Dic17—21no—
277Dic17—81no—
278Dic17—81no—
279Dic17C2×SD3212yes—
280C22×D8—171no—
281C22×D8—171no—
282#554—341no—
283D42—341no—
284C2×SD32—171no—
285C2×SD32—171no—
286C2×SD32—171no—
287C22×C16—171no—
288C22×C16—171no—
289#500—171no—
290#493—171no—
291#484—341no—
292#484—341no—
293#477—341no—
294#467—171no—
295#467—171no—
296#466—171no—
297#466C2×D1711yescharacteristic
298#459—341no—
299#447—341no—
300#447—341no—
301C4×Q16—171no—
302C4×D8—171no—
303C8×D4—171no—
304C2×#107—171no—
305C2×#107C2×D1711yescharacteristic, complemented
306#367—171no—
307#367C2×D1711yescharacteristic, complemented
308#367C2×D1711yescharacteristic, complemented
309C4⋊C16—171no—
310#358—171no—
311#357—171no—
312#357—171no—
313#357—171no—
314#357—171no—
315#357—171no—
316#357—171no—
317#357—171no—
318#357C2×D1711yescharacteristic, complemented
319#357C2×D1711yescharacteristic, complemented
320C22⋊C16—171no—
321C22⋊C16C2×D1711yescharacteristic, complemented
322C4×C16—171no—
323C2×C17—22no—
324C2×C17—161no—
325C2×C17#329411yescharacteristic
326C2×C17C22×SD3211yescharacteristic
327C2×C17#341511yescharacteristic
328D17—22no—
329D17—82no—
330D17—161no—
331C22×D4—341no—
332C22×D4—341no—
333C22×D4—681no—
334C2×Q16—171no—
335C2×D8—171no—
336C2×D8—171no—
337C2×D8—342no—
338C2×D8—341no—
339C2×D8—341no—
340C2×D8—342no—
341C2×D8—341no—
342C2×D8—681no—
343C2×D8—681no—
344C2×D8C22×D1711yescharacteristic
345C22×C8—171no—
346C22×C8C22×D1711yescharacteristic
347C4⋊Q8—341no—
348#127—341no—
349#125—681no—
350C22⋊Q8—341no—
351C22⋊Q8—341no—
352C22⋊Q8—681no—
353C22⋊Q8—681no—
354#121—21no—
355#121—341no—
356#121—681no—
357#120—681no—
358#120—681no—
359C4×Q8—341no—
360C4×D4—171no—
361C4×D4—341no—
362C4×D4—681no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—341no—
365SD32—342no—
366SD32—681no—
367SD32—681no—
368C2×C16—171no—
369C2×C16—171no—
370C2×C16—681no—
371C2×C16—681no—
372C2×C16C22×D1711yescharacteristic
373C2×C16C22×D1711yescharacteristic
374#107—21no—
375#107—171no—
376#107—341no—
377#107—341no—
378#107C22×D1711yescharacteristic
379#107C22×D1711yescharacteristic
380#107C22×D1711yescharacteristic
381C4⋊C8—171no—
382#103—341no—
383#103—341no—
384#103—341no—
385#102—21no—
386#102—341no—
387#102—341no—
388C22⋊C8—171no—
389C22⋊C8—171no—
390C4×C8—171no—
391C17#3386211yescharacteristic, complemented, Sylow
392C24—681no—
393C24—681no—
394C2×Q8—341no—
395C2×Q8—681no—
396C2×D4—21no—
397C2×D4—41no—
398C2×D4—171no—
399C2×D4—341no—
400C2×D4—341no—
401C2×D4—682no—
402C2×D4—681no—
403C2×D4—681no—
404C2×D4—681no—
405C2×D4—681no—
406C2×D4—682no—
407C2×D4—681no—
408C2×D4—1361no—
409C2×D4—1361no—
410C2×D4—1364no—
411C22×C4—41no—
412C22×C4—171no—
413C22×C4—681no—
414C22×C4—681no—
415C22×C4D4×D1711yescharacteristic
416Q16—341no—
417Q16—341no—
418D8—21no—
419D8—21no—
420D8—172no—
421D8—341no—
422D8—341no—
423D8—341no—
424D8—682no—
425D8—682no—
426D8—1361no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—171no—
430C2×C8—171no—
431C2×C8—341no—
432C2×C8—341no—
433C2×C8D4×D1711yescharacteristic
434C2×C8C23×D1711yescharacteristic
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—41no—
439C4⋊C4—171no—
440C4⋊C4—341no—
441C4⋊C4—681no—
442C4⋊C4—681no—
443C4⋊C4—681no—
444C4⋊C4—681no—
445C4⋊C4—681no—
446C22⋊C4—41no—
447C22⋊C4—171no—
448C22⋊C4—171no—
449C22⋊C4—681no—
450C22⋊C4—681no—
451C22⋊C4—681no—
452C22⋊C4—681no—
453C42—171no—
454C42—681no—
455C16—21no—
456C16—21no—
457C16—341no—
458C16—341no—
459C23—41no—
460C23—2721no—
461C23—2721no—
462C23—171no—
463C23—681no—
464C23—681no—
465C23—1362no—
466C23—1362no—
467C23—1362no—
468C23—1362no—
469C23D8×D1711yescharacteristic
470Q8—2721no—
471Q8—681no—
472Q8—681no—
473D4—41no—
474D4—41no—
475D4—82no—
476D4—2721no—
477D4—174no—
478D4—342no—
479D4—681no—
480D4—681no—
481D4—681no—
482D4—1362no—
483D4—1362no—
484D4—1362no—
485D4—1362no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—161no—
492C2×C4—2721no—
493C2×C4—171no—
494C2×C4—171no—
495C2×C4—171no—
496C2×C4—341no—
497C2×C4—341no—
498C2×C4—681no—
499C2×C4—681no—
500C2×C4—1362no—
501C2×C4D8×D1711yescharacteristic
502C2×C4C2×D4×D1711yescharacteristic
503C8—21no—
504C8—341no—
505C8—341no—
506C8#41×D1711yescharacteristic
507C8#1058769611yescharacteristic
508C22—21no—
509C22—21no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—2724no—
514C22—2724no—
515C22—172no—
516C22—341no—
517C22—341no—
518C22—682no—
519C22—1361no—
520C22—1361no—
521C22—1361no—
522C22—1361no—
523C22—1361no—
524C22SD32×D1712yes—
525C22C2×D8×D1711yescharacteristic
526C4—21no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—172no—
531C4—341no—
532C4—341no—
533C4—1361no—
534C4—1361no—
535C4#1176345111yescharacteristic
536C4#1176361811yescharacteristic
537C2—22no—
538C2—161no—
539C2—2721no—
540C2—342no—
541C2—1362no—
542C2#1454059511yescharacteristic
543C2C2×SD32×D1711yescharacteristic
544C2#1454192411yescharacteristic
subgroupG/H
centerC22C2×D8×D17
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#2233C2
socleC22×C17C22×D8
prime radical#14540458C2
hypercenter#2233D17
hypocenterC17#33862
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2233

Complements

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

NH ≅ G/Ntype
1. #14541174C2semidirect
3. #14531006C2semidirect
4. #14530971C2semidirect
5. #14530930C2semidirect
6. #14530881C2semidirect
7. #14530878C2semidirect
8. #14530823C2semidirect
9. #14530715C2semidirect
10. #14530714C2semidirect
11. #14530635C2semidirect
12. #14530601C2semidirect
13. C2×#11762609C2semidirect
14. D17×#357C2semidirect
15. C17×#2233C2semidirect
21. C2×#10587471C22semidirect
22. C2×#10587498C22semidirect
23. C22×#63214C22semidirect
24. C2×C16×D17C22semidirect
25. #11763213C22semidirect
30. #11763051C22semidirect
31. #11763050C22semidirect
36. #11762903C22semidirect
39. #107×D17C22semidirect
40. #11762636C22semidirect
41. #11762627C22semidirect
42. #11762612C22semidirect
44. #11762611C22semidirect
45. #11762610C22semidirect
48. #11762609C22semidirect
49. #11762609C22semidirect
50. #11762588C22semidirect
51. C16×Dic17C22semidirect
53. C2×C17×#107C22semidirect
54. C17×#367C22semidirect
55. C17×#367C22semidirect
56. C17×#357C22semidirect
57. C17×#357C22semidirect
58. C17×C22⋊C16C22semidirect
212. C22⋊D17SD32semidirect
240. #2233D17semidirect
305. C2×#107C2×D17semidirect
307. #367C2×D17semidirect
308. #367C2×D17semidirect
318. #357C2×D17semidirect
319. #357C2×D17semidirect
321. C22⋊C16C2×D17semidirect
391. C17#33862semidirect

Maximal subgroups

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

subgroupindex
#145411742
#145404582
#145310062
#145309712
#145309302
#145308812
#145308782
#145308232
#145307152
#145307142
#145306352
#145306012
C2×#117626092
D17×#3572
C17×#22332
#3386217