Finite Group Databasebeta

← #17191170

Subgroups — #17191170

Order 4352.

568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14541152C211yesmaximal, characteristic, complemented
2#14540463C211yesmaximal, characteristic
3#14531017C211yesmaximal, characteristic, complemented
4#14530980C211yesmaximal, characteristic, complemented
5#14530975C211yesmaximal, characteristic, complemented
6#14530943C211yesmaximal, characteristic, complemented
7#14530880C211yesmaximal, characteristic, complemented
8#14530874C211yesmaximal, characteristic, complemented
9#14530849C211yesmaximal, characteristic, complemented
10#14530724C211yesmaximal, characteristic, complemented
11#14530714C211yesmaximal, characteristic, complemented
12#14530639C211yesmaximal, characteristic, complemented
13C16×C22⋊D17C211yesmaximal, characteristic, complemented
14D17×#357C211yesmaximal, characteristic, complemented
15C17×#2138C211yesmaximal, characteristic, complemented
16C2×D8×D17C2211yescharacteristic
17#11763498—21no—
18#11763447—21no—
19C2×#10587500C2211yescharacteristic, complemented
20C2×#10587471C2211yescharacteristic
21C2×#10587498C2211yescharacteristic
22C2×C16×D17C2211yescharacteristic
23#11763210C2211yescharacteristic, complemented
24#11763194C2211yescharacteristic
25#11763150—21no—
26#11763141—21no—
27#11763119—21no—
28#11763051C2211yescharacteristic
29#11763050C2211yescharacteristic, complemented
30#11763047C2211yescharacteristic
31#11763045C2211yescharacteristic
32#11763026—21no—
33#11762958—21no—
34#11762954—21no—
35#11762903C2211yescharacteristic, complemented
36D8×Dic17C2211yescharacteristic
37C8×C22⋊D17C2211yescharacteristic
38#107×D17C2211yescharacteristic
39#11762637C2211yescharacteristic, complemented
40#11762636C2211yescharacteristic, complemented
41#11762634C2211yescharacteristic, complemented
42#11762627C2211yescharacteristic, complemented
43#11762614C2211yescharacteristic, complemented
44#11762613C2211yescharacteristic, complemented
45#11762611C2211yescharacteristic
46#11762610C2211yescharacteristic
47#11762609C2211yescharacteristic
48#11762609C2211yescharacteristic
49C22⋊#63214C2211yescharacteristic, complemented
50#11762588C2211yescharacteristic
51C16×Dic17C2211yescharacteristic, complemented
52C2×C17×#135C2211yescharacteristic
53C22×C16×C17C2211yescharacteristic, complemented
54C17×#427C2211yescharacteristic, complemented
55C17×#358C2211yescharacteristic, complemented
56C17×#358C2211yescharacteristic, complemented
57C17×#357C2211yescharacteristic
58C17×#357C2211yescharacteristic
59C2×#41×C17—21no—
60C2×D4×D17—21no—
61C17×#135—22no—
62C17×#135—41no—
63C2×Q16×C17C2311yescharacteristic
64C2×SD16×C17—21no—
65C2×D8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C17×#117—21no—
68#10587633—21no—
69#10587632—21no—
70C2×#63230—21no—
71#10587625—41no—
72#10587623—41no—
73D4×Dic17—21no—
74C2×#63228—21no—
75#10587616—21no—
76#10587615—41no—
77C4×C22⋊D17D411yescharacteristic
78C2×#63227—41no—
79#10587610—21no—
80D8×D17—22no—
81D8×D17—41no—
82C2×Dic68C2311yescharacteristic
83C2×D136C2311yescharacteristic
84C2×C8×D17C2311yescharacteristic
85#10587583—21no—
86C4⋊C4×D17—21no—
87#10587576—21no—
88#10587573—21no—
89#10587569—41no—
90#10587565—41no—
91#10587558—41no—
92#10587556—41no—
93C2×C16×C17—41no—
94C2×C16×C17C2311yescharacteristic
95C2×C16×C17C2311yescharacteristic
96C17×#107C2311yescharacteristic
97C17×#107C2311yescharacteristic
98C17×#106—21no—
99#10587506—21no—
100#10587503—21no—
101C22⋊#3622D411yescharacteristic
102#10587500—41no—
103#10587498—22no—
104#10587492—21no—
105#10587491D411yescharacteristic
106#10587489C2311yescharacteristic
107#10587489C2311yescharacteristic
108#10587487—21no—
109#10587485D411yescharacteristic
110C8×Dic17C2311yescharacteristic
111C2×#63214C2311yescharacteristic
112C2×#63214C2311yescharacteristic
113#10587481—21no—
114#10587481—21no—
115#10587479—21no—
116#10587478—21no—
117#10587471—41no—
118C16×D17—41no—
119C23×D17—41no—
120#41×C17—42no—
121#41×C17—81no—
122C2×Q8×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17—41no—
125C22×C4×C17—41no—
126C22×C4×C17C2×D411yescharacteristic
127C2×C22⋊D17—41no—
128C2×C22⋊D17—41no—
129C2×C22⋊D17D811yescharacteristic
130C22×Dic17—41no—
131D4×D17—42no—
132D4×D17—81no—
133C2×D68—21no—
134C2×C4×D17—41no—
135C2×C4×D17C2×D411yescharacteristic
136C2×Dic34—21no—
137Q16×C17—21no—
138Q16×C17—21no—
139SD16×C17—42no—
140D8×C17—21no—
141D8×C17#4112yes—
142C2×C8×C17—21no—
143C2×C8×C17—21no—
144C2×C8×C17C2×D411yescharacteristic
145C2×C8×C17C2411yescharacteristic
146C4⋊C4×C17—21no—
147C4⋊C4×C17—21no—
148C22⋊C4×C17—41no—
149C42×C17—41no—
150C22⋊Dic17—41no—
151C22⋊Dic17—41no—
152C22⋊Dic17D811yescharacteristic
153#63230—81no—
154#63228—42no—
155#63227—41no—
156#63227—41no—
157#63227—41no—
158#63227—41no—
159#63227—41no—
160#63227D811yescharacteristic
161C4⋊Dic17—21no—
162C4⋊Dic17—21no—
163C4⋊Dic17—41no—
164C4⋊Dic17—41no—
165#63225—41no—
166#63225—41no—
167#63225—41no—
168#63225—41no—
169#63225—41no—
170#63225D811yescharacteristic
171C4×Dic17—41no—
172C4×Dic17C2×D411yescharacteristic
173C2×#3622C2×D411yescharacteristic
174C2×#3622C2×D411yescharacteristic
175Dic68—21no—
176Dic68—21no—
177D136—21no—
178D136—21no—
179C8×D17—21no—
180C8×D17—21no—
181C16×C17—21no—
182C16×C17—21no—
183#63214—21no—
184#63214—21no—
185#33863—171nomaximal, Sylow
186C23×C17—41no—
187C23×C17C2×D811yescharacteristic
188C22×D17—41no—
189C22×D17—82no—
190C22×D17—82no—
191C22×D17—161no—
192C22×D17C2×D811yescharacteristic
193Q8×C17—41no—
194Q8×C17—41no—
195D4×C17—22no—
196D4×C17—41no—
197D4×C17—82no—
198C2×C4×C17—21no—
199C2×C4×C17—21no—
200C2×C4×C17—41no—
201C2×C4×C17—41no—
202C2×C4×C17—41no—
203C2×C4×C17—82no—
204C2×C4×C17C2×D811yescharacteristic
205C2×C4×C17C22×D411yescharacteristic
206C22⋊D17—22no—
207C22⋊D17—82no—
208C22⋊D17—161no—
209C2×Dic17—41no—
210C2×Dic17—41no—
211C2×Dic17—41no—
212C2×Dic17—161no—
213C2×Dic17C2×D811yescharacteristic
214C2×Dic17C2×D811yescharacteristic
215C2×Dic17C2×D811yescharacteristic
216D68—41no—
217D68—41no—
218C4×D17—21no—
219C4×D17—21no—
220C4×D17—82no—
221Dic34—41no—
222Dic34—41no—
223C8×C17—21no—
224C8×C17C2×#4111yescharacteristic
225C8×C17H(5,𝔽2)11yescharacteristic
226#3622—21no—
227#3622—21no—
228#3336—171no—
229#3280—171no—
230#2242—171no—
231#2235—171no—
232#2234—171no—
233#2228—171no—
234#2217—171no—
235#2216—171no—
236#2211—171no—
237#2189—171no—
238#2187—171no—
239C4×SD32—171no—
240D4×C16—171no—
241#2138D1711yescharacteristic, complemented
242C2×#357—171no—
243C22×C17—21no—
244C22×C17—21no—
245C22×C17—21no—
246C22×C17—42no—
247C22×C17—81no—
248C22×C17—81no—
249C22×C17C22×D811yescharacteristic
250C2×D17—21no—
251C2×D17—21no—
252C2×D17—81no—
253C2×D17—81no—
254C2×D17—81no—
255C2×D17—164no—
256C2×D17#50812yes—
257C4×C17—21no—
258C4×C17—81no—
259C4×C17—81no—
260C4×C17—81no—
261C4×C17#54611yescharacteristic
262C4×C17#57711yescharacteristic
263Dic17—21no—
264Dic17—21no—
265Dic17—21no—
266Dic17—81no—
267Dic17—81no—
268Dic17—81no—
269C2×#135C2×D1711yescharacteristic
270C22×D8—171no—
271#552—341no—
272#546—341no—
273C2×Q32—171no—
274C2×SD32—171no—
275C2×SD32—171no—
276C22×C16—171no—
277C22×C16C2×D1711yescharacteristic, complemented
278#499—171no—
279#495—171no—
280#485—341no—
281#483—341no—
282#479—341no—
283#467—171no—
284#467—171no—
285#467—171no—
286#466—171no—
287#463—341no—
288#450—341no—
289#449—341no—
290C4×Q16—171no—
291C4×D8—171no—
292C8×D4—171no—
293#427C2×D1711yescharacteristic, complemented
294C2×#107—171no—
295#367—171no—
296#367—171no—
297#366—171no—
298C4⋊C16—171no—
299#358—171no—
300#358—171no—
301#358C2×D1711yescharacteristic, complemented
302#358C2×D1711yescharacteristic, complemented
303#357—171no—
304#357—171no—
305#357—171no—
306#357—171no—
307#357C2×D1711yescharacteristic
308#357C2×D1711yescharacteristic
309C22⋊C16—171no—
310C22⋊C16—171no—
311C4×C16—171no—
312C2×C17—41no—
313C2×C17—82no—
314C2×C17#329411yescharacteristic
315C2×C17C2×#50811yescharacteristic
316C2×C17#341611yescharacteristic
317D17—22no—
318D17—161no—
319C2×#41—21no—
320C22×D4—341no—
321#135—22no—
322#135—41no—
323C2×Q16—171no—
324C2×Q16C22×D1711yescharacteristic
325C2×SD16—21no—
326C2×SD16—341no—
327C2×D8—171no—
328C2×D8—341no—
329C2×D8—342no—
330C2×D8—681no—
331C2×D8C22×D1711yescharacteristic
332C22×C8—171no—
333C22×C8C22×D1711yescharacteristic
334#126—681no—
335#125—341no—
336#125—681no—
337#124—341no—
338#123—681no—
339#123—681no—
340C22⋊Q8—341no—
341C22⋊Q8—341no—
342C22⋊Q8—341no—
343#121—341no—
344#121—681no—
345#120—681no—
346C4×Q8—341no—
347C4×D4—171no—
348C4×D4—341no—
349#117—21no—
350#117—681no—
351C2×C4⋊C4—341no—
352C2×C22⋊C4—681no—
353Q32—681no—
354SD32—342no—
355SD32—681no—
356C2×C16—41no—
357C2×C16—171no—
358C2×C16—171no—
359C2×C16—681no—
360C2×C16C22×D1711yescharacteristic
361C2×C16C22×D1711yescharacteristic
362#107—171no—
363#107—171no—
364#107—341no—
365#107C22×D1711yescharacteristic
366#107C22×D1711yescharacteristic
367#106—21no—
368#106—341no—
369C4⋊C8—171no—
370#103—341no—
371#103—341no—
372#103—341no—
373#103—341no—
374#102—341no—
375#102—341no—
376C22⋊C8—171no—
377C22⋊C8—171no—
378C4×C8—171no—
379C17#3386311yescharacteristic, complemented, Sylow
380C24—681no—
381#41—42no—
382#41—81no—
383C2×Q8—21no—
384C2×Q8—341no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—171no—
388C2×D4—341no—
389C2×D4—681no—
390C2×D4—682no—
391C2×D4—681no—
392C2×D4—1361no—
393C22×C4—41no—
394C22×C4—171no—
395C22×C4—681no—
396C22×C4—681no—
397C22×C4D4×D1711yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—341no—
401Q16—341no—
402SD16—42no—
403SD16—1361no—
404D8—21no—
405D8—341no—
406D8—341no—
407D8—682no—
408D8#6325212yes—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—171no—
412C2×C8—171no—
413C2×C8—341no—
414C2×C8—341no—
415C2×C8D4×D1711yescharacteristic
416C2×C8C23×D1711yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—171no—
420C4⋊C4—341no—
421C4⋊C4—341no—
422C4⋊C4—681no—
423C4⋊C4—681no—
424C4⋊C4—681no—
425C4⋊C4—681no—
426C4⋊C4—681no—
427C4⋊C4—681no—
428C4⋊C4—681no—
429C22⋊C4—41no—
430C22⋊C4—171no—
431C22⋊C4—171no—
432C22⋊C4—681no—
433C22⋊C4—681no—
434C22⋊C4—681no—
435C22⋊C4—681no—
436C22⋊C4—681no—
437C22⋊C4—681no—
438C22⋊C4—681no—
439C42—41no—
440C42—171no—
441C42—681no—
442C16—21no—
443C16—21no—
444C16—341no—
445C16—341no—
446C23—41no—
447C23—2721no—
448C23—171no—
449C23—681no—
450C23—1362no—
451C23—1362no—
452C23D8×D1711yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—681no—
456Q8—681no—
457D4—22no—
458D4—41no—
459D4—82no—
460D4—2721no—
461D4—342no—
462D4—681no—
463D4—681no—
464D4—1362no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—2721no—
472C2×C4—171no—
473C2×C4—171no—
474C2×C4—171no—
475C2×C4—341no—
476C2×C4—341no—
477C2×C4—681no—
478C2×C4—681no—
479C2×C4—681no—
480C2×C4—1362no—
481C2×C4D8×D1711yescharacteristic
482C2×C4C2×D4×D1711yescharacteristic
483C8—21no—
484C8—341no—
485C8—341no—
486C8C2×#6325211yescharacteristic
487C8#1058769111yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—2724no—
495C22—172no—
496C22—341no—
497C22—341no—
498C22—1361no—
499C22—1361no—
500C22—1361no—
501C22C2×D8×D1711yescharacteristic
502C4—21no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—341no—
507C4—341no—
508C4—341no—
509C4—1361no—
510C4—1361no—
511C4—1361no—
512C4#1176344911yescharacteristic
513C4#1176361511yescharacteristic
514C2—41no—
515C2—82no—
516C2—2721no—
517C2—342no—
518C2#1454059311yescharacteristic
519C2D17×#50811yescharacteristic
520C2#1454192711yescharacteristic
subgroupG/H
centerC22C2×D8×D17
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#2138C2
socleC22×C17C22×D8
prime radical#14540463C2
hypercenter#2138D17
hypocenterC17#33863
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #14541152C2semidirect
3. #14531017C2semidirect
4. #14530980C2semidirect
5. #14530975C2semidirect
6. #14530943C2semidirect
7. #14530880C2semidirect
8. #14530874C2semidirect
9. #14530849C2semidirect
10. #14530724C2semidirect
11. #14530714C2semidirect
12. #14530639C2semidirect
13. C16×C22⋊D17C2semidirect
14. D17×#357C2semidirect
15. C17×#2138C2semidirect
19. C2×#10587500C22semidirect
23. #11763210C22semidirect
29. #11763050C22semidirect
35. #11762903C22semidirect
39. #11762637C22semidirect
40. #11762636C22semidirect
41. #11762634C22semidirect
42. #11762627C22semidirect
43. #11762614C22semidirect
44. #11762613C22semidirect
49. C22⋊#63214C22semidirect
51. C16×Dic17C22semidirect
53. C22×C16×C17C22semidirect
54. C17×#427C22semidirect
55. C17×#358C22semidirect
56. C17×#358C22semidirect
241. #2138D17semidirect
277. C22×C16C2×D17semidirect
293. #427C2×D17semidirect
301. #358C2×D17semidirect
302. #358C2×D17semidirect
379. C17#33863semidirect

Maximal subgroups

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

subgroupindex
#145411522
#145404632
#145310172
#145309802
#145309752
#145309432
#145308802
#145308742
#145308492
#145307242
#145307142
#145306392
C16×C22⋊D172
D17×#3572
C17×#21382
#3386317