Finite Group Databasebeta

← #17191184

Subgroups — #17191184

Order 4352.

580 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14541187C211yesmaximal, characteristic, complemented
2#14531015C211yesmaximal, characteristic, complemented
3#14530984C211yesmaximal, characteristic, complemented
4#14530972C211yesmaximal, characteristic, complemented
5#14530941C211yesmaximal, characteristic, complemented
6#14530882C211yesmaximal, characteristic, complemented
7#14530868C211yesmaximal, characteristic, complemented
8#14530836C211yesmaximal, characteristic, complemented
9#14530707C211yesmaximal, characteristic, complemented
10#14530696C211yesmaximal, characteristic, complemented
11C4×#10587497C211yesmaximal, characteristic, complemented
12D4×#63214C211yesmaximal, characteristic, complemented
13#14530491C211yesmaximal, characteristic, complemented
14C2×#11762610C211yesmaximal, characteristic, complemented
15C17×#3280C211yesmaximal, characteristic, complemented
16#11763505—21no—
17C2×#10587499C2211yescharacteristic, complemented
18C2×#10587497C2211yescharacteristic
19C2×#10587497C2211yescharacteristic
20C22×#63214C2211yescharacteristic
21C22×#63214C2211yescharacteristic, complemented
22#11763211C2211yescharacteristic, complemented
23#11763134—21no—
24#11763128—21no—
25#11763110—21no—
26#11763045C2211yescharacteristic
27#11763045C2211yescharacteristic, complemented
28C4×D136C2211yescharacteristic, complemented
29#11762847C2211yescharacteristic, complemented
30C2×#10587489C2211yescharacteristic
31#11762636C2211yescharacteristic, complemented
32#11762634C2211yescharacteristic, complemented
33#11762634C2211yescharacteristic, complemented
34C4⋊#63214C2211yescharacteristic, complemented
35#11762613C2211yescharacteristic, complemented
36#11762613C2211yescharacteristic, complemented
37#11762613C2211yescharacteristic, complemented
38#11762610C2211yescharacteristic
39#11762610C2211yescharacteristic
40#11762610C2211yescharacteristic
41#11762610C2211yescharacteristic
42#11762610C2211yescharacteristic
43#11762610C2211yescharacteristic
44#11762609C2211yescharacteristic, complemented
45C22⋊#63214C2211yescharacteristic
46C22⋊#63214C2211yescharacteristic, complemented
47C4×#63214C2211yescharacteristic, complemented
48C2×C17×#135C2211yescharacteristic
49C22×D8×C17C2211yescharacteristic
50C17×#546—21no—
51C17×#495C2211yescharacteristic
52C17×#467C2211yescharacteristic
53C17×#466C2211yescharacteristic
54C17×#463—21no—
55C17×#450—21no—
56C17×#449—21no—
57C4×D8×C17C2211yescharacteristic
58C8×D4×C17C2211yescharacteristic
59C2×#41×C17—21no—
60C22×D4×C17—21no—
61C17×#135—22no—
62C17×#135—41no—
63C2×Q16×C17C2311yescharacteristic
64C2×SD16×C17—21no—
65C2×SD16×C17—21no—
66C2×D8×C17—22no—
67C2×D8×C17—21no—
68C2×D8×C17—41no—
69C2×D8×C17C2311yescharacteristic
70C2×D8×C17C2311yescharacteristic
71C22×C8×C17C2311yescharacteristic
72C22×C8×C17C2311yescharacteristic
73C17×#124—21no—
74C17×#123—41no—
75C17×C22⋊Q8—21no—
76C17×#121—21no—
77C17×#121—41no—
78C17×#120—41no—
79C4×D4×C17—21no—
80C4×D4×C17D411yescharacteristic
81C2×C22⋊C4×C17—41no—
82#10587625—41no—
83#10587616—21no—
84#10587616—21no—
85#10587612—21no—
86C2×C4⋊Dic17—21no—
87C2×D136C2311yescharacteristic
88#10587584—41no—
89#10587569—41no—
90#10587567—41no—
91C4×D68—21no—
92#10587556—21no—
93C17×#107C2311yescharacteristic
94C17×C4⋊C8D411yescharacteristic
95C17×#103—21no—
96C17×#102—21no—
97C17×#102—21no—
98C17×C22⋊C8D411yescharacteristic
99C17×C22⋊C8D411yescharacteristic
100C4×C8×C17C2311yescharacteristic
101#10587499—41no—
102#10587497—22no—
103#10587497—41no—
104#10587492—21no—
105#10587492—21no—
106#10587492—21no—
107#10587489—21no—
108#10587489C2311yescharacteristic
109#10587489C2311yescharacteristic
110#10587489C2311yescharacteristic
111#10587488—21no—
112#10587488—21no—
113C2×#63214—41no—
114C2×#63214—41no—
115C2×#63214C2311yescharacteristic
116C2×#63214C2311yescharacteristic
117C2×#63214C2311yescharacteristic
118C2×#63214C2311yescharacteristic
119C24×C17—41no—
120#41×C17—42no—
121#41×C17—81no—
122C2×Q8×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17—42no—
126C2×D4×C17—41no—
127C2×D4×C17—41no—
128C2×D4×C17—41no—
129C2×D4×C17—81no—
130C2×D4×C17D811yescharacteristic
131C22×C4×C17—41no—
132C22×C4×C17—41no—
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C2×C22⋊D17—41no—
136C2×C22⋊D17—41no—
137C22×Dic17—41no—
138C2×D68—21no—
139C2×C4×D17—41no—
140Q16×C17—21no—
141Q16×C17—21no—
142SD16×C17—42no—
143SD16×C17—81no—
144D8×C17—21no—
145D8×C17—21no—
146D8×C17—21no—
147D8×C17—42no—
148D8×C17#4112yes—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17C2×D411yescharacteristic
154C2×C8×C17C2×D411yescharacteristic
155C2×C8×C17C2×D411yescharacteristic
156C2×C8×C17C2411yescharacteristic
157C4⋊C4×C17—21no—
158C4⋊C4×C17—41no—
159C4⋊C4×C17D811yescharacteristic
160C22⋊C4×C17—41no—
161C22⋊C4×C17—41no—
162C22⋊C4×C17—41no—
163C22⋊C4×C17—41no—
164C22⋊C4×C17—41no—
165C22⋊C4×C17D811yescharacteristic
166C22⋊C4×C17D811yescharacteristic
167C42×C17C2×D411yescharacteristic
168C22⋊Dic17—41no—
169#63227—41no—
170#63227—41no—
171#63227—41no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—41no—
176#63225—41no—
177#63225—41no—
178C4×Dic17—41no—
179D136—21no—
180D136—21no—
181#63214—21no—
182#63214—21no—
183#63214—21no—
184#63214—21no—
185#33864—171nomaximal, Sylow
186C23×C17—41no—
187C23×C17—41no—
188C23×C17—82no—
189C23×C17—82no—
190C23×C17—161no—
191C23×C17C2×D811yescharacteristic
192C23×C17C2×D811yescharacteristic
193C22×D17—41no—
194Q8×C17—41no—
195Q8×C17—41no—
196D4×C17—22no—
197D4×C17—22no—
198D4×C17—41no—
199D4×C17—41no—
200D4×C17—41no—
201D4×C17—82no—
202D4×C17—82no—
203D4×C17—161no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—82no—
211C2×C4×C17—161no—
212C2×C4×C17C2×D811yescharacteristic
213C2×C4×C17C2×D811yescharacteristic
214C2×C4×C17C2×D811yescharacteristic
215C2×C4×C17C2×D811yescharacteristic
216C2×C4×C17C22×D411yescharacteristic
217C22⋊D17—82no—
218C22⋊D17—161no—
219C2×Dic17—41no—
220C2×Dic17—41no—
221C2×Dic17—41no—
222C2×Dic17—161no—
223D68—41no—
224D68—41no—
225C4×D17—161no—
226C8×C17—21no—
227C8×C17—21no—
228C8×C17—21no—
229C8×C17C2×#4111yescharacteristic
230C8×C17H(5,𝔽2)11yescharacteristic
231#3339—171no—
232#3280D1711yescharacteristic, complemented
233#2242—171no—
234#2236—171no—
235#2234—171no—
236#2228—171no—
237#2217—171no—
238#2215—171no—
239#2210—171no—
240#2186—171no—
241#2184—171no—
242C4×D16—171no—
243D4×C16—171no—
244#2138—171no—
245C2×#357—171no—
246C22×C17—21no—
247C22×C17—21no—
248C22×C17—21no—
249C22×C17—21no—
250C22×C17—21no—
251C22×C17—42no—
252C22×C17—81no—
253C22×C17—81no—
254C22×C17—81no—
255C22×C17—81no—
256C22×C17—81no—
257C22×C17—164no—
258C22×C17#50812yes—
259C22×C17C22×D811yescharacteristic
260C2×D17—81no—
261C2×D17—81no—
262C2×D17—81no—
263C4×C17—21no—
264C4×C17—21no—
265C4×C17—21no—
266C4×C17—21no—
267C4×C17—81no—
268C4×C17—81no—
269C4×C17#54611yescharacteristic
270C4×C17#57711yescharacteristic
271Dic17—81no—
272Dic17—81no—
273Dic17—81no—
274C2×#135C2×D1711yescharacteristic
275C22×D8C2×D1711yescharacteristic
276#553—341no—
277#546—21no—
278C2×SD32—171no—
279C2×D16—171no—
280C2×D16—171no—
281C22×C16—171no—
282C22×C16—171no—
283#499—171no—
284#495C2×D1711yescharacteristic
285#482—341no—
286#481—341no—
287#478—341no—
288#467C2×D1711yescharacteristic
289#466—171no—
290#466—171no—
291#466C2×D1711yescharacteristic
292#463—21no—
293#450—21no—
294#449—21no—
295C4×D8—171no—
296C4×D8C2×D1711yescharacteristic
297C8×D4C2×D1711yescharacteristic
298#427—171no—
299C2×#107—171no—
300#367—171no—
301#366—171no—
302#366—171no—
303C4⋊C16—171no—
304#358—171no—
305#358—171no—
306#358—171no—
307#357—171no—
308#357—171no—
309#357—171no—
310#357—171no—
311#357—171no—
312#357—171no—
313#357—171no—
314C22⋊C16—171no—
315C22⋊C16—171no—
316C4×C16—171no—
317C2×C17—22no—
318C2×C17—41no—
319C2×C17—82no—
320C2×C17—161no—
321C2×C17#329411yescharacteristic
322C2×C17C2×#50811yescharacteristic
323C2×C17#341511yescharacteristic
324D17—161no—
325C2×#41—21no—
326C22×D4—21no—
327#135—22no—
328#135—41no—
329C2×Q16C22×D1711yescharacteristic
330C2×SD16—21no—
331C2×SD16—21no—
332C2×D8—22no—
333C2×D8—21no—
334C2×D8—41no—
335C2×D8—171no—
336C2×D8C22×D1711yescharacteristic
337C2×D8C22×D1711yescharacteristic
338C22×C8C22×D1711yescharacteristic
339C22×C8C22×D1711yescharacteristic
340#126—681no—
341#125—341no—
342#124—21no—
343#123—41no—
344#123—681no—
345C22⋊Q8—21no—
346#121—21no—
347#121—41no—
348#121—341no—
349#121—341no—
350#121—681no—
351#120—41no—
352C4×D4—21no—
353C4×D4—341no—
354C4×D4—681no—
355C4×D4C22⋊D1711yescharacteristic
356#117—341no—
357C2×C4⋊C4—341no—
358C2×C22⋊C4—41no—
359SD32—681no—
360D16—342no—
361D16—681no—
362C2×C16—171no—
363C2×C16—171no—
364C2×C16—171no—
365C2×C16—171no—
366C2×C16—681no—
367C2×C16—681no—
368#107—171no—
369#107—171no—
370#107—171no—
371#107—341no—
372#107C22×D1711yescharacteristic
373#106—341no—
374#106—341no—
375C4⋊C8C22⋊D1711yescharacteristic
376#103—21no—
377#102—21no—
378#102—21no—
379#102—341no—
380#102—341no—
381#102—341no—
382C22⋊C8C22⋊D1711yescharacteristic
383C22⋊C8C22⋊D1711yescharacteristic
384C4×C8C22×D1711yescharacteristic
385C17#3386411yescharacteristic, complemented, Sylow
386C24—41no—
387#41—42no—
388#41—81no—
389C2×Q8—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—42no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—81no—
397C2×D4—341no—
398C2×D4—681no—
399C2×D4—681no—
400C2×D4#6322811yescharacteristic
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—681no—
404C22×C4—681no—
405C22×C4C2×C22⋊D1711yescharacteristic
406C22×C4C2×C22⋊D1711yescharacteristic
407Q16—21no—
408Q16—21no—
409SD16—42no—
410SD16—81no—
411D8—21no—
412D8—21no—
413D8—21no—
414D8—42no—
415D8—341no—
416D8—341no—
417D8#6325412yes—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8C2×C22⋊D1711yescharacteristic
423C2×C8C2×C22⋊D1711yescharacteristic
424C2×C8C2×C22⋊D1711yescharacteristic
425C2×C8C23×D1711yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—341no—
429C4⋊C4—341no—
430C4⋊C4—341no—
431C4⋊C4—681no—
432C4⋊C4—681no—
433C4⋊C4—681no—
434C4⋊C4#6322811yescharacteristic
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—681no—
441C22⋊C4—681no—
442C22⋊C4—681no—
443C22⋊C4—681no—
444C22⋊C4#6322811yescharacteristic
445C22⋊C4#6322811yescharacteristic
446C42—681no—
447C42C2×C22⋊D1711yescharacteristic
448C16—341no—
449C16—341no—
450C16—341no—
451C16—341no—
452C23—41no—
453C23—41no—
454C23—82no—
455C23—82no—
456C23—161no—
457C23—681no—
458C23C2×#6322811yescharacteristic
459C23C2×#6322811yescharacteristic
460Q8—41no—
461Q8—41no—
462D4—22no—
463D4—22no—
464D4—41no—
465D4—41no—
466D4—41no—
467D4—82no—
468D4—82no—
469D4—161no—
470D4—2721no—
471D4—681no—
472D4—681no—
473D4—1362no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—82no—
481C2×C4—161no—
482C2×C4—2721no—
483C2×C4—2721no—
484C2×C4—681no—
485C2×C4—681no—
486C2×C4—681no—
487C2×C4C2×#6322811yescharacteristic
488C2×C4C2×#6322811yescharacteristic
489C2×C4C2×#6322811yescharacteristic
490C2×C4C2×#6322811yescharacteristic
491C2×C4C22×C22⋊D1711yescharacteristic
492C8—21no—
493C8—21no—
494C8—21no—
495C8C2×#6325411yescharacteristic
496C8#1058769111yescharacteristic
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—42no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—164no—
509C22—1361no—
510C22—1361no—
511C22—1361no—
512C22#1176323912yes—
513C22C22×#6322811yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—81no—
519C4—81no—
520C4—1361no—
521C4—1361no—
522C4—1361no—
523C4#1176345311yescharacteristic
524C4#1176362011yescharacteristic
525C2—22no—
526C2—41no—
527C2—82no—
528C2—161no—
529C2—2721no—
530C2#1454059711yescharacteristic
531C2C2×#1176323911yescharacteristic
532C2#1454192311yescharacteristic
subgroupG/H
centerC22C22×#63228
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#3280C2
socleC22×C17C22×D8
hypercenter#3280D17
hypocenterC17#33864
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. #14541187C2semidirect
2. #14531015C2semidirect
3. #14530984C2semidirect
4. #14530972C2semidirect
5. #14530941C2semidirect
6. #14530882C2semidirect
7. #14530868C2semidirect
8. #14530836C2semidirect
9. #14530707C2semidirect
10. #14530696C2semidirect
11. C4×#10587497C2semidirect
12. D4×#63214C2semidirect
13. #14530491C2semidirect
14. C2×#11762610C2semidirect
15. C17×#3280C2semidirect
17. C2×#10587499C22semidirect
21. C22×#63214C22semidirect
22. #11763211C22semidirect
27. #11763045C22semidirect
28. C4×D136C22semidirect
29. #11762847C22semidirect
31. #11762636C22semidirect
32. #11762634C22semidirect
33. #11762634C22semidirect
34. C4⋊#63214C22semidirect
35. #11762613C22semidirect
36. #11762613C22semidirect
37. #11762613C22semidirect
44. #11762609C22semidirect
46. C22⋊#63214C22semidirect
47. C4×#63214C22semidirect
232. #3280D17semidirect
385. C17#33864semidirect

Maximal subgroups

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

subgroupindex
#145411872
#145310152
#145309842
#145309722
#145309412
#145308822
#145308682
#145308362
#145307072
#145306962
C4×#105874972
D4×#632142
#145304912
C2×#117626102
C17×#32802
#3386417