Finite Group Databasebeta

← #17191169

Subgroups — #17191169

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#14541153C211yesmaximal, characteristic, complemented
2#14531015C211yesmaximal, characteristic, complemented
3#14530979C211yesmaximal, characteristic, complemented
4#14530972C211yesmaximal, characteristic, complemented
5#14530941C211yesmaximal, characteristic, complemented
6#14530878C211yesmaximal, characteristic, complemented
7#14530873C211yesmaximal, characteristic, complemented
8#14530843C211yesmaximal, characteristic, complemented
9#14530724C211yesmaximal, characteristic, complemented
10#14530713C211yesmaximal, characteristic, complemented
11C4×#10587498C211yesmaximal, characteristic, complemented
12D4×#63214C211yesmaximal, characteristic, complemented
13#14530491C211yesmaximal, characteristic, complemented
14C2×#11762610C211yesmaximal, characteristic, complemented
15C17×#3280C211yesmaximal, characteristic
16#11763499—21no—
17C2×#10587500C2211yescharacteristic, complemented
18C2×#10587498C2211yescharacteristic
19C2×#10587498C2211yescharacteristic
20C22×#63214C2211yescharacteristic
21C22×#63214C2211yescharacteristic, complemented
22#11763211C2211yescharacteristic, complemented
23#11763151—21no—
24#11763140—21no—
25#11763117—21no—
26#11763050C2211yescharacteristic
27#11763050C2211yescharacteristic, complemented
28C4×Dic68C2211yescharacteristic, complemented
29#11762847C2211yescharacteristic, complemented
30C2×#10587489C2211yescharacteristic
31#11762636C2211yescharacteristic, complemented
32#11762636C2211yescharacteristic, complemented
33#11762634C2211yescharacteristic, complemented
34C4⋊#63214C2211yescharacteristic, complemented
35#11762613C2211yescharacteristic, complemented
36#11762613C2211yescharacteristic, complemented
37#11762613C2211yescharacteristic, complemented
38#11762612C2211yescharacteristic, complemented
39#11762610C2211yescharacteristic
40#11762610C2211yescharacteristic
41#11762610C2211yescharacteristic
42#11762610C2211yescharacteristic
43#11762610C2211yescharacteristic
44#11762610C2211yescharacteristic
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#10587621—41no—
83#10587612—21no—
84C2×C4⋊Dic17—21no—
85#10587610—21no—
86#10587610—21no—
87C2×Dic68C2311yescharacteristic
88#10587575—41no—
89#10587565—41no—
90#10587563—41no—
91#10587556—21no—
92C4×Dic34—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#10587500—41no—
102#10587498—22no—
103#10587498—41no—
104#10587489—21no—
105#10587489C2311yescharacteristic
106#10587489C2311yescharacteristic
107#10587489C2311yescharacteristic
108#10587488—21no—
109#10587488—21no—
110#10587487—21no—
111#10587487—21no—
112#10587487—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
135C22×Dic17—41no—
136C2×Dic34—21no—
137Q16×C17—21no—
138Q16×C17—21no—
139SD16×C17—42no—
140SD16×C17—81no—
141D8×C17—21no—
142D8×C17—21no—
143D8×C17—21no—
144D8×C17—42no—
145D8×C17#4112yes—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17C2×D411yescharacteristic
151C2×C8×C17C2×D411yescharacteristic
152C2×C8×C17C2×D411yescharacteristic
153C2×C8×C17C2411yescharacteristic
154C4⋊C4×C17—21no—
155C4⋊C4×C17—41no—
156C4⋊C4×C17D811yescharacteristic
157C22⋊C4×C17—41no—
158C22⋊C4×C17—41no—
159C22⋊C4×C17—41no—
160C22⋊C4×C17—41no—
161C22⋊C4×C17—41no—
162C22⋊C4×C17D811yescharacteristic
163C22⋊C4×C17D811yescharacteristic
164C42×C17C2×D411yescharacteristic
165C22⋊Dic17—41no—
166C22⋊Dic17—41no—
167C22⋊Dic17—41no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—41no—
172#63225—41no—
173#63225—41no—
174#63225—41no—
175#63225—41no—
176#63225—41no—
177C4×Dic17—41no—
178C4×Dic17—41no—
179Dic68—21no—
180Dic68—21no—
181#63214—21no—
182#63214—21no—
183#63214—21no—
184#63214—21no—
185#33863—171nomaximal, Sylow
186C23×C17—41no—
187C23×C17—41no—
188C23×C17—82no—
189C23×C17—82no—
190C23×C17—161no—
191C23×C17C2×D811yescharacteristic
192C23×C17C2×D811yescharacteristic
193Q8×C17—41no—
194Q8×C17—41no—
195D4×C17—22no—
196D4×C17—22no—
197D4×C17—41no—
198D4×C17—41no—
199D4×C17—41no—
200D4×C17—82no—
201D4×C17—82no—
202D4×C17—161no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—41no—
208C2×C4×C17—41no—
209C2×C4×C17—82no—
210C2×C4×C17—161no—
211C2×C4×C17C2×D811yescharacteristic
212C2×C4×C17C2×D811yescharacteristic
213C2×C4×C17C2×D811yescharacteristic
214C2×C4×C17C2×D811yescharacteristic
215C2×C4×C17C22×D411yescharacteristic
216C2×Dic17—41no—
217C2×Dic17—41no—
218C2×Dic17—41no—
219C2×Dic17—41no—
220C2×Dic17—82no—
221Dic34—41no—
222Dic34—41no—
223C8×C17—21no—
224C8×C17—21no—
225C8×C17—21no—
226C8×C17C2×#4111yescharacteristic
227C8×C17H(5,𝔽2)11yescharacteristic
228#3336—171no—
229#3280D1711yescharacteristic
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#2138—171no—
242C2×#357—171no—
243C22×C17—21no—
244C22×C17—21no—
245C22×C17—21no—
246C22×C17—21no—
247C22×C17—21no—
248C22×C17—42no—
249C22×C17—81no—
250C22×C17—81no—
251C22×C17—81no—
252C22×C17—81no—
253C22×C17—81no—
254C22×C17—164no—
255C22×C17#50812yes—
256C22×C17C22×D811yescharacteristic
257C4×C17—21no—
258C4×C17—21no—
259C4×C17—21no—
260C4×C17—21no—
261C4×C17—81no—
262C4×C17—81no—
263C4×C17#54611yescharacteristic
264C4×C17#57711yescharacteristic
265Dic17—81no—
266Dic17—81no—
267Dic17—81no—
268Dic17—81no—
269C2×#135C2×D1711yescharacteristic
270C22×D8C2×D1711yescharacteristic
271#552—341no—
272#546—21no—
273C2×Q32—171no—
274C2×SD32—171no—
275C2×SD32—171no—
276C22×C16—171no—
277C22×C16—171no—
278#499—171no—
279#495C2×D1711yescharacteristic
280#485—341no—
281#483—341no—
282#479—341no—
283#467—171no—
284#467—171no—
285#467C2×D1711yescharacteristic
286#466C2×D1711yescharacteristic
287#463—21no—
288#450—21no—
289#449—21no—
290C4×Q16—171no—
291C4×D8C2×D1711yescharacteristic
292C8×D4C2×D1711yescharacteristic
293#427—171no—
294C2×#107—171no—
295#367—171no—
296#367—171no—
297#366—171no—
298C4⋊C16—171no—
299#358—171no—
300#358—171no—
301#358—171no—
302#358—171no—
303#357—171no—
304#357—171no—
305#357—171no—
306#357—171no—
307#357—171no—
308#357—171no—
309C22⋊C16—171no—
310C22⋊C16—171no—
311C4×C16—171no—
312C2×C17—22no—
313C2×C17—41no—
314C2×C17—82no—
315C2×C17—161no—
316C2×C17#329411yescharacteristic
317C2×C17C2×#50811yescharacteristic
318C2×C17#341611yescharacteristic
319C2×#41—21no—
320C22×D4—21no—
321#135—22no—
322#135—41no—
323C2×Q16—171no—
324C2×Q16C22×D1711yescharacteristic
325C2×SD16—21no—
326C2×SD16—21no—
327C2×D8—22no—
328C2×D8—21no—
329C2×D8—41no—
330C2×D8C22×D1711yescharacteristic
331C2×D8C22×D1711yescharacteristic
332C22×C8C22×D1711yescharacteristic
333C22×C8C22×D1711yescharacteristic
334#126—681no—
335#125—341no—
336#125—681no—
337#124—21no—
338#123—41no—
339#123—681no—
340C22⋊Q8—21no—
341C22⋊Q8—341no—
342C22⋊Q8—341no—
343#121—21no—
344#121—41no—
345#120—41no—
346C4×Q8—341no—
347C4×D4—21no—
348C4×D4C22⋊D1711yescharacteristic
349#117—341no—
350#117—681no—
351C2×C4⋊C4—341no—
352C2×C22⋊C4—41no—
353Q32—681no—
354SD32—342no—
355SD32—681no—
356C2×C16—171no—
357C2×C16—171no—
358C2×C16—171no—
359C2×C16—171no—
360C2×C16—681no—
361C2×C16—681no—
362#107—171no—
363#107—171no—
364#107—171no—
365#107—341no—
366#107C22×D1711yescharacteristic
367#106—341no—
368#106—341no—
369C4⋊C8C22⋊D1711yescharacteristic
370#103—21no—
371#103—341no—
372#103—341no—
373#103—341no—
374#102—21no—
375#102—21no—
376C22⋊C8C22⋊D1711yescharacteristic
377C22⋊C8C22⋊D1711yescharacteristic
378C4×C8C22×D1711yescharacteristic
379C17#3386311yescharacteristic, complemented, Sylow
380C24—41no—
381#41—42no—
382#41—81no—
383C2×Q8—21no—
384C2×Q8—341no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—81no—
392C2×D4#6322811yescharacteristic
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—681no—
396C22×C4C2×C22⋊D1711yescharacteristic
397C22×C4C2×C22⋊D1711yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—341no—
401Q16—341no—
402SD16—42no—
403SD16—81no—
404D8—21no—
405D8—21no—
406D8—21no—
407D8—42no—
408D8#6325412yes—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×C22⋊D1711yescharacteristic
414C2×C8C2×C22⋊D1711yescharacteristic
415C2×C8C2×C22⋊D1711yescharacteristic
416C2×C8C23×D1711yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—41no—
419C4⋊C4—341no—
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#6322811yescharacteristic
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—681no—
435C22⋊C4—681no—
436C22⋊C4—681no—
437C22⋊C4#6322811yescharacteristic
438C22⋊C4#6322811yescharacteristic
439C42—681no—
440C42—681no—
441C42C2×C22⋊D1711yescharacteristic
442C16—341no—
443C16—341no—
444C16—341no—
445C16—341no—
446C23—41no—
447C23—41no—
448C23—82no—
449C23—82no—
450C23—161no—
451C23C2×#6322811yescharacteristic
452C23C2×#6322811yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—681no—
456Q8—681no—
457D4—22no—
458D4—22no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—82no—
463D4—82no—
464D4—161no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—161no—
473C2×C4—681no—
474C2×C4—681no—
475C2×C4—681no—
476C2×C4—681no—
477C2×C4—1362no—
478C2×C4C2×#6322811yescharacteristic
479C2×C4C2×#6322811yescharacteristic
480C2×C4C2×#6322811yescharacteristic
481C2×C4C2×#6322811yescharacteristic
482C2×C4C22×C22⋊D1711yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8C2×#6325411yescharacteristic
487C8#1058769111yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—164no—
500C22#1176323912yes—
501C22C22×#6322811yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—1361no—
509C4—1361no—
510C4—1361no—
511C4—1361no—
512C4#1176345311yescharacteristic
513C4#1176362011yescharacteristic
514C2—22no—
515C2—41no—
516C2—82no—
517C2—161no—
518C2#1454059711yescharacteristic
519C2C2×#1176323911yescharacteristic
520C2#1454193111yescharacteristic
subgroupG/H
centerC22C22×#63228
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
Fitting, prime radicalC17×#3280C2
socleC22×C17C22×D8
hypercenter#3280D17
hypocenterC17#33863
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. #14541153C2semidirect
2. #14531015C2semidirect
3. #14530979C2semidirect
4. #14530972C2semidirect
5. #14530941C2semidirect
6. #14530878C2semidirect
7. #14530873C2semidirect
8. #14530843C2semidirect
9. #14530724C2semidirect
10. #14530713C2semidirect
11. C4×#10587498C2semidirect
12. D4×#63214C2semidirect
13. #14530491C2semidirect
14. C2×#11762610C2semidirect
17. C2×#10587500C22semidirect
21. C22×#63214C22semidirect
22. #11763211C22semidirect
27. #11763050C22semidirect
28. C4×Dic68C22semidirect
29. #11762847C22semidirect
31. #11762636C22semidirect
32. #11762636C22semidirect
33. #11762634C22semidirect
34. C4⋊#63214C22semidirect
35. #11762613C22semidirect
36. #11762613C22semidirect
37. #11762613C22semidirect
38. #11762612C22semidirect
46. C22⋊#63214C22semidirect
47. C4×#63214C22semidirect
379. C17#33863semidirect

Maximal subgroups

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

subgroupindex
#145411532
#145310152
#145309792
#145309722
#145309412
#145308782
#145308732
#145308432
#145307242
#145307132
C4×#105874982
D4×#632142
#145304912
C2×#117626102
C17×#32802
#3386317