Finite Group Databasebeta

← #17191171

Subgroups — #17191171

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#14540471C211yesmaximal, characteristic
3#14531016C211yesmaximal, characteristic, complemented
4#14530982C211yesmaximal, characteristic, complemented
5#14530976C211yesmaximal, characteristic, complemented
6#14530942C211yesmaximal, characteristic, complemented
7#14530880C211yesmaximal, characteristic, complemented
8#14530875C211yesmaximal, characteristic, complemented
9#14530847C211yesmaximal, characteristic, complemented
10#14530726C211yesmaximal, characteristic, complemented
11#14530715C211yesmaximal, characteristic, complemented
12C4×#10587471C211yesmaximal, characteristic, complemented
13#14530492C211yesmaximal, characteristic, complemented
14C2×#11762611C211yesmaximal, characteristic, complemented
15D4×C16×C17C211yesmaximal, characteristic, complemented
16C2×#10587590C2211yescharacteristic
17C22×D136C2211yescharacteristic
18#11763499—21no—
19#11763455—21no—
20C2×Dic136C2211yescharacteristic, complemented
21C2×#10587471C2211yescharacteristic
22C2×#10587471C2211yescharacteristic
23#11763211C2211yescharacteristic, complemented
24#11763193C2211yescharacteristic
25#11763151—21no—
26#11763140—21no—
27#11763117—21no—
28#11763050C2211yescharacteristic
29#11763050C2211yescharacteristic
30#11763050C2211yescharacteristic, complemented
31#11763045C2211yescharacteristic
32#11763025—21no—
33#11762962—21no—
34#11762956—21no—
35C4×Dic68C2211yescharacteristic, complemented
36C4×D136C2211yescharacteristic
37#11762847C2211yescharacteristic, complemented
38C2×#10587489C2211yescharacteristic
39#11762637C2211yescharacteristic, complemented
40#11762637C2211yescharacteristic, complemented
41#11762635C2211yescharacteristic, complemented
42#11762614C2211yescharacteristic, complemented
43#11762614C2211yescharacteristic, complemented
44#11762614C2211yescharacteristic, complemented
45#11762614C2211yescharacteristic, complemented
46#11762611C2211yescharacteristic
47#11762611C2211yescharacteristic
48#11762611C2211yescharacteristic
49#11762611C2211yescharacteristic
50#11762611C2211yescharacteristic
51#11762611C2211yescharacteristic
52C22×C16×C17C2211yescharacteristic
53C22×C16×C17C2211yescharacteristic, complemented
54C8×D4×C17C2211yescharacteristic
55C17×C4⋊C16C2211yescharacteristic, complemented
56C17×C22⋊C16C2211yescharacteristic
57C17×C22⋊C16C2211yescharacteristic, complemented
58C4×C16×C17C2211yescharacteristic, complemented
59C2×#63252—21no—
60C22×D68—21no—
61C22×C8×C17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C4×D4×C17D411yescharacteristic
64#10587623—41no—
65#10587621—41no—
66#10587616—21no—
67#10587612—21no—
68C2×C4⋊Dic17—21no—
69#10587610—21no—
70#10587610—21no—
71#10587610—21no—
72C2×Dic68C2311yescharacteristic
73C2×Dic68C2311yescharacteristic
74#10587590—22no—
75#10587590—41no—
76C2×D136—22no—
77C2×D136—21no—
78C2×D136—41no—
79C2×D136C2311yescharacteristic
80C2×D136C2311yescharacteristic
81C2×#63219—21no—
82C2×#63219—21no—
83#10587580—41no—
84#10587575—41no—
85#10587570—41no—
86C22⋊C4×D17—41no—
87#10587565—41no—
88#10587563—41no—
89#10587561—21no—
90C4×D68—21no—
91#10587556—21no—
92C4×Dic34—21no—
93C2×C16×C17—41no—
94C2×C16×C17—41no—
95C2×C16×C17C2311yescharacteristic
96C2×C16×C17C2311yescharacteristic
97C2×C16×C17C2311yescharacteristic
98C2×C16×C17C2311yescharacteristic
99C17×C4⋊C8D411yescharacteristic
100C17×C22⋊C8D411yescharacteristic
101C17×C22⋊C8D411yescharacteristic
102C4×C8×C17C2311yescharacteristic
103#10587492—21no—
104#10587492—21no—
105#10587489—21no—
106#10587489C2311yescharacteristic
107#10587489C2311yescharacteristic
108#10587489C2311yescharacteristic
109#10587489C2311yescharacteristic
110#10587488—21no—
111#10587488—21no—
112#10587487—21no—
113#10587487—21no—
114#10587487—21no—
115#10587487—21no—
116Dic136—41no—
117#10587471—22no—
118#10587471—41no—
119C23×D17—41no—
120C2×D4×C17D811yescharacteristic
121C22×C4×C17C2×D411yescharacteristic
122C22×C4×C17C2×D411yescharacteristic
123C2×C22⋊D17—41no—
124C2×C22⋊D17—41no—
125C22×Dic17—41no—
126#63252—42no—
127#63252—81no—
128C2×D68—21no—
129C2×D68—21no—
130C2×D68—42no—
131C2×D68—41no—
132C2×D68—81no—
133C2×C4×D17—41no—
134C2×C4×D17—41no—
135C2×Dic34—21no—
136C2×Dic34—21no—
137C2×C8×C17—21no—
138C2×C8×C17—21no—
139C2×C8×C17—21no—
140C2×C8×C17—21no—
141C2×C8×C17C2×D411yescharacteristic
142C2×C8×C17C2×D411yescharacteristic
143C2×C8×C17C2×D411yescharacteristic
144C2×C8×C17C2411yescharacteristic
145C4⋊C4×C17D811yescharacteristic
146C22⋊C4×C17D811yescharacteristic
147C22⋊C4×C17D811yescharacteristic
148C42×C17C2×D411yescharacteristic
149C22⋊Dic17—41no—
150C22⋊Dic17—41no—
151C22⋊Dic17—41no—
152C22⋊Dic17—41no—
153#63227—41no—
154#63227—41no—
155#63227—41no—
156#63227—41no—
157C4⋊Dic17—21no—
158C4⋊Dic17—21no—
159C4⋊Dic17—21no—
160C4⋊Dic17—21no—
161C4⋊Dic17—41no—
162#63225—41no—
163#63225—41no—
164#63225—41no—
165#63225—41no—
166#63225—41no—
167#63225—41no—
168C4×Dic17—41no—
169C4×Dic17—41no—
170Dic68—21no—
171Dic68—21no—
172Dic68—21no—
173Dic68—21no—
174D136—21no—
175D136—21no—
176D136—21no—
177D136—42no—
178D136#4112yes—
179#63219—42no—
180#63219—81no—
181C16×C17—21no—
182C16×C17—21no—
183C16×C17—21no—
184C16×C17—21no—
185#33863—171nomaximal, Sylow
186C23×C17C2×D811yescharacteristic
187C23×C17C2×D811yescharacteristic
188C22×D17—41no—
189C22×D17—41no—
190C22×D17—82no—
191C22×D17—82no—
192C22×D17—161no—
193D4×C17—22no—
194C2×C4×C17—21no—
195C2×C4×C17—21no—
196C2×C4×C17—21no—
197C2×C4×C17—21no—
198C2×C4×C17C2×D811yescharacteristic
199C2×C4×C17C2×D811yescharacteristic
200C2×C4×C17C2×D811yescharacteristic
201C2×C4×C17C2×D811yescharacteristic
202C2×C4×C17C22×D411yescharacteristic
203C22⋊D17—82no—
204C22⋊D17—161no—
205C2×Dic17—41no—
206C2×Dic17—41no—
207C2×Dic17—41no—
208C2×Dic17—41no—
209C2×Dic17—41no—
210C2×Dic17—41no—
211C2×Dic17—82no—
212D68—22no—
213D68—41no—
214D68—41no—
215D68—41no—
216D68—82no—
217C4×D17—82no—
218C4×D17—161no—
219Dic34—41no—
220Dic34—41no—
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#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×C16D1711yescharacteristic, complemented
241#2138—171no—
242C2×#357—171no—
243C22×C17—21no—
244C22×C17—21no—
245C22×C17—21no—
246C22×C17—21no—
247C22×C17—21no—
248C22×C17#50812yes—
249C22×C17C22×D811yescharacteristic
250C2×D17—42no—
251C2×D17—81no—
252C2×D17—81no—
253C2×D17—81no—
254C2×D17—81no—
255C2×D17—81no—
256C2×D17—164no—
257C4×C17—21no—
258C4×C17—21no—
259C4×C17—21no—
260C4×C17—21no—
261C4×C17#54611yescharacteristic
262C4×C17#57711yescharacteristic
263Dic17—81no—
264Dic17—81no—
265Dic17—81no—
266Dic17—81no—
267Dic17—81no—
268Dic17—81no—
269C2×#135—171no—
270C22×D8—171no—
271#552—341no—
272#546—341no—
273C2×Q32—171no—
274C2×SD32—171no—
275C2×SD32—171no—
276C22×C16C2×D1711yescharacteristic
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×D4C2×D1711yescharacteristic
293#427—171no—
294C2×#107—171no—
295#367—171no—
296#367—171no—
297#366—171no—
298C4⋊C16C2×D1711yescharacteristic, complemented
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⋊C16C2×D1711yescharacteristic
310C22⋊C16C2×D1711yescharacteristic, complemented
311C4×C16C2×D1711yescharacteristic, complemented
312C2×C17—22no—
313C2×C17—41no—
314C2×C17#329411yescharacteristic
315C2×C17C2×#50811yescharacteristic
316C2×C17#341611yescharacteristic
317D17—82no—
318D17—161no—
319C2×#41—341no—
320C22×D4—341no—
321#135—342no—
322#135—681no—
323C2×Q16—171no—
324C2×Q16—171no—
325C2×SD16—341no—
326C2×SD16—341no—
327C2×D8—171no—
328C2×D8—171no—
329C2×D8—342no—
330C2×D8—341no—
331C2×D8—681no—
332C22×C8C22×D1711yescharacteristic
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—341no—
348C4×D4D6811yescharacteristic
349#117—341no—
350#117—681no—
351C2×C4⋊C4—341no—
352C2×C22⋊C4—681no—
353Q32—681no—
354SD32—342no—
355SD32—681no—
356C2×C16—41no—
357C2×C16—41no—
358C2×C16C22×D1711yescharacteristic
359C2×C16C22×D1711yescharacteristic
360C2×C16C22×D1711yescharacteristic
361C2×C16C22×D1711yescharacteristic
362#107—171no—
363#107—171no—
364#107—171no—
365#107—171no—
366#107—341no—
367#106—341no—
368#106—341no—
369C4⋊C8D6811yescharacteristic
370#103—341no—
371#103—341no—
372#103—341no—
373#103—341no—
374#102—341no—
375#102—341no—
376C22⋊C8D6811yescharacteristic
377C22⋊C8D6811yescharacteristic
378C4×C8C22×D1711yescharacteristic
379C17#3386311yescharacteristic, complemented, Sylow
380C24—681no—
381#41—682no—
382#41—1361no—
383C2×Q8—341no—
384C2×Q8—341no—
385C2×D4—341no—
386C2×D4—341no—
387C2×D4—682no—
388C2×D4—681no—
389C2×D4—681no—
390C2×D4—681no—
391C2×D4—1361no—
392C2×D4D13611yescharacteristic
393C22×C4—681no—
394C22×C4—681no—
395C22×C4—681no—
396C22×C4C2×D6811yescharacteristic
397C22×C4C2×D6811yescharacteristic
398Q16—341no—
399Q16—341no—
400Q16—341no—
401Q16—341no—
402SD16—682no—
403SD16—1361no—
404D8—172no—
405D8—341no—
406D8—341no—
407D8—341no—
408D8—682no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×D6811yescharacteristic
414C2×C8C2×D6811yescharacteristic
415C2×C8C2×D6811yescharacteristic
416C2×C8C23×D1711yescharacteristic
417C4⋊C4—341no—
418C4⋊C4—341no—
419C4⋊C4—341no—
420C4⋊C4—341no—
421C4⋊C4—681no—
422C4⋊C4—681no—
423C4⋊C4—681no—
424C4⋊C4—681no—
425C4⋊C4—681no—
426C4⋊C4—681no—
427C4⋊C4—681no—
428C4⋊C4D13611yescharacteristic
429C22⋊C4—681no—
430C22⋊C4—681no—
431C22⋊C4—681no—
432C22⋊C4—681no—
433C22⋊C4—681no—
434C22⋊C4—681no—
435C22⋊C4—681no—
436C22⋊C4—681no—
437C22⋊C4D13611yescharacteristic
438C22⋊C4D13611yescharacteristic
439C42—681no—
440C42—681no—
441C42C2×D6811yescharacteristic
442C16—21no—
443C16—21no—
444C16—21no—
445C16—21no—
446C23—2721no—
447C23—681no—
448C23—681no—
449C23—1362no—
450C23—1362no—
451C23C2×D13611yescharacteristic
452C23C2×D13611yescharacteristic
453Q8—681no—
454Q8—681no—
455Q8—681no—
456Q8—681no—
457D4—22no—
458D4—2721no—
459D4—342no—
460D4—681no—
461D4—681no—
462D4—681no—
463D4—1362no—
464D4—1362no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—2721no—
470C2×C4—681no—
471C2×C4—681no—
472C2×C4—681no—
473C2×C4—681no—
474C2×C4—681no—
475C2×C4—681no—
476C2×C4—1362no—
477C2×C4—1362no—
478C2×C4C2×D13611yescharacteristic
479C2×C4C2×D13611yescharacteristic
480C2×C4C2×D13611yescharacteristic
481C2×C4C2×D13611yescharacteristic
482C2×C4C22×D6811yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8#1058769111yescharacteristic
487C8#41×D1711yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—2724no—
494C22—682no—
495C22—1361no—
496C22—1361no—
497C22—1361no—
498C22—1361no—
499C22—1361no—
500C22#1176324112yes—
501C22C22×D13611yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—1361no—
507C4—1361no—
508C4—1361no—
509C4—1361no—
510C4—1361no—
511C4—1361no—
512C4#1176345411yescharacteristic
513C4#1176362111yescharacteristic
514C2—22no—
515C2—41no—
516C2—2721no—
517C2—1362no—
518C2#1454059811yescharacteristic
519C2C2×#1176324111yescharacteristic
520C2#1454193211yescharacteristic
subgroupG/H
centerC22C22×D136
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingD4×C16×C17C2
socleC22×C17C22×D8
prime radical#14540471C2
hypercenterD4×C16D17
hypocenterC17#33863
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
1. #14541153C2semidirect
3. #14531016C2semidirect
4. #14530982C2semidirect
5. #14530976C2semidirect
6. #14530942C2semidirect
7. #14530880C2semidirect
8. #14530875C2semidirect
9. #14530847C2semidirect
10. #14530726C2semidirect
11. #14530715C2semidirect
12. C4×#10587471C2semidirect
13. #14530492C2semidirect
14. C2×#11762611C2semidirect
15. D4×C16×C17C2semidirect
20. C2×Dic136C22semidirect
23. #11763211C22semidirect
30. #11763050C22semidirect
35. C4×Dic68C22semidirect
37. #11762847C22semidirect
39. #11762637C22semidirect
40. #11762637C22semidirect
41. #11762635C22semidirect
42. #11762614C22semidirect
43. #11762614C22semidirect
44. #11762614C22semidirect
45. #11762614C22semidirect
53. C22×C16×C17C22semidirect
55. C17×C4⋊C16C22semidirect
57. C17×C22⋊C16C22semidirect
58. C4×C16×C17C22semidirect
240. D4×C16D17semidirect
277. C22×C16C2×D17semidirect
298. C4⋊C16C2×D17semidirect
310. C22⋊C16C2×D17semidirect
311. C4×C16C2×D17semidirect
379. C17#33863semidirect

Maximal subgroups

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

subgroupindex
#145411532
#145404712
#145310162
#145309822
#145309762
#145309422
#145308802
#145308752
#145308472
#145307262
#145307152
C4×#105874712
#145304922
C2×#117626112
D4×C16×C172
#3386317