Finite Group Databasebeta

← #17191221

Subgroups — #17191221

Order 4352.

546 conjugacy classes of subgroups.

The subgroup lattice has 508 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#14530989C211yesmaximal, characteristic, complemented
4#14530962C211yesmaximal, characteristic, complemented
5#14530948C211yesmaximal, characteristic, complemented
6#14530878C211yesmaximal, characteristic, complemented
7#14530873C211yesmaximal, characteristic, complemented
8#14530843C211yesmaximal, characteristic, complemented
9#14530724C211yesmaximal, characteristic, complemented
10#14530718C211yesmaximal, characteristic, complemented
11C4×#10587500C211yesmaximal, characteristic, complemented
12D4×#63214C211yesmaximal, characteristic, complemented
13#14530491C211yesmaximal, characteristic, complemented
14C2×#11762613C211yesmaximal, characteristic, complemented
15C17×#3285C211yesmaximal, characteristic
16#11763499—21no—
17C2×#10587500C2211yescharacteristic
18C2×#10587500C2211yescharacteristic
19C2×#10587498C2211yescharacteristic
20C22×#63214C2211yescharacteristic
21C22×#63214C2211yescharacteristic
22#11763211C2211yescharacteristic
23#11763151—21no—
24#11763140—21no—
25#11763117—21no—
26#11763050C2211yescharacteristic
27#11763050C2211yescharacteristic
28C4×Dic68C2211yescharacteristic
29#11762847C2211yescharacteristic
30C2×#10587489C2211yescharacteristic
31#11762636C2211yescharacteristic
32#11762634C2211yescharacteristic
33#11762634C2211yescharacteristic
34C4⋊#63214C2211yescharacteristic
35#11762613C2211yescharacteristic
36#11762613C2211yescharacteristic
37#11762613C2211yescharacteristic
38#11762613C2211yescharacteristic
39#11762613C2211yescharacteristic
40#11762613C2211yescharacteristic
41#11762612C2211yescharacteristic
42#11762610C2211yescharacteristic
43#11762610C2211yescharacteristic
44#11762610C2211yescharacteristic
45C22⋊#63214C2211yescharacteristic
46C22⋊#63214C2211yescharacteristic
47C4×#63214C2211yescharacteristic
48C2×C17×#135C2211yescharacteristic
49C22×Q16×C17C2211yescharacteristic
50C17×#548—21no—
51C17×#495C2211yescharacteristic
52C17×#467C2211yescharacteristic
53C17×#466C2211yescharacteristic
54C17×#464—21no—
55C17×#452—21no—
56C17×#448—21no—
57C4×Q16×C17C2211yescharacteristic
58C8×D4×C17C2211yescharacteristic
59C2×#41×C17—21no—
60C22×Q8×C17—21no—
61C17×#135—22no—
62C17×#135—41no—
63C2×Q16×C17—22no—
64C2×Q16×C17—21no—
65C2×Q16×C17—41no—
66C2×Q16×C17C2311yescharacteristic
67C2×Q16×C17C2311yescharacteristic
68C2×SD16×C17—21no—
69C2×SD16×C17—21no—
70C2×D8×C17C2311yescharacteristic
71C22×C8×C17C2311yescharacteristic
72C22×C8×C17C2311yescharacteristic
73C17×#124—21no—
74C17×#124—41no—
75C17×C22⋊Q8—21no—
76C17×C22⋊Q8—41no—
77C17×#121—21no—
78C17×#121—41no—
79C4×Q8×C17—21no—
80C4×D4×C17—41no—
81C4×D4×C17D411yescharacteristic
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×#103—21no—
97C17×#102—21no—
98C17×C22⋊C8D411yescharacteristic
99C17×C22⋊C8D411yescharacteristic
100C4×C8×C17C2311yescharacteristic
101#10587500—22no—
102#10587500—41no—
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
119#41×C17—42no—
120#41×C17—81no—
121C2×Q8×C17—21no—
122C2×Q8×C17—21no—
123C2×Q8×C17—42no—
124C2×Q8×C17—41no—
125C2×Q8×C17—81no—
126C2×D4×C17—21no—
127C2×D4×C17—41no—
128C2×D4×C17—41no—
129C2×D4×C17D811yescharacteristic
130C22×C4×C17—41no—
131C22×C4×C17—41no—
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×Dic17—41no—
135C2×Dic34—21no—
136Q16×C17—21no—
137Q16×C17—21no—
138Q16×C17—21no—
139Q16×C17—42no—
140Q16×C17#4112yes—
141SD16×C17—42no—
142SD16×C17—81no—
143D8×C17—21no—
144D8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17—21no—
149C2×C8×C17C2×D411yescharacteristic
150C2×C8×C17C2×D411yescharacteristic
151C2×C8×C17C2×D411yescharacteristic
152C2×C8×C17C2411yescharacteristic
153C4⋊C4×C17—21no—
154C4⋊C4×C17—41no—
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×C17D811yescharacteristic
162C22⋊C4×C17D811yescharacteristic
163C42×C17—41no—
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#33867—171nomaximal, Sylow
186C23×C17—41no—
187C23×C17C2×D811yescharacteristic
188C23×C17C2×D811yescharacteristic
189Q8×C17—22no—
190Q8×C17—41no—
191Q8×C17—41no—
192Q8×C17—41no—
193Q8×C17—82no—
194D4×C17—22no—
195D4×C17—41no—
196D4×C17—41no—
197D4×C17—82no—
198D4×C17—82no—
199C2×C4×C17—21no—
200C2×C4×C17—21no—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—41no—
204C2×C4×C17—41no—
205C2×C4×C17—41no—
206C2×C4×C17—82no—
207C2×C4×C17—82no—
208C2×C4×C17C2×D811yescharacteristic
209C2×C4×C17C2×D811yescharacteristic
210C2×C4×C17C2×D811yescharacteristic
211C2×C4×C17C2×D811yescharacteristic
212C2×C4×C17C22×D411yescharacteristic
213C2×Dic17—41no—
214C2×Dic17—41no—
215C2×Dic17—41no—
216C2×Dic17—41no—
217C2×Dic17—82no—
218Dic34—41no—
219Dic34—41no—
220C8×C17—21no—
221C8×C17—21no—
222C8×C17—21no—
223C8×C17C2×#4111yescharacteristic
224C8×C17H(5,𝔽2)11yescharacteristic
225#3336—171no—
226#3285D1711yescharacteristic
227#2242—171no—
228#2237—171no—
229#2232—171no—
230#2229—171no—
231#2217—171no—
232#2216—171no—
233#2211—171no—
234#2189—171no—
235#2188—171no—
236C4×Q32—171no—
237D4×C16—171no—
238#2138—171no—
239C2×#358—171no—
240C22×C17—21no—
241C22×C17—21no—
242C22×C17—21no—
243C22×C17—21no—
244C22×C17—21no—
245C22×C17—81no—
246C22×C17—81no—
247C22×C17—81no—
248C22×C17#50812yes—
249C22×C17C22×D811yescharacteristic
250C4×C17—21no—
251C4×C17—21no—
252C4×C17—21no—
253C4×C17—21no—
254C4×C17—42no—
255C4×C17—81no—
256C4×C17—81no—
257C4×C17#54611yescharacteristic
258C4×C17#57711yescharacteristic
259Dic17—81no—
260Dic17—81no—
261Dic17—81no—
262Dic17—81no—
263C2×#135C2×D1711yescharacteristic
264C22×Q16C2×D1711yescharacteristic
265#552—341no—
266#548—21no—
267C2×Q32—171no—
268C2×Q32—171no—
269C2×SD32—171no—
270C22×C16—171no—
271C22×C16—171no—
272#499—171no—
273#495C2×D1711yescharacteristic
274#485—341no—
275#483—341no—
276#479—341no—
277#467—171no—
278#467—171no—
279#467C2×D1711yescharacteristic
280#466C2×D1711yescharacteristic
281#464—21no—
282#452—21no—
283#448—21no—
284C4×Q16—171no—
285C4×Q16C2×D1711yescharacteristic
286C8×D4C2×D1711yescharacteristic
287#427—171no—
288C2×#107—171no—
289#367—171no—
290#366—171no—
291#366—171no—
292C4⋊C16—171no—
293#358—171no—
294#358—171no—
295#358—171no—
296#358—171no—
297#358—171no—
298#358—171no—
299#358—171no—
300#357—171no—
301#357—171no—
302#357—171no—
303C22⋊C16—171no—
304C22⋊C16—171no—
305C4×C16—171no—
306C2×C17—22no—
307C2×C17—41no—
308C2×C17—161no—
309C2×C17#329411yescharacteristic
310C2×C17C2×#50811yescharacteristic
311C2×C17#341711yescharacteristic
312C2×#41—21no—
313C22×Q8—21no—
314#135—22no—
315#135—41no—
316C2×Q16—22no—
317C2×Q16—21no—
318C2×Q16—41no—
319C2×Q16—171no—
320C2×Q16C22×D1711yescharacteristic
321C2×Q16C22×D1711yescharacteristic
322C2×SD16—21no—
323C2×SD16—21no—
324C2×D8C22×D1711yescharacteristic
325C22×C8C22×D1711yescharacteristic
326C22×C8C22×D1711yescharacteristic
327#126—681no—
328#125—341no—
329#125—681no—
330#124—21no—
331#124—41no—
332#123—681no—
333C22⋊Q8—21no—
334C22⋊Q8—41no—
335C22⋊Q8—341no—
336C22⋊Q8—341no—
337#121—21no—
338#121—41no—
339C4×Q8—21no—
340C4×Q8—341no—
341C4×D4—41no—
342C4×D4C22⋊D1711yescharacteristic
343#117—341no—
344#117—681no—
345C2×C4⋊C4—341no—
346Q32—342no—
347Q32—681no—
348SD32—681no—
349C2×C16—171no—
350C2×C16—171no—
351C2×C16—171no—
352C2×C16—171no—
353C2×C16—681no—
354C2×C16—681no—
355#107—171no—
356#107—171no—
357#107—171no—
358#107—341no—
359#107C22×D1711yescharacteristic
360#106—341no—
361#106—341no—
362C4⋊C8C22⋊D1711yescharacteristic
363#103—21no—
364#103—21no—
365#103—341no—
366#103—341no—
367#103—341no—
368#102—21no—
369C22⋊C8C22⋊D1711yescharacteristic
370C22⋊C8C22⋊D1711yescharacteristic
371C4×C8C22×D1711yescharacteristic
372C17#3386711yescharacteristic, complemented, Sylow
373#41—42no—
374#41—81no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—42no—
378C2×Q8—41no—
379C2×Q8—81no—
380C2×Q8—341no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4#6322811yescharacteristic
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—681no—
388C22×C4C2×C22⋊D1711yescharacteristic
389C22×C4C2×C22⋊D1711yescharacteristic
390Q16—21no—
391Q16—21no—
392Q16—21no—
393Q16—42no—
394Q16—341no—
395Q16—341no—
396Q16#6325412yes—
397SD16—42no—
398SD16—81no—
399D8—21no—
400D8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8C2×C22⋊D1711yescharacteristic
406C2×C8C2×C22⋊D1711yescharacteristic
407C2×C8C2×C22⋊D1711yescharacteristic
408C2×C8C23×D1711yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—341no—
413C4⋊C4—341no—
414C4⋊C4—341no—
415C4⋊C4—681no—
416C4⋊C4—681no—
417C4⋊C4—681no—
418C4⋊C4—681no—
419C4⋊C4—681no—
420C4⋊C4—681no—
421C4⋊C4#6322811yescharacteristic
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—681no—
427C22⋊C4—681no—
428C22⋊C4—681no—
429C22⋊C4#6322811yescharacteristic
430C22⋊C4#6322811yescharacteristic
431C42—41no—
432C42—681no—
433C42—681no—
434C42C2×C22⋊D1711yescharacteristic
435C16—341no—
436C16—341no—
437C16—341no—
438C16—341no—
439C23—41no—
440C23C2×#6322811yescharacteristic
441C23C2×#6322811yescharacteristic
442Q8—22no—
443Q8—41no—
444Q8—41no—
445Q8—41no—
446Q8—82no—
447Q8—681no—
448Q8—681no—
449D4—22no—
450D4—41no—
451D4—41no—
452D4—82no—
453D4—82no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—681no—
464C2×C4—681no—
465C2×C4—681no—
466C2×C4—681no—
467C2×C4—1362no—
468C2×C4C2×#6322811yescharacteristic
469C2×C4C2×#6322811yescharacteristic
470C2×C4C2×#6322811yescharacteristic
471C2×C4C2×#6322811yescharacteristic
472C2×C4C22×C22⋊D1711yescharacteristic
473C8—21no—
474C8—21no—
475C8—21no—
476C8C2×#6325411yescharacteristic
477C8#1058769111yescharacteristic
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22#1176323912yes—
487C22C22×#6322811yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—42no—
493C4—81no—
494C4—81no—
495C4—1361no—
496C4—1361no—
497C4—1361no—
498C4—1361no—
499C4#1176345311yescharacteristic
500C4#1176362011yescharacteristic
501C2—22no—
502C2—41no—
503C2—161no—
504C2#1454059711yescharacteristic
505C2C2×#1176323911yescharacteristic
506C2#1454193511yescharacteristic
subgroupG/H
centerC22C22×#63228
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
Fitting, prime radicalC17×#3285C2
socleC22×C17C22×D8
hypercenter#3285D17
hypocenterC17#33867
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3285

Complements

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

NH ≅ G/Ntype
1. #14541153C2semidirect
2. #14531015C2semidirect
3. #14530989C2semidirect
4. #14530962C2semidirect
5. #14530948C2semidirect
6. #14530878C2semidirect
7. #14530873C2semidirect
8. #14530843C2semidirect
9. #14530724C2semidirect
10. #14530718C2semidirect
11. C4×#10587500C2semidirect
12. D4×#63214C2semidirect
13. #14530491C2semidirect
14. C2×#11762613C2semidirect
372. C17#33867semidirect

Maximal subgroups

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

subgroupindex
#145411532
#145310152
#145309892
#145309622
#145309482
#145308782
#145308732
#145308432
#145307242
#145307182
C4×#105875002
D4×#632142
#145304912
C2×#117626132
C17×#32852
#3386717