Finite Group Databasebeta

← #17107442

Subgroups — #17107442

Order 4352.

672 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531809C211yesmaximal, characteristic
2#14530203C211yesmaximal, characteristic, complemented
3#14530162C211yesmaximal, characteristic, complemented
4#14529987C211yesmaximal, characteristic
5#14529980C211yesmaximal, characteristic
6#14529550C211yesmaximal, characteristic, complemented
7#14529519C211yesmaximal, characteristic, complemented
8#14529399C211yesmaximal, characteristic
9#14529373C211yesmaximal, characteristic, complemented
10#14529369C211yesmaximal, characteristic, complemented
11#14529037C211yesmaximal, characteristic, complemented
12#14528981C211yesmaximal, characteristic
13#14528876C211yesmaximal, characteristic
14#14528617C211yesmaximal, characteristic, complemented
15C17×#2494C211yesmaximal, characteristic
16C2×#10587610C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×#10587612C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×#10587481C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×C22⋊#3622C2211yescharacteristic
29C2×C22⋊#3622C2211yescharacteristic
30C2×#10587474C2211yescharacteristic
31#11762715C2211yescharacteristic
32#11762707C2211yescharacteristic
33#11762703—21no—
34#11762697—21no—
35#11762691—21no—
36C4×C4⋊Dic17C2211yescharacteristic
37#11762660—21no—
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C17×#401—21no—
49C17×#397—21no—
50C17×#389C2211yescharacteristic
51C17×#389C2211yescharacteristic
52C17×#388—21no—
53C17×#387—21no—
54C17×#382C2211yescharacteristic
55C17×#380C2211yescharacteristic
56C17×#380C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C22×Dic34C2311yescharacteristic
61C2×C4⋊C4×C17—21no—
62C2×C4⋊C4×C17—21no—
63C2×C4⋊C4×C17C2311yescharacteristic
64C2×C4⋊C4×C17C2311yescharacteristic
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C22⋊C4×C17—21no—
68C2×C22⋊C4×C17—21no—
69C2×C22⋊C4×C17D411yescharacteristic
70C2×C22⋊C4×C17D411yescharacteristic
71C2×C42×C17—21no—
72C2×C42×C17C2311yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C22⋊Dic17—21no—
75C2×C22⋊Dic17—21no—
76#10587612—22no—
77#10587612—41no—
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81#10587610—22no—
82#10587610—22no—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C22×#3622C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C4×Dic34—22no—
94C4×Dic34—41no—
95C17×#95—21no—
96C17×#95—21no—
97C17×#95—21no—
98C17×#95—21no—
99C17×#95—21no—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95D411yescharacteristic
103C17×#95D411yescharacteristic
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108C22⋊#3622—22no—
109C22⋊#3622—41no—
110#10587481—22no—
111#10587481—22no—
112#10587481—22no—
113#10587481—41no—
114#10587479—22no—
115#10587479—41no—
116#10587478—22no—
117#10587478—22no—
118C4⋊#3622—22no—
119#10587474—41no—
120C24×C17C2×D411yescharacteristic
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2411yescharacteristic
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C2×Dic34—21no—
142C2×Dic34—21no—
143C2×Dic34—21no—
144C2×Dic34—21no—
145C2×Dic34—21no—
146C2×Dic34#4112yes—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—21no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—42no—
156C4⋊C4×C17—81no—
157C22⋊C4×C17—41no—
158C22⋊C4×C17—41no—
159C22⋊C4×C17—42no—
160C22⋊C4×C17—81no—
161C42×C17—21no—
162C42×C17—21no—
163C42×C17—21no—
164C42×C17—21no—
165C42×C17—81no—
166C22⋊Dic17—24no—
167C22⋊Dic17—42no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17#4112yes—
173C4⋊Dic17#4112yes—
174#63225—24no—
175#63225—42no—
176#63225—42no—
177C4×Dic17—42no—
178C4×Dic17—42no—
179C4×Dic17—42no—
180C2×#3622—21no—
181C2×#3622—21no—
182C2×#3622—21no—
183C2×#3622—21no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622#4112yes—
192#27061—171nomaximal, Sylow
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17C22×D411yescharacteristic
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17H(5,𝔽2)11yescharacteristic
237C2×C4×C172−1+411yescharacteristic
238C2×Dic17—22no—
239C2×Dic17—22no—
240C2×Dic17—22no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250Dic34—22no—
251Dic34—22no—
252Dic34—41no—
253Dic34—41no—
254#3622—22no—
255#3622—41no—
256#3622—41no—
257#3622—41no—
258#2494D1711yescharacteristic
259#2362—171no—
260#2074—171no—
261#2066—171no—
262#2036—171no—
263#2035—171no—
264#1949—171no—
265#1943—171no—
266#1924—171no—
267#1921—171no—
268#1920—171no—
269#1863—171no—
270#1854—171no—
271#1825—171no—
272#1763—171no—
273C22×C17—41no—
274C22×C17—41no—
275C22×C17—41no—
276C22×C17—41no—
277C22×C17—41no—
278C22×C17—41no—
279C22×C17—41no—
280C22×C17#52511yescharacteristic
281C22×C17#54611yescharacteristic
282C22×C17#54711yescharacteristic
283C22×C17#57511yescharacteristic
284C22×C17#57511yescharacteristic
285C22×C17#57711yescharacteristic
286C22×C17#57811yescharacteristic
287C4×C17—41no—
288C4×C17—41no—
289C4×C17—41no—
290C4×C17—41no—
291C4×C17—41no—
292C4×C17—81no—
293C4×C17—81no—
294C4×C17—81no—
295C4×C17—81no—
296C4×C17#52911yescharacteristic
297C4×C17#53911yescharacteristic
298C4×C17#55311yescharacteristic
299C4×C17#55611yescharacteristic
300Dic17—42no—
301Dic17—42no—
302Dic17—42no—
303Dic17—81no—
304C2×C22⋊Q8—171no—
305C2×C4×Q8—171no—
306C2×#117—171no—
307C2×#107—171no—
308C2×#107—171no—
309C2×#106—171no—
310C2×#106—171no—
311C2×C4⋊C8—171no—
312C2×#103—171no—
313C2×#103—171no—
314C2×#103—171no—
315C2×#103—171no—
316C2×C22⋊C8—171no—
317C2×C22⋊C8—171no—
318C2×#97—171no—
319#401—21no—
320#397—21no—
321#391—171no—
322#389—171no—
323#389C2×D1711yescharacteristic
324#389C2×D1711yescharacteristic
325#388—21no—
326#388—341no—
327#388—341no—
328#387—21no—
329#387—341no—
330#382C2×D1711yescharacteristic
331#380C2×D1711yescharacteristic
332#380C2×D1711yescharacteristic
333C4×C4⋊C4—171no—
334C4×C4⋊C4—341no—
335C4×C4⋊C4C2×D1711yescharacteristic
336C4×C22⋊C4C2×D1711yescharacteristic
337#340—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#340—171no—
342#340—171no—
343#340—171no—
344#340—171no—
345#336—171no—
346#336—171no—
347C2×C17—81no—
348C2×C17#246611yescharacteristic
349C2×C17#311111yescharacteristic
350C2×C17#312111yescharacteristic
351C2×C17#330711yescharacteristic
352C2×C17#332711yescharacteristic
353C2×C17#334911yescharacteristic
354C2×C17#335011yescharacteristic
355C22×Q8—171no—
356C23×C4C22×D1711yescharacteristic
357C22×C8—171no—
358C22×C8—171no—
359C22×C8—171no—
360C22×C8—171no—
361C22⋊Q8—342no—
362C22⋊Q8—342no—
363C4×Q8—342no—
364C4×Q8—681no—
365#117—342no—
366#117—681no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4C22×D1711yescharacteristic
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—341no—
382C2×C22⋊C4—341no—
383C2×C22⋊C4C22⋊D1711yescharacteristic
384C2×C22⋊C4C22⋊D1711yescharacteristic
385C2×C42—21no—
386C2×C42—341no—
387C2×C42—341no—
388C2×C42—341no—
389C2×C42C22×D1711yescharacteristic
390C2×C42C22×D1711yescharacteristic
391#107—342no—
392#107—342no—
393#106—342no—
394#106—681no—
395C4⋊C8—342no—
396#103—342no—
397#103—342no—
398#103—342no—
399#103—681no—
400C22⋊C8—342no—
401C22⋊C8—681no—
402#97—681no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—341no—
411#95—341no—
412#95—341no—
413#95—341no—
414#95C22⋊D1711yescharacteristic
415#95C22⋊D1711yescharacteristic
416C17#2706111yescharacteristic, complemented, Sylow
417C24C2×C22⋊D1711yescharacteristic
418C2×Q8—172no—
419C2×Q8—341no—
420C2×Q8—341no—
421C2×Q8—341no—
422C2×Q8—341no—
423C2×Q8—341no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—341no—
435C22×C4—341no—
436C22×C4—341no—
437C22×C4—341no—
438C22×C4C2×C22⋊D1711yescharacteristic
439C22×C4C2×C22⋊D1711yescharacteristic
440C22×C4C2×C22⋊D1711yescharacteristic
441C22×C4C2×C22⋊D1711yescharacteristic
442C22×C4C2×C22⋊D1711yescharacteristic
443C22×C4C23×D1711yescharacteristic
444C2×C8—172no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8—341no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—341no—
451C2×C8—341no—
452C2×C8—341no—
453C2×C8—341no—
454C2×C8—341no—
455C2×C8—341no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—42no—
465C4⋊C4—81no—
466C4⋊C4—172no—
467C4⋊C4—172no—
468C4⋊C4—344no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—682no—
474C4⋊C4—682no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—344no—
480C22⋊C4—682no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—81no—
486C42—682no—
487C42—682no—
488C42—682no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23C22×C22⋊D1711yescharacteristic
497Q8—342no—
498Q8—342no—
499Q8—681no—
500Q8—681no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—342no—
533C2×C4—342no—
534C2×C4—342no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4—681no—
541C2×C4—681no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4#41×D1711yescharacteristic
545C2×C4#41×D1711yescharacteristic
546C2×C4#41×D1711yescharacteristic
547C2×C4#41×D1711yescharacteristic
548C2×C4#1058769611yescharacteristic
549C2×C4#1058769711yescharacteristic
550C8—342no—
551C8—681no—
552C8—681no—
553C8—681no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22#1176331611yescharacteristic
562C22#1176344711yescharacteristic
563C22#1176345811yescharacteristic
564C22#1176360711yescharacteristic
565C22#1176360711yescharacteristic
566C22#1176362011yescharacteristic
567C22#1176362411yescharacteristic
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—682no—
578C4—682no—
579C4—682no—
580C4—1361no—
581C4#1176334211yescharacteristic
582C4#1176340011yescharacteristic
583C4#1176351011yescharacteristic
584C4#1176352611yescharacteristic
585C2—81no—
586C2#1453281611yescharacteristic
587C2#1453888711yescharacteristic
588C2#1453900711yescharacteristic
589C2#1454075411yescharacteristic
590C2#1454104611yescharacteristic
591C2#1454130811yescharacteristic
592C2#1454132311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2494C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2494D17
hypocenterC17#27061
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2494

Complements

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

NH ≅ G/Ntype
2. #14530203C2semidirect
3. #14530162C2semidirect
6. #14529550C2semidirect
7. #14529519C2semidirect
9. #14529373C2semidirect
10. #14529369C2semidirect
11. #14529037C2semidirect
14. #14528617C2semidirect
416. C17#27061semidirect

Maximal subgroups

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

subgroupindex
#145318092
#145302032
#145301622
#145299872
#145299802
#145295502
#145295192
#145293992
#145293732
#145293692
#145290372
#145289812
#145288762
#145286172
C17×#24942
#2706117