Finite Group Databasebeta

← #17107400

Subgroups — #17107400

Order 4352.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533202C211yesmaximal, characteristic
2#14531811C211yesmaximal, characteristic
3#14530196C211yesmaximal, characteristic, complemented
4#14530166C211yesmaximal, characteristic, complemented
5#14529988C211yesmaximal, characteristic
6#14529987C211yesmaximal, characteristic
7#14529541C211yesmaximal, characteristic, complemented
8#14529521C211yesmaximal, characteristic, complemented
9#14529394C211yesmaximal, characteristic
10#14529375C211yesmaximal, characteristic, complemented
11#14529353C211yesmaximal, characteristic, complemented
12#14529039C211yesmaximal, characteristic, complemented
13#14528983C211yesmaximal, characteristic
14#14528613C211yesmaximal, characteristic, complemented
15C17×#1825C211yesmaximal, characteristic
16C2×#10587565—21no—
17C2×#10587610C2211yescharacteristic
18C2×#10587573C2211yescharacteristic
19C2×#10587489C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×#10587487C2211yescharacteristic
25C2×#10587487C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×C22⋊#3622C2211yescharacteristic
29C2×#10587486C2211yescharacteristic
30#11762750—21no—
31#11762737—21no—
32#11762714C2211yescharacteristic
33#11762708C2211yescharacteristic
34#11762705C2211yescharacteristic
35#11762705C2211yescharacteristic
36#11762703—21no—
37#11762697—21no—
38#11762696—21no—
39#11762693—21no—
40#11762678C2211yescharacteristic
41#11762669C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4⋊C4×Dic17C2211yescharacteristic
44C4×#63225—21no—
45C4×C22⋊Dic17C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×#117C2211yescharacteristic
53C2×C17×C22⋊C8C2211yescharacteristic
54C17×#380C2211yescharacteristic
55C17×#340C2211yescharacteristic
56C17×#340C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C22×Dic34C2311yescharacteristic
61C22×C8×C17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C17×#117—22no—
64C17×#117—41no—
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C22⋊C4×C17—21no—
70C2×C22⋊C4×C17—21no—
71C2×C42×C17—21no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17D411yescharacteristic
75C2×C22⋊Dic17D411yescharacteristic
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80#10587610—22no—
81#10587610—22no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17C2311yescharacteristic
92C2×C4×Dic17C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95#10587573—22no—
96#10587573—41no—
97#10587565—44no—
98C17×C22⋊C8—41no—
99C17×#95—21no—
100#10587502—21no—
101#10587502—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502D411yescharacteristic
109#10587502D411yescharacteristic
110C22⋊#3622—22no—
111#10587489—22no—
112#10587487—22no—
113#10587487—41no—
114#10587486—41no—
115#10587485—22no—
116#10587481—22no—
117#10587481—22no—
118#10587479—41no—
119#10587478—22no—
120#10587478—22no—
121C24×C17C2×D411yescharacteristic
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×C17—21no—
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2411yescharacteristic
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17C2×D411yescharacteristic
139C22×Dic17C2×D411yescharacteristic
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142C2×Dic34—21no—
143C2×Dic34—21no—
144C2×Dic34—21no—
145C2×Dic34—21no—
146C2×Dic34—21no—
147C2×Dic34#4112yes—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17#4112yes—
162C22⋊C4×C17—42no—
163C22⋊C4×C17—42no—
164C42×C17—42no—
165C22⋊Dic17—24no—
166C22⋊Dic17—22no—
167C22⋊Dic17—41no—
168C22⋊Dic17—42no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—42no—
175C4⋊Dic17#4112yes—
176#63225—24no—
177#63225—42no—
178#63225—42no—
179#63225—42no—
180#63225—42no—
181#63225—42no—
182C4×Dic17—21no—
183C4×Dic17—21no—
184C4×Dic17—21no—
185C4×Dic17—21no—
186C4×Dic17—42no—
187C4×Dic17—42no—
188C4×Dic17—42no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622#4112yes—
195#27058—171nomaximal, Sylow
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17C22×D411yescharacteristic
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—22no—
207C2×C4×C17—21no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—41no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
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×C17C2×#4111yescharacteristic
224C2×C4×C17C2×#4111yescharacteristic
225C2×C4×C17C2×#4111yescharacteristic
226C2×C4×C17C2×#4111yescharacteristic
227C2×C4×C17H(5,𝔽2)11yescharacteristic
228C2×C4×C172−1+411yescharacteristic
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—22no—
232C2×Dic17—22no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17AGL(1,ℤ/8)12yes—
253Dic34—22no—
254Dic34—22no—
255Dic34—41no—
256Dic34—41no—
257C8×C17—41no—
258C8×C17—41no—
259#3622—22no—
260#3622—41no—
261#2496—171no—
262#2362—171no—
263#2072—171no—
264#2066—171no—
265#2036—171no—
266#2036—171no—
267#1947—171no—
268#1943—171no—
269#1924—171no—
270#1921—171no—
271#1918—171no—
272#1863—171no—
273#1854—171no—
274#1825D1711yescharacteristic
275#1763—171no—
276C22×C17—41no—
277C22×C17—41no—
278C22×C17—41no—
279C22×C17—41no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17—41no—
283C22×C17#52511yescharacteristic
284C22×C17#54611yescharacteristic
285C22×C17#54711yescharacteristic
286C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C17#57511yescharacteristic
288C22×C17#57811yescharacteristic
289C22×C17#57811yescharacteristic
290C4×C17—42no—
291C4×C17—41no—
292C4×C17—81no—
293C4×C17—81no—
294C4×C17—81no—
295C4×C17#52911yescharacteristic
296C4×C17#53911yescharacteristic
297C4×C17#55311yescharacteristic
298C4×C17#55611yescharacteristic
299Dic17—22no—
300Dic17—41no—
301Dic17—41no—
302Dic17—42no—
303Dic17—42no—
304Dic17—41no—
305Dic17—81no—
306Dic17—81no—
307C2×#126—341no—
308C2×C22⋊Q8—171no—
309C2×C4×Q8—171no—
310C2×#117C2×D1711yescharacteristic
311C2×#107—171no—
312C2×#107—171no—
313C2×#107—171no—
314C2×#106—171no—
315C2×C4⋊C8—171no—
316C2×#103—171no—
317C2×#103—171no—
318C2×#103—171no—
319C2×#103—171no—
320C2×C22⋊C8—171no—
321C2×C22⋊C8C2×D1711yescharacteristic
322C2×#97—171no—
323#400—341no—
324#397—341no—
325#391—171no—
326#389—171no—
327#389—171no—
328#389—171no—
329#388—341no—
330#388—341no—
331#387—341no—
332#387—341no—
333#384—171no—
334#380—171no—
335#380C2×D1711yescharacteristic
336C4×C4⋊C4—171no—
337C4×C4⋊C4—171no—
338C4×C4⋊C4—341no—
339C4×C22⋊C4—171no—
340#340—171no—
341#340—171no—
342#340—171no—
343#340—171no—
344#340C2×D1711yescharacteristic
345#340C2×D1711yescharacteristic
346#340C2×D1711yescharacteristic
347#340C2×D1711yescharacteristic
348#336—171no—
349#336—171no—
350C2×C17—81no—
351C2×C17#246611yescharacteristic
352C2×C17#311511yescharacteristic
353C2×C17#312711yescharacteristic
354C2×C17#331311yescharacteristic
355C2×C17#333011yescharacteristic
356C2×C17#334311yescharacteristic
357C2×C17#334911yescharacteristic
358C22×Q8—171no—
359C23×C4C22×D1711yescharacteristic
360C22×C8—171no—
361C22×C8—171no—
362C22×C8C22×D1711yescharacteristic
363C22×C8C22×D1711yescharacteristic
364#126—684no—
365C22⋊Q8—342no—
366C22⋊Q8—342no—
367C4×Q8—342no—
368C4×Q8—681no—
369#117—22no—
370#117—41no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4—341no—
377C2×C4⋊C4—341no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4C22×D1711yescharacteristic
382C2×C4⋊C4C22×D1711yescharacteristic
383C2×C4⋊C4C22×D1711yescharacteristic
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—171no—
388C2×C22⋊C4—171no—
389C2×C22⋊C4—341no—
390C2×C22⋊C4—341no—
391C2×C42—21no—
392C2×C42—171no—
393C2×C42—171no—
394C2×C42—341no—
395C2×C42—341no—
396C2×C42—341no—
397#107—342no—
398#107—342no—
399#107—342no—
400#106—681no—
401C4⋊C8—342no—
402#103—342no—
403#103—342no—
404#103—342no—
405#103—681no—
406C22⋊C8—41no—
407C22⋊C8—342no—
408#97—681no—
409#95—21no—
410#95—171no—
411#95—171no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95—341no—
417#95—341no—
418#95—341no—
419#95—341no—
420C17#2705811yescharacteristic, complemented, Sylow
421C24D4×D1711yescharacteristic
422C2×Q8—172no—
423C2×Q8—341no—
424C2×Q8—341no—
425C2×Q8—341no—
426C2×Q8—341no—
427C2×Q8—341no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—171no—
439C22×C4—171no—
440C22×C4—171no—
441C22×C4—171no—
442C22×C4—341no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—341no—
446C22×C4D4×D1711yescharacteristic
447C22×C4C23×D1711yescharacteristic
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—172no—
455C2×C8—341no—
456C2×C8—341no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8—341no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—172no—
468C4⋊C4—344no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—682no—
475C4⋊C4—682no—
476C4⋊C4—682no—
477C4⋊C4—682no—
478C4⋊C4—682no—
479C4⋊C4—682no—
480C4⋊C4#6325212yes—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—342no—
484C22⋊C4—344no—
485C22⋊C4—681no—
486C22⋊C4—682no—
487C42—42no—
488C42—341no—
489C42—341no—
490C42—341no—
491C42—341no—
492C42—682no—
493C42—682no—
494C42—682no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C2×D4×D1711yescharacteristic
503Q8—342no—
504Q8—342no—
505Q8—681no—
506Q8—681no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
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—172no—
527C2×C4—341no—
528C2×C4—341no—
529C2×C4—341no—
530C2×C4—342no—
531C2×C4—342no—
532C2×C4—341no—
533C2×C4—341no—
534C2×C4—341no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—681no—
541C2×C4—681no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4C2×#6325211yescharacteristic
551C2×C4#1058769111yescharacteristic
552C2×C4#41×D1711yescharacteristic
553C2×C4#41×D1711yescharacteristic
554C2×C4#41×D1711yescharacteristic
555C2×C4#1058769711yescharacteristic
556C8—41no—
557C8—41no—
558C8—342no—
559C8—681no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1176331711yescharacteristic
568C22#1176344911yescharacteristic
569C22#1176345711yescharacteristic
570C22AGL(1,ℤ/8)×D1711yescharacteristic
571C22#1176360611yescharacteristic
572C22#1176362311yescharacteristic
573C22#1176362511yescharacteristic
574C4—42no—
575C4—41no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—342no—
580C4—681no—
581C4—681no—
582C4—681no—
583C4—682no—
584C4—682no—
585C4—1361no—
586C4—1361no—
587C4#1176333711yescharacteristic
588C4#1176340211yescharacteristic
589C4#1176350411yescharacteristic
590C4#1176352711yescharacteristic
591C2—81no—
592C2#1453281911yescharacteristic
593C2#1453893011yescharacteristic
594C2#1453908011yescharacteristic
595C2#1454084611yescharacteristic
596C2#1454108511yescharacteristic
597C2#1454121711yescharacteristic
598C2#1454131211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1825C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1825D17
hypocenterC17#27058
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
3. #14530196C2semidirect
4. #14530166C2semidirect
7. #14529541C2semidirect
8. #14529521C2semidirect
10. #14529375C2semidirect
11. #14529353C2semidirect
12. #14529039C2semidirect
14. #14528613C2semidirect
420. C17#27058semidirect

Maximal subgroups

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

subgroupindex
#145332022
#145318112
#145301962
#145301662
#145299882
#145299872
#145295412
#145295212
#145293942
#145293752
#145293532
#145290392
#145289832
#145286132
C17×#18252
#2705817