Finite Group Databasebeta

← #17107040

Subgroups — #17107040

Order 4352.

728 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533202C211yesmaximal, characteristic
2#14531789C211yesmaximal, characteristic
3#14530196C211yesmaximal, characteristic, complemented
4#14530166C211yesmaximal, characteristic, complemented
5#14529981C211yesmaximal, characteristic
6#14529980C211yesmaximal, characteristic
7#14529546C211yesmaximal, characteristic, complemented
8#14529511C211yesmaximal, characteristic, complemented
9#14529387C211yesmaximal, characteristic
10#14529368C211yesmaximal, characteristic, complemented
11#14529360C211yesmaximal, characteristic, complemented
12#14529053C211yesmaximal, characteristic, complemented
13#14528977C211yesmaximal, characteristic
14#14528603C211yesmaximal, characteristic, complemented
15C17×#1824C211yesmaximal, characteristic
16C2×#10587565—21no—
17C2×#10587610C2211yescharacteristic
18C2×#10587573C2211yescharacteristic
19C2×#10587488C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×#10587487C2211yescharacteristic
25C2×#10587487C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×C22⋊#3622C2211yescharacteristic
29C2×C8×Dic17C2211yescharacteristic
30#11762750—21no—
31#11762737—21no—
32#11762714C2211yescharacteristic
33#11762708C2211yescharacteristic
34#11762705C2211yescharacteristic
35#11762705C2211yescharacteristic
36#11762704—21no—
37#11762696—21no—
38#11762690—21no—
39#11762682—21no—
40#11762678C2211yescharacteristic
41#11762669C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4⋊C4×Dic17C2211yescharacteristic
44C4×C22⋊Dic17—21no—
45C4×C22⋊Dic17C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C22×C4⋊C4×C17C2211yescharacteristic
53C2×C17×C22⋊C8C2211yescharacteristic
54C17×#380C2211yescharacteristic
55C17×#340C2211yescharacteristic
56C17×#340C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C23×C4×C17—21no—
60C23×C4×C17C2311yescharacteristic
61C22×Dic34C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C2×C4⋊C4×C17—22no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—41no—
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C22⋊C4×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#10587488—22no—
112#10587487—22no—
113#10587487—41no—
114#10587485—22no—
115C8×Dic17—41no—
116#10587481—22no—
117#10587481—22no—
118#10587479—22no—
119#10587479—22no—
120#10587479—41no—
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×C17—42no—
133C22×C4×C17—42no—
134C22×C4×C17—81no—
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2411yescharacteristic
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17C2×D411yescharacteristic
142C22×Dic17C2×D411yescharacteristic
143C22×Dic17C2×D411yescharacteristic
144C22×Dic17C2×D411yescharacteristic
145C2×Dic34—21no—
146C2×Dic34—21no—
147C2×Dic34—21no—
148C2×Dic34—21no—
149C2×Dic34—21no—
150C2×Dic34#4112yes—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17—21no—
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—42no—
165C4⋊C4×C17#4112yes—
166C22⋊C4×C17—42no—
167C22⋊Dic17—24no—
168C22⋊Dic17—41no—
169C22⋊Dic17—42no—
170C22⋊Dic17SD1614yes—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—42no—
177C4⋊Dic17#4112yes—
178#63225—24no—
179#63225—42no—
180#63225—42no—
181#63225—42no—
182#63225—42no—
183#63225—81no—
184C4×Dic17—21no—
185C4×Dic17—21no—
186C4×Dic17—21no—
187C4×Dic17—21no—
188C4×Dic17—42no—
189C4×Dic17—42no—
190C4×Dic17—81no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622#4112yes—
197#27034—171nomaximal, Sylow
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17—21no—
204C23×C17C2×SD1612yes—
205C23×C17C22×D411yescharacteristic
206C2×C4×C17—21no—
207C2×C4×C17—22no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
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×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—84no—
226C2×C4×C17C2×#4111yescharacteristic
227C2×C4×C17C2×#4111yescharacteristic
228C2×C4×C17C2×#4111yescharacteristic
229C2×C4×C17C2×#4111yescharacteristic
230C2×C4×C17H(5,𝔽2)11yescharacteristic
231C2×C4×C172−1+411yescharacteristic
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—22no—
237C2×Dic17—22no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
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×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17C2×SD1612yes—
255C2×Dic17C2×SD1612yes—
256Dic34—22no—
257Dic34—22no—
258Dic34—41no—
259Dic34—41no—
260C8×C17—41no—
261C8×C17—41no—
262#3622—22no—
263#3622—41no—
264#2496—171no—
265#2360—171no—
266#2072—171no—
267#2066—171no—
268#2035—171no—
269#2035—171no—
270#1948—171no—
271#1941—171no—
272#1923—171no—
273#1920—171no—
274#1919—171no—
275#1865—171no—
276#1853—171no—
277#1824D1711yescharacteristic
278C4×#103—171no—
279C22×C17—22no—
280C22×C17—22no—
281C22×C17—22no—
282C22×C17—41no—
283C22×C17—41no—
284C22×C17—41no—
285C22×C17—41no—
286C22×C17#52511yescharacteristic
287C22×C17#54611yescharacteristic
288C22×C17#54711yescharacteristic
289C22×C17C22×SD1611yescharacteristic
290C22×C17#57511yescharacteristic
291C22×C17#57711yescharacteristic
292C22×C17#57811yescharacteristic
293C4×C17—41no—
294C4×C17—42no—
295C4×C17—81no—
296C4×C17—81no—
297C4×C17—81no—
298C4×C17#52911yescharacteristic
299C4×C17#53911yescharacteristic
300C4×C17#55311yescharacteristic
301C4×C17#55611yescharacteristic
302Dic17—22no—
303Dic17—22no—
304Dic17—41no—
305Dic17—42no—
306Dic17—42no—
307Dic17—41no—
308Dic17—81no—
309Dic17—81no—
310C2×#126—341no—
311C2×C22⋊Q8—171no—
312C2×C4×Q8—171no—
313C22×C4⋊C4C2×D1711yescharacteristic
314C2×#106—171no—
315C2×#106—171no—
316C2×#106—171no—
317C2×#106—171no—
318C2×C4⋊C8—171no—
319C2×#103—171no—
320C2×#103—171no—
321C2×#103—171no—
322C2×#103—171no—
323C2×C22⋊C8—171no—
324C2×C22⋊C8C2×D1711yescharacteristic
325C2×C4×C8—171no—
326#400—341no—
327#397—341no—
328#391—171no—
329#389—171no—
330#389—171no—
331#389—171no—
332#389—341no—
333#387—341no—
334#387—341no—
335#385—341no—
336#384—171no—
337#380—171no—
338#380C2×D1711yescharacteristic
339C4×C4⋊C4—171no—
340C4×C4⋊C4—171no—
341C4×C22⋊C4—171no—
342C4×C22⋊C4—341no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340C2×D1711yescharacteristic
348#340C2×D1711yescharacteristic
349#340C2×D1711yescharacteristic
350#340C2×D1711yescharacteristic
351#336—171no—
352#336—171no—
353C2×C17—42no—
354C2×C17#246611yescharacteristic
355C2×C17#310311yescharacteristic
356C2×C17#312011yescharacteristic
357C2×C17#329811yescharacteristic
358C2×C17#331111yescharacteristic
359C2×C17#333511yescharacteristic
360C2×C17#335211yescharacteristic
361C22×Q8—171no—
362C23×C4—21no—
363C23×C4C22×D1711yescharacteristic
364C22×C8—171no—
365C22×C8—171no—
366C22×C8C22×D1711yescharacteristic
367C22×C8C22×D1711yescharacteristic
368#126—684no—
369C22⋊Q8—342no—
370C22⋊Q8—342no—
371C4×Q8—342no—
372C4×Q8—681no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—41no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—341no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4—341no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C4⋊C4C22×D1711yescharacteristic
388C2×C4⋊C4C22×D1711yescharacteristic
389C2×C4⋊C4C22×D1711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—171no—
392C2×C22⋊C4—171no—
393C2×C22⋊C4—341no—
394C2×C22⋊C4—341no—
395C2×C42—171no—
396C2×C42—171no—
397C2×C42—341no—
398C2×C42—341no—
399C2×C42—341no—
400#106—342no—
401#106—342no—
402#106—342no—
403#106—681no—
404C4⋊C8—342no—
405#103—342no—
406#103—342no—
407#103—342no—
408#103—681no—
409C22⋊C8—41no—
410C22⋊C8—342no—
411C4×C8—681no—
412#95—21no—
413#95—171no—
414#95—171no—
415#95—341no—
416#95—341no—
417#95—341no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422#95—341no—
423C17#2703411yescharacteristic, complemented, Sylow
424C24D4×D1711yescharacteristic
425C2×Q8—172no—
426C2×Q8—341no—
427C2×Q8—341no—
428C2×Q8—341no—
429C2×Q8—341no—
430C2×Q8—341no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—81no—
444C22×C4—171no—
445C22×C4—171no—
446C22×C4—171no—
447C22×C4—171no—
448C22×C4—341no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—341no—
452C22×C4D4×D1711yescharacteristic
453C22×C4C23×D1711yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—172no—
461C2×C8—341no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—42no—
474C4⋊C4—172no—
475C4⋊C4—344no—
476C4⋊C4—341no—
477C4⋊C4—341no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—682no—
482C4⋊C4—682no—
483C4⋊C4—682no—
484C4⋊C4—682no—
485C4⋊C4—682no—
486C4⋊C4—1361no—
487C4⋊C4#6325212yes—
488C22⋊C4—42no—
489C22⋊C4—174no—
490C22⋊C4—344no—
491C22⋊C4—681no—
492C22⋊C4—682no—
493C42—341no—
494C42—341no—
495C42—341no—
496C42—341no—
497C42—682no—
498C42—682no—
499C42—1361no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23SD16×D1712yes—
507C23C2×D4×D1711yescharacteristic
508Q8—342no—
509Q8—342no—
510Q8—681no—
511Q8—681no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
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—84no—
532C2×C4—172no—
533C2×C4—172no—
534C2×C4—341no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—342no—
540C2×C4—342no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4C2×#6325211yescharacteristic
557C2×C4#1058769111yescharacteristic
558C2×C4#41×D1711yescharacteristic
559C2×C4#41×D1711yescharacteristic
560C2×C4#41×D1711yescharacteristic
561C2×C4#1058769711yescharacteristic
562C8—41no—
563C8—41no—
564C8—342no—
565C8—681no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#1176331711yescharacteristic
574C22#1176344911yescharacteristic
575C22#1176345711yescharacteristic
576C22C2×SD16×D1711yescharacteristic
577C22#1176360511yescharacteristic
578C22#1176361911yescharacteristic
579C22#1176362211yescharacteristic
580C4—42no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—342no—
586C4—342no—
587C4—681no—
588C4—681no—
589C4—682no—
590C4—682no—
591C4—1361no—
592C4—1361no—
593C4#1176333711yescharacteristic
594C4#1176340211yescharacteristic
595C4#1176350411yescharacteristic
596C4#1176352711yescharacteristic
597C2—42no—
598C2#1453281911yescharacteristic
599C2#1453881211yescharacteristic
600C2#1453899311yescharacteristic
601C2#1454064511yescharacteristic
602C2#1454082111yescharacteristic
603C2#1454113911yescharacteristic
604C2#1454135911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1824C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1824D17
hypocenterC17#27034
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
3. #14530196C2semidirect
4. #14530166C2semidirect
7. #14529546C2semidirect
8. #14529511C2semidirect
10. #14529368C2semidirect
11. #14529360C2semidirect
12. #14529053C2semidirect
14. #14528603C2semidirect
423. C17#27034semidirect

Maximal subgroups

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

subgroupindex
#145332022
#145317892
#145301962
#145301662
#145299812
#145299802
#145295462
#145295112
#145293872
#145293682
#145293602
#145290532
#145289772
#145286032
C17×#18242
#2703417