Finite Group Databasebeta

← #17093291

Subgroups — #17093291

Order 4352.

714 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14534025C211yesmaximal, characteristic, complemented
2#14531732C211yesmaximal, characteristic
3#14530307C211yesmaximal, characteristic
4#14530305C211yesmaximal, characteristic
5#14530170C211yesmaximal, characteristic
6#14529554C211yesmaximal, characteristic, complemented
7#14529549C211yesmaximal, characteristic, complemented
8#14529546C211yesmaximal, characteristic, complemented
9#14529499C211yesmaximal, characteristic, complemented
10#14529374C211yesmaximal, characteristic, complemented
11#14529373C211yesmaximal, characteristic, complemented
12#14528942C211yesmaximal, characteristic, complemented
13#14528782C211yesmaximal, characteristic
14#102×Dic17C211yesmaximal, characteristic
15C17×#2067C211yesmaximal, characteristic
16C2×D4×Dic17C2211yescharacteristic
17C2×#10587563—21no—
18C2×#10587489C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587488C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×C8×Dic17C2211yescharacteristic
28#11762750—21no—
29#11762747—21no—
30#11762712C2211yescharacteristic
31#11762705C2211yescharacteristic
32#11762704—21no—
33#11762703—21no—
34#11762697—21no—
35#11762678C2211yescharacteristic
36#11762678C2211yescharacteristic
37#11762675C2211yescharacteristic
38#11762674C2211yescharacteristic
39#11762673—21no—
40#11762667—21no—
41C4⋊C4×Dic17C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4×#63225C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762564C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×#107C2211yescharacteristic
53C2×C17×C4⋊C8C2211yescharacteristic
54C2×C17×#102C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C17×#390C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C22×D4×C17C2311yescharacteristic
60C23×Dic17—21no—
61C22×C8×C17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17C2311yescharacteristic
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C22⋊C4×C17—21no—
68C2×C22⋊C4×C17—21no—
69C2×C42×C17C2311yescharacteristic
70C2×C22⋊Dic17—21no—
71C2×C22⋊Dic17—21no—
72C2×C22⋊Dic17—21no—
73D4×Dic17—22no—
74D4×Dic17Q814yes—
75C2×C4⋊Dic17C2311yescharacteristic
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×#63225—21no—
80C2×#63225—21no—
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225D411yescharacteristic
85C2×#63225D411yescharacteristic
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17C2311yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93#10587563—44no—
94C17×#107—41no—
95C17×C4⋊C8—22no—
96C17×#102—22no—
97C17×#102—41no—
98C17×#95—21no—
99#10587503—22no—
100#10587503—41no—
101#10587502—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502D411yescharacteristic
110#10587502D411yescharacteristic
111#10587489—41no—
112#10587488—41no—
113#10587485—22no—
114#10587485—22no—
115C8×Dic17—22no—
116#10587479—41no—
117#10587478—22no—
118#10587478—22no—
119C24×C17—21no—
120C2×D4×C17—21no—
121C2×D4×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17C2×Q812yes—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2411yescharacteristic
132C22×Dic17—21no—
133C22×Dic17—24no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—42no—
138C22×Dic17—42no—
139C22×Dic17C2×D411yescharacteristic
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142C22×Dic17C2×D411yescharacteristic
143C2×C8×C17—21no—
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17#4112yes—
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—81no—
156C22⋊C4×C17—42no—
157C22⋊C4×C17—81no—
158C42×C17—21no—
159C42×C17—21no—
160C22⋊Dic17—24no—
161C22⋊Dic17—42no—
162C22⋊Dic17—81no—
163C4⋊Dic17—21no—
164C4⋊Dic17—21no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17C2×Q812yes—
171#63225—22no—
172#63225—22no—
173#63225—42no—
174#63225—42no—
175#63225—81no—
176#63225—81no—
177#63225—81no—
178C4×Dic17—21no—
179C4×Dic17—21no—
180C4×Dic17—42no—
181C4×Dic17—42no—
182C4×Dic17—42no—
183C4×Dic17C2×Q812yes—
184C4×Dic17#4112yes—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622#4112yes—
191#26087—171nomaximal, Sylow
192C23×C17—22no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17C22×D411yescharacteristic
200D4×C17—22no—
201D4×C17—22no—
202D4×C17—41no—
203D4×C17—41no—
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—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×C17C22×Q811yescharacteristic
220C2×C4×C17C2×#4111yescharacteristic
221C2×C4×C17C2×#4111yescharacteristic
222C2×C4×C17C2×#4111yescharacteristic
223C2×C4×C17H(5,𝔽2)11yescharacteristic
224C2×C4×C172−1+411yescharacteristic
225C2×Dic17—21no—
226C2×Dic17—21no—
227C2×Dic17—22no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—41no—
235C2×Dic17—41no—
236C2×Dic17—48no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17C22⋊Q812yes—
247C2×Dic17C22⋊Q812yes—
248C2×Dic17#13512yes—
249C2×Dic17AGL(1,ℤ/8)12yes—
250C8×C17—22no—
251C8×C17—41no—
252#3622—22no—
253#3622—41no—
254#2571—171no—
255#2354—171no—
256#2092—171no—
257#2092—171no—
258#2067—171no—
259#2067D1711yescharacteristic
260#1949—171no—
261#1949—171no—
262#1948—171no—
263#1939—171no—
264#1921—171no—
265#1921—171no—
266#1845—171no—
267#1802—171no—
268C4×#102—171no—
269C22×C17—42no—
270C22×C17—42no—
271C22×C17—42no—
272C22×C17—81no—
273C22×C17—81no—
274C22×C17—81no—
275C22×C17—81no—
276C22×C17C2×C22⋊Q811yescharacteristic
277C22×C17#54711yescharacteristic
278C22×C17#55011yescharacteristic
279C22×C17C2×#13511yescharacteristic
280C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C17#57611yescharacteristic
282C22×C17#57811yescharacteristic
283C4×C17—41no—
284C4×C17—41no—
285C4×C17—81no—
286C4×C17—81no—
287C4×C17—81no—
288C4×C17#53311yescharacteristic
289C4×C17#54411yescharacteristic
290C4×C17#55211yescharacteristic
291C4×C17#55311yescharacteristic
292Dic17—22no—
293Dic17—42no—
294Dic17—41no—
295Dic17—41no—
296Dic17—81no—
297Dic17—81no—
298Dic17—81no—
299Dic17#47814yes—
300C2×C4×D4—171no—
301C2×#117—341no—
302C2×#107—171no—
303C2×#107—171no—
304C2×#107—171no—
305C2×#107C2×D1711yescharacteristic
306C2×#106—171no—
307C2×#106—171no—
308C2×C4⋊C8—171no—
309C2×C4⋊C8—171no—
310C2×C4⋊C8C2×D1711yescharacteristic
311C2×#102—171no—
312C2×#102—171no—
313C2×#102C2×D1711yescharacteristic
314C2×#102C2×D1711yescharacteristic
315C2×C4×C8—171no—
316#400—341no—
317#400—341no—
318#390—171no—
319#390C2×D1711yescharacteristic
320#389—171no—
321#389—341no—
322#389C2×D1711yescharacteristic
323#388—341no—
324#388—341no—
325#384—171no—
326#384—171no—
327#382—171no—
328#382—171no—
329#382—341no—
330#380—341no—
331C4×C4⋊C4—171no—
332C4×C4⋊C4—171no—
333C4×C4⋊C4—171no—
334C4×C4⋊C4—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340C2×D1711yescharacteristic
341#336—171no—
342#336—171no—
343C2×C17—82no—
344C2×C17#244111yescharacteristic
345C2×C17C2×#47811yescharacteristic
346C2×C17#308311yescharacteristic
347C2×C17#334111yescharacteristic
348C2×C17#335311yescharacteristic
349C2×C17#339311yescharacteristic
350C2×C17#339911yescharacteristic
351C22×D4C22×D1711yescharacteristic
352C23×C4—341no—
353C22×C8—171no—
354C22×C8—171no—
355C22×C8C22×D1711yescharacteristic
356C22×C8C22×D1711yescharacteristic
357C4×D4—174no—
358C4×D4—342no—
359#117—684no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—171no—
362C2×C4⋊C4—171no—
363C2×C4⋊C4—171no—
364C2×C4⋊C4—171no—
365C2×C4⋊C4—171no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—341no—
368C2×C4⋊C4—341no—
369C2×C4⋊C4—341no—
370C2×C4⋊C4—341no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4C22×D1711yescharacteristic
373C2×C4⋊C4C22×D1711yescharacteristic
374C2×C4⋊C4C22×D1711yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—341no—
378C2×C22⋊C4—341no—
379C2×C22⋊C4—341no—
380C2×C42—171no—
381C2×C42—171no—
382C2×C42—341no—
383C2×C42—341no—
384C2×C42—341no—
385C2×C42C22×D1711yescharacteristic
386#107—41no—
387#107—342no—
388#107—342no—
389#107—681no—
390#106—681no—
391#106—681no—
392C4⋊C8—22no—
393C4⋊C8—342no—
394C4⋊C8—342no—
395#102—22no—
396#102—41no—
397#102—342no—
398#102—681no—
399C4×C8—342no—
400#95—21no—
401#95—171no—
402#95—171no—
403#95—341no—
404#95—341no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411C17#2608711yescharacteristic, complemented, Sylow
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4Q8×D1712yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—171no—
423C22×C4—171no—
424C22×C4—171no—
425C22×C4—171no—
426C22×C4—344no—
427C22×C4—341no—
428C22×C4—341no—
429C22×C4—341no—
430C22×C4—341no—
431C22×C4—682no—
432C22×C4—682no—
433C22×C4D4×D1711yescharacteristic
434C22×C4D4×D1711yescharacteristic
435C22×C4C23×D1711yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—172no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8#6325412yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—81no—
455C4⋊C4—172no—
456C4⋊C4—342no—
457C4⋊C4—342no—
458C4⋊C4—341no—
459C4⋊C4—341no—
460C4⋊C4—341no—
461C4⋊C4—341no—
462C4⋊C4—341no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—682no—
466C4⋊C4—682no—
467C4⋊C4—1361no—
468C4⋊C4—1361no—
469C4⋊C4—1361no—
470C22⋊C4—42no—
471C22⋊C4—81no—
472C22⋊C4—344no—
473C22⋊C4—682no—
474C22⋊C4—1361no—
475C42—21no—
476C42—21no—
477C42—172no—
478C42—172no—
479C42—341no—
480C42—341no—
481C42—682no—
482C42—682no—
483C42—682no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×D4×D1711yescharacteristic
492D4—22no—
493D4—22no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—172no—
512C2×C4—172no—
513C2×C4—172no—
514C2×C4—172no—
515C2×C4—341no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—341no—
519C2×C4—342no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—688no—
525C2×C4—681no—
526C2×C4—681no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4C2×#6325411yescharacteristic
537C2×C4#1058769111yescharacteristic
538C2×C4C2×Q8×D1711yescharacteristic
539C2×C4#41×D1711yescharacteristic
540C2×C4#41×D1711yescharacteristic
541C2×C4#1058769711yescharacteristic
542C8—22no—
543C8—41no—
544C8—342no—
545C8—681no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C22⋊Q8×D1711yescharacteristic
554C22#1176345911yescharacteristic
555C22#1176347711yescharacteristic
556C22#135×D1711yescharacteristic
557C22AGL(1,ℤ/8)×D1711yescharacteristic
558C22#1176361011yescharacteristic
559C22#1176362511yescharacteristic
560C4—41no—
561C4—41no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—174no—
566C4—342no—
567C4—682no—
568C4—681no—
569C4—681no—
570C4—1361no—
571C4—1361no—
572C4—1361no—
573C4#1176336111yescharacteristic
574C4#1176343811yescharacteristic
575C4#1176349511yescharacteristic
576C4#1176350211yescharacteristic
577C2—82no—
578C2#1453249211yescharacteristic
579C2D17×#47811yescharacteristic
580C2#1453863011yescharacteristic
581C2#1454120311yescharacteristic
582C2#1454136311yescharacteristic
583C2#1454176711yescharacteristic
584C2#1454182911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2067C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2067D17
hypocenterC17#26087
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
1. #14534025C2semidirect
6. #14529554C2semidirect
7. #14529549C2semidirect
8. #14529546C2semidirect
9. #14529499C2semidirect
10. #14529374C2semidirect
11. #14529373C2semidirect
12. #14528942C2semidirect
411. C17#26087semidirect

Maximal subgroups

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

subgroupindex
#145340252
#145317322
#145303072
#145303052
#145301702
#145295542
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