Finite Group Databasebeta

← #17092497

Subgroups — #17092497

Order 4352.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533806C211yesmaximal, characteristic, complemented
2#14531757C211yesmaximal, characteristic
3#14530170C211yesmaximal, characteristic
4#14530156C211yesmaximal, characteristic
5#14529928C211yesmaximal, characteristic
6#14529918C211yesmaximal, characteristic
7#14529738C211yesmaximal, characteristic, complemented
8#14529734C211yesmaximal, characteristic, complemented
9#14529554C211yesmaximal, characteristic, complemented
10#14529500C211yesmaximal, characteristic, complemented
11#14529375C211yesmaximal, characteristic, complemented
12#14529368C211yesmaximal, characteristic, complemented
13#14528914C211yesmaximal, characteristic, complemented
14C4×#10587503C211yesmaximal, characteristic
15C17×#1804C211yesmaximal, characteristic
16C2×#10587574C2211yescharacteristic
17C2×#10587576C2211yescharacteristic
18C2×#10587489C2211yescharacteristic
19C2×#10587488C2211yescharacteristic
20C2×#10587485C2211yescharacteristic
21C2×#10587485C2211yescharacteristic
22C2×#10587503C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×C4×#3622C2211yescharacteristic
25#11762743—21no—
26#11762730—21no—
27#11762713—21no—
28#11762712C2211yescharacteristic
29#11762712C2211yescharacteristic
30#11762706—21no—
31#11762705C2211yescharacteristic
32#11762705C2211yescharacteristic
33#11762702—21no—
34#11762702—21no—
35#11762675C2211yescharacteristic
36#11762672—21no—
37C4⋊C4×Dic17C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40C4×C4⋊Dic17C2211yescharacteristic
41C4×C22⋊Dic17—21no—
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C4×D4×C17C2211yescharacteristic
53C2×C17×C4⋊C8C2211yescharacteristic
54C2×C17×#102C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C17×#382C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C22×D4×C17C2311yescharacteristic
60C23×C4×C17—21no—
61C22×Dic34—21no—
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C4×D4×C17—22no—
65C4×D4×C17Q814yescomplemented
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C22⋊C4×C17—21no—
70C2×C42×C17C2311yescharacteristic
71C2×C22⋊Dic17—21no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75C2×C4⋊Dic17—21no—
76C2×C4⋊Dic17D411yescharacteristic
77C2×C4⋊Dic17D411yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17C2311yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93#10587576—22no—
94#10587576—41no—
95#10587574—22no—
96#10587574—41no—
97C17×C4⋊C8—22no—
98C17×#102—41no—
99C17×#102—41no—
100C17×#95—21no—
101C17×#95—21no—
102#10587503—22no—
103#10587503—22no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502D411yescharacteristic
112#10587502D411yescharacteristic
113#10587489—22no—
114#10587488—22no—
115#10587485—41no—
116#10587485—41no—
117C4×#3622—22no—
118C24×C17—21no—
119C2×D4×C17—21no—
120C2×D4×C17—21no—
121C2×D4×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17C2×Q812yes—
125C22×C4×C17—21no—
126C22×C4×C17—24no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—81no—
130C22×C4×C17—81no—
131C22×C4×C17C2×D411yescharacteristic
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—42no—
143C2×Dic34—42no—
144C2×Dic34—81no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17C2×Q812yes—
157C22⋊C4×C17—24no—
158C42×C17C2×Q812yes—
159C42×C17#4112yes—
160C22⋊Dic17—42no—
161C22⋊Dic17—42no—
162C22⋊Dic17—81no—
163C22⋊Dic17—81no—
164C4⋊Dic17—21no—
165C4⋊Dic17—21no—
166C4⋊Dic17—22no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—41no—
172C4⋊Dic17—42no—
173C4⋊Dic17#4112yes—
174C4⋊Dic17#4112yes—
175#63225—42no—
176#63225—42no—
177#63225—42no—
178#63225—81no—
179C4×Dic17—21no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—21no—
183C4×Dic17—81no—
184C4×Dic17—81no—
185C4×Dic17—81no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192#26028—171nomaximal, Sylow
193C23×C17—22no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17C22×D411yescharacteristic
201D4×C17—22no—
202D4×C17—22no—
203D4×C17—41no—
204D4×C17—41no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—22no—
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—84no—
219C2×C4×C17C22⋊Q812yes—
220C2×C4×C17C22⋊Q812yes—
221C2×C4×C17C22×Q811yescharacteristic
222C2×C4×C17C2×#4111yescharacteristic
223C2×C4×C17C2×#4111yescharacteristic
224C2×C4×C17C2×#4111yescharacteristic
225C2×C4×C17H(5,𝔽2)11yescharacteristic
226C2×C4×C172−1+411yescharacteristic
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—22no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—22no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—41no—
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—
250C2×Dic17#13512yes—
251Dic34—84no—
252C8×C17—41no—
253C8×C17—41no—
254#3622—41no—
255#3622—41no—
256#2550—171no—
257#2357—171no—
258#2067—171no—
259#2065—171no—
260#2025—171no—
261#2023—171no—
262#1989—171no—
263#1988—171no—
264#1949—171no—
265#1939—171no—
266#1921—171no—
267#1920—171no—
268#1835—171no—
269#1804D1711yescharacteristic
270C4×#102—171no—
271C22×C17—42no—
272C22×C17—42no—
273C22×C17—42no—
274C22×C17—81no—
275C22×C17—81no—
276C22×C17—81no—
277C22×C17—81no—
278C22×C17C2×C22⋊Q811yescharacteristic
279C22×C17#54611yescharacteristic
280C22×C17#54811yescharacteristic
281C22×C17C2×#13511yescharacteristic
282C22×C17#57511yescharacteristic
283C22×C17#57611yescharacteristic
284C22×C17#57711yescharacteristic
285C4×C17—22no—
286C4×C17—22no—
287C4×C17—42no—
288C4×C17—81no—
289C4×C17—81no—
290C4×C17#53311yescharacteristic
291C4×C17#54411yescharacteristic
292C4×C17#55511yescharacteristic
293C4×C17#55611yescharacteristic
294Dic17—22no—
295Dic17—42no—
296Dic17—42no—
297Dic17—41no—
298Dic17—41no—
299Dic17—41no—
300Dic17—81no—
301Dic17—81no—
302C2×C4⋊Q8—171no—
303C2×#125—171no—
304C2×C4×D4C2×D1711yescharacteristic
305C2×#107—171no—
306C2×#106—171no—
307C2×C4⋊C8—171no—
308C2×C4⋊C8—171no—
309C2×C4⋊C8C2×D1711yescharacteristic
310C2×#102—171no—
311C2×#102—171no—
312C2×#102C2×D1711yescharacteristic
313C2×#102C2×D1711yescharacteristic
314C2×C4×C8—171no—
315#398—341no—
316#395—341no—
317#391—341no—
318#390—171no—
319#390—171no—
320#389—171no—
321#389—171no—
322#389—341no—
323#388—341no—
324#388—341no—
325#382—171no—
326#382C2×D1711yescharacteristic
327#381—341no—
328C4×C4⋊C4—171no—
329C4×C4⋊C4—171no—
330C4×C4⋊C4—171no—
331C4×C4⋊C4—171no—
332C4×C22⋊C4—341no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#340C2×D1711yescharacteristic
342#340C2×D1711yescharacteristic
343#336—171no—
344#336—171no—
345C2×C17—82no—
346C2×C17#243911yescharacteristic
347C2×C17#307611yescharacteristic
348C2×C17#308111yescharacteristic
349C2×C17#329711yescharacteristic
350C2×C17#332611yescharacteristic
351C2×C17#336511yescharacteristic
352C2×C17#337011yescharacteristic
353C22×Q8—341no—
354C22×D4C22×D1711yescharacteristic
355C23×C4—21no—
356C22×C8—171no—
357C22×C8—171no—
358C22×C8C22×D1711yescharacteristic
359C22×C8C22×D1711yescharacteristic
360C4⋊Q8—342no—
361C4⋊Q8—681no—
362#125—342no—
363#125—681no—
364C4×D4—22no—
365C4×D4Dic3414yescomplemented
366C2×C4⋊C4—171no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—171no—
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⋊C4—341no—
376C2×C4⋊C4—341no—
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—341no—
382C2×C22⋊C4—341no—
383C2×C22⋊C4—341no—
384C2×C22⋊C4—341no—
385C2×C42—171no—
386C2×C42—171no—
387C2×C42—341no—
388C2×C42—341no—
389C2×C42—341no—
390C2×C42C22×D1711yescharacteristic
391#107—342no—
392#106—342no—
393C4⋊C8—22no—
394C4⋊C8—681no—
395C4⋊C8—681no—
396#102—41no—
397#102—41no—
398#102—342no—
399#102—342no—
400C4×C8—342no—
401#95—21no—
402#95—21no—
403#95—171no—
404#95—171no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411#95—341no—
412C17#2602811yescharacteristic, complemented, Sylow
413C24—21no—
414C2×Q8—682no—
415C2×Q8—682no—
416C2×Q8—1361no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×Dic3412yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—171no—
430C22×C4—171no—
431C22×C4—171no—
432C22×C4—171no—
433C22×C4—341no—
434C22×C4—341no—
435C22×C4—341no—
436C22×C4—341no—
437C22×C4D4×D1711yescharacteristic
438C22×C4D4×D1711yescharacteristic
439C22×C4C23×D1711yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—341no—
447C2×C8—341no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—341no—
451C2×C8—341no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—172no—
458C4⋊C4—172no—
459C4⋊C4—342no—
460C4⋊C4—341no—
461C4⋊C4—341no—
462C4⋊C4—341no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—681no—
467C4⋊C4—682no—
468C4⋊C4—682no—
469C4⋊C4—682no—
470C4⋊C4—682no—
471C4⋊C4—1361no—
472C4⋊C4C2×Dic3412yes—
473C22⋊C4—24no—
474C22⋊C4—682no—
475C22⋊C4—682no—
476C22⋊C4—1361no—
477C22⋊C4—1361no—
478C42—341no—
479C42—341no—
480C42—341no—
481C42—341no—
482C42—1361no—
483C42—1361no—
484C42—1361no—
485C42C2×Dic3412yes—
486C42#6325412yes—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C2×D4×D1711yescharacteristic
495Q8—1364no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—84no—
514C2×C4—172no—
515C2×C4—341no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—341no—
519C2×C4—341no—
520C2×C4—342no—
521C2×C4—342no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—341no—
526C2×C4—341no—
527C2×C4—341no—
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×C4—681no—
537C2×C4—681no—
538C2×C4#1058756412yes—
539C2×C4#1058756412yes—
540C2×C4C22×Dic3411yescharacteristic
541C2×C4C2×#6325411yescharacteristic
542C2×C4#1058769111yescharacteristic
543C2×C4#1058769411yescharacteristic
544C2×C4#41×D1711yescharacteristic
545C2×C4#41×D1711yescharacteristic
546C8—41no—
547C8—41no—
548C8—681no—
549C8—681no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1058756411yescharacteristic
558C22#1176345611yescharacteristic
559C22#1176346911yescharacteristic
560C22#135×D1711yescharacteristic
561C22#1176360611yescharacteristic
562C22#1176361311yescharacteristic
563C22#1176361911yescharacteristic
564C4—22no—
565C4—22no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—342no—
570C4—681no—
571C4—682no—
572C4—682no—
573C4—681no—
574C4—681no—
575C4—1361no—
576C4—1361no—
577C4#1176336211yescharacteristic
578C4#1176344011yescharacteristic
579C4#1176352311yescharacteristic
580C4#1176353011yescharacteristic
581C2—82no—
582C2#1453247611yescharacteristic
583C2#1453856811yescharacteristic
584C2#1453861711yescharacteristic
585C2#1454064111yescharacteristic
586C2#1454104411yescharacteristic
587C2#1454148311yescharacteristic
588C2#1454153511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1804C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#1804D17
hypocenterC17#26028
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
1. #14533806C2semidirect
7. #14529738C2semidirect
8. #14529734C2semidirect
9. #14529554C2semidirect
10. #14529500C2semidirect
11. #14529375C2semidirect
12. #14529368C2semidirect
13. #14528914C2semidirect
65. C4×D4×C17Q8semidirect
365. C4×D4Dic34semidirect
412. C17#26028semidirect

Maximal subgroups

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

subgroupindex
#145338062
#145317572
#145301702
#145301562
#145299282
#145299182
#145297382
#145297342
#145295542
#145295002
#145293752
#145293682
#145289142
C4×#105875032
C17×#18042
#2602817