Finite Group Databasebeta

← #17092501

Subgroups — #17092501

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#14531759C211yesmaximal, characteristic
3#14530173C211yesmaximal, characteristic
4#14530159C211yesmaximal, characteristic
5#14529929C211yesmaximal, characteristic
6#14529919C211yesmaximal, characteristic
7#14529738C211yesmaximal, characteristic, complemented
8#14529734C211yesmaximal, characteristic, complemented
9#14529550C211yesmaximal, characteristic, complemented
10#14529502C211yesmaximal, characteristic, complemented
11#14529374C211yesmaximal, characteristic, complemented
12#14529367C211yesmaximal, characteristic, complemented
13#14528791C211yesmaximal, characteristic
14C4×#10587492C211yesmaximal, characteristic
15C17×#1835C211yesmaximal, characteristic, complemented
16C2×#10587574C2211yescharacteristic
17C2×#10587576C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×#10587485C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×#10587492C2211yescharacteristic
25C2×#10587492C2211yescharacteristic
26C2×#10587480C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28#11762743—21no—
29#11762730—21no—
30#11762713—21no—
31#11762710C2211yescharacteristic
32#11762710C2211yescharacteristic
33#11762706—21no—
34#11762705C2211yescharacteristic
35#11762705C2211yescharacteristic
36#11762701—21no—
37#11762701—21no—
38#11762671—21no—
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C4×C4⋊Dic17C2211yescharacteristic
42C4×C4⋊Dic17C2211yescharacteristic
43#11762656—21no—
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C4×C8×C17C2211yescharacteristic
53C17×#382C2211yescharacteristic
54C17×#382C2211yescharacteristic
55C17×#340C2211yescharacteristic
56C17×#340C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C22×D68C2311yescharacteristic
60C22×C4×D17—21no—
61C22×Dic34—21no—
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C2×C4⋊C4×C17C2311yescharacteristic
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C42×C17C2311yescharacteristic
69C2×#63227—21no—
70C2×#63227—21no—
71C2×#63227—21no—
72C2×#63227—21no—
73C2×#63227—21no—
74C2×C4⋊Dic17—21no—
75C2×C4⋊Dic17D411yescharacteristic
76C2×C4⋊Dic17D411yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×#63225—21no—
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×C4×Dic17—21no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17C2311yescharacteristic
88C2×C4×Dic17C2311yescharacteristic
89C22×#3622C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91#10587576—22no—
92#10587576—41no—
93#10587574—22no—
94#10587574—41no—
95C4×D68—22no—
96C4×D68Q814yescomplemented
97C4×C8×C17—22no—
98C17×#95—21no—
99C17×#95—21no—
100C17×#95—21no—
101C17×#95—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502D411yescharacteristic
108#10587502D411yescharacteristic
109#10587492—22no—
110#10587492—22no—
111#10587485—41no—
112#10587485—41no—
113#10587480—41no—
114#10587480—41no—
115#10587479—22no—
116#10587478—22no—
117C4⋊#3622—22no—
118C23×D17—21no—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17C2×D411yescharacteristic
124C22×C4×C17C2×D411yescharacteristic
125C22×C4×C17C2411yescharacteristic
126C22×Dic17—21no—
127C22×Dic17—21no—
128C22×Dic17—21no—
129C22×Dic17C2×D411yescharacteristic
130C22×Dic17C2×D411yescharacteristic
131C22×Dic17C2×D411yescharacteristic
132C22×Dic17C2×D411yescharacteristic
133C2×D68—21no—
134C2×D68—21no—
135C2×D68—21no—
136C2×D68—21no—
137C2×D68—21no—
138C2×D68C2×Q812yes—
139C2×C4×D17—24no—
140C2×C4×D17—81no—
141C2×C4×D17—81no—
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×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C42×C17C2×Q812yes—
160C42×C17#4112yes—
161#63227—24no—
162#63227—42no—
163#63227—42no—
164#63227—81no—
165#63227—81no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—22no—
170C4⋊Dic17—41no—
171C4⋊Dic17—42no—
172C4⋊Dic17C2×Q812yes—
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×C17C22×D411yescharacteristic
194C22×D17—22no—
195C22×D17—41no—
196C22×D17—41no—
197C22×D17—41no—
198C22×D17—41no—
199C22×D17—41no—
200C22×D17—41no—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—41no—
206C2×C4×C17—41no—
207C2×C4×C17—41no—
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×C17C22⋊Q812yes—
218C2×C4×C17C22⋊Q812yes—
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—22no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—22no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17#13512yes—
246D68—22no—
247D68—22no—
248D68—41no—
249D68—41no—
250C4×D17—84no—
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#1835D1711yescharacteristic, complemented
269#1804—171no—
270C4×#102—171no—
271C22×C17C2×C22⋊Q811yescharacteristic
272C22×C17#54611yescharacteristic
273C22×C17#54811yescharacteristic
274C22×C17C2×#13511yescharacteristic
275C22×C17#57511yescharacteristic
276C22×C17#57611yescharacteristic
277C22×C17#57711yescharacteristic
278C2×D17—42no—
279C2×D17—42no—
280C2×D17—42no—
281C2×D17—81no—
282C2×D17—81no—
283C2×D17—81no—
284C2×D17—81no—
285C4×C17—22no—
286C4×C17—22no—
287C4×C17—81no—
288C4×C17—81no—
289C4×C17—81no—
290C4×C17—81no—
291C4×C17#53311yescharacteristic
292C4×C17#54411yescharacteristic
293C4×C17#55511yescharacteristic
294C4×C17#55611yescharacteristic
295Dic17—22no—
296Dic17—42no—
297Dic17—42no—
298Dic17—42no—
299Dic17—41no—
300Dic17—41no—
301Dic17—41no—
302C2×C4⋊Q8—171no—
303C2×#125—171no—
304C2×C4×D4—171no—
305C2×#107—171no—
306C2×#106—171no—
307C2×C4⋊C8—171no—
308C2×C4⋊C8—171no—
309C2×C4⋊C8—171no—
310C2×#102—171no—
311C2×#102—171no—
312C2×#102—171no—
313C2×#102—171no—
314C2×C4×C8C2×D1711yescharacteristic
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#382C2×D1711yescharacteristic
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#340C2×D1711yescharacteristic
340#340C2×D1711yescharacteristic
341#340C2×D1711yescharacteristic
342#340C2×D1711yescharacteristic
343#336—171no—
344#336—171no—
345C2×C17#243911yescharacteristic
346C2×C17#307611yescharacteristic
347C2×C17#308111yescharacteristic
348C2×C17#329711yescharacteristic
349C2×C17#332611yescharacteristic
350C2×C17#336511yescharacteristic
351C2×C17#337011yescharacteristic
352D17—82no—
353C22×Q8—341no—
354C22×D4—171no—
355C23×C4—341no—
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—174no—
365C4×D4—342no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C22⋊C4—341no—
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—342no—
394C4⋊C8—681no—
395C4⋊C8—681no—
396#102—342no—
397#102—342no—
398#102—681no—
399#102—681no—
400C4×C8—22no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—171no—
406#95—171no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411#95—341no—
412C17#2602811yescharacteristic, complemented, Sylow
413C24—341no—
414C2×Q8—682no—
415C2×Q8—682no—
416C2×Q8—1361no—
417C2×D4—172no—
418C2×D4—341no—
419C2×D4—341no—
420C2×D4—341no—
421C2×D4—341no—
422C2×D4—341no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—171no—
428C22×C4—171no—
429C22×C4—171no—
430C22×C4—171no—
431C22×C4—341no—
432C22×C4—341no—
433C22×C4—341no—
434C22×C4—344no—
435C22×C4—1361no—
436C22×C4—1361no—
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—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—172no—
461C4⋊C4—172no—
462C4⋊C4—172no—
463C4⋊C4—342no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—681no—
468C4⋊C4—682no—
469C4⋊C4—682no—
470C4⋊C4—682no—
471C4⋊C4—682no—
472C4⋊C4—1361no—
473C22⋊C4—344no—
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—
485C42#6325212yes—
486C42Q8×D1712yes—
487C23—342no—
488C23—681no—
489C23—681no—
490C23—681no—
491C23—681no—
492C23—681no—
493C23—681no—
494C23C2×D4×D1711yescharacteristic
495Q8—1364no—
496D4—342no—
497D4—342no—
498D4—681no—
499D4—681no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
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—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—172no—
517C2×C4—341no—
518C2×C4—341no—
519C2×C4—342no—
520C2×C4—342no—
521C2×C4—341no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—341no—
526C2×C4—341no—
527C2×C4—341no—
528C2×C4—341no—
529C2×C4—341no—
530C2×C4—342no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—1364no—
538C2×C4#1058758212yes—
539C2×C4#1058758212yes—
540C2×C4C2×#6325211yescharacteristic
541C2×C4C2×Q8×D1711yescharacteristic
542C2×C4#41×D1711yescharacteristic
543C2×C4#41×D1711yescharacteristic
544C2×C4#1058769611yescharacteristic
545C2×C4#1058769711yescharacteristic
546C8—41no—
547C8—41no—
548C8—681no—
549C8—681no—
550C22—682no—
551C22—682no—
552C22—682no—
553C22—1361no—
554C22—1361no—
555C22—1361no—
556C22—1361no—
557C22C2×#1058758211yescharacteristic
558C22#1176345211yescharacteristic
559C22#1176346711yescharacteristic
560C22#135×D1711yescharacteristic
561C22#1176360611yescharacteristic
562C22#1176361111yescharacteristic
563C22#1176361611yescharacteristic
564C4—22no—
565C4—22no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—342no—
571C4—681no—
572C4—682no—
573C4—682no—
574C4—682no—
575C4—681no—
576C4—681no—
577C4#1176336511yescharacteristic
578C4#1176343911yescharacteristic
579C4#1176352211yescharacteristic
580C4#1176352711yescharacteristic
581C2—1362no—
582C2#1453247811yescharacteristic
583C2#1453857011yescharacteristic
584C2#1453861811yescharacteristic
585C2#1454063311yescharacteristic
586C2#1454103611yescharacteristic
587C2#1454147711yescharacteristic
588C2#1454154311yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1835C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenter#1835D17
hypocenterC17#26028
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1835

Complements

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

NH ≅ G/Ntype
1. #14533806C2semidirect
7. #14529738C2semidirect
8. #14529734C2semidirect
9. #14529550C2semidirect
10. #14529502C2semidirect
11. #14529374C2semidirect
12. #14529367C2semidirect
15. C17×#1835C2semidirect
96. C4×D68Q8semidirect
268. #1835D17semidirect
412. C17#26028semidirect

Maximal subgroups

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

subgroupindex
#145338062
#145317592
#145301732
#145301592
#145299292
#145299192
#145297382
#145297342
#145295502
#145295022
#145293742
#145293672
#145287912
C4×#105874922
C17×#18352
#2602817