Finite Group Databasebeta

← #17092489

Subgroups — #17092489

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#14533805C211yesmaximal, characteristic, complemented
2#14530172C211yesmaximal, characteristic
3#14530158C211yesmaximal, characteristic
4#14529930C211yesmaximal, characteristic
5#14529920C211yesmaximal, characteristic
6#14529741C211yesmaximal, characteristic, complemented
7#14529736C211yesmaximal, characteristic, complemented
8#14529550C211yesmaximal, characteristic, complemented
9#14529503C211yesmaximal, characteristic, complemented
10#14529376C211yesmaximal, characteristic, complemented
11#14529369C211yesmaximal, characteristic, complemented
12#14528914C211yesmaximal, characteristic, complemented
13#14528794C211yesmaximal, characteristic
14C4×#10587503C211yesmaximal, characteristic
15C17×#2357C211yesmaximal, characteristic
16C2×#10587555C2211yescharacteristic
17C2×#10587556C2211yescharacteristic
18C2×#10587478C2211yescharacteristic
19C2×#10587479C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×C4⋊#3622C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×C4×#3622C2211yescharacteristic
28#11762742—21no—
29#11762729—21no—
30#11762713—21no—
31#11762707C2211yescharacteristic
32#11762707C2211yescharacteristic
33#11762706—21no—
34#11762675C2211yescharacteristic
35C4×C4⋊Dic17C2211yescharacteristic
36#11762571C2211yescharacteristic
37#11762571C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C4×D4×C17C2211yescharacteristic
49C17×#390C2211yescharacteristic
50C17×#390C2211yescharacteristic
51C17×#388—21no—
52C17×#388—21no—
53C17×#382C2211yescharacteristic
54C17×#381—21no—
55C4×C4⋊C4×C17C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17—21no—
59C22×D4×C17C2311yescharacteristic
60C23×C4×C17—21no—
61C22×Dic34—21no—
62C4×D4×C17—22no—
63C4×D4×C17Q814yescomplemented
64C2×C4⋊C4×C17D411yescharacteristic
65C2×C4⋊C4×C17D411yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C22⋊C4×C17—21no—
70C2×C22⋊C4×C17—21no—
71C2×C22⋊C4×C17—21no—
72C2×C22⋊C4×C17—21no—
73C2×C22⋊C4×C17—21no—
74C2×C42×C17—21no—
75C2×C42×C17—21no—
76C2×C42×C17C2311yescharacteristic
77C2×C42×C17C2311yescharacteristic
78C2×C42×C17C2311yescharacteristic
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×C4×Dic17—21no—
89C22×#3622C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93#10587556—22no—
94#10587556—41no—
95#10587555—22no—
96#10587555—41no—
97C17×#95—21no—
98C17×#95—21no—
99C17×#95—21no—
100C17×#95—21no—
101C17×#95D411yescharacteristic
102C17×#95D411yescharacteristic
103#10587503—22no—
104#10587503—22no—
105#10587503—41no—
106#10587503—41no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587479—22no—
113#10587478—22no—
114C4⋊#3622—22no—
115C4⋊#3622—41no—
116C4⋊#3622—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×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2411yescharacteristic
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C2×Dic34—42no—
143C2×Dic34—42no—
144C2×Dic34—81no—
145C4⋊C4×C17—22no—
146C4⋊C4×C17—21no—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—21no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—41no—
152C4⋊C4×C17C2×Q812yes—
153C22⋊C4×C17—24no—
154C22⋊C4×C17—42no—
155C22⋊C4×C17—42no—
156C22⋊C4×C17—81no—
157C22⋊C4×C17—81no—
158C42×C17—21no—
159C42×C17—21no—
160C42×C17—21no—
161C42×C17—21no—
162C42×C17—81no—
163C42×C17—81no—
164C42×C17C2×Q812yes—
165C42×C17#4112yes—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—42no—
173C4⋊Dic17#4112yes—
174C4⋊Dic17#4112yes—
175#63225—42no—
176#63225—42no—
177#63225—42no—
178#63225—81no—
179C4×Dic17—81no—
180C2×#3622—21no—
181C2×#3622—21no—
182C2×#3622—21no—
183C2×#3622—21no—
184C2×#3622—21no—
185C2×#3622—21no—
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—21no—
210C2×C4×C17—21no—
211C2×C4×C17—22no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—84no—
230C2×C4×C17C22⋊Q812yes—
231C2×C4×C17C22⋊Q812yes—
232C2×C4×C17#13512yes—
233C2×C4×C17C22×Q811yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17H(5,𝔽2)11yescharacteristic
238C2×C4×C172−1+411yescharacteristic
239C2×Dic17—22no—
240C2×Dic17—22no—
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—41no—
251Dic34—84no—
252#3622—41no—
253#3622—41no—
254#3622—41no—
255#3622—41no—
256#2550—171no—
257#2357D1711yescharacteristic
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#1804—171no—
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—22no—
288C4×C17—42no—
289C4×C17—41no—
290C4×C17—41no—
291C4×C17—41no—
292C4×C17—81no—
293C4×C17—81no—
294C4×C17#53311yescharacteristic
295C4×C17#54411yescharacteristic
296C4×C17#55511yescharacteristic
297C4×C17#55611yescharacteristic
298Dic17—42no—
299Dic17—42no—
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⋊C8—171no—
310C2×#102—171no—
311C2×#102—171no—
312C2×#102—171no—
313C2×#102—171no—
314C2×C4×C8—171no—
315#398—341no—
316#395—341no—
317#391—341no—
318#390C2×D1711yescharacteristic
319#390C2×D1711yescharacteristic
320#389—171no—
321#389—171no—
322#389—341no—
323#388—21no—
324#388—21no—
325#382—171no—
326#382C2×D1711yescharacteristic
327#381—21no—
328C4×C4⋊C4—171no—
329C4×C4⋊C4C2×D1711yescharacteristic
330C4×C4⋊C4C2×D1711yescharacteristic
331C4×C4⋊C4C2×D1711yescharacteristic
332C4×C22⋊C4—21no—
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#340—171no—
342#340—171no—
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×C8—171no—
359C22×C8—171no—
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—341no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4C22⋊D1711yescharacteristic
376C2×C4⋊C4C22⋊D1711yescharacteristic
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—341no—
388C2×C42C22×D1711yescharacteristic
389C2×C42C22×D1711yescharacteristic
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—342no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95C22⋊D1711yescharacteristic
411#95C22⋊D1711yescharacteristic
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—341no—
430C22×C4—341no—
431C22×C4—341no—
432C22×C4—341no—
433C22×C4C2×C22⋊D1711yescharacteristic
434C22×C4C2×C22⋊D1711yescharacteristic
435C22×C4C2×C22⋊D1711yescharacteristic
436C22×C4C2×C22⋊D1711yescharacteristic
437C22×C4C2×C22⋊D1711yescharacteristic
438C22×C4C2×C22⋊D1711yescharacteristic
439C22×C4C23×D1711yescharacteristic
440C2×C8—341no—
441C2×C8—341no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8—341no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—341no—
451C2×C8—341no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—172no—
460C4⋊C4—172no—
461C4⋊C4—341no—
462C4⋊C4—341no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—682no—
468C4⋊C4—682no—
469C4⋊C4—682no—
470C4⋊C4—682no—
471C4⋊C4—1361no—
472C4⋊C4C2×Dic3412yes—
473C22⋊C4—24no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—81no—
483C42—81no—
484C42—1361no—
485C42C2×Dic3412yes—
486C42#6325212yes—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C22×C22⋊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—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
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—84no—
525C2×C4—342no—
526C2×C4—342no—
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×C4—681no—
537C2×C4#1058761012yes—
538C2×C4#1058761012yes—
539C2×C4#1058763612yes—
540C2×C4C22×Dic3411yescharacteristic
541C2×C4C2×#6325211yescharacteristic
542C2×C4#41×D1711yescharacteristic
543C2×C4#41×D1711yescharacteristic
544C2×C4#1058769611yescharacteristic
545C2×C4#1058769711yescharacteristic
546C8—681no—
547C8—681no—
548C8—681no—
549C8—681no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1058761011yescharacteristic
558C22#1176344711yescharacteristic
559C22#1176346611yescharacteristic
560C22C2×#1058763611yescharacteristic
561C22#1176360711yescharacteristic
562C22#1176361211yescharacteristic
563C22#1176362011yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—41no—
568C4—42no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—682no—
574C4—682no—
575C4—1361no—
576C4—1361no—
577C4#1176336211yescharacteristic
578C4#1176343711yescharacteristic
579C4#1176352211yescharacteristic
580C4#1176352711yescharacteristic
581C2—82no—
582C2#1453246911yescharacteristic
583C2#1453857211yescharacteristic
584C2#1453861511yescharacteristic
585C2#1454062811yescharacteristic
586C2#1454103111yescharacteristic
587C2#1454147611yescharacteristic
588C2#1454153111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2357C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2357D17
hypocenterC17#26028
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. #14533805C2semidirect
6. #14529741C2semidirect
7. #14529736C2semidirect
8. #14529550C2semidirect
9. #14529503C2semidirect
10. #14529376C2semidirect
11. #14529369C2semidirect
12. #14528914C2semidirect
63. 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
#145338052
#145301722
#145301582
#145299302
#145299202
#145297412
#145297362
#145295502
#145295032
#145293762
#145293692
#145289142
#145287942
C4×#105875032
C17×#23572
#2602817