Finite Group Databasebeta

← #17045492

Subgroups — #17045492

Order 4352.

1224 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762960C211yesmaximal, characteristic
2C4×#10587610C211yesmaximal, characteristic, complemented
3#14529637C212yesmaximal, complemented
4#14529470C212yesmaximal, complemented
5#14529408C212yesmaximal, complemented
6#14529142C212yesmaximal, complemented
7#14529091C211yesmaximal, characteristic, complemented
8#14528977C211yesmaximal, characteristic, complemented
9#14528715C211yesmaximal, characteristic
10C4×C22⋊#3622C211yesmaximal, characteristic, complemented
11C2×C17×#390C211yesmaximal, characteristic
12C22×#63229C2212yes—
13C2×#10587610C2211yescharacteristic
14C2×C4×Dic34C2211yescharacteristic, complemented
15#11762960C418yescomplemented
16C2×#10587479C2212yescomplemented
17C2×#10587481C2212yescomplemented
18C2×#10587503C2212yes—
19C2×#10587503C2212yes—
20C2×#10587503C2212yes—
21C2×C22⋊#3622C2211yescharacteristic
22C2×C22⋊#3622C2211yescharacteristic
23C2×C4×#3622C2212yescomplemented
24#11762715C2211yescharacteristic, complemented
25#11762707C2211yescharacteristic, complemented
26#11762693—21no—
27#11762682—21no—
28#11762669C2211yescharacteristic
29C4×#63225—21no—
30C4×C4⋊Dic17C2211yescharacteristic, complemented
31C4×C22⋊Dic17—21no—
32#11762571C2212yescomplemented
33#11762571C2212yescomplemented
34#11762564C2212yescomplemented
35C23×D4×C17C2211yescharacteristic
36C22×C4⋊C4×C17C2211yescharacteristic
37C22×C22⋊C4×C17—21no—
38C22×C22⋊C4×C17—21no—
39C22×C42×C17C2211yescharacteristic
40C17×#390—21no—
41C17×#390—21no—
42C17×#390C2212yes—
43C17×#390C2212yes—
44C25×C17—21no—
45C22×D4×C17—21no—
46C22×D4×C17—22no—
47C22×D4×C17—42no—
48C22×D4×C17—42no—
49C22×D4×C17C2×C414yes—
50C22×D4×C17C2312yes—
51C23×C4×C17—21no—
52C23×C4×C17D411yescharacteristic
53C23×C4×C17D411yescharacteristic
54C23×C4×C17C2311yescharacteristic
55C22×Dic34C2311yescharacteristic
56C2×C4⋊C4×C17—21no—
57C2×C4⋊C4×C17—42no—
58C2×C4⋊C4×C17C2312yes—
59C2×C22⋊C4×C17—22no—
60C2×C22⋊C4×C17—22no—
61C2×C22⋊C4×C17—22no—
62C2×C22⋊C4×C17—22no—
63C2×C22⋊C4×C17—44no—
64C2×C22⋊C4×C17—44no—
65C2×C42×C17—42no—
66C2×C42×C17D411yescharacteristic
67C2×C42×C17D411yescharacteristic
68C2×C42×C17C2311yescharacteristic
69C2×C42×C17C2311yescharacteristic
70C2×C22⋊Dic17—21no—
71C2×C22⋊Dic17—21no—
72C2×#63229—42no—
73C2×#63229—42no—
74C2×#63229C2×C418yescomplemented
75C2×C4⋊Dic17C2311yescharacteristic
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78#10587610—22no—
79#10587610C2×C414yes—
80C2×#63225—21no—
81C2×#63225—21no—
82C2×#63225—21no—
83C2×C4×Dic17—21no—
84C2×C4×Dic17—21no—
85C22×#3622C2312yes—
86C22×#3622C2312yes—
87C4×Dic34—42no—
88#10587503—42no—
89#10587503—42no—
90#10587503C2×C418yescomplemented
91#10587502—21no—
92#10587502—21no—
93#10587502—21no—
94#10587502—21no—
95#10587502—21no—
96C22⋊#3622—22no—
97C22⋊#3622C2×C414yes—
98#10587481—42no—
99#10587479—42no—
100C4×#3622—42no—
101C24×C17—24no—
102C24×C17—22no—
103C24×C17—42no—
104C24×C17—42no—
105C24×C17—42no—
106C24×C17—42no—
107C24×C17—42no—
108C24×C17—42no—
109C24×C17C2×D411yescharacteristic
110C2×D4×C17—24no—
111C2×D4×C17—22no—
112C2×D4×C17—22no—
113C2×D4×C17—22no—
114C2×D4×C17—22no—
115C2×D4×C17—22no—
116C2×D4×C17—48no—
117C2×D4×C17—42no—
118C2×D4×C17—42no—
119C2×D4×C17—48no—
120C2×D4×C17C22×C414yes—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—41no—
127C22×C4×C17—41no—
128C22×C4×C17—41no—
129C22×C4×C17—42no—
130C22×C4×C17—42no—
131C22×C4×C17—42no—
132C22×C4×C17SD1614yescomplemented
133C22×C4×C17C22×C412yes—
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17#4112yes—
140C22×C4×C17C2411yescharacteristic
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C2×Dic34—22no—
146C2×Dic34—22no—
147C2×Dic34—21no—
148C2×Dic34C22×C412yes—
149C4⋊C4×C17—22no—
150C4⋊C4×C17—22no—
151C4⋊C4×C17—42no—
152C22⋊C4×C17—44no—
153C22⋊C4×C17—44no—
154C22⋊C4×C17—44no—
155C22⋊C4×C17—44no—
156C42×C17—22no—
157C42×C17—22no—
158C42×C17—41no—
159C42×C17—41no—
160C22⋊Dic17—24no—
161C22⋊Dic17—42no—
162#63229—48no—
163C4⋊Dic17—21no—
164C4⋊Dic17—22no—
165C4⋊Dic17—22no—
166C4⋊Dic17C22×C412yes—
167#63225—24no—
168#63225—42no—
169#63225—42no—
170C4×Dic17—42no—
171C4×Dic17—42no—
172C2×#3622—22no—
173C2×#3622—22no—
174C2×#3622—22no—
175C2×#3622—22no—
176C2×#3622—22no—
177C2×#3622C22×C414yes—
178#22043—171nomaximal, Sylow
179C23×C17—21no—
180C23×C17—21no—
181C23×C17—21no—
182C23×C17—21no—
183C23×C17—24no—
184C23×C17—48no—
185C23×C17—48no—
186C23×C17—42no—
187C23×C17—42no—
188C23×C17—42no—
189C23×C17—42no—
190C23×C17—42no—
191C23×C17—42no—
192C23×C17—48no—
193C23×C17—84no—
194C23×C17—84no—
195C23×C17—84no—
196C23×C17—84no—
197C23×C17C4×D412yes—
198C23×C17C2×SD1612yes—
199C23×C17#13512yes—
200C23×C17C22×D411yescharacteristic
201D4×C17—24no—
202D4×C17—24no—
203D4×C17—48no—
204D4×C17—42no—
205D4×C17—42no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—24no—
215C2×C4×C17—24no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—44no—
220C2×C4×C17—44no—
221C2×C4×C17—42no—
222C2×C4×C17—42no—
223C2×C4×C17—42no—
224C2×C4×C17—84no—
225C2×C4×C17C4×D412yes—
226C2×C4×C17C2×SD1612yes—
227C2×C4×C17C2×SD1612yes—
228C2×C4×C17AGL(1,ℤ/8)12yes—
229C2×C4×C17C23×C411yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17H(5,𝔽2)12yes—
232C2×C4×C17H(5,𝔽2)12yes—
233C2×Dic17—22no—
234C2×Dic17—22no—
235C2×Dic17—41no—
236C2×Dic17—41no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245Dic34—24no—
246Dic34—42no—
247#3622—24no—
248#3622—42no—
249C2×#450—171no—
250C4×C22⋊Q8—171no—
251C2×#390D1711yescharacteristic
252#1968—172no—
253#1934—172no—
254#1926—172no—
255#1880—172no—
256#1872—171no—
257#1853—171no—
258#1786—171no—
259C4×C22⋊C8—171no—
260C22×C17—22no—
261C22×C17—22no—
262C22×C17—22no—
263C22×C17—22no—
264C22×C17—22no—
265C22×C17—22no—
266C22×C17—44no—
267C22×C17—44no—
268C22×C17—44no—
269C22×C17—816no—
270C22×C17—82no—
271C22×C17—82no—
272C22×C17—82no—
273C22×C17—82no—
274C22×C17C4×SD1614yes—
275C22×C17C2×C4×D411yescharacteristic
276C22×C17#53411yescharacteristic
277C22×C17#53511yescharacteristic
278C22×C17C22×SD1611yescharacteristic
279C22×C17C2×#13511yescharacteristic
280C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C17#57511yescharacteristic
282C4×C17—22no—
283C4×C17—22no—
284C4×C17—22no—
285C4×C17—22no—
286C4×C17—41no—
287C4×C17—82no—
288C4×C17#51812yes—
289C4×C17#53812yes—
290Dic17—42no—
291Dic17—42no—
292Dic17—81no—
293Dic17—81no—
294C23×D4C2×D1711yescharacteristic
295C22×SD16—172no—
296C2×C22⋊Q8—171no—
297C2×C4×Q8—171no—
298C22×C4⋊C4C2×D1711yescharacteristic
299C22×C22⋊C4—21no—
300C22×C22⋊C4—21no—
301C22×C42C2×D1711yescharacteristic
302#450—178no—
303C2×#106—172no—
304C2×#103—172no—
305C2×#102—172no—
306C2×#102—172no—
307C2×#102—172no—
308C2×C22⋊C8—171no—
309C2×C22⋊C8—171no—
310C2×C4×C8—172no—
311#391—171no—
312#390—21no—
313#390—21no—
314#390C2×D1712yes—
315#390C2×D1712yes—
316#389—171no—
317#387—341no—
318#385—341no—
319#380—171no—
320C4×C4⋊C4—171no—
321C4×C4⋊C4—341no—
322C4×C22⋊C4—341no—
323#340—172no—
324#340—172no—
325#336—172no—
326C2×C17—24no—
327C2×C17—84no—
328C2×C17#232111yescharacteristic
329C2×C17C2×C4×SD1611yescharacteristic
330C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
331C2×C17#318711yescharacteristic
332C2×C17#318811yescharacteristic
333C2×C17#320811yescharacteristic
334C2×C17#321111yescharacteristic
335C25—21no—
336C22×Q8—171no—
337C22×D4—22no—
338C22×D4—21no—
339C22×D4—42no—
340C22×D4—42no—
341C22×D4C4×D1714yes—
342C22×D4C22×D1712yes—
343C23×C4—21no—
344C23×C4C22⋊D1711yescharacteristic
345C23×C4C22⋊D1711yescharacteristic
346C23×C4C22×D1711yescharacteristic
347C2×SD16—178no—
348C2×SD16—682no—
349C2×SD16—682no—
350C22×C8—172no—
351C22×C8—172no—
352C22⋊Q8—174no—
353C22⋊Q8—342no—
354C4×Q8—682no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—42no—
357C2×C4⋊C4—171no—
358C2×C4⋊C4—171no—
359C2×C4⋊C4—171no—
360C2×C4⋊C4—341no—
361C2×C4⋊C4—341no—
362C2×C4⋊C4—341no—
363C2×C4⋊C4C22×D1712yes—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—44no—
369C2×C22⋊C4—44no—
370C2×C22⋊C4—341no—
371C2×C22⋊C4—341no—
372C2×C42—42no—
373C2×C42—341no—
374C2×C42—341no—
375C2×C42C22⋊D1711yescharacteristic
376C2×C42C22⋊D1711yescharacteristic
377C2×C42C22×D1711yescharacteristic
378C2×C42C22×D1711yescharacteristic
379#106—682no—
380#103—682no—
381#102—178no—
382#102—682no—
383#102—682no—
384C22⋊C8—174no—
385C22⋊C8—342no—
386C4×C8—682no—
387#95—341no—
388#95—341no—
389#95—341no—
390#95—341no—
391#95—341no—
392C17#2204311yescharacteristic, complemented, Sylow
393C24—22no—
394C24—24no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24C2×C22⋊D1711yescharacteristic
402C2×Q8—172no—
403C2×Q8—341no—
404C2×Q8—342no—
405C2×Q8—342no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4C2×C4×D1714yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—341no—
429C22×C4—341no—
430C22×C4—341no—
431C22×C4—341no—
432C22×C4#6322914yescomplemented
433C22×C4C2×C4×D1712yes—
434C22×C4#6325212yes—
435C22×C4C2×C22⋊D1711yescharacteristic
436C22×C4C2×C22⋊D1711yescharacteristic
437C22×C4C2×C22⋊D1711yescharacteristic
438C22×C4C2×C22⋊D1711yescharacteristic
439C22×C4C2×C22⋊D1711yescharacteristic
440C22×C4C23×D1711yescharacteristic
441SD16—688no—
442C2×C8—174no—
443C2×C8—342no—
444C2×C8—342no—
445C2×C8—342no—
446C2×C8—342no—
447C2×C8—342no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—172no—
452C4⋊C4—341no—
453C4⋊C4—344no—
454C4⋊C4—342no—
455C4⋊C4—342no—
456C4⋊C4—682no—
457C4⋊C4—682no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—344no—
463C22⋊C4—682no—
464C42—22no—
465C42—22no—
466C42—41no—
467C42—41no—
468C42—682no—
469C42—682no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—24no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—48no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23C4×C22⋊D1712yes—
489C23C2×#6322912yes—
490C23#1058763612yes—
491C23C22×C22⋊D1711yescharacteristic
492Q8—344no—
493Q8—682no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—84no—
518C2×C4—342no—
519C2×C4—342no—
520C2×C4—681no—
521C2×C4—681no—
522C2×C4—681no—
523C2×C4—681no—
524C2×C4—681no—
525C2×C4—681no—
526C2×C4—681no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4C4×C22⋊D1712yes—
531C2×C4#1058761812yes—
532C2×C4C2×#6322912yes—
533C2×C4C2×#6322912yes—
534C2×C4C22×C4×D1711yescharacteristic
535C2×C4C2×#6325211yescharacteristic
536C2×C4#1058769112yes—
537C2×C4#1058769112yes—
538C8—344no—
539C8—682no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22C4×#6322914yes—
555C22C2×C4×C22⋊D1711yescharacteristic
556C22#1176336811yescharacteristic
557C22#1176337611yescharacteristic
558C22C22×#6322911yescharacteristic
559C22C2×#1058763611yescharacteristic
560C22C2×#1058761811yescharacteristic
561C22#1176360711yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—82no—
568C4—682no—
569C4—682no—
570C4—1361no—
571C4—1361no—
572C4#1176328212yes—
573C4#1176339712yes—
574C2—24no—
575C2—84no—
576C2#1453144211yescharacteristic
577C2C2×C4×#6322911yescharacteristic
578C2C4×#1058761811yescharacteristic
579C2#1453954111yescharacteristic
580C2#1453954911yescharacteristic
581C2#1453972511yescharacteristic
582C2#1453974911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#390C2
socleC23×C17C22×D4
prime radicalC23×D4×C17C22
cosocleC22×C4×C17C24
hypercenterC2×#390D17
hypocenterC17#22043
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C4×#10587610C2semidirect
3. #14529637C2semidirect
4. #14529470C2semidirect
5. #14529408C2semidirect
6. #14529142C2semidirect
7. #14529091C2semidirect
8. #14528977C2semidirect
10. C4×C22⋊#3622C2semidirect
14. C2×C4×Dic34C22semidirect
15. #11762960C4semidirect
16. C2×#10587479C22semidirect
17. C2×#10587481C22semidirect
23. C2×C4×#3622C22semidirect
24. #11762715C22semidirect
25. #11762707C22semidirect
30. C4×C4⋊Dic17C22semidirect
32. #11762571C22semidirect
33. #11762571C22semidirect
34. #11762564C22semidirect
74. C2×#63229C2×C4semidirect
90. #10587503C2×C4semidirect
132. C22×C4×C17SD16semidirect
392. C17#22043semidirect
432. C22×C4#63229semidirect

Maximal subgroups

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

subgroupindex
C2×#117629602
C4×#105876102
#145296372
#145294702
#145294082
#145291422
#145290912
#145289772
#145287152
C4×C22⋊#36222
C2×C17×#3902
#2204317