Finite Group Databasebeta

← #17070412

Subgroups — #17070412

Order 4352.

1294 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D17C211yesmaximal, characteristic
2C2×#11763094C211yesmaximal, characteristic, complemented
3C4×#10587632C211yesmaximal, characteristic
4#14530115C212yesmaximal, complemented
5#14529923C212yesmaximal, complemented
6#14529737C211yesmaximal, characteristic, complemented
7#14529735C211yesmaximal, characteristic, complemented
8#14529088C212yesmaximal
9#14528975C212yesmaximal
10#14528761C211yesmaximal, characteristic
11C2×C17×#475C211yesmaximal, characteristic, complemented
12C22×Q8×D17—21no—
13C2×#10587555C2211yescharacteristic
14C2×#10587574—21no—
15C2×#10587574C2212yes—
16C2×#10587630—21no—
17C4⋊Q8×D17—24no—
18C4⋊Q8×D17—24no—
19C2×#10587632C2211yescharacteristic
20C2×C4⋊C4×D17—21no—
21C2×C4⋊C4×D17C2212yes—
22C2×C42×D17C2211yescharacteristic
23#11763094—24no—
24C2×#10587479C2212yes—
25C2×C4⋊#3622C2211yescharacteristic
26C2×#10587506C2212yes—
27C2×#10587491C2212yes—
28#11762716C2211yescharacteristic
29#11762705C2211yescharacteristic
30#11762690—21no—
31#11762680—21no—
32#11762666C2211yescharacteristic
33C4×#63225—21no—
34C4×C4⋊Dic17C2211yescharacteristic
35C4×#63227—21no—
36#11762572C2212yes—
37#11762572C2212yes—
38#11762571C2212yes—
39#11762571C2212yes—
40#11762563C2212yes—
41C2×C17×C4⋊Q8C2211yescharacteristic
42C2×C4×Q8×C17C2211yescharacteristic
43C17×#475—24no—
44C2×C17×#106C2212yes—
45C2×C17×C4⋊C8C2211yescharacteristic
46C2×C17×#103C2212yes—
47C22×Q8×C17—21no—
48C22×Q8×C17C2311yescharacteristic
49C2×Q8×D17—28no—
50C2×Q8×D17—44no—
51C2×Q8×D17—44no—
52C22×C4×D17—22no—
53C22×C4×D17D411yescharacteristic
54C22×C4×D17D411yescharacteristic
55C22×C4×D17C2311yescharacteristic
56C22×Dic34—21no—
57C22×Dic34—22no—
58C22×C8×C17C2312yes—
59C17×C4⋊Q8—22no—
60C17×C4⋊Q8D414yes—
61C4×Q8×C17—22no—
62C4×Q8×C17—22no—
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2312yes—
67C2×C42×C17—21no—
68C2×C42×C17C2311yescharacteristic
69#10587632—22no—
70#10587632Q814yescomplemented
71#10587630—28no—
72C2×#63227—21no—
73C2×#63227—21no—
74C2×C4⋊Dic17—21no—
75C2×C4⋊Dic17C2311yescharacteristic
76C2×C4⋊Dic17C2312yes—
77C2×C4⋊Dic17C2311yescharacteristic
78C2×#63225—21no—
79C2×#63225—22no—
80C2×#63225—22no—
81C2×#63225—21no—
82C2×C4×Dic17—21no—
83C2×C4×Dic17D411yescharacteristic
84C2×C4×Dic17D411yescharacteristic
85C2×C4×Dic17C2311yescharacteristic
86C22×#3622C2312yes—
87C4⋊C4×D17—28no—
88C4⋊C4×D17—24no—
89C4⋊C4×D17—24no—
90#10587574—28no—
91#10587574—24no—
92#10587574—24no—
93C42×D17—22no—
94C42×D17—22no—
95#10587555—22no—
96#10587555D414yes—
97C17×#106—24no—
98C17×C4⋊C8—22no—
99C17×#103—24no—
100#10587506—24no—
101#10587502—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587491—24no—
107#10587479—24no—
108C4⋊#3622—22no—
109C23×D17C2×D411yescharacteristic
110C2×Q8×C17—24no—
111C2×Q8×C17—21no—
112C2×Q8×C17—21no—
113C2×Q8×C17—21no—
114C2×Q8×C17—21no—
115C2×Q8×C17—21no—
116C2×Q8×C17—42no—
117C2×Q8×C17—42no—
118C2×Q8×C17C2×Q812yescomplemented
119C22×C4×C17—22no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17C2×D411yescharacteristic
123C22×C4×C17C2×D411yescharacteristic
124C22×C4×C17C2411yescharacteristic
125C22×Dic17—22no—
126C22×Dic17—21no—
127C22×Dic17—21no—
128C22×Dic17C2×D411yescharacteristic
129C22×Dic17C2×D411yescharacteristic
130C22×Dic17C2×D411yescharacteristic
131Q8×D17—432no—
132C2×C4×D17—22no—
133C2×C4×D17—22no—
134C2×C4×D17—22no—
135C2×C4×D17—22no—
136C2×C4×D17—21no—
137C2×C4×D17—21no—
138C2×C4×D17—21no—
139C2×C4×D17—21no—
140C2×C4×D17—22no—
141C2×C4×D17—28no—
142C2×C4×D17—44no—
143C2×C4×D17—44no—
144C2×C4×D17SD1614yes—
145C2×C4×D17C2×Q812yes—
146C2×C4×D17#4112yes—
147C2×Dic34—28no—
148C2×Dic34—24no—
149C2×Dic34—44no—
150C2×Dic34—42no—
151C2×Dic34—42no—
152C2×Dic34—44no—
153C2×C8×C17—22no—
154C2×C8×C17—22no—
155C2×C8×C17—22no—
156C4⋊C4×C17—24no—
157C4⋊C4×C17—22no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—42no—
161C4⋊C4×C17C2×D414yes—
162C42×C17—42no—
163C42×C17C2×D412yes—
164C42×C17C2×D412yes—
165#63227—24no—
166#63227—42no—
167C4⋊Dic17—24no—
168C4⋊Dic17—22no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17C2×D414yes—
173C4⋊Dic17C2×Q812yescomplemented
174#63225—28no—
175#63225—28no—
176#63225—24no—
177#63225—42no—
178C4×Dic17—22no—
179C4×Dic17—22no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—42no—
183C2×#3622—22no—
184C2×#3622—22no—
185C2×#3622—22no—
186#24301—171nomaximal, Sylow
187C23×C17C22×D411yescharacteristic
188C22×D17—21no—
189C22×D17—21no—
190C22×D17—21no—
191C22×D17—21no—
192C22×D17C22⋊Q812yes—
193C22×D17C2×SD1612yes—
194C22×D17#13712yes—
195Q8×C17—22no—
196Q8×C17—22no—
197Q8×C17—41no—
198Q8×C17—41no—
199Q8×C17—48no—
200C2×C4×C17—21no—
201C2×C4×C17—21no—
202C2×C4×C17—24no—
203C2×C4×C17—22no—
204C2×C4×C17—41no—
205C2×C4×C17—41no—
206C2×C4×C17—42no—
207C2×C4×C17—42no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—41no—
211C2×C4×C17#12012yes—
212C2×C4×C17#12012yes—
213C2×C4×C17C2×SD1612yes—
214C2×C4×C17AGL(1,ℤ/8)12yes—
215C2×C4×C17C22×D411yescharacteristic
216C2×C4×C17C22×D411yescharacteristic
217C2×C4×C17C22×Q811yescharacteristic
218C2×C4×C17C2×#4111yescharacteristic
219C2×C4×C172−1+411yescharacteristic
220C2×C4×C172−1+411yescharacteristic
221C2×Dic17—21no—
222C2×Dic17—21no—
223C2×Dic17—21no—
224C2×Dic17—21no—
225C2×Dic17—21no—
226C2×Dic17—21no—
227C2×Dic17—24no—
228C2×Dic17—22no—
229C2×Dic17—42no—
230C2×Dic17—42no—
231C2×Dic17—41no—
232C2×Dic17—41no—
233C2×Dic17—41no—
234C2×Dic17—41no—
235C2×Dic17—41no—
236C2×Dic17C22⋊Q812yes—
237C2×Dic17C2×SD1612yes—
238C2×Dic17#13712yes—
239C4×D17—28no—
240C4×D17—22no—
241C4×D17—22no—
242C4×D17—22no—
243C4×D17—22no—
244C4×D17—28no—
245C4×D17—416no—
246Dic34—416no—
247Dic34—48no—
248C8×C17—42no—
249#3622—42no—
250C22×C4⋊Q8—171no—
251C2×#475—171no—
252C2×#475D1711yescharacteristic, complemented
253C4×C22⋊Q8—171no—
254#2058—172no—
255#2024—172no—
256#1988—171no—
257#1988—171no—
258#1872—172no—
259#1853—172no—
260#1797—171no—
261C22×C17C2×#12011yescharacteristic
262C22×C17C2×C22⋊Q811yescharacteristic
263C22×C17#53611yescharacteristic
264C22×C17C22×SD1611yescharacteristic
265C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C17C2×#13711yescharacteristic
267C22×C17C2×#13711yescharacteristic
268C2×D17—22no—
269C2×D17—22no—
270C2×D17—22no—
271C2×D17—22no—
272C2×D17—22no—
273C2×D17—22no—
274C2×D17#47514yescomplemented
275C4×C17—22no—
276C4×C17—42no—
277C4×C17—44no—
278C4×C17—81no—
279C4×C17#45014yes—
280C4×C17#54811yescharacteristic
281C4×C17#54811yescharacteristic
282C4×C17D4×Q811yescharacteristic
283C4×C17D4×Q811yescharacteristic
284Dic17—22no—
285Dic17—22no—
286Dic17—22no—
287Dic17—42no—
288Dic17—44no—
289Dic17—81no—
290C23×Q8—341no—
291C2×C4⋊Q8—172no—
292C2×C4⋊Q8—171no—
293C2×C4⋊Q8—341no—
294C2×C4⋊Q8—341no—
295C2×C4⋊Q8—344no—
296C2×C4⋊Q8—344no—
297C2×C4⋊Q8C2×D1711yescharacteristic
298C2×C22⋊Q8—171no—
299C2×C4×Q8C2×D1711yescharacteristic
300C22×C4⋊C4—172no—
301C22×C4⋊C4—341no—
302C22×C42—171no—
303#475—24no—
304#475—344no—
305C2×#106—172no—
306C2×#106C2×D1712yes—
307C2×C4⋊C8—171no—
308C2×C4⋊C8C2×D1711yescharacteristic
309C2×#103—172no—
310C2×#103C2×D1712yes—
311C2×C22⋊C8—172no—
312#391—171no—
313#389—171no—
314#387—341no—
315#385—341no—
316#380—171no—
317C4×C4⋊C4—171no—
318C4×C4⋊C4—341no—
319C4×C22⋊C4—341no—
320#340—172no—
321#340—172no—
322#340—172no—
323#340—172no—
324#336—172no—
325C2×C17#240911yescharacteristic
326C2×C17C2×#45011yescharacteristic
327C2×C17#300411yescharacteristic
328C2×C17C2×#47511yescharacteristic
329C2×C17#307911yescharacteristic
330C2×C17#322911yescharacteristic
331C2×C17#323911yescharacteristic
332D17—24no—
333C22×Q8—21no—
334C22×Q8—342no—
335C22×Q8—348no—
336C22×Q8—341no—
337C22×Q8—684no—
338C22×Q8—684no—
339C22×Q8C22×D1711yescharacteristic
340C23×C4—171no—
341C23×C4—171no—
342C23×C4—171no—
343C23×C4—342no—
344C22×C8—172no—
345C22×C8C22×D1712yes—
346C4⋊Q8—22no—
347C4⋊Q8—174no—
348C4⋊Q8—344no—
349C4⋊Q8—348no—
350C4⋊Q8—348no—
351C4⋊Q8—344no—
352C4⋊Q8—342no—
353C4⋊Q8C22⋊D1714yes—
354C22⋊Q8—174no—
355C22⋊Q8—342no—
356C4×Q8—22no—
357C4×Q8—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—172no—
361C2×C4⋊C4—171no—
362C2×C4⋊C4—171no—
363C2×C4⋊C4—344no—
364C2×C4⋊C4—348no—
365C2×C4⋊C4—342no—
366C2×C4⋊C4—342no—
367C2×C4⋊C4—341no—
368C2×C4⋊C4—341no—
369C2×C4⋊C4—344no—
370C2×C4⋊C4—341no—
371C2×C4⋊C4C22×D1711yescharacteristic
372C2×C4⋊C4C22×D1712yes—
373C2×C22⋊C4—341no—
374C2×C22⋊C4—341no—
375C2×C42—21no—
376C2×C42—171no—
377C2×C42—171no—
378C2×C42—171no—
379C2×C42—342no—
380C2×C42—342no—
381C2×C42—341no—
382C2×C42C22×D1711yescharacteristic
383#106—24no—
384#106—344no—
385C4⋊C8—22no—
386C4⋊C8—342no—
387#103—24no—
388#103—344no—
389C22⋊C8—344no—
390#95—341no—
391#95—341no—
392#95—341no—
393#95—341no—
394#95—341no—
395C17#2430111yescharacteristic, complemented, Sylow
396C24—171no—
397C2×Q8—21no—
398C2×Q8—24no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—348no—
406C2×Q8—344no—
407C2×Q8—684no—
408C2×Q8—684no—
409C2×Q8—6832no—
410C2×Q8—682no—
411C2×Q8—682no—
412C2×Q8Q8×D1712yescomplemented
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—171no—
417C22×C4—171no—
418C22×C4—171no—
419C22×C4—172no—
420C22×C4—172no—
421C22×C4—174no—
422C22×C4—342no—
423C22×C4—342no—
424C22×C4—348no—
425C22×C4—341no—
426C22×C4—341no—
427C22×C4—341no—
428C22×C4—341no—
429C22×C4—342no—
430C22×C4—342no—
431C22×C4—342no—
432C22×C4—341no—
433C22×C4—342no—
434C22×C4—341no—
435C22×C4—684no—
436C22×C4—684no—
437C22×C4D4×D1711yescharacteristic
438C22×C4D4×D1711yescharacteristic
439C22×C4C23×D1711yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—342no—
444C2×C8—342no—
445C2×C8—342no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—42no—
451C4⋊C4—174no—
452C4⋊C4—172no—
453C4⋊C4—344no—
454C4⋊C4—348no—
455C4⋊C4—348no—
456C4⋊C4—341no—
457C4⋊C4—342no—
458C4⋊C4—344no—
459C4⋊C4—341no—
460C4⋊C4—341no—
461C4⋊C4—682no—
462C4⋊C4C2×C22⋊D1714yes—
463C22⋊C4—344no—
464C22⋊C4—682no—
465C42—42no—
466C42—341no—
467C42—341no—
468C42—342no—
469C42—342no—
470C42—682no—
471C42D4×D1712yes—
472C42C2×C22⋊D1712yes—
473C23—172no—
474C23—172no—
475C23—172no—
476C23—341no—
477C23—341no—
478C23—341no—
479C23—341no—
480C23C2×D4×D1711yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—41no—
484Q8—41no—
485Q8—48no—
486Q8—6816no—
487Q8—688no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—172no—
500C2×C4—172no—
501C2×C4—172no—
502C2×C4—348no—
503C2×C4—341no—
504C2×C4—341no—
505C2×C4—341no—
506C2×C4—341no—
507C2×C4—341no—
508C2×C4—341no—
509C2×C4—342no—
510C2×C4—342no—
511C2×C4—348no—
512C2×C4—342no—
513C2×C4—342no—
514C2×C4—344no—
515C2×C4—342no—
516C2×C4—6816no—
517C2×C4—682no—
518C2×C4—681no—
519C2×C4—682no—
520C2×C4—681no—
521C2×C4—681no—
522C2×C4—681no—
523C2×C4—681no—
524C2×C4SD16×D1712yes—
525C2×C4#1058760012yes—
526C2×C4#1058762312yes—
527C2×C4#1058762312yes—
528C2×C4C2×D4×D1711yescharacteristic
529C2×C4C2×Q8×D1711yescharacteristic
530C2×C4#1058769411yescharacteristic
531C2×C4#41×D1711yescharacteristic
532C2×C4#1058769711yescharacteristic
533C2×C4C22×C22⋊D1711yescharacteristic
534C8—42no—
535C8—682no—
536C22—174no—
537C22—342no—
538C22—342no—
539C22—342no—
540C22—342no—
541C22—342no—
542C22—342no—
543C22C2×#1058762311yescharacteristic
544C22C22⋊Q8×D1711yescharacteristic
545C22#1176338211yescharacteristic
546C22C2×SD16×D1711yescharacteristic
547C22C2×#1058760011yescharacteristic
548C22#137×D1711yescharacteristic
549C22#137×D1711yescharacteristic
550C4—22no—
551C4—42no—
552C4—44no—
553C4—81no—
554C4—342no—
555C4—342no—
556C4—342no—
557C4—682no—
558C4—684no—
559C4—1361no—
560C4#1176295814yes—
561C4#1176346611yescharacteristic
562C4#1176346911yescharacteristic
563C4#1176347211yescharacteristic
564C4Q8×C22⋊D1711yescharacteristic
565C2—344no—
566C2#1453214511yescharacteristic
567C2C2×#1176295811yescharacteristic
568C2#1453801611yescharacteristic
569C2D17×#47511yescharacteristic
570C2#1453859411yescharacteristic
571C2#1453992711yescharacteristic
572C2#1454001811yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#475C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
hypercenterC2×#475D17
hypocenterC17#24301
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
2. C2×#11763094C2semidirect
4. #14530115C2semidirect
5. #14529923C2semidirect
6. #14529737C2semidirect
7. #14529735C2semidirect
11. C2×C17×#475C2semidirect
70. #10587632Q8semidirect
118. C2×Q8×C17C2×Q8semidirect
173. C4⋊Dic17C2×Q8semidirect
252. C2×#475D17semidirect
274. C2×D17#475semidirect
395. C17#24301semidirect
412. C2×Q8Q8×D17semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D172
C2×#117630942
C4×#105876322
#145301152
#145299232
#145297372
#145297352
#145290882
#145289752
#145287612
C2×C17×#4752
#2430117