Finite Group Databasebeta

← #17066306

Subgroups — #17066306

Order 4352.

1382 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762959C211yesmaximal, characteristic, complemented
2C2×#11762942C211yesmaximal, characteristic
3#14531364C211yesmaximal, characteristic, complemented
4#14529917C212yesmaximal, complemented
5#14529861C212yesmaximal, complemented
6#14529139C212yesmaximal, complemented
7#14529132C212yesmaximal, complemented
8#14529107C211yesmaximal, characteristic, complemented
9#14529097C211yesmaximal, characteristic, complemented
10#14528855C211yesmaximal, characteristic, complemented
11C2×C17×#542C211yesmaximal, characteristic, complemented
12C23×D68C2211yescharacteristic
13C22×#63230C2212yes—
14C22×#63228C2212yes—
15C2×#10587612C2211yescharacteristic
16#11762959—24no—
17#11762942—24no—
18C2×#10587506C2212yescomplemented
19C2×#10587503C2212yescomplemented
20C2×#10587480C2212yes—
21C2×#10587480C2212yes—
22C2×C22⋊#3622C2211yescharacteristic
23C2×C22⋊#3622C2211yescharacteristic
24#11762711C2212yes—
25#11762711C2212yes—
26#11762710—21no—
27#11762710—21no—
28#11762664—21no—
29#11762664—21no—
30#11762571C2212yescomplemented
31#11762571C2212yescomplemented
32#11762564C2212yescomplemented
33#11762564C2212yescomplemented
34C17×#542—24no—
35C17×#542—24no—
36C2×C17×C4⋊Q8C2211yescharacteristic, complemented
37C2×C17×#126—21no—
38C2×C17×#124—21no—
39C2×C17×#124C2211yescharacteristic, complemented
40C2×C17×#123—21no—
41C2×C17×C22⋊Q8—21no—
42C2×C17×C22⋊Q8C2211yescharacteristic
43C2×C17×#121C2211yescharacteristic
44C2×C4×Q8×C17C2211yescharacteristic, complemented
45C2×C4×D4×C17C2211yescharacteristic, complemented
46C2×C17×#117C2211yescharacteristic
47C24×D17—21no—
48C22×Q8×C17—21no—
49C22×Q8×C17C2311yescharacteristic
50C22×D4×C17—21no—
51C22×D4×C17C2311yescharacteristic
52C23×C4×C17—21no—
53C23×C4×C17C2311yescharacteristic
54C22×D68—21no—
55C22×D68—24no—
56C22×D68—22no—
57C22×D68—22no—
58C22×D68—42no—
59C22×D68C2312yes—
60C17×C4⋊Q8—22no—
61C17×C4⋊Q8—22no—
62C17×#126—28no—
63C17×#124—28no—
64C17×#124—22no—
65C17×#124—22no—
66C17×#123—28no—
67C17×C22⋊Q8—28no—
68C17×C22⋊Q8—22no—
69C17×C22⋊Q8D414yes—
70C17×#121—22no—
71C17×#121D414yes—
72C4×Q8×C17—22no—
73C4×Q8×C17—22no—
74C4×D4×C17—22no—
75C4×D4×C17—22no—
76C17×#117—22no—
77C17×#117—22no—
78C2×C4⋊C4×C17—21no—
79C2×C4⋊C4×C17—21no—
80C2×C4⋊C4×C17—21no—
81C2×C4⋊C4×C17D411yescharacteristic
82C2×C4⋊C4×C17D411yescharacteristic
83C2×C4⋊C4×C17C2311yescharacteristic
84C2×C4⋊C4×C17C2311yescharacteristic
85C2×C22⋊C4×C17—21no—
86C2×C22⋊C4×C17—21no—
87C2×C22⋊C4×C17—21no—
88C2×C22⋊C4×C17—21no—
89C2×C22⋊C4×C17—21no—
90C2×C22⋊C4×C17D411yescharacteristic
91C2×C22⋊C4×C17D411yescharacteristic
92C2×C42×C17—21no—
93C2×C42×C17C2311yescharacteristic
94C2×C42×C17C2311yescharacteristic
95C2×C22⋊Dic17—21no—
96C2×#63230—24no—
97C2×#63230—24no—
98C2×#63230—42no—
99C2×#63228—24no—
100C2×#63228—24no—
101C2×#63228—42no—
102C2×#63227—22no—
103C2×#63227—22no—
104C2×#63227—22no—
105C2×#63227—22no—
106#10587612—42no—
107C2×C4⋊Dic17C2312yes—
108C2×C4×Dic17—21no—
109C22×#3622C2312yes—
110C22×#3622C2312yes—
111#10587506—42no—
112#10587503—42no—
113C22⋊#3622—22no—
114C22⋊#3622—22no—
115#10587480—24no—
116#10587480—24no—
117C24×C17—21no—
118C24×C17C2×D411yescharacteristic
119C23×D17—22no—
120C23×D17—24no—
121C23×D17—42no—
122C23×D17—42no—
123C23×D17—82no—
124C23×D17—82no—
125C2×Q8×C17—24no—
126C2×Q8×C17—21no—
127C2×Q8×C17—22no—
128C2×Q8×C17—22no—
129C2×Q8×C17—82no—
130C2×Q8×C17C2×D412yes—
131C2×D4×C17—24no—
132C2×D4×C17—21no—
133C2×D4×C17—22no—
134C2×D4×C17—22no—
135C2×D4×C17—82no—
136C2×D4×C17C2×D412yes—
137C22×C4×C17—22no—
138C22×C4×C17—22no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—24no—
143C22×C4×C17—82no—
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D412yes—
150C22×C4×C17C2×D412yes—
151C22×C4×C17C2411yescharacteristic
152C22×Dic17—22no—
153C2×D68—22no—
154C2×D68—22no—
155C2×D68—22no—
156C2×D68—22no—
157C2×D68—24no—
158C2×D68—22no—
159C2×D68—48no—
160C2×D68—48no—
161C2×D68—42no—
162C2×D68—42no—
163C2×D68#4114yes—
164C4⋊C4×C17—24no—
165C4⋊C4×C17—24no—
166C4⋊C4×C17—22no—
167C4⋊C4×C17—24no—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17C2×D412yes—
172C4⋊C4×C17C2×D412yes—
173C22⋊C4×C17—24no—
174C22⋊C4×C17—24no—
175C22⋊C4×C17—24no—
176C22⋊C4×C17—24no—
177C22⋊C4×C17—22no—
178C22⋊C4×C17—22no—
179C22⋊C4×C17—24no—
180C42×C17—24no—
181C42×C17—21no—
182C42×C17—21no—
183C42×C17—21no—
184C42×C17—21no—
185C22⋊Dic17—42no—
186#63230—48no—
187#63228—48no—
188#63227—82no—
189#63227—82no—
190#63227—82no—
191#63227—82no—
192C4⋊Dic17—22no—
193C4⋊Dic17—22no—
194C4×Dic17—42no—
195C2×#3622—22no—
196C2×#3622—22no—
197C2×#3622—22no—
198C2×#3622—22no—
199C2×#3622—22no—
200C2×#3622—22no—
201#23945—171nomaximal, Sylow
202C23×C17—21no—
203C23×C17—21no—
204C23×C17—22no—
205C23×C17—22no—
206C23×C17—22no—
207C23×C17—41no—
208C23×C17—41no—
209C23×C17—42no—
210C23×C17—42no—
211C23×C17#12012yes—
212C23×C17C22×D411yescharacteristic
213C22×D17—24no—
214C22×D17—48no—
215C22×D17—42no—
216C22×D17—42no—
217C22×D17—42no—
218C22×D17—42no—
219C22×D17—42no—
220C22×D17—42no—
221C22×D17—88no—
222C22×D17—84no—
223C22×D17—84no—
224C22×D17—88no—
225Q8×C17—24no—
226Q8×C17—42no—
227Q8×C17—84no—
228D4×C17—24no—
229D4×C17—42no—
230D4×C17—84no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—24no—
234C2×C4×C17—24no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—22no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—42no—
249C2×C4×C17—42no—
250C2×C4×C17—84no—
251C2×C4×C17#12012yes—
252C2×C4×C17C22×D411yescharacteristic
253C2×C4×C17C22×D411yescharacteristic
254C2×C4×C17C2×#4112yes—
255C2×C4×C17H(5,𝔽2)12yes—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259D68—24no—
260D68—24no—
261D68—48no—
262D68—42no—
263D68—42no—
264#3622—42no—
265#3622—42no—
266C2×#542D1711yescharacteristic, complemented
267C2×#450—171no—
268C2×#447—171no—
269#2309—171no—
270#2023—172no—
271#2012—172no—
272#1880—172no—
273#1879—172no—
274#1875—171no—
275#1873—171no—
276#1818—171no—
277C22×C17—22no—
278C22×C17—22no—
279C22×C17—22no—
280C22×C17—42no—
281C22×C17—42no—
282C22×C17—42no—
283C22×C17—42no—
284C22×C17—42no—
285C22×C17—82no—
286C22×C17—82no—
287C22×C17C2×#12011yescharacteristic
288C22×C17#52511yescharacteristic
289C22×C17#53511yescharacteristic
290C22×C17#57612yes—
291C22×C17#57712yes—
292C2×D17—44no—
293C2×D17—44no—
294C2×D17—44no—
295C2×D17—816no—
296C2×D17—82no—
297C2×D17—82no—
298C2×D17—82no—
299C2×D17—82no—
300C4×C17—22no—
301C4×C17—42no—
302C4×C17—41no—
303C4×C17—41no—
304C4×C17—42no—
305C4×C17—42no—
306C4×C17—41no—
307C4×C17—41no—
308C4×C17#54612yes—
309C4×C17#54612yes—
310Dic17—82no—
311C23×D4—171no—
312C22×SD16—172no—
313C22×D8—172no—
314#542—24no—
315#542—24no—
316C2×C4⋊Q8C2×D1711yescharacteristic, complemented
317C2×#126—21no—
318C2×#124—21no—
319C2×#124C2×D1711yescharacteristic, complemented
320C2×#123—21no—
321C2×C22⋊Q8—21no—
322C2×C22⋊Q8C2×D1711yescharacteristic
323C2×#121C2×D1711yescharacteristic
324C2×C4×Q8C2×D1711yescharacteristic, complemented
325C2×C4×D4C2×D1711yescharacteristic, complemented
326C2×#117—171no—
327C2×#117C2×D1711yescharacteristic
328#450—344no—
329#447—344no—
330C2×#103—172no—
331C2×#102—172no—
332C2×#102—172no—
333C2×#102—172no—
334C2×C22⋊C8—171no—
335C2×C22⋊C8—171no—
336#390—172no—
337#390—172no—
338#390—341no—
339#390—341no—
340#379—341no—
341#379—341no—
342#340—172no—
343#340—172no—
344#336—172no—
345#336—172no—
346C2×C17—42no—
347C2×C17—82no—
348C2×C17#240411yescharacteristic
349C2×C17#300811yescharacteristic
350C2×C17#301011yescharacteristic
351C2×C17#311812yes—
352C2×C17#322412yes—
353D17—84no—
354C25—341no—
355C22×Q8—21no—
356C22×Q8C22×D1711yescharacteristic
357C22×D4—21no—
358C22×D4—172no—
359C22×D4—344no—
360C22×D4—341no—
361C22×D4—342no—
362C22×D4—342no—
363C22×D4—682no—
364C22×D4C22×D1711yescharacteristic
365C23×C4—21no—
366C23×C4C22×D1711yescharacteristic
367C2×SD16—344no—
368C2×SD16—344no—
369C2×SD16—682no—
370C2×D8—344no—
371C2×D8—344no—
372C2×D8—682no—
373C22×C8—172no—
374C22×C8—172no—
375C4⋊Q8—22no—
376C4⋊Q8—22no—
377#126—28no—
378#124—28no—
379#124—22no—
380#124—22no—
381#123—28no—
382C22⋊Q8—28no—
383C22⋊Q8—22no—
384C22⋊Q8C22⋊D1714yes—
385#121—22no—
386#121C22⋊D1714yes—
387C4×Q8—22no—
388C4×Q8—22no—
389C4×D4—22no—
390C4×D4—22no—
391#117—22no—
392#117—22no—
393#117—682no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—172no—
398C2×C4⋊C4C22⋊D1711yescharacteristic
399C2×C4⋊C4C22⋊D1711yescharacteristic
400C2×C4⋊C4C22×D1711yescharacteristic
401C2×C4⋊C4C22×D1711yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—341no—
408C2×C22⋊C4—342no—
409C2×C22⋊C4—342no—
410C2×C22⋊C4—342no—
411C2×C22⋊C4—342no—
412C2×C22⋊C4C22⋊D1711yescharacteristic
413C2×C22⋊C4C22⋊D1711yescharacteristic
414C2×C42—21no—
415C2×C42—341no—
416C2×C42C22×D1711yescharacteristic
417C2×C42C22×D1711yescharacteristic
418#103—682no—
419#102—344no—
420#102—344no—
421#102—682no—
422C22⋊C8—342no—
423C22⋊C8—342no—
424C17#2394511yescharacteristic, complemented, Sylow
425C24—21no—
426C24—344no—
427C24—342no—
428C24—682no—
429C24—682no—
430C24—1362no—
431C24—1362no—
432C24C2×C22⋊D1711yescharacteristic
433C2×Q8—24no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—82no—
438C2×Q8C2×C22⋊D1712yes—
439C2×D4—24no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—82no—
444C2×D4—174no—
445C2×D4—344no—
446C2×D4—342no—
447C2×D4—342no—
448C2×D4—342no—
449C2×D4—342no—
450C2×D4—342no—
451C2×D4—688no—
452C2×D4—688no—
453C2×D4—682no—
454C2×D4—682no—
455C2×D4C2×C22⋊D1712yes—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—82no—
463C22×C4—342no—
464C22×C4C2×C22⋊D1711yescharacteristic
465C22×C4C2×C22⋊D1711yescharacteristic
466C22×C4C2×C22⋊D1711yescharacteristic
467C22×C4C2×C22⋊D1711yescharacteristic
468C22×C4C2×C22⋊D1712yes—
469C22×C4C2×C22⋊D1712yes—
470C22×C4C2×C22⋊D1711yescharacteristic
471C22×C4C23×D1711yescharacteristic
472SD16—688no—
473D8—688no—
474C2×C8—342no—
475C2×C8—342no—
476C2×C8—342no—
477C2×C8—342no—
478C2×C8—342no—
479C2×C8—342no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—342no—
488C4⋊C4—342no—
489C4⋊C4C2×C22⋊D1712yes—
490C4⋊C4C2×C22⋊D1712yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—24no—
498C22⋊C4—682no—
499C22⋊C4—1362no—
500C22⋊C4—1362no—
501C22⋊C4—1362no—
502C22⋊C4—1362no—
503C42—24no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—682no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—21no—
513C23—22no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—344no—
519C23—688no—
520C23—682no—
521C23—682no—
522C23—682no—
523C23—682no—
524C23—682no—
525C23—682no—
526C23—1364no—
527C23—1364no—
528C23—1368no—
529C23—1368no—
530C23#1058763912yes—
531C23C22×C22⋊D1711yescharacteristic
532Q8—24no—
533Q8—42no—
534Q8—84no—
535D4—24no—
536D4—42no—
537D4—84no—
538D4—344no—
539D4—344no—
540D4—688no—
541D4—682no—
542D4—682no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—84no—
563C2×C4—682no—
564C2×C4—682no—
565C2×C4—682no—
566C2×C4#1058763912yes—
567C2×C4#41×D1712yes—
568C2×C4#1058769612yes—
569C2×C4C22×C22⋊D1711yescharacteristic
570C2×C4C22×C22⋊D1711yescharacteristic
571C8—682no—
572C8—682no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—82no—
582C22—82no—
583C22—684no—
584C22—684no—
585C22—684no—
586C22—13616no—
587C22—1362no—
588C22—1362no—
589C22—1362no—
590C22—1362no—
591C22C2×#1058763911yescharacteristic
592C22#1176331611yescharacteristic
593C22#1176337311yescharacteristic
594C22#1176361212yes—
595C22#1176361712yes—
596C4—22no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—41no—
604C4—1362no—
605C4#1176344712yes—
606C4#1176344712yes—
607C2—42no—
608C2—82no—
609C2—1364no—
610C2#1453209911yescharacteristic
611C2#1453805211yescharacteristic
612C2#1453807011yescharacteristic
613C2#1453896212yes—
614C2#1453987212yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#542C2
socleC23×C17C22×D4
prime radicalC2×#11762942C2
hypercenterC2×#542D17
hypocenterC17#23945
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. C2×#11762959C2semidirect
3. #14531364C2semidirect
4. #14529917C2semidirect
5. #14529861C2semidirect
6. #14529139C2semidirect
7. #14529132C2semidirect
8. #14529107C2semidirect
9. #14529097C2semidirect
10. #14528855C2semidirect
11. C2×C17×#542C2semidirect
18. C2×#10587506C22semidirect
19. C2×#10587503C22semidirect
30. #11762571C22semidirect
31. #11762571C22semidirect
32. #11762564C22semidirect
33. #11762564C22semidirect
36. C2×C17×C4⋊Q8C22semidirect
39. C2×C17×#124C22semidirect
44. C2×C4×Q8×C17C22semidirect
45. C2×C4×D4×C17C22semidirect
266. C2×#542D17semidirect
316. C2×C4⋊Q8C2×D17semidirect
319. C2×#124C2×D17semidirect
324. C2×C4×Q8C2×D17semidirect
325. C2×C4×D4C2×D17semidirect
424. C17#23945semidirect

Maximal subgroups

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

subgroupindex
C2×#117629592
C2×#117629422
#145313642
#145299172
#145298612
#145291392
#145291322
#145291072
#145290972
#145288552
C2×C17×#5422
#2394517