Finite Group Databasebeta

← #17067544

Subgroups — #17067544

Order 4352.

1338 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×#11763137C211yesmaximal, characteristic, complemented
2C2×#11762810C211yesmaximal, characteristic, complemented
3#14533101C211yesmaximal, characteristic, complemented
4#14530027C212yesmaximal, complemented
5#14529977C212yesmaximal, complemented
6#14529218C211yesmaximal, characteristic, complemented
7#14529082C211yesmaximal, characteristic, complemented
8#14529045C212yesmaximal, complemented
9#14528724C211yesmaximal, characteristic, complemented
10#14528724C212yesmaximal, complemented
11C2×C17×#547C211yesmaximal, characteristic
12C22×#63223C2211yescharacteristic
13C22×C4⋊Dic17C2211yescharacteristic
14#11763137—24no—
15C2×#10587479C2212yes—
16#11762810—24no—
17C2×#10587506C2212yes—
18C2×#10587506C2212yes—
19C2×#10587506C2212yes—
20C2×#10587506C2212yes—
21C2×#10587503C2212yes—
22C2×#10587503C2212yes—
23C2×C22⋊#3622C2211yescharacteristic
24C2×C22⋊#3622C2212yes—
25#11762748—21no—
26#11762745—21no—
27#11762707C2212yes—
28#11762689—21no—
29#11762681—21no—
30#11762669C2212yes—
31#11762669C2211yescharacteristic
32#11762571C2212yes—
33#11762564C2212yes—
34C22×#41×C17C2211yescharacteristic
35C22×#41×C17C2211yescharacteristic
36C17×#547—24no—
37C17×#547—24no—
38C2×C17×C4⋊Q8C2211yescharacteristic
39C2×C17×#123—21no—
40C2×C17×#123—21no—
41C2×C17×C22⋊Q8C2211yescharacteristic
42C2×C17×C22⋊Q8C2211yescharacteristic
43C2×C17×#121—21no—
44C2×C4×D4×C17C2211yescharacteristic
45C2×C4×D4×C17C2211yescharacteristic
46C22×C4⋊C4×C17—21no—
47C2×#41×C17—22no—
48C2×#41×C17—22no—
49C2×#41×C17—22no—
50C2×#41×C17—24no—
51C2×#41×C17—24no—
52C2×#41×C17—42no—
53C2×#41×C17—42no—
54C2×#41×C17D414yes—
55C22×Q8×C17C2311yescharacteristic
56C22×Q8×C17C2311yescharacteristic
57C22×D4×C17—21no—
58C22×D4×C17—21no—
59C22×D4×C17D411yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17—21no—
63C23×C4×C17—21no—
64C23×C4×C17C2311yescharacteristic
65C23×C4×C17C2311yescharacteristic
66C23×Dic17—21no—
67C17×C4⋊Q8—22no—
68C17×C4⋊Q8—22no—
69C17×#123—28no—
70C17×#123—28no—
71C17×C22⋊Q8—22no—
72C17×C22⋊Q8—22no—
73C17×C22⋊Q8—22no—
74C17×C22⋊Q8D414yes—
75C17×#121—28no—
76C4×D4×C17—22no—
77C4×D4×C17—22no—
78C4×D4×C17—22no—
79C4×D4×C17—22no—
80C2×C4⋊C4×C17—28no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17—21no—
83C2×C4⋊C4×C17—21no—
84C2×C4⋊C4×C17—21no—
85C2×C4⋊C4×C17D411yescharacteristic
86C2×C4⋊C4×C17C2311yescharacteristic
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×C17C2311yescharacteristic
93C2×C22⋊Dic17—21no—
94C2×C22⋊Dic17—22no—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17—22no—
97C2×C4⋊Dic17—22no—
98C2×C4⋊Dic17C2312yes—
99C2×C4⋊Dic17C2312yes—
100C2×#63225—22no—
101C2×#63223—22no—
102C2×#63223—42no—
103C22×#3622C2312yes—
104C22×#3622C2312yes—
105#10587506—24no—
106#10587506—24no—
107#10587506—42no—
108#10587506—42no—
109#10587503—24no—
110#10587503—24no—
111#10587502—22no—
112#10587502—21no—
113#10587502—22no—
114C22⋊#3622—22no—
115C22⋊#3622—42no—
116#10587479—24no—
117C24×C17—21no—
118C24×C17C2×D411yescharacteristic
119C24×C17C2×D411yescharacteristic
120#41×C17—28no—
121#41×C17—44no—
122#41×C17—48no—
123#41×C17—48no—
124#41×C17—44no—
125#41×C17—44no—
126C2×Q8×C17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17—22no—
129C2×Q8×C17—22no—
130C2×Q8×C17—22no—
131C2×Q8×C17—22no—
132C2×Q8×C17C2×D412yes—
133C2×Q8×C17C2×D412yes—
134C2×D4×C17—24no—
135C2×D4×C17—24no—
136C2×D4×C17—22no—
137C2×D4×C17—22no—
138C2×D4×C17—21no—
139C2×D4×C17—22no—
140C2×D4×C17—41no—
141C2×D4×C17—44no—
142C2×D4×C17—44no—
143C2×D4×C17C2×D412yes—
144C2×D4×C17#4114yes—
145C22×C4×C17—22no—
146C22×C4×C17—22no—
147C22×C4×C17—24no—
148C22×C4×C17—24no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17—24no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—22no—
156C22×C4×C17—22no—
157C22×C4×C17—44no—
158C22×C4×C17—44no—
159C22×C4×C17—42no—
160C22×C4×C17—42no—
161C22×C4×C17C2×D411yescharacteristic
162C22×C4×C17C2×D411yescharacteristic
163C22×C4×C17C2×D411yescharacteristic
164C22×C4×C17C2×D412yes—
165C22×C4×C17C2×D412yes—
166C22×C4×C17C2×D411yescharacteristic
167C22×C4×C17C2411yescharacteristic
168C22×Dic17—22no—
169C22×Dic17—22no—
170C22×Dic17—42no—
171C22×Dic17—82no—
172C4⋊C4×C17—24no—
173C4⋊C4×C17—24no—
174C4⋊C4×C17—24no—
175C4⋊C4×C17—24no—
176C4⋊C4×C17—22no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17C2×D412yes—
179C22⋊C4×C17—24no—
180C22⋊C4×C17—24no—
181C22⋊C4×C17—24no—
182C22⋊C4×C17—22no—
183C22⋊C4×C17—22no—
184C42×C17—21no—
185C42×C17—21no—
186C22⋊Dic17—42no—
187C22⋊Dic17—82no—
188C4⋊Dic17—22no—
189C4⋊Dic17—22no—
190C4⋊Dic17—22no—
191C4⋊Dic17—22no—
192C4⋊Dic17—42no—
193#63225—82no—
194#63223—44no—
195C2×#3622—22no—
196C2×#3622—22no—
197C2×#3622—22no—
198C2×#3622—22no—
199C2×#3622—22no—
200C2×#3622—22no—
201#24055—171nomaximal, Sylow
202C23×C17—22no—
203C23×C17—22no—
204C23×C17—24no—
205C23×C17—21no—
206C23×C17—21no—
207C23×C17—22no—
208C23×C17—21no—
209C23×C17—22no—
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—22no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17—42no—
216C23×C17#12012yes—
217C23×C17C22×D411yescharacteristic
218Q8×C17—22no—
219Q8×C17—22no—
220Q8×C17—24no—
221Q8×C17—41no—
222Q8×C17—41no—
223Q8×C17—42no—
224D4×C17—22no—
225D4×C17—24no—
226D4×C17—44no—
227D4×C17—48no—
228D4×C17—48no—
229D4×C17#12314yes—
230C2×C4×C17—21no—
231C2×C4×C17—22no—
232C2×C4×C17—21no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—24no—
236C2×C4×C17—21no—
237C2×C4×C17—22no—
238C2×C4×C17—21no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—48no—
244C2×C4×C17—48no—
245C2×C4×C17—48no—
246C2×C4×C17—42no—
247C2×C4×C17—42no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—42no—
253C2×C4×C17#12012yes—
254C2×C4×C17#13712yes—
255C2×C4×C17C22×D411yescharacteristic
256C2×C4×C17C22×D411yescharacteristic
257C2×C4×C17C2×#4112yes—
258C2×C4×C17H(5,𝔽2)12yes—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—84no—
266#3622—42no—
267#3622—42no—
268C2×#547D1711yescharacteristic
269C2×#483—171no—
270C2×#417—171no—
271#2487—171no—
272#2043—172no—
273#2035—172no—
274#1894—171no—
275#1870—171no—
276#1864—172no—
277#1788—172no—
278#1788—171no—
279C22×C17—22no—
280C22×C17—24no—
281C22×C17—22no—
282C22×C17—22no—
283C22×C17—42no—
284C22×C17—44no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17—42no—
288C22×C17—44no—
289C22×C17—42no—
290C22×C17—41no—
291C22×C17—41no—
292C22×C17—41no—
293C22×C17—42no—
294C22×C17C2×#12011yescharacteristic
295C22×C17#54612yes—
296C22×C17C2×#13711yescharacteristic
297C22×C17#57511yescharacteristic
298C22×C17#57712yes—
299C4×C17—22no—
300C4×C17—22no—
301C4×C17—41no—
302C4×C17—42no—
303C4×C17—42no—
304C4×C17—42no—
305C4×C17—41no—
306C4×C17C2×#12311yescharacteristic
307C4×C17#53411yescharacteristic
308C4×C17#54612yes—
309Dic17—82no—
310Dic17—82no—
311C22×#41C2×D1711yescharacteristic
312C22×#41C2×D1711yescharacteristic
313C22×M16—171no—
314#547—24no—
315#547—24no—
316C2×C4⋊Q8C2×D1711yescharacteristic
317C2×#123—21no—
318C2×#123—21no—
319C2×C22⋊Q8C2×D1711yescharacteristic
320C2×C22⋊Q8C2×D1711yescharacteristic
321C2×#121—21no—
322C2×C4×D4C2×D1711yescharacteristic
323C2×C4×D4C2×D1711yescharacteristic
324C22×C4⋊C4—21no—
325C22×C4⋊C4—171no—
326#483—344no—
327C2×#106—172no—
328#417—344no—
329C2×#103—172no—
330C2×#103—172no—
331C2×#103—172no—
332C2×#103—172no—
333C2×#102—172no—
334C2×#102—172no—
335C2×C22⋊C8—171no—
336C2×C22⋊C8—172no—
337#400—341no—
338#399—341no—
339#389—172no—
340#386—341no—
341#385—341no—
342#380—171no—
343#380—172no—
344#340—172no—
345#336—172no—
346C2×C17—42no—
347C2×C17—44no—
348C2×C17—81no—
349C2×C17#240511yescharacteristic
350C2×C17#300711yescharacteristic
351C2×C17#301011yescharacteristic
352C2×C17#330712yes—
353C2×C17#333112yes—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—24no—
358C2×#41—24no—
359C2×#41—42no—
360C2×#41—42no—
361C2×#41C22⋊D1714yes—
362C22×Q8C22×D1711yescharacteristic
363C22×Q8C22×D1711yescharacteristic
364C22×D4—21no—
365C22×D4—21no—
366C22×D4C22⋊D1711yescharacteristic
367C22×D4C22×D1711yescharacteristic
368C23×C4—21no—
369C23×C4—21no—
370C23×C4—21no—
371C23×C4—341no—
372C23×C4C22×D1711yescharacteristic
373C23×C4C22×D1711yescharacteristic
374C2×M16—342no—
375C2×M16—682no—
376C22×C8—172no—
377C22×C8—172no—
378C4⋊Q8—22no—
379C4⋊Q8—22no—
380#123—28no—
381#123—28no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8C22⋊D1714yes—
386#121—28no—
387C4×D4—22no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C2×C4⋊C4—28no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—172no—
397C2×C4⋊C4—172no—
398C2×C4⋊C4—342no—
399C2×C4⋊C4—342no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4—342no—
402C2×C4⋊C4C22⋊D1711yescharacteristic
403C2×C4⋊C4C22×D1711yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—341no—
408C2×C22⋊C4—342no—
409C2×C22⋊C4C22⋊D1711yescharacteristic
410C2×C22⋊C4C22⋊D1711yescharacteristic
411C2×C42C22×D1711yescharacteristic
412#106—344no—
413#103—344no—
414#103—344no—
415#103—682no—
416#103—682no—
417#102—344no—
418#102—344no—
419C22⋊C8—342no—
420C22⋊C8—682no—
421#95—342no—
422#95—341no—
423#95—342no—
424C17#2405511yescharacteristic, complemented, Sylow
425C24—21no—
426C24C2×C22⋊D1711yescharacteristic
427C24C2×C22⋊D1711yescharacteristic
428#41—28no—
429#41—44no—
430#41—48no—
431#41—48no—
432#41—44no—
433#41—44no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8C2×C22⋊D1712yes—
441C2×Q8C2×C22⋊D1712yes—
442C2×D4—24no—
443C2×D4—22no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—41no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4#6325414yes—
452C2×D4C2×C22⋊D1712yes—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—682no—
472C22×C4—1362no—
473C22×C4C2×C22⋊D1711yescharacteristic
474C22×C4C2×C22⋊D1711yescharacteristic
475C22×C4C2×C22⋊D1712yes—
476C22×C4C2×C22⋊D1712yes—
477C22×C4C2×C22⋊D1711yescharacteristic
478C22×C4C2×C22⋊D1711yescharacteristic
479C22×C4C23×D1711yescharacteristic
480M16—684no—
481C2×C8—342no—
482C2×C8—342no—
483C2×C8—342no—
484C2×C8—342no—
485C2×C8—342no—
486C2×C8—342no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—342no—
494C4⋊C4—342no—
495C4⋊C4—342no—
496C4⋊C4—342no—
497C4⋊C4—682no—
498C4⋊C4—1362no—
499C4⋊C4C2×C22⋊D1712yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—682no—
506C22⋊C4—1362no—
507C42—21no—
508C42—21no—
509C23—22no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23#1058763912yes—
524C23C22×C22⋊D1711yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—24no—
528Q8—41no—
529Q8—41no—
530Q8—42no—
531D4—22no—
532D4—24no—
533D4—44no—
534D4—48no—
535D4—48no—
536D4#1058762114yes—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—682no—
561C2×C4—682no—
562C2×C4—682no—
563C2×C4—682no—
564C2×C4—682no—
565C2×C4—682no—
566C2×C4—1364no—
567C2×C4#1058763712yes—
568C2×C4#1058763912yes—
569C2×C4C2×#6325412yes—
570C2×C4#1058769112yes—
571C2×C4C22×C22⋊D1711yescharacteristic
572C2×C4C22×C22⋊D1711yescharacteristic
573C8—682no—
574C8—682no—
575C22—22no—
576C22—24no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—44no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—41no—
589C22—42no—
590C22C2×#1058763911yescharacteristic
591C22#1176345312yes—
592C22C2×#1058763711yescharacteristic
593C22#1176360711yescharacteristic
594C22#1176361712yes—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—1362no—
603C4—1362no—
604C4C2×#1058762111yescharacteristic
605C4#1176336811yescharacteristic
606C4#1176345312yes—
607C2—42no—
608C2—44no—
609C2—81no—
610C2#1453210711yescharacteristic
611C2#1453804411yescharacteristic
612C2#1453807011yescharacteristic
613C2#1454076012yes—
614C2#1454110012yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#547C2
socleC23×C17C22×D4
hypercenterC2×#547D17
hypocenterC17#24055
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#11763137C2semidirect
2. C2×#11762810C2semidirect
3. #14533101C2semidirect
4. #14530027C2semidirect
5. #14529977C2semidirect
6. #14529218C2semidirect
7. #14529082C2semidirect
8. #14529045C2semidirect
9. #14528724C2semidirect
10. #14528724C2semidirect
424. C17#24055semidirect

Maximal subgroups

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

subgroupindex
C2×#117631372
C2×#117628102
#145331012
#145300272
#145299772
#145292182
#145290822
#145290452
#145287242
C2×C17×#5472
#2405517