Finite Group Databasebeta

← #17067370

Subgroups — #17067370

Order 4352.

1338 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763137C211yesmaximal, characteristic, complemented
2C22×#10587506C211yesmaximal, characteristic, complemented
3#14533092C211yesmaximal, characteristic
4#14530027C212yesmaximal, complemented
5#14529977C212yesmaximal, complemented
6#14529208C211yesmaximal, characteristic, complemented
7#14529077C211yesmaximal, characteristic
8#14529052C212yesmaximal, complemented
9#14528719C211yesmaximal, characteristic, complemented
10#14528719C212yesmaximal, complemented
11C2×D4×Q8×C17C211yesmaximal, characteristic
12C23×#3622C2211yescharacteristic
13C2×#10587621—21no—
14C22×C4⋊Dic17C2211yescharacteristic
15#11763137—24no—
16C2×#10587479C2212yes—
17C2×#10587506—24no—
18C2×#10587506C2212yes—
19C2×#10587506C2212yes—
20C2×#10587506C2212yes—
21C2×#10587506C2212yes—
22C2×#10587506C2212yes—
23C2×#10587506C2212yes—
24C2×C22⋊#3622C2211yescharacteristic
25C2×C22⋊#3622C2212yes—
26#11762739—21no—
27#11762734—21no—
28#11762707C2212yes—
29#11762670—21no—
30#11762669C2211yescharacteristic
31#11762669C2212yes—
32#11762571C2212yes—
33#11762564C2212yes—
34C23×Q8×C17C2211yescharacteristic
35C23×Q8×C17C2211yescharacteristic
36D4×Q8×C17—24no—
37D4×Q8×C17—24no—
38C2×C17×C4⋊Q8—21no—
39C2×C17×C4⋊Q8C2211yescharacteristic
40C2×C17×C22⋊Q8—21no—
41C2×C17×C22⋊Q8—21no—
42C2×C17×C22⋊Q8C2211yescharacteristic
43C2×C17×C22⋊Q8C2211yescharacteristic
44C2×C4×Q8×C17C2211yescharacteristic
45C2×C4×D4×C17—21no—
46C2×C4×D4×C17C2211yescharacteristic
47C22×Q8×C17—22no—
48C22×Q8×C17—22no—
49C22×Q8×C17—21no—
50C22×Q8×C17—21no—
51C22×Q8×C17—22no—
52C22×Q8×C17—24no—
53C22×Q8×C17—24no—
54C22×Q8×C17—42no—
55C22×Q8×C17—42no—
56C22×Q8×C17D414yes—
57C22×Q8×C17C2311yescharacteristic
58C22×Q8×C17C2311yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C22×D4×C17D411yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17—21no—
63C23×C4×C17C2311yescharacteristic
64C23×C4×C17C2311yescharacteristic
65C23×Dic17—21no—
66C17×C4⋊Q8—28no—
67C17×C4⋊Q8—22no—
68C17×C4⋊Q8—22no—
69C17×C22⋊Q8—28no—
70C17×C22⋊Q8—28no—
71C17×C22⋊Q8—22no—
72C17×C22⋊Q8—22no—
73C17×C22⋊Q8—22no—
74C17×C22⋊Q8D414yes—
75C4×Q8×C17—22no—
76C4×Q8×C17—22no—
77C4×D4×C17—28no—
78C4×D4×C17—22no—
79C4×D4×C17—22no—
80C2×C4⋊C4×C17—21no—
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×C17D411yescharacteristic
90C2×C22⋊C4×C17D411yescharacteristic
91C2×C42×C17—21no—
92C2×C42×C17C2311yescharacteristic
93C2×C22⋊Dic17—22no—
94C2×C22⋊Dic17—21no—
95#10587621—44no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17—22no—
98C2×C4⋊Dic17—22no—
99C2×C4⋊Dic17C2312yes—
100C2×C4⋊Dic17C2312yes—
101C2×#63225—22no—
102C22×#3622—22no—
103C22×#3622—42no—
104C22×#3622C2312yes—
105C22×#3622C2312yes—
106#10587506—24no—
107#10587506—24no—
108#10587506—24no—
109#10587506—24no—
110#10587506—42no—
111#10587506—42no—
112#10587502—22no—
113#10587502—21no—
114#10587502—22no—
115C22⋊#3622—22no—
116C22⋊#3622—42no—
117#10587479—24no—
118C24×C17C2×D411yescharacteristic
119C24×C17C2×D411yescharacteristic
120C2×Q8×C17—24no—
121C2×Q8×C17—24no—
122C2×Q8×C17—21no—
123C2×Q8×C17—21no—
124C2×Q8×C17—21no—
125C2×Q8×C17—22no—
126C2×Q8×C17—28no—
127C2×Q8×C17—22no—
128C2×Q8×C17—22no—
129C2×Q8×C17—22no—
130C2×Q8×C17—22no—
131C2×Q8×C17—44no—
132C2×Q8×C17—44no—
133C2×Q8×C17—44no—
134C2×Q8×C17—48no—
135C2×Q8×C17—48no—
136C2×Q8×C17—44no—
137C2×Q8×C17—44no—
138C2×Q8×C17C2×D412yes—
139C2×Q8×C17C2×D412yes—
140C2×Q8×C17C2×D412yes—
141C2×Q8×C17#4114yes—
142C2×D4×C17—22no—
143C2×D4×C17—22no—
144C2×D4×C17SD1614yes—
145C22×C4×C17—21no—
146C22×C4×C17—22no—
147C22×C4×C17—22no—
148C22×C4×C17—24no—
149C22×C4×C17—24no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
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×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2×D411yescharacteristic
162C22×C4×C17C2×D412yes—
163C22×C4×C17C2×D412yes—
164C22×C4×C17C2×D411yescharacteristic
165C22×C4×C17C2411yescharacteristic
166C22×Dic17—22no—
167C22×Dic17—22no—
168C22×Dic17—42no—
169C22×Dic17—42no—
170C22×Dic17—42no—
171C4⋊C4×C17—24no—
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—22no—
181C22⋊C4×C17—24no—
182C22⋊C4×C17—22no—
183C42×C17—24no—
184C42×C17—21no—
185C42×C17—21no—
186C22⋊Dic17—42no—
187C22⋊Dic17—44no—
188C4⋊Dic17—22no—
189C4⋊Dic17—22no—
190C4⋊Dic17—22no—
191C4⋊Dic17—22no—
192C4⋊Dic17—42no—
193#63225—44no—
194C2×#3622—22no—
195C2×#3622—22no—
196C2×#3622—22no—
197C2×#3622—22no—
198C2×#3622—22no—
199C2×#3622—22no—
200C2×#3622—44no—
201#24041—171nomaximal, Sylow
202C23×C17—22no—
203C23×C17—22no—
204C23×C17—21no—
205C23×C17—22no—
206C23×C17—21no—
207C23×C17—22no—
208C23×C17#12012yes—
209C23×C17C2×SD1612yes—
210C23×C17C2×SD1612yes—
211C23×C17#13712yes—
212C23×C17C22×D411yescharacteristic
213Q8×C17—22no—
214Q8×C17—24no—
215Q8×C17—22no—
216Q8×C17—22no—
217Q8×C17—24no—
218Q8×C17—48no—
219Q8×C17—48no—
220Q8×C17—41no—
221Q8×C17—41no—
222Q8×C17—42no—
223Q8×C17#12314yescomplemented
224D4×C17—28no—
225C2×C4×C17—21no—
226C2×C4×C17—22no—
227C2×C4×C17—22no—
228C2×C4×C17—24no—
229C2×C4×C17—21no—
230C2×C4×C17—22no—
231C2×C4×C17—22no—
232C2×C4×C17—24no—
233C2×C4×C17—21no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—21no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—48no—
241C2×C4×C17—48no—
242C2×C4×C17—42no—
243C2×C4×C17—42no—
244C2×C4×C17—42no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—42no—
248C2×C4×C17#12012yes—
249C2×C4×C17C2×SD1612yes—
250C2×C4×C17C22×D411yescharacteristic
251C2×C4×C17C22×D411yescharacteristic
252C2×C4×C17C2×#4112yes—
253C2×C4×C17H(5,𝔽2)12yes—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17—84no—
261#3622—42no—
262#3622—42no—
263C2×D4×Q8D1711yescharacteristic
264C2×#483—171no—
265C22×#103—171no—
266#2486—171no—
267#2043—172no—
268#2035—172no—
269#1892—171no—
270#1869—171no—
271#1865—172no—
272#1787—172no—
273#1787—171no—
274C22×C17—22no—
275C22×C17—22no—
276C22×C17—24no—
277C22×C17—24no—
278C22×C17—22no—
279C22×C17—24no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17—42no—
283C22×C17#44814yes—
284C22×C17C2×#12011yescharacteristic
285C22×C17#54612yes—
286C22×C17C22×SD1611yescharacteristic
287C22×C17C2×#13711yescharacteristic
288C22×C17#57812yes—
289C4×C17—22no—
290C4×C17—24no—
291C4×C17—22no—
292C4×C17—41no—
293C4×C17—42no—
294C4×C17—42no—
295C4×C17—41no—
296C4×C17—42no—
297C4×C17C2×#12311yescharacteristic
298C4×C17#53411yescharacteristic
299C4×C17#54612yes—
300Dic17—82no—
301Dic17—82no—
302C23×Q8C2×D1711yescharacteristic
303C23×Q8C2×D1711yescharacteristic
304C23×C8—171no—
305D4×Q8—24no—
306D4×Q8—24no—
307C2×C4⋊Q8—21no—
308C2×C4⋊Q8C2×D1711yescharacteristic
309C2×#123—341no—
310C2×C22⋊Q8—21no—
311C2×C22⋊Q8—21no—
312C2×C22⋊Q8C2×D1711yescharacteristic
313C2×C22⋊Q8C2×D1711yescharacteristic
314C2×C4×Q8C2×D1711yescharacteristic
315C2×C4×D4—21no—
316C2×C4×D4C2×D1711yescharacteristic
317C22×C4⋊C4—171no—
318#483—344no—
319C2×#106—172no—
320C2×#103—172no—
321C2×#103—172no—
322C2×#103—172no—
323C2×#103—172no—
324C2×#103—172no—
325C2×#103—172no—
326C2×#103—344no—
327C2×C22⋊C8—171no—
328C2×C22⋊C8—172no—
329#398—341no—
330#397—341no—
331#389—172no—
332#381—341no—
333#380—171no—
334#380—172no—
335#340—172no—
336#336—172no—
337C2×C17—24no—
338C2×C17—42no—
339C2×C17#240511yescharacteristic
340C2×C17C2×#44811yescharacteristic
341C2×C17#301111yescharacteristic
342C2×C17#329812yes—
343C2×C17#331212yes—
344C22×Q8—22no—
345C22×Q8—22no—
346C22×Q8—21no—
347C22×Q8—21no—
348C22×Q8—22no—
349C22×Q8—24no—
350C22×Q8—24no—
351C22×Q8—42no—
352C22×Q8—42no—
353C22×Q8C22⋊D1714yes—
354C22×Q8C22×D1711yescharacteristic
355C22×Q8C22×D1711yescharacteristic
356C22×Q8C22×D1711yescharacteristic
357C22×D4C22⋊D1711yescharacteristic
358C23×C4—21no—
359C23×C4—21no—
360C23×C4—341no—
361C23×C4C22×D1711yescharacteristic
362C23×C4C22×D1711yescharacteristic
363C22×C8—172no—
364C22×C8—172no—
365C22×C8—342no—
366C22×C8—682no—
367C4⋊Q8—28no—
368C4⋊Q8—22no—
369C4⋊Q8—22no—
370#123—684no—
371C22⋊Q8—28no—
372C22⋊Q8—28no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376C22⋊Q8C22⋊D1714yes—
377C4×Q8—22no—
378C4×Q8—22no—
379C4×D4—28no—
380C4×D4—22no—
381C4×D4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—172no—
388C2×C4⋊C4—172no—
389C2×C4⋊C4—342no—
390C2×C4⋊C4—342no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4—342no—
393C2×C4⋊C4C22⋊D1711yescharacteristic
394C2×C4⋊C4C22×D1711yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—341no—
398C2×C22⋊C4—342no—
399C2×C22⋊C4C22⋊D1711yescharacteristic
400C2×C22⋊C4C22⋊D1711yescharacteristic
401C2×C42—21no—
402C2×C42C22×D1711yescharacteristic
403#106—344no—
404#103—344no—
405#103—344no—
406#103—344no—
407#103—344no—
408#103—682no—
409#103—682no—
410C22⋊C8—342no—
411C22⋊C8—682no—
412#95—342no—
413#95—341no—
414#95—342no—
415C17#2404111yescharacteristic, complemented, Sylow
416C24C2×C22⋊D1711yescharacteristic
417C24C2×C22⋊D1711yescharacteristic
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—24no—
424C2×Q8—28no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—48no—
433C2×Q8—48no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8#6325414yes—
437C2×Q8C2×C22⋊D1712yes—
438C2×Q8C2×C22⋊D1712yes—
439C2×Q8C2×C22⋊D1712yes—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4#6323014yes—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—342no—
458C22×C4—342no—
459C22×C4—682no—
460C22×C4—682no—
461C22×C4—682no—
462C22×C4C2×C22⋊D1711yescharacteristic
463C22×C4C2×C22⋊D1711yescharacteristic
464C22×C4C2×C22⋊D1712yes—
465C22×C4C2×C22⋊D1712yes—
466C22×C4C2×C22⋊D1711yescharacteristic
467C22×C4C2×C22⋊D1711yescharacteristic
468C22×C4C23×D1711yescharacteristic
469C2×C8—342no—
470C2×C8—342no—
471C2×C8—342no—
472C2×C8—342no—
473C2×C8—342no—
474C2×C8—342no—
475C2×C8—684no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—342no—
484C4⋊C4—342no—
485C4⋊C4—342no—
486C4⋊C4—342no—
487C4⋊C4—684no—
488C4⋊C4—682no—
489C4⋊C4C2×C22⋊D1712yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—682no—
495C22⋊C4—684no—
496C42—24no—
497C42—21no—
498C42—21no—
499C23—22no—
500C23—22no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—22no—
505C23C2×#6323012yes—
506C23C2×#6323012yes—
507C23#1058762812yes—
508C23#1058763912yes—
509C23C22×C22⋊D1711yescharacteristic
510Q8—22no—
511Q8—24no—
512Q8—22no—
513Q8—22no—
514Q8—24no—
515Q8—48no—
516Q8—48no—
517Q8—41no—
518Q8—41no—
519Q8—42no—
520Q8#1058762114yescomplemented
521D4—28no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—682no—
546C2×C4—682no—
547C2×C4—682no—
548C2×C4—682no—
549C2×C4—682no—
550C2×C4—682no—
551C2×C4—1364no—
552C2×C4C2×#6323012yes—
553C2×C4#1058763912yes—
554C2×C4C2×#6325412yes—
555C2×C4#1058769112yes—
556C2×C4C22×C22⋊D1711yescharacteristic
557C2×C4C22×C22⋊D1711yescharacteristic
558C8—682no—
559C8—682no—
560C22—22no—
561C22—24no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—22no—
566C22—41no—
567C22—42no—
568C22—41no—
569C22#1176294814yes—
570C22C2×#1058763911yescharacteristic
571C22#1176345312yes—
572C22C22×#6323011yescharacteristic
573C22C2×#1058762811yescharacteristic
574C22#1176362412yes—
575C4—22no—
576C4—24no—
577C4—22no—
578C4—41no—
579C4—42no—
580C4—42no—
581C4—41no—
582C4—42no—
583C4—1362no—
584C4—1362no—
585C4C2×#1058762111yescharacteristic
586C4#1176336811yescharacteristic
587C4#1176345312yes—
588C2—24no—
589C2—42no—
590C2#1453210711yescharacteristic
591C2C2×#1176294811yescharacteristic
592C2#1453807911yescharacteristic
593C2#1454064912yes—
594C2#1454083512yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×D4×Q8×C17C2
socleC23×C17C22×D4
prime radicalC22×D4×C17D4
hypercenterC2×D4×Q8D17
hypocenterC17#24041
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#11763137C2semidirect
2. C22×#10587506C2semidirect
4. #14530027C2semidirect
5. #14529977C2semidirect
6. #14529208C2semidirect
8. #14529052C2semidirect
9. #14528719C2semidirect
10. #14528719C2semidirect
223. Q8×C17#123semidirect
415. C17#24041semidirect
520. Q8#10587621semidirect

Maximal subgroups

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

subgroupindex
C2×#117631372
C22×#105875062
#145330922
#145300272
#145299772
#145292082
#145290772
#145290522
#145287192
C2×D4×Q8×C172
#2404117