Finite Group Databasebeta

← #17090862

Subgroups — #17090862

Order 4352.

1302 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763168C211yesmaximal, characteristic, complemented
2C2×#11763094C211yesmaximal, characteristic, complemented
3#14533957C211yesmaximal, characteristic, complemented
4#14529977C212yesmaximal, complemented
5#14529970C212yesmaximal, complemented
6#14529717C211yesmaximal, characteristic, complemented
7#14529583C211yesmaximal, characteristic, complemented
8#14528724C212yesmaximal, complemented
9#14528719C212yesmaximal, complemented
10#14528614C211yesmaximal, characteristic, complemented
11C2×C17×#548C211yesmaximal, characteristic
12C2×#10587555C2211yescharacteristic
13#11763168—24no—
14#11763094—24no—
15C2×#10587479C2212yes—
16C2×C4⋊#3622C2211yescharacteristic
17C2×#10587506C2212yes—
18C2×#10587506C2212yes—
19C2×#10587506C2212yes—
20C2×#10587506C2212yes—
21C2×#10587503C2212yes—
22C2×#10587503C2212yes—
23C2×C22⋊#3622C2212yes—
24C2×C22⋊#3622C2212yes—
25C2×#10587474C2211yescharacteristic
26#11762721—21no—
27#11762721—21no—
28#11762712—21no—
29#11762674—21no—
30#11762669C2212yes—
31#11762669C2212yes—
32C4×C4⋊Dic17C2211yescharacteristic
33#11762571C2212yes—
34C22×#41×C17C2211yescharacteristic
35C23×Q8×C17C2211yescharacteristic
36C17×#548—24no—
37C17×#548—24no—
38C2×C17×#124—21no—
39C2×C17×#124C2211yescharacteristic
40C2×C17×C22⋊Q8—21no—
41C2×C17×C22⋊Q8C2211yescharacteristic
42C2×C17×#121—21no—
43C2×C17×#121C2211yescharacteristic
44C2×C4×Q8×C17C2211yescharacteristic
45C2×C4×D4×C17—21no—
46C2×C4×D4×C17C2211yescharacteristic
47C2×#41×C17—22no—
48C2×#41×C17—22no—
49C2×#41×C17—24no—
50C2×#41×C17—42no—
51C22×Q8×C17—22no—
52C22×Q8×C17—22no—
53C22×Q8×C17—21no—
54C22×Q8×C17—24no—
55C22×Q8×C17—42no—
56C22×Q8×C17C2311yescharacteristic
57C22×Q8×C17C2311yescharacteristic
58C22×D4×C17—21no—
59C22×D4×C17—21no—
60C22×D4×C17D411yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×C4×C17—21no—
63C23×C4×C17—21no—
64C23×C4×C17C2311yescharacteristic
65C23×C4×C17C2311yescharacteristic
66C22×Dic34—21no—
67C17×#124—28no—
68C17×#124—22no—
69C17×#124D414yes—
70C17×C22⋊Q8—28no—
71C17×C22⋊Q8—22no—
72C17×C22⋊Q8—22no—
73C17×#121—28no—
74C17×#121—22no—
75C17×#121—22no—
76C4×Q8×C17—22no—
77C4×Q8×C17D414yes—
78C4×D4×C17—28no—
79C4×D4×C17—22no—
80C4×D4×C17—22no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17—21no—
83C2×C4⋊C4×C17D411yescharacteristic
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×C17D411yescharacteristic
90C2×C22⋊C4×C17D411yescharacteristic
91C2×C42×C17—21no—
92C2×C42×C17C2311yescharacteristic
93C2×C22⋊Dic17—22no—
94C2×C22⋊Dic17—22no—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17C2312yes—
97C2×C4⋊Dic17C2312yes—
98C2×C4×Dic17—21no—
99C22×#3622C2312yes—
100C22×#3622C2312yes—
101#10587555—22no—
102#10587555—22no—
103#10587506—24no—
104#10587506—24no—
105#10587506—24no—
106#10587506—42no—
107#10587503—24no—
108#10587503—42no—
109#10587502—21no—
110#10587502—22no—
111#10587502—22no—
112C22⋊#3622—42no—
113C22⋊#3622—42no—
114#10587479—24no—
115C4⋊#3622—22no—
116#10587474—22no—
117C24×C17—21no—
118C24×C17C2×D411yescharacteristic
119C24×C17C2×D411yescharacteristic
120#41×C17—48no—
121#41×C17—44no—
122#41×C17—44no—
123C2×Q8×C17—24no—
124C2×Q8×C17—21no—
125C2×Q8×C17—21no—
126C2×Q8×C17—22no—
127C2×Q8×C17—22no—
128C2×Q8×C17—22no—
129C2×Q8×C17—44no—
130C2×Q8×C17—48no—
131C2×Q8×C17—44no—
132C2×Q8×C17—44no—
133C2×Q8×C17C2×D412yes—
134C2×Q8×C17C2×D412yes—
135C2×Q8×C17#4114yes—
136C2×D4×C17—24no—
137C2×D4×C17—24no—
138C2×D4×C17—22no—
139C2×D4×C17—22no—
140C2×D4×C17—21no—
141C2×D4×C17—22no—
142C2×D4×C17—22no—
143C2×D4×C17—41no—
144C2×D4×C17—44no—
145C2×D4×C17—42no—
146C2×D4×C17—42no—
147C2×D4×C17C2×D412yes—
148C22×C4×C17—21no—
149C22×C4×C17—24no—
150C22×C4×C17—24no—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—21no—
157C22×C4×C17—22no—
158C22×C4×C17—22no—
159C22×C4×C17—22no—
160C22×C4×C17—22no—
161C22×C4×C17—44no—
162C22×C4×C17—44no—
163C22×C4×C17C2×D411yescharacteristic
164C22×C4×C17C2×D411yescharacteristic
165C22×C4×C17C2×D411yescharacteristic
166C22×C4×C17C2×D411yescharacteristic
167C22×C4×C17C2411yescharacteristic
168C22×Dic17—22no—
169C22×Dic17—22no—
170C2×Dic34—42no—
171C2×Dic34—82no—
172C4⋊C4×C17—24no—
173C4⋊C4×C17—22no—
174C4⋊C4×C17—24no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17C2×D412yes—
177C22⋊C4×C17—24no—
178C22⋊C4×C17—24no—
179C22⋊C4×C17—24no—
180C22⋊C4×C17—22no—
181C22⋊C4×C17—24no—
182C22⋊C4×C17—22no—
183C42×C17—24no—
184C42×C17C2×D412yes—
185C42×C17C2×D412yes—
186C22⋊Dic17—82no—
187C22⋊Dic17—82no—
188C4⋊Dic17—22no—
189C4⋊Dic17—22no—
190C4⋊Dic17—22no—
191C4⋊Dic17—22no—
192C4⋊Dic17—42no—
193C4×Dic17—42no—
194C2×#3622—22no—
195C2×#3622—22no—
196C2×#3622—22no—
197C2×#3622—22no—
198C2×#3622—22no—
199C2×#3622—22no—
200#25895—171nomaximal, Sylow
201C23×C17—21no—
202C23×C17—22no—
203C23×C17—21no—
204C23×C17—22no—
205C23×C17—21no—
206C23×C17—22no—
207C23×C17—22no—
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—22no—
211C23×C17—42no—
212C23×C17—41no—
213C23×C17—41no—
214C23×C17—42no—
215C23×C17#13712yes—
216C23×C17C22×D411yescharacteristic
217Q8×C17—22no—
218Q8×C17—24no—
219Q8×C17—24no—
220Q8×C17—48no—
221Q8×C17—41no—
222Q8×C17—41no—
223Q8×C17#12414yes—
224D4×C17—24no—
225D4×C17—44no—
226D4×C17—48no—
227D4×C17—48no—
228D4×C17—41no—
229D4×C17—41no—
230C2×C4×C17—22no—
231C2×C4×C17—24no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—48no—
242C2×C4×C17—48no—
243C2×C4×C17—42no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—42no—
251C2×C4×C17—42no—
252C2×C4×C17#12012yes—
253C2×C4×C17#12012yes—
254C2×C4×C17C22×D411yescharacteristic
255C2×C4×C17C22×D411yescharacteristic
256C2×C4×C17C2×#4112yes—
257C2×C4×C17H(5,𝔽2)12yes—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264Dic34—84no—
265#3622—42no—
266#3622—42no—
267C2×#548D1711yescharacteristic
268C2×#489—171no—
269C2×#475—171no—
270#2563—171no—
271#2035—172no—
272#2034—172no—
273#1983—171no—
274#1955—171no—
275#1788—172no—
276#1787—172no—
277#1763—171no—
278C22×C17—22no—
279C22×C17—24no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17—42no—
283C22×C17—42no—
284C22×C17—41no—
285C22×C17—41no—
286C22×C17—42no—
287C22×C17—41no—
288C22×C17—42no—
289C22×C17—42no—
290C22×C17—42no—
291C22×C17—82no—
292C22×C17—82no—
293C22×C17C2×#12011yescharacteristic
294C22×C17#54612yes—
295C22×C17C2×#13711yescharacteristic
296C22×C17#57511yescharacteristic
297C22×C17#57712yes—
298C4×C17—24no—
299C4×C17—22no—
300C4×C17—22no—
301C4×C17—41no—
302C4×C17—41no—
303C4×C17—42no—
304C4×C17—42no—
305C4×C17C2×#12411yescharacteristic
306C4×C17#53511yescharacteristic
307C4×C17#54612yes—
308Dic17—82no—
309Dic17—82no—
310C22×#41C2×D1711yescharacteristic
311C23×Q8C2×D1711yescharacteristic
312#548—24no—
313#548—24no—
314C2×C4⋊Q8—171no—
315C2×#124—21no—
316C2×#124C2×D1711yescharacteristic
317C2×C22⋊Q8—21no—
318C2×C22⋊Q8C2×D1711yescharacteristic
319C2×#121—21no—
320C2×#121C2×D1711yescharacteristic
321C2×C4×Q8C2×D1711yescharacteristic
322C2×C4×D4—21no—
323C2×C4×D4C2×D1711yescharacteristic
324#489—344no—
325#475—344no—
326C2×#106—172no—
327C2×C4⋊C8—171no—
328C2×#103—172no—
329C2×#103—172no—
330C2×#103—172no—
331C2×#103—172no—
332C2×#102—172no—
333C2×#102—172no—
334C2×C22⋊C8—172no—
335C2×C22⋊C8—172no—
336C2×#97—171no—
337#393—341no—
338#393—341no—
339#390—341no—
340#382—341no—
341#380—172no—
342#380—172no—
343C4×C4⋊C4—171no—
344#340—172no—
345C2×C17—42no—
346C2×C17—81no—
347C2×C17—82no—
348C2×C17#243511yescharacteristic
349C2×C17#300711yescharacteristic
350C2×C17#301011yescharacteristic
351C2×C17#331612yes—
352C2×C17#332112yes—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41—24no—
356C2×#41—42no—
357C22×Q8—22no—
358C22×Q8—22no—
359C22×Q8—21no—
360C22×Q8—24no—
361C22×Q8—42no—
362C22×Q8—341no—
363C22×Q8C22×D1711yescharacteristic
364C22×Q8C22×D1711yescharacteristic
365C22×D4—21no—
366C22×D4—21no—
367C22×D4C22⋊D1711yescharacteristic
368C22×D4C22×D1711yescharacteristic
369C23×C4—21no—
370C23×C4—21no—
371C23×C4C22×D1711yescharacteristic
372C23×C4C22×D1711yescharacteristic
373C22×C8—172no—
374C22×C8—172no—
375C4⋊Q8—342no—
376C4⋊Q8—342no—
377#124—28no—
378#124—22no—
379#124C22⋊D1714yes—
380C22⋊Q8—28no—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383#121—28no—
384#121—22no—
385#121—22no—
386C4×Q8—22no—
387C4×Q8C22⋊D1714yes—
388C4×D4—28no—
389C4×D4—22no—
390C4×D4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—172no—
394C2×C4⋊C4—172no—
395C2×C4⋊C4—341no—
396C2×C4⋊C4C22⋊D1711yescharacteristic
397C2×C4⋊C4C22×D1711yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—342no—
403C2×C22⋊C4—342no—
404C2×C22⋊C4C22⋊D1711yescharacteristic
405C2×C22⋊C4C22⋊D1711yescharacteristic
406C2×C42—21no—
407C2×C42—341no—
408C2×C42C22×D1711yescharacteristic
409#106—344no—
410C4⋊C8—342no—
411#103—344no—
412#103—344no—
413#103—344no—
414#103—682no—
415#102—344no—
416#102—682no—
417C22⋊C8—682no—
418C22⋊C8—682no—
419#97—342no—
420#95—341no—
421#95—342no—
422#95—342no—
423C17#2589511yescharacteristic, complemented, Sylow
424C24—21no—
425C24C2×C22⋊D1711yescharacteristic
426C24C2×C22⋊D1711yescharacteristic
427#41—48no—
428#41—44no—
429#41—44no—
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—44no—
437C2×Q8—48no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—682no—
441C2×Q8—1362no—
442C2×Q8#6325414yes—
443C2×Q8C2×C22⋊D1712yes—
444C2×Q8C2×C22⋊D1712yes—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—41no—
453C2×D4—44no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4C2×C22⋊D1712yes—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4C2×C22⋊D1711yescharacteristic
475C22×C4C2×C22⋊D1711yescharacteristic
476C22×C4C2×C22⋊D1711yescharacteristic
477C22×C4C2×C22⋊D1711yescharacteristic
478C22×C4C23×D1711yescharacteristic
479C2×C8—342no—
480C2×C8—342no—
481C2×C8—342no—
482C2×C8—342no—
483C2×C8—342no—
484C2×C8—342no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—342no—
490C4⋊C4—342no—
491C4⋊C4—342no—
492C4⋊C4—342no—
493C4⋊C4—682no—
494C4⋊C4C2×C22⋊D1712yes—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—1362no—
502C22⋊C4—1362no—
503C42—24no—
504C42—682no—
505C42C2×C22⋊D1712yes—
506C42C2×C22⋊D1712yes—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—22no—
511C23—21no—
512C23—22no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23#1058763712yes—
522C23C22×C22⋊D1711yescharacteristic
523Q8—22no—
524Q8—24no—
525Q8—24no—
526Q8—48no—
527Q8—41no—
528Q8—41no—
529Q8—1364no—
530Q8#1058762214yes—
531D4—24no—
532D4—44no—
533D4—48no—
534D4—48no—
535D4—41no—
536D4—41no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—682no—
560C2×C4—682no—
561C2×C4—682no—
562C2×C4—682no—
563C2×C4—682no—
564C2×C4—682no—
565C2×C4#1058763912yes—
566C2×C4#1058763912yes—
567C2×C4C2×#6325412yes—
568C2×C4#1058769112yes—
569C2×C4C22×C22⋊D1711yescharacteristic
570C2×C4C22×C22⋊D1711yescharacteristic
571C8—682no—
572C8—682no—
573C22—22no—
574C22—24no—
575C22—41no—
576C22—41no—
577C22—42no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—41no—
582C22—41no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—82no—
587C22—82no—
588C22C2×#1058763911yescharacteristic
589C22#1176345312yes—
590C22C2×#1058763711yescharacteristic
591C22#1176360711yescharacteristic
592C22#1176361712yes—
593C4—22no—
594C4—24no—
595C4—22no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—42no—
600C4—1362no—
601C4—1362no—
602C4C2×#1058762211yescharacteristic
603C4#1176337611yescharacteristic
604C4#1176345312yes—
605C2—42no—
606C2—81no—
607C2—82no—
608C2#1453243111yescharacteristic
609C2#1453804411yescharacteristic
610C2#1453807011yescharacteristic
611C2#1454089512yes—
612C2#1454097012yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#548C2
socleC23×C17C22×D4
hypercenterC2×#548D17
hypocenterC17#25895
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
1. C2×#11763168C2semidirect
2. C2×#11763094C2semidirect
3. #14533957C2semidirect
4. #14529977C2semidirect
5. #14529970C2semidirect
6. #14529717C2semidirect
7. #14529583C2semidirect
8. #14528724C2semidirect
9. #14528719C2semidirect
10. #14528614C2semidirect
423. C17#25895semidirect

Maximal subgroups

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

subgroupindex
C2×#117631682
C2×#117630942
#145339572
#145299772
#145299702
#145297172
#145295832
#145287242
#145287192
#145286142
C2×C17×#5482
#2589517