Finite Group Databasebeta

← #17066756

Subgroups — #17066756

Order 4352.

1228 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762964C211yesmaximal, characteristic, complemented
2C2×#11762817C211yesmaximal, characteristic, complemented
3#14532149C211yesmaximal, characteristic
4#14529932C212yesmaximal, complemented
5#14529792C212yesmaximal, complemented
6#14529218C211yesmaximal, characteristic, complemented
7#14529125C212yesmaximal, complemented
8#14529091C212yesmaximal, complemented
9#14529082C211yesmaximal, characteristic
10#14528879C211yesmaximal, characteristic, complemented
11C2×C17×#552C211yesmaximal, characteristic
12C23×Dic34C2211yescharacteristic
13C22×#63231C2212yes—
14C22×#63223C2211yescharacteristic
15C2×#10587610C2212yes—
16#11762964—24no—
17#11762817—24no—
18C2×#10587506C2212yes—
19C2×#10587481C2212yes—
20C2×#10587481C2212yes—
21C2×#10587481C2212yes—
22C2×C22⋊#3622C2211yescharacteristic
23C2×C22⋊#3622C2212yes—
24#11762729—21no—
25#11762721—21no—
26#11762715C2212yes—
27#11762689—21no—
28#11762681—21no—
29#11762669C2211yescharacteristic
30#11762571C2212yes—
31#11762571C2212yes—
32#11762571C2212yes—
33#11762564C2212yes—
34C17×#552—24no—
35C17×#552—24no—
36C2×C17×#126—21no—
37C2×C17×#125—21no—
38C2×C17×#125C2211yescharacteristic
39C2×C17×#123—21no—
40C2×C17×C22⋊Q8C2211yescharacteristic
41C2×C17×C22⋊Q8C2211yescharacteristic
42C2×C4×Q8×C17C2211yescharacteristic
43C2×C4×D4×C17C2211yescharacteristic
44C2×C17×#117—21no—
45C2×C17×#117C2211yescharacteristic
46C22×C4⋊C4×C17C2211yescharacteristic
47C22×Q8×C17C2311yescharacteristic
48C22×D4×C17D411yescharacteristic
49C23×C4×C17—21no—
50C23×C4×C17C2311yescharacteristic
51C23×C4×C17C2311yescharacteristic
52C23×Dic17—21no—
53C22×Dic34—22no—
54C22×Dic34—24no—
55C22×Dic34—22no—
56C22×Dic34—21no—
57C22×Dic34—42no—
58C22×Dic34C2312yes—
59C17×#126—28no—
60C17×#125—28no—
61C17×#125—22no—
62C17×#125—22no—
63C17×#123—28no—
64C17×C22⋊Q8—22no—
65C17×C22⋊Q8—22no—
66C17×C22⋊Q8—22no—
67C17×C22⋊Q8D414yes—
68C4×Q8×C17—22no—
69C4×Q8×C17—22no—
70C4×D4×C17—22no—
71C4×D4×C17—22no—
72C17×#117—28no—
73C17×#117—22no—
74C17×#117D414yes—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17—21no—
79C2×C4⋊C4×C17—21no—
80C2×C4⋊C4×C17—21no—
81C2×C4⋊C4×C17—22no—
82C2×C4⋊C4×C17—22no—
83C2×C4⋊C4×C17D411yescharacteristic
84C2×C4⋊C4×C17C2311yescharacteristic
85C2×C4⋊C4×C17C2311yescharacteristic
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×C17—21no—
93C2×C42×C17—21no—
94C2×C42×C17C2311yescharacteristic
95C2×C22⋊Dic17—22no—
96C2×C22⋊Dic17—21no—
97C2×#63231—24no—
98C2×#63231—24no—
99C2×#63231—42no—
100C2×C4⋊Dic17C2312yes—
101#10587610—24no—
102#10587610—42no—
103C2×#63225—22no—
104C2×#63223—22no—
105C2×#63223—42no—
106C22×#3622C2312yes—
107C22×#3622C2312yes—
108#10587506—42no—
109#10587502—22no—
110#10587502—22no—
111#10587502—21no—
112C22⋊#3622—22no—
113C22⋊#3622—42no—
114#10587481—24no—
115#10587481—24no—
116#10587481—24no—
117C24×C17C2×D411yescharacteristic
118C24×C17C2×D411yescharacteristic
119C2×Q8×C17—21no—
120C2×Q8×C17—22no—
121C2×Q8×C17—22no—
122C2×Q8×C17C2×D412yes—
123C2×D4×C17—22no—
124C2×D4×C17—22no—
125C2×D4×C17—41no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—24no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—42no—
138C22×C4×C17—42no—
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D412yes—
144C22×C4×C17C2×D412yes—
145C22×C4×C17C2411yescharacteristic
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—44no—
149C22×Dic17—42no—
150C2×Dic34—28no—
151C2×Dic34—22no—
152C2×Dic34—22no—
153C2×Dic34—22no—
154C2×Dic34—22no—
155C2×Dic34—22no—
156C2×Dic34—48no—
157C2×Dic34—44no—
158C2×Dic34—44no—
159C2×Dic34—42no—
160C2×Dic34#4114yes—
161C4⋊C4×C17—24no—
162C4⋊C4×C17—24no—
163C4⋊C4×C17—24no—
164C4⋊C4×C17—24no—
165C4⋊C4×C17—24no—
166C4⋊C4×C17—24no—
167C4⋊C4×C17—22no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17C2×D412yes—
172C4⋊C4×C17C2×D412yes—
173C4⋊C4×C17C2×D412yes—
174C22⋊C4×C17—24no—
175C22⋊C4×C17—24no—
176C22⋊C4×C17—24no—
177C22⋊C4×C17—22no—
178C22⋊C4×C17—22no—
179C42×C17—24no—
180C42×C17—24no—
181C42×C17—21no—
182C42×C17—21no—
183C22⋊Dic17—44no—
184C22⋊Dic17—42no—
185#63231—48no—
186C4⋊Dic17—22no—
187C4⋊Dic17—22no—
188#63225—44no—
189#63223—44no—
190C2×#3622—22no—
191C2×#3622—22no—
192C2×#3622—22no—
193C2×#3622—22no—
194C2×#3622—22no—
195C2×#3622—22no—
196#23983—171nomaximal, Sylow
197C23×C17—21no—
198C23×C17—22no—
199C23×C17—22no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17—22no—
204C23×C17—22no—
205C23×C17#12012yes—
206C23×C17#13712yes—
207C23×C17C22×D411yescharacteristic
208Q8×C17—24no—
209Q8×C17—41no—
210Q8×C17—41no—
211D4×C17—44no—
212C2×C4×C17—22no—
213C2×C4×C17—22no—
214C2×C4×C17—22no—
215C2×C4×C17—24no—
216C2×C4×C17—21no—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—22no—
227C2×C4×C17—48no—
228C2×C4×C17—42no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—42no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—42no—
237C2×C4×C17#12012yes—
238C2×C4×C17C22×D411yescharacteristic
239C2×C4×C17C22×D411yescharacteristic
240C2×C4×C17C2×#4112yes—
241C2×C4×C17H(5,𝔽2)12yes—
242C2×Dic17—24no—
243C2×Dic17—48no—
244C2×Dic17—42no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249Dic34—24no—
250Dic34—24no—
251Dic34—48no—
252Dic34—42no—
253Dic34—42no—
254#3622—42no—
255#3622—42no—
256C2×#552D1711yescharacteristic
257C2×#451—171no—
258C2×#419—171no—
259#2409—171no—
260#2026—172no—
261#2001—172no—
262#1894—171no—
263#1878—172no—
264#1872—172no—
265#1870—171no—
266#1825—171no—
267C22×C17—22no—
268C22×C17—22no—
269C22×C17—22no—
270C22×C17—22no—
271C22×C17—24no—
272C22×C17—41no—
273C22×C17—41no—
274C22×C17—42no—
275C22×C17—42no—
276C22×C17—42no—
277C22×C17C2×#12011yescharacteristic
278C22×C17#54612yes—
279C22×C17C2×#13711yescharacteristic
280C22×C17#57511yescharacteristic
281C22×C17#57812yes—
282C4×C17—22no—
283C4×C17—41no—
284C4×C17—42no—
285C4×C17—41no—
286C4×C17—42no—
287C4×C17—42no—
288C4×C17—41no—
289C4×C17—42no—
290C4×C17#52511yescharacteristic
291C4×C17#53511yescharacteristic
292C4×C17#54612yes—
293Dic17—44no—
294Dic17—82no—
295C23×Q8—171no—
296C22×Q16—172no—
297C22×M16—171no—
298#552—24no—
299#552—24no—
300C2×#126—21no—
301C2×#125—21no—
302C2×#125C2×D1711yescharacteristic
303C2×#123—21no—
304C2×C22⋊Q8—172no—
305C2×C22⋊Q8C2×D1711yescharacteristic
306C2×C22⋊Q8C2×D1711yescharacteristic
307C2×C4×Q8C2×D1711yescharacteristic
308C2×C4×D4C2×D1711yescharacteristic
309C2×#117—21no—
310C2×#117C2×D1711yescharacteristic
311C22×C4⋊C4C2×D1711yescharacteristic
312#451—344no—
313#419—344no—
314C2×#103—172no—
315C2×#103—172no—
316C2×#103—172no—
317C2×#103—172no—
318C2×C22⋊C8—171no—
319C2×C22⋊C8—172no—
320#395—341no—
321#393—341no—
322#391—172no—
323#386—341no—
324#385—341no—
325#380—171no—
326#340—172no—
327#340—172no—
328#340—172no—
329#336—172no—
330C2×C17—42no—
331C2×C17—42no—
332C2×C17#240411yescharacteristic
333C2×C17#300711yescharacteristic
334C2×C17#301111yescharacteristic
335C2×C17#331212yes—
336C2×C17#332712yes—
337C22×Q8—172no—
338C22×Q8—344no—
339C22×Q8—341no—
340C22×Q8—342no—
341C22×Q8—342no—
342C22×Q8—682no—
343C22×Q8C22×D1711yescharacteristic
344C22×D4C22⋊D1711yescharacteristic
345C23×C4—21no—
346C23×C4—341no—
347C23×C4C22×D1711yescharacteristic
348C23×C4C22×D1711yescharacteristic
349C2×Q16—344no—
350C2×Q16—344no—
351C2×Q16—682no—
352C2×M16—342no—
353C2×M16—682no—
354C22×C8—172no—
355C22×C8—172no—
356#126—28no—
357#125—28no—
358#125—22no—
359#125—22no—
360#123—28no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—344no—
365C22⋊Q8—682no—
366C22⋊Q8C22⋊D1714yes—
367C4×Q8—22no—
368C4×Q8—22no—
369C4×D4—22no—
370C4×D4—22no—
371#117—28no—
372#117—22no—
373#117C22⋊D1714yes—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—172no—
383C2×C4⋊C4—342no—
384C2×C4⋊C4C22⋊D1711yescharacteristic
385C2×C4⋊C4C22×D1711yescharacteristic
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C4⋊C4C22×D1711yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—342no—
392C2×C22⋊C4—341no—
393C2×C22⋊C4C22⋊D1711yescharacteristic
394C2×C22⋊C4C22⋊D1711yescharacteristic
395C2×C42—21no—
396C2×C42—21no—
397C2×C42C22×D1711yescharacteristic
398#103—344no—
399#103—344no—
400#103—344no—
401#103—682no—
402C22⋊C8—342no—
403C22⋊C8—682no—
404#95—341no—
405#95—342no—
406#95—342no—
407C17#2398311yescharacteristic, complemented, Sylow
408C24C2×C22⋊D1711yescharacteristic
409C24C2×C22⋊D1711yescharacteristic
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—174no—
414C2×Q8—348no—
415C2×Q8—342no—
416C2×Q8—342no—
417C2×Q8—342no—
418C2×Q8—342no—
419C2×Q8—342no—
420C2×Q8—684no—
421C2×Q8—688no—
422C2×Q8—684no—
423C2×Q8—682no—
424C2×Q8C2×C22⋊D1712yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—41no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—342no—
442C22×C4—342no—
443C22×C4—684no—
444C22×C4—682no—
445C22×C4C2×C22⋊D1711yescharacteristic
446C22×C4C2×C22⋊D1711yescharacteristic
447C22×C4C2×C22⋊D1711yescharacteristic
448C22×C4C2×C22⋊D1712yes—
449C22×C4C2×C22⋊D1712yes—
450C22×C4C2×C22⋊D1711yescharacteristic
451C22×C4C23×D1711yescharacteristic
452Q16—688no—
453M16—684no—
454C2×C8—342no—
455C2×C8—342no—
456C2×C8—342no—
457C2×C8—342no—
458C2×C8—342no—
459C2×C8—342no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—342no—
471C4⋊C4—342no—
472C4⋊C4—684no—
473C4⋊C4C2×C22⋊D1712yes—
474C4⋊C4C2×C22⋊D1712yes—
475C4⋊C4C2×C22⋊D1712yes—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—684no—
482C22⋊C4—682no—
483C42—24no—
484C42—21no—
485C42—21no—
486C42—24no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23#1058763712yes—
496C23#1058763912yes—
497C23C22×C22⋊D1711yescharacteristic
498Q8—24no—
499Q8—41no—
500Q8—41no—
501Q8—344no—
502Q8—344no—
503Q8—688no—
504Q8—682no—
505Q8—682no—
506D4—44no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—48no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—344no—
533C2×C4—688no—
534C2×C4—682no—
535C2×C4—682no—
536C2×C4—682no—
537C2×C4—682no—
538C2×C4—682no—
539C2×C4#1058763912yes—
540C2×C4#41×D1712yes—
541C2×C4#1058769612yes—
542C2×C4C22×C22⋊D1711yescharacteristic
543C2×C4C22×C22⋊D1711yescharacteristic
544C8—682no—
545C8—682no—
546C22—22no—
547C22—24no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—42no—
553C22—41no—
554C22—42no—
555C22—42no—
556C22C2×#1058763911yescharacteristic
557C22#1176344712yes—
558C22C2×#1058763711yescharacteristic
559C22#1176360711yescharacteristic
560C22#1176362412yes—
561C4—22no—
562C4—41no—
563C4—42no—
564C4—41no—
565C4—42no—
566C4—42no—
567C4—41no—
568C4—42no—
569C4—684no—
570C4—1362no—
571C4#1176331611yescharacteristic
572C4#1176337311yescharacteristic
573C4#1176344712yes—
574C2—42no—
575C2—42no—
576C2#1453209911yescharacteristic
577C2#1453804411yescharacteristic
578C2#1453807911yescharacteristic
579C2#1454082912yes—
580C2#1454104612yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#552C2
socleC23×C17C22×D4
prime radicalC22×D4×C17D4
hypercenterC2×#552D17
hypocenterC17#23983
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#11762964C2semidirect
2. C2×#11762817C2semidirect
4. #14529932C2semidirect
5. #14529792C2semidirect
6. #14529218C2semidirect
7. #14529125C2semidirect
8. #14529091C2semidirect
10. #14528879C2semidirect
407. C17#23983semidirect

Maximal subgroups

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

subgroupindex
C2×#117629642
C2×#117628172
#145321492
#145299322
#145297922
#145292182
#145291252
#145290912
#145290822
#145288792
C2×C17×#5522
#2398317