Finite Group Databasebeta

← #17067641

Subgroups — #17067641

Order 4352.

1176 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763137C211yesmaximal, characteristic, complemented
2C2×#11763120C211yesmaximal, characteristic
3C2×#11762707C211yesmaximal, characteristic, complemented
4#14530119C212yesmaximal, complemented
5#14530020C212yesmaximal, complemented
6#14529102C211yesmaximal, characteristic
7#14529092C211yesmaximal, characteristic, complemented
8#14529045C212yesmaximal, complemented
9#14529038C212yesmaximal, complemented
10#14528852C211yesmaximal, characteristic, complemented
11C2×C17×#529C211yesmaximal, characteristic
12C22×C4⋊Dic17C2211yescharacteristic
13C22×C4⋊Dic17C2211yescharacteristic
14C22×#63225—21no—
15#11763137—24no—
16#11763120—24no—
17C2×#10587478C2212yes—
18C2×#10587479C2212yes—
19C2×#10587506C2212yes—
20C2×#10587506C2212yes—
21C2×#10587503C2212yes—
22C2×#10587503C2212yes—
23C2×C22⋊#3622C2211yescharacteristic
24C2×C22⋊#3622C2211yescharacteristic
25#11762707—21no—
26#11762707—21no—
27#11762707C2212yes—
28#11762707C2212yes—
29C2×#10587502—21no—
30#11762571C2212yes—
31#11762571C2212yes—
32#11762564C2212yes—
33#11762564C2212yes—
34C17×#529—24no—
35C17×#529—24no—
36C2×C17×#126—21no—
37C2×C17×#125C2211yescharacteristic
38C2×C17×#124C2211yescharacteristic
39C2×C17×#123—21no—
40C2×C17×C22⋊Q8C2211yescharacteristic
41C2×C17×#121C2211yescharacteristic
42C2×C4×Q8×C17C2211yescharacteristic
43C2×C4×D4×C17—21no—
44C2×C4×D4×C17C2211yescharacteristic
45C2×C17×#117—21no—
46C22×C42×C17C2211yescharacteristic
47C22×Q8×C17C2311yescharacteristic
48C22×D4×C17—21no—
49C22×D4×C17C2311yescharacteristic
50C23×C4×C17—21no—
51C23×C4×C17D411yescharacteristic
52C23×C4×C17D411yescharacteristic
53C23×C4×C17C2311yescharacteristic
54C23×Dic17—21no—
55C23×Dic17—21no—
56C17×#126—28no—
57C17×#125—22no—
58C17×#125—22no—
59C17×#124—22no—
60C17×#124—22no—
61C17×#123—28no—
62C17×C22⋊Q8—22no—
63C17×C22⋊Q8D414yes—
64C17×#121—22no—
65C17×#121D414yes—
66C4×Q8×C17—22no—
67C4×Q8×C17—22no—
68C4×D4×C17—28no—
69C4×D4×C17—22no—
70C4×D4×C17—22no—
71C17×#117—28no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17—21no—
83C2×C42×C17—22no—
84C2×C42×C17—22no—
85C2×C42×C17—21no—
86C2×C42×C17D411yescharacteristic
87C2×C42×C17D411yescharacteristic
88C2×C42×C17C2311yescharacteristic
89C2×C42×C17C2311yescharacteristic
90C2×C4⋊Dic17—24no—
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17—22no—
94C2×C4⋊Dic17—22no—
95C2×C4⋊Dic17C2312yes—
96C2×C4⋊Dic17C2312yes—
97C2×#63225—22no—
98C2×#63225—22no—
99C2×#63225—44no—
100C22×#3622C2312yes—
101C22×#3622C2312yes—
102#10587506—24no—
103#10587506—24no—
104#10587503—24no—
105#10587503—24no—
106#10587502—22no—
107#10587502—22no—
108C22⋊#3622—22no—
109C22⋊#3622—22no—
110#10587479—24no—
111#10587478—24no—
112C24×C17—21no—
113C24×C17C2×D411yescharacteristic
114C2×Q8×C17—21no—
115C2×Q8×C17—22no—
116C2×Q8×C17—22no—
117C2×Q8×C17C2×D412yes—
118C2×D4×C17—24no—
119C2×D4×C17—21no—
120C2×D4×C17—22no—
121C2×D4×C17—22no—
122C2×D4×C17—42no—
123C2×D4×C17—42no—
124C2×D4×C17C2×D412yes—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—24no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—22no—
136C22×C4×C17—41no—
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×C17C2×D411yescharacteristic
146C22×C4×C17C2411yescharacteristic
147C22×Dic17—24no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—42no—
151C22×Dic17—42no—
152C22×Dic17—42no—
153C22×Dic17—42no—
154C22×Dic17—42no—
155C4⋊C4×C17—24no—
156C4⋊C4×C17—24no—
157C4⋊C4×C17—24no—
158C4⋊C4×C17—24no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17C2×D412yes—
162C4⋊C4×C17C2×D412yes—
163C22⋊C4×C17—24no—
164C22⋊C4×C17—24no—
165C22⋊C4×C17—24no—
166C22⋊C4×C17—24no—
167C22⋊C4×C17—24no—
168C42×C17—24no—
169C42×C17—22no—
170C42×C17—22no—
171C42×C17—21no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17—21no—
175C4⋊Dic17—24no—
176C4⋊Dic17—22no—
177C4⋊Dic17—22no—
178C4⋊Dic17—22no—
179C4⋊Dic17—42no—
180C4⋊Dic17#4114yes—
181#63225—44no—
182#63225—44no—
183C2×#3622—22no—
184C2×#3622—22no—
185C2×#3622—22no—
186C2×#3622—22no—
187C2×#3622—22no—
188C2×#3622—22no—
189#24066—171nomaximal, Sylow
190C23×C17—21no—
191C23×C17—21no—
192C23×C17—21no—
193C23×C17—21no—
194C23×C17—22no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17—41no—
201C23×C17#12012yes—
202C23×C17#13512yes—
203C23×C17AGL(1,ℤ/8)12yes—
204C23×C17C22×D411yescharacteristic
205Q8×C17—24no—
206Q8×C17—42no—
207D4×C17—24no—
208D4×C17—48no—
209D4×C17—42no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—24no—
215C2×C4×C17—24no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—22no—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—44no—
229C2×C4×C17—48no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—44no—
237C2×C4×C17#12012yes—
238C2×C4×C17#13712yes—
239C2×C4×C17C22×D411yescharacteristic
240C2×C4×C17C22×D411yescharacteristic
241C2×C4×C17C2×#4112yes—
242C2×C4×C172−1+412yes—
243C2×Dic17—24no—
244C2×Dic17—48no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17—42no—
250C2×Dic17—84no—
251#3622—42no—
252#3622—42no—
253C2×#529D1711yescharacteristic
254C2×#483—171no—
255C2×#480—171no—
256C2×#389—171no—
257#2059—172no—
258#2042—172no—
259#1874—171no—
260#1872—171no—
261#1864—172no—
262#1863—172no—
263#1817—171no—
264C22×C17—22no—
265C22×C17—22no—
266C22×C17—22no—
267C22×C17—22no—
268C22×C17—22no—
269C22×C17—22no—
270C22×C17—42no—
271C22×C17—42no—
272C22×C17—42no—
273C22×C17—81no—
274C22×C17—81no—
275C22×C17—81no—
276C22×C17—81no—
277C22×C17#44914yes—
278C22×C17C2×#12011yescharacteristic
279C22×C17#52511yescharacteristic
280C22×C17#53611yescharacteristic
281C22×C17C2×#13511yescharacteristic
282C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C17C2×#13711yescharacteristic
284C22×C17#57511yescharacteristic
285C4×C17—22no—
286C4×C17—22no—
287C4×C17—42no—
288C4×C17—41no—
289C4×C17—41no—
290C4×C17—42no—
291C4×C17—41no—
292C4×C17—42no—
293C4×C17#54712yes—
294C4×C17#54712yes—
295Dic17—44no—
296Dic17—82no—
297#529—24no—
298#529—24no—
299C2×#126—21no—
300C2×#125C2×D1711yescharacteristic
301C2×#124C2×D1711yescharacteristic
302C2×#123—21no—
303C2×C22⋊Q8C2×D1711yescharacteristic
304C2×#121C2×D1711yescharacteristic
305C2×C4×Q8C2×D1711yescharacteristic
306C2×C4×D4—21no—
307C2×C4×D4C2×D1711yescharacteristic
308C2×#117—21no—
309C22×C4⋊C4—171no—
310C22×C4⋊C4—171no—
311C22×C4⋊C4—341no—
312C22×C42C2×D1711yescharacteristic
313#483—344no—
314#480—344no—
315C2×#107—172no—
316C2×#106—172no—
317C2×#103—172no—
318C2×#103—172no—
319C2×#102—172no—
320C2×#102—172no—
321C2×C22⋊C8—171no—
322C2×C22⋊C8—171no—
323#389—172no—
324#389—172no—
325#389—341no—
326#389—341no—
327C2×#95—341no—
328#340—172no—
329#340—172no—
330#336—172no—
331#336—172no—
332C2×C17—24no—
333C2×C17—82no—
334C2×C17#240611yescharacteristic
335C2×C17C2×#44911yescharacteristic
336C2×C17#300711yescharacteristic
337C2×C17#310811yescharacteristic
338C2×C17#311411yescharacteristic
339C2×C17#323211yescharacteristic
340C2×C17#323811yescharacteristic
341C22×Q8C22×D1711yescharacteristic
342C22×D4—21no—
343C22×D4C22×D1711yescharacteristic
344C23×C4—21no—
345C23×C4—341no—
346C23×C4—341no—
347C23×C4C22⋊D1711yescharacteristic
348C23×C4C22⋊D1711yescharacteristic
349C23×C4C22×D1711yescharacteristic
350C22×C8—172no—
351C22×C8—172no—
352#126—28no—
353#125—22no—
354#125—22no—
355#124—22no—
356#124—22no—
357#123—28no—
358C22⋊Q8—22no—
359C22⋊Q8C22⋊D1714yes—
360#121—22no—
361#121C22⋊D1714yes—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×D4—28no—
365C4×D4—22no—
366C4×D4—22no—
367#117—28no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—172no—
373C2×C4⋊C4—172no—
374C2×C4⋊C4—344no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4—342no—
377C2×C4⋊C4—342no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4—342no—
380C2×C4⋊C4—342no—
381C2×C4⋊C4—684no—
382C2×C4⋊C4C22×D1711yescharacteristic
383C2×C4⋊C4C22×D1711yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C42—21no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42C22⋊D1711yescharacteristic
393C2×C42C22⋊D1711yescharacteristic
394C2×C42C22×D1711yescharacteristic
395C2×C42C22×D1711yescharacteristic
396#107—344no—
397#106—344no—
398#103—344no—
399#103—344no—
400#102—344no—
401#102—344no—
402C22⋊C8—342no—
403C22⋊C8—342no—
404#95—342no—
405#95—342no—
406C17#2406611yescharacteristic, complemented, Sylow
407C24—21no—
408C24C2×C22⋊D1711yescharacteristic
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8C2×C22⋊D1712yes—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4C2×C22⋊D1712yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—344no—
435C22×C4—342no—
436C22×C4—342no—
437C22×C4—682no—
438C22×C4—682no—
439C22×C4—682no—
440C22×C4—682no—
441C22×C4—682no—
442C22×C4C2×C22⋊D1711yescharacteristic
443C22×C4C2×C22⋊D1711yescharacteristic
444C22×C4C2×C22⋊D1711yescharacteristic
445C22×C4C2×C22⋊D1712yes—
446C22×C4C2×C22⋊D1712yes—
447C22×C4C2×C22⋊D1711yescharacteristic
448C22×C4C2×C22⋊D1711yescharacteristic
449C22×C4C23×D1711yescharacteristic
450C2×C8—342no—
451C2×C8—342no—
452C2×C8—342no—
453C2×C8—342no—
454C2×C8—342no—
455C2×C8—342no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—24no—
462C4⋊C4—174no—
463C4⋊C4—344no—
464C4⋊C4—342no—
465C4⋊C4—342no—
466C4⋊C4—342no—
467C4⋊C4—684no—
468C4⋊C4—684no—
469C4⋊C4—682no—
470C4⋊C4C2×C22⋊D1712yes—
471C4⋊C4C2×C22⋊D1712yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C42—24no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—22no—
483C42—22no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23#1058763512yes—
496C23#1058763612yes—
497C23#1058763912yes—
498C23C22×C22⋊D1711yescharacteristic
499Q8—24no—
500Q8—42no—
501D4—24no—
502D4—48no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—44no—
523C2×C4—48no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—344no—
532C2×C4—688no—
533C2×C4—682no—
534C2×C4—682no—
535C2×C4—682no—
536C2×C4—682no—
537C2×C4—682no—
538C2×C4—1364no—
539C2×C4#1058763712yes—
540C2×C4#1058763912yes—
541C2×C4#41×D1712yes—
542C2×C4#1058769712yes—
543C2×C4C22×C22⋊D1711yescharacteristic
544C2×C4C22×C22⋊D1711yescharacteristic
545C8—682no—
546C8—682no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22#1176295414yes—
561C22C2×#1058763911yescharacteristic
562C22#1176331611yescharacteristic
563C22#1176338011yescharacteristic
564C22C2×#1058763611yescharacteristic
565C22C2×#1058763511yescharacteristic
566C22C2×#1058763711yescharacteristic
567C22#1176360711yescharacteristic
568C4—22no—
569C4—22no—
570C4—41no—
571C4—41no—
572C4—42no—
573C4—42no—
574C4—41no—
575C4—42no—
576C4—684no—
577C4—1362no—
578C4#1176345812yes—
579C4#1176345812yes—
580C2—24no—
581C2—82no—
582C2#1453211511yescharacteristic
583C2C2×#1176295411yescharacteristic
584C2#1453804411yescharacteristic
585C2#1453886011yescharacteristic
586C2#1453891811yescharacteristic
587C2#1453994811yescharacteristic
588C2#1454000211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#529C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenterC2×#529D17
hypocenterC17#24066
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
1. C2×#11763137C2semidirect
3. C2×#11762707C2semidirect
4. #14530119C2semidirect
5. #14530020C2semidirect
7. #14529092C2semidirect
8. #14529045C2semidirect
9. #14529038C2semidirect
10. #14528852C2semidirect
406. C17#24066semidirect

Maximal subgroups

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

subgroupindex
C2×#117631372
C2×#117631202
C2×#117627072
#145301192
#145300202
#145291022
#145290922
#145290452
#145290382
#145288522
C2×C17×#5292
#2406617