Finite Group Databasebeta

← #17049975

Subgroups — #17049975

Order 4352.

1530 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763462C211yesmaximal, characteristic
2C2×#11762888C211yesmaximal, characteristic, complemented
3C2×#11762809C212yesmaximal, complemented
4#14531684C211yesmaximal, characteristic, complemented
5#14529708C211yesmaximal, characteristic, complemented
6#14529214C212yesmaximal, complemented
7#14529088C212yesmaximal, complemented
8#14528781C211yesmaximal, characteristic, complemented
9#14528723C212yesmaximal, complemented
10C4×#10587492C211yesmaximal, characteristic, complemented
11C17×#1956C211yesmaximal, characteristic, complemented
12C22×#63256C2212yes—
13C22×#63218C2212yes—
14#11763462—24no—
15#11763462C418yescomplemented
16C2×#10587580—22no—
17C2×#10587632C2212yes—
18C2×#10587581—21no—
19C2×C4×D68C2211yescharacteristic
20C2×C4×D68C2211yescharacteristic
21C2×#10587579—21no—
22C2×C4⋊C4×D17—21no—
23C2×C4⋊C4×D17C2212yes—
24#11762888—24no—
25C2×C4⋊#3622C2211yescharacteristic
26#11762809—28no—
27C2×#10587506C2212yes—
28C2×#10587506C2212yes—
29C2×#10587492C2211yescharacteristic
30C2×#10587492C2211yescharacteristic
31C2×#10587480C2212yes—
32C2×#10587491C2212yes—
33#11762705—21no—
34#11762680—22no—
35#11762666C2212yes—
36C4×C4⋊Dic17C2211yescharacteristic
37#11762572C2211yescharacteristic
38#11762572C2211yescharacteristic
39#11762571C2212yes—
40C2×C17×C4⋊Q8C2211yescharacteristic
41C2×C17×#103C2212yes—
42C2×C17×#103C2212yes—
43C2×C4×C8×C17C2211yescharacteristic
44C17×#384C2211yescharacteristic
45C22×Q8×C17C2312yes—
46C2×#63256—24no—
47C2×#63256—24no—
48C2×#63256—24no—
49C2×#63256—42no—
50C2×#63256—42no—
51C2×#63256C2×C418yescomplemented
52C22×D68—22no—
53C22×D68D411yescharacteristic
54C22×D68C2311yescharacteristic
55C22×C4×D17—21no—
56C22×C4×D17—22no—
57C22×C4×D17—22no—
58C22×C4×D17C2312yes—
59C22×C8×C17C2311yescharacteristic
60C22×C8×C17C2311yescharacteristic
61C17×C4⋊Q8—22no—
62C17×C4⋊Q8C2×C414yescomplemented
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17C2312yes—
68C2×C42×C17C2311yescharacteristic
69#10587632—24no—
70#10587632C2×C418yescomplemented
71C2×#63227—22no—
72C2×#63227—22no—
73C2×#63227—21no—
74C2×#63227D412yes—
75C2×C4⋊Dic17D411yescharacteristic
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×#63225—22no—
80C2×#63225—22no—
81C2×C4×Dic17—21no—
82C22×#3622C2312yes—
83C2×#63218—24no—
84C2×#63218—24no—
85C2×#63218—42no—
86#10587581—28no—
87#10587580—216no—
88#10587579—44no—
89C4⋊C4×D17—28no—
90C4⋊C4×D17—24no—
91C4⋊C4×D17—42no—
92C4×D68—22no—
93C4×D68C2×C414yes—
94C4×D68C2×C414yescomplemented
95C4×D68D414yes—
96C17×#103—24no—
97C17×#103—42no—
98C4×C8×C17—22no—
99#10587506—24no—
100#10587506—42no—
101#10587502—21no—
102#10587502—22no—
103#10587492—22no—
104#10587492D414yescomplemented
105#10587491—24no—
106#10587480—24no—
107C4⋊#3622—22no—
108C23×D17—21no—
109C23×D17C2×D412yes—
110C2×Q8×C17—22no—
111C2×Q8×C17—22no—
112C2×Q8×C17—22no—
113C2×Q8×C17—22no—
114C2×Q8×C17—22no—
115C2×Q8×C17C22×C414yes—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17C2×D411yescharacteristic
119C22×C4×C17C2×D411yescharacteristic
120C22×C4×C17C2411yescharacteristic
121C22×Dic17—21no—
122C22×Dic17—21no—
123C22×Dic17—21no—
124C22×Dic17C2×D412yes—
125#63256—216no—
126#63256—48no—
127#63256—48no—
128#63256—48no—
129C2×D68—22no—
130C2×D68—22no—
131C2×D68—21no—
132C2×D68—21no—
133C2×D68—21no—
134C2×D68—28no—
135C2×D68—44no—
136C2×D68—44no—
137C2×D68C22⋊C414yes—
138C2×D68C22×C412yes—
139C2×D68C2×D412yes—
140C2×D68#4112yes—
141C2×C4×D17—22no—
142C2×C4×D17—22no—
143C2×C4×D17—22no—
144C2×C4×D17—22no—
145C2×C4×D17—24no—
146C2×C4×D17—28no—
147C2×C4×D17—44no—
148C2×C4×D17—44no—
149C2×C4×D17—42no—
150C2×C4×D17—42no—
151C2×C4×D17—44no—
152C2×C4×D17—44no—
153C2×C4×D17—82no—
154C2×C4×D17C22×C414yes—
155C2×C4×D17C2×D414yes—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17C2×D412yes—
162C4⋊C4×C17—24no—
163C4⋊C4×C17—24no—
164C4⋊C4×C17—22no—
165C4⋊C4×C17—22no—
166C4⋊C4×C17—42no—
167C4⋊C4×C17—81no—
168C42×C17C22×C412yes—
169C42×C17C2×D412yes—
170#63227—28no—
171#63227—24no—
172#63227—44no—
173#63227C22⋊C418yes—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17C22⋊C414yes—
179C4⋊Dic17C22×C412yes—
180C4⋊Dic17C2×D412yes—
181#63225—28no—
182#63225—44no—
183C4×Dic17—42no—
184C2×#3622—22no—
185C2×#3622—22no—
186C2×#3622—22no—
187#63218—48no—
188#22431—171nomaximal, Sylow
189C23×C17C22×D411yescharacteristic
190C22×D17—22no—
191C22×D17—22no—
192C22×D17—22no—
193C22×D17—22no—
194C22×D17—22no—
195C22×D17—22no—
196C22×D17—22no—
197C22×D17—22no—
198C22×D17—22no—
199C22×D17—41no—
200C22×D17—41no—
201C22×D17—41no—
202C22×D17—41no—
203C22×D17C2×C22⋊C414yes—
204Q8×C17—24no—
205Q8×C17—24no—
206Q8×C17—42no—
207Q8×C17—42no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—24no—
211C2×C4×C17—42no—
212C2×C4×C17—42no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—42no—
216C2×C4×C17C2×C22⋊C412yes—
217C2×C4×C17C2×C22⋊C412yes—
218C2×C4×C17AGL(1,ℤ/8)12yes—
219C2×C4×C17#13712yes—
220C2×C4×C17C23×C411yescharacteristic
221C2×C4×C17C22×D411yescharacteristic
222C2×C4×C17C22×D411yescharacteristic
223C2×C4×C17C2×#4111yescharacteristic
224C2×C4×C17H(5,𝔽2)11yescharacteristic
225C2×C4×C17H(5,𝔽2)11yescharacteristic
226C2×Dic17—22no—
227C2×Dic17—22no—
228C2×Dic17—22no—
229C2×Dic17—22no—
230C2×Dic17—41no—
231C2×Dic17—41no—
232C2×Dic17—41no—
233C2×Dic17—41no—
234C2×Dic17—41no—
235C2×Dic17—41no—
236C2×Dic17—41no—
237C2×Dic17C2×C22⋊C414yes—
238D68—22no—
239D68—22no—
240D68—22no—
241D68—28no—
242D68—416no—
243D68C4×D414yescomplemented
244C4×D17—24no—
245C4×D17—24no—
246C4×D17—24no—
247C4×D17—24no—
248C4×D17—416no—
249C4×D17—48no—
250C4×D17—88no—
251C8×C17—22no—
252C8×C17—41no—
253#3622—42no—
254C2×#547—171no—
255C2×#436—171no—
256C2×#417—172no—
257D4×C4⋊C4—171no—
258#1981—171no—
259#1956D1711yescharacteristic, complemented
260#1894—172no—
261#1872—172no—
262#1802—171no—
263#1788—172no—
264C4×#102—171no—
265C22×C17C22×C22⋊C411yescharacteristic
266C22×C17D4211yescharacteristic
267C22×C17#54611yescharacteristic
268C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C17C2×#13711yescharacteristic
270C22×C17#57711yescharacteristic
271C22×C17#57811yescharacteristic
272C2×D17—24no—
273C2×D17—24no—
274C2×D17—24no—
275C2×D17—24no—
276C2×D17—42no—
277C2×D17—42no—
278C2×D17—42no—
279C2×D17—42no—
280C2×D17—42no—
281C2×D17—42no—
282C2×D17—42no—
283C2×D17—42no—
284C2×D17—42no—
285C2×D17—42no—
286C4×C17—22no—
287C4×C17—44no—
288C4×C17—82no—
289C4×C17#41714yescomplemented
290C4×C17C2×C4×D411yescharacteristic
291C4×C17#51811yescharacteristic
292C4×C17D4211yescharacteristic
293C4×C17#54611yescharacteristic
294Dic17—24no—
295Dic17—42no—
296Dic17—42no—
297Dic17—81no—
298C22×#41—172no—
299C22×M16—172no—
300#547—178no—
301#547—344no—
302C2×C4⋊Q8C2×D1711yescharacteristic
303C2×#123—342no—
304C2×C22⋊Q8—172no—
305C2×#121—341no—
306C2×C4×D4—171no—
307C2×C4×D4—171no—
308C2×C4×D4—341no—
309C22×C4⋊C4—172no—
310C22×C4⋊C4—341no—
311#436—344no—
312C2×C4⋊C8—171no—
313#417—348no—
314C2×#103—172no—
315C2×#103—172no—
316C2×#103C2×D1712yes—
317C2×#103C2×D1712yes—
318C2×#102—171no—
319C2×#102—171no—
320C2×#102—172no—
321C2×C22⋊C8—172no—
322C2×C4×C8C2×D1711yescharacteristic
323#389—341no—
324#385—342no—
325#384C2×D1711yescharacteristic
326#380—172no—
327C4×C4⋊C4—171no—
328#340—171no—
329#340—171no—
330#340—172no—
331C2×C17D4×C22⋊C411yescharacteristic
332C2×C17C2×#41711yescharacteristic
333C2×C17#289811yescharacteristic
334C2×C17#327211yescharacteristic
335C2×C17#327611yescharacteristic
336C2×C17#333011yescharacteristic
337C2×C17#333111yescharacteristic
338D17—44no—
339D17—44no—
340C2×#41—178no—
341C2×#41—344no—
342C2×#41—344no—
343C2×#41—344no—
344C2×#41—682no—
345C2×#41—682no—
346C22×Q8C22×D1712yes—
347C22×D4—171no—
348C22×D4—171no—
349C22×D4—342no—
350C23×C4—172no—
351C23×C4—342no—
352C23×C4—342no—
353C23×C4—341no—
354C2×M16—344no—
355C2×M16—344no—
356C2×M16—682no—
357C22×C8—172no—
358C22×C8C22×D1711yescharacteristic
359C22×C8C22×D1711yescharacteristic
360C4⋊Q8—22no—
361C4⋊Q8C4×D1714yescomplemented
362#123—3416no—
363C22⋊Q8—178no—
364C22⋊Q8—344no—
365#121—348no—
366C4×D4—174no—
367C4×D4—174no—
368C4×D4—174no—
369C4×D4—342no—
370C4×D4—684no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—348no—
380C2×C4⋊C4—344no—
381C2×C4⋊C4—342no—
382C2×C4⋊C4—342no—
383C2×C4⋊C4—682no—
384C2×C4⋊C4C22×D1712yes—
385C2×C22⋊C4—172no—
386C2×C22⋊C4—342no—
387C2×C22⋊C4—341no—
388C2×C22⋊C4—342no—
389C2×C42—341no—
390C2×C42C22×D1711yescharacteristic
391C4⋊C8—342no—
392#103—24no—
393#103—42no—
394#103—344no—
395#103—682no—
396#102—174no—
397#102—344no—
398#102—342no—
399C22⋊C8—344no—
400C4×C8—22no—
401#95—342no—
402#95—341no—
403C17#2243111yescharacteristic, complemented, Sylow
404C24—172no—
405C24—341no—
406#41—3416no—
407#41—688no—
408#41—688no—
409#41—688no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8C2×C4×D1714yes—
416C2×D4—172no—
417C2×D4—172no—
418C2×D4—172no—
419C2×D4—174no—
420C2×D4—348no—
421C2×D4—341no—
422C2×D4—341no—
423C2×D4—341no—
424C2×D4—342no—
425C2×D4—342no—
426C2×D4—684no—
427C2×D4—684no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—172no—
431C22×C4—174no—
432C22×C4—174no—
433C22×C4—348no—
434C22×C4—342no—
435C22×C4—342no—
436C22×C4—342no—
437C22×C4—342no—
438C22×C4—341no—
439C22×C4—341no—
440C22×C4—344no—
441C22×C4—341no—
442C22×C4—684no—
443C22×C4—684no—
444C22×C4—684no—
445C22×C4—684no—
446C22×C4—682no—
447C22×C4—682no—
448C22×C4—1362no—
449C22×C4D4×D1711yescharacteristic
450C22×C4D4×D1711yescharacteristic
451C22×C4C23×D1711yescharacteristic
452M16—688no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—342no—
459C2×C8—342no—
460C2×C8—342no—
461C2×C8D4×D1712yes—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—81no—
468C4⋊C4—174no—
469C4⋊C4—172no—
470C4⋊C4—172no—
471C4⋊C4—348no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—684no—
477C22⋊C4—178no—
478C22⋊C4—348no—
479C22⋊C4—344no—
480C22⋊C4—684no—
481C42—682no—
482C42C2×C4×D1712yes—
483C42D4×D1712yes—
484C23—174no—
485C23—342no—
486C23—342no—
487C23—342no—
488C23—342no—
489C23—342no—
490C23—342no—
491C23—342no—
492C23—342no—
493C23—342no—
494C23—681no—
495C23—681no—
496C23—681no—
497C23—681no—
498C23C2×D4×D1711yescharacteristic
499Q8—24no—
500Q8—24no—
501Q8—42no—
502Q8—42no—
503D4—174no—
504D4—342no—
505D4—342no—
506D4—348no—
507D4—342no—
508D4—6816no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—174no—
518C2×C4—342no—
519C2×C4—342no—
520C2×C4—344no—
521C2×C4—344no—
522C2×C4—344no—
523C2×C4—344no—
524C2×C4—342no—
525C2×C4—342no—
526C2×C4—6816no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—688no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—1368no—
536C2×C4C22⋊C4×D1712yes—
537C2×C4C22⋊C4×D1712yes—
538C2×C4#1058760012yes—
539C2×C4#1058760412yes—
540C2×C4C22×C4×D1711yescharacteristic
541C2×C4C2×D4×D1711yescharacteristic
542C2×C4C2×D4×D1711yescharacteristic
543C2×C4#1058769111yescharacteristic
544C2×C4#41×D1711yescharacteristic
545C2×C4#1058769611yescharacteristic
546C8—22no—
547C8—41no—
548C8—682no—
549C22—344no—
550C22—344no—
551C22—344no—
552C22—344no—
553C22—682no—
554C22—682no—
555C22—682no—
556C22—682no—
557C22—682no—
558C22—682no—
559C22—682no—
560C22—682no—
561C22—682no—
562C22—682no—
563C22C2×C22⋊C4×D1711yescharacteristic
564C22#1176344411yescharacteristic
565C22#1176345211yescharacteristic
566C22C2×#1058760011yescharacteristic
567C22C2×#1058760411yescharacteristic
568C22#1176361511yescharacteristic
569C22#1176362211yescharacteristic
570C4—22no—
571C4—44no—
572C4—82no—
573C4—344no—
574C4—682no—
575C4—682no—
576C4—1361no—
577C4#1176280914yescomplemented
578C4C4×D4×D1711yescharacteristic
579C4#1176328111yescharacteristic
580C4#1176344311yescharacteristic
581C4#1176345611yescharacteristic
582C2—684no—
583C2—684no—
584C2#1453160511yescharacteristic
585C2C2×#1176280911yescharacteristic
586C2#1453731211yescharacteristic
587C2#1454038311yescharacteristic
588C2#1454042911yescharacteristic
589C2#1454108811yescharacteristic
590C2#1454109911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1956C2
socleC23×C17C22×D4
prime radicalC2×#11763462C2
cosocleC22×C4×C17C24
hypercenter#1956D17
hypocenterC17#22431
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
2. C2×#11762888C2semidirect
3. C2×#11762809C2semidirect
4. #14531684C2semidirect
5. #14529708C2semidirect
6. #14529214C2semidirect
7. #14529088C2semidirect
8. #14528781C2semidirect
9. #14528723C2semidirect
10. C4×#10587492C2semidirect
11. C17×#1956C2semidirect
15. #11763462C4semidirect
51. C2×#63256C2×C4semidirect
62. C17×C4⋊Q8C2×C4semidirect
70. #10587632C2×C4semidirect
94. C4×D68C2×C4semidirect
104. #10587492D4semidirect
243. D68C4×D4semidirect
259. #1956D17semidirect
289. C4×C17#417semidirect
361. C4⋊Q8C4×D17semidirect
403. C17#22431semidirect
577. C4#11762809semidirect

Maximal subgroups

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

subgroupindex
C2×#117634622
C2×#117628882
C2×#117628092
#145316842
#145297082
#145292142
#145290882
#145287812
#145287232
C4×#105874922
C17×#19562
#2243117