Finite Group Databasebeta

← #17044061

Subgroups — #17044061

Order 4352.

1234 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763346C211yesmaximal, characteristic
2C2×#11762865C211yesmaximal, characteristic, complemented
3D17×#384C211yesmaximal, characteristic, complemented
4#14529751C212yesmaximal, complemented
5#14529597C212yesmaximal, complemented
6#14529100C212yesmaximal
7#14529088C212yesmaximal, complemented
8#14528789C211yesmaximal, characteristic, complemented
9#14528785C211yesmaximal, characteristic, complemented
10#14528781C211yesmaximal, characteristic, complemented
11C17×#1803C211yesmaximal, characteristic, complemented
12#11763346—24no—
13#11763346C418yescomplemented
14C2×#10587584—21no—
15C2×#10587576C2211yescharacteristic
16C2×#10587633C2211yescharacteristic
17C2×#10587580—21no—
18C2×#10587632C2211yescharacteristic
19C2×#10587581C2211yescharacteristic
20C2×C4×D68—21no—
21C2×C4×D68C2211yescharacteristic
22C2×#10587558—21no—
23C2×C4⋊C4×D17—21no—
24C2×C4⋊C4×D17—21no—
25C2×C4⋊C4×D17C2212yes—
26C2×C42×D17C2211yescharacteristic
27#11762865—24no—
28C2×C4⋊#3622C2211yescharacteristic
29C2×#10587506C2212yes—
30C2×#10587492C2212yes—
31C2×#10587480C2212yes—
32C2×#10587491C2212yes—
33C2×#10587486C2212yes—
34#11762679C2211yescharacteristic
35#11762678—21no—
36#11762678—21no—
37#11762678C2212yes—
38#11762572C2212yes—
39#11762571C2212yes—
40#11762571C2212yes—
41C2×C4×Q8×C17C2211yescharacteristic
42C2×C17×C4⋊C8C2211yescharacteristic
43C2×C17×#103C2212yes—
44C17×#384C2211yescharacteristic
45C17×#340C2212yes—
46C22×Q8×C17C2311yescharacteristic
47C22×D68—21no—
48C22×D68C2311yescharacteristic
49C22×C4×D17—21no—
50C22×C4×D17—22no—
51C22×C4×D17D411yescharacteristic
52C22×C4×D17D411yescharacteristic
53C22×C4×D17C2311yescharacteristic
54C22×C8×C17C2312yes—
55C4×Q8×C17—22no—
56C4×Q8×C17C2×C414yescomplemented
57C2×C4⋊C4×C17—21no—
58C2×C4⋊C4×C17—21no—
59C2×C4⋊C4×C17—21no—
60C2×C4⋊C4×C17C2312yes—
61C2×C4⋊C4×C17C2311yescharacteristic
62C2×C42×C17—21no—
63C2×C42×C17C2311yescharacteristic
64#10587633—22no—
65#10587633C2×C414yescomplemented
66#10587632—22no—
67#10587632C2×C414yes—
68C2×#63227—21no—
69C2×#63227—21no—
70C2×#63227—21no—
71C2×#63227—21no—
72C2×#63227—21no—
73C2×C4⋊Dic17—21no—
74C2×C4⋊Dic17—21no—
75C2×C4⋊Dic17—21no—
76C2×C4⋊Dic17C2312yes—
77C2×C4⋊Dic17C2311yescharacteristic
78C2×#63225—21no—
79C2×#63225—21no—
80C2×#63225—22no—
81C2×#63225—22no—
82C2×#63225—22no—
83C2×C4×Dic17D411yescharacteristic
84C2×C4×Dic17D411yescharacteristic
85C2×C4×Dic17C2311yescharacteristic
86C22×#3622C2312yes—
87#10587584—28no—
88#10587581—22no—
89#10587581C2×C414yes—
90#10587580—28no—
91C4⋊C4×D17—42no—
92C4⋊C4×D17—42no—
93C4⋊C4×D17—44no—
94C4⋊C4×D17—44no—
95#10587576—22no—
96#10587576C2×C414yescomplemented
97C4×D68—28no—
98C4×D68—22no—
99C4×D68C2×C414yescomplemented
100#10587558—28no—
101C42×D17C2×C414yes—
102C42×D17D414yes—
103C17×C4⋊C8—22no—
104C17×#103—42no—
105#10587506—42no—
106#10587492—42no—
107#10587491—24no—
108#10587486—24no—
109#10587480—42no—
110C4⋊#3622—22no—
111C23×D17—21no—
112C23×D17C2×D411yescharacteristic
113C2×Q8×C17—21no—
114C2×Q8×C17—21no—
115C2×Q8×C17—21no—
116C2×Q8×C17—21no—
117C2×Q8×C17—21no—
118C2×Q8×C17C22×C412yes—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—22no—
122C22×C4×C17C2×D411yescharacteristic
123C22×C4×C17C2×D411yescharacteristic
124C22×C4×C17C2411yescharacteristic
125C22×Dic17—21no—
126C22×Dic17—22no—
127C22×Dic17C2×D411yescharacteristic
128C22×Dic17C2×D411yescharacteristic
129C22×Dic17C2×D411yescharacteristic
130C2×D68—21no—
131C2×D68—21no—
132C2×D68—21no—
133C2×D68—21no—
134C2×D68—24no—
135C2×D68—21no—
136C2×D68—42no—
137C2×D68—42no—
138C2×D68C22×C412yes—
139C2×C4×D17—22no—
140C2×C4×D17—22no—
141C2×C4×D17—22no—
142C2×C4×D17—22no—
143C2×C4×D17—21no—
144C2×C4×D17—21no—
145C2×C4×D17—21no—
146C2×C4×D17—21no—
147C2×C4×D17—24no—
148C2×C4×D17—42no—
149C2×C4×D17—42no—
150C2×C4×D17—44no—
151C2×C4×D17—44no—
152C2×C4×D17—44no—
153C2×C4×D17C22⋊C414yes—
154C2×C4×D17C22⋊C414yes—
155C2×C4×D17C22×C412yes—
156C2×C4×D17C2×D412yes—
157C2×C8×C17—22no—
158C2×C8×C17—22no—
159C2×C8×C17—22no—
160C4⋊C4×C17—24no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—22no—
163C4⋊C4×C17—22no—
164C4⋊C4×C17—42no—
165C4⋊C4×C17—42no—
166C4⋊C4×C17C22×C412yes—
167C42×C17—24no—
168C42×C17C22×C412yes—
169C42×C17C2×D412yes—
170#63227—24no—
171#63227—24no—
172#63227—24no—
173#63227—24no—
174#63227—24no—
175C4⋊Dic17—21no—
176C4⋊Dic17—24no—
177C4⋊Dic17—22no—
178C4⋊Dic17—22no—
179C4⋊Dic17—42no—
180C4⋊Dic17—42no—
181C4⋊Dic17C22×C412yes—
182#63225—24no—
183#63225—24no—
184#63225—44no—
185#63225—44no—
186#63225—44no—
187C4×Dic17C22⋊C414yes—
188C4×Dic17C22⋊C414yes—
189C4×Dic17C22×C412yes—
190C4×Dic17C2×D412yes—
191C2×#3622—22no—
192C2×#3622—22no—
193C2×#3622—22no—
194#21867—171nomaximal, Sylow
195C23×C17C22×D411yescharacteristic
196C22×D17—21no—
197C22×D17—21no—
198C22×D17—21no—
199C22×D17—21no—
200C22×D17—22no—
201C22×D17—41no—
202C22×D17—41no—
203C22×D17—41no—
204C22×D17—41no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17C2×C22⋊C412yes—
208C22×D17AGL(1,ℤ/8)12yes—
209C22×D17#13712yes—
210Q8×C17—22no—
211Q8×C17—22no—
212Q8×C17—41no—
213Q8×C17—41no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—22no—
217C2×C4×C17—22no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—42no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17C2×C22⋊C412yes—
226C2×C4×C17C2×C22⋊C412yes—
227C2×C4×C17AGL(1,ℤ/8)12yes—
228C2×C4×C17#13712yes—
229C2×C4×C17C23×C411yescharacteristic
230C2×C4×C17C22×D411yescharacteristic
231C2×C4×C17H(5,𝔽2)11yescharacteristic
232C2×C4×C17H(5,𝔽2)11yescharacteristic
233C2×C4×C172−1+411yescharacteristic
234C2×C4×C172−1+411yescharacteristic
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—22no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—42no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17C2×C22⋊C412yes—
248C2×Dic17C2×C22⋊C412yes—
249C2×Dic17C2×C22⋊C412yes—
250D68—22no—
251D68—22no—
252D68—48no—
253D68—41no—
254D68—41no—
255C4×D17—28no—
256C4×D17—22no—
257C4×D17—22no—
258C4×D17—22no—
259C4×D17—22no—
260C4×D17—28no—
261C4×D17—48no—
262C4×D17—88no—
263C8×C17—42no—
264#3622—42no—
265C2×#529—171no—
266C2×#432—171no—
267C2×#384—171no—
268#1992—172no—
269#1959—172no—
270#1874—172no—
271#1872—172no—
272#1803—171no—
273#1803D1711yescharacteristic, complemented
274#1802—171no—
275#1802—171no—
276C22×C17C22×C22⋊C411yescharacteristic
277C22×C17#53511yescharacteristic
278C22×C17#53611yescharacteristic
279C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C17C2×#13711yescharacteristic
282C22×C17C2×#13711yescharacteristic
283C2×D17—22no—
284C2×D17—22no—
285C2×D17—22no—
286C2×D17—22no—
287C2×D17—22no—
288C2×D17—22no—
289C2×D17—42no—
290C2×D17—42no—
291C2×D17—42no—
292C2×D17—81no—
293C2×D17—81no—
294C2×D17—81no—
295C2×D17—81no—
296C2×D17#41714yes—
297C4×C17—22no—
298C4×C17—42no—
299C4×C17—42no—
300C4×C17—82no—
301C4×C17#41714yes—
302C4×C17#52011yescharacteristic
303C4×C17#52011yescharacteristic
304C4×C17#53711yescharacteristic
305C4×C17#53711yescharacteristic
306Dic17—22no—
307Dic17—22no—
308Dic17—22no—
309Dic17—42no—
310Dic17—82no—
311#529—178no—
312#529—344no—
313C2×#126—341no—
314C2×#125—171no—
315C2×#124—171no—
316C2×#123—341no—
317C2×C22⋊Q8—171no—
318C2×#121—171no—
319C2×C4×Q8C2×D1711yescharacteristic
320C2×C4×D4—171no—
321C2×C4×D4—341no—
322C2×#117—341no—
323C22×C4⋊C4—172no—
324C22×C4⋊C4—341no—
325C22×C4⋊C4—341no—
326C22×C42—171no—
327#432—344no—
328C2×C4⋊C8—171no—
329C2×C4⋊C8C2×D1711yescharacteristic
330C2×#103—172no—
331C2×#103C2×D1712yes—
332C2×#102—172no—
333C2×#102—172no—
334C2×C22⋊C8—172no—
335C2×#97—172no—
336#384—172no—
337#384—171no—
338#384—341no—
339#384—341no—
340#384C2×D1711yescharacteristic
341#340—172no—
342#340—172no—
343#340—172no—
344#340C2×D1712yes—
345C2×C17#231111yescharacteristic
346C2×C17C2×#41711yescharacteristic
347C2×C17C2×#41711yescharacteristic
348C2×C17#321711yescharacteristic
349C2×C17#321811yescharacteristic
350C2×C17#323811yescharacteristic
351C2×C17#323911yescharacteristic
352D17—24no—
353D17—82no—
354C22×Q8C22×D1711yescharacteristic
355C22×D4—171no—
356C22×D4—341no—
357C23×C4—171no—
358C23×C4—171no—
359C23×C4—171no—
360C23×C4—342no—
361C23×C4—341no—
362C22×C8—172no—
363C22×C8C22×D1712yes—
364#126—348no—
365#125—174no—
366#125—342no—
367#124—174no—
368#124—342no—
369#123—348no—
370C22⋊Q8—174no—
371C22⋊Q8—342no—
372#121—174no—
373#121—342no—
374C4×Q8—22no—
375C4×Q8C4×D1714yescomplemented
376C4×D4—174no—
377C4×D4—348no—
378C4×D4—342no—
379#117—348no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—172no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—342no—
389C2×C4⋊C4—342no—
390C2×C4⋊C4—342no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—682no—
394C2×C4⋊C4—682no—
395C2×C4⋊C4—684no—
396C2×C4⋊C4—684no—
397C2×C4⋊C4C22×D1712yes—
398C2×C4⋊C4C22×D1711yescharacteristic
399C2×C22⋊C4—341no—
400C2×C22⋊C4—341no—
401C2×C22⋊C4—341no—
402C2×C22⋊C4—341no—
403C2×C22⋊C4—341no—
404C2×C42—21no—
405C2×C42—171no—
406C2×C42—171no—
407C2×C42—171no—
408C2×C42—174no—
409C2×C42—174no—
410C2×C42C22×D1711yescharacteristic
411C4⋊C8—22no—
412C4⋊C8—342no—
413#103—42no—
414#103—682no—
415#102—682no—
416#102—682no—
417C22⋊C8—344no—
418#97—344no—
419C17#2186711yescharacteristic, complemented, Sylow
420C24—171no—
421C24—341no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8C2×C4×D1712yes—
428C2×D4—172no—
429C2×D4—344no—
430C2×D4—341no—
431C2×D4—341no—
432C2×D4—341no—
433C2×D4—341no—
434C2×D4—341no—
435C2×D4—682no—
436C2×D4—682no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—171no—
441C22×C4—171no—
442C22×C4—171no—
443C22×C4—172no—
444C22×C4—172no—
445C22×C4—174no—
446C22×C4—174no—
447C22×C4—342no—
448C22×C4—342no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—341no—
452C22×C4—341no—
453C22×C4—342no—
454C22×C4—342no—
455C22×C4—341no—
456C22×C4—344no—
457C22×C4—342no—
458C22×C4—684no—
459C22×C4—684no—
460C22×C4—684no—
461C22×C4—682no—
462C22×C4—682no—
463C22×C4D4×D1711yescharacteristic
464C22×C4D4×D1711yescharacteristic
465C22×C4C23×D1711yescharacteristic
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—342no—
470C2×C8—342no—
471C2×C8—342no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—172no—
479C4⋊C4—344no—
480C4⋊C4—344no—
481C4⋊C4—344no—
482C4⋊C4—341no—
483C4⋊C4—342no—
484C4⋊C4—342no—
485C4⋊C4—682no—
486C4⋊C4—684no—
487C4⋊C4—684no—
488C4⋊C4—684no—
489C4⋊C4—682no—
490C4⋊C4C2×C4×D1712yes—
491C22⋊C4—344no—
492C22⋊C4—344no—
493C22⋊C4—344no—
494C22⋊C4—344no—
495C22⋊C4—344no—
496C42—24no—
497C42—172no—
498C42—172no—
499C42—174no—
500C42—174no—
501C42C2×C4×D1712yes—
502C42D4×D1712yes—
503C23—172no—
504C23—172no—
505C23—172no—
506C23—341no—
507C23—341no—
508C23—341no—
509C23—341no—
510C23—342no—
511C23—681no—
512C23—681no—
513C23—681no—
514C23—681no—
515C23—681no—
516C23—681no—
517C23C2×D4×D1711yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—342no—
523D4—342no—
524D4—688no—
525D4—681no—
526D4—681no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—172no—
539C2×C4—172no—
540C2×C4—172no—
541C2×C4—348no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—342no—
549C2×C4—342no—
550C2×C4—348no—
551C2×C4—342no—
552C2×C4—342no—
553C2×C4—342no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—688no—
557C2×C4—682no—
558C2×C4—682no—
559C2×C4—682no—
560C2×C4—1368no—
561C2×C4C22⋊C4×D1712yes—
562C2×C4C22⋊C4×D1712yes—
563C2×C4#1058760012yes—
564C2×C4#1058760412yes—
565C2×C4C22×C4×D1711yescharacteristic
566C2×C4C2×D4×D1711yescharacteristic
567C2×C4#1058769111yescharacteristic
568C2×C4#1058769411yescharacteristic
569C2×C4#1058769611yescharacteristic
570C2×C4#1058769711yescharacteristic
571C8—42no—
572C8—682no—
573C22—174no—
574C22—342no—
575C22—342no—
576C22—342no—
577C22—342no—
578C22—342no—
579C22—342no—
580C22—682no—
581C22—682no—
582C22—682no—
583C22—1361no—
584C22—1361no—
585C22—1361no—
586C22—1361no—
587C22C2×C22⋊C4×D1711yescharacteristic
588C22#1176337511yescharacteristic
589C22#1176338211yescharacteristic
590C22AGL(1,ℤ/8)×D1711yescharacteristic
591C22C2×#1058760011yescharacteristic
592C22#137×D1711yescharacteristic
593C22C2×#1058760411yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—82no—
598C4—342no—
599C4—342no—
600C4—342no—
601C4—682no—
602C4—1362no—
603C4#1176280914yes—
604C4#1176328911yescharacteristic
605C4#1176329011yescharacteristic
606C4#1176338811yescharacteristic
607C4#1176338911yescharacteristic
608C2—344no—
609C2—1362no—
610C2#1453137311yescharacteristic
611C2D17×#41711yescharacteristic
612C2C2×#1176280911yescharacteristic
613C2#1453980611yescharacteristic
614C2#1453981511yescharacteristic
615C2#1454000911yescharacteristic
616C2#1454001811yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1803C2
socleC23×C17C22×D4
prime radicalC2×#10587581C22
cosocleC22×C4×C17C24
hypercenter#1803D17
hypocenterC17#21867
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#11762865C2semidirect
3. D17×#384C2semidirect
4. #14529751C2semidirect
5. #14529597C2semidirect
7. #14529088C2semidirect
8. #14528789C2semidirect
9. #14528785C2semidirect
10. #14528781C2semidirect
11. C17×#1803C2semidirect
13. #11763346C4semidirect
56. C4×Q8×C17C2×C4semidirect
65. #10587633C2×C4semidirect
96. #10587576C2×C4semidirect
99. C4×D68C2×C4semidirect
273. #1803D17semidirect
375. C4×Q8C4×D17semidirect
419. C17#21867semidirect

Maximal subgroups

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

subgroupindex
C2×#117633462
C2×#117628652
D17×#3842
#145297512
#145295972
#145291002
#145290882
#145287892
#145287852
#145287812
C17×#18032
#2186717