Finite Group Databasebeta

← #17044083

Subgroups — #17044083

Order 4352.

1086 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763363C211yesmaximal, characteristic
2C2×#11762865C211yesmaximal, characteristic
3#14531189C211yesmaximal, characteristic
4#14529755C212yesmaximal, complemented
5#14529602C212yesmaximal, complemented
6#14529095C212yesmaximal
7#14529093C212yesmaximal
8#14528790C211yesmaximal, characteristic, complemented
9#14528789C211yesmaximal, characteristic, complemented
10#14528786C211yesmaximal, characteristic, complemented
11C17×#1803C211yesmaximal, characteristic, complemented
12#11763363—24no—
13#11763363C418yescomplemented
14C2×#10587574C2211yescharacteristic
15C2×#10587630C2211yescharacteristic
16C2×#10587575—21no—
17C2×#10587583C2211yescharacteristic
18C2×#10587582—21no—
19C2×#10587632C2211yescharacteristic
20C2×C4×Dic34—21no—
21C2×C4×Dic34C2211yescharacteristic
22C2×#10587578C2212yes—
23C2×#10587558—21no—
24C2×C42×D17C2211yescharacteristic
25#11762865—24no—
26C2×C4⋊#3622C2211yescharacteristic
27C2×#10587487C2212yes—
28C2×#10587506C2212yes—
29C2×#10587481C2212yes—
30C2×#10587491C2212yes—
31C2×#10587486C2212yes—
32#11762679C2211yescharacteristic
33#11762674C2212yes—
34#11762666—21no—
35#11762666—21no—
36C4⋊C4×Dic17—21no—
37C4⋊C4×Dic17—21no—
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×C4×D17D411yescharacteristic
48C22×C4×D17D411yescharacteristic
49C22×C4×D17C2311yescharacteristic
50C22×Dic34—21no—
51C22×Dic34C2311yescharacteristic
52C22×C8×C17C2312yes—
53C4×Q8×C17—22no—
54C4×Q8×C17C2×C414yes—
55C2×C4⋊C4×C17—21no—
56C2×C4⋊C4×C17—21no—
57C2×C4⋊C4×C17—21no—
58C2×C4⋊C4×C17C2312yes—
59C2×C4⋊C4×C17C2311yescharacteristic
60C2×C42×C17—21no—
61C2×C42×C17C2311yescharacteristic
62#10587632—22no—
63#10587632C2×C414yes—
64#10587630—22no—
65#10587630C2×C414yes—
66C2×#63227—21no—
67C2×#63227—21no—
68C2×#63227—22no—
69C2×C4⋊Dic17—21no—
70C2×C4⋊Dic17—21no—
71C2×C4⋊Dic17—21no—
72C2×C4⋊Dic17C2312yes—
73C2×C4⋊Dic17C2311yescharacteristic
74C2×#63225—21no—
75C2×#63225—21no—
76C2×#63225—21no—
77C2×#63225—21no—
78C2×#63225—21no—
79C2×C4×Dic17—21no—
80C2×C4×Dic17—22no—
81C2×C4×Dic17D411yescharacteristic
82C2×C4×Dic17D411yescharacteristic
83C2×C4×Dic17C2311yescharacteristic
84C22×#3622C2312yes—
85#10587583—22no—
86#10587583C2×C414yes—
87#10587582—28no—
88#10587578—42no—
89#10587578—42no—
90#10587575—28no—
91#10587574—22no—
92#10587574C2×C414yes—
93#10587558—28no—
94C42×D17C2×C414yes—
95C42×D17D414yes—
96C4×Dic34—28no—
97C4×Dic34—22no—
98C4×Dic34C2×C414yes—
99C17×C4⋊C8—22no—
100C17×#103—42no—
101#10587506—42no—
102#10587502—22no—
103#10587502—22no—
104#10587491—24no—
105#10587487—42no—
106#10587486—24no—
107#10587481—42no—
108C4⋊#3622—22no—
109C23×D17C2×D411yescharacteristic
110C2×Q8×C17—21no—
111C2×Q8×C17—21no—
112C2×Q8×C17—21no—
113C2×Q8×C17—21no—
114C2×Q8×C17—21no—
115C2×Q8×C17C22×C412yes—
116C22×C4×C17—21no—
117C22×C4×C17—21no—
118C22×C4×C17—22no—
119C22×C4×C17C2×D411yescharacteristic
120C22×C4×C17C2×D411yescharacteristic
121C22×C4×C17C2411yescharacteristic
122C22×Dic17—21no—
123C22×Dic17—21no—
124C22×Dic17—22no—
125C22×Dic17C2×D411yescharacteristic
126C22×Dic17C2×D411yescharacteristic
127C22×Dic17C2×D411yescharacteristic
128C2×C4×D17—22no—
129C2×C4×D17—22no—
130C2×C4×D17—21no—
131C2×C4×D17—21no—
132C2×C4×D17—21no—
133C2×C4×D17—21no—
134C2×C4×D17—41no—
135C2×C4×D17—41no—
136C2×C4×D17C22⋊C414yes—
137C2×C4×D17C22⋊C414yes—
138C2×C4×D17C22×C412yes—
139C2×C4×D17C2×D412yes—
140C2×Dic34—21no—
141C2×Dic34—21no—
142C2×Dic34—21no—
143C2×Dic34—21no—
144C2×Dic34—21no—
145C2×Dic34—24no—
146C2×Dic34—42no—
147C2×Dic34—42no—
148C2×Dic34C22×C412yes—
149C2×C8×C17—22no—
150C2×C8×C17—22no—
151C2×C8×C17—22no—
152C4⋊C4×C17—24no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—22no—
155C4⋊C4×C17—22no—
156C4⋊C4×C17—42no—
157C4⋊C4×C17—42no—
158C4⋊C4×C17C22×C412yes—
159C42×C17—24no—
160C42×C17C22×C412yes—
161C42×C17C2×D412yes—
162#63227—24no—
163#63227—24no—
164#63227—44no—
165C4⋊Dic17—21no—
166C4⋊Dic17—24no—
167C4⋊Dic17—22no—
168C4⋊Dic17—22no—
169C4⋊Dic17—42no—
170C4⋊Dic17—42no—
171C4⋊Dic17C22×C412yes—
172#63225—24no—
173#63225—24no—
174#63225—24no—
175#63225—24no—
176#63225—24no—
177C4×Dic17—24no—
178C4×Dic17—44no—
179C4×Dic17C22⋊C414yes—
180C4×Dic17C22⋊C414yes—
181C4×Dic17C22×C412yes—
182C4×Dic17C2×D412yes—
183C2×#3622—22no—
184C2×#3622—22no—
185C2×#3622—22no—
186#21870—171nomaximal, Sylow
187C23×C17C22×D411yescharacteristic
188C22×D17—21no—
189C22×D17—21no—
190C22×D17—21no—
191C22×D17—21no—
192C22×D17—21no—
193C22×D17—21no—
194C22×D17C2×C22⋊C412yes—
195Q8×C17—22no—
196Q8×C17—22no—
197Q8×C17—41no—
198Q8×C17—41no—
199C2×C4×C17—21no—
200C2×C4×C17—21no—
201C2×C4×C17—22no—
202C2×C4×C17—22no—
203C2×C4×C17—41no—
204C2×C4×C17—41no—
205C2×C4×C17—41no—
206C2×C4×C17—41no—
207C2×C4×C17—42no—
208C2×C4×C17—42no—
209C2×C4×C17—42no—
210C2×C4×C17C2×C22⋊C412yes—
211C2×C4×C17C2×C22⋊C412yes—
212C2×C4×C17#13712yes—
213C2×C4×C17#13712yes—
214C2×C4×C17C23×C411yescharacteristic
215C2×C4×C17C22×D411yescharacteristic
216C2×C4×C17H(5,𝔽2)11yescharacteristic
217C2×C4×C17H(5,𝔽2)11yescharacteristic
218C2×C4×C172−1+411yescharacteristic
219C2×C4×C172−1+411yescharacteristic
220C2×Dic17—21no—
221C2×Dic17—21no—
222C2×Dic17—21no—
223C2×Dic17—21no—
224C2×Dic17—21no—
225C2×Dic17—21no—
226C2×Dic17—22no—
227C2×Dic17—22no—
228C2×Dic17—41no—
229C2×Dic17—41no—
230C2×Dic17—41no—
231C2×Dic17—41no—
232C2×Dic17—42no—
233C2×Dic17—42no—
234C2×Dic17—42no—
235C2×Dic17C2×C22⋊C412yes—
236C2×Dic17C2×C22⋊C412yes—
237C2×Dic17C2×C22⋊C412yes—
238C4×D17—22no—
239C4×D17—22no—
240C4×D17—22no—
241C4×D17—22no—
242C4×D17—28no—
243C4×D17—44no—
244Dic34—22no—
245Dic34—22no—
246Dic34—48no—
247Dic34—41no—
248Dic34—41no—
249C8×C17—42no—
250#3622—42no—
251C2×#533—171no—
252C2×#432—171no—
253#2285—171no—
254#1993—172no—
255#1960—172no—
256#1873—172no—
257#1873—172no—
258#1803—171no—
259#1803—171no—
260#1803—171no—
261#1803D1711yescharacteristic, complemented
262C22×C17C22×C22⋊C411yescharacteristic
263C22×C17#53511yescharacteristic
264C22×C17#53611yescharacteristic
265C22×C17C2×#13711yescharacteristic
266C22×C17C2×#13711yescharacteristic
267C22×C17#57511yescharacteristic
268C22×C17#57511yescharacteristic
269C2×D17—22no—
270C2×D17—22no—
271C2×D17—22no—
272C2×D17—41no—
273C2×D17—41no—
274C2×D17—41no—
275C2×D17—41no—
276C4×C17—22no—
277C4×C17—42no—
278C4×C17—42no—
279C4×C17—82no—
280C4×C17#41914yes—
281C4×C17#52011yescharacteristic
282C4×C17#52011yescharacteristic
283C4×C17#53711yescharacteristic
284C4×C17#53711yescharacteristic
285Dic17—22no—
286Dic17—22no—
287Dic17—22no—
288Dic17—42no—
289Dic17—42no—
290Dic17—82no—
291#533—178no—
292#533—344no—
293C2×C4⋊Q8—171no—
294C2×C4⋊Q8—171no—
295C2×#125—341no—
296C2×C22⋊Q8—171no—
297C2×C22⋊Q8—171no—
298C2×C22⋊Q8—341no—
299C2×C4×Q8—171no—
300C2×C4×Q8—341no—
301C2×C4×Q8C2×D1711yescharacteristic
302C2×#117—172no—
303C2×#117—341no—
304C22×C42—171no—
305#432—344no—
306C2×C4⋊C8—171no—
307C2×C4⋊C8C2×D1711yescharacteristic
308C2×#103—172no—
309C2×#103—172no—
310C2×#103—172no—
311C2×#103C2×D1712yes—
312C2×C22⋊C8—172no—
313C2×#97—172no—
314#384—171no—
315#384C2×D1711yescharacteristic
316#382—172no—
317#380—341no—
318#380—341no—
319C4×C4⋊C4—341no—
320C4×C4⋊C4—341no—
321#340—172no—
322#340—172no—
323#340—172no—
324#340C2×D1712yes—
325C2×C17#231111yescharacteristic
326C2×C17C2×#41911yescharacteristic
327C2×C17#289611yescharacteristic
328C2×C17#321611yescharacteristic
329C2×C17#321611yescharacteristic
330C2×C17#323711yescharacteristic
331C2×C17#323711yescharacteristic
332D17—42no—
333C22×Q8—171no—
334C22×Q8—341no—
335C22×Q8C22×D1711yescharacteristic
336C23×C4—171no—
337C23×C4—171no—
338C23×C4—171no—
339C22×C8—172no—
340C22×C8C22×D1712yes—
341C4⋊Q8—174no—
342C4⋊Q8—174no—
343C4⋊Q8—342no—
344C4⋊Q8—342no—
345#125—348no—
346C22⋊Q8—174no—
347C22⋊Q8—174no—
348C22⋊Q8—342no—
349C22⋊Q8—348no—
350C22⋊Q8—342no—
351C4×Q8—22no—
352C4×Q8—174no—
353C4×Q8—348no—
354C4×Q8—342no—
355C4×Q8C4×D1714yes—
356#117—348no—
357#117—682no—
358#117—682no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—171no—
363C2×C4⋊C4—172no—
364C2×C4⋊C4—341no—
365C2×C4⋊C4—341no—
366C2×C4⋊C4—341no—
367C2×C4⋊C4—341no—
368C2×C4⋊C4—341no—
369C2×C4⋊C4—341no—
370C2×C4⋊C4—341no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4C22×D1712yes—
373C2×C4⋊C4C22×D1711yescharacteristic
374C2×C22⋊C4—341no—
375C2×C22⋊C4—341no—
376C2×C22⋊C4—342no—
377C2×C42—21no—
378C2×C42—171no—
379C2×C42—171no—
380C2×C42—171no—
381C2×C42—174no—
382C2×C42—174no—
383C2×C42—341no—
384C2×C42—342no—
385C2×C42C22×D1711yescharacteristic
386C4⋊C8—22no—
387C4⋊C8—342no—
388#103—42no—
389#103—682no—
390#103—682no—
391#103—682no—
392C22⋊C8—344no—
393#97—344no—
394#95—342no—
395#95—342no—
396C17#2187011yescharacteristic, complemented, Sylow
397C24—171no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—172no—
404C2×Q8—344no—
405C2×Q8—341no—
406C2×Q8—341no—
407C2×Q8—341no—
408C2×Q8—341no—
409C2×Q8—341no—
410C2×Q8—682no—
411C2×Q8—682no—
412C2×Q8C2×C4×D1712yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—171no—
417C22×C4—171no—
418C22×C4—171no—
419C22×C4—172no—
420C22×C4—172no—
421C22×C4—174no—
422C22×C4—174no—
423C22×C4—341no—
424C22×C4—341no—
425C22×C4—341no—
426C22×C4—341no—
427C22×C4—342no—
428C22×C4—342no—
429C22×C4—341no—
430C22×C4—341no—
431C22×C4—342no—
432C22×C4—681no—
433C22×C4—681no—
434C22×C4D4×D1711yescharacteristic
435C22×C4D4×D1711yescharacteristic
436C22×C4C23×D1711yescharacteristic
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—342no—
441C2×C8—342no—
442C2×C8—342no—
443C4⋊C4—21no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—172no—
450C4⋊C4—344no—
451C4⋊C4—344no—
452C4⋊C4—344no—
453C4⋊C4—344no—
454C4⋊C4—344no—
455C4⋊C4—341no—
456C4⋊C4—344no—
457C4⋊C4—342no—
458C4⋊C4—342no—
459C4⋊C4—682no—
460C4⋊C4—682no—
461C4⋊C4C2×C4×D1712yes—
462C22⋊C4—344no—
463C22⋊C4—344no—
464C22⋊C4—684no—
465C42—24no—
466C42—172no—
467C42—172no—
468C42—174no—
469C42—174no—
470C42—344no—
471C42—684no—
472C42C2×C4×D1712yes—
473C42D4×D1712yes—
474C23—172no—
475C23—341no—
476C23—341no—
477C23—341no—
478C23—341no—
479C23—341no—
480C23—341no—
481C23C2×D4×D1711yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—41no—
485Q8—41no—
486Q8—342no—
487Q8—342no—
488Q8—688no—
489Q8—681no—
490Q8—681no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—172no—
503C2×C4—172no—
504C2×C4—172no—
505C2×C4—341no—
506C2×C4—341no—
507C2×C4—341no—
508C2×C4—341no—
509C2×C4—341no—
510C2×C4—341no—
511C2×C4—342no—
512C2×C4—342no—
513C2×C4—348no—
514C2×C4—342no—
515C2×C4—342no—
516C2×C4—342no—
517C2×C4—342no—
518C2×C4—684no—
519C2×C4—681no—
520C2×C4—681no—
521C2×C4—681no—
522C2×C4—681no—
523C2×C4—682no—
524C2×C4—682no—
525C2×C4—682no—
526C2×C4C22⋊C4×D1712yes—
527C2×C4C22⋊C4×D1712yes—
528C2×C4#1058760112yes—
529C2×C4#1058760412yes—
530C2×C4C22×C4×D1711yescharacteristic
531C2×C4C2×D4×D1711yescharacteristic
532C2×C4#1058769111yescharacteristic
533C2×C4#1058769411yescharacteristic
534C2×C4#1058769611yescharacteristic
535C2×C4#1058769711yescharacteristic
536C8—42no—
537C8—682no—
538C22—342no—
539C22—342no—
540C22—342no—
541C22—681no—
542C22—681no—
543C22—681no—
544C22—681no—
545C22C2×C22⋊C4×D1711yescharacteristic
546C22#1176337511yescharacteristic
547C22#1176338211yescharacteristic
548C22C2×#1058760111yescharacteristic
549C22C2×#1058760411yescharacteristic
550C22#1176360511yescharacteristic
551C22#1176360811yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—342no—
557C4—342no—
558C4—342no—
559C4—682no—
560C4—682no—
561C4—1362no—
562C4#1176281614yes—
563C4#1176328911yescharacteristic
564C4#1176329011yescharacteristic
565C4#1176338811yescharacteristic
566C4#1176338911yescharacteristic
567C2—682no—
568C2#1453137311yescharacteristic
569C2C2×#1176281611yescharacteristic
570C2#1453730011yescharacteristic
571C2#1453979311yescharacteristic
572C2#1453979411yescharacteristic
573C2#1453999611yescharacteristic
574C2#1453999711yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1803C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1803D17
hypocenterC17#21870
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
4. #14529755C2semidirect
5. #14529602C2semidirect
8. #14528790C2semidirect
9. #14528789C2semidirect
10. #14528786C2semidirect
11. C17×#1803C2semidirect
13. #11763363C4semidirect
261. #1803D17semidirect
396. C17#21870semidirect

Maximal subgroups

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

subgroupindex
C2×#117633632
C2×#117628652
#145311892
#145297552
#145296022
#145290952
#145290932
#145287902
#145287892
#145287862
C17×#18032
#2187017