Finite Group Databasebeta

← #17044277

Subgroups — #17044277

Order 4352.

1130 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763363C211yesmaximal, characteristic
2C2×#11762831C211yesmaximal, characteristic
3#14531189C211yesmaximal, characteristic
4#14529734C212yesmaximal, complemented
5#14529705C212yesmaximal, complemented
6#14529216C212yesmaximal
7#14529214C212yesmaximal
8#14528790C211yesmaximal, characteristic, complemented
9#14528789C211yesmaximal, characteristic, complemented
10#14528786C211yesmaximal, characteristic, complemented
11C17×#1803C211yesmaximal, characteristic, complemented
12C22×#63218C2212yes—
13#11763363—24no—
14#11763363C418yescomplemented
15C2×#10587574C2211yescharacteristic
16C2×#10587630C2211yescharacteristic
17C2×#10587575—21no—
18C2×#10587583C2211yescharacteristic
19C2×#10587582—21no—
20C2×#10587632C2211yescharacteristic
21C2×C4×Dic34—21no—
22C2×C4×Dic34C2211yescharacteristic
23C2×#10587578—21no—
24C2×#10587558—21no—
25C2×C42×D17C2211yescharacteristic
26#11762831—24no—
27C2×C4⋊#3622C2211yescharacteristic
28C2×#10587485C2212yes—
29C2×#10587487C2212yes—
30C2×#10587506C2212yes—
31C2×#10587481C2212yes—
32#11762679C2211yescharacteristic
33#11762674—21no—
34#11762666—21no—
35#11762666C2212yes—
36C4⋊C4×Dic17—21no—
37C4⋊C4×Dic17C2212yes—
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—
85C2×#63218—24no—
86C2×#63218—42no—
87C2×#63218—42no—
88#10587583—22no—
89#10587583C2×C414yes—
90#10587582—28no—
91#10587578—44no—
92#10587575—28no—
93#10587574—22no—
94#10587574C2×C414yes—
95#10587558—28no—
96C42×D17C2×C414yes—
97C42×D17D414yes—
98C4×Dic34—28no—
99C4×Dic34—22no—
100C4×Dic34C2×C414yes—
101C17×C4⋊C8—22no—
102C17×#103—42no—
103#10587506—42no—
104#10587502—22no—
105#10587502—22no—
106#10587487—42no—
107#10587485—24no—
108#10587481—42no—
109C4⋊#3622—22no—
110C23×D17C2×D411yescharacteristic
111C2×Q8×C17—21no—
112C2×Q8×C17—21no—
113C2×Q8×C17—21no—
114C2×Q8×C17—21no—
115C2×Q8×C17—21no—
116C2×Q8×C17C22×C412yes—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—22no—
120C22×C4×C17C2×D411yescharacteristic
121C22×C4×C17C2×D411yescharacteristic
122C22×C4×C17C2411yescharacteristic
123C22×Dic17—21no—
124C22×Dic17—21no—
125C22×Dic17—22no—
126C22×Dic17C2×D411yescharacteristic
127C22×Dic17C2×D411yescharacteristic
128C22×Dic17C2×D411yescharacteristic
129C2×C4×D17—22no—
130C2×C4×D17—22no—
131C2×C4×D17—21no—
132C2×C4×D17—21no—
133C2×C4×D17—21no—
134C2×C4×D17—21no—
135C2×C4×D17—22no—
136C2×C4×D17—22no—
137C2×C4×D17C22⋊C414yes—
138C2×C4×D17C22⋊C414yes—
139C2×C4×D17C22×C412yes—
140C2×C4×D17C2×D412yes—
141C2×Dic34—21no—
142C2×Dic34—21no—
143C2×Dic34—21no—
144C2×Dic34—21no—
145C2×Dic34—21no—
146C2×Dic34—24no—
147C2×Dic34—42no—
148C2×Dic34—42no—
149C2×Dic34C22×C412yes—
150C2×C8×C17—22no—
151C2×C8×C17—22no—
152C2×C8×C17—22no—
153C4⋊C4×C17—24no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—22no—
156C4⋊C4×C17—22no—
157C4⋊C4×C17—42no—
158C4⋊C4×C17—42no—
159C4⋊C4×C17C22×C412yes—
160C42×C17—24no—
161C42×C17C22×C412yes—
162C42×C17C2×D412yes—
163#63227—24no—
164#63227—24no—
165#63227—44no—
166C4⋊Dic17—21no—
167C4⋊Dic17—24no—
168C4⋊Dic17—22no—
169C4⋊Dic17—22no—
170C4⋊Dic17—42no—
171C4⋊Dic17—42no—
172C4⋊Dic17C22×C412yes—
173#63225—24no—
174#63225—24no—
175#63225—24no—
176#63225—24no—
177#63225—24no—
178C4×Dic17—24no—
179C4×Dic17—44no—
180C4×Dic17C22⋊C414yes—
181C4×Dic17C22⋊C414yes—
182C4×Dic17C22×C412yes—
183C4×Dic17C2×D412yes—
184C2×#3622—22no—
185C2×#3622—22no—
186C2×#3622—22no—
187#63218—48no—
188#21894—171nomaximal, Sylow
189C23×C17C22×D411yescharacteristic
190C22×D17—21no—
191C22×D17—21no—
192C22×D17—21no—
193C22×D17—21no—
194C22×D17—21no—
195C22×D17—21no—
196C22×D17C2×C22⋊C412yes—
197Q8×C17—22no—
198Q8×C17—22no—
199Q8×C17—41no—
200Q8×C17—41no—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—22no—
204C2×C4×C17—22no—
205C2×C4×C17—41no—
206C2×C4×C17—41no—
207C2×C4×C17—41no—
208C2×C4×C17—41no—
209C2×C4×C17—42no—
210C2×C4×C17—42no—
211C2×C4×C17—42no—
212C2×C4×C17C2×C22⋊C412yes—
213C2×C4×C17C2×C22⋊C412yes—
214C2×C4×C17#13712yes—
215C2×C4×C17#13712yes—
216C2×C4×C17C23×C411yescharacteristic
217C2×C4×C17C22×D411yescharacteristic
218C2×C4×C17H(5,𝔽2)11yescharacteristic
219C2×C4×C17H(5,𝔽2)11yescharacteristic
220C2×C4×C172−1+411yescharacteristic
221C2×C4×C172−1+411yescharacteristic
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—
238C2×Dic17AGL(1,ℤ/8)12yes—
239C2×Dic17#13712yes—
240C4×D17—22no—
241C4×D17—22no—
242C4×D17—22no—
243C4×D17—22no—
244C4×D17—28no—
245C4×D17—28no—
246Dic34—22no—
247Dic34—22no—
248Dic34—41no—
249Dic34—41no—
250Dic34—48no—
251C8×C17—42no—
252#3622—42no—
253C2×#533—171no—
254C2×#423—171no—
255#2285—171no—
256#1988—172no—
257#1981—172no—
258#1894—172no—
259#1894—172no—
260#1803—171no—
261#1803—171no—
262#1803—171no—
263#1803D1711yescharacteristic, complemented
264C22×C17C22×C22⋊C411yescharacteristic
265C22×C17#53711yescharacteristic
266C22×C17#53711yescharacteristic
267C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C17C2×#13711yescharacteristic
269C22×C17C2×#13711yescharacteristic
270C22×C17C2×#13711yescharacteristic
271C2×D17—22no—
272C2×D17—22no—
273C2×D17—22no—
274C2×D17—41no—
275C2×D17—41no—
276C2×D17—41no—
277C2×D17—41no—
278C4×C17—22no—
279C4×C17—42no—
280C4×C17—42no—
281C4×C17—82no—
282C4×C17#41914yes—
283C4×C17#52011yescharacteristic
284C4×C17#52011yescharacteristic
285C4×C17#53511yescharacteristic
286C4×C17#53611yescharacteristic
287Dic17—22no—
288Dic17—22no—
289Dic17—42no—
290Dic17—42no—
291Dic17—82no—
292Dic17#41714yes—
293C22×M16—172no—
294#533—178no—
295#533—344no—
296C2×C4⋊Q8—171no—
297C2×C4⋊Q8—171no—
298C2×#125—341no—
299C2×C22⋊Q8—171no—
300C2×C22⋊Q8—171no—
301C2×C22⋊Q8—341no—
302C2×C4×Q8—171no—
303C2×C4×Q8—341no—
304C2×C4×Q8C2×D1711yescharacteristic
305C2×#117—341no—
306C2×#117—341no—
307C22×C42—171no—
308#423—344no—
309C2×C4⋊C8—172no—
310C2×C4⋊C8—171no—
311C2×C4⋊C8C2×D1711yescharacteristic
312C2×#103—172no—
313C2×#103—172no—
314C2×#103—172no—
315C2×#103C2×D1712yes—
316#384—171no—
317#384C2×D1711yescharacteristic
318#382—341no—
319#380—172no—
320#380—341no—
321C4×C4⋊C4—172no—
322C4×C4⋊C4—341no—
323#340—172no—
324#340—172no—
325#340—172no—
326#340C2×D1712yes—
327C2×C17#231111yescharacteristic
328C2×C17C2×#41911yescharacteristic
329C2×C17C2×#41711yescharacteristic
330C2×C17#325411yescharacteristic
331C2×C17#325411yescharacteristic
332C2×C17#325511yescharacteristic
333C2×C17#325511yescharacteristic
334D17—42no—
335C22×Q8—171no—
336C22×Q8—341no—
337C22×Q8C22×D1711yescharacteristic
338C23×C4—171no—
339C23×C4—171no—
340C23×C4—171no—
341C2×M16—344no—
342C2×M16—682no—
343C2×M16—682no—
344C22×C8—172no—
345C22×C8C22×D1712yes—
346C4⋊Q8—174no—
347C4⋊Q8—174no—
348C4⋊Q8—342no—
349C4⋊Q8—342no—
350#125—348no—
351C22⋊Q8—174no—
352C22⋊Q8—174no—
353C22⋊Q8—342no—
354C22⋊Q8—348no—
355C22⋊Q8—342no—
356C4×Q8—22no—
357C4×Q8—174no—
358C4×Q8—348no—
359C4×Q8—342no—
360C4×Q8C4×D1714yes—
361#117—348no—
362#117—684no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—172no—
368C2×C4⋊C4—341no—
369C2×C4⋊C4—341no—
370C2×C4⋊C4—341no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4C22×D1712yes—
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C22⋊C4—341no—
379C2×C22⋊C4—341no—
380C2×C22⋊C4—342no—
381C2×C42—21no—
382C2×C42—171no—
383C2×C42—171no—
384C2×C42—171no—
385C2×C42—174no—
386C2×C42—174no—
387C2×C42—341no—
388C2×C42—342no—
389C2×C42C22×D1711yescharacteristic
390C4⋊C8—22no—
391C4⋊C8—344no—
392C4⋊C8—342no—
393#103—42no—
394#103—682no—
395#103—682no—
396#103—682no—
397#95—342no—
398#95—342no—
399C17#2189411yescharacteristic, complemented, Sylow
400C24—171no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—172no—
407C2×Q8—344no—
408C2×Q8—341no—
409C2×Q8—341no—
410C2×Q8—341no—
411C2×Q8—341no—
412C2×Q8—341no—
413C2×Q8—682no—
414C2×Q8—682no—
415C2×Q8C2×C4×D1712yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—171no—
420C22×C4—171no—
421C22×C4—171no—
422C22×C4—172no—
423C22×C4—172no—
424C22×C4—174no—
425C22×C4—174no—
426C22×C4—342no—
427C22×C4—342no—
428C22×C4—341no—
429C22×C4—341no—
430C22×C4—341no—
431C22×C4—341no—
432C22×C4—342no—
433C22×C4—342no—
434C22×C4—341no—
435C22×C4—341no—
436C22×C4—342no—
437C22×C4D4×D1711yescharacteristic
438C22×C4D4×D1711yescharacteristic
439C22×C4C23×D1711yescharacteristic
440M16—688no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—342no—
445C2×C8—342no—
446C2×C8—342no—
447C4⋊C4—21no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—172no—
454C4⋊C4—344no—
455C4⋊C4—344no—
456C4⋊C4—344no—
457C4⋊C4—344no—
458C4⋊C4—344no—
459C4⋊C4—341no—
460C4⋊C4—344no—
461C4⋊C4—342no—
462C4⋊C4—342no—
463C4⋊C4—682no—
464C4⋊C4—682no—
465C4⋊C4C2×C4×D1712yes—
466C22⋊C4—344no—
467C22⋊C4—344no—
468C22⋊C4—684no—
469C42—24no—
470C42—172no—
471C42—172no—
472C42—174no—
473C42—174no—
474C42—344no—
475C42—684no—
476C42C2×C4×D1712yes—
477C42D4×D1712yes—
478C23—172no—
479C23—341no—
480C23—341no—
481C23—341no—
482C23—341no—
483C23—341no—
484C23—341no—
485C23C2×D4×D1711yescharacteristic
486Q8—22no—
487Q8—22no—
488Q8—41no—
489Q8—41no—
490Q8—342no—
491Q8—342no—
492Q8—688no—
493Q8—681no—
494Q8—681no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—172no—
507C2×C4—172no—
508C2×C4—172no—
509C2×C4—172no—
510C2×C4—172no—
511C2×C4—348no—
512C2×C4—341no—
513C2×C4—341no—
514C2×C4—341no—
515C2×C4—341no—
516C2×C4—342no—
517C2×C4—342no—
518C2×C4—348no—
519C2×C4—342no—
520C2×C4—342no—
521C2×C4—342no—
522C2×C4—342no—
523C2×C4—681no—
524C2×C4—681no—
525C2×C4—681no—
526C2×C4—681no—
527C2×C4—682no—
528C2×C4—682no—
529C2×C4—682no—
530C2×C4C22⋊C4×D1712yes—
531C2×C4C22⋊C4×D1712yes—
532C2×C4#1058760112yes—
533C2×C4#1058760412yes—
534C2×C4C22×C4×D1711yescharacteristic
535C2×C4C2×D4×D1711yescharacteristic
536C2×C4#1058769111yescharacteristic
537C2×C4#1058769411yescharacteristic
538C2×C4#1058769611yescharacteristic
539C2×C4#1058769711yescharacteristic
540C8—42no—
541C8—682no—
542C22—342no—
543C22—342no—
544C22—342no—
545C22—681no—
546C22—681no—
547C22—681no—
548C22—681no—
549C22C2×C22⋊C4×D1711yescharacteristic
550C22#1176338811yescharacteristic
551C22#1176338911yescharacteristic
552C22AGL(1,ℤ/8)×D1711yescharacteristic
553C22#137×D1711yescharacteristic
554C22C2×#1058760111yescharacteristic
555C22C2×#1058760411yescharacteristic
556C4—22no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—174no—
561C4—342no—
562C4—342no—
563C4—682no—
564C4—682no—
565C4—1362no—
566C4#1176281614yes—
567C4#1176328911yescharacteristic
568C4#1176329011yescharacteristic
569C4#1176337511yescharacteristic
570C4#1176338211yescharacteristic
571C2—682no—
572C2#1453137311yescharacteristic
573C2C2×#1176281611yescharacteristic
574C2D17×#41711yescharacteristic
575C2#1454016911yescharacteristic
576C2#1454017011yescharacteristic
577C2#1454017611yescharacteristic
578C2#1454017711yescharacteristic
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#21894
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. #14529734C2semidirect
5. #14529705C2semidirect
8. #14528790C2semidirect
9. #14528789C2semidirect
10. #14528786C2semidirect
11. C17×#1803C2semidirect
14. #11763363C4semidirect
263. #1803D17semidirect
399. C17#21894semidirect

Maximal subgroups

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

subgroupindex
C2×#117633632
C2×#117628312
#145311892
#145297342
#145297052
#145292162
#145292142
#145287902
#145287892
#145287862
C17×#18032
#2189417