Finite Group Databasebeta

← #17900594

Subgroups — #17900594

Order 4352.

1744 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14542401C211yesmaximal, characteristic
2#14537936C213yesmaximal, complemented
3#14537929C213yesmaximal, complemented
4#14537915C211yesmaximal, characteristic, complemented
5#14537914C213yesmaximal, complemented
6#14537882C213yesmaximal, complemented
7#14537865C211yesmaximal, characteristic, complemented
8#14537863C211yesmaximal, characteristic, complemented
9#14537790C213yesmaximal, complemented
10#14537788C213yesmaximal, complemented
11#14537746C213yesmaximal, complemented
12C8×#63256C211yesmaximal, characteristic, complemented
13Q8×#63218C211yesmaximal, characteristic, complemented
14#14537715C213yesmaximal, complemented
15Q8×M16×C17C211yesmaximal, characteristic, complemented
16#11763535C411yescharacteristic
17#11763531C413yes—
18#11763500C413yes—
19#11763494C413yes—
20Q8×C22⋊D17C411yescharacteristic
21#11763471C411yescharacteristic
22#11763470C411yescharacteristic
23#11763460C413yes—
24#11763363C2213yes—
25#11763346C2213yes—
26#11763343C2213yes—
27C4×#63256C2211yescharacteristic
28C4×#63252C2213yes—
29C4×Q8×D17C2211yescharacteristic
30#11762939C2213yes—
31#11762937C2213yes—
32#11762937C2213yes—
33#11762936C2213yes—
34Q8×#3622C2211yescharacteristic
35Q8×#3622C2211yescharacteristic
36Q8×#3622C2213yes—
37Q8×#3622C2213yes—
38C8×Dic34C2213yes—
39#11762933C2213yes—
40#11762933C2216yes—
41#11762888C2213yes—
42#11762886C2216yes—
43#11762886C2213yes—
44#11762885C2211yescharacteristic
45#11762885C2213yes—
46#11762885C2216yes—
47#11762882C2213yes—
48#11762882C2213yes—
49#11762878C2213yes—
50#11762878C2213yes—
51C8×D68C2213yes—
52#11762874C2213yes—
53#11762871C2216yes—
54#11762869C2216yes—
55#11762862C2213yes—
56#11762862C2213yes—
57#11762861C2213yes—
58#11762861C2213yes—
59#11762860C2213yes—
60#11762860C2213yes—
61#11762834C2216yes—
62#11762831C2213yes—
63#11762831C2213yes—
64#11762766C2216yes—
65C4×#63218C2213yes—
66C4×C8×D17C2213yes—
67C2×C4×Q8×C17C2211yescharacteristic
68C17×#446C2213yes—
69C17×#446C2213yes—
70C8×Q8×C17C2211yescharacteristic
71C8×Q8×C17C2211yescharacteristic
72C17×#423C2213yes—
73C4×M16×C17C2213yes—
74C22×Q8×C17C2×C411yescharacteristic
75C2×#63256C2×C411yescharacteristic
76C2×Q8×D17C2×C411yescharacteristic
77C2×#63252C2×C413yes—
78C2×M16×C17C2311yescharacteristic
79C2×M16×C17C2313yes—
80C4×Q8×C17C2×C411yescharacteristic
81C4×Q8×C17C2×C413yes—
82C4×Q8×C17C2311yescharacteristic
83C4×Q8×C17C2313yes—
84C2×C4⋊C4×C17C2×C413yes—
85C2×C42×C17C2313yes—
86#10587633C2×C413yes—
87#10587632C2×C413yes—
88#10587632C2×C413yes—
89Q8×Dic17C2×C411yescharacteristic
90Q8×Dic17C2×C411yescharacteristic
91#10587630C2×C413yes—
92#10587616C2×C413yes—
93#10587615C2×C416yes—
94C4×C22⋊D17C2×C413yes—
95C4×C22⋊D17C2×C413yes—
96C4×C22⋊D17C2311yescharacteristic
97C4×C22⋊D17C2313yes—
98#10587612C2×C413yes—
99#10587610C2×C413yes—
100C2×#63218C2311yescharacteristic
101C2×#63218C2313yes—
102C2×#63218C2316yes—
103C2×C8×D17C2313yes—
104C2×C8×D17C2313yes—
105#10587584C2×C416yes—
106#10587583C2×C413yes—
107#10587582C2×C416yes—
108#10587581C2×C413yes—
109#10587580C2×C416yes—
110#10587579C2×C413yes—
111#10587578C2×C413yes—
112C4⋊C4×D17C2×C413yes—
113#10587576C2×C413yes—
114#10587575C2×C416yes—
115#10587574C2×C413yes—
116#10587573C2×C413yes—
117#10587562C2×C416yes—
118#10587561C2×C413yes—
119C4×D68C2×C413yes—
120C4×D68C2313yes—
121C4×D68C2313yes—
122#10587558C2×C413yes—
123#10587558C2×C413yes—
124#10587558C2316yes—
125C42×D17C2313yes—
126C42×D17C2313yes—
127#10587556C2×C413yes—
128C4×Dic34C2×C413yes—
129C4×Dic34C2313yes—
130C4×Dic34C2313yes—
131C17×C4⋊C8C2313yes—
132C17×C4⋊C8C2313yes—
133C17×C4⋊C8C2316yes—
134C17×#97C2313yes—
135C17×#97C2313yes—
136C4×C8×C17C2313yes—
137C4×C8×C17C2313yes—
138C22⋊#3622C2311yescharacteristic
139C22⋊#3622C2313yes—
140#10587491C2311yescharacteristic
141#10587491C2313yes—
142#10587491C2313yes—
143#10587491C2313yes—
144#10587491C2316yes—
145#10587486C2313yes—
146#10587486C2313yes—
147#10587485C2313yes—
148#10587485C2316yes—
149#10587485C2313yes—
150#10587485C2311yescharacteristic
151#10587485C2313yes—
152C8×Dic17C2311yescharacteristic
153C8×Dic17C2313yes—
154C8×Dic17C2316yes—
155C4⋊#3622C2313yes—
156C4⋊#3622C2313yes—
157C4⋊#3622C2313yes—
158C4⋊#3622C2313yes—
159C4×#3622C2313yes—
160C4×#3622C2313yes—
161C4×#3622C2316yes—
162C2×Q8×C17—24no—
163C2×Q8×C17C22×C413yes—
164C2×Q8×C17C22×C411yescharacteristic
165C22×C4×C17C22×C413yes—
166C22×C4×C17C22×C413yes—
167C22×C4×C17C2411yescharacteristic
168C2×C22⋊D17C22×C411yescharacteristic
169C2×C22⋊D17C22×C413yes—
170#63256—24no—
171Q8×D17—24no—
172#63252—46no—
173C2×D68C22×C413yes—
174C2×D68C22×C413yes—
175C2×C4×D17C22×C413yes—
176C2×C4×D17C22×C413yes—
177C2×C4×D17C22×C413yes—
178C2×C4×D17C22×C413yes—
179C2×C4×D17C2411yescharacteristic
180C2×C4×D17C2413yes—
181C2×Dic34C22×C413yes—
182C2×Dic34C22×C413yes—
183M16×C17—22no—
184M16×C17—43no—
185C2×C8×C17C2413yes—
186C2×C8×C17C2413yes—
187C2×C8×C17C2411yescharacteristic
188C2×C8×C17C2411yescharacteristic
189C4⋊C4×C17C22×C413yes—
190C4⋊C4×C17C22×C413yes—
191C4⋊C4×C17C22×C416yes—
192C42×C17C22×C413yes—
193C42×C17C22×C413yes—
194C42×C17C2413yes—
195C42×C17C2413yes—
196C22⋊Dic17C22×C413yes—
197C22⋊Dic17C22×C411yescharacteristic
198#63227C22×C411yescharacteristic
199#63227C22×C413yes—
200#63227C22×C413yes—
201#63227C22×C413yes—
202#63227C22×C416yes—
203#63227C22×C416yes—
204#63227C22×C416yes—
205C4⋊Dic17C22×C413yes—
206C4⋊Dic17C22×C413yes—
207C4⋊Dic17C22×C413yes—
208C4⋊Dic17C22×C413yes—
209#63225C22×C411yescharacteristic
210#63225C22×C413yes—
211#63225C22×C413yes—
212#63225C22×C413yes—
213#63225C22×C416yes—
214#63225C22×C416yes—
215#63225C22×C416yes—
216C4×Dic17C22×C413yes—
217C4×Dic17C22×C413yes—
218C4×Dic17C22×C413yes—
219C4×Dic17C22×C413yes—
220C4×Dic17C2411yescharacteristic
221C4×Dic17C2413yes—
222C2×#3622C2413yes—
223C2×#3622C2411yescharacteristic
224C2×#3622C2413yes—
225C2×#3622C2411yescharacteristic
226#63218—22no—
227#63218—43no—
228#63218—46no—
229C8×D17—26no—
230C8×D17—43no—
231#60345—171nomaximal, Sylow
232C23×C17C23×C411yescharacteristic
233C22×D17C23×C411yescharacteristic
234C22×D17C23×C413yes—
235Q8×C17—26no—
236Q8×C17#13114yes—
237C2×C4×C17—26no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—43no—
241C2×C4×C17C23×C411yescharacteristic
242C2×C4×C17C23×C413yes—
243C2×C4×C17C23×C413yes—
244C2×C4×C17C23×C413yes—
245C2×C4×C17C23×C413yes—
246C2×C4×C17C2511yescharacteristic
247C22⋊D17—22no—
248C22⋊D17—43no—
249C2×Dic17C23×C413yes—
250C2×Dic17C23×C411yescharacteristic
251C2×Dic17C23×C411yescharacteristic
252C2×Dic17C23×C413yes—
253C2×Dic17C23×C413yes—
254C2×Dic17C23×C411yescharacteristic
255D68—26no—
256D68—43no—
257C4×D17—26no—
258C4×D17—26no—
259C4×D17—23no—
260C4×D17—21no—
261C4×D17—23no—
262C4×D17—21no—
263C4×D17—26no—
264C4×D17—43no—
265Dic34—26no—
266Dic34—26no—
267C8×C17—21no—
268C8×C17—23no—
269C8×C17—23no—
270C8×C172−1+412yes—
271#3622—23no—
272#3622—21no—
273#3622—21no—
274#3622—23no—
275#3482—171no—
276#2993—173no—
277#2993—173no—
278#2991—171no—
279#2991—173no—
280#2986—173no—
281#2982—171no—
282#2982—171no—
283#2969—173no—
284#2969—173no—
285#2965—173no—
286C8×#41—171no—
287Q8×M16—171no—
288Q8×M16D1711yescharacteristic, complemented
289#2960—173no—
290C22×C17—21no—
291C22×C17—21no—
292C22×C17—21no—
293C22×C17C24×C411yescharacteristic
294C2×D17—23no—
295C2×D17—21no—
296C2×D17—21no—
297C2×D17—23no—
298C2×D17—23no—
299C2×D17—21no—
300C4×C17—21no—
301C4×C17—23no—
302C4×C17—23no—
303C4×C17—23no—
304C4×C17C2×#13116yes—
305C4×C17C2×2−1+411yescharacteristic
306C4×C17#58511yescharacteristic
307Dic17—23no—
308Dic17—21no—
309Dic17—23no—
310Dic17—21no—
311Dic17—23no—
312Dic17#56812yes—
313#557—171no—
314#556—173no—
315#552—173no—
316#552—173no—
317D4×Q8—171no—
318#548—171no—
319#548—171no—
320#547—173no—
321#533—173no—
322#529—173no—
323#529—173no—
324C4×#41—171no—
325C4×#41—173no—
326C2×C4×Q8—171no—
327C2×C4×Q8C2×D1711yescharacteristic
328#446—173no—
329#446—173no—
330#446—173no—
331#446—173no—
332#446C2×D1713yes—
333#446C2×D1713yes—
334C8×Q8—173no—
335C8×Q8—171no—
336C8×Q8—171no—
337C8×Q8—173no—
338C8×Q8—173no—
339C8×Q8—176no—
340C8×Q8—173no—
341C8×Q8C2×D1711yescharacteristic
342C8×Q8C2×D1711yescharacteristic
343#436—171no—
344#436—173no—
345#436—176no—
346#436—173no—
347#436—176no—
348#436—173no—
349#435—173no—
350#435—173no—
351#435—173no—
352#435—173no—
353C8×D4—173no—
354C8×D4—173no—
355#433—176no—
356#433—176no—
357#431—173no—
358#431—173no—
359#431—173no—
360#431—173no—
361#431—173no—
362#431—173no—
363#424—176no—
364#423—173no—
365#423—173no—
366#423C2×D1713yes—
367#405—176no—
368C4×M16—173no—
369C4×M16C2×D1713yes—
370C2×C4×C8—173no—
371C2×C17—41no—
372C2×C17C4×2−1+411yescharacteristic
373C2×C17C22×#13111yescharacteristic
374C2×C17C2×#56811yescharacteristic
375D17—41no—
376D17—43no—
377C2×#41—171no—
378C2×#41—173no—
379C22×Q8—171no—
380C22×Q8C4×D1711yescharacteristic
381C2×M16—171no—
382C2×M16—173no—
383C2×M16—176no—
384C2×M16C22×D1711yescharacteristic
385C2×M16C22×D1713yes—
386C22×C8—173no—
387C22×C8—173no—
388C4⋊Q8—173no—
389C4⋊Q8—173no—
390#126—176no—
391#126—176no—
392#125—173no—
393#125—176no—
394#125—173no—
395#124—173no—
396#124—173no—
397#123—176no—
398#123—176no—
399C22⋊Q8—173no—
400C22⋊Q8—173no—
401C22⋊Q8—173no—
402C22⋊Q8—176no—
403C22⋊Q8—173no—
404#121—173no—
405#121—173no—
406C4×Q8—171no—
407C4×Q8—171no—
408C4×Q8—173no—
409C4×Q8—173no—
410C4×Q8—173no—
411C4×Q8—173no—
412C4×Q8C4×D1711yescharacteristic
413C4×Q8C4×D1713yes—
414C4×Q8C22×D1711yescharacteristic
415C4×Q8C22×D1713yes—
416C4×D4—173no—
417C4×D4—173no—
418C4×D4—171no—
419C4×D4—173no—
420C4×D4—173no—
421C4×D4—173no—
422C4×D4—173no—
423C4×D4—173no—
424#117—173no—
425#117—173no—
426#117—173no—
427#117—173no—
428#117—176no—
429C2×C4⋊C4—173no—
430C2×C4⋊C4C4×D1713yes—
431C2×C42—173no—
432C2×C42—173no—
433C2×C42C22×D1713yes—
434C4⋊C8—173no—
435C4⋊C8—176no—
436C4⋊C8—173no—
437C4⋊C8—171no—
438C4⋊C8—173no—
439C4⋊C8—173no—
440C4⋊C8—173no—
441C4⋊C8—173no—
442C4⋊C8—173no—
443C4⋊C8C22×D1713yes—
444C4⋊C8C22×D1713yes—
445C4⋊C8C22×D1716yes—
446C22⋊C8—171no—
447C22⋊C8—173no—
448C22⋊C8—171no—
449C22⋊C8—173no—
450C22⋊C8—176no—
451C22⋊C8—173no—
452C22⋊C8—173no—
453#97—173no—
454#97—173no—
455#97C22×D1713yes—
456#97C22×D1713yes—
457C4×C8—171no—
458C4×C8—173no—
459C4×C8—176no—
460C4×C8—173no—
461C4×C8—173no—
462C4×C8—176no—
463C4×C8C22×D1713yes—
464C4×C8C22×D1713yes—
465C17#6034511yescharacteristic, complemented, Sylow
466#41—344no—
467#41—686no—
468C2×Q8—24no—
469C2×Q8—173no—
470C2×Q8—173no—
471C2×Q8—344no—
472C2×Q8C2×C4×D1713yes—
473C2×Q8C2×C4×D1711yescharacteristic
474C2×D4—171no—
475C2×D4—173no—
476C2×D4—173no—
477C2×D4—173no—
478C22×C4—173no—
479C22×C4—173no—
480C22×C4—173no—
481C22×C4—171no—
482C22×C4—173no—
483C22×C4—173no—
484C22×C4C2×C4×D1713yes—
485C22×C4C2×C4×D1713yes—
486C22×C4C23×D1711yescharacteristic
487M16—22no—
488M16—43no—
489M16—342no—
490M16—683no—
491M16—686no—
492C2×C8—173no—
493C2×C8—171no—
494C2×C8—171no—
495C2×C8—173no—
496C2×C8—346no—
497C2×C8—683no—
498C2×C8C23×D1713yes—
499C2×C8C23×D1713yes—
500C2×C8C23×D1711yescharacteristic
501C2×C8C23×D1711yescharacteristic
502C4⋊C4—173no—
503C4⋊C4—176no—
504C4⋊C4—171no—
505C4⋊C4—173no—
506C4⋊C4—173no—
507C4⋊C4—173no—
508C4⋊C4—173no—
509C4⋊C4—173no—
510C4⋊C4—173no—
511C4⋊C4—176no—
512C4⋊C4—176no—
513C4⋊C4C2×C4×D1713yes—
514C4⋊C4C2×C4×D1713yes—
515C4⋊C4C2×C4×D1716yes—
516C22⋊C4—171no—
517C22⋊C4—173no—
518C22⋊C4—171no—
519C22⋊C4—173no—
520C22⋊C4—176no—
521C22⋊C4—173no—
522C22⋊C4—173no—
523C22⋊C4—176no—
524C22⋊C4—176no—
525C42—173no—
526C42—173no—
527C42—173no—
528C42—173no—
529C42—171no—
530C42—173no—
531C42C2×C4×D1713yes—
532C42C2×C4×D1713yes—
533C42C23×D1713yes—
534C42C23×D1713yes—
535C23—173no—
536C23—171no—
537C23C22×C4×D1711yescharacteristic
538Q8—26no—
539Q8—346no—
540Q8—346no—
541Q8#1058759314yes—
542D4—342no—
543D4—346no—
544D4—683no—
545D4—683no—
546C2×C4—26no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—43no—
550C2×C4—171no—
551C2×C4—171no—
552C2×C4—171no—
553C2×C4—173no—
554C2×C4—173no—
555C2×C4—173no—
556C2×C4—346no—
557C2×C4—346no—
558C2×C4—346no—
559C2×C4—343no—
560C2×C4—341no—
561C2×C4—343no—
562C2×C4—341no—
563C2×C4—683no—
564C2×C4C22×C4×D1713yes—
565C2×C4C22×C4×D1711yescharacteristic
566C2×C4C22×C4×D1713yes—
567C2×C4C22×C4×D1713yes—
568C2×C4C22×C4×D1713yes—
569C2×C4C24×D1711yescharacteristic
570C8—21no—
571C8—23no—
572C8—23no—
573C8—343no—
574C8—341no—
575C8—341no—
576C8—343no—
577C8#1058769412yes—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—341no—
582C22—343no—
583C22—343no—
584C22—341no—
585C22—343no—
586C22—341no—
587C22C23×C4×D1711yescharacteristic
588C4—23no—
589C4—23no—
590C4—23no—
591C4—21no—
592C4—172no—
593C4—341no—
594C4—343no—
595C4—343no—
596C4—341no—
597C4—343no—
598C4C2×#1058759316yes—
599C4C2×#1058769411yescharacteristic
600C4#1176364511yescharacteristic
601C2—41no—
602C2—681no—
603C2—683no—
604C2C4×#1058769411yescharacteristic
605C2C22×#1058759311yescharacteristic
606C2D17×#56811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D17
commutator, socleC22×C17C24×C4
FittingQ8×M16×C17C2
prime radical#14542401C2
cosocleC2×C4×C17C25
hypercenterQ8×M16D17
hypocenterC17#60345
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #14537936C2semidirect
3. #14537929C2semidirect
4. #14537915C2semidirect
5. #14537914C2semidirect
6. #14537882C2semidirect
7. #14537865C2semidirect
8. #14537863C2semidirect
9. #14537790C2semidirect
10. #14537788C2semidirect
11. #14537746C2semidirect
12. C8×#63256C2semidirect
13. Q8×#63218C2semidirect
14. #14537715C2semidirect
15. Q8×M16×C17C2semidirect
288. Q8×M16D17semidirect
465. C17#60345semidirect

Maximal subgroups

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

subgroupindex
#145424012
#145379362
#145379292
#145379152
#145379142
#145378822
#145378652
#145378632
#145377902
#145377882
#145377462
C8×#632562
Q8×#632182
#145377152
Q8×M16×C172
#6034517