Finite Group Databasebeta

← #17005355

Subgroups — #17005355

Order 4352.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538835C211yesmaximal, characteristic
2#14537787C211yesmaximal, characteristic, complemented
3D4×#63218C211yesmaximal, characteristic, complemented
4#14527699C211yesmaximal, characteristic, complemented
5#14527671C211yesmaximal, characteristic, complemented
6#14527662C211yesmaximal, characteristic, complemented
7#14527648C211yesmaximal, characteristic, complemented
8#14527621C211yesmaximal, characteristic, complemented
9#14527612C211yesmaximal, characteristic, complemented
10#14527576C211yesmaximal, characteristic, complemented
11#14527574C211yesmaximal, characteristic, complemented
12#14527248C211yesmaximal, characteristic, complemented
13#14527173C211yesmaximal, characteristic, complemented
14#14527080C211yesmaximal, characteristic, complemented
15C17×SD16⋊C8C211yesmaximal, characteristic, complemented
16C22×#63218—21no—
17#11763346C2211yescharacteristic
18C4×D4×D17C2211yescharacteristic
19#11763143C411yescharacteristic
20#11763129C411yescharacteristic
21#11763108C411yescharacteristic
22#11763107C411yescharacteristic
23#11763027C411yescharacteristic
24#11763026C411yescharacteristic
25#11762958C411yescharacteristic
26#11762944C411yescharacteristic
27#11762940C2211yescharacteristic, complemented
28C4×#63230C2211yescharacteristic
29C4×D136C2211yescharacteristic
30C4×#63228C2211yescharacteristic
31#11762888—21no—
32#11762888C2211yescharacteristic
33#11762888C2211yescharacteristic, complemented
34#11762886C2211yescharacteristic
35#11762880—21no—
36#11762879—21no—
37#11762877—21no—
38D4×#3622C2211yescharacteristic
39#11762874C2211yescharacteristic, complemented
40#11762869—21no—
41#11762861C2211yescharacteristic
42#11762860C2211yescharacteristic
43#11762860C2211yescharacteristic
44#11762831C2211yescharacteristic
45#11762771—21no—
46#11762763—21no—
47C4×#63218C2211yescharacteristic
48#11762561C2211yescharacteristic, complemented
49#11762560C2211yescharacteristic, complemented
50#11762559C2211yescharacteristic, complemented
51Q8⋊#3622C2211yescharacteristic, complemented
52#11762538C2211yescharacteristic
53#11762538C2211yescharacteristic, complemented
54D4⋊#3622C2211yescharacteristic
55#11762536C2211yescharacteristic
56#11762536C2211yescharacteristic, complemented
57#11762531C2211yescharacteristic, complemented
58#11762531C2211yescharacteristic, complemented
59C8×#3622C2211yescharacteristic, complemented
60C8×Q8×C17C2211yescharacteristic, complemented
61C4×SD16×C17C2211yescharacteristic
62C8×D4×C17C2211yescharacteristic
63C17×#335C2211yescharacteristic, complemented
64C17×Q8⋊C8C2211yescharacteristic, complemented
65C17×D4⋊C8C2211yescharacteristic
66C17×#322C2211yescharacteristic, complemented
67C2×D4×D17C2×C411yescharacteristic
68C22×C4×D17—21no—
69C2×SD16×C17C2×C411yescharacteristic
70C22×C8×C17—21no—
71C4×Q8×C17C2311yescharacteristic
72C4×D4×C17C2311yescharacteristic
73#10587633C2×C411yescharacteristic
74#10587632C2×C411yescharacteristic
75C2×#63230C2×C411yescharacteristic
76D4×Dic17C2×C411yescharacteristic
77C2×#63228C2×C411yescharacteristic
78C4×C22⋊D17—21no—
79C22×#3622—21no—
80C2×D136C2×C411yescharacteristic
81C2×#63218—21no—
82C2×#63218—21no—
83C2×#63218—21no—
84C2×#63218—21no—
85C2×#63218—44no—
86C2×#63218D411yescharacteristic
87C2×#63218D411yescharacteristic
88C2×C8×D17—21no—
89#10587584—21no—
90#10587581C2×C411yescharacteristic
91#10587580—21no—
92#10587579C2×C411yescharacteristic
93C4⋊C4×D17C2×C411yescharacteristic
94#10587576C2×C411yescharacteristic
95#10587567—21no—
96C22⋊C4×D17—21no—
97C4×D68—21no—
98C4×D68C2311yescharacteristic
99C4×D68C2311yescharacteristic
100#10587558—21no—
101C42×D17C2311yescharacteristic
102C17×#106C2×C411yescharacteristic
103C17×C4⋊C8—21no—
104C17×C4⋊C8C2311yescharacteristic
105C17×C4⋊C8C2311yescharacteristic
106C17×#103C2×C411yescharacteristic
107C17×#102C2×C411yescharacteristic
108C17×C22⋊C8—21no—
109C4×C8×C17—21no—
110C4×C8×C17C2311yescharacteristic
111C4×C8×C17C2311yescharacteristic
112C4×C8×C17C2311yescharacteristic
113#10587506C2×C411yescharacteristic
114#10587503C2×C411yescharacteristic
115C22⋊#3622—21no—
116#10587492C2×C411yescharacteristic
117#10587492C2×C411yescharacteristic
118#10587491—21no—
119#10587491—21no—
120#10587491—21no—
121#10587491—21no—
122#10587491—21no—
123#10587491—21no—
124#10587491C2×C411yescharacteristic
125#10587491C2×C411yescharacteristic
126#10587489C2×C411yescharacteristic
127#10587486D411yescharacteristic
128#10587486D411yescharacteristic
129#10587485—21no—
130#10587485—21no—
131#10587485C2×C411yescharacteristic
132#10587485C2×C411yescharacteristic
133C8×Dic17—21no—
134#10587480C2×C411yescharacteristic
135#10587480C2×C411yescharacteristic
136#10587479C2×C411yescharacteristic
137#10587478C2×C411yescharacteristic
138C4⋊#3622—21no—
139C4⋊#3622C2311yescharacteristic
140C4⋊#3622C2311yescharacteristic
141#10587474—21no—
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C4×#3622C2311yescharacteristic
145C23×D17—21no—
146C2×Q8×C17C22×C411yescharacteristic
147C2×D4×C17C22×C411yescharacteristic
148C22×C4×C17—21no—
149C2×C22⋊D17—21no—
150C22×Dic17—21no—
151D4×D17—22no—
152D4×D17—41no—
153C2×D68—21no—
154C2×D68C22×C411yescharacteristic
155C2×D68C22×C411yescharacteristic
156C2×C4×D17—21no—
157C2×C4×D17—21no—
158C2×C4×D17—21no—
159C2×C4×D17—21no—
160C2×C4×D17—42no—
161C2×C4×D17—42no—
162C2×C4×D17C22×C411yescharacteristic
163C2×C4×D17C2×D411yescharacteristic
164C2×C4×D17#4111yescharacteristic
165SD16×C17—22no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17—42no—
170C2×C8×C17C22×C411yescharacteristic
171C2×C8×C17C22×C411yescharacteristic
172C2×C8×C17C2×D411yescharacteristic
173C2×C8×C17C2×D411yescharacteristic
174C4⋊C4×C17—21no—
175C4⋊C4×C17C22×C411yescharacteristic
176C4⋊C4×C17C22×C411yescharacteristic
177C22⋊C4×C17—21no—
178C42×C17—21no—
179C42×C17C2411yescharacteristic
180C22⋊Dic17—21no—
181#63230—41no—
182#63228—22no—
183#63227—21no—
184#63227—21no—
185#63227—21no—
186#63227—21no—
187#63227—21no—
188#63227—21no—
189#63227—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17C22×C411yescharacteristic
192C4⋊Dic17C22×C411yescharacteristic
193#63225—21no—
194#63225—21no—
195#63225—21no—
196C4×Dic17C22×C411yescharacteristic
197C4×Dic17C2×D411yescharacteristic
198C4×Dic17#4111yescharacteristic
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—42no—
203C2×#3622C22×C411yescharacteristic
204C2×#3622C22×C411yescharacteristic
205C2×#3622C2×D411yescharacteristic
206C2×#3622C2×D411yescharacteristic
207D136—41no—
208#63218—22no—
209#63218—41no—
210#63218—42no—
211#63218—42no—
212#63218—42no—
213#63218—42no—
214C8×D17—42no—
215#18006—171nomaximal, Sylow
216C23×C17—21no—
217C22×D17—21no—
218C22×D17—21no—
219C22×D17—42no—
220C22×D17—42no—
221C22×D17—81no—
222C22×D17C4×D411yescharacteristic
223Q8×C17—21no—
224Q8×C17—21no—
225D4×C17—21no—
226D4×C17#13112yes—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17C23×C411yescharacteristic
233C2×C4×C17C22×D411yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C22⋊D17—42no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—42no—
239C2×Dic17C4×D411yescharacteristic
240C2×Dic17C4×D411yescharacteristic
241C2×Dic17C4×D411yescharacteristic
242D68—21no—
243D68—21no—
244D68—21no—
245D68—21no—
246D68—81no—
247C4×D17—21no—
248C4×D17—21no—
249C4×D17—21no—
250C4×D17—21no—
251C4×D17—41no—
252C4×D17—42no—
253C4×D17—42no—
254C4×D17—41no—
255C4×D17—41no—
256C4×D17—41no—
257C4×D17—41no—
258C8×C17—21no—
259C8×C17—21no—
260C8×C17—21no—
261C8×C17—21no—
262C8×C17—41no—
263C8×C17—41no—
264C8×C17—41no—
265#3622—21no—
266#3622—21no—
267#3622—21no—
268#3622—41no—
269#3622—41no—
270#3622—41no—
271#3622AGL(1,ℤ/8)12yes—
272#3105—171no—
273#2969—171no—
274D4×M16—171no—
275#1594—171no—
276#1590—171no—
277#1588—171no—
278#1586—171no—
279#1582—171no—
280#1581—171no—
281SD16⋊C8D1711yescharacteristic, complemented
282C8×D8—171no—
283#1574—171no—
284#1495—171no—
285#1478—171no—
286#1453—171no—
287C22×C17—22no—
288C22×C17—41no—
289C22×C17—41no—
290C22×C17C2×C4×D411yescharacteristic
291C2×D17—21no—
292C2×D17—21no—
293C2×D17—21no—
294C2×D17—41no—
295C2×D17—41no—
296C2×D17—41no—
297C2×D17—41no—
298C2×D17—41no—
299C2×D17—41no—
300C2×D17—84no—
301C4×C17—21no—
302C4×C17—41no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17C2×#13111yescharacteristic
306C4×C17#56811yescharacteristic
307C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
308C4×C17#57511yescharacteristic
309Dic17—21no—
310Dic17—21no—
311Dic17—21no—
312Dic17—41no—
313Dic17—41no—
314C22×M16—341no—
315#529—171no—
316C2×C4×D4—171no—
317#484—171no—
318#481—171no—
319#478—171no—
320#478—171no—
321#463—171no—
322#463—171no—
323#450—171no—
324#447—171no—
325#446—171no—
326C8×Q8C2×D1711yescharacteristic, complemented
327C4×SD16—171no—
328C4×SD16C2×D1711yescharacteristic
329C4×D8—171no—
330C4×D8—171no—
331#436—171no—
332#436—171no—
333#436—171no—
334#436—341no—
335#435—341no—
336#435—341no—
337#435—341no—
338C8×D4—171no—
339C8×D4—171no—
340C8×D4C2×D1711yescharacteristic
341#433—341no—
342#431—171no—
343#431—171no—
344#431—171no—
345#423—171no—
346#407—341no—
347#405—341no—
348C4×M16—171no—
349#335—171no—
350#335C2×D1711yescharacteristic, complemented
351#334—171no—
352#334—171no—
353Q8⋊C8—171no—
354Q8⋊C8C2×D1711yescharacteristic, complemented
355D4⋊C8—171no—
356D4⋊C8—171no—
357D4⋊C8—171no—
358D4⋊C8—171no—
359D4⋊C8—171no—
360D4⋊C8C2×D1711yescharacteristic
361#322—171no—
362#322—171no—
363#322C2×D1711yescharacteristic, complemented
364C82—171no—
365C2×C17—42no—
366C2×C17#293311yescharacteristic
367C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
368C2×C17#295711yescharacteristic
369D17—41no—
370D17—81no—
371D17—81no—
372C22×D4—171no—
373C23×C4—341no—
374C2×SD16—171no—
375C2×SD16C4×D1711yescharacteristic
376C2×D8—171no—
377C2×D8—171no—
378C2×M16—171no—
379C2×M16—171no—
380C2×M16—341no—
381C2×M16—341no—
382C2×M16—341no—
383C2×M16—341no—
384C2×M16—684no—
385C22×C8—21no—
386C22×C8—341no—
387C22×C8—341no—
388#126—341no—
389#125—171no—
390#124—171no—
391#123—341no—
392C22⋊Q8—171no—
393#121—171no—
394C4×Q8C22×D1711yescharacteristic
395C4×D4—171no—
396C4×D4—171no—
397C4×D4—171no—
398C4×D4—171no—
399C4×D4—341no—
400C4×D4—341no—
401C4×D4—341no—
402C4×D4C22×D1711yescharacteristic
403#117—341no—
404C2×C4⋊C4—171no—
405C2×C22⋊C4—341no—
406C2×C42—171no—
407#107—171no—
408#107—171no—
409#106—171no—
410#106C4×D1711yescharacteristic
411C4⋊C8—21no—
412C4⋊C8—171no—
413C4⋊C8—171no—
414C4⋊C8—171no—
415C4⋊C8—171no—
416C4⋊C8—341no—
417C4⋊C8—341no—
418C4⋊C8—341no—
419C4⋊C8C22×D1711yescharacteristic
420C4⋊C8C22×D1711yescharacteristic
421#103—171no—
422#103C4×D1711yescharacteristic
423#102—171no—
424#102—171no—
425#102—171no—
426#102—171no—
427#102—171no—
428#102C4×D1711yescharacteristic
429C22⋊C8—21no—
430C22⋊C8—171no—
431C22⋊C8—171no—
432C22⋊C8—341no—
433C22⋊C8—341no—
434C22⋊C8—341no—
435C22⋊C8—341no—
436C22⋊C8—341no—
437C22⋊C8—341no—
438C22⋊C8—341no—
439#97—171no—
440#97—171no—
441#97—341no—
442C4×C8—21no—
443C4×C8—171no—
444C4×C8—171no—
445C4×C8—171no—
446C4×C8—341no—
447C4×C8C22×D1711yescharacteristic
448C4×C8C22×D1711yescharacteristic
449C4×C8C22×D1711yescharacteristic
450C17#1800611yescharacteristic, complemented, Sylow
451C24—341no—
452C2×Q8C2×C4×D1711yescharacteristic
453C2×D4—171no—
454C2×D4—171no—
455C2×D4—341no—
456C2×D4—341no—
457C2×D4—342no—
458C2×D4—681no—
459C2×D4C2×C4×D1711yescharacteristic
460C22×C4—21no—
461C22×C4—171no—
462C22×C4—171no—
463C22×C4—171no—
464C22×C4—341no—
465C22×C4—341no—
466C22×C4—341no—
467C22×C4—341no—
468C22×C4—341no—
469C22×C4—682no—
470C22×C4—682no—
471SD16—22no—
472SD16—681no—
473D8—342no—
474D8—681no—
475M16—342no—
476M16—681no—
477M16—682no—
478M16—682no—
479M16—682no—
480M16—682no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—42no—
485C2×C8—171no—
486C2×C8—171no—
487C2×C8—171no—
488C2×C8—171no—
489C2×C8—341no—
490C2×C8—341no—
491C2×C8—341no—
492C2×C8—682no—
493C2×C8—682no—
494C2×C8C2×C4×D1711yescharacteristic
495C2×C8C2×C4×D1711yescharacteristic
496C2×C8D4×D1711yescharacteristic
497C2×C8D4×D1711yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—171no—
500C4⋊C4—171no—
501C4⋊C4—341no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4C2×C4×D1711yescharacteristic
506C4⋊C4C2×C4×D1711yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—341no—
509C22⋊C4—341no—
510C22⋊C4—341no—
511C22⋊C4—341no—
512C22⋊C4—341no—
513C22⋊C4—341no—
514C22⋊C4—341no—
515C22⋊C4—341no—
516C42—21no—
517C42—171no—
518C42—171no—
519C42—171no—
520C42C23×D1711yescharacteristic
521C23—21no—
522C23—171no—
523C23—341no—
524C23—341no—
525C23—682no—
526C23—682no—
527C23—1361no—
528Q8—21no—
529Q8—21no—
530D4—21no—
531D4—341no—
532D4—341no—
533D4—341no—
534D4—341no—
535D4—682no—
536D4—1361no—
537D4#1058758712yes—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—171no—
544C2×C4—171no—
545C2×C4—171no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—681no—
553C2×C4—682no—
554C2×C4—682no—
555C2×C4—682no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4C22×C4×D1711yescharacteristic
561C2×C4C2×D4×D1711yescharacteristic
562C2×C4#41×D1711yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—41no—
568C8—41no—
569C8—41no—
570C8—172no—
571C8—341no—
572C8—341no—
573C8—341no—
574C8—681no—
575C8—681no—
576C8—681no—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—341no—
581C22—341no—
582C22—341no—
583C22—681no—
584C22—681no—
585C22—681no—
586C22—681no—
587C22—681no—
588C22—681no—
589C22—1364no—
590C22C4×D4×D1711yescharacteristic
591C4—21no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—341no—
596C4—341no—
597C4—341no—
598C4—681no—
599C4—681no—
600C4C2×#1058758711yescharacteristic
601C4#1176357411yescharacteristic
602C4AGL(1,ℤ/8)×D1711yescharacteristic
603C4#1176360811yescharacteristic
604C2—42no—
605C2—681no—
606C2—1361no—
607C2—1361no—
608C2#1453750411yescharacteristic
609C2#1453758911yescharacteristic
610C2#1453769811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×SD16⋊C8C2
socleC22×C17C2×C4×D4
prime radical#11762944C4
cosocleC42×C17C24
hypercenterSD16⋊C8D17
hypocenterC17#18006
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
2. #14537787C2semidirect
3. D4×#63218C2semidirect
4. #14527699C2semidirect
5. #14527671C2semidirect
6. #14527662C2semidirect
7. #14527648C2semidirect
8. #14527621C2semidirect
9. #14527612C2semidirect
10. #14527576C2semidirect
11. #14527574C2semidirect
12. #14527248C2semidirect
13. #14527173C2semidirect
14. #14527080C2semidirect
15. C17×SD16⋊C8C2semidirect
27. #11762940C22semidirect
33. #11762888C22semidirect
39. #11762874C22semidirect
48. #11762561C22semidirect
49. #11762560C22semidirect
50. #11762559C22semidirect
51. Q8⋊#3622C22semidirect
53. #11762538C22semidirect
56. #11762536C22semidirect
57. #11762531C22semidirect
58. #11762531C22semidirect
59. C8×#3622C22semidirect
60. C8×Q8×C17C22semidirect
63. C17×#335C22semidirect
64. C17×Q8⋊C8C22semidirect
66. C17×#322C22semidirect
281. SD16⋊C8D17semidirect
326. C8×Q8C2×D17semidirect
350. #335C2×D17semidirect
354. Q8⋊C8C2×D17semidirect
363. #322C2×D17semidirect
450. C17#18006semidirect

Maximal subgroups

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

subgroupindex
#145388352
#145377872
D4×#632182
#145276992
#145276712
#145276622
#145276482
#145276212
#145276122
#145275762
#145275742
#145272482
#145271732
#145270802
C17×SD16⋊C82
#1800617