Finite Group Databasebeta

← #17005222

Subgroups — #17005222

Order 4352.

680 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538790C211yesmaximal, characteristic
2#14537732C211yesmaximal, characteristic, complemented
3D4×#63218C211yesmaximal, characteristic
4#14527674C211yesmaximal, characteristic, complemented
5#14527673C211yesmaximal, characteristic, complemented
6#14527629C211yesmaximal, characteristic, complemented
7#14527624C211yesmaximal, characteristic, complemented
8#14527617C211yesmaximal, characteristic, complemented
9SD16×#3622C211yesmaximal, characteristic, complemented
10C8×#63229C211yesmaximal, characteristic, complemented
11#14527553C211yesmaximal, characteristic, complemented
12#14527237C211yesmaximal, characteristic, complemented
13D17×D4⋊C8C211yesmaximal, characteristic
14#14527063C211yesmaximal, characteristic, complemented
15C17×#1582C211yesmaximal, characteristic, complemented
16C22×#63218—21no—
17#11763363C2211yescharacteristic
18C4×D4×D17C2211yescharacteristic
19#11763141C411yescharacteristic
20#11763138C411yescharacteristic
21#11763098C411yescharacteristic
22#11763097C411yescharacteristic
23#11763013C411yescharacteristic
24#11763012C411yescharacteristic
25#11762961C411yescharacteristic
26#11762958C411yescharacteristic
27#11762937—21no—
28#11762936—21no—
29#11762936C2211yescharacteristic, complemented
30Q8×#3622C2211yescharacteristic, complemented
31C8×Dic34C2211yescharacteristic, complemented
32C4×#63219C2211yescharacteristic
33C4×#63230C2211yescharacteristic
34C4×#63229C2211yescharacteristic
35#11762888C2211yescharacteristic
36#11762886C2211yescharacteristic
37#11762880—21no—
38#11762877—21no—
39D4×#3622C2211yescharacteristic
40#11762860C2211yescharacteristic
41#11762834—21no—
42#11762831C2211yescharacteristic
43C4⋊C8×D17C2211yescharacteristic
44#11762771—21no—
45#11762766—21no—
46C4×#63218C2211yescharacteristic
47C4×C8×D17C2211yescharacteristic
48#11762560C2211yescharacteristic, complemented
49#11762559C2211yescharacteristic, complemented
50#11762559C2211yescharacteristic, complemented
51#11762541C2211yescharacteristic, complemented
52Q8⋊#3622C2211yescharacteristic, complemented
53#11762539C2211yescharacteristic, complemented
54#11762538C2211yescharacteristic
55D4⋊#3622C2211yescharacteristic
56#11762536C2211yescharacteristic
57#11762531C2211yescharacteristic, complemented
58C8×#3622C2211yescharacteristic, complemented
59C8×#3622C2211yescharacteristic, complemented
60C17×#446C2211yescharacteristic, complemented
61C4×SD16×C17C2211yescharacteristic
62C8×D4×C17C2211yescharacteristic
63C17×#334C2211yescharacteristic, 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#10587632C2×C411yescharacteristic
74#10587630C2×C411yescharacteristic
75C2×#63230C2×C411yescharacteristic
76D4×Dic17C2×C411yescharacteristic
77C2×#63229C2×C411yescharacteristic
78C4×C22⋊D17—21no—
79C22×#3622—21no—
80C2×#63219C2×C411yescharacteristic
81C2×#63218—21no—
82C2×#63218—21no—
83C2×#63218—21no—
84C2×#63218—44no—
85C2×#63218D411yescharacteristic
86C2×#63218D411yescharacteristic
87C2×C8×D17—21no—
88C2×C8×D17—21no—
89#10587583C2×C411yescharacteristic
90#10587582—21no—
91#10587579C2×C411yescharacteristic
92C4⋊C4×D17C2×C411yescharacteristic
93#10587575—21no—
94#10587574C2×C411yescharacteristic
95#10587567—21no—
96C22⋊C4×D17—21no—
97C4×D68C2311yescharacteristic
98#10587558—21no—
99C42×D17C2311yescharacteristic
100C4×Dic34—21no—
101C4×Dic34C2311yescharacteristic
102C17×#106C2×C411yescharacteristic
103C17×C4⋊C8—21no—
104C17×C4⋊C8C2311yescharacteristic
105C17×C4⋊C8C2311yescharacteristic
106C17×#103C2×C411yescharacteristic
107C17×#102C2×C411yescharacteristic
108C17×C22⋊C8—21no—
109C17×#97—21no—
110C4×C8×C17C2311yescharacteristic
111C4×C8×C17C2311yescharacteristic
112C4×C8×C17C2311yescharacteristic
113#10587506C2×C411yescharacteristic
114#10587503C2×C411yescharacteristic
115C22⋊#3622—21no—
116#10587492C2×C411yescharacteristic
117#10587491—21no—
118#10587491—21no—
119#10587491C2×C411yescharacteristic
120#10587491C2×C411yescharacteristic
121#10587488C2×C411yescharacteristic
122#10587487C2×C411yescharacteristic
123#10587486—21no—
124#10587486D411yescharacteristic
125#10587486D411yescharacteristic
126#10587485—21no—
127#10587485—21no—
128#10587485—21no—
129#10587485—21no—
130#10587485C2×C411yescharacteristic
131#10587485C2×C411yescharacteristic
132C8×Dic17—21no—
133C8×Dic17—21no—
134#10587481C2×C411yescharacteristic
135#10587480C2×C411yescharacteristic
136#10587479C2×C411yescharacteristic
137#10587479C2×C411yescharacteristic
138C4⋊#3622—21no—
139C4⋊#3622C2311yescharacteristic
140C4⋊#3622C2311yescharacteristic
141C4×#3622—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×D68C22×C411yescharacteristic
154C2×C4×D17—21no—
155C2×C4×D17—21no—
156C2×C4×D17—21no—
157C2×C4×D17—42no—
158C2×C4×D17—42no—
159C2×C4×D17C22×C411yescharacteristic
160C2×C4×D17C2×D411yescharacteristic
161C2×C4×D17#4111yescharacteristic
162C2×Dic34—21no—
163C2×Dic34C22×C411yescharacteristic
164SD16×C17—22no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—42no—
169C2×C8×C17C22×C411yescharacteristic
170C2×C8×C17C22×C411yescharacteristic
171C2×C8×C17C2×D411yescharacteristic
172C2×C8×C17C2×D411yescharacteristic
173C4⋊C4×C17—21no—
174C4⋊C4×C17C22×C411yescharacteristic
175C4⋊C4×C17C22×C411yescharacteristic
176C22⋊C4×C17—21no—
177C42×C17—21no—
178C42×C17C2411yescharacteristic
179C22⋊Dic17—21no—
180#63230—41no—
181#63229—22no—
182#63227—21no—
183#63227—21no—
184#63227—21no—
185#63227—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17C22×C411yescharacteristic
188C4⋊Dic17C22×C411yescharacteristic
189#63225—21no—
190#63225—21no—
191#63225—21no—
192#63225—21no—
193#63225—21no—
194#63225—21no—
195C4×Dic17—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
207#63219—41no—
208#63218—22no—
209#63218—42no—
210#63218—42no—
211#63218—42no—
212#63218SD1614yescomplemented
213C8×D17—42no—
214C8×D17—42no—
215#17995—171nomaximal, Sylow
216C23×C17—21no—
217C22×D17—21no—
218C22×D17—42no—
219C22×D17—42no—
220C22×D17—81no—
221C22×D17C4×D411yescharacteristic
222Q8×C17—21no—
223Q8×C17—21no—
224D4×C17—21no—
225D4×C17#13112yes—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17C23×C411yescharacteristic
232C2×C4×C17C22×D411yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C22⋊D17—42no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—42no—
239C2×Dic17C4×D411yescharacteristic
240C2×Dic17C4×D411yescharacteristic
241C2×Dic17C4×D411yescharacteristic
242D68—21no—
243D68—21no—
244C4×D17—21no—
245C4×D17—21no—
246C4×D17—41no—
247C4×D17—42no—
248C4×D17—41no—
249C4×D17—41no—
250C4×D17—42no—
251C4×D17C2×SD1612yes—
252C4×D17#13512yes—
253Dic34—21no—
254Dic34—21no—
255Dic34—81no—
256C8×C17—21no—
257C8×C17—21no—
258C8×C17—21no—
259C8×C17—41no—
260C8×C17—41no—
261C8×C17—41no—
262C8×C17C2×SD1612yes—
263#3622—22no—
264#3622—21no—
265#3622—21no—
266#3622—21no—
267#3622—41no—
268#3622—41no—
269#3622C2×SD1612yes—
270#3101—171no—
271#2962—171no—
272D4×M16—171no—
273#1590—171no—
274#1590—171no—
275#1583—171no—
276#1583—171no—
277#1582—171no—
278#1582D1711yescharacteristic, complemented
279C8×SD16—171no—
280C8×SD16—171no—
281#1570—171no—
282#1493—171no—
283C2×D4⋊C8—171no—
284C8×M16—171no—
285C22×C17—22no—
286C22×C17—41no—
287C22×C17—41no—
288C22×C17C2×C4×D411yescharacteristic
289C2×D17—21no—
290C2×D17—21no—
291C2×D17—41no—
292C2×D17—41no—
293C2×D17—41no—
294C2×D17—84no—
295C2×D17C4×SD1612yes—
296C4×C17—21no—
297C4×C17—41no—
298C4×C17—41no—
299C4×C17—41no—
300C4×C17C2×#13111yescharacteristic
301C4×C17#56811yescharacteristic
302C4×C17C22×SD1611yescharacteristic
303C4×C17C2×#13511yescharacteristic
304Dic17—21no—
305Dic17—21no—
306Dic17—41no—
307Dic17—41no—
308Dic17—41no—
309Dic17C4×SD1612yes—
310C22×M16—341no—
311#533—171no—
312C2×C4×D4—171no—
313#483—171no—
314#483—171no—
315#476—171no—
316#476—171no—
317#461—171no—
318#461—171no—
319#450—171no—
320#450—171no—
321#446—171no—
322#446—341no—
323#446—341no—
324#446C2×D1711yescharacteristic, complemented
325C8×Q8—171no—
326C8×Q8—171no—
327C4×SD16—171no—
328C4×SD16—171no—
329C4×SD16—171no—
330C4×SD16C2×D1711yescharacteristic
331#436—171no—
332#436—171no—
333#435—341no—
334#435—341no—
335C8×D4—171no—
336C8×D4C2×D1711yescharacteristic
337#431—171no—
338#424—341no—
339#423—171no—
340C2×C4⋊C8—171no—
341#407—341no—
342#405—341no—
343C4×M16—171no—
344C2×C4×C8—171no—
345#334—171no—
346#334—171no—
347#334—171no—
348#334C2×D1711yescharacteristic, complemented
349Q8⋊C8—171no—
350Q8⋊C8—171no—
351Q8⋊C8—171no—
352Q8⋊C8C2×D1711yescharacteristic, complemented
353D4⋊C8—171no—
354D4⋊C8—171no—
355D4⋊C8—171no—
356D4⋊C8C2×D1711yescharacteristic
357#322—171no—
358#322C2×D1711yescharacteristic, complemented
359C82—171no—
360C82—171no—
361C2×C17—42no—
362C2×C17#293311yescharacteristic
363C2×C17C2×C4×SD1611yescharacteristic
364C2×C17#295611yescharacteristic
365D17—22no—
366D17—81no—
367C22×D4—171no—
368C23×C4—341no—
369C2×SD16—171no—
370C2×SD16—171no—
371C2×SD16—171no—
372C2×SD16C4×D1711yescharacteristic
373C2×M16—171no—
374C2×M16—171no—
375C2×M16—341no—
376C2×M16—341no—
377C2×M16—341no—
378C2×M16—684no—
379C22×C8—21no—
380C22×C8—341no—
381C22×C8—341no—
382C22×C8—341no—
383C4⋊Q8—171no—
384C4⋊Q8—171no—
385#125—341no—
386C22⋊Q8—171no—
387C22⋊Q8—171no—
388C22⋊Q8—341no—
389C4×Q8—171no—
390C4×Q8—341no—
391C4×Q8C22×D1711yescharacteristic
392C4×D4—171no—
393C4×D4—171no—
394C4×D4—171no—
395C4×D4—341no—
396C4×D4—341no—
397C4×D4C22×D1711yescharacteristic
398#117—341no—
399C2×C4⋊C4—171no—
400C2×C22⋊C4—341no—
401C2×C42—171no—
402#106—171no—
403#106—171no—
404#106—171no—
405#106C4×D1711yescharacteristic
406C4⋊C8—21no—
407C4⋊C8—171no—
408C4⋊C8—171no—
409C4⋊C8—171no—
410C4⋊C8—171no—
411C4⋊C8—341no—
412C4⋊C8—341no—
413C4⋊C8—341no—
414C4⋊C8—341no—
415C4⋊C8—341no—
416C4⋊C8C22×D1711yescharacteristic
417C4⋊C8C22×D1711yescharacteristic
418#103—171no—
419#103—171no—
420#103—171no—
421#103C4×D1711yescharacteristic
422#102—171no—
423#102—171no—
424#102—171no—
425#102C4×D1711yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—171no—
428C22⋊C8—171no—
429C22⋊C8—341no—
430C22⋊C8—341no—
431C22⋊C8—341no—
432#97—21no—
433#97—171no—
434#97—171no—
435#97—341no—
436C4×C8—171no—
437C4×C8—171no—
438C4×C8—171no—
439C4×C8—341no—
440C4×C8—341no—
441C4×C8—341no—
442C4×C8C22×D1711yescharacteristic
443C4×C8C22×D1711yescharacteristic
444C4×C8C22×D1711yescharacteristic
445C17#1799511yescharacteristic, complemented, Sylow
446C24—341no—
447C2×Q8—171no—
448C2×Q8—341no—
449C2×Q8C2×C4×D1711yescharacteristic
450C2×D4—171no—
451C2×D4—341no—
452C2×D4—342no—
453C2×D4—681no—
454C2×D4C2×C4×D1711yescharacteristic
455C22×C4—21no—
456C22×C4—171no—
457C22×C4—171no—
458C22×C4—171no—
459C22×C4—341no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—682no—
464C22×C4—682no—
465SD16—22no—
466SD16—342no—
467SD16—681no—
468SD16—681no—
469M16—174no—
470M16—342no—
471M16—682no—
472M16—682no—
473M16—682no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—42no—
478C2×C8—171no—
479C2×C8—171no—
480C2×C8—171no—
481C2×C8—171no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—341no—
485C2×C8—682no—
486C2×C8—682no—
487C2×C8—682no—
488C2×C8C2×C4×D1711yescharacteristic
489C2×C8C2×C4×D1711yescharacteristic
490C2×C8D4×D1711yescharacteristic
491C2×C8D4×D1711yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—171no—
494C4⋊C4—171no—
495C4⋊C4—341no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4C2×C4×D1711yescharacteristic
503C4⋊C4C2×C4×D1711yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—341no—
506C22⋊C4—341no—
507C22⋊C4—341no—
508C22⋊C4—341no—
509C22⋊C4—341no—
510C42—21no—
511C42—171no—
512C42—171no—
513C42—171no—
514C42—341no—
515C42C23×D1711yescharacteristic
516C23—21no—
517C23—171no—
518C23—341no—
519C23—682no—
520C23—682no—
521C23—1361no—
522Q8—21no—
523Q8—21no—
524Q8—341no—
525Q8—341no—
526Q8—1361no—
527D4—21no—
528D4—341no—
529D4—341no—
530D4—682no—
531D4#1058759312yes—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—171no—
538C2×C4—172no—
539C2×C4—172no—
540C2×C4—171no—
541C2×C4—171no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—681no—
548C2×C4—682no—
549C2×C4—682no—
550C2×C4—682no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4C22×C4×D1711yescharacteristic
554C2×C4C2×D4×D1711yescharacteristic
555C2×C4#41×D1711yescharacteristic
556C8—21no—
557C8—21no—
558C8—21no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—172no—
563C8—342no—
564C8—341no—
565C8—341no—
566C8—341no—
567C8—681no—
568C8—681no—
569C8SD16×D1712yes—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—172no—
574C22—341no—
575C22—341no—
576C22—681no—
577C22—681no—
578C22—681no—
579C22—1364no—
580C22C4×D4×D1711yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—172no—
586C4—341no—
587C4—341no—
588C4—681no—
589C4—681no—
590C4—681no—
591C4C2×#1058759311yescharacteristic
592C4#1176357411yescharacteristic
593C4C2×SD16×D1711yescharacteristic
594C4#135×D1711yescharacteristic
595C2—42no—
596C2—342no—
597C2—1361no—
598C2#1453751011yescharacteristic
599C2C4×SD16×D1711yescharacteristic
600C2#1453769211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1582C2
socleC22×C17C2×C4×D4
prime radicalC2×D4×D17C2×C4
cosocleC42×C17C24
hypercenter#1582D17
hypocenterC17#17995
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1582

Complements

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

NH ≅ G/Ntype
2. #14537732C2semidirect
4. #14527674C2semidirect
5. #14527673C2semidirect
6. #14527629C2semidirect
7. #14527624C2semidirect
8. #14527617C2semidirect
9. SD16×#3622C2semidirect
10. C8×#63229C2semidirect
11. #14527553C2semidirect
12. #14527237C2semidirect
14. #14527063C2semidirect
15. C17×#1582C2semidirect
29. #11762936C22semidirect
30. Q8×#3622C22semidirect
31. C8×Dic34C22semidirect
48. #11762560C22semidirect
49. #11762559C22semidirect
50. #11762559C22semidirect
51. #11762541C22semidirect
52. Q8⋊#3622C22semidirect
53. #11762539C22semidirect
57. #11762531C22semidirect
58. C8×#3622C22semidirect
59. C8×#3622C22semidirect
60. C17×#446C22semidirect
63. C17×#334C22semidirect
64. C17×Q8⋊C8C22semidirect
66. C17×#322C22semidirect
212. #63218SD16semidirect
278. #1582D17semidirect
324. #446C2×D17semidirect
348. #334C2×D17semidirect
352. Q8⋊C8C2×D17semidirect
358. #322C2×D17semidirect
445. C17#17995semidirect

Maximal subgroups

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

subgroupindex
#145387902
#145377322
D4×#632182
#145276742
#145276732
#145276292
#145276242
#145276172
SD16×#36222
C8×#632292
#145275532
#145272372
D17×D4⋊C82
#145270632
C17×#15822
#1799517