Finite Group Databasebeta

← #17190798

Subgroups — #17190798

Order 4352.

738 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763244C211yesmaximal, characteristic, complemented
2C22×#10587499C211yesmaximal, characteristic, complemented
3D4×D136C211yesmaximal, characteristic, complemented
4#14531051C211yesmaximal, characteristic, complemented
5#14530899C211yesmaximal, characteristic, complemented
6#14530868C211yesmaximal, characteristic, complemented
7#14530836C211yesmaximal, characteristic, complemented
8#14530824C211yesmaximal, characteristic, complemented
9#14530712C211yesmaximal, characteristic, complemented
10#14530707C211yesmaximal, characteristic, complemented
11#14530701C211yesmaximal, characteristic, complemented
12#14530695C211yesmaximal, characteristic, complemented
13#14530646C211yesmaximal, characteristic, complemented
14#14530610C211yesmaximal, characteristic, complemented
15C17×#3285C211yesmaximal, characteristic, complemented
16C22×D136C2211yescharacteristic
17C22×D136C2211yescharacteristic
18D4×D68—21no—
19#11763244—24no—
20C2×#10587499—24no—
21C2×#10587499C2211yescharacteristic
22C2×#10587499C2211yescharacteristic
23C2×#10587499C2211yescharacteristic, complemented
24C2×#10587499C2211yescharacteristic, complemented
25C2×#10587499C2211yescharacteristic, complemented
26C2×#10587499C2211yescharacteristic, complemented
27C2×#10587497C2211yescharacteristic
28C2×#10587497C2211yescharacteristic, complemented
29C2×#10587497C2211yescharacteristic, complemented
30C2×#10587483C2211yescharacteristic, complemented
31C22×#63214C2211yescharacteristic, complemented
32#11763186C2211yescharacteristic
33#11763045C2211yescharacteristic
34#11763045C2211yescharacteristic
35#11763005—21no—
36#11762945—21no—
37#11762945—21no—
38C4×D136C2211yescharacteristic
39#11762634C2211yescharacteristic, complemented
40C4⋊#63214C2211yescharacteristic, complemented
41#11762613C2211yescharacteristic, complemented
42#11762613C2211yescharacteristic, complemented
43#11762610C2211yescharacteristic, complemented
44#11762609C2211yescharacteristic
45#11762609C2211yescharacteristic, complemented
46#11762609C2211yescharacteristic, complemented
47C22⋊#63214C2211yescharacteristic, complemented
48C22⋊#63214C2211yescharacteristic, complemented
49#11762585C2211yescharacteristic, complemented
50C2×C17×#135C2211yescharacteristic, complemented
51C22×Q16×C17C2211yescharacteristic, complemented
52C17×#548—21no—
53C17×#495C2211yescharacteristic, complemented
54C17×#467C2211yescharacteristic, complemented
55C17×#466C2211yescharacteristic, complemented
56C17×#464—21no—
57C17×#452—21no—
58C17×#448—21no—
59C4×Q16×C17C2211yescharacteristic, complemented
60C8×D4×C17C2211yescharacteristic
61C2×#41×C17—21no—
62C22×Q8×C17—21no—
63C2×D4×D17—41no—
64C22×D68—21no—
65C22×D68—21no—
66C17×#135—22no—
67C17×#135—41no—
68C2×Q16×C17—22no—
69C2×Q16×C17—21no—
70C2×Q16×C17—41no—
71C2×Q16×C17C2311yescharacteristic
72C2×Q16×C17C2311yescharacteristic
73C2×SD16×C17—21no—
74C2×SD16×C17—21no—
75C2×D8×C17C2311yescharacteristic
76C22×C8×C17C2311yescharacteristic
77C22×C8×C17C2311yescharacteristic
78C17×#124—21no—
79C17×#124—41no—
80C17×C22⋊Q8—21no—
81C17×C22⋊Q8—41no—
82C17×#121—21no—
83C17×#121—41no—
84C4×Q8×C17—21no—
85C4×D4×C17—41no—
86C4×D4×C17D411yescharacteristic
87#10587616—21no—
88#10587616—21no—
89C2×D136—22no—
90C2×D136—22no—
91C2×D136—21no—
92C2×D136—21no—
93C2×D136—21no—
94C2×D136—41no—
95C2×D136—41no—
96C2×D136C2311yescharacteristic
97C2×D136C2311yescharacteristic
98C2×D136C2311yescharacteristic
99#10587581—41no—
100#10587568—41no—
101#10587568—41no—
102#10587560—21no—
103C4×D68—21no—
104C17×#107C2311yescharacteristic
105C17×C4⋊C8D411yescharacteristic
106C17×#103—21no—
107C17×#103—21no—
108C17×#102—21no—
109C17×C22⋊C8D411yescharacteristic
110C17×C22⋊C8D411yescharacteristic
111C4×C8×C17C2311yescharacteristic
112#10587499—22no—
113#10587499—22no—
114#10587499—22no—
115#10587499—22no—
116#10587499—41no—
117#10587499D414yescomplemented
118#10587497—22no—
119#10587497—22no—
120#10587497—41no—
121#10587492—21no—
122#10587492—21no—
123#10587492—21no—
124#10587489C2311yescharacteristic
125#10587483—22no—
126C2×#63214—22no—
127C2×#63214C2311yescharacteristic
128C2×#63214C2311yescharacteristic
129C2×#63214C2311yescharacteristic
130C2×#63214C2311yescharacteristic
131C23×D17—41no—
132C23×D17—41no—
133#41×C17—42no—
134#41×C17—81no—
135C2×Q8×C17—21no—
136C2×Q8×C17—21no—
137C2×Q8×C17—42no—
138C2×Q8×C17—41no—
139C2×Q8×C17—81no—
140C2×D4×C17—21no—
141C2×D4×C17—41no—
142C2×D4×C17—41no—
143C2×D4×C17D811yescharacteristic
144C22×C4×C17—41no—
145C22×C4×C17—41no—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C2×C22⋊D17—41no—
149C2×C22⋊D17—41no—
150D4×D17—84no—
151C2×D68—21no—
152C2×D68—21no—
153C2×D68—21no—
154C2×D68—42no—
155C2×D68—42no—
156C2×D68—41no—
157C2×D68—41no—
158C2×D68—41no—
159C2×D68—41no—
160C2×D68—81no—
161C2×D68—81no—
162C2×C4×D17—41no—
163Q16×C17—21no—
164Q16×C17—21no—
165Q16×C17—21no—
166Q16×C17—42no—
167Q16×C17C2×D412yescomplemented
168SD16×C17—42no—
169SD16×C17—81no—
170D8×C17—21no—
171D8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17C2×D411yescharacteristic
177C2×C8×C17C2×D411yescharacteristic
178C2×C8×C17C2×D411yescharacteristic
179C2×C8×C17C2411yescharacteristic
180C4⋊C4×C17—21no—
181C4⋊C4×C17—41no—
182C4⋊C4×C17—41no—
183C4⋊C4×C17D811yescharacteristic
184C22⋊C4×C17—41no—
185C22⋊C4×C17—41no—
186C22⋊C4×C17—41no—
187C22⋊C4×C17—41no—
188C22⋊C4×C17D811yescharacteristic
189C22⋊C4×C17D811yescharacteristic
190C42×C17—41no—
191C42×C17C2×D411yescharacteristic
192#63227—41no—
193#63227—41no—
194#63227—41no—
195C4⋊Dic17—21no—
196D136—21no—
197D136—21no—
198D136—21no—
199D136—21no—
200D136—21no—
201D136—42no—
202D136—42no—
203D136—81no—
204D136C2×D412yes—
205#63214—21no—
206#63214—21no—
207#63214—21no—
208#63214C2×D412yescomplemented
209#33831—171nomaximal, Sylow
210C23×C17—41no—
211C23×C17C2×D811yescharacteristic
212C23×C17C2×D811yescharacteristic
213C22×D17—41no—
214C22×D17—41no—
215C22×D17—41no—
216C22×D17—82no—
217C22×D17—82no—
218C22×D17—82no—
219C22×D17—82no—
220C22×D17—161no—
221C22×D17—161no—
222Q8×C17—22no—
223Q8×C17—41no—
224Q8×C17—41no—
225Q8×C17—41no—
226Q8×C17—82no—
227D4×C17—22no—
228D4×C17—41no—
229D4×C17—41no—
230D4×C17—82no—
231D4×C17—82no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—82no—
240C2×C4×C17—82no—
241C2×C4×C17C2×D811yescharacteristic
242C2×C4×C17C2×D811yescharacteristic
243C2×C4×C17C2×D811yescharacteristic
244C2×C4×C17C2×D811yescharacteristic
245C2×C4×C17C22×D411yescharacteristic
246C22⋊D17—82no—
247C22⋊D17—82no—
248C2×Dic17—41no—
249D68—22no—
250D68—41no—
251D68—41no—
252D68—41no—
253D68—41no—
254D68—41no—
255D68—82no—
256D68—82no—
257D68—82no—
258D68—161no—
259C4×D17—82no—
260C8×C17—21no—
261C8×C17—21no—
262C8×C17—21no—
263C8×C17C22×D411yescharacteristic
264C8×C17H(5,𝔽2)11yescharacteristic
265C2×#509—171no—
266C22×SD32—171no—
267#3285D1711yescharacteristic, complemented
268D4×D8—171no—
269#2250—171no—
270#2220—171no—
271#2215—171no—
272#2210—171no—
273#2207—171no—
274#2187—171no—
275#2186—171no—
276#2185—171no—
277#2184—171no—
278#2175—171no—
279#2168—171no—
280C22×C17—21no—
281C22×C17—21no—
282C22×C17—21no—
283C22×C17—21no—
284C22×C17—21no—
285C22×C17—81no—
286C22×C17—81no—
287C22×C17—81no—
288C22×C17#50912yescomplemented
289C22×C17C22×D811yescharacteristic
290C2×D17—42no—
291C2×D17—81no—
292C2×D17—81no—
293C2×D17—81no—
294C2×D17—81no—
295C2×D17—81no—
296C2×D17—81no—
297C2×D17—81no—
298C2×D17—81no—
299C2×D17—164no—
300C2×D17—164no—
301C4×C17—21no—
302C4×C17—21no—
303C4×C17—21no—
304C4×C17—21no—
305C4×C17—42no—
306C4×C17—81no—
307C4×C17—81no—
308C4×C17D4211yescharacteristic
309C4×C17#57611yescharacteristic
310Dic17—81no—
311C2×#135C2×D1711yescharacteristic, complemented
312C22×Q16C2×D1711yescharacteristic, complemented
313C22×D8—171no—
314C22×D8—171no—
315#548—21no—
316D42—341no—
317#509—344no—
318C2×SD32—171no—
319C2×SD32—171no—
320C2×SD32—171no—
321C2×SD32—171no—
322C2×SD32—171no—
323C2×SD32—171no—
324C2×SD32—344no—
325C2×D16—171no—
326C2×D16—171no—
327C2×D16—171no—
328C2×M32—171no—
329C22×C16—171no—
330#495C2×D1711yescharacteristic, complemented
331#493—171no—
332#467C2×D1711yescharacteristic, complemented
333#466—171no—
334#466—171no—
335#466C2×D1711yescharacteristic, complemented
336#464—21no—
337#459—341no—
338#452—21no—
339#448—21no—
340#447—341no—
341#447—341no—
342C4×Q16C2×D1711yescharacteristic, complemented
343C4×D8—171no—
344C8×D4C2×D1711yescharacteristic
345#366—171no—
346C4⋊C16—171no—
347#358—171no—
348#358—171no—
349#357—171no—
350#357—171no—
351#357—171no—
352#357—171no—
353C22⋊C16—171no—
354C22⋊C16—171no—
355#346—171no—
356C2×C17—22no—
357C2×C17—41no—
358C2×C17—161no—
359C2×C17D4×D811yescharacteristic
360C2×C17C2×#50911yescharacteristic
361C2×C17#341611yescharacteristic
362D17—82no—
363D17—161no—
364D17—161no—
365C2×#41—21no—
366C22×Q8—21no—
367C22×D4—341no—
368C22×D4—341no—
369C22×D4—681no—
370#135—22no—
371#135—41no—
372C2×Q16—22no—
373C2×Q16—21no—
374C2×Q16—41no—
375C2×Q16C22×D1711yescharacteristic
376C2×Q16C22×D1711yescharacteristic
377C2×SD16—21no—
378C2×SD16—21no—
379C2×D8—171no—
380C2×D8—171no—
381C2×D8—171no—
382C2×D8—342no—
383C2×D8—342no—
384C2×D8—341no—
385C2×D8—341no—
386C2×D8—341no—
387C2×D8—681no—
388C2×D8—681no—
389C2×D8C22×D1711yescharacteristic
390C22×C8C22×D1711yescharacteristic
391C22×C8C22×D1711yescharacteristic
392#127—341no—
393#124—21no—
394#124—41no—
395C22⋊Q8—21no—
396C22⋊Q8—41no—
397#121—21no—
398#121—41no—
399#121—341no—
400#121—341no—
401#121—681no—
402#120—681no—
403#120—681no—
404C4×Q8—21no—
405C4×D4—41no—
406C4×D4—341no—
407C4×D4C22⋊D1711yescharacteristic
408SD32—174no—
409SD32—342no—
410SD32—342no—
411SD32—342no—
412SD32—342no—
413SD32—681no—
414D16—342no—
415D16—342no—
416D16—681no—
417M32—342no—
418C2×C16—171no—
419C2×C16—171no—
420C2×C16—171no—
421C2×C16—171no—
422C2×C16—342no—
423#107—171no—
424#107C22×D1711yescharacteristic
425C4⋊C8C22⋊D1711yescharacteristic
426#103—21no—
427#103—21no—
428#102—21no—
429#102—341no—
430#102—341no—
431#102—341no—
432C22⋊C8C22⋊D1711yescharacteristic
433C22⋊C8C22⋊D1711yescharacteristic
434C4×C8C22×D1711yescharacteristic
435C17#3383111yescharacteristic, complemented, Sylow
436C24—681no—
437C24—681no—
438#41—42no—
439#41—81no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—42no—
443C2×Q8—41no—
444C2×Q8—81no—
445C2×D4—21no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—341no—
449C2×D4—341no—
450C2×D4—341no—
451C2×D4—682no—
452C2×D4—682no—
453C2×D4—681no—
454C2×D4—681no—
455C2×D4—681no—
456C2×D4—681no—
457C2×D4—681no—
458C2×D4—681no—
459C2×D4—1361no—
460C2×D4—1361no—
461C2×D4—1364no—
462C2×D4#6322811yescharacteristic
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—681no—
466C22×C4C2×C22⋊D1711yescharacteristic
467C22×C4C2×C22⋊D1711yescharacteristic
468Q16—21no—
469Q16—21no—
470Q16—21no—
471Q16—42no—
472Q16D4×D1712yescomplemented
473SD16—42no—
474SD16—81no—
475D8—21no—
476D8—21no—
477D8—172no—
478D8—341no—
479D8—341no—
480D8—341no—
481D8—341no—
482D8—341no—
483D8—682no—
484D8—682no—
485D8—1361no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8C2×C22⋊D1711yescharacteristic
491C2×C8C2×C22⋊D1711yescharacteristic
492C2×C8C2×C22⋊D1711yescharacteristic
493C2×C8C23×D1711yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—341no—
498C4⋊C4#6322811yescharacteristic
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—681no—
504C22⋊C4—681no—
505C22⋊C4—681no—
506C22⋊C4#6322811yescharacteristic
507C22⋊C4#6322811yescharacteristic
508C42—41no—
509C42C2×C22⋊D1711yescharacteristic
510C16—172no—
511C16—341no—
512C16—341no—
513C16—341no—
514C23—41no—
515C23—2721no—
516C23—2721no—
517C23—681no—
518C23—681no—
519C23—681no—
520C23—1362no—
521C23—1362no—
522C23—1362no—
523C23—1362no—
524C23C2×#6322811yescharacteristic
525C23C2×#6322811yescharacteristic
526Q8—22no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—82no—
531D4—22no—
532D4—41no—
533D4—41no—
534D4—82no—
535D4—82no—
536D4—2721no—
537D4—342no—
538D4—681no—
539D4—681no—
540D4—681no—
541D4—681no—
542D4—681no—
543D4—1362no—
544D4—1362no—
545D4—1362no—
546D4—1362no—
547D4—1362no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—681no—
558C2×C4—1362no—
559C2×C4C2×#6322811yescharacteristic
560C2×C4C2×#6322811yescharacteristic
561C2×C4C2×#6322811yescharacteristic
562C2×C4C2×#6322811yescharacteristic
563C2×C4C22×C22⋊D1711yescharacteristic
564C8—21no—
565C8—21no—
566C8—21no—
567C8C2×D4×D1711yescharacteristic
568C8#1058769111yescharacteristic
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—2724no—
578C22—2724no—
579C22—682no—
580C22—1361no—
581C22—1361no—
582C22—1361no—
583C22—1361no—
584C22—1361no—
585C22—1361no—
586C22—1361no—
587C22—1361no—
588C22#1176324412yescomplemented
589C22C22×#6322811yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—1361no—
598C4D4×C22⋊D1711yescharacteristic
599C4#1176361211yescharacteristic
600C2—22no—
601C2—41no—
602C2—161no—
603C2—2721no—
604C2—2721no—
605C2—1362no—
606C2D4×#6322811yescharacteristic
607C2C2×#1176324411yescharacteristic
608C2#1454192811yescharacteristic
subgroupG/H
centerC22C22×#63228
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#3285C2
socleC22×C17C22×D8
hypercenter#3285D17
hypocenterC17#33831
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3285

Complements

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

NH ≅ G/Ntype
1. C2×#11763244C2semidirect
2. C22×#10587499C2semidirect
3. D4×D136C2semidirect
4. #14531051C2semidirect
5. #14530899C2semidirect
6. #14530868C2semidirect
7. #14530836C2semidirect
8. #14530824C2semidirect
9. #14530712C2semidirect
10. #14530707C2semidirect
11. #14530701C2semidirect
12. #14530695C2semidirect
13. #14530646C2semidirect
14. #14530610C2semidirect
15. C17×#3285C2semidirect
23. C2×#10587499C22semidirect
24. C2×#10587499C22semidirect
25. C2×#10587499C22semidirect
26. C2×#10587499C22semidirect
28. C2×#10587497C22semidirect
29. C2×#10587497C22semidirect
30. C2×#10587483C22semidirect
31. C22×#63214C22semidirect
39. #11762634C22semidirect
40. C4⋊#63214C22semidirect
41. #11762613C22semidirect
42. #11762613C22semidirect
43. #11762610C22semidirect
45. #11762609C22semidirect
46. #11762609C22semidirect
47. C22⋊#63214C22semidirect
48. C22⋊#63214C22semidirect
49. #11762585C22semidirect
50. C2×C17×#135C22semidirect
51. C22×Q16×C17C22semidirect
53. C17×#495C22semidirect
54. C17×#467C22semidirect
55. C17×#466C22semidirect
59. C4×Q16×C17C22semidirect
117. #10587499D4semidirect
167. Q16×C17C2×D4semidirect
208. #63214C2×D4semidirect
267. #3285D17semidirect
288. C22×C17#509semidirect
311. C2×#135C2×D17semidirect
312. C22×Q16C2×D17semidirect
330. #495C2×D17semidirect
332. #467C2×D17semidirect
335. #466C2×D17semidirect
342. C4×Q16C2×D17semidirect
435. C17#33831semidirect
472. Q16D4×D17semidirect
588. C22#11763244semidirect

Maximal subgroups

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

subgroupindex
C2×#117632442
C22×#105874992
D4×D1362
#145310512
#145308992
#145308682
#145308362
#145308242
#145307122
#145307072
#145307012
#145306952
#145306462
#145306102
C17×#32852
#3383117