Finite Group Databasebeta

← D17×#2401 #17200619

Subgroups — D17×#2401 #17200619

Order 4352.

2294 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14532087C211yesmaximal, characteristic, complemented
2#14532086C211yesmaximal, characteristic, complemented
3#14532085C213yesmaximal, complemented
4#14532084C213yesmaximal, complemented
5#14532083C213yesmaximal, complemented
6#14532082C211yesmaximal, characteristic, complemented
7#14532081C213yesmaximal, complemented
8D17×#401C211yesmaximal, characteristic, complemented
9D17×#400C213yesmaximal, complemented
10D17×#397C213yesmaximal
11D17×#396C211yesmaximal, characteristic
12C4×C4⋊C4×D17C213yesmaximal, complemented
13C4×C22⋊C4×D17C213yesmaximal
14D17×#376C211yesmaximal, characteristic, complemented
15C17×#2401C211yesmaximal, characteristic, complemented
16C23×C4×D17C2211yescharacteristic
17C2×C4⋊C4×D17C2213yes—
18C2×C4⋊C4×D17C2213yes—
19C2×C22⋊C4×D17C2213yes—
20C2×C22⋊C4×D17C2213yes—
21C2×C42×D17C2213yes—
22C2×C42×D17C2213yes—
23C17⋊#401C2213yescomplemented
24C17⋊#401C2211yescharacteristic, complemented
25C17⋊#401C2213yescomplemented
26#11762751C2213yescomplemented
27#11762751C2213yescomplemented
28#11762750C2213yescomplemented
29#11762750C2213yescomplemented
30#11762749C2213yescomplemented
31#11762748C2213yescomplemented
32#11762747C2213yescomplemented
33#11762738C2213yes—
34#11762738C2213yes—
35#11762737C2213yes—
36#11762737C2213yes—
37#11762736C2213yes—
38#11762735C2213yes—
39#11762734C2213yes—
40#11762733C2211yescharacteristic
41#11762732C2213yes—
42#11762731C2213yes—
43C4⋊C4×Dic17C2213yescomplemented
44C4⋊C4×Dic17C2213yescomplemented
45C4×#63225C2213yescomplemented
46C4×#63225C2213yescomplemented
47C4×C4⋊Dic17C2213yescomplemented
48#11762660C2213yescomplemented
49#11762660C2213yescomplemented
50C22⋊C4×Dic17C2213yes—
51C22⋊C4×Dic17C2213yes—
52C4×#63227C2213yes—
53C4×#63227C2213yes—
54C4×C22⋊Dic17C2213yes—
55#11762656C2213yes—
56#11762656C2213yes—
57#11762655C2213yescomplemented
58#11762654C2213yescomplemented
59#11762654C2211yescharacteristic, complemented
60#95×D17C2211yescharacteristic
61#95×D17C2213yes—
62#95×D17C2213yes—
63#95×D17C2213yes—
64#95×D17C2213yes—
65#95×D17C2213yes—
66C17×#401C2211yescharacteristic, complemented
67C17×#400C2213yescomplemented
68C17×#397C2213yes—
69C17×#396C2211yescharacteristic
70C4×C4⋊C4×C17C2213yescomplemented
71C4×C22⋊C4×C17C2213yes—
72C17×#376C2211yescharacteristic, complemented
73C24×D17C2311yescharacteristic
74C23×C4×C17C2311yescharacteristic
75C23×Dic17C2311yescharacteristic
76C22×C4×D17—212no—
77C22×C4×D17C2313yes—
78C22×C4×D17C2313yes—
79C22×C4×D17C2313yes—
80C22×C4×D17C2311yescharacteristic
81C22×C4×D17C2311yescharacteristic
82C22×C4×D17C2311yescharacteristic
83C22×C4×D17C2313yes—
84C22×C4×D17C2311yescharacteristic
85C2×C4⋊C4×C17C2313yes—
86C2×C4⋊C4×C17C2313yes—
87C2×C22⋊C4×C17C2313yes—
88C2×C22⋊C4×C17C2313yes—
89C2×C42×C17C2313yes—
90C2×C42×C17C2313yes—
91C2×C22⋊Dic17C2313yes—
92C2×C22⋊Dic17C2313yes—
93C2×#63227C2313yes—
94C2×#63227C2313yes—
95C2×#63227C2313yes—
96C2×#63227C2313yes—
97C2×C4⋊Dic17C2313yes—
98C2×C4⋊Dic17C2313yes—
99C2×#63225C2313yes—
100C2×#63225C2313yes—
101C2×#63225C2313yes—
102C2×#63225C2313yes—
103C2×C4×Dic17C2313yes—
104C2×C4×Dic17C2313yes—
105C2×C4×Dic17C2313yes—
106C2×C4×Dic17C2313yes—
107C2×C4×Dic17C2313yes—
108C2×C4×Dic17C2313yes—
109C4⋊C4×D17—212no—
110C4⋊C4×D17—212no—
111C22⋊C4×D17—212no—
112C22⋊C4×D17—212no—
113C42×D17—212no—
114C42×D17—212no—
115C17×#95C2311yescharacteristic
116C17×#95C2313yes—
117C17×#95C2313yes—
118C17×#95C2313yes—
119C17×#95C2313yes—
120C17×#95C2313yes—
121#10587502C2313yes—
122#10587502C2313yes—
123#10587502C2313yes—
124#10587502C2313yes—
125#10587502C2313yes—
126#10587502C2313yes—
127#10587502C2313yes—
128#10587502C2313yes—
129#10587502C2313yes—
130#10587502C2313yes—
131#10587502C2313yes—
132#10587502C2313yes—
133#10587502C2313yes—
134#10587502C2313yes—
135#10587502C2313yes—
136#10587502C2313yes—
137C24×C17C2411yescharacteristic
138C23×D17—26no—
139C23×D17—26no—
140C23×D17#4114yes—
141C23×D17C2411yescharacteristic
142C23×D17C2411yescharacteristic
143C22×C4×C17—26no—
144C22×C4×C17C2411yescharacteristic
145C22×C4×C17C2413yes—
146C22×C4×C17C2413yes—
147C22×C4×C17C2413yes—
148C22×C4×C17C2413yes—
149C22×C4×C17C2411yescharacteristic
150C22×Dic17—26no—
151C22×Dic17C2411yescharacteristic
152C22×Dic17C2413yes—
153C22×Dic17C2413yes—
154C22×Dic17C2413yes—
155C22×Dic17C2413yes—
156C22×Dic17C2411yescharacteristic
157C2×C4×D17—26no—
158C2×C4×D17—26no—
159C2×C4×D17—26no—
160C2×C4×D17—26no—
161C2×C4×D17—26no—
162C2×C4×D17—26no—
163C2×C4×D17—26no—
164C2×C4×D17—26no—
165C2×C4×D17—26no—
166C2×C4×D17—26no—
167C2×C4×D17—26no—
168C2×C4×D17—26no—
169C2×C4×D17—26no—
170C2×C4×D17—26no—
171C2×C4×D17—416no—
172C2×C4×D17#41112yes—
173C2×C4×D17#41112yes—
174C4⋊C4×C17—26no—
175C4⋊C4×C17—26no—
176C22⋊C4×C17—26no—
177C22⋊C4×C17—26no—
178C42×C17—26no—
179C42×C17—26no—
180C22⋊Dic17—26no—
181C22⋊Dic17—26no—
182#63227—26no—
183#63227—26no—
184#63227—26no—
185#63227—26no—
186C4⋊Dic17—26no—
187C4⋊Dic17—26no—
188#63225—26no—
189#63225—26no—
190#63225—26no—
191#63225—26no—
192C4×Dic17—26no—
193C4×Dic17—26no—
194C4×Dic17—26no—
195C4×Dic17—26no—
196C4×Dic17—26no—
197C4×Dic17—26no—
198C2×#2401—171nomaximal, Sylow
199C23×C17—23no—
200C23×C17—23no—
201C23×C17C2×#4112yes—
202C23×C17C2511yescharacteristic
203C22×D17—23no—
204C22×D17—23no—
205C22×D17—224no—
206C22×D17—44no—
207C22×D17—412no—
208C22×D17C2×#4112yes—
209C22×D17C2×#4116yes—
210C22×D17C2×#4112yes—
211C22×D17C2×#4116yes—
212C2×C4×C17—23no—
213C2×C4×C17—23no—
214C2×C4×C17—23no—
215C2×C4×C17—23no—
216C2×C4×C17—23no—
217C2×C4×C17—23no—
218C2×C4×C17—23no—
219C2×C4×C17—23no—
220C2×C4×C17—23no—
221C2×C4×C17—23no—
222C2×C4×C17—23no—
223C2×C4×C17—23no—
224C2×C4×C17—44no—
225C2×C4×C17C2×#4116yes—
226C2×C4×C17C2×#4116yes—
227C2×Dic17—23no—
228C2×Dic17—23no—
229C2×Dic17—23no—
230C2×Dic17—23no—
231C2×Dic17—23no—
232C2×Dic17—23no—
233C2×Dic17—23no—
234C2×Dic17—23no—
235C2×Dic17—23no—
236C2×Dic17—23no—
237C2×Dic17—23no—
238C2×Dic17—23no—
239C2×Dic17—44no—
240C2×Dic17C2×#4116yes—
241C2×Dic17C2×#4116yes—
242C4×D17—224no—
243C4×D17—44no—
244C4×D17—412no—
245C4×D17—412no—
246C4×D17—412no—
247C4×D17—44no—
248C4×D17—44no—
249C4×D17#126116yes—
250#2401—171no—
251#2401—173no—
252#2401—173no—
253#2401—171no—
254#2401—173no—
255#2401—173no—
256#2401—171no—
257#2401D1711yescharacteristic, direct product factor
258C2×#401—171no—
259C2×#400—173no—
260C2×#397—173no—
261C2×#396—171no—
262C2×C4×C4⋊C4—173no—
263C2×C4×C22⋊C4—173no—
264C2×#376—171no—
265C22×C17—26no—
266C22×C17—41no—
267C22×C17—43no—
268C22×C17C22×#4113yes—
269C22×C17C22×#4111yescharacteristic
270C22×C17C22×#4113yes—
271C2×D17—26no—
272C2×D17—41no—
273C2×D17—43no—
274C2×D17—416no—
275C2×D17C2×#12614yes—
276C2×D17#529112yes—
277C2×D17#529112yes—
278C4×C17—26no—
279C4×C17—41no—
280C4×C17—43no—
281C4×C17—43no—
282C4×C17—43no—
283C4×C17C2×#12614yes—
284Dic17—26no—
285Dic17—41no—
286Dic17—43no—
287Dic17—43no—
288Dic17—43no—
289Dic17C2×#12614yes—
290C24×C4—171no—
291C22×C4⋊C4—173no—
292C22×C4⋊C4—173no—
293C22×C22⋊C4—173no—
294C22×C22⋊C4—173no—
295C22×C42—173no—
296C22×C42—173no—
297#401—173no—
298#401—173no—
299#401—171no—
300#401C2×D1711yescharacteristic, complemented
301#400—173no—
302#400—173no—
303#400—173no—
304#400—173no—
305#400—173no—
306#400—173no—
307#400—173no—
308#400C2×D1713yescomplemented
309#397—173no—
310#397—173no—
311#397—173no—
312#397—173no—
313#397—173no—
314#397—173no—
315#397—173no—
316#397C2×D1713yes—
317#396—173no—
318#396—173no—
319#396—171no—
320#396C2×D1711yescharacteristic
321C4×C4⋊C4—173no—
322C4×C4⋊C4—173no—
323C4×C4⋊C4—173no—
324C4×C4⋊C4—173no—
325C4×C4⋊C4—173no—
326C4×C4⋊C4—173no—
327C4×C4⋊C4—173no—
328C4×C4⋊C4C2×D1713yescomplemented
329C4×C22⋊C4—173no—
330C4×C22⋊C4—173no—
331C4×C22⋊C4—173no—
332C4×C22⋊C4—173no—
333C4×C22⋊C4—173no—
334C4×C22⋊C4—173no—
335C4×C22⋊C4—173no—
336C4×C22⋊C4C2×D1713yes—
337#376—173no—
338#376—173no—
339#376—171no—
340#376C2×D1711yescharacteristic, complemented
341C2×#95—171no—
342C2×#95—173no—
343C2×#95—173no—
344C2×#95—173no—
345C2×#95—173no—
346C2×#95—173no—
347C2×C17—42no—
348C2×C17C22×#12611yescharacteristic
349C2×C17C2×#52913yes—
350C2×C17C2×#52913yes—
351D17—42no—
352D17#240118yesdirect product factor
353C25—171no—
354C23×C4—171no—
355C23×C4—173no—
356C23×C4—171no—
357C23×C4—173no—
358C23×C4—173no—
359C23×C4—173no—
360C23×C4—171no—
361C23×C4—171no—
362C23×C4—171no—
363C23×C4—3412no—
364C23×C4C22×D1711yescharacteristic
365C2×C4⋊C4—173no—
366C2×C4⋊C4—173no—
367C2×C4⋊C4—173no—
368C2×C4⋊C4—173no—
369C2×C4⋊C4—173no—
370C2×C4⋊C4—173no—
371C2×C4⋊C4—3412no—
372C2×C4⋊C4—3412no—
373C2×C4⋊C4C22×D1713yes—
374C2×C4⋊C4C22×D1713yes—
375C2×C22⋊C4—173no—
376C2×C22⋊C4—173no—
377C2×C22⋊C4—173no—
378C2×C22⋊C4—173no—
379C2×C22⋊C4—173no—
380C2×C22⋊C4—173no—
381C2×C22⋊C4—3412no—
382C2×C22⋊C4—3412no—
383C2×C22⋊C4C22×D1713yes—
384C2×C22⋊C4C22×D1713yes—
385C2×C42—173no—
386C2×C42—173no—
387C2×C42—173no—
388C2×C42—173no—
389C2×C42—173no—
390C2×C42—173no—
391C2×C42—3412no—
392C2×C42—3412no—
393C2×C42C22×D1713yes—
394C2×C42C22×D1713yes—
395#95—173no—
396#95—173no—
397#95—173no—
398#95—173no—
399#95—173no—
400#95—173no—
401#95—173no—
402#95—173no—
403#95—173no—
404#95—173no—
405#95—173no—
406#95—173no—
407#95—173no—
408#95—173no—
409#95—173no—
410#95—173no—
411#95C22×D1711yescharacteristic
412#95C22×D1713yes—
413#95C22×D1713yes—
414#95C22×D1713yes—
415#95C22×D1713yes—
416#95C22×D1713yes—
417C17C2×#240111yescharacteristic, complemented, Sylow
418C24—171no—
419C24—174no—
420C24—171no—
421C24—346no—
422C24—346no—
423C24C23×D1711yescharacteristic
424C22×C4—26no—
425C22×C4—171no—
426C22×C4—1712no—
427C22×C4—173no—
428C22×C4—1712no—
429C22×C4—171no—
430C22×C4—173no—
431C22×C4—173no—
432C22×C4—173no—
433C22×C4—346no—
434C22×C4—346no—
435C22×C4—346no—
436C22×C4—346no—
437C22×C4—346no—
438C22×C4—346no—
439C22×C4—346no—
440C22×C4—346no—
441C22×C4—346no—
442C22×C4—346no—
443C22×C4—346no—
444C22×C4—346no—
445C22×C4—346no—
446C22×C4—346no—
447C22×C4—346no—
448C22×C4—6816no—
449C22×C4C23×D1711yescharacteristic
450C22×C4C23×D1713yes—
451C22×C4C23×D1713yes—
452C22×C4C23×D1713yes—
453C22×C4C23×D1713yes—
454C22×C4C23×D1711yescharacteristic
455C4⋊C4—26no—
456C4⋊C4—26no—
457C4⋊C4—346no—
458C4⋊C4—346no—
459C4⋊C4—346no—
460C4⋊C4—346no—
461C4⋊C4—346no—
462C4⋊C4—346no—
463C22⋊C4—26no—
464C22⋊C4—26no—
465C22⋊C4—346no—
466C22⋊C4—346no—
467C22⋊C4—346no—
468C22⋊C4—346no—
469C22⋊C4—346no—
470C22⋊C4—346no—
471C42—26no—
472C42—26no—
473C42—346no—
474C42—346no—
475C42—346no—
476C42—346no—
477C42—346no—
478C42—346no—
479C23—23no—
480C23—23no—
481C23—176no—
482C23—176no—
483C23—172no—
484C23—172no—
485C23—3424no—
486C23—343no—
487C23—343no—
488C23—684no—
489C23—6812no—
490C23#41×D1712yes—
491C23C24×D1711yescharacteristic
492C2×C4—23no—
493C2×C4—23no—
494C2×C4—23no—
495C2×C4—23no—
496C2×C4—23no—
497C2×C4—23no—
498C2×C4—23no—
499C2×C4—23no—
500C2×C4—23no—
501C2×C4—23no—
502C2×C4—23no—
503C2×C4—23no—
504C2×C4—44no—
505C2×C4—176no—
506C2×C4—1716no—
507C2×C4—176no—
508C2×C4—3424no—
509C2×C4—343no—
510C2×C4—343no—
511C2×C4—343no—
512C2×C4—343no—
513C2×C4—343no—
514C2×C4—343no—
515C2×C4—343no—
516C2×C4—343no—
517C2×C4—343no—
518C2×C4—343no—
519C2×C4—343no—
520C2×C4—343no—
521C2×C4—684no—
522C2×C4—684no—
523C2×C4—6812no—
524C2×C4—6812no—
525C2×C4—6812no—
526C2×C4—684no—
527C2×C4—684no—
528C2×C4#41×D1716yes—
529C2×C4#41×D1716yes—
530C22—26no—
531C22—41no—
532C22—43no—
533C22—174no—
534C22—1712no—
535C22—1712no—
536C22—346no—
537C22—6816no—
538C22—681no—
539C22—683no—
540C22C2×#41×D1713yes—
541C22C2×#41×D1711yescharacteristic
542C22C2×#41×D1713yes—
543C4—26no—
544C4—41no—
545C4—43no—
546C4—43no—
547C4—43no—
548C4—174no—
549C4—346no—
550C4—681no—
551C4—683no—
552C4—683no—
553C4—683no—
554C4#126×D1714yes—
555C2—42no—
556C2—178no—
557C2—682no—
558C2C2×#126×D1711yescharacteristic
559C2D17×#52913yes—
560C2D17×#52913yes—
subgroupG/H
centerC22×C4C23×D17
commutator, socle, cosocleC23×C17C25
FrattiniC23C24×D17
FittingC17×#2401C2
prime radicalC24×D17C23
hypercenter#2401D17
hypocenterC17C2×#2401
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #2401

Complements

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

NH ≅ G/Ntype
1. #14532087C2semidirect
2. #14532086C2semidirect
3. #14532085C2semidirect
4. #14532084C2semidirect
5. #14532083C2semidirect
6. #14532082C2semidirect
7. #14532081C2semidirect
8. D17×#401C2semidirect
9. D17×#400C2semidirect
12. C4×C4⋊C4×D17C2semidirect
14. D17×#376C2semidirect
15. C17×#2401C2semidirect
23. C17⋊#401C22semidirect
24. C17⋊#401C22semidirect
25. C17⋊#401C22semidirect
26. #11762751C22semidirect
27. #11762751C22semidirect
28. #11762750C22semidirect
29. #11762750C22semidirect
30. #11762749C22semidirect
31. #11762748C22semidirect
32. #11762747C22semidirect
43. C4⋊C4×Dic17C22semidirect
44. C4⋊C4×Dic17C22semidirect
45. C4×#63225C22semidirect
46. C4×#63225C22semidirect
47. C4×C4⋊Dic17C22semidirect
48. #11762660C22semidirect
49. #11762660C22semidirect
57. #11762655C22semidirect
58. #11762654C22semidirect
59. #11762654C22semidirect
66. C17×#401C22semidirect
67. C17×#400C22semidirect
70. C4×C4⋊C4×C17C22semidirect
72. C17×#376C22semidirect
257. #2401D17direct (N × H)
300. #401C2×D17semidirect
308. #400C2×D17semidirect
328. C4×C4⋊C4C2×D17semidirect
340. #376C2×D17semidirect
352. D17#2401direct (N × H)
417. C17C2×#2401semidirect

Maximal subgroups

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

subgroupindex
#145320872
#145320862
#145320852
#145320842
#145320832
#145320822
#145320812
D17×#4012
D17×#4002
D17×#3972
D17×#3962
C4×C4⋊C4×D172
C4×C22⋊C4×D172
D17×#3762
C17×#24012
C2×#240117