Finite Group Databasebeta

← C4×#126×D17 #17192801

Subgroups — C4×#126×D17 #17192801

Order 4352.

2838 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#126×D17C211yesmaximal, characteristic
2#14531338C213yesmaximal, complemented
3C4×#10587584C213yesmaximal, complemented
4#126×Dic17C211yesmaximal, characteristic, complemented
5C4×#10587562C211yesmaximal, characteristic, complemented
6#14531334C213yesmaximal, complemented
7C4×#10587565C213yesmaximal, complemented
8#14531332C211yesmaximal, characteristic, complemented
9D17×#388C213yesmaximal
10D17×#387C213yesmaximal, complemented
11D17×#383C211yesmaximal, characteristic, complemented
12C4×C4⋊C4×D17C213yesmaximal, complemented
13C4×C22⋊C4×D17C213yesmaximal
14C43×D17C211yesmaximal, characteristic, complemented
15C4×C17×#126C211yesmaximal, characteristic, complemented
16C23×C4×D17C2211yescharacteristic
17C2×#10587584C2213yes—
18C2×#10587562C2211yescharacteristic
19C2×#10587565C2213yes—
20#126×D17C4116yesdirect product factor
21C2×C4⋊C4×D17C2213yes—
22C2×C4⋊C4×D17C2213yes—
23C2×C22⋊C4×D17C2213yes—
24C2×C22⋊C4×D17C2213yes—
25C2×C42×D17C2213yes—
26C2×C42×D17C2213yes—
27C2×C42×D17C2213yes—
28C2×C42×D17C2211yescharacteristic
29#11762703C2213yes—
30#11762702C2213yes—
31#11762701C2213yes—
32#11762700C2213yes—
33#11762699C2213yes—
34#11762698C2213yes—
35#11762697C2213yes—
36#11762696C2213yescomplemented
37#11762695C2213yescomplemented
38#11762694C2213yescomplemented
39#11762693C2213yescomplemented
40#11762692C2213yescomplemented
41#11762691C2213yescomplemented
42#11762690C2213yescomplemented
43#11762677C2211yescharacteristic, complemented
44#11762676C2213yescomplemented
45#11762676C2213yescomplemented
46C4⋊C4×Dic17C2213yescomplemented
47C4⋊C4×Dic17C2213yescomplemented
48C4×#63225C2213yescomplemented
49C4×#63225C2213yescomplemented
50C4×C4⋊Dic17C2213yescomplemented
51#11762660C2213yescomplemented
52#11762660C2213yescomplemented
53C22⋊C4×Dic17C2213yes—
54C22⋊C4×Dic17C2213yes—
55C4×#63227C2213yes—
56C4×#63227C2213yes—
57C4×C22⋊Dic17C2213yes—
58#11762656C2213yes—
59#11762656C2213yes—
60#95×D17C2213yes—
61#95×D17C2213yes—
62#95×D17C2213yes—
63#95×D17C2213yes—
64C42×Dic17C2211yescharacteristic, complemented
65C42×Dic17C2213yescomplemented
66C42×Dic17C2213yescomplemented
67C2×C17×#126C2211yescharacteristic
68C17×#388C2213yes—
69C17×#387C2213yescomplemented
70C17×#383C2211yescharacteristic, complemented
71C4×C4⋊C4×C17C2213yescomplemented
72C4×C22⋊C4×C17C2213yes—
73C43×C17C2211yescharacteristic, complemented
74C24×D17C2311yescharacteristic
75C23×C4×C17C2311yescharacteristic
76C23×Dic17C2311yescharacteristic
77C22×C4×D17—212no—
78C22×C4×D17C2313yes—
79C22×C4×D17C2311yescharacteristic
80C22×C4×D17C2311yescharacteristic
81C22×C4×D17C2311yescharacteristic
82C22×C4×D17C2313yes—
83C22×C4×D17C2313yes—
84C22×C4×D17C2313yes—
85C22×C4×D17C2311yescharacteristic
86C17×#126C2×C418yescomplemented
87C2×C4⋊C4×C17C2313yes—
88C2×C4⋊C4×C17C2313yes—
89C2×C22⋊C4×C17C2313yes—
90C2×C22⋊C4×C17C2313yes—
91C2×C42×C17C2313yes—
92C2×C42×C17C2313yes—
93C2×C42×C17C2313yes—
94C2×C42×C17C2311yescharacteristic
95C2×C22⋊Dic17C2313yes—
96C2×C22⋊Dic17C2313yes—
97C2×#63227C2313yes—
98C2×#63227C2313yes—
99C2×#63227C2313yes—
100C2×#63227C2313yes—
101C2×C4⋊Dic17C2313yes—
102C2×C4⋊Dic17C2313yes—
103C2×#63225C2313yes—
104C2×#63225C2313yes—
105C2×#63225C2313yes—
106C2×#63225C2313yes—
107C2×C4×Dic17C2313yes—
108C2×C4×Dic17C2313yes—
109C2×C4×Dic17C2313yes—
110C2×C4×Dic17C2313yes—
111C2×C4×Dic17C2313yes—
112C2×C4×Dic17C2313yes—
113C2×C4×Dic17C2313yes—
114C2×C4×Dic17C2313yes—
115C2×C4×Dic17C2313yes—
116C2×C4×Dic17C2313yes—
117#10587584C2×C4124yescomplemented
118C4⋊C4×D17—212no—
119C4⋊C4×D17C2×C4124yescomplemented
120C22⋊C4×D17—212no—
121C22⋊C4×D17C2×C4124yes—
122#10587565C2×C4124yescomplemented
123#10587562C2×C418yescomplemented
124C42×D17—212no—
125C42×D17—212no—
126C42×D17—212no—
127C42×D17C2×C418yescomplemented
128C17×#95C2313yes—
129C17×#95C2313yes—
130C17×#95C2313yes—
131C17×#95C2313yes—
132#10587502C2313yes—
133#10587502C2313yes—
134#10587502C2313yes—
135#10587502C2313yes—
136#10587502C2313yes—
137#10587502C2313yes—
138#10587502C2313yes—
139#10587502C2313yes—
140#10587502C2313yes—
141#10587502C2313yes—
142#10587502C2313yes—
143#10587502C2313yes—
144C24×C17C2411yescharacteristic
145C23×D17—26no—
146C23×D17—26no—
147C23×D17C22×C414yes—
148C23×D17C2411yescharacteristic
149C23×D17C2411yescharacteristic
150C22×C4×C17—26no—
151C22×C4×C17C2411yescharacteristic
152C22×C4×C17C2413yes—
153C22×C4×C17C2413yes—
154C22×C4×C17C2413yes—
155C22×C4×C17C2413yes—
156C22×C4×C17C2411yescharacteristic
157C22×Dic17—26no—
158C22×Dic17C2411yescharacteristic
159C22×Dic17C2413yes—
160C22×Dic17C2413yes—
161C22×Dic17C2413yes—
162C22×Dic17C2413yes—
163C22×Dic17C2411yescharacteristic
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—26no—
172C2×C4×D17—26no—
173C2×C4×D17—26no—
174C2×C4×D17—26no—
175C2×C4×D17—26no—
176C2×C4×D17—416no—
177C2×C4×D17C22×C4112yes—
178C2×C4×D17C22×C4112yes—
179C2×C4×D17#41112yes—
180C2×C4×D17#41112yes—
181C4⋊C4×C17—26no—
182C4⋊C4×C17C22×C4112yescomplemented
183C22⋊C4×C17—26no—
184C22⋊C4×C17C22×C4112yes—
185C42×C17—26no—
186C42×C17—26no—
187C42×C17—26no—
188C42×C17C22×C414yescomplemented
189C22⋊Dic17—26no—
190C22⋊Dic17C22×C4112yes—
191#63227—26no—
192#63227—26no—
193#63227C22×C4112yes—
194#63227C22×C4112yes—
195C4⋊Dic17—26no—
196C4⋊Dic17C22×C4112yescomplemented
197#63225—26no—
198#63225—26no—
199#63225C22×C4112yescomplemented
200#63225C22×C4112yescomplemented
201C4×Dic17—26no—
202C4×Dic17—26no—
203C4×Dic17—26no—
204C4×Dic17—26no—
205C4×Dic17—26no—
206C4×Dic17—26no—
207C4×Dic17—26no—
208C4×Dic17—26no—
209C4×Dic17—26no—
210C4×Dic17C22×C4112yescomplemented
211C2×C4×#126—171nomaximal, Sylow
212C23×C17—23no—
213C23×C17—23no—
214C23×C17C23×C412yes—
215C23×C17C2511yescharacteristic
216C22×D17—23no—
217C22×D17—23no—
218C22×D17—224no—
219C22×D17—44no—
220C22×D17—412no—
221C22×D17C23×C412yes—
222C22×D17C23×C412yes—
223C22×D17C2×#4116yes—
224C22×D17C2×#4116yes—
225C2×C4×C17—23no—
226C2×C4×C17—23no—
227C2×C4×C17—23no—
228C2×C4×C17—23no—
229C2×C4×C17—23no—
230C2×C4×C17—23no—
231C2×C4×C17—23no—
232C2×C4×C17—23no—
233C2×C4×C17—23no—
234C2×C4×C17—23no—
235C2×C4×C17—44no—
236C2×C4×C17C23×C416yes—
237C2×C4×C17C23×C416yes—
238C2×C4×C17C2×#4116yes—
239C2×C4×C17C2×#4116yes—
240C2×Dic17—23no—
241C2×Dic17—23no—
242C2×Dic17—23no—
243C2×Dic17—23no—
244C2×Dic17—23no—
245C2×Dic17—23no—
246C2×Dic17—23no—
247C2×Dic17—23no—
248C2×Dic17—23no—
249C2×Dic17—23no—
250C2×Dic17—44no—
251C2×Dic17C23×C416yes—
252C2×Dic17C23×C416yes—
253C2×Dic17C2×#4116yes—
254C2×Dic17C2×#4116yes—
255C4×D17—224no—
256C4×D17—224no—
257C4×D17—224no—
258C4×D17—44no—
259C4×D17—412no—
260C4×D17—44no—
261C4×D17—44no—
262C4×D17#126116yesdirect product factor
263C22×#126—171no—
264C4×#126—173no—
265C4×#126—173no—
266C4×#126—171no—
267C4×#126—173no—
268C4×#126—173no—
269C4×#126—171no—
270C4×#126—171no—
271C4×#126D1711yescharacteristic, direct product factor
272C2×#388—173no—
273C2×#387—173no—
274C2×#383—171no—
275C2×C4×C4⋊C4—173no—
276C2×C4×C22⋊C4—173no—
277C2×C43—171no—
278C22×C17—26no—
279C22×C17—41no—
280C22×C17—43no—
281C22×C17C24×C411yescharacteristic
282C22×C17C22×#4113yes—
283C22×C17C22×#4113yes—
284C2×D17—26no—
285C2×D17—41no—
286C2×D17—43no—
287C2×D17—416no—
288C2×D17C4×#41112yes—
289C2×D17C2×#12614yes—
290C2×D17#529112yes—
291C4×C17—26no—
292C4×C17—26no—
293C4×C17—26no—
294C4×C17—41no—
295C4×C17—43no—
296C4×C17C2×#12614yescomplemented
297Dic17—26no—
298Dic17—26no—
299Dic17—26no—
300Dic17—41no—
301Dic17—43no—
302Dic17C2×#12614yescomplemented
303C24×C4—171no—
304C2×#126—1716no—
305C2×#126—173no—
306C2×#126—173no—
307C2×#126—171no—
308C2×#126C2×D1711yescharacteristic
309C22×C4⋊C4—173no—
310C22×C4⋊C4—173no—
311C22×C22⋊C4—173no—
312C22×C22⋊C4—173no—
313C22×C42—171no—
314C22×C42—173no—
315C22×C42—173no—
316C22×C42—173no—
317#388—173no—
318#388—173no—
319#388—173no—
320#388—173no—
321#388—173no—
322#388—173no—
323#388—173no—
324#388C2×D1713yes—
325#387—173no—
326#387—173no—
327#387—173no—
328#387—173no—
329#387—173no—
330#387—173no—
331#387—173no—
332#387C2×D1713yescomplemented
333#383—171no—
334#383—173no—
335#383—173no—
336#383C2×D1711yescharacteristic, complemented
337C4×C4⋊C4—173no—
338C4×C4⋊C4—173no—
339C4×C4⋊C4—173no—
340C4×C4⋊C4—173no—
341C4×C4⋊C4—173no—
342C4×C4⋊C4—173no—
343C4×C4⋊C4—173no—
344C4×C4⋊C4C2×D1713yescomplemented
345C4×C22⋊C4—173no—
346C4×C22⋊C4—173no—
347C4×C22⋊C4—173no—
348C4×C22⋊C4—173no—
349C4×C22⋊C4—173no—
350C4×C22⋊C4—173no—
351C4×C22⋊C4—173no—
352C4×C22⋊C4C2×D1713yes—
353C2×#95—173no—
354C2×#95—173no—
355C2×#95—173no—
356C2×#95—173no—
357C43—173no—
358C43—173no—
359C43—171no—
360C43C2×D1711yescharacteristic, complemented
361C2×C17—42no—
362C2×C17C2×C4×#4113yes—
363C2×C17C22×#12611yescharacteristic
364C2×C17C2×#52913yes—
365D17—42no—
366D17C4×#12618yesdirect product factor
367C25—171no—
368C23×C4—171no—
369C23×C4—173no—
370C23×C4—173no—
371C23×C4—173no—
372C23×C4—171no—
373C23×C4—173no—
374C23×C4—171no—
375C23×C4—171no—
376C23×C4—171no—
377C23×C4—3412no—
378C23×C4C22×D1711yescharacteristic
379#126—1724no—
380#126—1724no—
381#126—178no—
382#126C4×D1718yesdirect product factor
383C2×C4⋊C4—173no—
384C2×C4⋊C4—173no—
385C2×C4⋊C4—173no—
386C2×C4⋊C4—1724no—
387C2×C4⋊C4—173no—
388C2×C4⋊C4—173no—
389C2×C4⋊C4—173no—
390C2×C4⋊C4—3412no—
391C2×C4⋊C4C22×D1713yes—
392C2×C4⋊C4C22×D1713yes—
393C2×C22⋊C4—173no—
394C2×C22⋊C4—173no—
395C2×C22⋊C4—173no—
396C2×C22⋊C4—1724no—
397C2×C22⋊C4—173no—
398C2×C22⋊C4—173no—
399C2×C22⋊C4—173no—
400C2×C22⋊C4—3412no—
401C2×C22⋊C4C22×D1713yes—
402C2×C22⋊C4C22×D1713yes—
403C2×C42—173no—
404C2×C42—178no—
405C2×C42—173no—
406C2×C42—173no—
407C2×C42—173no—
408C2×C42—173no—
409C2×C42—173no—
410C2×C42—173no—
411C2×C42—173no—
412C2×C42—173no—
413C2×C42—173no—
414C2×C42—3412no—
415C2×C42—3412no—
416C2×C42—3412no—
417C2×C42C22×D1713yes—
418C2×C42C22×D1713yes—
419C2×C42C22×D1713yes—
420C2×C42C22×D1711yescharacteristic
421#95—173no—
422#95—173no—
423#95—173no—
424#95—173no—
425#95—173no—
426#95—173no—
427#95—173no—
428#95—173no—
429#95—173no—
430#95—173no—
431#95—173no—
432#95—173no—
433#95C22×D1713yes—
434#95C22×D1713yes—
435#95C22×D1713yes—
436#95C22×D1713yes—
437C17C2×C4×#12611yescharacteristic, complemented, Sylow
438C24—171no—
439C24—174no—
440C24—171no—
441C24—346no—
442C24—346no—
443C24C23×D1711yescharacteristic
444C22×C4—26no—
445C22×C4—171no—
446C22×C4—1712no—
447C22×C4—173no—
448C22×C4—173no—
449C22×C4—173no—
450C22×C4—1712no—
451C22×C4—1712no—
452C22×C4—1712no—
453C22×C4—171no—
454C22×C4—173no—
455C22×C4—346no—
456C22×C4—346no—
457C22×C4—346no—
458C22×C4—346no—
459C22×C4—346no—
460C22×C4—346no—
461C22×C4—346no—
462C22×C4—346no—
463C22×C4—346no—
464C22×C4—346no—
465C22×C4—346no—
466C22×C4—346no—
467C22×C4—346no—
468C22×C4—6816no—
469C22×C4C23×D1711yescharacteristic
470C22×C4C23×D1713yes—
471C22×C4C23×D1713yes—
472C22×C4C23×D1713yes—
473C22×C4C23×D1713yes—
474C22×C4C23×D1711yescharacteristic
475C4⋊C4—26no—
476C4⋊C4—1712no—
477C4⋊C4—1712no—
478C4⋊C4—1712no—
479C4⋊C4—346no—
480C4⋊C4—346no—
481C4⋊C4—346no—
482C4⋊C4C2×C4×D17112yescomplemented
483C22⋊C4—26no—
484C22⋊C4—1712no—
485C22⋊C4—1712no—
486C22⋊C4—1712no—
487C22⋊C4—346no—
488C22⋊C4—346no—
489C22⋊C4—346no—
490C22⋊C4C2×C4×D17112yes—
491C42—26no—
492C42—26no—
493C42—26no—
494C42—1712no—
495C42—346no—
496C42—346no—
497C42—346no—
498C42—346no—
499C42—346no—
500C42—346no—
501C42—346no—
502C42—346no—
503C42—346no—
504C42C2×C4×D1714yescomplemented
505C23—23no—
506C23—23no—
507C23—172no—
508C23—176no—
509C23—176no—
510C23—172no—
511C23—3424no—
512C23—343no—
513C23—343no—
514C23—684no—
515C23—6812no—
516C23C22×C4×D1712yes—
517C23C24×D1711yescharacteristic
518C2×C4—23no—
519C2×C4—23no—
520C2×C4—23no—
521C2×C4—23no—
522C2×C4—23no—
523C2×C4—23no—
524C2×C4—23no—
525C2×C4—23no—
526C2×C4—23no—
527C2×C4—23no—
528C2×C4—44no—
529C2×C4—176no—
530C2×C4—1716no—
531C2×C4—176no—
532C2×C4—176no—
533C2×C4—176no—
534C2×C4—3424no—
535C2×C4—3424no—
536C2×C4—3424no—
537C2×C4—343no—
538C2×C4—343no—
539C2×C4—343no—
540C2×C4—343no—
541C2×C4—343no—
542C2×C4—343no—
543C2×C4—343no—
544C2×C4—343no—
545C2×C4—343no—
546C2×C4—343no—
547C2×C4—684no—
548C2×C4—684no—
549C2×C4—6812no—
550C2×C4—684no—
551C2×C4—684no—
552C2×C4C22×C4×D1716yes—
553C2×C4C22×C4×D1716yes—
554C2×C4#41×D1716yes—
555C2×C4#41×D1716yes—
556C22—26no—
557C22—41no—
558C22—43no—
559C22—1712no—
560C22—174no—
561C22—1712no—
562C22—346no—
563C22—6816no—
564C22—681no—
565C22—683no—
566C22C23×C4×D1711yescharacteristic
567C22C2×#41×D1713yes—
568C22C2×#41×D1713yes—
569C4—26no—
570C4—26no—
571C4—26no—
572C4—41no—
573C4—43no—
574C4—174no—
575C4—346no—
576C4—346no—
577C4—346no—
578C4—681no—
579C4—683no—
580C4#126×D1714yesdirect product factor
581C2—42no—
582C2—178no—
583C2—682no—
584C2C4×#41×D1713yes—
585C2C2×#126×D1711yescharacteristic
586C2D17×#52913yes—
subgroupG/H
centerC22×C4C23×D17
commutatorC22×C17C24×C4
FrattiniC23C24×D17
FittingC4×C17×#126C2
socle, cosocleC23×C17C25
prime radicalC24×D17C23
hypercenterC4×#126D17
hypocenterC17C2×C4×#126
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C4×#126

Complements

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

NH ≅ G/Ntype
2. #14531338C2semidirect
3. C4×#10587584C2semidirect
4. #126×Dic17C2semidirect
5. C4×#10587562C2semidirect
6. #14531334C2semidirect
7. C4×#10587565C2semidirect
8. #14531332C2semidirect
10. D17×#387C2semidirect
11. D17×#383C2semidirect
12. C4×C4⋊C4×D17C2semidirect
14. C43×D17C2semidirect
15. C4×C17×#126C2semidirect
20. #126×D17C4direct (N × H)
36. #11762696C22semidirect
37. #11762695C22semidirect
38. #11762694C22semidirect
39. #11762693C22semidirect
40. #11762692C22semidirect
41. #11762691C22semidirect
42. #11762690C22semidirect
43. #11762677C22semidirect
44. #11762676C22semidirect
45. #11762676C22semidirect
46. C4⋊C4×Dic17C22semidirect
47. C4⋊C4×Dic17C22semidirect
48. C4×#63225C22semidirect
49. C4×#63225C22semidirect
50. C4×C4⋊Dic17C22semidirect
51. #11762660C22semidirect
52. #11762660C22semidirect
64. C42×Dic17C22semidirect
65. C42×Dic17C22semidirect
66. C42×Dic17C22semidirect
69. C17×#387C22semidirect
70. C17×#383C22semidirect
71. C4×C4⋊C4×C17C22semidirect
73. C43×C17C22semidirect
86. C17×#126C2×C4semidirect
117. #10587584C2×C4semidirect
119. C4⋊C4×D17C2×C4semidirect
122. #10587565C2×C4semidirect
123. #10587562C2×C4semidirect
127. C42×D17C2×C4semidirect
182. C4⋊C4×C17C22×C4semidirect
188. C42×C17C22×C4semidirect
196. C4⋊Dic17C22×C4semidirect
199. #63225C22×C4semidirect
200. #63225C22×C4semidirect
210. C4×Dic17C22×C4semidirect
262. C4×D17#126direct (N × H)
271. C4×#126D17direct (N × H)
296. C4×C17C2×#126semidirect
302. Dic17C2×#126semidirect
332. #387C2×D17semidirect
336. #383C2×D17semidirect
344. C4×C4⋊C4C2×D17semidirect
360. C43C2×D17semidirect
366. D17C4×#126direct (N × H)
382. #126C4×D17direct (N × H)
437. C17C2×C4×#126semidirect
482. C4⋊C4C2×C4×D17semidirect
504. C42C2×C4×D17semidirect
580. C4#126×D17direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#126×D172
#145313382
C4×#105875842
#126×Dic172
C4×#105875622
#145313342
C4×#105875652
#145313322
D17×#3882
D17×#3872
D17×#3832
C4×C4⋊C4×D172
C4×C22⋊C4×D172
C43×D172
C4×C17×#1262
C2×C4×#12617