Finite Group Databasebeta

← #17044595

Subgroups — #17044595

Order 4352.

1194 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C8×D17C211yesmaximal, characteristic
2#14531166C211yesmaximal, characteristic
3C2×#11762666C211yesmaximal, characteristic
4#14528984C212yesmaximal, complemented
5#14528982C212yesmaximal, complemented
6#14528979C211yesmaximal, characteristic, complemented
7#14528972C211yesmaximal, characteristic, complemented
8#14528971C211yesmaximal, characteristic, complemented
9#14528812C212yesmaximal, complemented
10#14528806C212yesmaximal, complemented
11C17×#1853C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C22×C8×D17C2212yes—
14C2×#10587578C2211yescharacteristic
15C2×#10587578C2211yescharacteristic
16C22×C4⋊Dic17C2211yescharacteristic
17C22×#63227—21no—
18C2×#10587478C2212yescomplemented
19C2×#10587478C2212yescomplemented
20C2×#10587488C2212yescomplemented
21C2×C22⋊#3622C2211yescharacteristic
22C2×#10587491C2212yes—
23C22⋊C8×D17C418yescomplemented
24#11762671—21no—
25#11762669C2211yescharacteristic
26#11762666—21no—
27#11762666—21no—
28#11762666C2211yescharacteristic
29#11762666C2212yes—
30#11762666C2211yescharacteristic
31#11762666C2212yes—
32#11762664—21no—
33C22⋊C4×Dic17—21no—
34#11762656—21no—
35C2×#10587502—21no—
36#11762572C2212yescomplemented
37#11762571C2212yescomplemented
38#11762571C2212yescomplemented
39C22×C4⋊C4×C17C2211yescharacteristic
40C2×C17×#106C2212yescomplemented
41C2×C17×C22⋊C8C2211yescharacteristic
42C17×#380C2211yescharacteristic
43C17×#340C2212yescomplemented
44C24×D17D411yescharacteristic
45C23×C4×C17—21no—
46C23×C4×C17C2311yescharacteristic
47C23×Dic17—21no—
48C23×Dic17D411yescharacteristic
49C22×C4×D17—42no—
50C22×C4×D17—42no—
51C22×C4×D17C2×C414yes—
52C22×C4×D17D411yescharacteristic
53C22×C4×D17D411yescharacteristic
54C22×C4×D17Q814yes—
55C22×C4×D17C2311yescharacteristic
56C22×C4×D17C2311yescharacteristic
57C22×C8×C17C2312yes—
58C2×C4⋊C4×C17—21no—
59C2×C4⋊C4×C17—42no—
60C2×C4⋊C4×C17C2312yes—
61C2×C4⋊C4×C17C2311yescharacteristic
62C2×C4⋊C4×C17C2311yescharacteristic
63C2×C22⋊C4×C17—21no—
64C2×C22⋊Dic17—21no—
65C2×#63227—22no—
66C2×#63227—22no—
67C2×#63227—21no—
68C2×#63227—21no—
69C2×#63227—21no—
70C2×#63227—21no—
71C2×#63227—44no—
72C2×C4⋊Dic17—21no—
73C2×C4⋊Dic17—42no—
74C2×C4⋊Dic17C2312yes—
75C2×C4⋊Dic17C2311yescharacteristic
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4×Dic17—21no—
78C2×C4×Dic17—21no—
79C22×#3622C2312yes—
80C2×C8×D17—42no—
81C2×C8×D17—42no—
82C2×C8×D17C2×C418yescomplemented
83#10587578—42no—
84#10587578—42no—
85C17×#106—42no—
86C17×C22⋊C8C2×C414yes—
87C17×#95—21no—
88#10587502—21no—
89#10587502—22no—
90#10587502—22no—
91#10587502—21no—
92#10587502—21no—
93C22⋊#3622C2×C414yes—
94#10587491C2×C418yescomplemented
95#10587488—42no—
96#10587478—42no—
97#10587478—42no—
98C24×C17C2×D411yescharacteristic
99C23×D17—22no—
100C23×D17—22no—
101C23×D17—22no—
102C23×D17—22no—
103C23×D17—41no—
104C23×D17—41no—
105C23×D17C4⋊C414yes—
106C23×D17C2×D411yescharacteristic
107C23×D17C2×D411yescharacteristic
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—21no—
112C22×C4×C17—21no—
113C22×C4×C17—42no—
114C22×C4×C17—42no—
115C22×C4×C17—42no—
116C22×C4×C17C22×C412yes—
117C22×C4×C17C2×D411yescharacteristic
118C22×C4×C17C2×Q812yes—
119C22×C4×C17C2411yescharacteristic
120C22×Dic17—22no—
121C22×Dic17—21no—
122C22×Dic17—21no—
123C22×Dic17—41no—
124C22×Dic17—41no—
125C22×Dic17—42no—
126C22×Dic17—42no—
127C22×Dic17—42no—
128C22×Dic17C4⋊C414yes—
129C22×Dic17C2×D411yescharacteristic
130C22×Dic17C2×D411yescharacteristic
131C2×C4×D17—22no—
132C2×C4×D17—22no—
133C2×C4×D17—22no—
134C2×C4×D17—22no—
135C2×C4×D17—48no—
136C2×C4×D17—48no—
137C2×C4×D17—41no—
138C2×C4×D17—41no—
139C2×C4×D17—41no—
140C2×C4×D17—41no—
141C2×C4×D17C4⋊C414yes—
142C2×C4×D17C4⋊C414yes—
143C2×C4×D17C22×C412yes—
144C2×C4×D17C22×C412yes—
145C2×C4×D17C2×Q812yes—
146C2×C4×D17C2×Q812yes—
147C2×C8×C17—22no—
148C2×C8×C17—22no—
149C2×C8×C17C22×C414yes—
150C4⋊C4×C17—22no—
151C4⋊C4×C17—22no—
152C4⋊C4×C17—22no—
153C4⋊C4×C17—22no—
154C4⋊C4×C17—42no—
155C22⋊C4×C17—42no—
156C22⋊Dic17—42no—
157#63227—42no—
158#63227—42no—
159#63227—44no—
160#63227—42no—
161#63227—44no—
162#63227—42no—
163C4⋊Dic17—22no—
164C4⋊Dic17—22no—
165C4⋊Dic17—22no—
166C4⋊Dic17—22no—
167C4⋊Dic17—42no—
168C4×Dic17—42no—
169C4×Dic17—42no—
170C2×#3622—22no—
171C2×#3622—22no—
172C2×#3622C22×C414yes—
173C8×D17—48no—
174#21941—171nomaximal, Sylow
175C23×C17—21no—
176C23×C17—21no—
177C23×C17—21no—
178C23×C17—21no—
179C23×C17C2×C4⋊C412yes—
180C23×C17#13512yes—
181C23×C17#13512yes—
182C23×C17C22×D411yescharacteristic
183C22×D17—21no—
184C22×D17—21no—
185C22×D17—21no—
186C22×D17—21no—
187C22×D17—21no—
188C22×D17—21no—
189C22×D17—21no—
190C22×D17—21no—
191C22×D17—21no—
192C22×D17—21no—
193C22×D17—28no—
194C22×D17—28no—
195C22×D17—44no—
196C22×D17—44no—
197C22×D17—44no—
198C22×D17—44no—
199C22×D17—44no—
200C22×D17C2×C4⋊C412yes—
201C22×D17C2×C4⋊C412yes—
202C22×D17AGL(1,ℤ/8)12yes—
203C22×D17#13712yes—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—24no—
207C2×C4×C17—24no—
208C2×C4×C17—42no—
209C2×C4×C17—41no—
210C2×C4×C17—41no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—84no—
218C2×C4×C17C2×C4⋊C412yes—
219C2×C4×C17C23×C411yescharacteristic
220C2×C4×C17C22×Q811yescharacteristic
221C2×C4×C17H(5,𝔽2)12yes—
222C2×C4×C17H(5,𝔽2)12yes—
223C2×Dic17—21no—
224C2×Dic17—21no—
225C2×Dic17—21no—
226C2×Dic17—21no—
227C2×Dic17—44no—
228C2×Dic17—42no—
229C2×Dic17—41no—
230C2×Dic17—41no—
231C2×Dic17—41no—
232C2×Dic17—41no—
233C2×Dic17—42no—
234C2×Dic17—42no—
235C2×Dic17—41no—
236C2×Dic17—41no—
237C2×Dic17—84no—
238C2×Dic17C2×C4⋊C412yes—
239C2×Dic17C2×C4⋊C412yes—
240C4×D17—24no—
241C4×D17—24no—
242C4×D17—24no—
243C4×D17—24no—
244C4×D17—44no—
245C4×D17—44no—
246C8×C17—24no—
247#3622—24no—
248C22×C22⋊C8—171no—
249#2280—171no—
250C2×#380—171no—
251#1854—172no—
252#1854—172no—
253#1853—171no—
254#1853D1711yescharacteristic, complemented
255#1852—171no—
256#1852—171no—
257#1808—172no—
258#1807—172no—
259C22×C17—22no—
260C22×C17—22no—
261C22×C17—22no—
262C22×C17—22no—
263C22×C17—22no—
264C22×C17—22no—
265C22×C17#42714yes—
266C22×C17C22×C4⋊C411yescharacteristic
267C22×C17#53411yescharacteristic
268C22×C17#53411yescharacteristic
269C22×C17C2×#13511yescharacteristic
270C22×C17C2×#13511yescharacteristic
271C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C17C2×#13711yescharacteristic
273C2×D17—22no—
274C2×D17—22no—
275C2×D17—22no—
276C2×D17—22no—
277C2×D17—22no—
278C2×D17—22no—
279C2×D17—22no—
280C2×D17—22no—
281C2×D17—22no—
282C2×D17—416no—
283C2×D17—41no—
284C2×D17—41no—
285C2×D17—41no—
286C2×D17—41no—
287C2×D17#42814yes—
288C4×C17—22no—
289C4×C17—81no—
290C4×C17—82no—
291C4×C17—81no—
292C4×C17#51812yes—
293C4×C17#54312yes—
294Dic17—22no—
295Dic17—22no—
296Dic17—81no—
297Dic17—82no—
298Dic17—81no—
299C24×C4—171no—
300C23×C8—172no—
301C2×#117—171no—
302C2×#117—171no—
303C22×C4⋊C4—171no—
304C22×C4⋊C4C2×D1711yescharacteristic
305C22×C22⋊C4—341no—
306C2×#107—172no—
307C2×#107—172no—
308C2×#106—172no—
309C2×#106C2×D1712yescomplemented
310C2×C22⋊C8—171no—
311C2×C22⋊C8—172no—
312C2×C22⋊C8—178no—
313C2×C22⋊C8C2×D1711yescharacteristic
314#381—341no—
315#380—171no—
316#380—171no—
317#380—172no—
318#380—171no—
319#380—172no—
320#380—341no—
321#380—341no—
322#380C2×D1711yescharacteristic
323#379—341no—
324C4×C22⋊C4—341no—
325C4×C22⋊C4—341no—
326C2×#95—341no—
327#340—172no—
328#340—172no—
329#340—172no—
330#340C2×D1712yescomplemented
331C2×C17—24no—
332C2×C17#231511yescharacteristic
333C2×C17C2×#42711yescharacteristic
334C2×C17C2×#42811yescharacteristic
335C2×C17#318811yescharacteristic
336C2×C17#318811yescharacteristic
337C2×C17#319011yescharacteristic
338C2×C17#319011yescharacteristic
339D17—24no—
340D17—42no—
341C25—171no—
342C23×C4—21no—
343C23×C4—171no—
344C23×C4—171no—
345C23×C4—171no—
346C23×C4—174no—
347C23×C4—171no—
348C23×C4—171no—
349C23×C4—174no—
350C23×C4—341no—
351C23×C4—682no—
352C23×C4—682no—
353C23×C4C22×D1711yescharacteristic
354C22×C8—172no—
355C22×C8—178no—
356C22×C8—682no—
357C22×C8—682no—
358C22×C8C22×D1712yes—
359#117—682no—
360#117—682no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—42no—
363C2×C4⋊C4—172no—
364C2×C4⋊C4—171no—
365C2×C4⋊C4—171no—
366C2×C4⋊C4—341no—
367C2×C4⋊C4—682no—
368C2×C4⋊C4C22×D1712yes—
369C2×C4⋊C4C22×D1711yescharacteristic
370C2×C4⋊C4C22×D1711yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—342no—
373C2×C22⋊C4—342no—
374C2×C22⋊C4—341no—
375C2×C22⋊C4—341no—
376C2×C22⋊C4—341no—
377C2×C22⋊C4—341no—
378C2×C22⋊C4—341no—
379C2×C22⋊C4—684no—
380C2×C42—341no—
381C2×C42—341no—
382#107—682no—
383#107—682no—
384#106—42no—
385#106—682no—
386C22⋊C8—174no—
387C22⋊C8—178no—
388C22⋊C8C4×D1714yes—
389#95—21no—
390#95—342no—
391#95—342no—
392#95—341no—
393#95—341no—
394#95—341no—
395C17#2194111yescharacteristic, complemented, Sylow
396C24—171no—
397C24—171no—
398C24—174no—
399C24—342no—
400C24—342no—
401C24—342no—
402C24—342no—
403C24—681no—
404C24—681no—
405C24D4×D1711yescharacteristic
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—171no—
415C22×C4—171no—
416C22×C4—174no—
417C22×C4—172no—
418C22×C4—172no—
419C22×C4—172no—
420C22×C4—172no—
421C22×C4—174no—
422C22×C4—174no—
423C22×C4—342no—
424C22×C4—342no—
425C22×C4—342no—
426C22×C4—342no—
427C22×C4—341no—
428C22×C4—341no—
429C22×C4—342no—
430C22×C4—688no—
431C22×C4—688no—
432C22×C4—681no—
433C22×C4—681no—
434C22×C4—681no—
435C22×C4—681no—
436C22×C4—681no—
437C22×C4—681no—
438C22×C4—682no—
439C22×C4—682no—
440C22×C4—682no—
441C22×C4C2×C4×D1712yes—
442C22×C4D4×D1711yescharacteristic
443C22×C4Q8×D1712yes—
444C22×C4C23×D1711yescharacteristic
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—174no—
448C2×C8—342no—
449C2×C8—342no—
450C2×C8—688no—
451C2×C8C2×C4×D1714yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—342no—
458C4⋊C4—342no—
459C4⋊C4—342no—
460C4⋊C4—342no—
461C4⋊C4—682no—
462C22⋊C4—42no—
463C22⋊C4—684no—
464C22⋊C4—682no—
465C22⋊C4—682no—
466C22⋊C4—684no—
467C22⋊C4—682no—
468C22⋊C4—682no—
469C22⋊C4—682no—
470C42—682no—
471C42—682no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—172no—
477C23—172no—
478C23—172no—
479C23—172no—
480C23—348no—
481C23—341no—
482C23—341no—
483C23—341no—
484C23—341no—
485C23—341no—
486C23—341no—
487C23—341no—
488C23—341no—
489C23—348no—
490C23—341no—
491C23—341no—
492C23—684no—
493C23—684no—
494C23—684no—
495C23—684no—
496C23—684no—
497C23C4⋊C4×D1712yes—
498C23#1058760212yes—
499C23#1058760212yes—
500C23C2×D4×D1711yescharacteristic
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—84no—
515C2×C4—172no—
516C2×C4—172no—
517C2×C4—341no—
518C2×C4—341no—
519C2×C4—341no—
520C2×C4—341no—
521C2×C4—344no—
522C2×C4—344no—
523C2×C4—344no—
524C2×C4—344no—
525C2×C4—684no—
526C2×C4—684no—
527C2×C4—684no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—682no—
531C2×C4—682no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—682no—
537C2×C4—1364no—
538C2×C4C4⋊C4×D1712yes—
539C2×C4C22×C4×D1711yescharacteristic
540C2×C4#1058769112yes—
541C2×C4C2×Q8×D1711yescharacteristic
542C2×C4#1058769612yes—
543C8—24no—
544C8—344no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—174no—
552C22—342no—
553C22—342no—
554C22—342no—
555C22—342no—
556C22—342no—
557C22—342no—
558C22—342no—
559C22—342no—
560C22—342no—
561C22—6816no—
562C22—681no—
563C22—681no—
564C22—681no—
565C22—681no—
566C22#1176284414yes—
567C22C2×C4⋊C4×D1711yescharacteristic
568C22#1176336911yescharacteristic
569C22#1176337011yescharacteristic
570C22C2×#1058760211yescharacteristic
571C22C2×#1058760211yescharacteristic
572C22AGL(1,ℤ/8)×D1711yescharacteristic
573C22#137×D1711yescharacteristic
574C4—22no—
575C4—81no—
576C4—82no—
577C4—81no—
578C4—342no—
579C4—342no—
580C4—1362no—
581C4—1361no—
582C4—1361no—
583C4#1176328112yes—
584C4#1176343412yes—
585C2—24no—
586C2—344no—
587C2—682no—
588C2#1453139511yescharacteristic
589C2C2×#1176284411yescharacteristic
590C2D17×#42811yescharacteristic
591C2#1453954811yescharacteristic
592C2#1453955211yescharacteristic
593C2#1453956211yescharacteristic
594C2#1453957011yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1853C2
socleC23×C17C22×D4
prime radicalC24×D17D4
cosocleC22×C4×C17C24
hypercenter#1853D17
hypocenterC17#21941
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
4. #14528984C2semidirect
5. #14528982C2semidirect
6. #14528979C2semidirect
7. #14528972C2semidirect
8. #14528971C2semidirect
9. #14528812C2semidirect
10. #14528806C2semidirect
11. C17×#1853C2semidirect
18. C2×#10587478C22semidirect
19. C2×#10587478C22semidirect
20. C2×#10587488C22semidirect
23. C22⋊C8×D17C4semidirect
36. #11762572C22semidirect
37. #11762571C22semidirect
38. #11762571C22semidirect
40. C2×C17×#106C22semidirect
43. C17×#340C22semidirect
82. C2×C8×D17C2×C4semidirect
94. #10587491C2×C4semidirect
254. #1853D17semidirect
309. C2×#106C2×D17semidirect
330. #340C2×D17semidirect
395. C17#21941semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C8×D172
#145311662
C2×#117626662
#145289842
#145289822
#145289792
#145289722
#145289712
#145288122
#145288062
C17×#18532
#2194117