Finite Group Databasebeta

← #17045495

Subgroups — #17045495

Order 4352.

1224 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×Dic17C211yesmaximal, characteristic, complemented
2C2×#11762712C211yesmaximal, characteristic
3#14529640C212yesmaximal, complemented
4#14529469C212yesmaximal, complemented
5#14529413C212yesmaximal, complemented
6#14529141C212yesmaximal, complemented
7#14529092C211yesmaximal, characteristic, complemented
8#14528979C211yesmaximal, characteristic, complemented
9#14528715C211yesmaximal, characteristic
10C22⋊C8×Dic17C211yesmaximal, characteristic, complemented
11C2×C17×#450C211yesmaximal, characteristic
12C2×Q8×Dic17C2211yescharacteristic, complemented
13C22×C4⋊Dic17C2211yescharacteristic
14C22×C22⋊Dic17—21no—
15C22×C22⋊Dic17—21no—
16C22×C4×Dic17C2211yescharacteristic
17C2×#10587488C2212yescomplemented
18C2×#10587506C2212yescomplemented
19C2×#10587503C2212yes—
20C2×#10587503C2212yes—
21C2×C22⋊#3622C2211yescharacteristic
22C2×C8×Dic17C2212yescomplemented
23#11762716C2211yescharacteristic, complemented
24#11762712—21no—
25#11762712—21no—
26#11762712C2212yes—
27#11762712C2212yes—
28#11762708C2211yescharacteristic, complemented
29#11762695—21no—
30#11762684—21no—
31#11762669C2211yescharacteristic
32C4⋊C4×Dic17—21no—
33C4⋊C4×Dic17C2211yescharacteristic, complemented
34C22⋊C4×Dic17—21no—
35#11762572C2212yescomplemented
36#11762571C2212yescomplemented
37#11762563C2212yescomplemented
38C23×D4×C17C2211yescharacteristic
39C22×SD16×C17C2212yes—
40C2×C17×C22⋊Q8C2211yescharacteristic
41C17×#450C418yescomplemented
42C2×C17×#102C2212yes—
43C2×C17×C22⋊C8C2211yescharacteristic
44C25×C17—21no—
45C22×Q8×C17C2311yescharacteristic
46C22×D4×C17—21no—
47C22×D4×C17—22no—
48C22×D4×C17—42no—
49C22×D4×C17—42no—
50C22×D4×C17C2×C414yes—
51C22×D4×C17C2312yes—
52C23×C4×C17C2311yescharacteristic
53C23×Dic17—21no—
54C23×Dic17D411yescharacteristic
55C23×Dic17D411yescharacteristic
56C2×SD16×C17—42no—
57C2×SD16×C17—42no—
58C2×SD16×C17C2×C418yescomplemented
59C22×C8×C17C2312yes—
60C17×C22⋊Q8—22no—
61C17×C22⋊Q8C2×C414yes—
62C2×C4⋊C4×C17—21no—
63C2×C4⋊C4×C17C2311yescharacteristic
64C2×C22⋊C4×C17—21no—
65C2×C22⋊Dic17—21no—
66C2×C22⋊Dic17—22no—
67C2×C22⋊Dic17—22no—
68C2×C22⋊Dic17—22no—
69C2×C22⋊Dic17—22no—
70C2×C22⋊Dic17—44no—
71C2×C22⋊Dic17—44no—
72Q8×Dic17—42no—
73C2×C4⋊Dic17—21no—
74C2×C4⋊Dic17—21no—
75C2×C4⋊Dic17—21no—
76C2×C4⋊Dic17—42no—
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2312yes—
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4×Dic17—21no—
81C2×C4×Dic17—21no—
82C2×C4×Dic17—42no—
83C2×C4×Dic17D411yescharacteristic
84C2×C4×Dic17D411yescharacteristic
85C2×C4×Dic17C2311yescharacteristic
86C2×C4×Dic17C2311yescharacteristic
87C22×#3622C2312yes—
88C17×#102C2×C418yescomplemented
89C17×C22⋊C8C2×C414yes—
90#10587506—42no—
91#10587503—42no—
92#10587503—42no—
93#10587502—21no—
94#10587502—21no—
95#10587502—21no—
96#10587502—21no—
97#10587502—21no—
98C22⋊#3622—22no—
99#10587488—42no—
100C8×Dic17—42no—
101C24×C17—24no—
102C24×C17—22no—
103C24×C17—42no—
104C24×C17—42no—
105C24×C17—42no—
106C24×C17—42no—
107C24×C17—42no—
108C24×C17—42no—
109C24×C17C2×D411yescharacteristic
110C2×Q8×C17—21no—
111C2×Q8×C17—22no—
112C2×Q8×C17—22no—
113C2×Q8×C17C22×C412yes—
114C2×D4×C17—24no—
115C2×D4×C17—22no—
116C2×D4×C17—22no—
117C2×D4×C17—22no—
118C2×D4×C17—22no—
119C2×D4×C17—22no—
120C2×D4×C17—48no—
121C2×D4×C17—42no—
122C2×D4×C17—42no—
123C2×D4×C17—48no—
124C2×D4×C17C22×C414yes—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17C22×C412yes—
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17#4112yes—
132C22×C4×C17C2411yescharacteristic
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—41no—
139C22×Dic17—41no—
140C22×Dic17—41no—
141C22×Dic17—42no—
142C22×Dic17—42no—
143C22×Dic17—42no—
144C22×Dic17SD1614yescomplemented
145C22×Dic17C2×D411yescharacteristic
146C22×Dic17C2×D411yescharacteristic
147C22×Dic17C2×D411yescharacteristic
148C22×Dic17C2×D411yescharacteristic
149SD16×C17—48no—
150C2×C8×C17—22no—
151C2×C8×C17—22no—
152C2×C8×C17C22×C414yes—
153C4⋊C4×C17—24no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17C22×C412yes—
156C22⋊C4×C17—24no—
157C22⋊Dic17—44no—
158C22⋊Dic17—42no—
159C22⋊Dic17—44no—
160C22⋊Dic17—44no—
161C22⋊Dic17—44no—
162C4⋊Dic17—22no—
163C4⋊Dic17—22no—
164C4⋊Dic17—22no—
165C4⋊Dic17—22no—
166C4⋊Dic17—42no—
167C4⋊Dic17—42no—
168C4⋊Dic17—42no—
169C4×Dic17—22no—
170C4×Dic17—22no—
171C4×Dic17—41no—
172C4×Dic17—41no—
173C4×Dic17—42no—
174C4×Dic17—42no—
175C2×#3622—22no—
176C2×#3622—22no—
177C2×#3622—22no—
178#22043—171nomaximal, Sylow
179C23×C17—21no—
180C23×C17—21no—
181C23×C17—21no—
182C23×C17—21no—
183C23×C17—24no—
184C23×C17—48no—
185C23×C17—48no—
186C23×C17—42no—
187C23×C17—42no—
188C23×C17—42no—
189C23×C17—42no—
190C23×C17—42no—
191C23×C17—42no—
192C23×C17—48no—
193C23×C17—84no—
194C23×C17—84no—
195C23×C17—84no—
196C23×C17—84no—
197C23×C17C4×D412yes—
198C23×C17C2×SD1612yes—
199C23×C17#13512yes—
200C23×C17C22×D411yescharacteristic
201Q8×C17—24no—
202Q8×C17—42no—
203D4×C17—24no—
204D4×C17—24no—
205D4×C17—48no—
206D4×C17—42no—
207D4×C17—42no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—24no—
211C2×C4×C17—24no—
212C2×C4×C17—22no—
213C2×C4×C17—22no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17C4×D412yes—
219C2×C4×C17C23×C411yescharacteristic
220C2×C4×C17C2×#4111yescharacteristic
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—21no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—44no—
233C2×Dic17—44no—
234C2×Dic17—41no—
235C2×Dic17—42no—
236C2×Dic17—41no—
237C2×Dic17—42no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—42no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—84no—
244C2×Dic17C2×SD1612yes—
245C2×Dic17C2×SD1612yes—
246C2×Dic17AGL(1,ℤ/8)12yes—
247C8×C17—24no—
248#3622—42no—
249C2×#450D1711yescharacteristic
250C4×C22⋊Q8—171no—
251C2×#390—171no—
252#1968—172no—
253#1934—172no—
254#1926—172no—
255#1880—172no—
256#1872—171no—
257#1853—171no—
258#1786—171no—
259C4×C22⋊C8—171no—
260C22×C17—22no—
261C22×C17—22no—
262C22×C17—22no—
263C22×C17—22no—
264C22×C17—22no—
265C22×C17—22no—
266C22×C17—44no—
267C22×C17—44no—
268C22×C17—44no—
269C22×C17—816no—
270C22×C17—82no—
271C22×C17—82no—
272C22×C17—82no—
273C22×C17—82no—
274C22×C17C4×SD1614yes—
275C22×C17C2×C4×D411yescharacteristic
276C22×C17#53411yescharacteristic
277C22×C17#53511yescharacteristic
278C22×C17C22×SD1611yescharacteristic
279C22×C17C2×#13511yescharacteristic
280C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C17#57511yescharacteristic
282C4×C17—22no—
283C4×C17—42no—
284C4×C17—42no—
285C4×C17#51812yes—
286C4×C17#53812yes—
287Dic17—22no—
288Dic17—22no—
289Dic17—22no—
290Dic17—41no—
291Dic17—81no—
292Dic17—82no—
293Dic17—81no—
294C23×D4C2×D1711yescharacteristic
295C22×SD16C2×D1712yes—
296C2×C22⋊Q8C2×D1711yescharacteristic
297C2×C4×Q8—171no—
298C22×C4⋊C4—171no—
299C22×C22⋊C4—341no—
300C22×C22⋊C4—341no—
301C22×C42—171no—
302#450Dic1718yescomplemented
303C2×#106—172no—
304C2×#103—172no—
305C2×#102—172no—
306C2×#102—172no—
307C2×#102C2×D1712yes—
308C2×C22⋊C8—171no—
309C2×C22⋊C8C2×D1711yescharacteristic
310C2×C4×C8—172no—
311#391—171no—
312#390—172no—
313#390—172no—
314#390—341no—
315#390—341no—
316#389—171no—
317#387—341no—
318#385—341no—
319#380—171no—
320C4×C4⋊C4—171no—
321C4×C4⋊C4—341no—
322C4×C22⋊C4—341no—
323#340—172no—
324#340—172no—
325#336—172no—
326C2×C17—24no—
327C2×C17—84no—
328C2×C17#232111yescharacteristic
329C2×C17C2×C4×SD1611yescharacteristic
330C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
331C2×C17#318711yescharacteristic
332C2×C17#318811yescharacteristic
333C2×C17#320811yescharacteristic
334C2×C17#321111yescharacteristic
335C25—21no—
336C22×Q8C22×D1711yescharacteristic
337C22×D4—21no—
338C22×D4—22no—
339C22×D4—42no—
340C22×D4—42no—
341C22×D4C2×Dic1714yes—
342C22×D4C22×D1712yes—
343C23×C4—171no—
344C23×C4—171no—
345C23×C4—341no—
346C23×C4C22×D1711yescharacteristic
347C2×SD16—42no—
348C2×SD16—42no—
349C2×SD16C2×Dic1718yescomplemented
350C22×C8—172no—
351C22×C8C22×D1712yes—
352C22⋊Q8—22no—
353C22⋊Q8C2×Dic1714yes—
354C4×Q8—682no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—172no—
357C2×C4⋊C4—171no—
358C2×C4⋊C4—171no—
359C2×C4⋊C4—341no—
360C2×C4⋊C4—341no—
361C2×C4⋊C4—341no—
362C2×C4⋊C4—682no—
363C2×C4⋊C4C22×D1711yescharacteristic
364C2×C22⋊C4—21no—
365C2×C22⋊C4—342no—
366C2×C22⋊C4—342no—
367C2×C22⋊C4—342no—
368C2×C22⋊C4—342no—
369C2×C22⋊C4—341no—
370C2×C22⋊C4—684no—
371C2×C22⋊C4—684no—
372C2×C42—171no—
373C2×C42—171no—
374C2×C42—171no—
375C2×C42—171no—
376C2×C42—341no—
377C2×C42—341no—
378C2×C42—682no—
379#106—682no—
380#103—682no—
381#102—682no—
382#102—682no—
383#102C2×Dic1718yescomplemented
384C22⋊C8—342no—
385C22⋊C8C2×Dic1714yes—
386C4×C8—682no—
387#95—341no—
388#95—341no—
389#95—341no—
390#95—341no—
391#95—341no—
392C17#2204311yescharacteristic, complemented, Sylow
393C24—22no—
394C24—24no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24D4×D1711yescharacteristic
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8C22×Dic1712yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4C22×Dic1714yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—171no—
422C22×C4—171no—
423C22×C4—171no—
424C22×C4—174no—
425C22×C4—171no—
426C22×C4—342no—
427C22×C4—342no—
428C22×C4—341no—
429C22×C4—341no—
430C22×C4—342no—
431C22×C4—681no—
432C22×C4—681no—
433C22×C4—681no—
434C22×C4—682no—
435C22×C4—682no—
436C22×C4—682no—
437C22×C4D4×D1711yescharacteristic
438C22×C4#6325412yes—
439C22×C4C22×Dic1712yes—
440C22×C4C23×D1711yescharacteristic
441SD16—48no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—342no—
445C2×C8—342no—
446C2×C8—342no—
447C2×C8C22×Dic1714yes—
448C4⋊C4—21no—
449C4⋊C4—24no—
450C4⋊C4—342no—
451C4⋊C4—342no—
452C4⋊C4—342no—
453C4⋊C4—342no—
454C4⋊C4—682no—
455C4⋊C4—682no—
456C4⋊C4—682no—
457C4⋊C4C22×Dic1712yes—
458C22⋊C4—24no—
459C22⋊C4—684no—
460C22⋊C4—684no—
461C22⋊C4—684no—
462C22⋊C4—684no—
463C22⋊C4—682no—
464C42—342no—
465C42—342no—
466C42—681no—
467C42—681no—
468C42—682no—
469C42—682no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—24no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—48no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23SD16×D1712yes—
489C23#1058760212yes—
490C23D4×Dic1712yes—
491C23C2×D4×D1711yescharacteristic
492Q8—24no—
493Q8—42no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—172no—
510C2×C4—172no—
511C2×C4—172no—
512C2×C4—341no—
513C2×C4—341no—
514C2×C4—341no—
515C2×C4—341no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—341no—
519C2×C4—341no—
520C2×C4—341no—
521C2×C4—684no—
522C2×C4—684no—
523C2×C4—682no—
524C2×C4—681no—
525C2×C4—681no—
526C2×C4—682no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—682no—
531C2×C4—681no—
532C2×C4—1364no—
533C2×C4D4×Dic1712yes—
534C2×C4C2×#6325411yescharacteristic
535C2×C4#1058769112yes—
536C2×C4#1058769112yes—
537C2×C4C23×Dic1711yescharacteristic
538C8—24no—
539C8—682no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22SD16×Dic1714yes—
555C22C2×D4×Dic1711yescharacteristic
556C22#1176336911yescharacteristic
557C22#1176337411yescharacteristic
558C22C2×SD16×D1711yescharacteristic
559C22C2×#1058760211yescharacteristic
560C22AGL(1,ℤ/8)×D1711yescharacteristic
561C22#1176360511yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—342no—
566C4—342no—
567C4—342no—
568C4—681no—
569C4—1361no—
570C4—1362no—
571C4—1361no—
572C4#1176328412yes—
573C4#1176339412yes—
574C2—24no—
575C2—84no—
576C2#1453144511yescharacteristic
577C2C2×SD16×Dic1711yescharacteristic
578C2AGL(1,ℤ/8)×Dic1711yescharacteristic
579C2#1453953811yescharacteristic
580C2#1453955211yescharacteristic
581C2#1453972311yescharacteristic
582C2#1453975211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#450C2
socleC23×C17C22×D4
prime radicalC23×D4×C17C22
cosocleC22×C4×C17C24
hypercenterC2×#450D17
hypocenterC17#22043
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×Dic17C2semidirect
3. #14529640C2semidirect
4. #14529469C2semidirect
5. #14529413C2semidirect
6. #14529141C2semidirect
7. #14529092C2semidirect
8. #14528979C2semidirect
10. C22⋊C8×Dic17C2semidirect
12. C2×Q8×Dic17C22semidirect
17. C2×#10587488C22semidirect
18. C2×#10587506C22semidirect
22. C2×C8×Dic17C22semidirect
23. #11762716C22semidirect
28. #11762708C22semidirect
33. C4⋊C4×Dic17C22semidirect
35. #11762572C22semidirect
36. #11762571C22semidirect
37. #11762563C22semidirect
41. C17×#450C4semidirect
58. C2×SD16×C17C2×C4semidirect
88. C17×#102C2×C4semidirect
144. C22×Dic17SD16semidirect
302. #450Dic17semidirect
349. C2×SD16C2×Dic17semidirect
383. #102C2×Dic17semidirect
392. C17#22043semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×Dic172
C2×#117627122
#145296402
#145294692
#145294132
#145291412
#145290922
#145289792
#145287152
C22⋊C8×Dic172
C2×C17×#4502
#2204317