Finite Group Databasebeta

← #17049976

Subgroups — #17049976

Order 4352.

1530 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762887C211yesmaximal, characteristic, complemented
2C2×#11762810C212yesmaximal, complemented
3D4×C4⋊Dic17C211yesmaximal, characteristic, complemented
4#14529708C211yesmaximal, characteristic, complemented
5#14529587C211yesmaximal, characteristic, complemented
6#14529218C212yesmaximal, complemented
7#14529092C212yesmaximal, complemented
8#14528784C211yesmaximal, characteristic, complemented
9#14528724C212yesmaximal, complemented
10C4×#10587503C211yesmaximal, characteristic, complemented
11C2×C17×#547C211yesmaximal, characteristic
12C22×#63223C2212yes—
13C2×D4×Dic17—21no—
14C22×C4⋊Dic17C2212yes—
15#11762887—24no—
16C2×C4⋊#3622C2211yescharacteristic
17#11762810—28no—
18C2×#10587506C2212yes—
19C2×#10587506C2212yes—
20C2×#10587506C2212yes—
21C2×#10587506C2212yes—
22C2×#10587503C2212yes—
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×C22⋊#3622C2212yes—
26C2×C4×#3622C2211yescharacteristic
27#11762708—21no—
28#11762684—22no—
29#11762679C2211yescharacteristic
30#11762669C2212yes—
31C4×C4⋊Dic17C2211yescharacteristic
32#11762571C2212yes—
33#11762571C2211yescharacteristic
34#11762571C2211yescharacteristic
35C22×#41×C17C2212yes—
36C17×#547—24no—
37C17×#547C418yescomplemented
38C2×C17×C4⋊Q8C2211yescharacteristic
39C2×C17×#123—22no—
40C2×C17×C22⋊Q8C2212yes—
41C2×C17×#121—21no—
42C2×C4×D4×C17C2211yescharacteristic
43C2×C4×D4×C17C2211yescharacteristic
44C22×C4⋊C4×C17—21no—
45C2×#41×C17—24no—
46C2×#41×C17—24no—
47C2×#41×C17—24no—
48C2×#41×C17—42no—
49C2×#41×C17—42no—
50C2×#41×C17C2×C418yescomplemented
51C22×Q8×C17C2312yes—
52C22×D4×C17—22no—
53C22×D4×C17D411yescharacteristic
54C22×D4×C17C2311yescharacteristic
55C23×C4×C17—21no—
56C23×C4×C17—22no—
57C23×C4×C17C2312yes—
58C23×Dic17—22no—
59C17×C4⋊Q8—22no—
60C17×C4⋊Q8C2×C414yescomplemented
61C17×#123—216no—
62C17×C22⋊Q8—24no—
63C17×C22⋊Q8C2×C418yescomplemented
64C17×#121—28no—
65C4×D4×C17—22no—
66C4×D4×C17C2×C414yes—
67C4×D4×C17C2×C414yescomplemented
68C4×D4×C17D414yes—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—22no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—28no—
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—22no—
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17D412yes—
78C2×C42×C17C2311yescharacteristic
79C2×C22⋊Dic17—22no—
80D4×Dic17—44no—
81C2×C4⋊Dic17—22no—
82C2×C4⋊Dic17—21no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—24no—
85C2×C4⋊Dic17—42no—
86C2×C4⋊Dic17C2312yes—
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4×Dic17—21no—
90C2×#63223—24no—
91C2×#63223—24no—
92C2×#63223—42no—
93C22×#3622C2311yescharacteristic
94C22×#3622C2312yes—
95C22×#3622C2311yescharacteristic
96#10587506—24no—
97#10587506—24no—
98#10587506—42no—
99#10587506—42no—
100#10587503—24no—
101#10587503—22no—
102#10587503D414yescomplemented
103#10587502—22no—
104#10587502—21no—
105C22⋊#3622—24no—
106C4⋊#3622—22no—
107C4×#3622—22no—
108C24×C17—21no—
109C24×C17C2×D412yes—
110#41×C17—216no—
111#41×C17—48no—
112#41×C17—48no—
113#41×C17—48no—
114C2×Q8×C17—22no—
115C2×Q8×C17—22no—
116C2×Q8×C17—22no—
117C2×Q8×C17—22no—
118C2×Q8×C17—22no—
119C2×Q8×C17C22×C414yes—
120C2×D4×C17—28no—
121C2×D4×C17—22no—
122C2×D4×C17—22no—
123C2×D4×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17—44no—
127C2×D4×C17—44no—
128C2×D4×C17C22⋊C414yes—
129C2×D4×C17C22×C412yes—
130C2×D4×C17C2×D412yes—
131C2×D4×C17#4112yes—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—21no—
138C22×C4×C17—28no—
139C22×C4×C17—24no—
140C22×C4×C17—44no—
141C22×C4×C17—44no—
142C22×C4×C17—42no—
143C22×C4×C17—42no—
144C22×C4×C17C22×C414yes—
145C22×C4×C17C2×D412yes—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D414yes—
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2411yescharacteristic
150C22×Dic17—21no—
151C22×Dic17—22no—
152C22×Dic17—21no—
153C22×Dic17—44no—
154C22×Dic17—44no—
155C22×Dic17—82no—
156C4⋊C4×C17—28no—
157C4⋊C4×C17—24no—
158C4⋊C4×C17—24no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17C22⋊C414yes—
161C4⋊C4×C17C22×C412yes—
162C22⋊C4×C17—28no—
163C22⋊C4×C17—24no—
164C22⋊C4×C17C22⋊C418yes—
165C42×C17C22×C412yes—
166C42×C17C2×D412yes—
167C22⋊Dic17—44no—
168C4⋊Dic17—21no—
169C4⋊Dic17—22no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—22no—
173C4⋊Dic17—44no—
174C4⋊Dic17—42no—
175C4⋊Dic17—81no—
176C4⋊Dic17C2×D412yes—
177C4×Dic17—42no—
178#63223—48no—
179C2×#3622—21no—
180C2×#3622—21no—
181C2×#3622—21no—
182C2×#3622—22no—
183C2×#3622—22no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—22no—
187C2×#3622C2×D412yes—
188#22431—171nomaximal, Sylow
189C23×C17—22no—
190C23×C17—22no—
191C23×C17—22no—
192C23×C17—22no—
193C23×C17—22no—
194C23×C17—22no—
195C23×C17—22no—
196C23×C17—22no—
197C23×C17—22no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17—41no—
201C23×C17—41no—
202C23×C17C2×C22⋊C414yes—
203C23×C17C22×D411yescharacteristic
204Q8×C17—24no—
205Q8×C17—24no—
206Q8×C17—42no—
207Q8×C17—42no—
208D4×C17—22no—
209D4×C17—22no—
210D4×C17—22no—
211D4×C17—28no—
212D4×C17—416no—
213D4×C17C4×D414yescomplemented
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—24no—
219C2×C4×C17—24no—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
222C2×C4×C17—22no—
223C2×C4×C17—24no—
224C2×C4×C17—416no—
225C2×C4×C17—42no—
226C2×C4×C17—42no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—48no—
230C2×C4×C17C2×C22⋊C412yes—
231C2×C4×C17C2×C22⋊C414yes—
232C2×C4×C17C2×C22⋊C412yes—
233C2×C4×C17AGL(1,ℤ/8)12yes—
234C2×C4×C17#13712yes—
235C2×C4×C17C23×C411yescharacteristic
236C2×C4×C17C22×D411yescharacteristic
237C2×C4×C17C22×D411yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17H(5,𝔽2)11yescharacteristic
240C2×C4×C17H(5,𝔽2)11yescharacteristic
241C2×Dic17—22no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—42no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17—41no—
250C2×Dic17—88no—
251#3622—22no—
252#3622—42no—
253#3622—41no—
254C2×#547D1711yescharacteristic
255C2×#436—171no—
256C2×#417—172no—
257D4×C4⋊C4—171no—
258#1981—171no—
259#1956—171no—
260#1894—172no—
261#1872—172no—
262#1802—171no—
263#1788—172no—
264C4×#102—171no—
265C22×C17—24no—
266C22×C17—24no—
267C22×C17—24no—
268C22×C17—24no—
269C22×C17—42no—
270C22×C17—42no—
271C22×C17—42no—
272C22×C17—42no—
273C22×C17—42no—
274C22×C17—42no—
275C22×C17—42no—
276C22×C17—42no—
277C22×C17—42no—
278C22×C17—42no—
279C22×C17C22×C22⋊C411yescharacteristic
280C22×C17D4211yescharacteristic
281C22×C17#54611yescharacteristic
282C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C17C2×#13711yescharacteristic
284C22×C17#57711yescharacteristic
285C22×C17#57811yescharacteristic
286C4×C17—24no—
287C4×C17—22no—
288C4×C17—44no—
289C4×C17—42no—
290C4×C17#41714yescomplemented
291C4×C17C2×C4×D411yescharacteristic
292C4×C17#51811yescharacteristic
293C4×C17D4211yescharacteristic
294C4×C17#54611yescharacteristic
295Dic17—42no—
296Dic17—81no—
297Dic17—82no—
298C22×#41C2×D1712yes—
299C22×M16—172no—
300#547—24no—
301#547Dic1718yescomplemented
302C2×C4⋊Q8C2×D1711yescharacteristic
303C2×#123—22no—
304C2×C22⋊Q8C2×D1712yes—
305C2×#121—21no—
306C2×C4×D4—341no—
307C2×C4×D4C2×D1711yescharacteristic
308C2×C4×D4C2×D1711yescharacteristic
309C22×C4⋊C4—21no—
310C22×C4⋊C4—172no—
311#436—344no—
312C2×C4⋊C8—171no—
313#417—348no—
314C2×#103—172no—
315C2×#103—172no—
316C2×#103—172no—
317C2×#103—172no—
318C2×#102—171no—
319C2×#102—172no—
320C2×#102—171no—
321C2×C22⋊C8—172no—
322C2×C4×C8—171no—
323#389—341no—
324#385—342no—
325#384—171no—
326#380—172no—
327C4×C4⋊C4—171no—
328#340—172no—
329#340—171no—
330#340—171no—
331C2×C17—44no—
332C2×C17—44no—
333C2×C17D4×C22⋊C411yescharacteristic
334C2×C17C2×#41711yescharacteristic
335C2×C17#289811yescharacteristic
336C2×C17#327211yescharacteristic
337C2×C17#327611yescharacteristic
338C2×C17#333011yescharacteristic
339C2×C17#333111yescharacteristic
340C2×#41—24no—
341C2×#41—24no—
342C2×#41—24no—
343C2×#41—42no—
344C2×#41—42no—
345C2×#41C2×Dic1718yescomplemented
346C22×Q8C22×D1712yes—
347C22×D4—22no—
348C22×D4C22⋊D1711yescharacteristic
349C22×D4C22×D1711yescharacteristic
350C23×C4—21no—
351C23×C4—22no—
352C23×C4—342no—
353C23×C4C22×D1712yes—
354C2×M16—344no—
355C2×M16—344no—
356C2×M16—682no—
357C22×C8—171no—
358C22×C8—172no—
359C22×C8—171no—
360C4⋊Q8—22no—
361C4⋊Q8C2×Dic1714yescomplemented
362#123—216no—
363C22⋊Q8—24no—
364C22⋊Q8C2×Dic1718yescomplemented
365#121—28no—
366C4×D4—22no—
367C4×D4—684no—
368C4×D4C2×Dic1714yes—
369C4×D4C2×Dic1714yescomplemented
370C4×D4C22⋊D1714yes—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—28no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—172no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—344no—
379C2×C4⋊C4—342no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—341no—
382C2×C4⋊C4—682no—
383C2×C4⋊C4C22⋊D1711yescharacteristic
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C22⋊C4—22no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—342no—
388C2×C22⋊C4C22⋊D1712yes—
389C2×C42—341no—
390C2×C42C22×D1711yescharacteristic
391C4⋊C8—342no—
392#103—344no—
393#103—344no—
394#103—682no—
395#103—682no—
396#102—174no—
397#102—344no—
398#102—342no—
399C22⋊C8—344no—
400C4×C8—342no—
401#95—341no—
402#95—342no—
403C17#2243111yescharacteristic, complemented, Sylow
404C24—21no—
405C24C2×C22⋊D1712yes—
406#41—216no—
407#41—48no—
408#41—48no—
409#41—48no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8C22×Dic1714yes—
416C2×D4—28no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4C22⋊Dic1714yes—
425C2×D4D4×D1712yes—
426C2×D4#6325412yes—
427C2×D4C22×Dic1712yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—341no—
441C22×C4—341no—
442C22×C4—342no—
443C22×C4—684no—
444C22×C4—684no—
445C22×C4—1362no—
446C22×C4C22×Dic1714yes—
447C22×C4C2×C22⋊D1714yes—
448C22×C4C2×C22⋊D1711yescharacteristic
449C22×C4C2×C22⋊D1712yes—
450C22×C4C2×C22⋊D1711yescharacteristic
451C22×C4C23×D1711yescharacteristic
452M16—688no—
453C2×C8—172no—
454C2×C8—341no—
455C2×C8—341no—
456C2×C8—342no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8—342no—
460C2×C8—341no—
461C2×C8—342no—
462C4⋊C4—21no—
463C4⋊C4—24no—
464C4⋊C4—28no—
465C4⋊C4—24no—
466C4⋊C4—172no—
467C4⋊C4—341no—
468C4⋊C4—342no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—342no—
472C4⋊C4—684no—
473C4⋊C4—682no—
474C4⋊C4—1361no—
475C4⋊C4C22⋊Dic1714yes—
476C4⋊C4C22×Dic1712yes—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—684no—
480C22⋊C4C22⋊Dic1718yes—
481C42—682no—
482C42C22×Dic1712yes—
483C42C2×C22⋊D1712yes—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×C22⋊Dic1714yes—
498C23C22×C22⋊D1711yescharacteristic
499Q8—24no—
500Q8—24no—
501Q8—42no—
502Q8—42no—
503D4—22no—
504D4—28no—
505D4—22no—
506D4—22no—
507D4—416no—
508D4D4×Dic1714yescomplemented
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—416no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—48no—
525C2×C4—342no—
526C2×C4—681no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—682no—
530C2×C4—682no—
531C2×C4—681no—
532C2×C4—682no—
533C2×C4—681no—
534C2×C4—1368no—
535C2×C4#1058763512yes—
536C2×C4#1058763712yes—
537C2×C4C2×C22⋊Dic1712yes—
538C2×C4C2×C22⋊Dic1714yes—
539C2×C4C2×C22⋊Dic1712yes—
540C2×C4C2×D4×D1711yescharacteristic
541C2×C4C2×#6325411yescharacteristic
542C2×C4#1058769111yescharacteristic
543C2×C4#1058769111yescharacteristic
544C2×C4C23×Dic1711yescharacteristic
545C2×C4C22×C22⋊D1711yescharacteristic
546C8—342no—
547C8—682no—
548C8—681no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22C22×C22⋊Dic1711yescharacteristic
564C22D4×C22⋊D1711yescharacteristic
565C22#1176345311yescharacteristic
566C22C2×#1058763511yescharacteristic
567C22C2×#1058763711yescharacteristic
568C22#1176361711yescharacteristic
569C22#1176362411yescharacteristic
570C4—24no—
571C4—22no—
572C4—44no—
573C4—42no—
574C4—682no—
575C4—1361no—
576C4—1362no—
577C4#1176281014yescomplemented
578C4C2×D4×Dic1711yescharacteristic
579C4#1176328411yescharacteristic
580C4D4×C22⋊D1711yescharacteristic
581C4#1176345311yescharacteristic
582C2—44no—
583C2—44no—
584C2D4×C22⋊Dic1711yescharacteristic
585C2C2×#1176281011yescharacteristic
586C2#1453731711yescharacteristic
587C2#1454038411yescharacteristic
588C2#1454042611yescharacteristic
589C2#1454108911yescharacteristic
590C2#1454110011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#547C2
socleC23×C17C22×D4
cosocleC22×C4×C17C24
hypercenterC2×#547D17
hypocenterC17#22431
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#11762887C2semidirect
2. C2×#11762810C2semidirect
3. D4×C4⋊Dic17C2semidirect
4. #14529708C2semidirect
5. #14529587C2semidirect
6. #14529218C2semidirect
7. #14529092C2semidirect
8. #14528784C2semidirect
9. #14528724C2semidirect
10. C4×#10587503C2semidirect
37. C17×#547C4semidirect
50. C2×#41×C17C2×C4semidirect
60. C17×C4⋊Q8C2×C4semidirect
63. C17×C22⋊Q8C2×C4semidirect
67. C4×D4×C17C2×C4semidirect
102. #10587503D4semidirect
213. D4×C17C4×D4semidirect
290. C4×C17#417semidirect
301. #547Dic17semidirect
345. C2×#41C2×Dic17semidirect
361. C4⋊Q8C2×Dic17semidirect
364. C22⋊Q8C2×Dic17semidirect
369. C4×D4C2×Dic17semidirect
403. C17#22431semidirect
508. D4D4×Dic17semidirect
577. C4#11762810semidirect

Maximal subgroups

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

subgroupindex
C2×#117628872
C2×#117628102
D4×C4⋊Dic172
#145297082
#145295872
#145292182
#145290922
#145287842
#145287242
C4×#105875032
C2×C17×#5472
#2243117