Finite Group Databasebeta

← #17049973

Subgroups — #17049973

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×#11763463C211yesmaximal, characteristic
2C2×#11762886C211yesmaximal, characteristic, complemented
3C2×#11762807C212yesmaximal, complemented
4#14531684C211yesmaximal, characteristic, complemented
5#14529706C211yesmaximal, characteristic, complemented
6#14529585C211yesmaximal, characteristic, complemented
7#14529216C212yesmaximal, complemented
8#14529090C212yesmaximal, complemented
9#14528721C212yesmaximal, complemented
10C4×#10587503C211yesmaximal, characteristic, complemented
11C17×#1802C211yesmaximal, characteristic, complemented
12C22×#63254C2212yes—
13C22×#63218C2212yes—
14#11763463—24no—
15#11763463C418yescomplemented
16C2×#10587555C2211yescharacteristic
17C2×#10587572—22no—
18C2×#10587583C2212yes—
19C2×#10587616—21no—
20C2×C4×D68C2211yescharacteristic
21C2×#10587579—21no—
22C22×C4⋊Dic17—21no—
23C2×C4⋊C4×D17C2212yes—
24#11762886—24no—
25#11762807—28no—
26C2×#10587487C2212yes—
27C2×#10587487C2212yes—
28C2×#10587481C2212yes—
29C2×#10587481C2212yes—
30C2×#10587503C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587491C2212yes—
33C2×C4×#3622C2211yescharacteristic
34#11762705—21no—
35#11762680—22no—
36#11762666C2212yes—
37C4×C4⋊Dic17C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40C2×C4×D4×C17C2211yescharacteristic
41C2×C17×C4⋊C8C2211yescharacteristic
42C2×C17×#102C2212yes—
43C17×#384C2211yescharacteristic
44C17×#340C2212yes—
45C22×D4×C17C2311yescharacteristic
46C23×C4×C17—21no—
47C22×C22⋊D17—22no—
48C23×Dic17—22no—
49C2×#63254—24no—
50C2×#63254—24no—
51C2×#63254—24no—
52C2×#63254—42no—
53C2×#63254—42no—
54C2×#63254C2×C418yescomplemented
55C22×D68D411yescharacteristic
56C22×C4×D17—22no—
57C22×C4×D17C2312yes—
58C22×Dic34C2312yes—
59C22×C8×C17C2312yes—
60C4×D4×C17—22no—
61C4×D4×C17C2×C414yescomplemented
62C2×C4⋊C4×C17—21no—
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17C2312yes—
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C22⋊C4×C17—21no—
67C2×C42×C17C2311yescharacteristic
68#10587616—28no—
69C2×#63227—22no—
70C2×#63227—22no—
71C2×#63227D412yes—
72C2×C4⋊Dic17—28no—
73C2×C4⋊Dic17—21no—
74C2×C4⋊Dic17—21no—
75C2×C4⋊Dic17—22no—
76C2×C4⋊Dic17D411yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×#63225—22no—
80C2×C4×Dic17—21no—
81C22×#3622C2311yescharacteristic
82C22×#3622C2311yescharacteristic
83C2×#63218—24no—
84C2×#63218—24no—
85C2×#63218—42no—
86#10587583—24no—
87#10587583C2×C418yescomplemented
88#10587579—44no—
89C4⋊C4×D17—24no—
90C4⋊C4×D17—42no—
91#10587572—216no—
92C4×D68C2×C414yes—
93C4×D68D414yes—
94#10587555—22no—
95#10587555C2×C414yescomplemented
96C17×C4⋊C8—22no—
97C17×#102—24no—
98#10587503—22no—
99#10587503D414yescomplemented
100#10587502—21no—
101#10587502—22no—
102#10587491—24no—
103#10587487—24no—
104#10587487—42no—
105#10587481—24no—
106#10587481—42no—
107C4×#3622—22no—
108C24×C17—21no—
109C23×D17C2×D412yes—
110C2×D4×C17—21no—
111C2×D4×C17—21no—
112C2×D4×C17—21no—
113C2×D4×C17C22×C412yes—
114C2×D4×C17C2×D412yes—
115C2×D4×C17#4112yes—
116C22×C4×C17—21no—
117C22×C4×C17—24no—
118C22×C4×C17—22no—
119C22×C4×C17—42no—
120C22×C4×C17—42no—
121C22×C4×C17C2×D411yescharacteristic
122C22×C4×C17C2×D411yescharacteristic
123C22×C4×C17C2411yescharacteristic
124C2×C22⋊D17—28no—
125C2×C22⋊D17—44no—
126C2×C22⋊D17—44no—
127C22×Dic17—22no—
128C22×Dic17—21no—
129C22×Dic17—28no—
130C22×Dic17—21no—
131C22×Dic17—44no—
132C22×Dic17—44no—
133C22×Dic17C2×D412yes—
134#63254—216no—
135#63254—48no—
136#63254—48no—
137#63254—48no—
138C2×D68—22no—
139C2×D68—22no—
140C2×D68C22⋊C414yes—
141C2×C4×D17—22no—
142C2×C4×D17—22no—
143C2×C4×D17—22no—
144C2×C4×D17—22no—
145C2×C4×D17—44no—
146C2×C4×D17—44no—
147C2×C4×D17—82no—
148C2×C4×D17C22×C414yes—
149C2×C4×D17C2×D414yes—
150C2×Dic34—22no—
151C2×Dic34—22no—
152C2×Dic34—22no—
153C2×Dic34—22no—
154C2×Dic34—22no—
155C2×Dic34C22×C414yes—
156C2×C8×C17—22no—
157C2×C8×C17—22no—
158C2×C8×C17—22no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—22no—
161C4⋊C4×C17—22no—
162C4⋊C4×C17—42no—
163C4⋊C4×C17—81no—
164C4⋊C4×C17C22×C412yes—
165C22⋊C4×C17—24no—
166C42×C17C22×C412yes—
167C42×C17C2×D412yes—
168#63227—28no—
169#63227—44no—
170#63227C22⋊C418yes—
171C4⋊Dic17—24no—
172C4⋊Dic17—24no—
173C4⋊Dic17—21no—
174C4⋊Dic17—28no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17C22⋊C414yes—
178C4⋊Dic17C2×D412yes—
179#63225—44no—
180C4×Dic17—42no—
181C2×#3622—21no—
182C2×#3622—21no—
183C2×#3622—21no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622C2×D412yes—
187#63218—48no—
188#22431—171nomaximal, Sylow
189C23×C17—22no—
190C23×C17—22no—
191C23×C17—22no—
192C23×C17—41no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17C22×D411yescharacteristic
197C22×D17—22no—
198C22×D17—22no—
199C22×D17—22no—
200C22×D17—22no—
201C22×D17—22no—
202C22×D17—22no—
203C22×D17C2×C22⋊C414yes—
204D4×C17—22no—
205D4×C17—22no—
206D4×C17—22no—
207D4×C17C4×D414yescomplemented
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—22no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—48no—
214C2×C4×C17—42no—
215C2×C4×C17—42no—
216C2×C4×C17—42no—
217C2×C4×C17C2×C22⋊C412yes—
218C2×C4×C17C2×C22⋊C412yes—
219C2×C4×C17AGL(1,ℤ/8)12yes—
220C2×C4×C17#13712yes—
221C2×C4×C17C23×C411yescharacteristic
222C2×C4×C17C22×D411yescharacteristic
223C2×C4×C17C22×D411yescharacteristic
224C2×C4×C17C2×#4111yescharacteristic
225C2×C4×C17H(5,𝔽2)11yescharacteristic
226C2×C4×C17H(5,𝔽2)11yescharacteristic
227C22⋊D17—416no—
228C2×Dic17—22no—
229C2×Dic17—22no—
230C2×Dic17—24no—
231C2×Dic17—22no—
232C2×Dic17—416no—
233C2×Dic17—42no—
234C2×Dic17—42no—
235C2×Dic17—41no—
236C2×Dic17—41no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17C2×C22⋊C414yes—
241D68—28no—
242C4×D17—24no—
243C4×D17—24no—
244C4×D17—24no—
245C4×D17—24no—
246C4×D17—88no—
247Dic34—24no—
248Dic34—24no—
249Dic34—42no—
250Dic34—42no—
251C8×C17—42no—
252#3622—22no—
253#3622—41no—
254C2×#547—171no—
255C2×#436—171no—
256C2×#417—172no—
257D4×C4⋊C4—171no—
258#1981—171no—
259#1956—171no—
260#1894—172no—
261#1872—172no—
262#1802D1711yescharacteristic, complemented
263#1788—172no—
264C4×#102—171no—
265C22×C17—24no—
266C22×C17—42no—
267C22×C17—42no—
268C22×C17—42no—
269C22×C17—42no—
270C22×C17—42no—
271C22×C17—42no—
272C22×C17C22×C22⋊C411yescharacteristic
273C22×C17D4211yescharacteristic
274C22×C17#54611yescharacteristic
275C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C17C2×#13711yescharacteristic
277C22×C17#57711yescharacteristic
278C22×C17#57811yescharacteristic
279C2×D17—24no—
280C2×D17—24no—
281C2×D17—24no—
282C2×D17—42no—
283C2×D17—42no—
284C2×D17—42no—
285C2×D17—42no—
286C4×C17—22no—
287C4×C17—42no—
288C4×C17—82no—
289C4×C17#41714yescomplemented
290C4×C17C2×C4×D411yescharacteristic
291C4×C17#51811yescharacteristic
292C4×C17D4211yescharacteristic
293C4×C17#54611yescharacteristic
294Dic17—24no—
295Dic17—44no—
296Dic17—42no—
297Dic17—81no—
298C22×#41—172no—
299C22×M16—172no—
300#547—178no—
301#547—344no—
302C2×C4⋊Q8—171no—
303C2×#123—342no—
304C2×C22⋊Q8—172no—
305C2×#121—341no—
306C2×C4×D4—171no—
307C2×C4×D4—341no—
308C2×C4×D4C2×D1711yescharacteristic
309C22×C4⋊C4—172no—
310C22×C4⋊C4—341no—
311#436—344no—
312C2×C4⋊C8C2×D1711yescharacteristic
313#417—348no—
314C2×#103—172no—
315C2×#103—172no—
316C2×#103—172no—
317C2×#103—172no—
318C2×#102—171no—
319C2×#102—171no—
320C2×#102C2×D1712yes—
321C2×C22⋊C8—172no—
322C2×C4×C8—171no—
323#389—341no—
324#385—342no—
325#384C2×D1711yescharacteristic
326#380—172no—
327C4×C4⋊C4—171no—
328#340—171no—
329#340—171no—
330#340C2×D1712yes—
331C2×C17—44no—
332C2×C17D4×C22⋊C411yescharacteristic
333C2×C17C2×#41711yescharacteristic
334C2×C17#289811yescharacteristic
335C2×C17#327211yescharacteristic
336C2×C17#327611yescharacteristic
337C2×C17#333011yescharacteristic
338C2×C17#333111yescharacteristic
339D17—44no—
340C2×#41—178no—
341C2×#41—344no—
342C2×#41—344no—
343C2×#41—344no—
344C2×#41—682no—
345C2×#41—682no—
346C22×Q8—172no—
347C22×D4—171no—
348C22×D4—342no—
349C22×D4C22×D1711yescharacteristic
350C23×C4—21no—
351C23×C4—172no—
352C23×C4—342no—
353C23×C4—342no—
354C2×M16—344no—
355C2×M16—344no—
356C2×M16—682no—
357C22×C8—171no—
358C22×C8—171no—
359C22×C8C22×D1712yes—
360C4⋊Q8—174no—
361C4⋊Q8—342no—
362#123—3416no—
363C22⋊Q8—178no—
364C22⋊Q8—344no—
365#121—348no—
366C4×D4—22no—
367C4×D4—174no—
368C4×D4—174no—
369C4×D4—684no—
370C4×D4C4×D1714yescomplemented
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—344no—
377C2×C4⋊C4—348no—
378C2×C4⋊C4—342no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—342no—
382C2×C4⋊C4—682no—
383C2×C4⋊C4C22×D1712yes—
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—172no—
387C2×C22⋊C4—342no—
388C2×C22⋊C4—342no—
389C2×C42—341no—
390C2×C42C22×D1711yescharacteristic
391C4⋊C8—22no—
392#103—344no—
393#103—344no—
394#103—682no—
395#103—682no—
396#102—24no—
397#102—174no—
398#102—342no—
399C22⋊C8—344no—
400C4×C8—342no—
401#95—342no—
402#95—341no—
403C17#2243111yescharacteristic, complemented, Sylow
404C24—21no—
405C24—172no—
406#41—3416no—
407#41—688no—
408#41—688no—
409#41—688no—
410C2×Q8—174no—
411C2×Q8—342no—
412C2×Q8—342no—
413C2×Q8—342no—
414C2×Q8—342no—
415C2×Q8—342no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—174no—
420C2×D4—348no—
421C2×D4—342no—
422C2×D4—342no—
423C2×D4—684no—
424C2×D4—684no—
425C2×D4C2×C4×D1712yes—
426C2×D4D4×D1712yes—
427C2×D4#6325612yes—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—172no—
434C22×C4—174no—
435C22×C4—174no—
436C22×C4—348no—
437C22×C4—342no—
438C22×C4—342no—
439C22×C4—342no—
440C22×C4—342no—
441C22×C4—341no—
442C22×C4—342no—
443C22×C4—341no—
444C22×C4—684no—
445C22×C4—684no—
446C22×C4—684no—
447C22×C4—684no—
448C22×C4—1362no—
449C22×C4D4×D1711yescharacteristic
450C22×C4D4×D1711yescharacteristic
451C22×C4C23×D1711yescharacteristic
452M16—688no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—172no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8—341no—
460C2×C8—341no—
461C2×C8—341no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—81no—
467C4⋊C4—174no—
468C4⋊C4—172no—
469C4⋊C4—348no—
470C4⋊C4—341no—
471C4⋊C4—344no—
472C4⋊C4—344no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4—684no—
476C4⋊C4C2×C4×D1712yes—
477C22⋊C4—24no—
478C22⋊C4—178no—
479C22⋊C4—348no—
480C22⋊C4—684no—
481C42—682no—
482C42C2×C4×D1712yes—
483C42D4×D1712yes—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—174no—
492C23—342no—
493C23—342no—
494C23—342no—
495C23—342no—
496C23—342no—
497C23—342no—
498C23C2×D4×D1711yescharacteristic
499Q8—344no—
500Q8—344no—
501Q8—682no—
502Q8—682no—
503D4—22no—
504D4—22no—
505D4—22no—
506D4—348no—
507D4—6816no—
508D4#1058757914yescomplemented
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—174no—
519C2×C4—342no—
520C2×C4—342no—
521C2×C4—344no—
522C2×C4—344no—
523C2×C4—344no—
524C2×C4—344no—
525C2×C4—344no—
526C2×C4—342no—
527C2×C4—6816no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—682no—
531C2×C4—682no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—1368no—
536C2×C4C22⋊C4×D1712yes—
537C2×C4C22⋊C4×D1712yes—
538C2×C4#1058759712yes—
539C2×C4#1058760112yes—
540C2×C4C22×C4×D1711yescharacteristic
541C2×C4C2×D4×D1711yescharacteristic
542C2×C4C2×D4×D1711yescharacteristic
543C2×C4C2×#6325611yescharacteristic
544C2×C4#1058769611yescharacteristic
545C2×C4#1058769611yescharacteristic
546C8—42no—
547C8—342no—
548C8—681no—
549C22—24no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—344no—
557C22—344no—
558C22—344no—
559C22—682no—
560C22—682no—
561C22—682no—
562C22—682no—
563C22C2×C22⋊C4×D1711yescharacteristic
564C22#1176344311yescharacteristic
565C22#1176345011yescharacteristic
566C22C2×#1058759711yescharacteristic
567C22C2×#1058760111yescharacteristic
568C22#1176361911yescharacteristic
569C22#1176362511yescharacteristic
570C4—22no—
571C4—42no—
572C4—82no—
573C4—344no—
574C4—684no—
575C4—682no—
576C4—1361no—
577C4#1176280714yescomplemented
578C4C2×#1058757911yescharacteristic
579C4#1176328311yescharacteristic
580C4#1176344311yescharacteristic
581C4#1176345011yescharacteristic
582C2—44no—
583C2—684no—
584C2#1453160411yescharacteristic
585C2C2×#1176280711yescharacteristic
586C2#1453731511yescharacteristic
587C2#1454037911yescharacteristic
588C2#1454043311yescharacteristic
589C2#1454108611yescharacteristic
590C2#1454109711yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1802C2
socleC23×C17C22×D4
prime radicalC2×#11763463C2
cosocleC22×C4×C17C24
hypercenter#1802D17
hypocenterC17#22431
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#11762886C2semidirect
3. C2×#11762807C2semidirect
4. #14531684C2semidirect
5. #14529706C2semidirect
6. #14529585C2semidirect
7. #14529216C2semidirect
8. #14529090C2semidirect
9. #14528721C2semidirect
10. C4×#10587503C2semidirect
11. C17×#1802C2semidirect
15. #11763463C4semidirect
54. C2×#63254C2×C4semidirect
61. C4×D4×C17C2×C4semidirect
87. #10587583C2×C4semidirect
95. #10587555C2×C4semidirect
99. #10587503D4semidirect
207. D4×C17C4×D4semidirect
262. #1802D17semidirect
289. C4×C17#417semidirect
370. C4×D4C4×D17semidirect
403. C17#22431semidirect
508. D4#10587579semidirect
577. C4#11762807semidirect

Maximal subgroups

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

subgroupindex
C2×#117634632
C2×#117628862
C2×#117628072
#145316842
#145297062
#145295852
#145292162
#145290902
#145287212
C4×#105875032
C17×#18022
#2243117