Finite Group Databasebeta

← #17068643

Subgroups — #17068643

Order 4352.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763366C211yesmaximal, characteristic, complemented
2C2×#11763100C211yesmaximal, characteristic, complemented
3C2×#11763015C211yesmaximal, characteristic, complemented
4C4×#10587616C211yesmaximal, characteristic
5#14529981C212yesmaximal, complemented
6#14529789C212yesmaximal
7#14529731C211yesmaximal, characteristic, complemented
8#14529706C211yesmaximal, characteristic, complemented
9#14529138C212yesmaximal, complemented
10#14529055C212yesmaximal, complemented
11C17×#1797C211yesmaximal, characteristic, complemented
12C22×#63219C2212yes—
13#11763366—24no—
14#11763366—24no—
15C2×#10587555C2211yescharacteristic
16C2×#10587555C2211yescharacteristic
17C2×#10587556—21no—
18C2×#10587610—21no—
19C2×#10587610C2212yes—
20C2×#10587616C2211yescharacteristic
21C2×C4×Dic34—21no—
22C2×C4×Dic34C2212yes—
23C2×C4×D68C2211yescharacteristic
24C2×C4×C22⋊D17—21no—
25C2×#10587612—21no—
26#11763100—24no—
27#11763015—24no—
28C2×#10587488C2212yes—
29C2×#10587487C2212yes—
30C2×#10587487C2212yes—
31C2×#10587487C2212yes—
32C2×#10587492C2212yes—
33#11762711C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762700—21no—
36#11762688—21no—
37#11762669C2211yescharacteristic
38C4×C4⋊Dic17C2211yescharacteristic
39C4×#63227—21no—
40#11762572C2212yes—
41#11762572C2212yes—
42C22×C42×C17C2211yescharacteristic
43C2×C17×C4⋊C8C2211yescharacteristic
44C2×C17×C4⋊C8C2211yescharacteristic
45C2×C17×C22⋊C8C2212yes—
46C17×#336C2212yes—
47C23×C4×C17D411yescharacteristic
48C23×C4×C17D411yescharacteristic
49C23×C4×C17C2311yescharacteristic
50C22×C22⋊D17—21no—
51C22×D68C2311yescharacteristic
52C22×C4×D17—21no—
53C22×Dic34—21no—
54C22×Dic34C2312yes—
55C22×C8×C17C2312yes—
56C22×C8×C17C2312yes—
57C2×C42×C17—22no—
58C2×C42×C17—22no—
59C2×C42×C17D411yescharacteristic
60C2×C42×C17D411yescharacteristic
61C2×C42×C17C2311yescharacteristic
62C2×C42×C17C2311yescharacteristic
63C2×C22⋊Dic17—22no—
64C2×C22⋊Dic17—21no—
65#10587616—22no—
66#10587616—22no—
67C4×C22⋊D17—44no—
68C2×#63227—21no—
69C2×#63227—21no—
70C2×#63227—21no—
71C2×#63227—21no—
72#10587612—28no—
73C2×C4⋊Dic17—21no—
74C2×C4⋊Dic17—21no—
75C2×C4⋊Dic17—21no—
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2312yes—
79C2×C4⋊Dic17C2311yescharacteristic
80#10587610—28no—
81#10587610—24no—
82#10587610—24no—
83C2×#63225—22no—
84C2×#63225—22no—
85C2×#63225—21no—
86C2×C4×Dic17—22no—
87C2×C4×Dic17—21no—
88C2×#63219—24no—
89C2×#63219—24no—
90C2×#63219—42no—
91C4×D68—22no—
92C4×D68—22no—
93#10587556—28no—
94#10587555—22no—
95#10587555—22no—
96#10587555D414yescomplemented
97#10587555D414yescomplemented
98C4×Dic34—28no—
99C4×Dic34—24no—
100C4×Dic34—24no—
101C17×C4⋊C8—22no—
102C17×C4⋊C8—22no—
103C17×C22⋊C8—42no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587492—24no—
108#10587488—24no—
109#10587487—24no—
110#10587487—24no—
111#10587487—42no—
112C24×C17C2×D411yescharacteristic
113C23×D17—21no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—41no—
124C22×C4×C17—41no—
125C22×C4×C17SD1614yescomplemented
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2411yescharacteristic
132C2×C22⋊D17—24no—
133C2×C22⋊D17—42no—
134C2×C22⋊D17—42no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C2×D68—21no—
141C2×D68—21no—
142C2×D68—21no—
143C2×D68—21no—
144C2×D68—21no—
145C2×D68#4112yes—
146C2×C4×D17—42no—
147C2×C4×D17—42no—
148C2×C4×D17—42no—
149C2×Dic34—22no—
150C2×Dic34—22no—
151C2×Dic34—22no—
152C2×Dic34—22no—
153C2×Dic34—24no—
154C2×Dic34—22no—
155C2×Dic34—42no—
156C2×Dic34—42no—
157C2×Dic34C2×D414yes—
158C2×C8×C17—22no—
159C2×C8×C17—22no—
160C2×C8×C17—22no—
161C2×C8×C17—22no—
162C2×C8×C17—22no—
163C2×C8×C17—22no—
164C42×C17—22no—
165C42×C17—22no—
166C42×C17—21no—
167C42×C17—21no—
168C42×C17C2×D412yes—
169C42×C17C2×D412yes—
170C22⋊Dic17—28no—
171C22⋊Dic17—42no—
172#63227—24no—
173#63227—42no—
174#63227—42no—
175#63227—42no—
176C4⋊Dic17—24no—
177C4⋊Dic17—24no—
178C4⋊Dic17—24no—
179C4⋊Dic17—22no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17C2×D414yes—
186C4⋊Dic17#4112yes—
187#63225—28no—
188#63225—28no—
189#63225—42no—
190C4×Dic17—28no—
191C4×Dic17—42no—
192#63219—48no—
193#24151—171nomaximal, Sylow
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17C2×SD1612yes—
201C23×C17C22×D411yescharacteristic
202C22×D17—22no—
203C22×D17—41no—
204C22×D17—41no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17—41no—
208C22×D17—41no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—28no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—44no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17#12012yes—
226C2×C4×C17#12012yes—
227C2×C4×C17C2×SD1612yes—
228C2×C4×C17C2×SD1612yes—
229C2×C4×C17#13712yes—
230C2×C4×C17C22×D411yescharacteristic
231C2×C4×C17C22×D411yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17H(5,𝔽2)11yescharacteristic
235C2×C4×C172−1+411yescharacteristic
236C22⋊D17—48no—
237C2×Dic17—24no—
238C2×Dic17—24no—
239C2×Dic17—22no—
240C2×Dic17—42no—
241C2×Dic17—42no—
242C2×Dic17—41no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252D68—22no—
253D68—22no—
254D68—41no—
255D68—41no—
256C4×D17—84no—
257Dic34—24no—
258Dic34—24no—
259Dic34—48no—
260Dic34—42no—
261Dic34—42no—
262C8×C17—42no—
263C8×C17—42no—
264C2×#533—171no—
265C2×#476—171no—
266C2×#461—171no—
267C4×#121—171no—
268#2035—172no—
269#2000—172no—
270#1987—171no—
271#1981—171no—
272#1880—172no—
273#1865—172no—
274#1797D1711yescharacteristic, complemented
275C22×C17—22no—
276C22×C17—22no—
277C22×C17—22no—
278C22×C17—41no—
279C22×C17—41no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17C2×#12011yescharacteristic
283C22×C17#52511yescharacteristic
284C22×C17#53711yescharacteristic
285C22×C17C22×SD1611yescharacteristic
286C22×C17C2×#13711yescharacteristic
287C22×C17#57511yescharacteristic
288C22×C17#57511yescharacteristic
289C2×D17—42no—
290C2×D17—42no—
291C2×D17—42no—
292C2×D17—81no—
293C2×D17—81no—
294C2×D17—81no—
295C2×D17—81no—
296C4×C17—22no—
297C4×C17—22no—
298C4×C17—41no—
299C4×C17—41no—
300C4×C17#44814yes—
301C4×C17#54611yescharacteristic
302C4×C17#54711yescharacteristic
303C4×C17#54811yescharacteristic
304C4×C17#55011yescharacteristic
305Dic17—42no—
306Dic17—44no—
307Dic17—44no—
308Dic17—81no—
309Dic17—81no—
310C22×SD16—172no—
311#533—344no—
312#533—344no—
313C2×C4⋊Q8—171no—
314C2×C4⋊Q8—171no—
315C2×#125—341no—
316C2×C22⋊Q8—172no—
317C2×C22⋊Q8—341no—
318C2×#121—171no—
319C2×C4×Q8—172no—
320C2×C4×Q8—341no—
321C2×C4×D4—171no—
322C2×C4×D4—341no—
323C2×#117—341no—
324C22×C42C2×D1711yescharacteristic
325#476—344no—
326#461—344no—
327C2×#106—172no—
328C2×C4⋊C8C2×D1711yescharacteristic
329C2×C4⋊C8C2×D1711yescharacteristic
330C2×#103—172no—
331C2×#103—172no—
332C2×#103—172no—
333C2×#102—172no—
334C2×C22⋊C8C2×D1712yes—
335#390—171no—
336#389—171no—
337#388—341no—
338#386—341no—
339#380—171no—
340C4×C4⋊C4—171no—
341C4×C22⋊C4—341no—
342#340—172no—
343#340—172no—
344#336C2×D1712yes—
345C2×C17—42no—
346C2×C17#240711yescharacteristic
347C2×C17C2×#44811yescharacteristic
348C2×C17#300511yescharacteristic
349C2×C17#310111yescharacteristic
350C2×C17#311611yescharacteristic
351C2×C17#324611yescharacteristic
352C2×C17#325211yescharacteristic
353D17—82no—
354C22×Q8—172no—
355C22×Q8—341no—
356C22×D4—171no—
357C22×D4—341no—
358C23×C4—341no—
359C23×C4D6811yescharacteristic
360C23×C4D6811yescharacteristic
361C23×C4C22×D1711yescharacteristic
362C2×SD16—344no—
363C2×SD16—344no—
364C2×SD16—682no—
365C22×C8C22×D1712yes—
366C22×C8C22×D1712yes—
367C4⋊Q8—174no—
368C4⋊Q8—174no—
369C4⋊Q8—342no—
370C4⋊Q8—342no—
371#125—348no—
372C22⋊Q8—344no—
373C22⋊Q8—344no—
374C22⋊Q8—348no—
375#121—342no—
376#121—342no—
377C4×Q8—344no—
378C4×Q8—344no—
379C4×Q8—348no—
380C4×D4—342no—
381C4×D4—342no—
382C4×D4—684no—
383#117—348no—
384C2×C4⋊C4—172no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4—342no—
391C2×C4⋊C4—342no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—341no—
394C2×C22⋊C4—342no—
395C2×C22⋊C4—341no—
396C2×C22⋊C4—341no—
397C2×C22⋊C4—341no—
398C2×C22⋊C4—341no—
399C2×C22⋊C4—341no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42—342no—
403C2×C42—341no—
404C2×C42D6811yescharacteristic
405C2×C42D6811yescharacteristic
406C2×C42C22×D1711yescharacteristic
407C2×C42C22×D1711yescharacteristic
408#106—344no—
409C4⋊C8—22no—
410C4⋊C8—22no—
411#103—344no—
412#103—344no—
413#103—682no—
414#102—344no—
415C22⋊C8—42no—
416#95—341no—
417#95—341no—
418#95—341no—
419C17#2415111yescharacteristic, complemented, Sylow
420C24—341no—
421C24C2×D6811yescharacteristic
422C2×Q8—174no—
423C2×Q8—342no—
424C2×Q8—344no—
425C2×Q8—342no—
426C2×Q8—342no—
427C2×Q8—342no—
428C2×Q8—342no—
429C2×Q8—682no—
430C2×Q8—682no—
431C2×D4—172no—
432C2×D4—344no—
433C2×D4—341no—
434C2×D4—341no—
435C2×D4—341no—
436C2×D4—341no—
437C2×D4—341no—
438C2×D4—682no—
439C2×D4—682no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—341no—
452C22×C4—342no—
453C22×C4—342no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4—682no—
457C22×C4—682no—
458C22×C4—682no—
459C22×C4#6321914yescomplemented
460C22×C4C2×D6811yescharacteristic
461C22×C4C2×D6811yescharacteristic
462C22×C4C2×D6811yescharacteristic
463C22×C4C2×D6811yescharacteristic
464C22×C4C2×D6811yescharacteristic
465C22×C4C23×D1711yescharacteristic
466SD16—688no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C4⋊C4—174no—
474C4⋊C4—172no—
475C4⋊C4—348no—
476C4⋊C4—348no—
477C4⋊C4—342no—
478C4⋊C4—341no—
479C4⋊C4—344no—
480C4⋊C4—344no—
481C4⋊C4—344no—
482C4⋊C4—341no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—682no—
487C22⋊C4—344no—
488C22⋊C4—348no—
489C22⋊C4—682no—
490C22⋊C4—682no—
491C22⋊C4—682no—
492C22⋊C4—682no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—22no—
497C42—348no—
498C42—682no—
499C42D4×D1712yes—
500C42D4×D1712yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—342no—
508C23—681no—
509C23—681no—
510C23—681no—
511C23—681no—
512C23—681no—
513C23—681no—
514C23C2×#6321912yes—
515C23C22×D6811yescharacteristic
516Q8—344no—
517Q8—344no—
518Q8—688no—
519Q8—682no—
520Q8—682no—
521D4—342no—
522D4—342no—
523D4—688no—
524D4—681no—
525D4—681no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—344no—
543C2×C4—344no—
544C2×C4—342no—
545C2×C4—682no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—682no—
549C2×C4—682no—
550C2×C4—682no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—1364no—
558C2×C4#1058756812yes—
559C2×C4#1058756812yes—
560C2×C4C2×#6321912yes—
561C2×C4C2×#6321912yes—
562C2×C4#1058759512yes—
563C2×C4C2×D4×D1711yescharacteristic
564C2×C4C2×D4×D1711yescharacteristic
565C2×C4#1058769111yescharacteristic
566C2×C4#1058769411yescharacteristic
567C2×C4#41×D1711yescharacteristic
568C2×C4#41×D1711yescharacteristic
569C8—42no—
570C8—42no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—682no—
579C22—682no—
580C22—682no—
581C22—1361no—
582C22—1361no—
583C22—1361no—
584C22—1361no—
585C22C2×#1058756811yescharacteristic
586C22#1176332111yescharacteristic
587C22#1176339011yescharacteristic
588C22C22×#6321911yescharacteristic
589C22C2×#1058759511yescharacteristic
590C22#1176360911yescharacteristic
591C22#1176360911yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—684no—
597C4—682no—
598C4—684no—
599C4—1361no—
600C4—1361no—
601C4#1176295014yes—
602C4#1176345611yescharacteristic
603C4#1176346111yescharacteristic
604C4#1176346911yescharacteristic
605C4#1176347711yescharacteristic
606C2—42no—
607C2—1362no—
608C2#1453213011yescharacteristic
609C2C2×#1176295011yescharacteristic
610C2#1453802911yescharacteristic
611C2#1453879311yescharacteristic
612C2#1453894711yescharacteristic
613C2#1454010111yescharacteristic
614C2#1454015811yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1797C2
socleC23×C17C22×D4
prime radicalC2×#10587616C22
hypercenter#1797D17
hypocenterC17#24151
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
1. C2×#11763366C2semidirect
2. C2×#11763100C2semidirect
3. C2×#11763015C2semidirect
5. #14529981C2semidirect
7. #14529731C2semidirect
8. #14529706C2semidirect
9. #14529138C2semidirect
10. #14529055C2semidirect
11. C17×#1797C2semidirect
96. #10587555D4semidirect
97. #10587555D4semidirect
125. C22×C4×C17SD16semidirect
274. #1797D17semidirect
419. C17#24151semidirect
459. C22×C4#63219semidirect

Maximal subgroups

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

subgroupindex
C2×#117633662
C2×#117631002
C2×#117630152
C4×#105876162
#145299812
#145297892
#145297312
#145297062
#145291382
#145290552
C17×#17972
#2415117