Finite Group Databasebeta

← #17137420

Subgroups — #17137420

Order 4352.

746 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531248C211yesmaximal, characteristic
2#14530285C211yesmaximal, characteristic, complemented
3#14530179C211yesmaximal, characteristic, complemented
4#14530169C211yesmaximal, characteristic
5#14530099C211yesmaximal, characteristic
6#14529641C211yesmaximal, characteristic
7#14529514C211yesmaximal, characteristic, complemented
8#14529466C211yesmaximal, characteristic
9#14529463C211yesmaximal, characteristic, complemented
10#14529422C211yesmaximal, characteristic, complemented
11#14529359C211yesmaximal, characteristic, complemented
12#14529121C211yesmaximal, characteristic
13#14529030C211yesmaximal, characteristic
14C4×C4⋊#3622C211yesmaximal, characteristic, complemented
15C17×#2863C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18C2×#10587558—21no—
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×C4⋊#3622C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×#10587480C2211yescharacteristic
30C2×C4×#3622C2211yescharacteristic
31C2×C4×#3622C2211yescharacteristic
32#11762711C2211yescharacteristic
33#11762711C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762700—21no—
37#11762700—21no—
38C4×C4⋊Dic17C2211yescharacteristic
39C4×#63227—21no—
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C17×#391C2211yescharacteristic
49C17×#391C2211yescharacteristic
50C17×#391C2211yescharacteristic
51C17×#389C2211yescharacteristic
52C17×#389C2211yescharacteristic
53C17×#389C2211yescharacteristic
54C17×#387—21no—
55C17×#387—21no—
56C17×#387—21no—
57C17×#383—21no—
58C43×C17C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C22×D68C2311yescharacteristic
61C22×C4×D17—21no—
62C2×C4⋊C4×C17—21no—
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C42×C17D411yescharacteristic
69C2×C42×C17D411yescharacteristic
70C2×C42×C17D411yescharacteristic
71C2×C42×C17D411yescharacteristic
72C2×C42×C17C2311yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×#63230—22no—
76C2×#63230—22no—
77C2×#63230—41no—
78C2×#63227—21no—
79C2×#63227—21no—
80C2×#63227—21no—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×#63225—21no—
87C2×#63225—21no—
88C2×C4×Dic17—21no—
89C22×#3622C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C4×D68—22no—
94C4×D68—22no—
95#10587558—44no—
96C17×#95—21no—
97C17×#95—21no—
98C17×#95—21no—
99C17×#95—21no—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104C17×#95—21no—
105#10587506—22no—
106#10587506—22no—
107#10587506—41no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587480—22no—
112#10587480—22no—
113#10587480—41no—
114#10587479—22no—
115#10587478—22no—
116#10587478—41no—
117C4⋊#3622—22no—
118C4⋊#3622—22no—
119C4×#3622—41no—
120C4×#3622—41no—
121C23×D17—21no—
122C2×Q8×C17—21no—
123C2×Q8×C17—21no—
124C2×Q8×C17—21no—
125C2×Q8×C17—21no—
126C2×Q8×C17—21no—
127C2×Q8×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2411yescharacteristic
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C2×D68—21no—
143C2×D68—21no—
144C2×D68—21no—
145C2×D68—21no—
146C2×D68—21no—
147C2×D68#4112yes—
148C2×C4×D17—24no—
149C2×C4×D17—42no—
150C2×C4×D17—42no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—42no—
158C4⋊C4×C17—42no—
159C4⋊C4×C17—42no—
160C42×C17—22no—
161C42×C17—22no—
162C42×C17—22no—
163C42×C17—22no—
164C42×C17—21no—
165C42×C17—21no—
166C42×C17—21no—
167C42×C17—21no—
168C42×C17—21no—
169C42×C17—21no—
170#63230—44no—
171#63227—24no—
172#63227—42no—
173#63227—42no—
174#63227—42no—
175#63227—42no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17#4112yes—
182#63225—42no—
183#63225—42no—
184C4×Dic17—42no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622#4112yes—
196C2×#3622#4112yes—
197#29433—171nomaximal, Sylow
198C23×C17C22×D411yescharacteristic
199C22×D17—22no—
200C22×D17—41no—
201C22×D17—41no—
202C22×D17—41no—
203C22×D17—41no—
204C22×D17—41no—
205C22×D17—41no—
206Q8×C17—41no—
207Q8×C17—41no—
208Q8×C17—41no—
209Q8×C17—41no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
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—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17#13512yes—
237C2×C4×C17#13512yes—
238C2×C4×C17AGL(1,ℤ/8)12yes—
239C2×C4×C17#13712yes—
240C2×C4×C17C2×#4111yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×C4×C17C2×#4111yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17H(5,𝔽2)11yescharacteristic
245C2×C4×C172−1+411yescharacteristic
246C2×Dic17—22no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255D68—22no—
256D68—22no—
257D68—41no—
258D68—41no—
259C4×D17—48no—
260#3622—22no—
261#3622—22no—
262#3622—41no—
263#3622—41no—
264#2863D1711yescharacteristic, complemented
265#2293—171no—
266#2088—171no—
267#2068—171no—
268#2067—171no—
269#2055—171no—
270#1968—171no—
271#1942—171no—
272#1934—171no—
273#1933—171no—
274#1928—171no—
275#1919—171no—
276#1877—171no—
277#1862—171no—
278C4×C4⋊C8—171no—
279C22×C17#52511yescharacteristic
280C22×C17#53211yescharacteristic
281C22×C17#53711yescharacteristic
282C22×C17C2×#13511yescharacteristic
283C22×C17C2×#13511yescharacteristic
284C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C17C2×#13711yescharacteristic
286C2×D17—42no—
287C2×D17—42no—
288C2×D17—42no—
289C2×D17—81no—
290C2×D17—81no—
291C2×D17—81no—
292C2×D17—81no—
293C4×C17—22no—
294C4×C17—22no—
295C4×C17—22no—
296C4×C17—22no—
297C4×C17—41no—
298C4×C17—41no—
299C4×C17—81no—
300C4×C17—81no—
301C4×C17—81no—
302C4×C17—81no—
303C4×C17#55111yescharacteristic
304C4×C17#55211yescharacteristic
305C4×C17#55311yescharacteristic
306C4×C17#55611yescharacteristic
307Dic17—42no—
308Dic17—81no—
309Dic17—81no—
310C22×SD16—171no—
311C2×C4×D4—171no—
312C2×#117—341no—
313C2×#107—171no—
314C2×#107—171no—
315C2×#106—171no—
316C2×C4⋊C8—171no—
317C2×C4⋊C8—171no—
318C2×#103—171no—
319C2×#103—171no—
320C2×#103—171no—
321C2×#102—171no—
322C2×#102—171no—
323C2×#102—171no—
324C2×C4×C8—171no—
325C2×C4×C8—171no—
326#391C2×D1711yescharacteristic
327#391C2×D1711yescharacteristic
328#391C2×D1711yescharacteristic
329#390—171no—
330#390—171no—
331#389—171no—
332#389—171no—
333#389C2×D1711yescharacteristic
334#389C2×D1711yescharacteristic
335#389C2×D1711yescharacteristic
336#388—341no—
337#388—341no—
338#387—21no—
339#387—21no—
340#387—21no—
341#383—21no—
342C4×C4⋊C4—171no—
343C4×C22⋊C4—341no—
344C43C2×D1711yescharacteristic
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340—171no—
351#336—171no—
352#336—171no—
353C2×C17#260111yescharacteristic
354C2×C17#310511yescharacteristic
355C2×C17#310711yescharacteristic
356C2×C17#316811yescharacteristic
357C2×C17#318211yescharacteristic
358C2×C17#324711yescharacteristic
359C2×C17#324811yescharacteristic
360D17—82no—
361C22×Q8C22×D1711yescharacteristic
362C22×D4—171no—
363C23×C4—341no—
364C2×SD16—342no—
365C2×SD16—342no—
366C2×SD16—681no—
367C22×C8—171no—
368C22×C8—171no—
369C22×C8—171no—
370C22×C8—171no—
371C4×D4—342no—
372C4×D4—342no—
373#117—684no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—171no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—341no—
382C2×C4⋊C4C22×D1711yescharacteristic
383C2×C4⋊C4C22×D1711yescharacteristic
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C22⋊C4—341no—
386C2×C22⋊C4—341no—
387C2×C22⋊C4—341no—
388C2×C22⋊C4—341no—
389C2×C22⋊C4—341no—
390C2×C42—341no—
391C2×C42C22⋊D1711yescharacteristic
392C2×C42C22⋊D1711yescharacteristic
393C2×C42C22⋊D1711yescharacteristic
394C2×C42C22⋊D1711yescharacteristic
395C2×C42C22×D1711yescharacteristic
396C2×C42C22×D1711yescharacteristic
397C2×C42C22×D1711yescharacteristic
398#107—342no—
399#107—681no—
400#106—342no—
401C4⋊C8—342no—
402C4⋊C8—342no—
403#103—342no—
404#103—342no—
405#103—681no—
406#102—342no—
407#102—342no—
408#102—681no—
409C4×C8—681no—
410C4×C8—681no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—341no—
421#95—341no—
422#95—341no—
423C17#2943311yescharacteristic, complemented, Sylow
424C24—341no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×D4—172no—
432C2×D4—341no—
433C2×D4—341no—
434C2×D4—341no—
435C2×D4—341no—
436C2×D4—341no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—344no—
442C22×C4—341no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—682no—
446C22×C4—682no—
447C22×C4C2×C22⋊D1711yescharacteristic
448C22×C4C2×C22⋊D1711yescharacteristic
449C22×C4C2×C22⋊D1711yescharacteristic
450C22×C4C2×C22⋊D1711yescharacteristic
451C22×C4C2×C22⋊D1711yescharacteristic
452C22×C4C2×C22⋊D1711yescharacteristic
453C22×C4C23×D1711yescharacteristic
454SD16—684no—
455C2×C8—172no—
456C2×C8—172no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8—341no—
460C2×C8—341no—
461C2×C8—341no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—172no—
477C4⋊C4—341no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—682no—
483C4⋊C4—682no—
484C22⋊C4—344no—
485C22⋊C4—682no—
486C22⋊C4—682no—
487C22⋊C4—682no—
488C22⋊C4—682no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—682no—
500C23—342no—
501C23—681no—
502C23—681no—
503C23—681no—
504C23—681no—
505C23—681no—
506C23—681no—
507C23C22×C22⋊D1711yescharacteristic
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512D4—342no—
513D4—342no—
514D4—681no—
515D4—681no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—342no—
543C2×C4—688no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4#1058762812yes—
553C2×C4#1058763512yes—
554C2×C4#1058763612yes—
555C2×C4#1058763612yes—
556C2×C4#41×D1711yescharacteristic
557C2×C4#41×D1711yescharacteristic
558C2×C4#41×D1711yescharacteristic
559C2×C4#41×D1711yescharacteristic
560C2×C4#1058769611yescharacteristic
561C2×C4#1058769711yescharacteristic
562C8—342no—
563C8—342no—
564C8—681no—
565C8—681no—
566C22—682no—
567C22—682no—
568C22—682no—
569C22—1361no—
570C22—1361no—
571C22—1361no—
572C22—1361no—
573C22#1176331611yescharacteristic
574C22#1176335511yescharacteristic
575C22#1176338711yescharacteristic
576C22C2×#1058763611yescharacteristic
577C22C2×#1058763611yescharacteristic
578C22C2×#1058763511yescharacteristic
579C22C2×#1058762811yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—682no—
591C4—1361no—
592C4—1361no—
593C4#1176348411yescharacteristic
594C4#1176349511yescharacteristic
595C4#1176351011yescharacteristic
596C4#1176352611yescharacteristic
597C2—1362no—
598C2#1453438311yescharacteristic
599C2#1453882711yescharacteristic
600C2#1453884911yescharacteristic
601C2#1453934811yescharacteristic
602C2#1453950311yescharacteristic
603C2#1454011011yescharacteristic
604C2#1454011511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2863C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenter#2863D17
hypocenterC17#29433
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C43 ◁ #2863

Complements

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

NH ≅ G/Ntype
2. #14530285C2semidirect
3. #14530179C2semidirect
7. #14529514C2semidirect
9. #14529463C2semidirect
10. #14529422C2semidirect
11. #14529359C2semidirect
14. C4×C4⋊#3622C2semidirect
15. C17×#2863C2semidirect
264. #2863D17semidirect
423. C17#29433semidirect

Maximal subgroups

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

subgroupindex
#145312482
#145302852
#145301792
#145301692
#145300992
#145296412
#145295142
#145294662
#145294632
#145294222
#145293592
#145291212
#145290302
C4×C4⋊#36222
C17×#28632
#2943317