Finite Group Databasebeta

← #17052105

Subgroups — #17052105

Order 4352.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762851C211yesmaximal, characteristic
2#14531671C211yesmaximal, characteristic
3#14531655C211yesmaximal, characteristic
4#14529525C211yesmaximal, characteristic, complemented
5#14529510C211yesmaximal, characteristic, complemented
6#14529375C211yesmaximal, characteristic, complemented
7#14529368C211yesmaximal, characteristic, complemented
8#14529361C211yesmaximal, characteristic, complemented
9#14529354C211yesmaximal, characteristic, complemented
10#14528876C211yesmaximal, characteristic
11#14528872C211yesmaximal, characteristic
12#14528684C211yesmaximal, characteristic, complemented
13#107×Dic17C211yesmaximal, characteristic, complemented
14#14528565C211yesmaximal, characteristic
15C17×#1809C211yesmaximal, characteristic
16C2×#10587612C2211yescharacteristic
17C2×#10587563—21no—
18C22×C4⋊Dic17C2211yescharacteristic
19#11762851C418yescomplemented
20C2×#10587489C2211yescharacteristic
21C2×#10587489C2211yescharacteristic
22C2×#10587488C2211yescharacteristic
23C2×#10587488C2211yescharacteristic
24C2×C22⋊#3622C2211yescharacteristic
25C2×#10587486C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27#11762705C2211yescharacteristic
28#11762705C2211yescharacteristic
29#11762705C2211yescharacteristic
30#11762705C2211yescharacteristic
31#11762703—21no—
32#11762703—21no—
33#11762692—21no—
34#11762692—21no—
35#11762673—21no—
36#11762670—21no—
37C4⋊C4×Dic17C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C4×C22⋊Dic17C2211yescharacteristic
42#11762654—21no—
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53C22×M16×C17C2211yescharacteristic
54C2×C17×#107C2211yescharacteristic
55C2×C17×#106C2211yescharacteristic
56C17×#380C2211yescharacteristic
57C17×#380C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C23×Dic17—21no—
62C2×M16×C17—22no—
63C2×M16×C17—22no—
64C2×M16×C17C2×C414yes—
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C22⋊C4×C17—21no—
72C2×C22⋊C4×C17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17D411yescharacteristic
75C2×C22⋊Dic17D411yescharacteristic
76#10587612—22no—
77#10587612C2×C414yes—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—22no—
80C2×C4⋊Dic17C2×C414yes—
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17C2311yescharacteristic
95C2×C4×Dic17C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98#10587563—44no—
99C17×#107—22no—
100C17×#106—22no—
101C17×#95—21no—
102C17×#95—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502D411yescharacteristic
112#10587502D411yescharacteristic
113C22⋊#3622—22no—
114#10587489C2×C414yes—
115#10587489C2×C414yes—
116#10587488C2×C414yes—
117#10587488C2×C414yes—
118#10587486—22no—
119C8×Dic17—22no—
120C24×C17C2×D411yescharacteristic
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
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—24no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—42no—
139C22×Dic17—42no—
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142C22×Dic17C2×D411yescharacteristic
143C22×Dic17C2×D411yescharacteristic
144M16×C17—28no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17C22×C412yes—
148C2×C8×C17C22×C412yes—
149C2×C8×C17#4112yes—
150C2×C8×C17#4112yes—
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—21no—
158C4⋊C4×C17—21no—
159C22⋊C4×C17—42no—
160C22⋊C4×C17—81no—
161C22⋊Dic17—24no—
162C22⋊Dic17—22no—
163C22⋊Dic17—41no—
164C4⋊Dic17—24no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17C22×C412yes—
170C4⋊Dic17C22×C412yes—
171C4⋊Dic17C22×C412yes—
172C4⋊Dic17C22×C412yes—
173#63225—42no—
174#63225—42no—
175#63225—81no—
176#63225—81no—
177C4×Dic17—24no—
178C4×Dic17—21no—
179C4×Dic17—21no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—42no—
183C4×Dic17—42no—
184C4×Dic17—42no—
185C4×Dic17—42no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192#22624—171nomaximal, Sylow
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17C4×D412yes—
200C23×C17C22×D411yescharacteristic
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—22no—
204C2×C4×C17—22no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—41no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—41no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17C4×D412yes—
220C2×C4×C17C23×C411yescharacteristic
221C2×C4×C17C2×#4111yescharacteristic
222C2×C4×C17C2×#4111yescharacteristic
223C2×C4×C17C2×#4111yescharacteristic
224C2×C4×C172−1+411yescharacteristic
225C2×C4×C172−1+411yescharacteristic
226C2×Dic17—21no—
227C2×Dic17—21no—
228C2×Dic17—22no—
229C2×Dic17—22no—
230C2×Dic17—22no—
231C2×Dic17—22no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—48no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17AGL(1,ℤ/8)12yes—
251C8×C17—22no—
252C8×C17#11714yescomplemented
253#3622—41no—
254#3622—41no—
255C2×#428—171no—
256#2346—171no—
257#2344—171no—
258#1944—171no—
259#1941—171no—
260#1921—171no—
261#1920—171no—
262#1919—171no—
263#1918—171no—
264#1825—171no—
265#1824—171no—
266#1809D1711yescharacteristic
267#1776—171no—
268C4×#107—171no—
269#1753—171no—
270C22×C17—22no—
271C22×C17—22no—
272C22×C17—22no—
273C22×C17—41no—
274C22×C17—41no—
275C22×C17—41no—
276C22×C17—41no—
277C22×C17C2×C4×D411yescharacteristic
278C22×C17#54711yescharacteristic
279C22×C17#54711yescharacteristic
280C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C17#57511yescharacteristic
282C22×C17#57711yescharacteristic
283C22×C17#57811yescharacteristic
284C4×C17—22no—
285C4×C17—81no—
286C4×C17—81no—
287C4×C17—81no—
288C4×C17—81no—
289C4×C17C2×#11711yescharacteristic
290C4×C17#51911yescharacteristic
291C4×C17#55211yescharacteristic
292C4×C17#55211yescharacteristic
293Dic17—22no—
294Dic17—41no—
295Dic17—41no—
296Dic17—41no—
297Dic17—42no—
298Dic17—42no—
299Dic17—42no—
300Dic17—42no—
301C22×M16C2×D1711yescharacteristic
302C2×#117—171no—
303C2×#117—341no—
304C22×C4⋊C4—171no—
305#428—178no—
306C2×#107—171no—
307C2×#107—171no—
308C2×#107C2×D1711yescharacteristic
309C2×#106—171no—
310C2×#106—171no—
311C2×#106C2×D1711yescharacteristic
312C2×C22⋊C8—171no—
313C2×#97—171no—
314C2×C4×C8—171no—
315#389—171no—
316#389—171no—
317#389—171no—
318#389—171no—
319#388—341no—
320#388—341no—
321#387—341no—
322#387—341no—
323#382—341no—
324#381—341no—
325#380C2×D1711yescharacteristic
326#380C2×D1711yescharacteristic
327C4×C4⋊C4—171no—
328C4×C4⋊C4—171no—
329C4×C4⋊C4—171no—
330C4×C4⋊C4—171no—
331C4×C22⋊C4—171no—
332#376—341no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#340C2×D1711yescharacteristic
342#340C2×D1711yescharacteristic
343#336—171no—
344#336—171no—
345C2×C17—42no—
346C2×C17#234411yescharacteristic
347C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
348C2×C17#294711yescharacteristic
349C2×C17#334311yescharacteristic
350C2×C17#334811yescharacteristic
351C2×C17#335211yescharacteristic
352C2×C17#335411yescharacteristic
353C23×C4—341no—
354C23×C4C22×D1711yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C4×D1714yes—
358C22×C8—171no—
359C22×C8—171no—
360C22×C8C22×D1711yescharacteristic
361C22×C8C22×D1711yescharacteristic
362#117—174no—
363#117—342no—
364#117—684no—
365C2×C4⋊C4—174no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—341no—
371C2×C4⋊C4—342no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—171no—
383C2×C22⋊C4—171no—
384C2×C22⋊C4—341no—
385C2×C42—171no—
386C2×C42—171no—
387C2×C42—341no—
388C2×C42—341no—
389C2×C42—341no—
390C2×C42—341no—
391C2×C42—341no—
392#107—22no—
393#107—174no—
394#107—174no—
395#106—22no—
396#106—174no—
397#106—174no—
398C22⋊C8—342no—
399#97—342no—
400C4×C8—342no—
401#95—21no—
402#95—21no—
403#95—171no—
404#95—171no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411#95—341no—
412#95—341no—
413C17#2262411yescharacteristic, complemented, Sylow
414C24D4×D1711yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—171no—
424C22×C4—171no—
425C22×C4—171no—
426C22×C4—171no—
427C22×C4—341no—
428C22×C4—341no—
429C22×C4—341no—
430C22×C4—341no—
431C22×C4—344no—
432C22×C4—682no—
433C22×C4—682no—
434C22×C4C2×C4×D1712yes—
435C22×C4D4×D1711yescharacteristic
436C22×C4#6325612yes—
437C22×C4C23×D1711yescharacteristic
438M16—28no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—341no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8C2×C4×D1712yes—
448C2×C8C2×C4×D1712yes—
449C2×C8#6325412yes—
450C2×C8#6325412yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—172no—
460C4⋊C4—172no—
461C4⋊C4—172no—
462C4⋊C4—172no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—344no—
468C4⋊C4—682no—
469C4⋊C4—682no—
470C4⋊C4—1361no—
471C4⋊C4—1361no—
472C22⋊C4—42no—
473C22⋊C4—81no—
474C22⋊C4—342no—
475C22⋊C4—344no—
476C22⋊C4—681no—
477C42—341no—
478C42—341no—
479C42—341no—
480C42—341no—
481C42—344no—
482C42—682no—
483C42—682no—
484C42—682no—
485C42—682no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#1058757912yes—
493C23C2×D4×D1711yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—172no—
513C2×C4—341no—
514C2×C4—341no—
515C2×C4—341no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—342no—
519C2×C4—342no—
520C2×C4—342no—
521C2×C4—342no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—341no—
526C2×C4—341no—
527C2×C4—341no—
528C2×C4—688no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4#1058757912yes—
538C2×C4C22×C4×D1711yescharacteristic
539C2×C4C2×#6325411yescharacteristic
540C2×C4C2×#6325411yescharacteristic
541C2×C4C2×#6325611yescharacteristic
542C2×C4#1058769711yescharacteristic
543C2×C4#1058769711yescharacteristic
544C8—22no—
545C8—681no—
546C8—681no—
547C8#1058756314yescomplemented
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×#1058757911yescharacteristic
556C22#1176345911yescharacteristic
557C22#1176345911yescharacteristic
558C22AGL(1,ℤ/8)×D1711yescharacteristic
559C22#1176360811yescharacteristic
560C22#1176361911yescharacteristic
561C22#1176362511yescharacteristic
562C4—22no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—342no—
568C4—681no—
569C4—681no—
570C4—682no—
571C4—682no—
572C4—682no—
573C4—682no—
574C4—681no—
575C4C2×#1058756311yescharacteristic
576C4#1176328611yescharacteristic
577C4#1176350111yescharacteristic
578C4#1176350111yescharacteristic
579C2—42no—
580C2#1453165511yescharacteristic
581C2#1453759111yescharacteristic
582C2#1453762511yescharacteristic
583C2#1454121911yescharacteristic
584C2#1454129411yescharacteristic
585C2#1454135411yescharacteristic
586C2#1454137411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1809C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#1809D17
hypocenterC17#22624
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
4. #14529525C2semidirect
5. #14529510C2semidirect
6. #14529375C2semidirect
7. #14529368C2semidirect
8. #14529361C2semidirect
9. #14529354C2semidirect
12. #14528684C2semidirect
13. #107×Dic17C2semidirect
19. #11762851C4semidirect
252. C8×C17#117semidirect
413. C17#22624semidirect
547. C8#10587563semidirect

Maximal subgroups

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

subgroupindex
C2×#117628512
#145316712
#145316552
#145295252
#145295102
#145293752
#145293682
#145293612
#145293542
#145288762
#145288722
#145286842
#107×Dic172
#145285652
C17×#18092
#2262417