Finite Group Databasebeta

← #17034898

Subgroups — #17034898

Order 4352.

810 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762767C211yesmaximal, characteristic
2#14531391C211yesmaximal, characteristic
3#14531391C211yesmaximal, characteristic
4#14528951C211yesmaximal, characteristic, complemented
5#14528949C211yesmaximal, characteristic, complemented
6#14528913C211yesmaximal, characteristic, complemented
7#14528910C211yesmaximal, characteristic, complemented
8#14528817C211yesmaximal, characteristic
9#14528811C211yesmaximal, characteristic
10#14528807C211yesmaximal, characteristic
11#14528680C211yesmaximal, characteristic, complemented
12#14528680C211yesmaximal, characteristic, complemented
13#107×Dic17C211yesmaximal, characteristic, complemented
14#106×Dic17C211yesmaximal, characteristic, complemented
15C17×#1809C211yesmaximal, characteristic
16C23×#3622C2211yescharacteristic
17C2×#10587612C2211yescharacteristic
18C2×#10587489C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587488C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24#11762767C418yescomplemented
25C2×#10587486C2211yescharacteristic
26C2×#10587486C2211yescharacteristic
27C2×C8×Dic17C2211yescharacteristic
28C2×C8×Dic17C2211yescharacteristic
29#11762703—21no—
30#11762703—21no—
31#11762697—21no—
32#11762697—21no—
33#11762694—21no—
34#11762694—21no—
35#11762692—21no—
36#11762692—21no—
37#11762678C2211yescharacteristic
38#11762678C2211yescharacteristic
39#11762674C2211yescharacteristic
40#11762674C2211yescharacteristic
41#11762669C2211yescharacteristic
42#11762669C2211yescharacteristic
43C4⋊C4×Dic17C2211yescharacteristic
44C4⋊C4×Dic17C2211yescharacteristic
45C4⋊C4×Dic17C2211yescharacteristic
46C4⋊C4×Dic17C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53C22×M16×C17C2211yescharacteristic
54C2×C17×#107C2211yescharacteristic
55C2×C17×#106C2211yescharacteristic
56C17×#380C2211yescharacteristic
57C17×#380C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C2×M16×C17—22no—
62C2×M16×C17—22no—
63C2×M16×C17C2×C414yes—
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C22⋊C4×C17—21no—
71C2×C22⋊C4×C17—21no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17D411yescharacteristic
75C2×C22⋊Dic17D411yescharacteristic
76#10587612C2×C414yes—
77#10587612Q814yes—
78C2×C4⋊Dic17D411yescharacteristic
79C2×C4⋊Dic17D411yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17C2311yescharacteristic
93C2×C4×Dic17C2311yescharacteristic
94C22×#3622—22no—
95C22×#3622—22no—
96C22×#3622C2×C414yes—
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
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#10587502—21no—
112#10587502—21no—
113#10587489—22no—
114#10587488—22no—
115#10587486C2×C414yes—
116#10587486C2×C414yes—
117C8×Dic17C2×C414yes—
118C8×Dic17C2×C414yes—
119#10587479—22no—
120#10587479—22no—
121#10587478—22no—
122#10587478—22no—
123C24×C17C2×D411yescharacteristic
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17C22×C412yes—
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×Q812yes—
135C22×C4×C17C2411yescharacteristic
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
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—81no—
160C22⋊C4×C17—81no—
161C22⋊Dic17—81no—
162C22⋊Dic17—81no—
163C22⋊Dic17C4⋊C414yes—
164C22⋊Dic17C4⋊C414yes—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17C4⋊C414yes—
174C4⋊Dic17C4⋊C414yes—
175#63225—81no—
176#63225—81no—
177#63225—81no—
178#63225—81no—
179C4×Dic17—81no—
180C4×Dic17—81no—
181C4×Dic17—81no—
182C4×Dic17—81no—
183C4×Dic17C22×C412yes—
184C4×Dic17C22×C412yes—
185C4×Dic17C2×Q812yes—
186C4×Dic17C2×Q812yes—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—28no—
190C2×#3622C22×C412yes—
191C2×#3622C22×C412yes—
192C2×#3622#4112yes—
193C2×#3622#4112yes—
194#20822—171nomaximal, Sylow
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17C2×C4⋊C412yes—
202C23×C17C22×D411yescharacteristic
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
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×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17C2×C4⋊C412yes—
222C2×C4×C17C23×C411yescharacteristic
223C2×C4×C17C22×Q811yescharacteristic
224C2×C4×C17C2×#4111yescharacteristic
225C2×C4×C17C2×#4111yescharacteristic
226C2×C4×C17C2×#4111yescharacteristic
227C2×C4×C17C2×#4111yescharacteristic
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—41no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17C2×C4⋊C412yes—
249C2×Dic17C2×C4⋊C412yes—
250C2×Dic17C2×C4⋊C412yes—
251C2×Dic17C2×C4⋊C412yes—
252C8×C17—22no—
253C8×C17#11714yes—
254#3622—22no—
255#3622#11714yes—
256C2×#405—171no—
257#2314—171no—
258#2314—171no—
259#1847—171no—
260#1847—171no—
261#1835—171no—
262#1834—171no—
263#1809—171no—
264#1809D1711yescharacteristic
265#1808—171no—
266#1807—171no—
267#1776—171no—
268#1776—171no—
269C4×#107—171no—
270C4×#106—171no—
271C22×C17—22no—
272C22×C17—22no—
273C22×C17—22no—
274C22×C17—41no—
275C22×C17—41no—
276C22×C17—41no—
277C22×C17—41no—
278C22×C17C22×C4⋊C411yescharacteristic
279C22×C17#53211yescharacteristic
280C22×C17#53211yescharacteristic
281C22×C17#57611yescharacteristic
282C22×C17#57711yescharacteristic
283C22×C17#57711yescharacteristic
284C22×C17#57811yescharacteristic
285C4×C17—22no—
286C4×C17—81no—
287C4×C17—81no—
288C4×C17—81no—
289C4×C17—81no—
290C4×C17C2×#11711yescharacteristic
291C4×C17C2×#11711yescharacteristic
292C4×C17#53311yescharacteristic
293C4×C17#53311yescharacteristic
294Dic17—22no—
295Dic17—22no—
296Dic17—22no—
297Dic17—22no—
298Dic17—81no—
299Dic17—81no—
300Dic17—81no—
301Dic17—81no—
302C22×M16C2×D1711yescharacteristic
303C23×C8—171no—
304C2×#117—171no—
305C2×#107—171no—
306C2×#107—171no—
307C2×#107—171no—
308C2×#107C2×D1711yescharacteristic
309C2×#106—171no—
310C2×#106—171no—
311C2×#106—171no—
312C2×#106C2×D1711yescharacteristic
313#405—178no—
314C2×#97—171no—
315C2×#97—171no—
316C2×C4×C8—171no—
317C2×C4×C8—171no—
318#388—341no—
319#388—341no—
320#388—341no—
321#388—341no—
322#387—341no—
323#387—341no—
324#387—341no—
325#387—341no—
326#384—171no—
327#384—171no—
328#382—171no—
329#382—171no—
330#380—171no—
331#380—171no—
332#380C2×D1711yescharacteristic
333#380C2×D1711yescharacteristic
334C4×C4⋊C4—171no—
335C4×C4⋊C4—171no—
336C4×C4⋊C4—171no—
337C4×C4⋊C4—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#340—171no—
342#340—171no—
343#340—171no—
344#340C2×D1711yescharacteristic
345#340C2×D1711yescharacteristic
346C2×C17—42no—
347C2×C17#228511yescharacteristic
348C2×C17#291211yescharacteristic
349C2×C17#291311yescharacteristic
350C2×C17#317811yescharacteristic
351C2×C17#317811yescharacteristic
352C2×C17#317911yescharacteristic
353C2×C17#317911yescharacteristic
354C23×C4C22×D1711yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C4×D1714yes—
358C22×C8—171no—
359C22×C8—174no—
360C22×C8—171no—
361C22×C8—342no—
362C22×C8—342no—
363C22×C8C22×D1711yescharacteristic
364C22×C8C22×D1711yescharacteristic
365#117—174no—
366#117—174no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4—341no—
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C4⋊C4C22×D1711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—171no—
384C2×C22⋊C4—171no—
385C2×C22⋊C4—341no—
386C2×C22⋊C4—341no—
387C2×C42—171no—
388C2×C42—171no—
389C2×C42—341no—
390C2×C42—341no—
391C2×C42—341no—
392C2×C42—341no—
393#107—22no—
394#107—342no—
395#107—342no—
396#107—342no—
397#106—22no—
398#106—342no—
399#106—342no—
400#106—342no—
401#97—174no—
402#97—174no—
403C4×C8—174no—
404C4×C8—174no—
405#95—21no—
406#95—21no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411#95—341no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95—341no—
417C17#2082211yescharacteristic, complemented, Sylow
418C24D4×D1711yescharacteristic
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—171no—
428C22×C4—171no—
429C22×C4—171no—
430C22×C4—171no—
431C22×C4—341no—
432C22×C4—341no—
433C22×C4—341no—
434C22×C4—341no—
435C22×C4C2×C4×D1712yes—
436C22×C4D4×D1711yescharacteristic
437C22×C4Q8×D1712yes—
438C22×C4C23×D1711yescharacteristic
439M16—28no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—172no—
443C2×C8—172no—
444C2×C8—172no—
445C2×C8—172no—
446C2×C8—348no—
447C2×C8—341no—
448C2×C8—341no—
449C2×C8C2×C4×D1712yes—
450C2×C8C2×C4×D1712yes—
451C2×C8#6325412yes—
452C2×C8#6325412yes—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—174no—
462C4⋊C4—174no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—341no—
468C4⋊C4—341no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—1361no—
472C4⋊C4—1361no—
473C4⋊C4—1361no—
474C4⋊C4—1361no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4—174no—
478C22⋊C4—174no—
479C22⋊C4—1361no—
480C22⋊C4—1361no—
481C42—172no—
482C42—172no—
483C42—172no—
484C42—172no—
485C42—1361no—
486C42—1361no—
487C42—1361no—
488C42—1361no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23C4⋊C4×D1712yes—
496C23C2×D4×D1711yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
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—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—172no—
516C2×C4—172no—
517C2×C4—172no—
518C2×C4—172no—
519C2×C4—341no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—341no—
526C2×C4—341no—
527C2×C4—681no—
528C2×C4—681no—
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—681no—
538C2×C4—681no—
539C2×C4C4⋊C4×D1712yes—
540C2×C4C22×C4×D1711yescharacteristic
541C2×C4C2×#6325411yescharacteristic
542C2×C4C2×#6325411yescharacteristic
543C2×C4C2×Q8×D1711yescharacteristic
544C2×C4#41×D1711yescharacteristic
545C2×C4#41×D1711yescharacteristic
546C8—22no—
547C8—174no—
548C8—342no—
549C8#1058756314yes—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×C4⋊C4×D1711yescharacteristic
558C22#1176335411yescharacteristic
559C22#1176335411yescharacteristic
560C22#1176361311yescharacteristic
561C22#1176361811yescharacteristic
562C22#1176361911yescharacteristic
563C22#1176362511yescharacteristic
564C4—22no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—342no—
570C4—342no—
571C4—342no—
572C4—342no—
573C4—1361no—
574C4—1361no—
575C4—1361no—
576C4—1361no—
577C4#117×D1711yescharacteristic
578C4C2×#1058756311yescharacteristic
579C4#1176336111yescharacteristic
580C4#1176336111yescharacteristic
581C2—42no—
582C2#1453118611yescharacteristic
583C2#1453737811yescharacteristic
584C2#1453738311yescharacteristic
585C2#1453944511yescharacteristic
586C2#1453945311yescharacteristic
587C2#1453946011yescharacteristic
588C2#1453946811yescharacteristic
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#20822
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. #14528951C2semidirect
5. #14528949C2semidirect
6. #14528913C2semidirect
7. #14528910C2semidirect
11. #14528680C2semidirect
12. #14528680C2semidirect
13. #107×Dic17C2semidirect
14. #106×Dic17C2semidirect
24. #11762767C4semidirect
417. C17#20822semidirect

Maximal subgroups

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

subgroupindex
C2×#117627672
#145313912
#145289512
#145289492
#145289132
#145289102
#145288172
#145288112
#145288072
#145286802
#107×Dic172
#106×Dic172
C17×#18092
#2082217