Finite Group Databasebeta

← #17092383

Subgroups — #17092383

Order 4352.

704 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
1#14532498C211yesmaximal, characteristic
2#14531663C211yesmaximal, characteristic
3#14530349C211yesmaximal, characteristic, complemented
4#14530283C211yesmaximal, characteristic, complemented
5#14530276C211yesmaximal, characteristic, complemented
6#14530120C211yesmaximal, characteristic, complemented
7#14529740C211yesmaximal, characteristic, complemented
8#14529539C211yesmaximal, characteristic, complemented
9#14529371C211yesmaximal, characteristic, complemented
10#14529360C211yesmaximal, characteristic, complemented
11#14528981C211yesmaximal, characteristic
12#14528967C211yesmaximal, characteristic
13#14528868C211yesmaximal, characteristic
14#14528504C211yesmaximal, characteristic
15C17×#1809C211yesmaximal, characteristic
16C2×#10587576C2211yescharacteristic
17C2×#10587612C2211yescharacteristic
18C22×C4⋊Dic17C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×#10587485C2211yescharacteristic
22C2×#10587485C2211yescharacteristic
23C2×C22⋊#3622C2211yescharacteristic
24#11762745—21no—
25#11762732—21no—
26#11762708C2211yescharacteristic
27#11762706—21no—
28#11762705C2211yescharacteristic
29#11762705C2211yescharacteristic
30#11762705C2211yescharacteristic
31#11762696—21no—
32#11762694—21no—
33#11762694—21no—
34#11762678C2211yescharacteristic
35#11762667—21no—
36C4⋊C4×Dic17C2211yescharacteristic
37C4⋊C4×Dic17C2211yescharacteristic
38C22⋊C4×Dic17—21no—
39C4×C22⋊Dic17C2211yescharacteristic
40#11762572C2211yescharacteristic
41#11762572C2211yescharacteristic
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C22×M16×C17C2211yescharacteristic
53C2×C17×#107C2211yescharacteristic
54C2×C17×#106C2211yescharacteristic
55C17×#380C2211yescharacteristic
56C17×#380C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C23×Dic17—21no—
61C2×M16×C17—22no—
62C2×M16×C17—22no—
63C2×M16×C17—41no—
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⋊Dic17D411yescharacteristic
74C2×C22⋊Dic17D411yescharacteristic
75#10587612—22no—
76#10587612—41no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17—22no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17Q814yes—
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×#63225—21no—
91C2×#63225—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17C2311yescharacteristic
96C2×C4×Dic17C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99#10587576—22no—
100#10587576—41no—
101C17×#107—41no—
102C17×#106—22no—
103C17×#95—21no—
104C17×#95—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#10587485—22no—
115#10587485—41no—
116#10587479—22no—
117#10587478—41no—
118C24×C17C2×D411yescharacteristic
119C22×C4×C17—21no—
120C22×C4×C17—21no—
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×C17C2×D411yescharacteristic
128C22×C4×C17C2×Q812yes—
129C22×C4×C17#4112yes—
130C22×C4×C17C2411yescharacteristic
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—24no—
135C22×Dic17—21no—
136C22×Dic17—81no—
137C22×Dic17—81no—
138C22×Dic17C2×D411yescharacteristic
139C22×Dic17C2×D411yescharacteristic
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142M16×C17—44no—
143C2×C8×C17—21no—
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17#4112yes—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C22⋊C4×C17—42no—
158C22⋊C4×C17—81no—
159C22⋊Dic17—22no—
160C22⋊Dic17—41no—
161C22⋊Dic17—42no—
162C4⋊Dic17—24no—
163C4⋊Dic17—21no—
164C4⋊Dic17—21no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—42no—
169C4⋊Dic17—81no—
170C4⋊Dic17C2×Q812yes—
171C4⋊Dic17C2×Q812yes—
172C4⋊Dic17#4112yes—
173#63225—42no—
174#63225—42no—
175#63225—42no—
176#63225—81no—
177#63225—81no—
178#63225—81no—
179C4×Dic17—21no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—21no—
183C4×Dic17—42no—
184C4×Dic17—81no—
185C4×Dic17—81no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192#26018—171nomaximal, Sylow
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17C22⋊Q812yes—
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×C17C22⋊Q812yes—
220C2×C4×C17C22×Q811yescharacteristic
221C2×C4×C17C2×#4111yescharacteristic
222C2×C4×C17C2×#4111yescharacteristic
223C2×C4×C17C2×#4111yescharacteristic
224C2×C4×C17H(5,𝔽2)11yescharacteristic
225C2×C4×C172−1+411yescharacteristic
226C2×Dic17—21no—
227C2×Dic17—21no—
228C2×Dic17—22no—
229C2×Dic17—22no—
230C2×Dic17—22no—
231C2×Dic17—21no—
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—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—84no—
250C2×Dic17#13712yes—
251C8×C17—22no—
252C8×C17—41no—
253#3622—41no—
254#3622—41no—
255#2441—171no—
256#2345—171no—
257#2100—171no—
258#2088—171no—
259#2086—171no—
260#2059—171no—
261#1989—171no—
262#1947—171no—
263#1920—171no—
264#1919—171no—
265#1854—171no—
266#1851—171no—
267#1823—171no—
268#1809D1711yescharacteristic
269#1736—171no—
270C22×C17—22no—
271C22×C17—22no—
272C22×C17—22no—
273C22×C17—41no—
274C22×C17—41no—
275C22×C17—41no—
276C22×C17—41no—
277C22×C17C2×C22⋊Q811yescharacteristic
278C22×C17#54611yescharacteristic
279C22×C17#54711yescharacteristic
280C22×C17C2×#13711yescharacteristic
281C22×C17#57511yescharacteristic
282C22×C17#57611yescharacteristic
283C22×C17#57711yescharacteristic
284C4×C17—22no—
285C4×C17—81no—
286C4×C17—81no—
287C4×C17—81no—
288C4×C17—81no—
289C4×C17#53311yescharacteristic
290C4×C17#54411yescharacteristic
291C4×C17#55111yescharacteristic
292C4×C17#55211yescharacteristic
293Dic17—22no—
294Dic17—41no—
295Dic17—42no—
296Dic17—42no—
297Dic17—42no—
298Dic17—41no—
299Dic17—41no—
300Dic17—81no—
301C22×M16C2×D1711yescharacteristic
302C2×#125—171no—
303C2×#117—171no—
304C22×C4⋊C4—171no—
305C2×#107—171no—
306C2×#107C2×D1711yescharacteristic
307C2×#106—171no—
308C2×#106C2×D1711yescharacteristic
309C2×C4⋊C8—171no—
310C2×C4⋊C8—171no—
311C2×C22⋊C8—171no—
312#399—341no—
313#396—341no—
314#389—171no—
315#389—171no—
316#389—171no—
317#389—171no—
318#389—341no—
319#387—341no—
320#387—341no—
321#387—341no—
322#384—171no—
323#380—341no—
324#380C2×D1711yescharacteristic
325#380C2×D1711yescharacteristic
326C4×C4⋊C4—171no—
327C4×C4⋊C4—171no—
328C4×C22⋊C4—171no—
329C4×C22⋊C4—341no—
330#340—171no—
331#340—171no—
332#340—171no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340C2×D1711yescharacteristic
341#340C2×D1711yescharacteristic
342#336—171no—
343#336—171no—
344C2×C17—42no—
345C2×C17#243811yescharacteristic
346C2×C17#307811yescharacteristic
347C2×C17#308111yescharacteristic
348C2×C17#330911yescharacteristic
349C2×C17#333111yescharacteristic
350C2×C17#334611yescharacteristic
351C2×C17#335111yescharacteristic
352C23×C4—341no—
353C23×C4C22×D1711yescharacteristic
354C2×M16—22no—
355C2×M16—22no—
356C2×M16—41no—
357C22×C8—171no—
358C22×C8—171no—
359C22×C8C22×D1711yescharacteristic
360C22×C8C22×D1711yescharacteristic
361#125—342no—
362#125—681no—
363#117—342no—
364#117—681no—
365C2×C4⋊C4—174no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—342no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4—341no—
377C2×C4⋊C4—341no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4C22×D1711yescharacteristic
381C2×C4⋊C4C22×D1711yescharacteristic
382C2×C4⋊C4C22×D1711yescharacteristic
383C2×C4⋊C4C22×D1711yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—171no—
387C2×C22⋊C4—171no—
388C2×C22⋊C4—341no—
389C2×C42—171no—
390C2×C42—171no—
391C2×C42—341no—
392C2×C42—341no—
393C2×C42—341no—
394#107—41no—
395#107—681no—
396#106—22no—
397#106—342no—
398C4⋊C8—342no—
399C4⋊C8—681no—
400C22⋊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—
411C17#2601811yescharacteristic, complemented, Sylow
412C24D4×D1711yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—171no—
422C22×C4—171no—
423C22×C4—171no—
424C22×C4—171no—
425C22×C4—341no—
426C22×C4—341no—
427C22×C4—341no—
428C22×C4—344no—
429C22×C4—341no—
430C22×C4—1361no—
431C22×C4—1361no—
432C22×C4#6325212yes—
433C22×C4D4×D1711yescharacteristic
434C22×C4Q8×D1712yes—
435C22×C4C23×D1711yescharacteristic
436M16—44no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8—341no—
448C2×C8#6325412yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—172no—
458C4⋊C4—172no—
459C4⋊C4—172no—
460C4⋊C4—341no—
461C4⋊C4—341no—
462C4⋊C4—341no—
463C4⋊C4—344no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—682no—
467C4⋊C4—682no—
468C4⋊C4—682no—
469C4⋊C4—682no—
470C4⋊C4—1361no—
471C4⋊C4—1361no—
472C4⋊C4—1361no—
473C4⋊C4—1361no—
474C22⋊C4—42no—
475C22⋊C4—81no—
476C22⋊C4—342no—
477C22⋊C4—681no—
478C22⋊C4—682no—
479C42—341no—
480C42—341no—
481C42—341no—
482C42—341no—
483C42—682no—
484C42—1361no—
485C42—1361no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#1058758212yes—
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—342no—
517C2×C4—342no—
518C2×C4—341no—
519C2×C4—341no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—341no—
526C2×C4—342no—
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—1364no—
537C2×C4#1058758212yes—
538C2×C4C2×#6325211yescharacteristic
539C2×C4C2×#6325411yescharacteristic
540C2×C4#1058769111yescharacteristic
541C2×C4C2×Q8×D1711yescharacteristic
542C2×C4#41×D1711yescharacteristic
543C2×C4#1058769711yescharacteristic
544C8—22no—
545C8—41no—
546C8—681no—
547C8—681no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×#1058758211yescharacteristic
556C22#1176344811yescharacteristic
557C22#1176345911yescharacteristic
558C22#137×D1711yescharacteristic
559C22#1176360511yescharacteristic
560C22#1176361311yescharacteristic
561C22#1176361811yescharacteristic
562C4—22no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—342no—
568C4—681no—
569C4—681no—
570C4—682no—
571C4—682no—
572C4—682no—
573C4—681no—
574C4—1361no—
575C4#1176336111yescharacteristic
576C4#1176343811yescharacteristic
577C4#1176348511yescharacteristic
578C4#1176349211yescharacteristic
579C2—42no—
580C2#1453246611yescharacteristic
581C2#1453858811yescharacteristic
582C2#1453861311yescharacteristic
583C2#1454078511yescharacteristic
584C2#1454109511yescharacteristic
585C2#1454126411yescharacteristic
586C2#1454133911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1809C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1809D17
hypocenterC17#26018
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
3. #14530349C2semidirect
4. #14530283C2semidirect
5. #14530276C2semidirect
6. #14530120C2semidirect
7. #14529740C2semidirect
8. #14529539C2semidirect
9. #14529371C2semidirect
10. #14529360C2semidirect
411. C17#26018semidirect

Maximal subgroups

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

subgroupindex
#145324982
#145316632
#145303492
#145302832
#145302762
#145301202
#145297402
#145295392
#145293712
#145293602
#145289812
#145289672
#145288682
#145285042
C17×#18092
#2601817