Finite Group Databasebeta

← #17005354

Subgroups — #17005354

Order 4352.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538831C211yesmaximal, characteristic
2#14537782C211yesmaximal, characteristic, complemented
3D4×#63223C211yesmaximal, characteristic, complemented
4#14527695C211yesmaximal, characteristic, complemented
5#14527677C211yesmaximal, characteristic, complemented
6#14527665C211yesmaximal, characteristic, complemented
7#14527645C211yesmaximal, characteristic, complemented
8#14527617C211yesmaximal, characteristic, complemented
9#14527616C211yesmaximal, characteristic, complemented
10#14527593C211yesmaximal, characteristic, complemented
11#14527576C211yesmaximal, characteristic, complemented
12#14527572C211yesmaximal, characteristic, complemented
13#14527250C211yesmaximal, characteristic, complemented
14#14527079C211yesmaximal, characteristic, complemented
15C17×#1478C211yesmaximal, characteristic, complemented
16C22×#63223—21no—
17#11763343C2211yescharacteristic
18#11763144C411yescharacteristic
19#11763127C411yescharacteristic
20#11763109C411yescharacteristic
21#11763109C411yescharacteristic
22#11763024C411yescharacteristic
23#11763024C411yescharacteristic
24#11762960C411yescharacteristic
25#11762943C411yescharacteristic
26#11762936C2211yescharacteristic, complemented
27#11762933C2211yescharacteristic, complemented
28C4×#63229C2211yescharacteristic
29C4×#63229C2211yescharacteristic
30C4×#63228C2211yescharacteristic
31C4×#63228C2211yescharacteristic
32#11762887C2211yescharacteristic
33#11762887C2211yescharacteristic
34#11762886C2211yescharacteristic, complemented
35#11762884—21no—
36#11762883—21no—
37#11762881—21no—
38#11762881—21no—
39D4×#3622C2211yescharacteristic
40D4×#3622C2211yescharacteristic
41#11762874C2211yescharacteristic, complemented
42#11762868—21no—
43#11762862C2211yescharacteristic
44#11762833C2211yescharacteristic
45#11762773—21no—
46#11762763—21no—
47C4×#63223C2211yescharacteristic
48#11762561C2211yescharacteristic, complemented
49#11762561C2211yescharacteristic, complemented
50#11762559C2211yescharacteristic, complemented
51#11762559C2211yescharacteristic, complemented
52#11762541C2211yescharacteristic, complemented
53#11762541C2211yescharacteristic, complemented
54#11762538C2211yescharacteristic, complemented
55#11762538C2211yescharacteristic, complemented
56#11762531C2211yescharacteristic, complemented
57#11762531C2211yescharacteristic, complemented
58#11762530C2211yescharacteristic, complemented
59C8×#3622C2211yescharacteristic, complemented
60C2×C4×D4×C17C2211yescharacteristic
61C17×#431C2211yescharacteristic
62C17×#431C2211yescharacteristic
63C17×D4⋊C8C2211yescharacteristic
64C17×D4⋊C8C2211yescharacteristic
65C17×D4⋊C8C2211yescharacteristic
66C17×D4⋊C8C2211yescharacteristic
67C22×D4×C17C2×C411yescharacteristic
68C23×C4×C17—21no—
69C4×D4×C17—21no—
70C4×D4×C17—21no—
71C4×D4×C17C2×C411yescharacteristic
72C4×D4×C17C2×C411yescharacteristic
73C4×D4×C17C2311yescharacteristic
74C4×D4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2×C411yescharacteristic
76C2×C22⋊C4×C17—21no—
77C2×C42×C17C2311yescharacteristic
78C2×#63229C2×C411yescharacteristic
79C2×#63229C2×C411yescharacteristic
80C2×#63228C2×C411yescharacteristic
81C2×#63228C2×C411yescharacteristic
82#10587616C2×C411yescharacteristic
83#10587615—21no—
84C4×C22⋊D17—21no—
85#10587610C2×C411yescharacteristic
86C2×#63223—21no—
87C2×#63223—21no—
88C2×#63223—21no—
89C2×#63223—44no—
90C2×#63223D411yescharacteristic
91C2×#63223D411yescharacteristic
92C22×#3622—21no—
93C22×#3622—21no—
94C2×#63218—21no—
95C2×C8×D17—21no—
96#10587562—21no—
97#10587561C2×C411yescharacteristic
98C4×D68C2311yescharacteristic
99#10587558—21no—
100#10587556C2×C411yescharacteristic
101C4×Dic34C2311yescharacteristic
102C17×C4⋊C8—21no—
103C17×C4⋊C8C2311yescharacteristic
104C17×C4⋊C8C2311yescharacteristic
105C17×C22⋊C8—21no—
106C17×C22⋊C8—21no—
107C4×C8×C17—21no—
108C4×C8×C17C2311yescharacteristic
109C4×C8×C17C2311yescharacteristic
110#10587503C2×C411yescharacteristic
111#10587503C2×C411yescharacteristic
112#10587503C2×C411yescharacteristic
113#10587503C2×C411yescharacteristic
114C22⋊#3622—21no—
115C22⋊#3622—21no—
116C22⋊#3622—21no—
117C22⋊#3622—21no—
118C22⋊#3622C2×C411yescharacteristic
119C22⋊#3622C2×C411yescharacteristic
120#10587491—21no—
121#10587491—21no—
122#10587486—21no—
123#10587485—21no—
124#10587485—21no—
125C8×Dic17—21no—
126#10587481C2×C411yescharacteristic
127#10587481C2×C411yescharacteristic
128#10587480C2×C411yescharacteristic
129#10587480C2×C411yescharacteristic
130#10587479C2×C411yescharacteristic
131#10587479C2×C411yescharacteristic
132#10587478C2×C411yescharacteristic
133#10587478C2×C411yescharacteristic
134C4⋊#3622—21no—
135C4⋊#3622C2×C411yescharacteristic
136C4⋊#3622C2×C411yescharacteristic
137C4⋊#3622C2311yescharacteristic
138C4⋊#3622C2311yescharacteristic
139#10587474D411yescharacteristic
140#10587474D411yescharacteristic
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C4×#3622C2311yescharacteristic
145C24×C17—21no—
146C2×D4×C17—22no—
147C2×D4×C17—21no—
148C2×D4×C17—41no—
149C2×D4×C17C22×C411yescharacteristic
150C2×D4×C17C22×C411yescharacteristic
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—42no—
157C22×C4×C17—42no—
158C22×C4×C17C22×C411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17#4111yescharacteristic
161C2×C22⋊D17—21no—
162C2×D68C22×C411yescharacteristic
163C2×C4×D17—21no—
164C2×Dic34C22×C411yescharacteristic
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17C22×C411yescharacteristic
171C4⋊C4×C17C22×C411yescharacteristic
172C22⋊C4×C17—21no—
173C22⋊C4×C17—21no—
174C22⋊C4×C17—21no—
175C22⋊C4×C17—21no—
176C42×C17C22×C411yescharacteristic
177C42×C17C2×D411yescharacteristic
178C42×C17#4111yescharacteristic
179C42×C17C2411yescharacteristic
180C22⋊Dic17—21no—
181#63229—22no—
182#63229—41no—
183#63228—22no—
184#63228—41no—
185#63227—21no—
186#63227—21no—
187#63227—21no—
188#63227—21no—
189C4⋊Dic17C22×C411yescharacteristic
190C4⋊Dic17C22×C411yescharacteristic
191#63225—21no—
192#63225—21no—
193#63225—21no—
194#63225—21no—
195C4×Dic17—21no—
196#63223—22no—
197#63223—41no—
198#63223—42no—
199#63223—42no—
200#63223—42no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—42no—
204C2×#3622—42no—
205C2×#3622C22×C411yescharacteristic
206C2×#3622C22×C411yescharacteristic
207C2×#3622C22×C411yescharacteristic
208C2×#3622C22×C411yescharacteristic
209C2×#3622C2×D411yescharacteristic
210C2×#3622C2×D411yescharacteristic
211C2×#3622C2×D411yescharacteristic
212C2×#3622C2×D411yescharacteristic
213#63218—42no—
214C8×D17—42no—
215#18006—171nomaximal, Sylow
216C23×C17—21no—
217C23×C17—21no—
218C23×C17—42no—
219C23×C17—42no—
220C23×C17—81no—
221C23×C17C4×D411yescharacteristic
222C22×D17—21no—
223D4×C17—21no—
224D4×C17—21no—
225D4×C17—21no—
226D4×C17—42no—
227D4×C17#13112yes—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—41no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17C4×D411yescharacteristic
243C2×C4×C17C4×D411yescharacteristic
244C2×C4×C17C4×D411yescharacteristic
245C2×C4×C17C23×C411yescharacteristic
246C2×C4×C17C22×D411yescharacteristic
247C2×C4×C17C2×#4111yescharacteristic
248C22⋊D17—81no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—21no—
252D68—21no—
253D68—21no—
254C4×D17—41no—
255C4×D17—41no—
256Dic34—21no—
257Dic34—21no—
258C8×C17—41no—
259C8×C17—41no—
260C8×C17—41no—
261C8×C17—41no—
262#3622—21no—
263#3622—21no—
264#3622—21no—
265#3622—21no—
266#3622—21no—
267#3622—21no—
268#3622—21no—
269#3622—41no—
270#3622—41no—
271#3622AGL(1,ℤ/8)12yes—
272#3105—171no—
273#2969—171no—
274D4×M16—171no—
275#1594—171no—
276#1590—171no—
277#1588—171no—
278#1586—171no—
279#1582—171no—
280#1581—171no—
281SD16⋊C8—171no—
282C8×D8—171no—
283#1574—171no—
284#1495—171no—
285#1478D1711yescharacteristic, complemented
286#1453—171no—
287C22×C17—21no—
288C22×C17—22no—
289C22×C17—21no—
290C22×C17—21no—
291C22×C17—41no—
292C22×C17—41no—
293C22×C17—41no—
294C22×C17—41no—
295C22×C17—41no—
296C22×C17—84no—
297C22×C17C2×C4×D411yescharacteristic
298C2×D17—41no—
299C2×D17—41no—
300C2×D17—41no—
301C4×C17—21no—
302C4×C17—21no—
303C4×C17—21no—
304C4×C17—21no—
305C4×C17—41no—
306C4×C17—41no—
307C4×C17C2×#13111yescharacteristic
308C4×C17#56811yescharacteristic
309C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
310C4×C17#57511yescharacteristic
311Dic17—41no—
312Dic17—41no—
313Dic17—41no—
314C22×M16—341no—
315#529—171no—
316C2×C4×D4C2×D1711yescharacteristic
317#484—171no—
318#481—171no—
319#478—171no—
320#478—171no—
321#463—171no—
322#463—171no—
323#450—171no—
324#447—171no—
325#446—171no—
326C8×Q8—171no—
327C4×SD16—171no—
328C4×SD16—171no—
329C4×D8—171no—
330C4×D8—171no—
331#436—171no—
332#436—171no—
333#436—171no—
334#436—341no—
335#435—341no—
336#435—341no—
337#435—341no—
338C8×D4—171no—
339C8×D4—171no—
340C8×D4—171no—
341#433—341no—
342#431—171no—
343#431C2×D1711yescharacteristic
344#431C2×D1711yescharacteristic
345#423—171no—
346#407—341no—
347#405—341no—
348C4×M16—171no—
349#335—171no—
350#335—171no—
351#334—171no—
352#334—171no—
353Q8⋊C8—171no—
354Q8⋊C8—171no—
355D4⋊C8—171no—
356D4⋊C8—171no—
357D4⋊C8C2×D1711yescharacteristic
358D4⋊C8C2×D1711yescharacteristic
359D4⋊C8C2×D1711yescharacteristic
360D4⋊C8C2×D1711yescharacteristic
361#322—171no—
362#322—171no—
363#322—171no—
364C82—171no—
365C2×C17—41no—
366C2×C17—42no—
367C2×C17—81no—
368C2×C17#293311yescharacteristic
369C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
370C2×C17#295711yescharacteristic
371D17—81no—
372C22×D4C4×D1711yescharacteristic
373C23×C4—21no—
374C2×SD16—171no—
375C2×SD16—171no—
376C2×D8—171no—
377C2×D8—171no—
378C2×M16—171no—
379C2×M16—171no—
380C2×M16—341no—
381C2×M16—341no—
382C2×M16—341no—
383C2×M16—341no—
384C2×M16—684no—
385C22×C8—341no—
386C22×C8—341no—
387C22×C8—341no—
388#126—341no—
389#125—171no—
390#124—171no—
391#123—341no—
392C22⋊Q8—171no—
393#121—171no—
394C4×Q8—171no—
395C4×D4—21no—
396C4×D4—21no—
397C4×D4—171no—
398C4×D4—341no—
399C4×D4C4×D1711yescharacteristic
400C4×D4C4×D1711yescharacteristic
401C4×D4C22×D1711yescharacteristic
402C4×D4C22×D1711yescharacteristic
403#117—341no—
404C2×C4⋊C4C4×D1711yescharacteristic
405C2×C22⋊C4—21no—
406C2×C42C22×D1711yescharacteristic
407#107—171no—
408#107—171no—
409#106—171no—
410#106—171no—
411C4⋊C8—21no—
412C4⋊C8—171no—
413C4⋊C8—171no—
414C4⋊C8—171no—
415C4⋊C8—171no—
416C4⋊C8—341no—
417C4⋊C8—341no—
418C4⋊C8—341no—
419C4⋊C8C22×D1711yescharacteristic
420C4⋊C8C22×D1711yescharacteristic
421#103—171no—
422#103—171no—
423#102—171no—
424#102—171no—
425#102—171no—
426#102—171no—
427#102—171no—
428#102—171no—
429C22⋊C8—21no—
430C22⋊C8—21no—
431C22⋊C8—171no—
432C22⋊C8—171no—
433C22⋊C8—341no—
434C22⋊C8—341no—
435C22⋊C8—341no—
436C22⋊C8—341no—
437C22⋊C8—341no—
438C22⋊C8—341no—
439#97—171no—
440#97—171no—
441#97—341no—
442C4×C8—21no—
443C4×C8—171no—
444C4×C8—171no—
445C4×C8—171no—
446C4×C8—171no—
447C4×C8—341no—
448C4×C8C22×D1711yescharacteristic
449C4×C8C22×D1711yescharacteristic
450C17#1800611yescharacteristic, complemented, Sylow
451C24—21no—
452C2×Q8—171no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—41no—
456C2×D4—171no—
457C2×D4—341no—
458C2×D4C2×C4×D1711yescharacteristic
459C2×D4C2×C4×D1711yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—341no—
468C22×C4C2×C4×D1711yescharacteristic
469C22×C4D4×D1711yescharacteristic
470C22×C4#6325411yescharacteristic
471SD16—342no—
472SD16—681no—
473D8—342no—
474D8—681no—
475M16—342no—
476M16—682no—
477M16—682no—
478M16—681no—
479M16—682no—
480M16—682no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—171no—
486C2×C8—171no—
487C2×C8—171no—
488C2×C8—171no—
489C2×C8—171no—
490C2×C8—171no—
491C2×C8—171no—
492C2×C8—171no—
493C2×C8—341no—
494C2×C8—341no—
495C2×C8—682no—
496C2×C8—682no—
497C2×C8—682no—
498C4⋊C4—21no—
499C4⋊C4—171no—
500C4⋊C4—171no—
501C4⋊C4—341no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4C2×C4×D1711yescharacteristic
506C4⋊C4C2×C4×D1711yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—341no—
512C22⋊C4—341no—
513C22⋊C4—341no—
514C22⋊C4—341no—
515C22⋊C4—341no—
516C42—341no—
517C42C2×C4×D1711yescharacteristic
518C42D4×D1711yescharacteristic
519C42#6325411yescharacteristic
520C42C23×D1711yescharacteristic
521C23—21no—
522C23—21no—
523C23—42no—
524C23—42no—
525C23—81no—
526C23—341no—
527C23#1058756711yescharacteristic
528Q8—341no—
529Q8—341no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—42no—
534D4—341no—
535D4—341no—
536D4—1361no—
537D4#1058758712yes—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—341no—
553C2×C4—341no—
554C2×C4—341no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4#1058756711yescharacteristic
558C2×C4#1058756711yescharacteristic
559C2×C4#1058756711yescharacteristic
560C2×C4C22×C4×D1711yescharacteristic
561C2×C4C2×D4×D1711yescharacteristic
562C2×C4C2×#6325411yescharacteristic
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—172no—
568C8—341no—
569C8—341no—
570C8—341no—
571C8—341no—
572C8—341no—
573C8—341no—
574C8—341no—
575C8—681no—
576C8—681no—
577C22—21no—
578C22—22no—
579C22—21no—
580C22—21no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—84no—
587C22—681no—
588C22—681no—
589C22—681no—
590C22C2×#1058756711yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—681no—
598C4—681no—
599C4—681no—
600C4C2×#1058758711yescharacteristic
601C4#1176357511yescharacteristic
602C4AGL(1,ℤ/8)×D1711yescharacteristic
603C4#1176360511yescharacteristic
604C2—41no—
605C2—42no—
606C2—81no—
607C2—1361no—
608C2#1453750511yescharacteristic
609C2#1453758811yescharacteristic
610C2#1453769711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1478C2
socleC22×C17C2×C4×D4
prime radical#11762943C4
cosocleC42×C17C24
hypercenter#1478D17
hypocenterC17#18006
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
2. #14537782C2semidirect
3. D4×#63223C2semidirect
4. #14527695C2semidirect
5. #14527677C2semidirect
6. #14527665C2semidirect
7. #14527645C2semidirect
8. #14527617C2semidirect
9. #14527616C2semidirect
10. #14527593C2semidirect
11. #14527576C2semidirect
12. #14527572C2semidirect
13. #14527250C2semidirect
14. #14527079C2semidirect
15. C17×#1478C2semidirect
26. #11762936C22semidirect
27. #11762933C22semidirect
34. #11762886C22semidirect
41. #11762874C22semidirect
48. #11762561C22semidirect
49. #11762561C22semidirect
50. #11762559C22semidirect
51. #11762559C22semidirect
52. #11762541C22semidirect
53. #11762541C22semidirect
54. #11762538C22semidirect
55. #11762538C22semidirect
56. #11762531C22semidirect
57. #11762531C22semidirect
58. #11762530C22semidirect
59. C8×#3622C22semidirect
285. #1478D17semidirect
450. C17#18006semidirect

Maximal subgroups

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

subgroupindex
#145388312
#145377822
D4×#632232
#145276952
#145276772
#145276652
#145276452
#145276172
#145276162
#145275932
#145275762
#145275722
#145272502
#145270792
C17×#14782
#1800617