Finite Group Databasebeta

← #17035327

Subgroups — #17035327

Order 4352.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763289C211yesmaximal, characteristic
2C2×#11762852C211yesmaximal, characteristic, complemented
3C2×#11762808C211yesmaximal, characteristic
4C2×#11762803C211yesmaximal, characteristic
5#14529051C212yesmaximal, complemented
6#14529044C212yesmaximal, complemented
7#14529037C212yesmaximal, complemented
8#14529030C212yesmaximal, complemented
9C2×#11762571C211yesmaximal, characteristic, complemented
10#14528481C211yesmaximal, characteristic, complemented
11C17×#2316C211yesmaximal, characteristic, complemented
12C22×#63252C2211yescharacteristic
13C22×#63223C2211yescharacteristic
14C23×#3622C2211yescharacteristic
15#11763289—24no—
16#11763289—24no—
17C2×C4×Dic34C2212yes—
18C2×C4×D68C2212yes—
19C2×#10587567—22no—
20C2×#10587578—21no—
21C22×C4⋊Dic17C2211yescharacteristic
22C2×C4⋊C4×D17—21no—
23#11762852—24no—
24C2×#10587478C2212yes—
25C2×#10587479C2212yes—
26#11762808C418yescomplemented
27#11762803—24no—
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587481C2212yes—
31C2×#10587480C2211yescharacteristic
32C2×#10587480C2211yescharacteristic
33C2×#10587480C2212yes—
34#11762571C2212yes—
35#11762571C2212yes—
36#11762564C2212yes—
37#11762564C2212yes—
38C2×C17×#117C2211yescharacteristic
39C2×C17×#117C2211yescharacteristic
40C22×C4⋊C4×C17C2211yescharacteristic
41C17×#389—21no—
42C17×#389—21no—
43C17×#389C2212yes—
44C17×#389C2212yes—
45C17×#380—22no—
46C23×C4×C17—21no—
47C23×C4×C17C2311yescharacteristic
48C22×C22⋊D17—21no—
49C23×Dic17—21no—
50C2×#63252—22no—
51C2×#63252—22no—
52C2×#63252—22no—
53C2×#63252C2×C414yes—
54C2×#63252D414yes—
55C2×#63252Q814yes—
56C22×D68C2311yescharacteristic
57C22×C4×D17—21no—
58C22×C4×D17—22no—
59C22×Dic34C2311yescharacteristic
60C17×#117—22no—
61C17×#117—22no—
62C17×#117—22no—
63C17×#117—22no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—22no—
66C2×C4⋊C4×C17—22no—
67C2×C4⋊C4×C17—22no—
68C2×C4⋊C4×C17C2×C414yescomplemented
69C2×C4⋊C4×C17D411yescharacteristic
70C2×C4⋊C4×C17D411yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2312yes—
74C2×C22⋊C4×C17—21no—
75C2×C22⋊C4×C17D412yes—
76C2×C42×C17—21no—
77C2×C42×C17C2312yes—
78C2×#63227—22no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—22no—
81C2×C4⋊Dic17D414yes—
82C2×C4⋊Dic17C2312yes—
83C2×#63225—22no—
84C2×C4×Dic17—22no—
85C2×#63223—22no—
86C2×#63223—22no—
87C2×#63223C2×C414yescomplemented
88C22×#3622—22no—
89C22×#3622—22no—
90C22×#3622—22no—
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2312yes—
94#10587578—28no—
95C4⋊C4×D17—28no—
96#10587567—216no—
97C4×D68—24no—
98C4×D68—24no—
99C4×Dic34—24no—
100C4×Dic34—24no—
101C17×#95—22no—
102C17×#95—22no—
103#10587481—22no—
104#10587481—22no—
105#10587481C2×C418yescomplemented
106#10587480—22no—
107#10587480—22no—
108#10587480C2×C418yescomplemented
109#10587479—24no—
110#10587478—24no—
111C24×C17C2×D411yescharacteristic
112C23×D17—21no—
113C22×C4×C17—22no—
114C22×C4×C17—24no—
115C22×C4×C17—21no—
116C22×C4×C17—21no—
117C22×C4×C17—81no—
118C22×C4×C17—81no—
119C22×C4×C17C22×C412yes—
120C22×C4×C17C2×D412yes—
121C22×C4×C17C2×D412yes—
122C22×C4×C17C2×D412yes—
123C22×C4×C17C2×D412yes—
124C22×C4×C17C2×D411yescharacteristic
125C22×C4×C17C2×Q812yes—
126C22×C4×C17#4112yes—
127C22×C4×C17#4112yes—
128C22×C4×C17C2411yescharacteristic
129C2×C22⋊D17—24no—
130C2×C22⋊D17—24no—
131C2×C22⋊D17—24no—
132C22×Dic17—21no—
133C22×Dic17—22no—
134C22×Dic17—24no—
135C22×Dic17—81no—
136C22×Dic17—81no—
137#63252—28no—
138#63252—28no—
139#63252—28no—
140#63252C4⋊C4116yescomplemented
141C2×D68—21no—
142C2×D68—21no—
143C2×D68—21no—
144C2×D68C22×C412yes—
145C2×D68C2×D412yes—
146C2×D68C2×Q812yes—
147C2×C4×D17—24no—
148C2×C4×D17—24no—
149C2×C4×D17—24no—
150C2×C4×D17—28no—
151C2×C4×D17—82no—
152C2×C4×D17—82no—
153C2×Dic34—21no—
154C2×Dic34—21no—
155C2×Dic34—21no—
156C2×Dic34C22×C412yes—
157C2×Dic34C2×D412yes—
158C2×Dic34C2×Q812yes—
159C4⋊C4×C17—24no—
160C4⋊C4×C17—22no—
161C4⋊C4×C17—22no—
162C4⋊C4×C17—22no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—82no—
168C4⋊C4×C17—82no—
169C4⋊C4×C17C22×C414yes—
170C22⋊C4×C17—24no—
171C22⋊C4×C17—42no—
172C42×C17—22no—
173C42×C17—22no—
174C42×C17—42no—
175#63227—28no—
176C4⋊Dic17—24no—
177C4⋊Dic17—22no—
178C4⋊Dic17C2×D414yes—
179#63225—28no—
180C4×Dic17—28no—
181#63223—28no—
182C2×#3622—22no—
183C2×#3622—22no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—44no—
191C2×#3622C22×C414yes—
192#20878—171nomaximal, Sylow
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17C4×D412yes—
198C23×C17#12012yes—
199C23×C17C22⋊Q812yes—
200C23×C17C22×D411yescharacteristic
201C22×D17—22no—
202C22×D17—22no—
203C22×D17—22no—
204C22×D17—41no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17—41no—
208C2×C4×C17—24no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—22no—
213C2×C4×C17—22no—
214C2×C4×C17—22no—
215C2×C4×C17—42no—
216C2×C4×C17—42no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—84no—
224C2×C4×C17C2×C4⋊C414yes—
225C2×C4×C17C4×D414yes—
226C2×C4×C17C4×D412yes—
227C2×C4×C17#12012yes—
228C2×C4×C17C22⋊Q814yes—
229C2×C4×C17C22⋊Q812yes—
230C2×C4×C17#12314yes—
231C2×C4×C17C23×C411yescharacteristic
232C2×C4×C17C22×D411yescharacteristic
233C2×C4×C17C22×D411yescharacteristic
234C2×C4×C17C22×Q811yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C22⋊D17—216no—
238C2×Dic17—22no—
239C2×Dic17—22no—
240C2×Dic17—22no—
241C2×Dic17—24no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—84no—
245D68—22no—
246D68—22no—
247D68—22no—
248D68C2×C4⋊C414yes—
249C4×D17—216no—
250C4×D17—88no—
251Dic34—22no—
252Dic34—22no—
253Dic34—22no—
254Dic34C2×C4⋊C414yes—
255#3622—24no—
256#3622—41no—
257#3622—41no—
258C2×#520—171no—
259C2×#428—171no—
260C2×#417—171no—
261C2×#416—171no—
262#2316D1711yescharacteristic, complemented
263#1865—172no—
264#1864—172no—
265#1863—172no—
266#1862—172no—
267C2×#340—171no—
268#1729—171no—
269C22×C17—22no—
270C22×C17—22no—
271C22×C17—22no—
272C22×C17—22no—
273C22×C17—22no—
274C22×C17—22no—
275C22×C17#38514yes—
276C22×C17C2×C4×D411yescharacteristic
277C22×C17C2×#12011yescharacteristic
278C22×C17C2×C22⋊Q811yescharacteristic
279C22×C17#57611yescharacteristic
280C22×C17#57711yescharacteristic
281C22×C17#57711yescharacteristic
282C22×C17#57811yescharacteristic
283C2×D17—24no—
284C2×D17—42no—
285C2×D17—42no—
286C2×D17—42no—
287C2×D17—42no—
288C2×D17—42no—
289C2×D17—42no—
290C4×C17—44no—
291C4×C17—42no—
292C4×C17—42no—
293C4×C17—81no—
294C4×C17—81no—
295C4×C17#38514yes—
296C4×C17C22×C4⋊C411yescharacteristic
297C4×C17C2×C4×D411yescharacteristic
298C4×C17C2×C22⋊Q811yescharacteristic
299C4×C17C2×#12311yescharacteristic
300Dic17—24no—
301Dic17—44no—
302C22×#41—171no—
303C22×M16—171no—
304C23×C8—171no—
305#520—344no—
306#520—344no—
307C2×C4×Q8—172no—
308C2×C4×D4—172no—
309C2×C4×D4—342no—
310C2×#117—341no—
311C2×#117C2×D1711yescharacteristic
312C2×#117C2×D1711yescharacteristic
313C22×C4⋊C4—171no—
314C22×C4⋊C4—341no—
315C22×C4⋊C4C2×D1711yescharacteristic
316#428—344no—
317C2×#107—172no—
318C2×#106—172no—
319#417—178no—
320#416—344no—
321C2×#103—172no—
322C2×#103—171no—
323C2×#103—171no—
324C2×#102—172no—
325C2×#102—171no—
326C2×#102—171no—
327#389—21no—
328#389—21no—
329#389C2×D1712yes—
330#389C2×D1712yes—
331#380—22no—
332#340—172no—
333#340—172no—
334#336—172no—
335#336—172no—
336C2×C17—24no—
337C2×C17C2×#38511yescharacteristic
338C2×C17#294611yescharacteristic
339C2×C17#294711yescharacteristic
340C2×C17#301211yescharacteristic
341C2×C17#301311yescharacteristic
342C2×C17#308311yescharacteristic
343C2×C17#308411yescharacteristic
344D17—44no—
345C2×#41—174no—
346C2×#41—174no—
347C2×#41—174no—
348C2×#41—342no—
349C2×#41—342no—
350C2×#41—342no—
351C22×Q8—171no—
352C22×D4—171no—
353C22×D4—341no—
354C23×C4—21no—
355C23×C4—341no—
356C23×C4—341no—
357C23×C4—342no—
358C23×C4C22×D1711yescharacteristic
359C2×M16—174no—
360C2×M16—342no—
361C2×M16—342no—
362C22×C8—172no—
363C22×C8—171no—
364C22×C8—171no—
365C22×C8—342no—
366C22×C8—342no—
367C22×C8—342no—
368C4×Q8—344no—
369C4×Q8—344no—
370C4×D4—344no—
371C4×D4—344no—
372C4×D4—3416no—
373#117—22no—
374#117—22no—
375#117—22no—
376#117—22no—
377#117—348no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—174no—
383C2×C4⋊C4—172no—
384C2×C4⋊C4—348no—
385C2×C4⋊C4—342no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—342no—
388C2×C4⋊C4C4×D1714yescomplemented
389C2×C4⋊C4C22⋊D1711yescharacteristic
390C2×C4⋊C4C22⋊D1711yescharacteristic
391C2×C4⋊C4C22×D1711yescharacteristic
392C2×C4⋊C4C22×D1711yescharacteristic
393C2×C4⋊C4C22×D1712yes—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—342no—
396C2×C22⋊C4C22⋊D1712yes—
397C2×C42—21no—
398C2×C42—342no—
399C2×C42C22×D1712yes—
400#107—344no—
401#106—344no—
402#103—178no—
403#103—342no—
404#103—342no—
405#102—178no—
406#102—342no—
407#102—342no—
408#95—22no—
409#95—22no—
410C17#2087811yescharacteristic, complemented, Sylow
411C24—341no—
412C24C2×C22⋊D1711yescharacteristic
413#41—1716no—
414#41—348no—
415#41—348no—
416#41—348no—
417C2×Q8—172no—
418C2×Q8—172no—
419C2×Q8—172no—
420C2×Q8—341no—
421C2×Q8—341no—
422C2×Q8—341no—
423C2×D4—172no—
424C2×D4—172no—
425C2×D4—172no—
426C2×D4—344no—
427C2×D4—344no—
428C2×D4—344no—
429C2×D4—341no—
430C2×D4—341no—
431C2×D4—341no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—348no—
439C22×C4—341no—
440C22×C4—344no—
441C22×C4—344no—
442C22×C4—344no—
443C22×C4—342no—
444C22×C4—344no—
445C22×C4—1362no—
446C22×C4—1362no—
447C22×C4—1361no—
448C22×C4—1361no—
449C22×C4C2×C4×D1712yes—
450C22×C4#6325212yes—
451C22×C4D4×D1712yes—
452C22×C4D4×D1712yes—
453C22×C4Q8×D1712yes—
454C22×C4#6325612yes—
455C22×C4C2×C22⋊D1712yes—
456C22×C4C2×C22⋊D1712yes—
457C22×C4C2×C22⋊D1711yescharacteristic
458C22×C4C23×D1711yescharacteristic
459M16—348no—
460C2×C8—174no—
461C2×C8—342no—
462C2×C8—342no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—684no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—174no—
481C4⋊C4—348no—
482C4⋊C4—342no—
483C4⋊C4—344no—
484C4⋊C4C2×C4×D1714yes—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—348no—
488C42—22no—
489C42—22no—
490C42—42no—
491C42—348no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—342no—
497C23—342no—
498C23—342no—
499C23—681no—
500C23—681no—
501C23—681no—
502C23—681no—
503C23C4×C22⋊D1712yes—
504C23#1058762312yes—
505C23#1058763212yes—
506C23C22×C22⋊D1711yescharacteristic
507Q8—174no—
508Q8—342no—
509Q8—342no—
510Q8—342no—
511D4—174no—
512D4—3416no—
513D4—342no—
514D4—342no—
515D4—342no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—342no—
533C2×C4—342no—
534C2×C4—342no—
535C2×C4—3416no—
536C2×C4—344no—
537C2×C4—682no—
538C2×C4—682no—
539C2×C4—1368no—
540C2×C4—1364no—
541C2×C4C4⋊C4×D1714yes—
542C2×C4#1058757914yes—
543C2×C4#1058758014yes—
544C2×C4#1058758214yes—
545C2×C4C4×C22⋊D1712yes—
546C2×C4#1058762312yes—
547C2×C4#1058763212yes—
548C2×C4C22×C4×D1711yescharacteristic
549C2×C4C2×#6325211yescharacteristic
550C2×C4C2×D4×D1711yescharacteristic
551C2×C4C2×D4×D1711yescharacteristic
552C2×C4C2×Q8×D1711yescharacteristic
553C2×C4C2×#6325611yescharacteristic
554C8—344no—
555C8—681no—
556C8—681no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—344no—
564C22—682no—
565C22—682no—
566C22—682no—
567C22—682no—
568C22—682no—
569C22—682no—
570C22#1176268014yes—
571C22C2×C4×C22⋊D1711yescharacteristic
572C22C2×#1058762311yescharacteristic
573C22C2×#1058763211yescharacteristic
574C22#1176361211yescharacteristic
575C22#1176361711yescharacteristic
576C22#1176362011yescharacteristic
577C22#1176362411yescharacteristic
578C4—42no—
579C4—42no—
580C4—44no—
581C4—81no—
582C4—81no—
583C4—344no—
584C4—684no—
585C4#1176268014yes—
586C4C2×C4⋊C4×D1711yescharacteristic
587C4C2×#1058757911yescharacteristic
588C4C2×#1058758211yescharacteristic
589C4C2×#1058758011yescharacteristic
590C2—24no—
591C2—684no—
592C2C2×#1176268011yescharacteristic
593C2#1453761611yescharacteristic
594C2#1453762711yescharacteristic
595C2#1453808511yescharacteristic
596C2#1453809611yescharacteristic
597C2#1453863111yescharacteristic
598C2#1453864211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2316C2
socleC23×C17C22×D4
prime radicalC22×#63252C22
cosocleC22×C4×C17C24
hypercenter#2316D17
hypocenterC17#20878
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2316

Complements

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

NH ≅ G/Ntype
2. C2×#11762852C2semidirect
5. #14529051C2semidirect
6. #14529044C2semidirect
7. #14529037C2semidirect
8. #14529030C2semidirect
9. C2×#11762571C2semidirect
10. #14528481C2semidirect
11. C17×#2316C2semidirect
26. #11762808C4semidirect
68. C2×C4⋊C4×C17C2×C4semidirect
87. C2×#63223C2×C4semidirect
105. #10587481C2×C4semidirect
108. #10587480C2×C4semidirect
140. #63252C4⋊C4semidirect
262. #2316D17semidirect
388. C2×C4⋊C4C4×D17semidirect
410. C17#20878semidirect

Maximal subgroups

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

subgroupindex
C2×#117632892
C2×#117628522
C2×#117628082
C2×#117628032
#145290512
#145290442
#145290372
#145290302
C2×#117625712
#145284812
C17×#23162
#2087817