Finite Group Databasebeta

← #17037344

Subgroups — #17037344

Order 4352.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531262C211yesmaximal, characteristic, complemented
2C4×D4×Dic17C211yesmaximal, characteristic
3#14529427C211yesmaximal, characteristic, complemented
4#14529420C211yesmaximal, characteristic, complemented
5#14529378C211yesmaximal, characteristic, complemented
6#14529357C211yesmaximal, characteristic, complemented
7#14529120C211yesmaximal, characteristic
8#14529120C211yesmaximal, characteristic
9#14529047C211yesmaximal, characteristic
10#14529033C211yesmaximal, characteristic
11#14528681C211yesmaximal, characteristic, complemented
12#14528614C211yesmaximal, characteristic, complemented
13#14528609C211yesmaximal, characteristic
14#14528532C211yesmaximal, characteristic, complemented
15C2×C17×#440C211yesmaximal, characteristic
16C2×D4×Dic17C2211yescharacteristic
17C2×D4×Dic17C2211yescharacteristic
18C22×C4×Dic17—21no—
19C2×#10587489C2211yescharacteristic
20C2×#10587488C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587474C2211yescharacteristic
26#11762716C2211yescharacteristic
27#11762716C2211yescharacteristic
28#11762708C2211yescharacteristic
29#11762708C2211yescharacteristic
30#11762695—21no—
31#11762695—21no—
32C4⋊C4×Dic17C2211yescharacteristic
33C4⋊C4×Dic17C2211yescharacteristic
34C4×C4⋊Dic17—21no—
35C4×C4⋊Dic17C2211yescharacteristic
36C4×C4⋊Dic17C2211yescharacteristic
37C22⋊C4×Dic17—21no—
38C22⋊C4×Dic17—21no—
39C4×C22⋊Dic17—21no—
40#11762655—21no—
41C42×Dic17C2211yescharacteristic
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762563C2211yescharacteristic
49#11762563C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C22×SD16×C17C2211yescharacteristic
53C2×C4×Q8×C17C2211yescharacteristic
54C2×C4×D4×C17C2211yescharacteristic
55C17×#440C418yescomplemented
56C2×C17×#107C2211yescharacteristic
57C2×C17×#103C2211yescharacteristic
58C2×C17×#102C2211yescharacteristic
59C2×C17×#97C2211yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×C4×C17—21no—
63C23×Dic17—21no—
64C23×Dic17—21no—
65C2×SD16×C17—22no—
66C2×SD16×C17—22no—
67C2×SD16×C17C2×C414yes—
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C4×Q8×C17—22no—
71C4×Q8×C17C2×C414yescomplemented
72C4×D4×C17—22no—
73C4×D4×C17C2×C414yes—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C22⋊C4×C17—21no—
78C2×C42×C17—21no—
79C2×C42×C17C2311yescharacteristic
80C2×C22⋊Dic17—21no—
81C2×C22⋊Dic17—21no—
82D4×Dic17—22no—
83D4×Dic17—22no—
84D4×Dic17—22no—
85D4×Dic17—22no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—44no—
97C2×C4×Dic17D411yescharacteristic
98C2×C4×Dic17D411yescharacteristic
99C2×C4×Dic17D411yescharacteristic
100C2×C4×Dic17D411yescharacteristic
101C2×C4×Dic17C2311yescharacteristic
102C2×C4×Dic17C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C17×#107C2×C414yescomplemented
106C17×#103C2×C414yescomplemented
107C17×#102C2×C414yes—
108C17×#97C2×C414yescomplemented
109#10587506—22no—
110#10587506—22no—
111#10587503—22no—
112#10587503—22no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587489—22no—
124#10587488—22no—
125#10587474—22no—
126C24×C17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17—21no—
132C2×Q8×C17C22×C412yes—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17C22×C412yes—
137C2×D4×C17#4112yes—
138C2×D4×C17#4112yes—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—24no—
143C22×C4×C17—42no—
144C22×C4×C17—42no—
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2411yescharacteristic
148C22×Dic17—24no—
149C22×Dic17—24no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—42no—
155C22×Dic17—42no—
156C22×Dic17—42no—
157C22×Dic17—42no—
158C22×Dic17C2×D411yescharacteristic
159C22×Dic17C2×D411yescharacteristic
160C22×Dic17C2×D411yescharacteristic
161C22×Dic17C2×D411yescharacteristic
162SD16×C17—28no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17C22×C412yes—
168C2×C8×C17C22×C412yes—
169C4⋊C4×C17—24no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17C22×C412yes—
173C4⋊C4×C17C22×C412yes—
174C22⋊C4×C17—24no—
175C42×C17—24no—
176C42×C17C22×C412yes—
177C42×C17#4112yes—
178C22⋊Dic17—24no—
179C22⋊Dic17—24no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—42no—
187C4⋊Dic17—42no—
188C4⋊Dic17#4112yes—
189C4⋊Dic17#4112yes—
190C4×Dic17—21no—
191C4×Dic17—21no—
192C4×Dic17—21no—
193C4×Dic17—21no—
194C4×Dic17—22no—
195C4×Dic17—22no—
196C4×Dic17—41no—
197C4×Dic17—41no—
198C4×Dic17—42no—
199C4×Dic17—42no—
200C4×Dic17—42no—
201C4×Dic17—42no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208#21121—171nomaximal, Sylow
209C23×C17—22no—
210C23×C17—22no—
211C23×C17—22no—
212C23×C17—41no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17—41no—
216C23×C17C22×D411yescharacteristic
217Q8×C17—22no—
218Q8×C17—22no—
219Q8×C17—41no—
220Q8×C17—41no—
221D4×C17—22no—
222D4×C17—22no—
223D4×C17—22no—
224D4×C17#11714yes—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—22no—
230C2×C4×C17—22no—
231C2×C4×C17—22no—
232C2×C4×C17—48no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17C4×D412yes—
240C2×C4×C17C4×D412yes—
241C2×C4×C17C23×C411yescharacteristic
242C2×C4×C17C2×#4111yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17C2×#4111yescharacteristic
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—22no—
254C2×Dic17—22no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—48no—
260C2×Dic17—48no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17AGL(1,ℤ/8)12yes—
272C2×Dic17#13712yes—
273C8×C17—22no—
274C8×C17—22no—
275#3622—41no—
276#3622—41no—
277C2×#440D1711yescharacteristic
278#2295—171no—
279C42×D4—171no—
280#1928—171no—
281#1927—171no—
282#1921—171no—
283#1918—171no—
284#1877—171no—
285#1877—171no—
286#1864—171no—
287#1862—171no—
288#1776—171no—
289#1763—171no—
290#1762—171no—
291#1745—171no—
292C22×C17—24no—
293C22×C17—42no—
294C22×C17—42no—
295C22×C17—42no—
296C22×C17—42no—
297C22×C17—42no—
298C22×C17—42no—
299C22×C17C2×C4×D411yescharacteristic
300C22×C17#52511yescharacteristic
301C22×C17#52511yescharacteristic
302C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C17C2×#13711yescharacteristic
304C22×C17#57511yescharacteristic
305C22×C17#57511yescharacteristic
306C4×C17—22no—
307C4×C17—22no—
308C4×C17—42no—
309C4×C17—42no—
310C4×C17—42no—
311C4×C17C2×#11711yescharacteristic
312C4×C17C4×#4111yescharacteristic
313C4×C17#52911yescharacteristic
314C4×C17#52911yescharacteristic
315Dic17—22no—
316Dic17—22no—
317Dic17—42no—
318Dic17—41no—
319Dic17—42no—
320Dic17—41no—
321Dic17—81no—
322Dic17—81no—
323C22×SD16C2×D1711yescharacteristic
324C2×C4×Q8C2×D1711yescharacteristic
325C2×C4×D4—171no—
326C2×C4×D4—171no—
327C2×C4×D4C2×D1711yescharacteristic
328C22×C42—341no—
329#440Dic1718yescomplemented
330C2×#107—171no—
331C2×#107C2×D1711yescharacteristic
332C2×#106—171no—
333C2×#103—171no—
334C2×#103—171no—
335C2×#103C2×D1711yescharacteristic
336C2×#102—171no—
337C2×#102—171no—
338C2×#102C2×D1711yescharacteristic
339C2×#97—171no—
340C2×#97C2×D1711yescharacteristic
341#391—171no—
342#391—171no—
343#389—171no—
344#389—171no—
345#387—341no—
346#387—341no—
347C4×C4⋊C4—171no—
348C4×C4⋊C4—171no—
349C4×C4⋊C4—171no—
350C4×C4⋊C4—171no—
351C4×C4⋊C4—341no—
352C4×C22⋊C4—341no—
353C4×C22⋊C4—341no—
354C4×C22⋊C4—341no—
355#376—341no—
356C43—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#336—171no—
364#336—171no—
365#336—171no—
366#336—171no—
367C2×C17—44no—
368C2×C17#229011yescharacteristic
369C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
370C2×C17C4×#13711yescharacteristic
371C2×C17#311011yescharacteristic
372C2×C17#311111yescharacteristic
373C2×C17#311411yescharacteristic
374C2×C17#311611yescharacteristic
375C22×Q8C22×D1711yescharacteristic
376C22×D4C22×D1711yescharacteristic
377C23×C4—21no—
378C23×C4—341no—
379C23×C4—341no—
380C2×SD16—22no—
381C2×SD16—22no—
382C2×SD16C2×Dic1714yes—
383C22×C8—171no—
384C22×C8—171no—
385C22×C8C22×D1711yescharacteristic
386C22×C8C22×D1711yescharacteristic
387C4×Q8—22no—
388C4×Q8C2×Dic1714yescomplemented
389C4×D4—22no—
390C4×D4—342no—
391C4×D4—342no—
392C4×D4—342no—
393C4×D4—342no—
394C4×D4C2×Dic1714yes—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—171no—
399C2×C4⋊C4—171no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4C22×D1711yescharacteristic
403C2×C4⋊C4C22×D1711yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—341no—
406C2×C22⋊C4—341no—
407C2×C42—21no—
408C2×C42—171no—
409C2×C42—171no—
410C2×C42—171no—
411C2×C42—171no—
412C2×C42—171no—
413C2×C42—171no—
414C2×C42—341no—
415C2×C42—341no—
416C2×C42—341no—
417C2×C42—341no—
418C2×C42—684no—
419C2×C42C22×D1711yescharacteristic
420#107—342no—
421#107C2×Dic1714yescomplemented
422#106—342no—
423#103—342no—
424#103—342no—
425#103C2×Dic1714yescomplemented
426#102—342no—
427#102—342no—
428#102C2×Dic1714yes—
429#97—342no—
430#97C2×Dic1714yescomplemented
431#95—341no—
432#95—341no—
433#95—341no—
434#95—341no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95—341no—
440#95—341no—
441C17#2112111yescharacteristic, complemented, Sylow
442C24—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8C22×Dic1712yes—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4#6325212yes—
453C2×D4#6325212yes—
454C2×D4C22×Dic1712yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—171no—
462C22×C4—171no—
463C22×C4—171no—
464C22×C4—171no—
465C22×C4—344no—
466C22×C4—344no—
467C22×C4—341no—
468C22×C4—341no—
469C22×C4—341no—
470C22×C4—341no—
471C22×C4—682no—
472C22×C4—682no—
473C22×C4—682no—
474C22×C4—682no—
475C22×C4D4×D1711yescharacteristic
476C22×C4D4×D1711yescharacteristic
477C22×C4C23×D1711yescharacteristic
478SD16—28no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—341no—
484C2×C8—341no—
485C2×C8—341no—
486C2×C8—341no—
487C2×C8—341no—
488C2×C8—341no—
489C2×C8C22×Dic1712yes—
490C2×C8C22×Dic1712yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—24no—
494C4⋊C4—172no—
495C4⋊C4—172no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4—682no—
503C4⋊C4—682no—
504C4⋊C4C22×Dic1712yes—
505C4⋊C4C22×Dic1712yes—
506C22⋊C4—24no—
507C22⋊C4—344no—
508C22⋊C4—344no—
509C42—24no—
510C42—342no—
511C42—342no—
512C42—341no—
513C42—341no—
514C42—341no—
515C42—341no—
516C42—681no—
517C42—681no—
518C42—682no—
519C42—682no—
520C42—682no—
521C42—682no—
522C42#6325412yes—
523C42C22×Dic1712yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×D4×D1711yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#1058761214yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—48no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—172no—
555C2×C4—172no—
556C2×C4—341no—
557C2×C4—341no—
558C2×C4—341no—
559C2×C4—341no—
560C2×C4—341no—
561C2×C4—341no—
562C2×C4—342no—
563C2×C4—342no—
564C2×C4—341no—
565C2×C4—341no—
566C2×C4—341no—
567C2×C4—341no—
568C2×C4—688no—
569C2×C4—688no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4—681no—
579C2×C4—681no—
580C2×C4D4×Dic1712yes—
581C2×C4D4×Dic1712yes—
582C2×C4C2×#6325211yescharacteristic
583C2×C4C2×#6325211yescharacteristic
584C2×C4C2×#6325411yescharacteristic
585C2×C4#41×D1711yescharacteristic
586C2×C4#41×D1711yescharacteristic
587C2×C4C23×Dic1711yescharacteristic
588C8—22no—
589C8—22no—
590C8—681no—
591C8—681no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C2×D4×Dic1711yescharacteristic
600C22#1176331811yescharacteristic
601C22#1176331811yescharacteristic
602C22AGL(1,ℤ/8)×D1711yescharacteristic
603C22#137×D1711yescharacteristic
604C22#1176360511yescharacteristic
605C22#1176360811yescharacteristic
606C4—22no—
607C4—22no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—342no—
612C4—342no—
613C4—682no—
614C4—681no—
615C4—682no—
616C4—681no—
617C4—1361no—
618C4—1361no—
619C4C2×#1058761211yescharacteristic
620C4#41×Dic1711yescharacteristic
621C4#1176333911yescharacteristic
622C4#1176333911yescharacteristic
623C2—44no—
624C2#1453122911yescharacteristic
625C2AGL(1,ℤ/8)×Dic1711yescharacteristic
626C2#137×Dic1711yescharacteristic
627C2#1453888111yescharacteristic
628C2#1453889711yescharacteristic
629C2#1453892611yescharacteristic
630C2#1453894211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#440C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenterC2×#440D17
hypocenterC17#21121
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. #14531262C2semidirect
3. #14529427C2semidirect
4. #14529420C2semidirect
5. #14529378C2semidirect
6. #14529357C2semidirect
11. #14528681C2semidirect
12. #14528614C2semidirect
14. #14528532C2semidirect
55. C17×#440C4semidirect
71. C4×Q8×C17C2×C4semidirect
105. C17×#107C2×C4semidirect
106. C17×#103C2×C4semidirect
108. C17×#97C2×C4semidirect
329. #440Dic17semidirect
388. C4×Q8C2×Dic17semidirect
421. #107C2×Dic17semidirect
425. #103C2×Dic17semidirect
430. #97C2×Dic17semidirect
441. C17#21121semidirect

Maximal subgroups

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

subgroupindex
#145312622
C4×D4×Dic172
#145294272
#145294202
#145293782
#145293572
#145291202
#145290472
#145290332
#145286812
#145286142
#145286092
#145285322
C2×C17×#4402
#2112117