Finite Group Databasebeta

← #17054188

Subgroups — #17054188

Order 4352.

810 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531757C211yesmaximal, characteristic
2#14531741C211yesmaximal, characteristic, complemented
3#14529667C211yesmaximal, characteristic
4#14529640C211yesmaximal, characteristic
5#14529427C211yesmaximal, characteristic, complemented
6#14529420C211yesmaximal, characteristic, complemented
7#14529413C211yesmaximal, characteristic
8#14529406C211yesmaximal, characteristic
9#14528952C211yesmaximal, characteristic, complemented
10#14528799C211yesmaximal, characteristic, complemented
11#14528784C211yesmaximal, characteristic
12#14528683C211yesmaximal, characteristic, complemented
13#107×Dic17C211yesmaximal, characteristic, complemented
14#14528656C211yesmaximal, characteristic, complemented
15C2×C17×#440C211yesmaximal, characteristic
16C2×#10587489C2211yescharacteristic
17C2×#10587489C2211yescharacteristic
18C2×#10587488C2211yescharacteristic
19C2×#10587488C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587486C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27#11762716C2211yescharacteristic
28#11762716C2211yescharacteristic
29#11762712C2211yescharacteristic
30#11762712C2211yescharacteristic
31#11762702—21no—
32#11762702—21no—
33#11762695—21no—
34#11762695—21no—
35#11762679C2211yescharacteristic
36#11762675—21no—
37#11762675C2211yescharacteristic
38#11762672—21no—
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4×C4⋊Dic17—21no—
44C4×C4⋊Dic17C2211yescharacteristic
45C4×C22⋊Dic17—21no—
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
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—
63C2×SD16×C17—22no—
64C2×SD16×C17—22no—
65C2×SD16×C17C2×C414yes—
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C4×Q8×C17—22no—
69C4×Q8×C17C2×C414yes—
70C4×D4×C17—22no—
71C4×D4×C17C2×C414yes—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—21no—
76C2×C42×C17—21no—
77C2×C42×C17C2311yescharacteristic
78C2×C22⋊Dic17—21no—
79C2×C22⋊Dic17—21no—
80C2×C22⋊Dic17—21no—
81C2×C22⋊Dic17—21no—
82C2×C4⋊Dic17—21no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17D411yescharacteristic
85C2×C4⋊Dic17D411yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17C2311yescharacteristic
95C2×C4×Dic17C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C17×#107C2×C414yes—
99C17×#103C2×C414yes—
100C17×#102C2×C414yes—
101C17×#97C2×C414yes—
102#10587506—41no—
103#10587506—41no—
104#10587503—41no—
105#10587503—41no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502D411yescharacteristic
117#10587502D411yescharacteristic
118#10587489—22no—
119#10587489—41no—
120#10587488—22no—
121#10587488—41no—
122#10587486—22no—
123C8×Dic17—22no—
124C4⋊#3622—22no—
125C24×C17—21no—
126C2×Q8×C17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17C22×C412yes—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17C22×C412yes—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—24no—
142C22×C4×C17—81no—
143C22×C4×C17—81no—
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2411yescharacteristic
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17C2×D411yescharacteristic
152C22×Dic17C2×D411yescharacteristic
153C22×Dic17C2×D411yescharacteristic
154C22×Dic17C2×D411yescharacteristic
155SD16×C17—28no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17C22×C412yes—
159C2×C8×C17C22×C412yes—
160C2×C8×C17#4112yes—
161C2×C8×C17#4112yes—
162C4⋊C4×C17—24no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17C22×C412yes—
166C4⋊C4×C17C22×C412yes—
167C22⋊C4×C17—24no—
168C42×C17—24no—
169C42×C17C22×C412yes—
170C42×C17#4112yes—
171C22⋊Dic17—42no—
172C22⋊Dic17—42no—
173C22⋊Dic17—81no—
174C22⋊Dic17—81no—
175C4⋊Dic17—22no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—41no—
185C4⋊Dic17—42no—
186C4⋊Dic17—81no—
187C4×Dic17—21no—
188C4×Dic17—21no—
189C4×Dic17—21no—
190C4×Dic17—21no—
191C4×Dic17—81no—
192C4×Dic17—81no—
193C4×Dic17—81no—
194C4×Dic17—81no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201#22819—171nomaximal, Sylow
202C23×C17—22no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17—41no—
208C23×C17—41no—
209C23×C17C22×D411yescharacteristic
210Q8×C17—22no—
211Q8×C17—22no—
212Q8×C17—41no—
213Q8×C17—41no—
214D4×C17—22no—
215D4×C17—22no—
216D4×C17—41no—
217D4×C17—41no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—84no—
232C2×C4×C17C4×D412yes—
233C2×C4×C17C4×D412yes—
234C2×C4×C17C23×C411yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17H(5,𝔽2)11yescharacteristic
239C2×C4×C172−1+411yescharacteristic
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17#13712yes—
264C8×C17—22no—
265C8×C17#11714yes—
266#3622—41no—
267#3622—41no—
268C2×#440D1711yescharacteristic
269#2357—171no—
270#2355—171no—
271#1973—171no—
272#1968—171no—
273#1928—171no—
274#1927—171no—
275#1926—171no—
276#1925—171no—
277#1847—171no—
278#1805—171no—
279#1802—171no—
280#1776—171no—
281C4×#107—171no—
282#1771—171no—
283C22×C17—42no—
284C22×C17—42no—
285C22×C17—42no—
286C22×C17—81no—
287C22×C17—81no—
288C22×C17—81no—
289C22×C17—81no—
290C22×C17C2×C4×D411yescharacteristic
291C22×C17#54711yescharacteristic
292C22×C17#55011yescharacteristic
293C22×C17C2×#13711yescharacteristic
294C22×C17#57511yescharacteristic
295C22×C17#57611yescharacteristic
296C22×C17#57711yescharacteristic
297C4×C17—22no—
298C4×C17—22no—
299C4×C17—42no—
300C4×C17—42no—
301C4×C17—42no—
302C4×C17C2×#11711yescharacteristic
303C4×C17#52011yescharacteristic
304C4×C17#55511yescharacteristic
305C4×C17#55611yescharacteristic
306Dic17—22no—
307Dic17—41no—
308Dic17—41no—
309Dic17—41no—
310Dic17—81no—
311Dic17—81no—
312Dic17—81no—
313Dic17—81no—
314C22×SD16C2×D1711yescharacteristic
315C2×C4×Q8C2×D1711yescharacteristic
316C2×C4×D4C2×D1711yescharacteristic
317#440Dic1718yescomplemented
318C2×#107—171no—
319C2×#107—171no—
320C2×#107C2×D1711yescharacteristic
321C2×#106—171no—
322C2×#106—171no—
323C2×C4⋊C8—171no—
324C2×#103—171no—
325C2×#103—171no—
326C2×#103C2×D1711yescharacteristic
327C2×#102—171no—
328C2×#102—171no—
329C2×#102C2×D1711yescharacteristic
330C2×#97—171no—
331C2×#97C2×D1711yescharacteristic
332C2×C4×C8—171no—
333#391—171no—
334#391—171no—
335#390—171no—
336#390—171no—
337#388—341no—
338#388—341no—
339#387—341no—
340#387—341no—
341#384—171no—
342#382—171no—
343#382—341no—
344#381—341no—
345C4×C4⋊C4—171no—
346C4×C4⋊C4—171no—
347C4×C4⋊C4—171no—
348C4×C4⋊C4—171no—
349C4×C4⋊C4—171no—
350C4×C4⋊C4—341no—
351C4×C22⋊C4—341no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#336—171no—
357#336—171no—
358C2×C17—82no—
359C2×C17#235411yescharacteristic
360C2×C17C4×#13711yescharacteristic
361C2×C17#294611yescharacteristic
362C2×C17#334611yescharacteristic
363C2×C17#335011yescharacteristic
364C2×C17#340011yescharacteristic
365C2×C17#340311yescharacteristic
366C22×Q8C22×D1711yescharacteristic
367C22×D4C22×D1711yescharacteristic
368C23×C4—21no—
369C2×SD16—22no—
370C2×SD16—22no—
371C2×SD16C2×Dic1714yes—
372C22×C8—171no—
373C22×C8—171no—
374C22×C8C22×D1711yescharacteristic
375C22×C8C22×D1711yescharacteristic
376C4×Q8—22no—
377C4×Q8C2×Dic1714yes—
378C4×D4—22no—
379C4×D4C2×Dic1714yes—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4C22×D1711yescharacteristic
390C2×C4⋊C4C22×D1711yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—341no—
393C2×C22⋊C4—341no—
394C2×C22⋊C4—341no—
395C2×C22⋊C4—341no—
396C2×C42—21no—
397C2×C42—171no—
398C2×C42—171no—
399C2×C42—341no—
400C2×C42—341no—
401C2×C42—341no—
402C2×C42—341no—
403C2×C42C22×D1711yescharacteristic
404#107—342no—
405#107—681no—
406#107C2×Dic1714yes—
407#106—342no—
408#106—681no—
409C4⋊C8—342no—
410#103—681no—
411#103—681no—
412#103C2×Dic1714yes—
413#102—681no—
414#102—681no—
415#102C2×Dic1714yes—
416#97—342no—
417#97C2×Dic1714yes—
418C4×C8—342no—
419#95—171no—
420#95—171no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429#95—341no—
430#95—341no—
431C17#2281911yescharacteristic, complemented, Sylow
432C24—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C22×Dic1712yes—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C22×Dic1712yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—171no—
452C22×C4—171no—
453C22×C4—171no—
454C22×C4—171no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4D4×D1711yescharacteristic
460C22×C4D4×D1711yescharacteristic
461C22×C4C23×D1711yescharacteristic
462SD16—28no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8#6325212yes—
472C2×C8#6325212yes—
473C2×C8C22×Dic1712yes—
474C2×C8C22×Dic1712yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—342no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—681no—
488C4⋊C4—682no—
489C4⋊C4—1361no—
490C4⋊C4C22×Dic1712yes—
491C4⋊C4C22×Dic1712yes—
492C22⋊C4—24no—
493C22⋊C4—682no—
494C22⋊C4—682no—
495C22⋊C4—1361no—
496C22⋊C4—1361no—
497C42—24no—
498C42—341no—
499C42—341no—
500C42—341no—
501C42—341no—
502C42—1361no—
503C42—1361no—
504C42—1361no—
505C42—1361no—
506C42#6325412yes—
507C42C22×Dic1712yes—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C2×D4×D1711yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—172no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4D4×Dic1712yes—
563C2×C4D4×Dic1712yes—
564C2×C4C2×#6325211yescharacteristic
565C2×C4C2×#6325211yescharacteristic
566C2×C4C2×#6325411yescharacteristic
567C2×C4#1058769611yescharacteristic
568C2×C4#1058769711yescharacteristic
569C2×C4C23×Dic1711yescharacteristic
570C8—22no—
571C8—681no—
572C8—681no—
573C8#1058761214yes—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×D4×Dic1711yescharacteristic
582C22#1176346411yescharacteristic
583C22#1176348011yescharacteristic
584C22#137×D1711yescharacteristic
585C22#1176360811yescharacteristic
586C22#1176361311yescharacteristic
587C22#1176361911yescharacteristic
588C4—22no—
589C4—22no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—342no—
594C4—681no—
595C4—681no—
596C4—681no—
597C4—1361no—
598C4—1361no—
599C4—1361no—
600C4—1361no—
601C4C2×#1058761211yescharacteristic
602C4#1176329211yescharacteristic
603C4#1176352411yescharacteristic
604C4#1176352911yescharacteristic
605C2—82no—
606C2#1453173311yescharacteristic
607C2#137×Dic1711yescharacteristic
608C2#1453761711yescharacteristic
609C2#1454127411yescharacteristic
610C2#1454133411yescharacteristic
611C2#1454183811yescharacteristic
612C2#1454187311yescharacteristic
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#22819
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
2. #14531741C2semidirect
5. #14529427C2semidirect
6. #14529420C2semidirect
9. #14528952C2semidirect
10. #14528799C2semidirect
12. #14528683C2semidirect
13. #107×Dic17C2semidirect
14. #14528656C2semidirect
55. C17×#440C4semidirect
317. #440Dic17semidirect
431. C17#22819semidirect

Maximal subgroups

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

subgroupindex
#145317572
#145317412
#145296672
#145296402
#145294272
#145294202
#145294132
#145294062
#145289522
#145287992
#145287842
#145286832
#107×Dic172
#145286562
C2×C17×#4402
#2281917