Finite Group Databasebeta

← #17054185

Subgroups — #17054185

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
1C2×#11762907C211yesmaximal, characteristic
2#14531759C211yesmaximal, characteristic
3#14531741C211yesmaximal, characteristic, complemented
4#14529662C211yesmaximal, characteristic
5#14529635C211yesmaximal, characteristic
6#14529428C211yesmaximal, characteristic, complemented
7#14529421C211yesmaximal, characteristic, complemented
8#14529409C211yesmaximal, characteristic
9#14529402C211yesmaximal, characteristic
10#14528953C211yesmaximal, characteristic, complemented
11#14528785C211yesmaximal, characteristic
12#14528680C211yesmaximal, characteristic, complemented
13#14528676C211yesmaximal, characteristic, complemented
14#14528661C211yesmaximal, characteristic, complemented
15C17×#1805C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18#11762907C418yescomplemented
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587492C2211yescharacteristic
26C2×#10587492C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587474C2211yescharacteristic
29C2×#10587486C2211yescharacteristic
30C2×C8×Dic17C2211yescharacteristic
31#11762716C2211yescharacteristic
32#11762716C2211yescharacteristic
33#11762710C2211yescharacteristic
34#11762710C2211yescharacteristic
35#11762701—21no—
36#11762701—21no—
37#11762695—21no—
38#11762695—21no—
39#11762679C2211yescharacteristic
40#11762675—21no—
41#11762671—21no—
42C4⋊C4×Dic17C2211yescharacteristic
43C4⋊C4×Dic17C2211yescharacteristic
44C4⋊C4×Dic17C2211yescharacteristic
45C4⋊C4×Dic17C2211yescharacteristic
46C4×C4⋊Dic17—21no—
47C4×C4⋊Dic17C2211yescharacteristic
48#11762656—21no—
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53C2×C4×Q8×C17C2211yescharacteristic
54C2×C17×C4⋊C8C2211yescharacteristic
55C2×C17×#103C2211yescharacteristic
56C2×C17×#103C2211yescharacteristic
57C17×#382C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C22×D68C2311yescharacteristic
62C22×C4×D17—21no—
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C4×Q8×C17—22no—
66C4×Q8×C17C2×C414yes—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C42×C17—21no—
72C2×C42×C17C2311yescharacteristic
73C2×#63230—22no—
74C2×#63230—22no—
75C2×#63230C2×C414yes—
76C2×#63227—21no—
77C2×#63227—21no—
78C2×#63227—21no—
79C2×#63227—21no—
80C2×#63227—21no—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17—21no—
83C2×C4⋊Dic17D411yescharacteristic
84C2×C4⋊Dic17D411yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4×Dic17—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17C2311yescharacteristic
93C2×C4×Dic17C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C4×D68—22no—
97C4×D68C2×C414yes—
98C17×C4⋊C8—22no—
99C17×#103—41no—
100C17×#103—41no—
101C17×#95—21no—
102C17×#95—21no—
103#10587506C2×C414yes—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502D411yescharacteristic
113#10587502D411yescharacteristic
114#10587492—41no—
115#10587492—41no—
116#10587486—22no—
117C8×Dic17—22no—
118#10587480C2×C414yes—
119#10587479—22no—
120#10587479—41no—
121#10587478—22no—
122#10587478—41no—
123#10587478C2×C414yes—
124#10587474C2×C414yes—
125C23×D17—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—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2411yescharacteristic
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17C2×D411yescharacteristic
143C22×Dic17C2×D411yescharacteristic
144C22×Dic17C2×D411yescharacteristic
145C22×Dic17C2×D411yescharacteristic
146C2×D68—21no—
147C2×D68—21no—
148C2×D68—21no—
149C2×D68—21no—
150C2×D68—21no—
151C2×D68C22×C412yes—
152C2×C4×D17—24no—
153C2×C4×D17—81no—
154C2×C4×D17—81no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C4⋊C4×C17—24no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17C22×C412yes—
168C42×C17—24no—
169C42×C17C22×C412yes—
170C42×C17#4112yes—
171#63230—28no—
172#63227—24no—
173#63227—42no—
174#63227—42no—
175#63227—81no—
176#63227—81no—
177C4⋊Dic17—21no—
178C4⋊Dic17—22no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—41no—
184C4⋊Dic17—42no—
185C4⋊Dic17—81no—
186C4⋊Dic17C22×C412yes—
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×#3622C22×C412yes—
198C2×#3622C22×C412yes—
199C2×#3622#4112yes—
200C2×#3622#4112yes—
201#22819—171nomaximal, Sylow
202C23×C17C22×D411yescharacteristic
203C22×D17—22no—
204C22×D17—41no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17—41no—
208C22×D17—41no—
209C22×D17—41no—
210Q8×C17—22no—
211Q8×C17—22no—
212Q8×C17—41no—
213Q8×C17—41no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17C4×D412yes—
231C2×C4×C17C4×D412yes—
232C2×C4×C17C23×C411yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17H(5,𝔽2)11yescharacteristic
237C2×C4×C172−1+411yescharacteristic
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—22no—
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—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17#13712yes—
259D68—22no—
260D68—22no—
261D68—41no—
262D68—41no—
263C4×D17—84no—
264C8×C17—41no—
265C8×C17—41no—
266#3622—22no—
267#3622#11714yes—
268C2×#440—171no—
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#1805D1711yescharacteristic, complemented
279#1802—171no—
280#1776—171no—
281C4×#107—171no—
282#1771—171no—
283C22×C17C2×C4×D411yescharacteristic
284C22×C17#54711yescharacteristic
285C22×C17#55011yescharacteristic
286C22×C17C2×#13711yescharacteristic
287C22×C17#57511yescharacteristic
288C22×C17#57611yescharacteristic
289C22×C17#57711yescharacteristic
290C2×D17—42no—
291C2×D17—42no—
292C2×D17—42no—
293C2×D17—81no—
294C2×D17—81no—
295C2×D17—81no—
296C2×D17—81no—
297C4×C17—22no—
298C4×C17—22no—
299C4×C17—42no—
300C4×C17—42no—
301C4×C17—81no—
302C4×C17—81no—
303C4×C17C2×#11711yescharacteristic
304C4×C17#52011yescharacteristic
305C4×C17#55511yescharacteristic
306C4×C17#55611yescharacteristic
307Dic17—22no—
308Dic17—41no—
309Dic17—41no—
310Dic17—41no—
311Dic17—42no—
312Dic17—81no—
313Dic17—81no—
314C22×SD16—171no—
315C2×C4×Q8C2×D1711yescharacteristic
316C2×C4×D4—171no—
317#440—178no—
318C2×#107—171no—
319C2×#107—171no—
320C2×#107—171no—
321C2×#106—171no—
322C2×#106—171no—
323C2×C4⋊C8C2×D1711yescharacteristic
324C2×#103—171no—
325C2×#103C2×D1711yescharacteristic
326C2×#103C2×D1711yescharacteristic
327C2×#102—171no—
328C2×#102—171no—
329C2×#102—171no—
330C2×#97—171no—
331C2×#97—171no—
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—341no—
343#382C2×D1711yescharacteristic
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#340C2×D1711yescharacteristic
355#340C2×D1711yescharacteristic
356#336—171no—
357#336—171no—
358C2×C17#235411yescharacteristic
359C2×C17C4×#13711yescharacteristic
360C2×C17#294611yescharacteristic
361C2×C17#334611yescharacteristic
362C2×C17#335011yescharacteristic
363C2×C17#340011yescharacteristic
364C2×C17#340311yescharacteristic
365D17—82no—
366C22×Q8C22×D1711yescharacteristic
367C22×D4—171no—
368C23×C4—341no—
369C2×SD16—174no—
370C2×SD16—342no—
371C2×SD16—342no—
372C22×C8—171no—
373C22×C8—171no—
374C22×C8C22×D1711yescharacteristic
375C22×C8C22×D1711yescharacteristic
376C4×Q8—22no—
377C4×Q8C4×D1714yes—
378C4×D4—174no—
379C4×D4—342no—
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—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4C22×D1711yescharacteristic
389C2×C4⋊C4C22×D1711yescharacteristic
390C2×C4⋊C4C22×D1711yescharacteristic
391C2×C22⋊C4—341no—
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—174no—
405#107—342no—
406#107—681no—
407#106—342no—
408#106—681no—
409C4⋊C8—22no—
410#103—41no—
411#103—41no—
412#103—174no—
413#102—174no—
414#102—681no—
415#102—681no—
416#97—174no—
417#97—342no—
418C4×C8—342no—
419#95—21no—
420#95—21no—
421#95—171no—
422#95—171no—
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—341no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×C4×D1712yes—
439C2×D4—172no—
440C2×D4—341no—
441C2×D4—341no—
442C2×D4—341no—
443C2×D4—341no—
444C2×D4—341no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—171no—
450C22×C4—171no—
451C22×C4—171no—
452C22×C4—171no—
453C22×C4—341no—
454C22×C4—344no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4—1361no—
458C22×C4—1361no—
459C22×C4D4×D1711yescharacteristic
460C22×C4D4×D1711yescharacteristic
461C22×C4C23×D1711yescharacteristic
462SD16—348no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—172no—
470C2×C8—172no—
471C2×C8—172no—
472C2×C8—172no—
473C2×C8—341no—
474C2×C8—341no—
475C4⋊C4—21no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—172no—
482C4⋊C4—342no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—341no—
488C4⋊C4—681no—
489C4⋊C4—682no—
490C4⋊C4—1361no—
491C4⋊C4C2×C4×D1712yes—
492C22⋊C4—344no—
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—
506C42C2×C4×D1712yes—
507C42#6325412yes—
508C23—342no—
509C23—681no—
510C23—681no—
511C23—681no—
512C23—681no—
513C23—681no—
514C23—681no—
515C23C2×D4×D1711yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520D4—342no—
521D4—342no—
522D4—681no—
523D4—681no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—172no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—342no—
547C2×C4—341no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
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—1364no—
562C2×C4#1058756712yes—
563C2×C4#1058756712yes—
564C2×C4C22×C4×D1711yescharacteristic
565C2×C4C2×#6325411yescharacteristic
566C2×C4#1058769111yescharacteristic
567C2×C4#1058769411yescharacteristic
568C2×C4#41×D1711yescharacteristic
569C2×C4#41×D1711yescharacteristic
570C8—41no—
571C8—41no—
572C8—174no—
573C8—342no—
574C22—682no—
575C22—682no—
576C22—682no—
577C22—1361no—
578C22—1361no—
579C22—1361no—
580C22—1361no—
581C22C2×#1058756711yescharacteristic
582C22#1176346111yescharacteristic
583C22#1176347711yescharacteristic
584C22#137×D1711yescharacteristic
585C22#1176360811yescharacteristic
586C22#1176361111yescharacteristic
587C22#1176361611yescharacteristic
588C4—22no—
589C4—22no—
590C4—42no—
591C4—42no—
592C4—81no—
593C4—81no—
594C4—342no—
595C4—681no—
596C4—681no—
597C4—682no—
598C4—681no—
599C4—1361no—
600C4—1361no—
601C4#117×D1711yescharacteristic
602C4#1176329011yescharacteristic
603C4#1176352311yescharacteristic
604C4#1176353011yescharacteristic
605C2—1362no—
606C2#1453172811yescharacteristic
607C2#1453759911yescharacteristic
608C2#1453761911yescharacteristic
609C2#1454127611yescharacteristic
610C2#1454132611yescharacteristic
611C2#1454184011yescharacteristic
612C2#1454187111yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1805C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1805D17
hypocenterC17#22819
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
3. #14531741C2semidirect
6. #14529428C2semidirect
7. #14529421C2semidirect
10. #14528953C2semidirect
12. #14528680C2semidirect
13. #14528676C2semidirect
14. #14528661C2semidirect
15. C17×#1805C2semidirect
18. #11762907C4semidirect
278. #1805D17semidirect
431. C17#22819semidirect

Maximal subgroups

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

subgroupindex
C2×#117629072
#145317592
#145317412
#145296622
#145296352
#145294282
#145294212
#145294092
#145294022
#145289532
#145287852
#145286802
#145286762
#145286612
C17×#18052
#2281917