Finite Group Databasebeta

← #17053550

Subgroups — #17053550

Order 4352.

830 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762894C211yesmaximal, characteristic
2#14531753C211yesmaximal, characteristic
3#14531702C211yesmaximal, characteristic, complemented
4#14529481C211yesmaximal, characteristic
5#14529479C211yesmaximal, characteristic
6#14529369C211yesmaximal, characteristic, complemented
7#14529367C211yesmaximal, characteristic, complemented
8#14529139C211yesmaximal, characteristic
9#14529137C211yesmaximal, characteristic
10#14528913C211yesmaximal, characteristic, complemented
11#14528797C211yesmaximal, characteristic, complemented
12#14528793C211yesmaximal, characteristic
13#14528652C211yesmaximal, characteristic, complemented
14#14528616C211yesmaximal, characteristic, complemented
15C17×#1763C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C2×Q8×Dic17C2211yescharacteristic
19C2×#10587579C2211yescharacteristic
20#11762894C418yescomplemented
21C2×#10587479C2211yescharacteristic
22C2×#10587485C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587492C2211yescharacteristic
25C2×#10587492C2211yescharacteristic
26C2×#10587480C2211yescharacteristic
27C2×#10587486C2211yescharacteristic
28C2×C8×Dic17C2211yescharacteristic
29#11762711C2211yescharacteristic
30#11762710C2211yescharacteristic
31#11762707—21no—
32#11762707C2211yescharacteristic
33#11762705—21no—
34#11762705C2211yescharacteristic
35#11762700—21no—
36#11762698—21no—
37#11762674—21no—
38#11762674C2211yescharacteristic
39#11762674C2211yescharacteristic
40#11762671—21no—
41C4⋊C4×Dic17C2211yescharacteristic
42C4×#63225—21no—
43C4×#63225C2211yescharacteristic
44C4×#63227—21no—
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762564C2211yescharacteristic
52#11762563C2211yescharacteristic
53C2×C4×Q8×C17C2211yescharacteristic
54C2×C17×#103C2211yescharacteristic
55C2×C17×#103C2211yescharacteristic
56C2×C17×#97C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
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×C17—41no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C42×C17—21no—
72C2×C42×C17—21no—
73C2×C42×C17C2311yescharacteristic
74Q8×Dic17—22no—
75Q8×Dic17—22no—
76Q8×Dic17—41no—
77Q8×Dic17C2×C414yes—
78C2×#63230—22no—
79C2×#63230—22no—
80C2×#63230C2×C414yes—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×#63227—21no—
84C2×#63227—21no—
85C2×#63227—21no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225D411yescharacteristic
94C2×#63225D411yescharacteristic
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17C2311yescharacteristic
99C2×C4×Dic17C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102#10587579—22no—
103#10587579C2×C414yes—
104C17×#103—22no—
105C17×#103C2×C414yes—
106C17×#97—41no—
107C17×#95—21no—
108#10587506—22no—
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#10587492—41no—
119#10587492C2×C414yes—
120#10587486—41no—
121#10587485—22no—
122C8×Dic17C2×C414yes—
123#10587480—41no—
124#10587479C2×C414yes—
125C23×D17—21no—
126C2×Q8×C17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17C22×C412yes—
130C2×Q8×C17#4112yes—
131C2×Q8×C17#4112yes—
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×C17C22×C412yes—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—42no—
167C4⋊C4×C17C22×C412yes—
168C42×C17—21no—
169C42×C17—21no—
170C42×C17—42no—
171C42×C17—81no—
172#63230—28no—
173#63227—24no—
174#63227—42no—
175#63227—42no—
176#63227—81no—
177#63227—81no—
178C4⋊Dic17—21no—
179C4⋊Dic17—24no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—42no—
185C4⋊Dic17C22×C412yes—
186#63225—41no—
187#63225—41no—
188#63225—42no—
189#63225—42no—
190C4×Dic17—21no—
191C4×Dic17—21no—
192C4×Dic17—24no—
193C4×Dic17—42no—
194C4×Dic17—81no—
195C4×Dic17C22×C412yes—
196C4×Dic17#4112yes—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622C22×C412yes—
203#22765—171nomaximal, Sylow
204C23×C17C22×D411yescharacteristic
205C22×D17—22no—
206C22×D17—41no—
207C22×D17—41no—
208C22×D17—41no—
209C22×D17—41no—
210C22×D17—41no—
211C22×D17—41no—
212Q8×C17—22no—
213Q8×C17—22no—
214Q8×C17—22no—
215Q8×C17#11714yes—
216C2×C4×C17—22no—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
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—22no—
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×Dic17C4×D412yes—
260C2×Dic17C4×D412yes—
261D68—22no—
262D68—22no—
263D68—41no—
264D68—41no—
265C4×D17—84no—
266C8×C17—22no—
267C8×C17—41no—
268#3622—22no—
269#3622—41no—
270C2×C4×SD16—171no—
271#2357—171no—
272#2351—171no—
273#1936—171no—
274#1936—171no—
275#1920—171no—
276#1920—171no—
277#1880—171no—
278#1880—171no—
279#1835—171no—
280#1805—171no—
281#1804—171no—
282#1770—171no—
283#1763—171no—
284#1763D1711yescharacteristic, complemented
285C22×C17C2×C4×D411yescharacteristic
286C22×C17#54611yescharacteristic
287C22×C17#54811yescharacteristic
288C22×C17#57511yescharacteristic
289C22×C17#57511yescharacteristic
290C22×C17#57711yescharacteristic
291C22×C17#57711yescharacteristic
292C2×D17—42no—
293C2×D17—42no—
294C2×D17—42no—
295C2×D17—81no—
296C2×D17—81no—
297C2×D17—81no—
298C2×D17—81no—
299C4×C17—24no—
300C4×C17—42no—
301C4×C17—41no—
302C4×C17—41no—
303C4×C17—81no—
304C4×C17—81no—
305C4×C17C2×#11711yescharacteristic
306C4×C17#52011yescharacteristic
307C4×C17#55211yescharacteristic
308C4×C17#55311yescharacteristic
309Dic17—22no—
310Dic17—22no—
311Dic17—42no—
312Dic17—41no—
313Dic17—41no—
314Dic17—81no—
315Dic17—81no—
316C22×SD16—171no—
317C2×C4×Q8—171no—
318C2×C4×Q8—171no—
319C2×C4×Q8C2×D1711yescharacteristic
320C2×C4×D4—171no—
321C4×SD16—178no—
322C2×#106—171no—
323C2×C4⋊C8—171no—
324C2×#103—171no—
325C2×#103C2×D1711yescharacteristic
326C2×#103C2×D1711yescharacteristic
327C2×#102—171no—
328C2×#102—171no—
329C2×#102—171no—
330C2×#97—171no—
331C2×#97C2×D1711yescharacteristic
332C2×C4×C8—171no—
333#390—171no—
334#390—171no—
335#389—171no—
336#389—171no—
337#389—341no—
338#389—341no—
339#388—341no—
340#388—341no—
341#382—171no—
342#382—171no—
343#382—341no—
344#381—341no—
345C4×C4⋊C4—171no—
346C4×C4⋊C4—171no—
347C4×C4⋊C4—341no—
348C4×C4⋊C4C2×D1711yescharacteristic
349C4×C22⋊C4—341no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#340C2×D1711yescharacteristic
357#340C2×D1711yescharacteristic
358#336—171no—
359#336—171no—
360C2×C17#235211yescharacteristic
361C2×C17#294311yescharacteristic
362C2×C17#294911yescharacteristic
363C2×C17#332611yescharacteristic
364C2×C17#332611yescharacteristic
365C2×C17#337111yescharacteristic
366C2×C17#337111yescharacteristic
367D17—82no—
368C22×Q8C22×D1711yescharacteristic
369C22×D4—171no—
370C23×C4—341no—
371C2×SD16—174no—
372C2×SD16—342no—
373C2×SD16—342no—
374C22×C8—171no—
375C22×C8—171no—
376C22×C8C22×D1711yescharacteristic
377C22×C8C22×D1711yescharacteristic
378C4×Q8—22no—
379C4×Q8—41no—
380C4×Q8—174no—
381C4×Q8—342no—
382C4×Q8—342no—
383C4×Q8—681no—
384C4×D4—174no—
385C4×D4—342no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—341no—
394C2×C4⋊C4—341no—
395C2×C4⋊C4—341no—
396C2×C4⋊C4C22×D1711yescharacteristic
397C2×C4⋊C4C22×D1711yescharacteristic
398C2×C4⋊C4C22×D1711yescharacteristic
399C2×C22⋊C4—341no—
400C2×C22⋊C4—341no—
401C2×C22⋊C4—341no—
402C2×C22⋊C4—341no—
403C2×C22⋊C4—341no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—171no—
407C2×C42—171no—
408C2×C42—341no—
409C2×C42—341no—
410C2×C42—341no—
411C2×C42C22×D1711yescharacteristic
412#106—174no—
413C4⋊C8—342no—
414#103—22no—
415#103—342no—
416#103C4×D1714yes—
417#102—174no—
418#102—681no—
419#102—681no—
420#97—41no—
421#97—681no—
422C4×C8—174no—
423#95—21no—
424#95—171no—
425#95—171no—
426#95—341no—
427#95—341no—
428#95—341no—
429#95—341no—
430#95—341no—
431#95—341no—
432#95—341no—
433C17#2276511yescharacteristic, complemented, Sylow
434C24—341no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×C4×D1712yes—
439C2×Q8#6325212yes—
440C2×Q8#6325212yes—
441C2×D4—172no—
442C2×D4—341no—
443C2×D4—341no—
444C2×D4—341no—
445C2×D4—341no—
446C2×D4—341no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—171no—
452C22×C4—171no—
453C22×C4—171no—
454C22×C4—171no—
455C22×C4—344no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—1361no—
460C22×C4—1361no—
461C22×C4D4×D1711yescharacteristic
462C22×C4D4×D1711yescharacteristic
463C22×C4C23×D1711yescharacteristic
464SD16—348no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—172no—
471C2×C8—341no—
472C2×C8—341no—
473C2×C8—341no—
474C2×C8—341no—
475C2×C8—341no—
476C2×C8C2×C4×D1712yes—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—172no—
484C4⋊C4—344no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—341no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—681no—
491C4⋊C4—681no—
492C4⋊C4—682no—
493C4⋊C4—682no—
494C4⋊C4—682no—
495C4⋊C4C2×C4×D1712yes—
496C22⋊C4—344no—
497C22⋊C4—682no—
498C22⋊C4—682no—
499C22⋊C4—1361no—
500C22⋊C4—1361no—
501C42—21no—
502C42—21no—
503C42—42no—
504C42—81no—
505C42—172no—
506C42—172no—
507C42—344no—
508C42—341no—
509C42—341no—
510C42—682no—
511C42—1361no—
512C23—342no—
513C23—681no—
514C23—681no—
515C23—681no—
516C23—681no—
517C23—681no—
518C23—681no—
519C23C2×D4×D1711yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8#1058755814yes—
524D4—342no—
525D4—342no—
526D4—681no—
527D4—681no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—172no—
547C2×C4—172no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—342no—
553C2×C4—341no—
554C2×C4—341no—
555C2×C4—341no—
556C2×C4—341no—
557C2×C4—341no—
558C2×C4—341no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—1364no—
568C2×C4C22×C4×D1711yescharacteristic
569C2×C4C2×#6325211yescharacteristic
570C2×C4C2×#6325211yescharacteristic
571C2×C4#1058769111yescharacteristic
572C2×C4#41×D1711yescharacteristic
573C2×C4#1058769711yescharacteristic
574C8—22no—
575C8—41no—
576C8—342no—
577C8—681no—
578C22—682no—
579C22—682no—
580C22—682no—
581C22—1361no—
582C22—1361no—
583C22—1361no—
584C22—1361no—
585C22C4×D4×D1711yescharacteristic
586C22#1176344911yescharacteristic
587C22#1176346811yescharacteristic
588C22#1176360811yescharacteristic
589C22#1176360811yescharacteristic
590C22#1176361811yescharacteristic
591C22#1176361911yescharacteristic
592C4—24no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—81no—
597C4—81no—
598C4—342no—
599C4—342no—
600C4—682no—
601C4—681no—
602C4—681no—
603C4—1361no—
604C4—1361no—
605C4C2×#1058755811yescharacteristic
606C4#1176328811yescharacteristic
607C4#1176349211yescharacteristic
608C4#1176350811yescharacteristic
609C2—1362no—
610C2#1453170711yescharacteristic
611C2#1453758311yescharacteristic
612C2#1453764011yescharacteristic
613C2#1454103311yescharacteristic
614C2#1454103811yescharacteristic
615C2#1454155311yescharacteristic
616C2#1454155711yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1763C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1763D17
hypocenterC17#22765
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
3. #14531702C2semidirect
6. #14529369C2semidirect
7. #14529367C2semidirect
10. #14528913C2semidirect
11. #14528797C2semidirect
13. #14528652C2semidirect
14. #14528616C2semidirect
15. C17×#1763C2semidirect
20. #11762894C4semidirect
284. #1763D17semidirect
433. C17#22765semidirect

Maximal subgroups

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

subgroupindex
C2×#117628942
#145317532
#145317022
#145294812
#145294792
#145293692
#145293672
#145291392
#145291372
#145289132
#145287972
#145287932
#145286522
#145286162
C17×#17632
#2276517