Finite Group Databasebeta

← #17052011

Subgroups — #17052011

Order 4352.

766 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762844C211yesmaximal, characteristic
2#14531676C211yesmaximal, characteristic
3#14531668C211yesmaximal, characteristic
4#14529528C211yesmaximal, characteristic, complemented
5#14529379C211yesmaximal, characteristic, complemented
6#14529373C211yesmaximal, characteristic, complemented
7#14529358C211yesmaximal, characteristic, complemented
8#14529352C211yesmaximal, characteristic, complemented
9#14528878C211yesmaximal, characteristic
10#14528877C211yesmaximal, characteristic
11#14528810C211yesmaximal, characteristic
12#14528683C211yesmaximal, characteristic, complemented
13#14528681C211yesmaximal, characteristic, complemented
14#14528561C211yesmaximal, characteristic
15C17×#1944C211yesmaximal, characteristic, complemented
16C22×C8×D17C2211yescharacteristic
17C2×#10587578C2211yescharacteristic
18C2×#10587578C2211yescharacteristic
19#11762844C418yescomplemented
20C2×#10587489C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587488C2211yescharacteristic
24C2×#10587491C2211yescharacteristic
25C2×#10587486C2211yescharacteristic
26#11762708C2211yescharacteristic
27#11762708C2211yescharacteristic
28#11762701—21no—
29#11762701—21no—
30#11762699—21no—
31#11762699—21no—
32#11762676—21no—
33#11762674—21no—
34#11762666C2211yescharacteristic
35#11762666C2211yescharacteristic
36C4⋊C4×Dic17—21no—
37C4⋊C4×Dic17C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40C4×C4⋊Dic17C2211yescharacteristic
41C4×#63227C2211yescharacteristic
42#11762654—21no—
43#11762572C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762564C2211yescharacteristic
52#11762563C2211yescharacteristic
53C2×C17×#107C2211yescharacteristic
54C2×C17×#106C2211yescharacteristic
55C2×C17×#97C2211yescharacteristic
56C17×#389C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C22×C4×D17C2311yescharacteristic
61C22×C8×C17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C42×C17C2311yescharacteristic
70C2×#63227—21no—
71C2×#63227—21no—
72C2×#63227—21no—
73C2×#63227—21no—
74C2×#63227D411yescharacteristic
75C2×#63227D411yescharacteristic
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4×Dic17—21no—
83C2×C4×Dic17—21no—
84C2×C4×Dic17—21no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17C2311yescharacteristic
89C22×#3622C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C2×C8×D17—22no—
92C2×C8×D17—22no—
93C2×C8×D17C2×C414yes—
94#10587578—22no—
95#10587578—22no—
96#10587578C2×C414yes—
97#10587578C2×C414yes—
98C17×#107—22no—
99C17×#106C2×C414yes—
100C17×#97—22no—
101C17×#95—21no—
102C17×#95—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502D411yescharacteristic
112#10587502D411yescharacteristic
113#10587491—22no—
114#10587489—22no—
115#10587488C2×C414yes—
116#10587486—22no—
117#10587478C2×C414yes—
118#10587478C2×C414yes—
119C23×D17C2×D411yescharacteristic
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17C2×D411yescharacteristic
125C22×C4×C17C2×D411yescharacteristic
126C22×C4×C17C2411yescharacteristic
127C22×Dic17—21no—
128C22×Dic17—21no—
129C22×Dic17—21no—
130C22×Dic17—21no—
131C22×Dic17C2×D411yescharacteristic
132C22×Dic17C2×D411yescharacteristic
133C22×Dic17C2×D411yescharacteristic
134C2×C4×D17—21no—
135C2×C4×D17—21no—
136C2×C4×D17—21no—
137C2×C4×D17—21no—
138C2×C4×D17C22×C412yes—
139C2×C4×D17#4112yes—
140C2×C8×C17—21no—
141C2×C8×C17—21no—
142C2×C8×C17—21no—
143C2×C8×C17C22×C412yes—
144C2×C8×C17#4112yes—
145C2×C8×C17#4112yes—
146C4⋊C4×C17—21no—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—21no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—42no—
153C4⋊C4×C17—81no—
154C4⋊C4×C17C22×C412yes—
155C4⋊C4×C17C22×C412yes—
156C42×C17—21no—
157C42×C17—21no—
158#63227—24no—
159#63227—24no—
160#63227—41no—
161#63227—41no—
162#63227—81no—
163#63227—81no—
164C4⋊Dic17—21no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—42no—
171C4⋊Dic17—81no—
172C4⋊Dic17C22×C412yes—
173C4⋊Dic17C22×C412yes—
174C4×Dic17—21no—
175C4×Dic17—21no—
176C4×Dic17—24no—
177C4×Dic17—24no—
178C4×Dic17—42no—
179C4×Dic17—42no—
180C4×Dic17—81no—
181C4×Dic17—81no—
182C2×#3622—21no—
183C2×#3622—21no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622C22×C412yes—
188C8×D17—28no—
189#22616—171nomaximal, Sylow
190C23×C17C22×D411yescharacteristic
191C22×D17—21no—
192C22×D17—21no—
193C22×D17—21no—
194C22×D17—21no—
195C22×D17—21no—
196C22×D17—21no—
197C22×D17C4×D412yes—
198C2×C4×C17—22no—
199C2×C4×C17—22no—
200C2×C4×C17—21no—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—41no—
207C2×C4×C17—41no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—41no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17C23×C411yescharacteristic
217C2×C4×C17C2×#4111yescharacteristic
218C2×C4×C17C2×#4111yescharacteristic
219C2×C4×C17C2×#4111yescharacteristic
220C2×C4×C172−1+411yescharacteristic
221C2×C4×C172−1+411yescharacteristic
222C2×Dic17—21no—
223C2×Dic17—21no—
224C2×Dic17—22no—
225C2×Dic17—22no—
226C2×Dic17—21no—
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—41no—
233C2×Dic17—41no—
234C2×Dic17—41no—
235C2×Dic17—41no—
236C2×Dic17—41no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17C4×D412yes—
243C4×D17—22no—
244C4×D17—22no—
245C4×D17—22no—
246C4×D17—22no—
247C8×C17—41no—
248C8×C17#11714yes—
249#3622—22no—
250#3622—41no—
251C2×#427—171no—
252#2347—171no—
253#2346—171no—
254#1944—171no—
255#1944D1711yescharacteristic, complemented
256#1921—171no—
257#1921—171no—
258#1918—171no—
259#1918—171no—
260#1825—171no—
261#1825—171no—
262#1808—171no—
263#1776—171no—
264#1776—171no—
265#1752—171no—
266C22×C17C2×C4×D411yescharacteristic
267C22×C17#54711yescharacteristic
268C22×C17#54711yescharacteristic
269C22×C17#57511yescharacteristic
270C22×C17#57511yescharacteristic
271C22×C17#57611yescharacteristic
272C22×C17#57811yescharacteristic
273C2×D17—22no—
274C2×D17—22no—
275C2×D17—22no—
276C2×D17—41no—
277C2×D17—41no—
278C2×D17—41no—
279C2×D17—41no—
280C4×C17—41no—
281C4×C17—41no—
282C4×C17—42no—
283C4×C17—42no—
284C4×C17—81no—
285C4×C17—81no—
286C4×C17C2×#11711yescharacteristic
287C4×C17#51911yescharacteristic
288C4×C17#55211yescharacteristic
289C4×C17#55211yescharacteristic
290Dic17—22no—
291Dic17—41no—
292Dic17—41no—
293Dic17—42no—
294Dic17—42no—
295Dic17—81no—
296Dic17—81no—
297C23×C8—171no—
298C2×#117—171no—
299C2×#117—171no—
300#427—178no—
301C2×#107—171no—
302C2×#107—171no—
303C2×#107—171no—
304C2×#107C2×D1711yescharacteristic
305C2×#106—171no—
306C2×#106C2×D1711yescharacteristic
307C2×C22⋊C8—171no—
308C2×#97—171no—
309C2×#97C2×D1711yescharacteristic
310#389—171no—
311#389—171no—
312#389C2×D1711yescharacteristic
313#389C2×D1711yescharacteristic
314#388—341no—
315#388—341no—
316#388—341no—
317#388—341no—
318#383—341no—
319#382—341no—
320#380—171no—
321#380—171no—
322C4×C4⋊C4—171no—
323C4×C4⋊C4—171no—
324C4×C4⋊C4—171no—
325C4×C4⋊C4—171no—
326C4×C4⋊C4—341no—
327C4×C22⋊C4—171no—
328#376—341no—
329#340—171no—
330#340—171no—
331#340—171no—
332#340—171no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340C2×D1711yescharacteristic
338#340C2×D1711yescharacteristic
339#336—171no—
340#336—171no—
341C2×C17#234411yescharacteristic
342C2×C17#294311yescharacteristic
343C2×C17#295111yescharacteristic
344C2×C17#334711yescharacteristic
345C2×C17#334711yescharacteristic
346C2×C17#334911yescharacteristic
347C2×C17#334911yescharacteristic
348D17—42no—
349C23×C4—171no—
350C22×C8—171no—
351C22×C8—174no—
352C22×C8—171no—
353C22×C8—342no—
354C22×C8—342no—
355C22×C8C22×D1711yescharacteristic
356C22×C8C22×D1711yescharacteristic
357#117—174no—
358#117—174no—
359#117—342no—
360#117—342no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—171no—
364C2×C4⋊C4—171no—
365C2×C4⋊C4—171no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—341no—
368C2×C4⋊C4—341no—
369C2×C4⋊C4C22×D1711yescharacteristic
370C2×C4⋊C4C22×D1711yescharacteristic
371C2×C4⋊C4C22×D1711yescharacteristic
372C2×C4⋊C4C22×D1711yescharacteristic
373C2×C22⋊C4—171no—
374C2×C22⋊C4—171no—
375C2×C22⋊C4—341no—
376C2×C22⋊C4—341no—
377C2×C22⋊C4—341no—
378C2×C22⋊C4—341no—
379C2×C42—171no—
380C2×C42—341no—
381C2×C42—341no—
382C2×C42—341no—
383C2×C42—341no—
384C2×C42—341no—
385C2×C42—341no—
386C2×C42C22×D1711yescharacteristic
387#107—22no—
388#107—174no—
389#107—174no—
390#107—342no—
391#106—174no—
392#106C4×D1714yes—
393C22⋊C8—342no—
394#97—22no—
395#97—342no—
396#95—21no—
397#95—21no—
398#95—171no—
399#95—171no—
400#95—341no—
401#95—341no—
402#95—341no—
403#95—341no—
404#95—341no—
405#95—341no—
406#95—341no—
407#95—341no—
408C17#2261611yescharacteristic, complemented, Sylow
409C24—171no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—171no—
415C22×C4—172no—
416C22×C4—172no—
417C22×C4—171no—
418C22×C4—171no—
419C22×C4—341no—
420C22×C4—341no—
421C22×C4—341no—
422C22×C4—341no—
423C22×C4—341no—
424C22×C4—341no—
425C22×C4—341no—
426C22×C4—341no—
427C22×C4D4×D1711yescharacteristic
428C22×C4D4×D1711yescharacteristic
429C22×C4C23×D1711yescharacteristic
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—172no—
434C2×C8—341no—
435C2×C8—341no—
436C2×C8—348no—
437C2×C8—341no—
438C2×C8—341no—
439C2×C8—341no—
440C2×C8C2×C4×D1712yes—
441C2×C8#6325412yes—
442C2×C8#6325612yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—42no—
450C4⋊C4—81no—
451C4⋊C4—172no—
452C4⋊C4—172no—
453C4⋊C4—341no—
454C4⋊C4—341no—
455C4⋊C4—341no—
456C4⋊C4—341no—
457C4⋊C4—341no—
458C4⋊C4—341no—
459C4⋊C4—682no—
460C4⋊C4—1361no—
461C4⋊C4C2×C4×D1712yes—
462C4⋊C4C2×C4×D1712yes—
463C22⋊C4—344no—
464C22⋊C4—344no—
465C22⋊C4—681no—
466C22⋊C4—681no—
467C22⋊C4—1361no—
468C22⋊C4—1361no—
469C42—21no—
470C42—21no—
471C42—344no—
472C42—344no—
473C42—341no—
474C42—341no—
475C42—682no—
476C42—682no—
477C42—1361no—
478C42—1361no—
479C23—172no—
480C23—341no—
481C23—341no—
482C23—341no—
483C23—341no—
484C23—341no—
485C23—341no—
486C23C2×D4×D1711yescharacteristic
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—172no—
506C2×C4—341no—
507C2×C4—341no—
508C2×C4—342no—
509C2×C4—342no—
510C2×C4—342no—
511C2×C4—342no—
512C2×C4—341no—
513C2×C4—341no—
514C2×C4—342no—
515C2×C4—342no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—341no—
519C2×C4—341no—
520C2×C4—681no—
521C2×C4—681no—
522C2×C4—681no—
523C2×C4—681no—
524C2×C4—681no—
525C2×C4—681no—
526C2×C4—681no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4C22×C4×D1711yescharacteristic
531C2×C4C2×#6325411yescharacteristic
532C2×C4C2×#6325611yescharacteristic
533C2×C4#1058769411yescharacteristic
534C2×C4#41×D1711yescharacteristic
535C2×C4#1058769711yescharacteristic
536C8—41no—
537C8—342no—
538C8—681no—
539C8#1058757814yes—
540C22—342no—
541C22—342no—
542C22—342no—
543C22—681no—
544C22—681no—
545C22—681no—
546C22—681no—
547C22C4×D4×D1711yescharacteristic
548C22#1176345911yescharacteristic
549C22#1176346211yescharacteristic
550C22#1176360511yescharacteristic
551C22#1176360811yescharacteristic
552C22#1176361311yescharacteristic
553C22#1176362511yescharacteristic
554C4—42no—
555C4—42no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—342no—
561C4—681no—
562C4—682no—
563C4—682no—
564C4—681no—
565C4—1361no—
566C4—1361no—
567C4C2×#1058757811yescharacteristic
568C4#1176328511yescharacteristic
569C4#1176349311yescharacteristic
570C4#1176349711yescharacteristic
571C2—682no—
572C2#1453165211yescharacteristic
573C2#1453757911yescharacteristic
574C2#1453765611yescharacteristic
575C2#1454127911yescharacteristic
576C2#1454128311yescharacteristic
577C2#1454130911yescharacteristic
578C2#1454131311yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1944C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1944D17
hypocenterC17#22616
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
4. #14529528C2semidirect
5. #14529379C2semidirect
6. #14529373C2semidirect
7. #14529358C2semidirect
8. #14529352C2semidirect
12. #14528683C2semidirect
13. #14528681C2semidirect
15. C17×#1944C2semidirect
19. #11762844C4semidirect
255. #1944D17semidirect
408. C17#22616semidirect

Maximal subgroups

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

subgroupindex
C2×#117628442
#145316762
#145316682
#145295282
#145293792
#145293732
#145293582
#145293522
#145288782
#145288772
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