Finite Group Databasebeta

← #17052092

Subgroups — #17052092

Order 4352.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762848C211yesmaximal, characteristic
2#14531676C211yesmaximal, characteristic
3#14531660C211yesmaximal, characteristic
4#14529513C211yesmaximal, characteristic, complemented
5#14529379C211yesmaximal, characteristic, complemented
6#14529366C211yesmaximal, characteristic, complemented
7#14529365C211yesmaximal, characteristic, complemented
8#14529352C211yesmaximal, characteristic, complemented
9#14528878C211yesmaximal, characteristic
10#14528869C211yesmaximal, characteristic
11#14528818C211yesmaximal, characteristic
12#14528681C211yesmaximal, characteristic, complemented
13#107×Dic17C211yesmaximal, characteristic, complemented
14#14528569C211yesmaximal, characteristic
15C17×#1944C211yesmaximal, characteristic, complemented
16C22×#63218C2211yescharacteristic
17C2×#10587578C2211yescharacteristic
18C2×C4⋊C4×D17C2211yescharacteristic
19#11762848C418yescomplemented
20C2×#10587489C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587488C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587491C2211yescharacteristic
25C2×C8×Dic17C2211yescharacteristic
26#11762708C2211yescharacteristic
27#11762708C2211yescharacteristic
28#11762701—21no—
29#11762699—21no—
30#11762696—21no—
31#11762690—21no—
32#11762676—21no—
33#11762666—21no—
34#11762666C2211yescharacteristic
35#11762666C2211yescharacteristic
36C4⋊C4×Dic17—21no—
37C4⋊C4×Dic17C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40C4×C4⋊Dic17C2211yescharacteristic
41C4×#63227C2211yescharacteristic
42#11762656—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×D17—21no—
61C22×C4×D17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C42×C17C2311yescharacteristic
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×#63225—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×#63218—22no—
92C2×#63218—22no—
93C2×#63218C2×C414yes—
94#10587578—22no—
95#10587578C2×C414yes—
96C4⋊C4×D17—22no—
97C4⋊C4×D17C2×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#10587489C2×C414yes—
115#10587488—22no—
116C8×Dic17—22no—
117#10587479C2×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×D17—24no—
139C2×C4×D17—81no—
140C2×C4×D17—81no—
141C2×C4×D17C22×C412yes—
142C2×C4×D17#4112yes—
143C2×C8×C17—21no—
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17C22×C412yes—
147C2×C8×C17#4112yes—
148C2×C8×C17#4112yes—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—42no—
156C4⋊C4×C17—81no—
157C4⋊C4×C17C22×C412yes—
158C4⋊C4×C17C22×C412yes—
159C42×C17—21no—
160C42×C17—21no—
161#63227—24no—
162#63227—22no—
163#63227—41no—
164#63227—42no—
165#63227—81no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—42no—
173C4⋊Dic17—81no—
174C4⋊Dic17C22×C412yes—
175C4⋊Dic17C22×C412yes—
176#63225—24no—
177C4×Dic17—21no—
178C4×Dic17—21no—
179C4×Dic17—24no—
180C4×Dic17—81no—
181C4×Dic17—81no—
182C4×Dic17—81no—
183C4×Dic17—81no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622C22×C412yes—
190#63218—28no—
191#22623—171nomaximal, Sylow
192C23×C17C22×D411yescharacteristic
193C22×D17—21no—
194C22×D17—21no—
195C22×D17—21no—
196C22×D17—21no—
197C22×D17—21no—
198C22×D17—21no—
199C22×D17C4×D412yes—
200C2×C4×C17—22no—
201C2×C4×C17—22no—
202C2×C4×C17—21no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
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×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17C23×C411yescharacteristic
219C2×C4×C17C2×#4111yescharacteristic
220C2×C4×C17C2×#4111yescharacteristic
221C2×C4×C17C2×#4111yescharacteristic
222C2×C4×C172−1+411yescharacteristic
223C2×C4×C172−1+411yescharacteristic
224C2×Dic17—21no—
225C2×Dic17—21no—
226C2×Dic17—21no—
227C2×Dic17—21no—
228C2×Dic17—22no—
229C2×Dic17—22no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
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×Dic17—41no—
243C2×Dic17C4×D412yes—
244C2×Dic17#13712yes—
245C4×D17—22no—
246C4×D17—22no—
247C4×D17—22no—
248C4×D17—22no—
249C4×D17—84no—
250C8×C17—41no—
251C8×C17#11714yes—
252#3622—22no—
253#3622—41no—
254C2×#428—171no—
255#2347—171no—
256#2345—171no—
257#1944D1711yescharacteristic, complemented
258#1941—171no—
259#1921—171no—
260#1920—171no—
261#1919—171no—
262#1918—171no—
263#1825—171no—
264#1823—171no—
265#1809—171no—
266#1776—171no—
267C4×#107—171no—
268#1753—171no—
269C22×C17C2×C4×D411yescharacteristic
270C22×C17#54711yescharacteristic
271C22×C17#54711yescharacteristic
272C22×C17C2×#13711yescharacteristic
273C22×C17#57511yescharacteristic
274C22×C17#57611yescharacteristic
275C22×C17#57711yescharacteristic
276C2×D17—22no—
277C2×D17—22no—
278C2×D17—22no—
279C2×D17—41no—
280C2×D17—41no—
281C2×D17—41no—
282C2×D17—41no—
283C4×C17—41no—
284C4×C17—41no—
285C4×C17—42no—
286C4×C17—42no—
287C4×C17—81no—
288C4×C17—81no—
289C4×C17C2×#11711yescharacteristic
290C4×C17#51911yescharacteristic
291C4×C17#55211yescharacteristic
292C4×C17#55211yescharacteristic
293Dic17—22no—
294Dic17—22no—
295Dic17—41no—
296Dic17—42no—
297Dic17—42no—
298Dic17—81no—
299Dic17—81no—
300C22×M16—171no—
301C2×#117—171no—
302C22×C4⋊C4—171no—
303#428—178no—
304C2×#107—171no—
305C2×#107—171no—
306C2×#107C2×D1711yescharacteristic
307C2×#106—171no—
308C2×#106—171no—
309C2×#106C2×D1711yescharacteristic
310C2×C22⋊C8—171no—
311C2×#97C2×D1711yescharacteristic
312C2×C4×C8—171no—
313#389—171no—
314#389—171no—
315#389C2×D1711yescharacteristic
316#389C2×D1711yescharacteristic
317#388—341no—
318#388—341no—
319#387—341no—
320#387—341no—
321#383—341no—
322#380—171no—
323#380—171no—
324#380—341no—
325C4×C4⋊C4—171no—
326C4×C4⋊C4—171no—
327C4×C4⋊C4—171no—
328C4×C4⋊C4—171no—
329C4×C4⋊C4—341no—
330C4×C22⋊C4—171no—
331C4×C22⋊C4—341no—
332#340—171no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340C2×D1711yescharacteristic
341#340C2×D1711yescharacteristic
342#336—171no—
343#336—171no—
344C2×C17#234411yescharacteristic
345C2×C17C4×#13711yescharacteristic
346C2×C17#294611yescharacteristic
347C2×C17#334611yescharacteristic
348C2×C17#335011yescharacteristic
349C2×C17#335511yescharacteristic
350C2×C17#335611yescharacteristic
351D17—42no—
352C23×C4—171no—
353C23×C4—341no—
354C2×M16—174no—
355C2×M16—342no—
356C2×M16—342no—
357C22×C8—171no—
358C22×C8—171no—
359C22×C8C22×D1711yescharacteristic
360C22×C8C22×D1711yescharacteristic
361#117—174no—
362#117—342no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—171no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—174no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—342no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4C22×D1711yescharacteristic
375C2×C4⋊C4C22×D1711yescharacteristic
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C22⋊C4—171no—
379C2×C22⋊C4—171no—
380C2×C22⋊C4—341no—
381C2×C22⋊C4—341no—
382C2×C22⋊C4—341no—
383C2×C42—171no—
384C2×C42—341no—
385C2×C42—341no—
386C2×C42—341no—
387C2×C42—341no—
388C2×C42—341no—
389C2×C42C22×D1711yescharacteristic
390#107—22no—
391#107—174no—
392#107—174no—
393#106—174no—
394#106—342no—
395#106C4×D1714yes—
396C22⋊C8—342no—
397#97—22no—
398C4×C8—342no—
399#95—21no—
400#95—21no—
401#95—171no—
402#95—171no—
403#95—341no—
404#95—341no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411C17#2262311yescharacteristic, complemented, Sylow
412C24—171no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—171no—
418C22×C4—171no—
419C22×C4—172no—
420C22×C4—172no—
421C22×C4—171no—
422C22×C4—344no—
423C22×C4—341no—
424C22×C4—341no—
425C22×C4—341no—
426C22×C4—341no—
427C22×C4—341no—
428C22×C4—341no—
429C22×C4—341no—
430C22×C4—341no—
431C22×C4—1361no—
432C22×C4—1361no—
433C22×C4D4×D1711yescharacteristic
434C22×C4D4×D1711yescharacteristic
435C22×C4C23×D1711yescharacteristic
436M16—348no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—172no—
441C2×C8—341no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8C2×C4×D1712yes—
447C2×C8#6325412yes—
448C2×C8#6325612yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—42no—
456C4⋊C4—81no—
457C4⋊C4—172no—
458C4⋊C4—172no—
459C4⋊C4—344no—
460C4⋊C4—341no—
461C4⋊C4—341no—
462C4⋊C4—341no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—682no—
467C4⋊C4—1361no—
468C4⋊C4C2×C4×D1712yes—
469C4⋊C4C2×C4×D1712yes—
470C22⋊C4—342no—
471C22⋊C4—344no—
472C22⋊C4—681no—
473C22⋊C4—682no—
474C22⋊C4—1361no—
475C42—21no—
476C42—21no—
477C42—344no—
478C42—341no—
479C42—341no—
480C42—1361no—
481C42—1361no—
482C42—1361no—
483C42—1361no—
484C23—172no—
485C23—341no—
486C23—341no—
487C23—341no—
488C23—341no—
489C23—341no—
490C23—341no—
491C23C2×D4×D1711yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—172no—
511C2×C4—172no—
512C2×C4—341no—
513C2×C4—341no—
514C2×C4—341no—
515C2×C4—341no—
516C2×C4—342no—
517C2×C4—342no—
518C2×C4—342no—
519C2×C4—342no—
520C2×C4—341no—
521C2×C4—342no—
522C2×C4—342no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—681no—
526C2×C4—681no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—1364no—
536C2×C4C22×C4×D1711yescharacteristic
537C2×C4C2×#6325411yescharacteristic
538C2×C4C2×#6325611yescharacteristic
539C2×C4#1058769411yescharacteristic
540C2×C4#41×D1711yescharacteristic
541C2×C4#1058769711yescharacteristic
542C8—41no—
543C8—342no—
544C8—681no—
545C8#1058757814yes—
546C22—342no—
547C22—342no—
548C22—342no—
549C22—681no—
550C22—681no—
551C22—681no—
552C22—681no—
553C22C4×D4×D1711yescharacteristic
554C22#1176345911yescharacteristic
555C22#1176346211yescharacteristic
556C22#137×D1711yescharacteristic
557C22#1176360511yescharacteristic
558C22#1176361311yescharacteristic
559C22#1176361811yescharacteristic
560C4—41no—
561C4—41no—
562C4—42no—
563C4—42no—
564C4—81no—
565C4—81no—
566C4—342no—
567C4—342no—
568C4—681no—
569C4—682no—
570C4—682no—
571C4—1361no—
572C4—1361no—
573C4C2×#1058757811yescharacteristic
574C4#1176328511yescharacteristic
575C4#1176349311yescharacteristic
576C4#1176349711yescharacteristic
577C2—682no—
578C2#1453165211yescharacteristic
579C2#1453760611yescharacteristic
580C2#1453761511yescharacteristic
581C2#1454126811yescharacteristic
582C2#1454132811yescharacteristic
583C2#1454138511yescharacteristic
584C2#1454139411yescharacteristic
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#22623
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. #14529513C2semidirect
5. #14529379C2semidirect
6. #14529366C2semidirect
7. #14529365C2semidirect
8. #14529352C2semidirect
12. #14528681C2semidirect
13. #107×Dic17C2semidirect
15. C17×#1944C2semidirect
19. #11762848C4semidirect
257. #1944D17semidirect
411. C17#22623semidirect

Maximal subgroups

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

subgroupindex
C2×#117628482
#145316762
#145316602
#145295132
#145293792
#145293662
#145293652
#145293522
#145288782
#145288692
#145288182
#145286812
#107×Dic172
#145285692
C17×#19442
#2262317