Finite Group Databasebeta

← #17107083

Subgroups — #17107083

Order 4352.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533186C211yesmaximal, characteristic
2#14531789C211yesmaximal, characteristic
3#14530206C211yesmaximal, characteristic, complemented
4#14530160C211yesmaximal, characteristic, complemented
5#14529987C211yesmaximal, characteristic
6#14529981C211yesmaximal, characteristic
7#14529551C211yesmaximal, characteristic, complemented
8#14529510C211yesmaximal, characteristic, complemented
9#14529393C211yesmaximal, characteristic
10#14529374C211yesmaximal, characteristic, complemented
11#14529371C211yesmaximal, characteristic, complemented
12#14529049C211yesmaximal, characteristic, complemented
13#14528870C211yesmaximal, characteristic
14#14528603C211yesmaximal, characteristic, complemented
15C17×#1853C211yesmaximal, characteristic
16C2×#10587610C2211yescharacteristic
17C2×#10587573C2211yescharacteristic
18C2×#10587478C2211yescharacteristic
19C2×#10587488C2211yescharacteristic
20C2×#10587485C2211yescharacteristic
21C2×#10587487C2211yescharacteristic
22C2×#10587487C2211yescharacteristic
23C2×#10587481C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27C17⋊#401—21no—
28#11762734—21no—
29#11762714C2211yescharacteristic
30#11762708C2211yescharacteristic
31#11762705C2211yescharacteristic
32#11762705C2211yescharacteristic
33#11762704—21no—
34#11762697—21no—
35#11762696—21no—
36#11762690—21no—
37#11762682—21no—
38#11762674C2211yescharacteristic
39#11762669C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42C4×C22⋊Dic17—21no—
43C4×C22⋊Dic17C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C22×C4⋊C4×C17C2211yescharacteristic
53C2×C17×#106C2211yescharacteristic
54C2×C17×#106C2211yescharacteristic
55C2×C17×C22⋊C8C2211yescharacteristic
56C17×#380C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C23×C4×C17—21no—
60C23×C4×C17C2311yescharacteristic
61C22×Dic34C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C2×C4⋊C4×C17—22no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—41no—
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C22⋊C4×C17—21no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17D411yescharacteristic
75C2×C22⋊Dic17D411yescharacteristic
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80#10587610—22no—
81#10587610—22no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17C2311yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93#10587573—22no—
94#10587573—41no—
95C17×#106—22no—
96C17×#106—41no—
97C17×C22⋊C8—22no—
98C17×#95—21no—
99#10587502—21no—
100#10587502—21no—
101#10587502—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502D411yescharacteristic
110#10587502D411yescharacteristic
111C22⋊#3622—41no—
112#10587488—22no—
113#10587487—22no—
114#10587487—22no—
115#10587485—22no—
116C8×Dic17—41no—
117#10587481—22no—
118#10587481—41no—
119#10587478—22no—
120C24×C17C2×D411yescharacteristic
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—42no—
132C22×C4×C17—42no—
133C22×C4×C17—81no—
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2411yescharacteristic
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142C22×Dic17C2×D411yescharacteristic
143C22×Dic17C2×D411yescharacteristic
144C2×Dic34—21no—
145C2×Dic34—21no—
146C2×Dic34—21no—
147C2×Dic34—21no—
148C2×Dic34—21no—
149C2×Dic34#4112yes—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17—21no—
154C2×C8×C17—21no—
155C2×C8×C17#4112yes—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—42no—
164C4⋊C4×C17#4112yes—
165C22⋊C4×C17—81no—
166C22⋊Dic17—24no—
167C22⋊Dic17—22no—
168C22⋊Dic17—22no—
169C22⋊Dic17—42no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—81no—
176C4⋊Dic17#4112yes—
177#63225—24no—
178#63225—42no—
179#63225—42no—
180#63225—81no—
181C4×Dic17—21no—
182C4×Dic17—21no—
183C4×Dic17—21no—
184C4×Dic17—21no—
185C4×Dic17—42no—
186C4×Dic17—81no—
187C4×Dic17—81no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194#27037—171nomaximal, Sylow
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17#13512yes—
202C23×C17C22×D411yescharacteristic
203C2×C4×C17—21no—
204C2×C4×C17—22no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
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×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—84no—
223C2×C4×C17C2×#4111yescharacteristic
224C2×C4×C17C2×#4111yescharacteristic
225C2×C4×C17C2×#4111yescharacteristic
226C2×C4×C17C2×#4111yescharacteristic
227C2×C4×C17H(5,𝔽2)11yescharacteristic
228C2×C4×C172−1+411yescharacteristic
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—22no—
232C2×Dic17—22no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17#13712yes—
253Dic34—22no—
254Dic34—22no—
255Dic34—41no—
256Dic34—41no—
257C8×C17—22no—
258C8×C17—41no—
259#3622—41no—
260#3622—41no—
261#2494—171no—
262#2360—171no—
263#2074—171no—
264#2066—171no—
265#2036—171no—
266#2035—171no—
267#1949—171no—
268#1941—171no—
269#1923—171no—
270#1921—171no—
271#1920—171no—
272#1865—171no—
273#1853D1711yescharacteristic
274#1823—171no—
275C4×#103—171no—
276C22×C17—22no—
277C22×C17—22no—
278C22×C17—22no—
279C22×C17—41no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17—41no—
283C22×C17#52511yescharacteristic
284C22×C17#54611yescharacteristic
285C22×C17#54711yescharacteristic
286C22×C17C2×#13511yescharacteristic
287C22×C17C2×#13711yescharacteristic
288C22×C17#57711yescharacteristic
289C22×C17#57711yescharacteristic
290C4×C17—41no—
291C4×C17—42no—
292C4×C17—81no—
293C4×C17—81no—
294C4×C17—81no—
295C4×C17#52911yescharacteristic
296C4×C17#53911yescharacteristic
297C4×C17#55311yescharacteristic
298C4×C17#55611yescharacteristic
299Dic17—22no—
300Dic17—41no—
301Dic17—41no—
302Dic17—42no—
303Dic17—42no—
304Dic17—41no—
305Dic17—81no—
306Dic17—81no—
307C2×C22⋊Q8—171no—
308C2×C4×Q8—171no—
309C22×C4⋊C4C2×D1711yescharacteristic
310C2×#107—171no—
311C2×#106—171no—
312C2×#106C2×D1711yescharacteristic
313C2×#106C2×D1711yescharacteristic
314C2×C4⋊C8—171no—
315C2×#103—171no—
316C2×#103—171no—
317C2×#103—171no—
318C2×#103—171no—
319C2×C22⋊C8—171no—
320C2×C22⋊C8C2×D1711yescharacteristic
321C2×C4×C8—171no—
322#401—341no—
323#397—341no—
324#391—171no—
325#389—171no—
326#389—171no—
327#389—171no—
328#389—341no—
329#388—341no—
330#387—341no—
331#387—341no—
332#385—341no—
333#382—171no—
334#380—171no—
335#380C2×D1711yescharacteristic
336C4×C4⋊C4—171no—
337C4×C4⋊C4—171no—
338C4×C22⋊C4—171no—
339C4×C22⋊C4—341no—
340#340—171no—
341#340—171no—
342#340—171no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340C2×D1711yescharacteristic
347#340C2×D1711yescharacteristic
348#336—171no—
349#336—171no—
350C2×C17—42no—
351C2×C17#246611yescharacteristic
352C2×C17#310811yescharacteristic
353C2×C17#312511yescharacteristic
354C2×C17#329611yescharacteristic
355C2×C17#332011yescharacteristic
356C2×C17#333711yescharacteristic
357C2×C17#335611yescharacteristic
358C22×Q8—171no—
359C23×C4—21no—
360C23×C4C22×D1711yescharacteristic
361C22×C8—171no—
362C22×C8—171no—
363C22×C8C22×D1711yescharacteristic
364C22×C8C22×D1711yescharacteristic
365C22⋊Q8—342no—
366C22⋊Q8—342no—
367C4×Q8—342no—
368C4×Q8—681no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—41no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4—341no—
377C2×C4⋊C4—341no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4C22×D1711yescharacteristic
381C2×C4⋊C4C22×D1711yescharacteristic
382C2×C4⋊C4C22×D1711yescharacteristic
383C2×C4⋊C4C22×D1711yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—171no—
386C2×C22⋊C4—171no—
387C2×C22⋊C4—341no—
388C2×C22⋊C4—341no—
389C2×C42—171no—
390C2×C42—171no—
391C2×C42—341no—
392C2×C42—341no—
393C2×C42—341no—
394#107—342no—
395#106—22no—
396#106—41no—
397#106—342no—
398C4⋊C8—342no—
399#103—342no—
400#103—342no—
401#103—342no—
402#103—681no—
403C22⋊C8—22no—
404C22⋊C8—681no—
405C4×C8—681no—
406#95—21no—
407#95—171no—
408#95—171no—
409#95—341no—
410#95—341no—
411#95—341no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95—341no—
417#95—341no—
418#95—341no—
419C17#2703711yescharacteristic, complemented, Sylow
420C24D4×D1711yescharacteristic
421C2×Q8—172no—
422C2×Q8—341no—
423C2×Q8—341no—
424C2×Q8—341no—
425C2×Q8—341no—
426C2×Q8—341no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—171no—
441C22×C4—171no—
442C22×C4—171no—
443C22×C4—171no—
444C22×C4—341no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4—341no—
448C22×C4D4×D1711yescharacteristic
449C22×C4C23×D1711yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—341no—
456C2×C8—341no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8—341no—
460C2×C8—341no—
461C2×C8#6325212yes—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—172no—
471C4⋊C4—344no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—341no—
477C4⋊C4—682no—
478C4⋊C4—682no—
479C4⋊C4—1361no—
480C4⋊C4—1361no—
481C4⋊C4#6325412yes—
482C22⋊C4—81no—
483C22⋊C4—342no—
484C22⋊C4—344no—
485C22⋊C4—342no—
486C22⋊C4—682no—
487C42—341no—
488C42—341no—
489C42—341no—
490C42—341no—
491C42—682no—
492C42—1361no—
493C42—1361no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23#1058760212yes—
501C23C2×D4×D1711yescharacteristic
502Q8—342no—
503Q8—342no—
504Q8—681no—
505Q8—681no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—172no—
527C2×C4—341no—
528C2×C4—341no—
529C2×C4—341no—
530C2×C4—341no—
531C2×C4—342no—
532C2×C4—342no—
533C2×C4—341no—
534C2×C4—341no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—681no—
541C2×C4—681no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4C2×#6325211yescharacteristic
551C2×C4C2×#6325411yescharacteristic
552C2×C4#1058769111yescharacteristic
553C2×C4#1058769411yescharacteristic
554C2×C4#41×D1711yescharacteristic
555C2×C4#41×D1711yescharacteristic
556C8—22no—
557C8—41no—
558C8—681no—
559C8—681no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1176331711yescharacteristic
568C22#1176344611yescharacteristic
569C22#1176346011yescharacteristic
570C22C2×#1058760211yescharacteristic
571C22#137×D1711yescharacteristic
572C22#1176361511yescharacteristic
573C22#1176361911yescharacteristic
574C4—42no—
575C4—41no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—342no—
580C4—682no—
581C4—681no—
582C4—681no—
583C4—681no—
584C4—682no—
585C4—1361no—
586C4—1361no—
587C4#1176334111yescharacteristic
588C4#1176340511yescharacteristic
589C4#1176350811yescharacteristic
590C4#1176353011yescharacteristic
591C2—42no—
592C2#1453281711yescharacteristic
593C2#1453886711yescharacteristic
594C2#1453906411yescharacteristic
595C2#1454062311yescharacteristic
596C2#1454094811yescharacteristic
597C2#1454115911yescharacteristic
598C2#1454139511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1853C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1853D17
hypocenterC17#27037
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
3. #14530206C2semidirect
4. #14530160C2semidirect
7. #14529551C2semidirect
8. #14529510C2semidirect
10. #14529374C2semidirect
11. #14529371C2semidirect
12. #14529049C2semidirect
14. #14528603C2semidirect
419. C17#27037semidirect

Maximal subgroups

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

subgroupindex
#145331862
#145317892
#145302062
#145301602
#145299872
#145299812
#145295512
#145295102
#145293932
#145293742
#145293712
#145290492
#145288702
#145286032
C17×#18532
#2703717