Finite Group Databasebeta

← #17052103

Subgroups — #17052103

Order 4352.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762850C211yesmaximal, characteristic
2#14531665C211yesmaximal, characteristic
3#14531649C211yesmaximal, characteristic
4#14529527C211yesmaximal, characteristic, complemented
5#14529510C211yesmaximal, characteristic, complemented
6#14529376C211yesmaximal, characteristic, complemented
7#14529367C211yesmaximal, characteristic, complemented
8#14529362C211yesmaximal, characteristic, complemented
9#14529353C211yesmaximal, characteristic, complemented
10#14528878C211yesmaximal, characteristic
11#14528873C211yesmaximal, characteristic
12#14528814C211yesmaximal, characteristic
13#107×Dic17C211yesmaximal, characteristic, complemented
14#14528569C211yesmaximal, characteristic
15C17×#1776C211yesmaximal, characteristic, complemented
16C22×#63218C2211yescharacteristic
17C2×#10587578C2211yescharacteristic
18C2×#10587558—21no—
19C2×C4⋊C4×D17C2211yescharacteristic
20#11762850C418yescomplemented
21C2×#10587489C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587488C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587491C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27#11762707C2211yescharacteristic
28#11762707C2211yescharacteristic
29#11762705C2211yescharacteristic
30#11762705C2211yescharacteristic
31#11762700—21no—
32#11762698—21no—
33#11762693—21no—
34#11762691—21no—
35#11762673—21no—
36#11762671—21no—
37#11762666C2211yescharacteristic
38#11762666C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C4×#63227C2211yescharacteristic
42#11762655—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
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C22×C4×D17—21no—
61C22×C4×D17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C2×C4⋊C4×C17C2311yescharacteristic
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C42×C17—21no—
69C2×C42×C17—21no—
70C2×C42×C17C2311yescharacteristic
71C2×#63227—21no—
72C2×#63227—21no—
73C2×#63227—21no—
74C2×#63227D411yescharacteristic
75C2×#63227D411yescharacteristic
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×#63225—21no—
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225—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—
98#10587558—44no—
99C17×#107C2×C414yes—
100C17×#106—22no—
101C17×#97—22no—
102C17×#95—21no—
103C17×#95—21no—
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#10587491—22no—
115#10587489—22no—
116#10587488C2×C414yes—
117C8×Dic17—22no—
118#10587479C2×C414yes—
119#10587478C2×C414yes—
120C23×D17C2×D411yescharacteristic
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17C2×D411yescharacteristic
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2411yescharacteristic
128C22×Dic17—21no—
129C22×Dic17—21no—
130C22×Dic17—21no—
131C22×Dic17—21no—
132C22×Dic17C2×D411yescharacteristic
133C22×Dic17C2×D411yescharacteristic
134C22×Dic17C2×D411yescharacteristic
135C2×C4×D17—21no—
136C2×C4×D17—21no—
137C2×C4×D17—21no—
138C2×C4×D17—21no—
139C2×C4×D17—24no—
140C2×C4×D17—42no—
141C2×C4×D17—42no—
142C2×C4×D17C22×C412yes—
143C2×C4×D17#4112yes—
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17C22×C412yes—
148C2×C8×C17#4112yes—
149C2×C8×C17#4112yes—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17C22×C412yes—
157C4⋊C4×C17C22×C412yes—
158C42×C17—21no—
159C42×C17—21no—
160C42×C17—42no—
161C42×C17—42no—
162#63227—24no—
163#63227—22no—
164#63227—41no—
165#63227—42no—
166#63227—81no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17C22×C412yes—
174C4⋊Dic17C22×C412yes—
175#63225—24no—
176#63225—42no—
177#63225—42no—
178#63225—81no—
179#63225—81no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—24no—
183C4×Dic17—42no—
184C4×Dic17—42no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622C22×C412yes—
191#63218—28no—
192#22624—171nomaximal, Sylow
193C23×C17C22×D411yescharacteristic
194C22×D17—21no—
195C22×D17—21no—
196C22×D17—21no—
197C22×D17—21no—
198C22×D17—21no—
199C22×D17—21no—
200C22×D17C4×D412yes—
201C2×C4×C17—22no—
202C2×C4×C17—22no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
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×C17C23×C411yescharacteristic
220C2×C4×C17C2×#4111yescharacteristic
221C2×C4×C17C2×#4111yescharacteristic
222C2×C4×C17C2×#4111yescharacteristic
223C2×C4×C172−1+411yescharacteristic
224C2×C4×C172−1+411yescharacteristic
225C2×Dic17—21no—
226C2×Dic17—21no—
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—22no—
230C2×Dic17—22no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
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×Dic17—41no—
244C2×Dic17C4×D412yes—
245C2×Dic17AGL(1,ℤ/8)12yes—
246C4×D17—22no—
247C4×D17—22no—
248C4×D17—22no—
249C4×D17—22no—
250C4×D17—48no—
251C8×C17—41no—
252C8×C17#11714yescomplemented
253#3622—22no—
254#3622—41no—
255C2×#428—171no—
256#2346—171no—
257#2344—171no—
258#1944—171no—
259#1941—171no—
260#1921—171no—
261#1920—171no—
262#1919—171no—
263#1918—171no—
264#1825—171no—
265#1824—171no—
266#1809—171no—
267#1776D1711yescharacteristic, complemented
268C4×#107—171no—
269#1753—171no—
270C22×C17C2×C4×D411yescharacteristic
271C22×C17#54711yescharacteristic
272C22×C17#54711yescharacteristic
273C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C17#57511yescharacteristic
275C22×C17#57711yescharacteristic
276C22×C17#57811yescharacteristic
277C2×D17—22no—
278C2×D17—22no—
279C2×D17—22no—
280C2×D17—41no—
281C2×D17—41no—
282C2×D17—41no—
283C2×D17—41no—
284C4×C17—41no—
285C4×C17—41no—
286C4×C17—42no—
287C4×C17—42no—
288C4×C17—81no—
289C4×C17—81no—
290C4×C17C2×#11711yescharacteristic
291C4×C17#51911yescharacteristic
292C4×C17#55211yescharacteristic
293C4×C17#55211yescharacteristic
294Dic17—22no—
295Dic17—22no—
296Dic17—41no—
297Dic17—42no—
298Dic17—42no—
299Dic17—81no—
300Dic17—81no—
301C22×M16—171no—
302C2×#117—171no—
303C2×#117—341no—
304C22×C4⋊C4—171no—
305#428—178no—
306C2×#107—171no—
307C2×#107—171no—
308C2×#107C2×D1711yescharacteristic
309C2×#106—171no—
310C2×#106—171no—
311C2×#106C2×D1711yescharacteristic
312C2×C22⋊C8—171no—
313C2×#97C2×D1711yescharacteristic
314C2×C4×C8—171no—
315#389—171no—
316#389—171no—
317#389—171no—
318#389—171no—
319#388—341no—
320#388—341no—
321#387—341no—
322#387—341no—
323#382—341no—
324#381—341no—
325#380—171no—
326#380—171no—
327C4×C4⋊C4—171no—
328C4×C4⋊C4—171no—
329C4×C4⋊C4C2×D1711yescharacteristic
330C4×C4⋊C4C2×D1711yescharacteristic
331C4×C22⋊C4—171no—
332#376—341no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#340C2×D1711yescharacteristic
342#340C2×D1711yescharacteristic
343#336—171no—
344#336—171no—
345C2×C17#234411yescharacteristic
346C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
347C2×C17#294711yescharacteristic
348C2×C17#334311yescharacteristic
349C2×C17#334811yescharacteristic
350C2×C17#335211yescharacteristic
351C2×C17#335411yescharacteristic
352D17—42no—
353C23×C4—171no—
354C23×C4—341no—
355C2×M16—174no—
356C2×M16—342no—
357C2×M16—342no—
358C22×C8—171no—
359C22×C8—171no—
360C22×C8C22×D1711yescharacteristic
361C22×C8C22×D1711yescharacteristic
362#117—174no—
363#117—342no—
364#117—684no—
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⋊C4—341no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C22⋊C4—171no—
381C2×C22⋊C4—171no—
382C2×C22⋊C4—341no—
383C2×C22⋊C4—341no—
384C2×C22⋊C4—341no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—171no—
388C2×C42—341no—
389C2×C42—341no—
390C2×C42—341no—
391C2×C42C22×D1711yescharacteristic
392#107—174no—
393#107—342no—
394#107C4×D1714yes—
395#106—22no—
396#106—174no—
397#106—174no—
398C22⋊C8—342no—
399#97—22no—
400C4×C8—342no—
401#95—21no—
402#95—21no—
403#95—171no—
404#95—171no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411#95—341no—
412#95—341no—
413C17#2262411yescharacteristic, complemented, Sylow
414C24—171no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—171no—
420C22×C4—171no—
421C22×C4—172no—
422C22×C4—172no—
423C22×C4—171no—
424C22×C4—344no—
425C22×C4—341no—
426C22×C4—341no—
427C22×C4—341no—
428C22×C4—341no—
429C22×C4—341no—
430C22×C4—341no—
431C22×C4—341no—
432C22×C4—341no—
433C22×C4—682no—
434C22×C4—682no—
435C22×C4D4×D1711yescharacteristic
436C22×C4D4×D1711yescharacteristic
437C22×C4C23×D1711yescharacteristic
438M16—348no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—172no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8—341no—
448C2×C8C2×C4×D1712yes—
449C2×C8#6325212yes—
450C2×C8#6325212yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
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—682no—
468C4⋊C4—1361no—
469C4⋊C4—1361no—
470C4⋊C4C2×C4×D1712yes—
471C4⋊C4C2×C4×D1712yes—
472C22⋊C4—342no—
473C22⋊C4—344no—
474C22⋊C4—681no—
475C22⋊C4—682no—
476C22⋊C4—1361no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42—42no—
481C42—344no—
482C42—341no—
483C42—341no—
484C42—682no—
485C42—682no—
486C23—172no—
487C23—341no—
488C23—341no—
489C23—341no—
490C23—341no—
491C23—341no—
492C23—341no—
493C23C2×D4×D1711yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
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—41no—
511C2×C4—41no—
512C2×C4—172no—
513C2×C4—172no—
514C2×C4—341no—
515C2×C4—341no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—342no—
519C2×C4—342no—
520C2×C4—342no—
521C2×C4—342no—
522C2×C4—341no—
523C2×C4—342no—
524C2×C4—342no—
525C2×C4—341no—
526C2×C4—341no—
527C2×C4—688no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4C22×C4×D1711yescharacteristic
539C2×C4C2×#6325211yescharacteristic
540C2×C4C2×#6325211yescharacteristic
541C2×C4#1058769411yescharacteristic
542C2×C4#41×D1711yescharacteristic
543C2×C4#1058769711yescharacteristic
544C8—41no—
545C8—342no—
546C8—681no—
547C8#1058755814yescomplemented
548C22—342no—
549C22—342no—
550C22—342no—
551C22—681no—
552C22—681no—
553C22—681no—
554C22—681no—
555C22C4×D4×D1711yescharacteristic
556C22#1176346011yescharacteristic
557C22#1176346411yescharacteristic
558C22AGL(1,ℤ/8)×D1711yescharacteristic
559C22#1176360511yescharacteristic
560C22#1176361811yescharacteristic
561C22#1176362511yescharacteristic
562C4—41no—
563C4—41no—
564C4—42no—
565C4—42no—
566C4—81no—
567C4—81no—
568C4—342no—
569C4—342no—
570C4—682no—
571C4—682no—
572C4—681no—
573C4—1361no—
574C4—1361no—
575C4C2×#1058755811yescharacteristic
576C4#1176328511yescharacteristic
577C4#1176349811yescharacteristic
578C4#1176350011yescharacteristic
579C2—682no—
580C2#1453164911yescharacteristic
581C2#1453759511yescharacteristic
582C2#1453762611yescharacteristic
583C2#1454122911yescharacteristic
584C2#1454130411yescharacteristic
585C2#1454135511yescharacteristic
586C2#1454137511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1776C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1776D17
hypocenterC17#22624
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #14529527C2semidirect
5. #14529510C2semidirect
6. #14529376C2semidirect
7. #14529367C2semidirect
8. #14529362C2semidirect
9. #14529353C2semidirect
13. #107×Dic17C2semidirect
15. C17×#1776C2semidirect
20. #11762850C4semidirect
252. C8×C17#117semidirect
267. #1776D17semidirect
413. C17#22624semidirect
547. C8#10587558semidirect

Maximal subgroups

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

subgroupindex
C2×#117628502
#145316652
#145316492
#145295272
#145295102
#145293762
#145293672
#145293622
#145293532
#145288782
#145288732
#145288142
#107×Dic172
#145285692
C17×#17762
#2262417