Finite Group Databasebeta

← #17052052

Subgroups — #17052052

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×#11762851C211yesmaximal, characteristic
2#14531667C211yesmaximal, characteristic
3#14531659C211yesmaximal, characteristic
4#14529522C211yesmaximal, characteristic, complemented
5#14529517C211yesmaximal, characteristic, complemented
6#14529377C211yesmaximal, characteristic, complemented
7#14529370C211yesmaximal, characteristic, complemented
8#14529363C211yesmaximal, characteristic, complemented
9#14529356C211yesmaximal, characteristic, complemented
10#14528880C211yesmaximal, characteristic
11#14528871C211yesmaximal, characteristic
12#14528817C211yesmaximal, characteristic
13#14528681C211yesmaximal, characteristic, complemented
14C4×#10587488C211yesmaximal, characteristic, complemented
15C17×#1753C211yesmaximal, characteristic
16C2×#10587612C2211yescharacteristic
17C22×C4⋊Dic17C2211yescharacteristic
18#11762851C418yescomplemented
19C2×#10587489C2211yescharacteristic
20C2×#10587489C2211yescharacteristic
21C2×#10587489C2211yescharacteristic
22C2×#10587488C2211yescharacteristic
23C2×#10587488C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25#11762707C2211yescharacteristic
26#11762707C2211yescharacteristic
27#11762707C2211yescharacteristic
28#11762707C2211yescharacteristic
29#11762697—21no—
30#11762697—21no—
31#11762692—21no—
32#11762692—21no—
33#11762673—21no—
34#11762669C2211yescharacteristic
35#11762669C2211yescharacteristic
36#11762667—21no—
37C4×C4⋊Dic17C2211yescharacteristic
38C4×C4⋊Dic17C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40C4×C4⋊Dic17C2211yescharacteristic
41C22⋊C4×Dic17—21no—
42#11762654—21no—
43#11762572C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53C22×M16×C17C2211yescharacteristic
54C2×C17×C22⋊C8C2211yescharacteristic
55C2×C17×#97C2211yescharacteristic
56C2×C4×C8×C17C2211yescharacteristic
57C4×C22⋊C4×C17C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C23×Dic17—21no—
62C2×M16×C17—22no—
63C2×M16×C17—22no—
64C2×M16×C17C2×C414yes—
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C2×C22⋊C4×C17D411yescharacteristic
70C2×C22⋊C4×C17D411yescharacteristic
71C2×C42×C17C2311yescharacteristic
72C2×C42×C17C2311yescharacteristic
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75C2×C22⋊Dic17—21no—
76#10587612—22no—
77#10587612C2×C414yes—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—22no—
80C2×C4⋊Dic17C2×C414yes—
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×#63225—21no—
90C2×#63225—21no—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C17×C22⋊C8—22no—
99C17×#97—22no—
100C4×C8×C17—22no—
101C17×#95D411yescharacteristic
102C17×#95D411yescharacteristic
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#10587502—21no—
112#10587502—21no—
113#10587489—22no—
114#10587489C2×C414yes—
115#10587489C2×C414yes—
116#10587488—22no—
117#10587488C2×C414yes—
118#10587488C2×C414yes—
119C24×C17C2×D411yescharacteristic
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17C22×C412yes—
125C22×C4×C17C2×D411yescharacteristic
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17#4112yes—
131C22×C4×C17C2411yescharacteristic
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17—24no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—81no—
142C22×Dic17—81no—
143M16×C17—28no—
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17C22×C412yes—
153C2×C8×C17C22×C412yes—
154C2×C8×C17#4112yes—
155C2×C8×C17#4112yes—
156C22⋊C4×C17—22no—
157C22⋊C4×C17—41no—
158C42×C17—21no—
159C42×C17—21no—
160C42×C17—21no—
161C42×C17—21no—
162C22⋊Dic17—24no—
163C22⋊Dic17—42no—
164C22⋊Dic17—81no—
165C4⋊Dic17—24no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17C22×C412yes—
179C4⋊Dic17C22×C412yes—
180C4⋊Dic17C22×C412yes—
181C4⋊Dic17C22×C412yes—
182#63225—81no—
183#63225—81no—
184#63225—81no—
185#63225—81no—
186C4×Dic17—24no—
187C4×Dic17—42no—
188C4×Dic17—42no—
189C4×Dic17—81no—
190C4×Dic17—81no—
191#22621—171nomaximal, Sylow
192C23×C17—21no—
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17C4×D412yes—
199C23×C17C22×D411yescharacteristic
200C2×C4×C17—21no—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—21no—
204C2×C4×C17—22no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17C4×D412yes—
218C2×C4×C17#13712yes—
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—22no—
226C2×Dic17—22no—
227C2×Dic17—22no—
228C2×Dic17—22no—
229C2×Dic17—41no—
230C2×Dic17—41no—
231C2×Dic17—41no—
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×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—84no—
250C8×C17—22no—
251C8×C17—41no—
252C8×C17—41no—
253C8×C17#11714yes—
254C2×#428—171no—
255#2346—171no—
256#2345—171no—
257#1943—171no—
258#1942—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×#106—171no—
268#1753D1711yescharacteristic
269C22×C17—22no—
270C22×C17—22no—
271C22×C17—22no—
272C22×C17—41no—
273C22×C17—41no—
274C22×C17—41no—
275C22×C17—41no—
276C22×C17C2×C4×D411yescharacteristic
277C22×C17#54711yescharacteristic
278C22×C17#54711yescharacteristic
279C22×C17C2×#13711yescharacteristic
280C22×C17#57511yescharacteristic
281C22×C17#57611yescharacteristic
282C22×C17#57711yescharacteristic
283C4×C17—22no—
284C4×C17—22no—
285C4×C17—41no—
286C4×C17—41no—
287C4×C17—41no—
288C4×C17C2×#11711yescharacteristic
289C4×C17#51911yescharacteristic
290C4×C17#55211yescharacteristic
291C4×C17#55211yescharacteristic
292Dic17—42no—
293Dic17—42no—
294Dic17—42no—
295Dic17—42no—
296Dic17—81no—
297Dic17—81no—
298Dic17—81no—
299Dic17—81no—
300C22×M16C2×D1711yescharacteristic
301C2×#117—171no—
302C22×C4⋊C4—171no—
303#428—178no—
304C2×#107—171no—
305C2×#107—171no—
306C2×#107—171no—
307C2×#106—171no—
308C2×#106—171no—
309C2×#106—171no—
310C2×C22⋊C8C2×D1711yescharacteristic
311C2×#97C2×D1711yescharacteristic
312C2×C4×C8C2×D1711yescharacteristic
313#389—171no—
314#389—171no—
315#389—171no—
316#389—171no—
317#388—341no—
318#388—341no—
319#387—341no—
320#387—341no—
321#382—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×C22⋊C4—341no—
330C4×C22⋊C4C2×D1711yescharacteristic
331#376—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#340—171no—
341#340—171no—
342#336C2×D1711yescharacteristic
343#336C2×D1711yescharacteristic
344C2×C17—42no—
345C2×C17#234411yescharacteristic
346C2×C17C4×#13711yescharacteristic
347C2×C17#294611yescharacteristic
348C2×C17#334411yescharacteristic
349C2×C17#334711yescharacteristic
350C2×C17#335111yescharacteristic
351C2×C17#335711yescharacteristic
352C23×C4—341no—
353C23×C4C22×D1711yescharacteristic
354C2×M16—22no—
355C2×M16—22no—
356C2×M16C4×D1714yes—
357C22×C8C22×D1711yescharacteristic
358C22×C8C22×D1711yescharacteristic
359C22×C8C22×D1711yescharacteristic
360C22×C8C22×D1711yescharacteristic
361#117—174no—
362#117—342no—
363C2×C4⋊C4—174no—
364C2×C4⋊C4—171no—
365C2×C4⋊C4—171no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—342no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4—341no—
377C2×C4⋊C4—341no—
378C2×C22⋊C4—341no—
379C2×C22⋊C4—341no—
380C2×C22⋊C4—341no—
381C2×C22⋊C4D6811yescharacteristic
382C2×C22⋊C4D6811yescharacteristic
383C2×C42—341no—
384C2×C42—341no—
385C2×C42—341no—
386C2×C42—341no—
387C2×C42—341no—
388C2×C42C22×D1711yescharacteristic
389C2×C42C22×D1711yescharacteristic
390#107—174no—
391#107—174no—
392#107—342no—
393#106—174no—
394#106—174no—
395#106—342no—
396C22⋊C8—22no—
397#97—22no—
398C4×C8—22no—
399#95—341no—
400#95—341no—
401#95—341no—
402#95—341no—
403#95—341no—
404#95—341no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95D6811yescharacteristic
410#95D6811yescharacteristic
411C17#2262111yescharacteristic, complemented, Sylow
412C24C2×D6811yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—341no—
418C22×C4—341no—
419C22×C4—341no—
420C22×C4—341no—
421C22×C4—344no—
422C22×C4—341no—
423C22×C4—341no—
424C22×C4—341no—
425C22×C4—341no—
426C22×C4—1361no—
427C22×C4—1361no—
428C22×C4C2×C4×D1712yes—
429C22×C4C2×D6811yescharacteristic
430C22×C4C2×D6811yescharacteristic
431C22×C4C2×D6811yescharacteristic
432C22×C4C2×D6811yescharacteristic
433C22×C4C2×D6811yescharacteristic
434C22×C4#6325212yes—
435C22×C4C23×D1711yescharacteristic
436M16—28no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8C2×C4×D1712yes—
446C2×C8C2×C4×D1712yes—
447C2×C8#6325412yes—
448C2×C8#6325412yes—
449C4⋊C4—172no—
450C4⋊C4—172no—
451C4⋊C4—172no—
452C4⋊C4—172no—
453C4⋊C4—341no—
454C4⋊C4—341no—
455C4⋊C4—341no—
456C4⋊C4—341no—
457C4⋊C4—344no—
458C4⋊C4—341no—
459C4⋊C4—341no—
460C4⋊C4—341no—
461C4⋊C4—341no—
462C4⋊C4—341no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—1361no—
467C4⋊C4—1361no—
468C4⋊C4—1361no—
469C4⋊C4—1361no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4—344no—
473C22⋊C4—682no—
474C22⋊C4—1361no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—344no—
480C42—682no—
481C42—682no—
482C42—1361no—
483C42—1361no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23C4×D6812yes—
491C23C22×D6811yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—342no—
510C2×C4—342no—
511C2×C4—342no—
512C2×C4—342no—
513C2×C4—681no—
514C2×C4—681no—
515C2×C4—681no—
516C2×C4—681no—
517C2×C4—681no—
518C2×C4—681no—
519C2×C4—681no—
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×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—1364no—
534C2×C4C4×D6812yes—
535C2×C4#1058759512yes—
536C2×C4C22×C4×D1711yescharacteristic
537C2×C4C2×#6325211yescharacteristic
538C2×C4C2×#6325411yescharacteristic
539C2×C4C2×#6325411yescharacteristic
540C2×C4#1058769711yescharacteristic
541C2×C4#1058769711yescharacteristic
542C8—22no—
543C8—41no—
544C8—41no—
545C8#1058756314yes—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22C2×C4×D6811yescharacteristic
554C22#1176346311yescharacteristic
555C22#1176346311yescharacteristic
556C22C2×#1058759511yescharacteristic
557C22#1176360911yescharacteristic
558C22#1176361411yescharacteristic
559C22#1176362111yescharacteristic
560C4—22no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—682no—
566C4—682no—
567C4—682no—
568C4—682no—
569C4—1361no—
570C4—1361no—
571C4—1361no—
572C4—1361no—
573C4C2×#1058756311yescharacteristic
574C4#1176328611yescharacteristic
575C4#1176349911yescharacteristic
576C4#1176349911yescharacteristic
577C2—42no—
578C2#1453165111yescharacteristic
579C2C4×#1058759511yescharacteristic
580C2#1453761811yescharacteristic
581C2#1454124111yescharacteristic
582C2#1454128611yescharacteristic
583C2#1454134611yescharacteristic
584C2#1454141111yescharacteristic
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1753C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#1753D17
hypocenterC17#22621
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
4. #14529522C2semidirect
5. #14529517C2semidirect
6. #14529377C2semidirect
7. #14529370C2semidirect
8. #14529363C2semidirect
9. #14529356C2semidirect
13. #14528681C2semidirect
14. C4×#10587488C2semidirect
18. #11762851C4semidirect
411. C17#22621semidirect

Maximal subgroups

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

subgroupindex
C2×#117628512
#145316672
#145316592
#145295222
#145295172
#145293772
#145293702
#145293632
#145293562
#145288802
#145288712
#145288172
#145286812
C4×#105874882
C17×#17532
#2262117