Finite Group Databasebeta

← #17046571

Subgroups — #17046571

Order 4352.

732 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×#11763109C211yesmaximal, characteristic
2#14531525C211yesmaximal, characteristic, complemented
3#14529741C211yesmaximal, characteristic, complemented
4#14529542C211yesmaximal, characteristic, complemented
5#14529524C211yesmaximal, characteristic, complemented
6#14529514C211yesmaximal, characteristic, complemented
7#14529484C211yesmaximal, characteristic
8#14529470C211yesmaximal, characteristic
9#14529401C211yesmaximal, characteristic
10#14529401C211yesmaximal, characteristic
11#14529373C211yesmaximal, characteristic, complemented
12#14529359C211yesmaximal, characteristic, complemented
13#14528784C211yesmaximal, characteristic
14#14528641C211yesmaximal, characteristic, complemented
15C17×#2319C211yesmaximal, characteristic
16C2×#10587556C2211yescharacteristic
17#11763109C418yescomplemented
18C2×#10587478C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×C4⋊#3622C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587474C2211yescharacteristic
29C2×C4×#3622C2211yescharacteristic
30#11762707C2211yescharacteristic
31#11762707C2211yescharacteristic
32#11762706—21no—
33#11762706—21no—
34#11762696—21no—
35#11762696—21no—
36#11762679C2211yescharacteristic
37C4×C4⋊Dic17C2211yescharacteristic
38C4×C4⋊Dic17C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C2×C4×D4×C17C2211yescharacteristic
50C17×#390C2211yescharacteristic
51C17×#390C2211yescharacteristic
52C17×#389C2211yescharacteristic
53C17×#389C2211yescharacteristic
54C17×#388—21no—
55C17×#388—21no—
56C17×#381—21no—
57C17×#380—21no—
58C4×C4⋊C4×C17C2211yescharacteristic
59C17×#376C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C4×D4×C17—22no—
63C4×D4×C17C2×C414yes—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C22⋊C4×C17—21no—
70C2×C22⋊C4×C17—21no—
71C2×C22⋊C4×C17—21no—
72C2×C22⋊C4×C17—21no—
73C2×C22⋊C4×C17—21no—
74C2×C42×C17—21no—
75C2×C42×C17C2311yescharacteristic
76C2×C42×C17C2311yescharacteristic
77C2×C42×C17C2311yescharacteristic
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17—21no—
90C22×#3622C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94#10587556—22no—
95#10587556C2×C414yes—
96C17×#95—21no—
97C17×#95—21no—
98C17×#95—21no—
99C17×#95D411yescharacteristic
100C17×#95D411yescharacteristic
101C17×#95D411yescharacteristic
102C17×#95D411yescharacteristic
103#10587503—41no—
104#10587503—41no—
105#10587503C2×C414yes—
106#10587503C2×C414yes—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587479C2×C414yes—
112#10587478—41no—
113#10587478—41no—
114#10587478C2×C414yes—
115C4⋊#3622—22no—
116C4⋊#3622C2×C414yes—
117#10587474—41no—
118C4×#3622—41no—
119C24×C17—21no—
120C2×D4×C17—21no—
121C2×D4×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17C22×C412yes—
126C22×C4×C17—21no—
127C22×C4×C17—24no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—81no—
131C22×C4×C17—81no—
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2411yescharacteristic
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C4⋊C4×C17—21no—
144C4⋊C4×C17—21no—
145C4⋊C4×C17—21no—
146C4⋊C4×C17—21no—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—81no—
149C4⋊C4×C17—81no—
150C4⋊C4×C17C22×C412yes—
151C22⋊C4×C17—24no—
152C22⋊C4×C17—81no—
153C22⋊C4×C17—81no—
154C22⋊C4×C17—81no—
155C22⋊C4×C17—81no—
156C42×C17—21no—
157C42×C17—21no—
158C42×C17—21no—
159C42×C17—21no—
160C42×C17—81no—
161C42×C17C22×C412yes—
162C42×C17#4112yes—
163C4⋊Dic17—21no—
164C4⋊Dic17—21no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—81no—
170C4⋊Dic17—81no—
171C4⋊Dic17C22×C412yes—
172C4⋊Dic17C22×C412yes—
173#63225—24no—
174#63225—24no—
175#63225—81no—
176#63225—81no—
177C4×Dic17—81no—
178C4×Dic17—81no—
179C2×#3622—21no—
180C2×#3622—21no—
181C2×#3622—21no—
182C2×#3622—21no—
183C2×#3622—21no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622C22×C412yes—
190C2×#3622C22×C412yes—
191#22126—171nomaximal, Sylow
192C23×C17—22no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17C22×D411yescharacteristic
200D4×C17—22no—
201D4×C17—22no—
202D4×C17—41no—
203D4×C17—41no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—22no—
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×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—84no—
230C2×C4×C17C4×D412yes—
231C2×C4×C17C4×D412yes—
232C2×C4×C17C23×C411yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17H(5,𝔽2)11yescharacteristic
235C2×C4×C172−1+411yescharacteristic
236C2×C4×C172−1+411yescharacteristic
237C2×C4×C172−1+411yescharacteristic
238C2×Dic17—22no—
239C2×Dic17—22no—
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—41no—
250#3622—22no—
251#3622—22no—
252#3622—41no—
253#3622—41no—
254C2×#478—171no—
255#2330—171no—
256#2319D1711yescharacteristic
257#1989—171no—
258#1947—171no—
259#1944—171no—
260#1942—171no—
261#1936—171no—
262#1934—171no—
263#1925—171no—
264#1925—171no—
265#1921—171no—
266#1919—171no—
267#1802—171no—
268#1768—171no—
269C22×C17—42no—
270C22×C17—42no—
271C22×C17—42no—
272C22×C17—81no—
273C22×C17—81no—
274C22×C17—81no—
275C22×C17—81no—
276C22×C17C2×C4×D411yescharacteristic
277C22×C17#53611yescharacteristic
278C22×C17#53711yescharacteristic
279C22×C17#57611yescharacteristic
280C22×C17#57611yescharacteristic
281C22×C17#57711yescharacteristic
282C22×C17#57811yescharacteristic
283C4×C17—22no—
284C4×C17—22no—
285C4×C17—42no—
286C4×C17—41no—
287C4×C17—41no—
288C4×C17—41no—
289C4×C17—41no—
290C4×C17—81no—
291C4×C17—81no—
292C4×C17#51911yescharacteristic
293C4×C17#52011yescharacteristic
294C4×C17#54111yescharacteristic
295C4×C17#54211yescharacteristic
296Dic17—42no—
297Dic17—42no—
298Dic17—81no—
299Dic17—81no—
300C2×#125—171no—
301C2×C4×D4C2×D1711yescharacteristic
302#478—178no—
303C2×#107—171no—
304C2×#107—171no—
305C2×#107—171no—
306C2×#106—171no—
307C2×C4⋊C8—171no—
308C2×C4⋊C8—171no—
309C2×#102—171no—
310C2×#102—171no—
311C2×#102—171no—
312C2×#102—171no—
313C2×#97—171no—
314C2×C4×C8—171no—
315#390C2×D1711yescharacteristic
316#390C2×D1711yescharacteristic
317#389—171no—
318#389—171no—
319#389—341no—
320#389—341no—
321#389C2×D1711yescharacteristic
322#389C2×D1711yescharacteristic
323#388—21no—
324#388—21no—
325#387—341no—
326#387—341no—
327#384—171no—
328#381—21no—
329#380—21no—
330C4×C4⋊C4—171no—
331C4×C4⋊C4—171no—
332C4×C4⋊C4C2×D1711yescharacteristic
333#376C2×D1711yescharacteristic
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#336—171no—
343#336—171no—
344C2×C17—82no—
345C2×C17#232311yescharacteristic
346C2×C17#294611yescharacteristic
347C2×C17#295111yescharacteristic
348C2×C17#324111yescharacteristic
349C2×C17#324311yescharacteristic
350C2×C17#325811yescharacteristic
351C2×C17#325911yescharacteristic
352C22×D4C22×D1711yescharacteristic
353C23×C4—21no—
354C22×C8—171no—
355C22×C8—171no—
356C22×C8—171no—
357C22×C8—171no—
358#125—174no—
359#125—342no—
360C4×D4—22no—
361C4×D4C4×D1714yes—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—171no—
365C2×C4⋊C4—171no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—341no—
369C2×C4⋊C4—341no—
370C2×C4⋊C4—341no—
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×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C42—21no—
383C2×C42—341no—
384C2×C42—341no—
385C2×C42C22×D1711yescharacteristic
386C2×C42C22×D1711yescharacteristic
387C2×C42C22×D1711yescharacteristic
388#107—174no—
389#107—681no—
390#107—681no—
391#106—174no—
392C4⋊C8—174no—
393C4⋊C8—342no—
394#102—174no—
395#102—174no—
396#102—681no—
397#102—681no—
398#97—681no—
399C4×C8—681no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—341no—
404#95—341no—
405#95—341no—
406#95—341no—
407#95C22⋊D1711yescharacteristic
408#95C22⋊D1711yescharacteristic
409#95C22⋊D1711yescharacteristic
410#95C22⋊D1711yescharacteristic
411C17#2212611yescharacteristic, complemented, Sylow
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×C4×D1712yes—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—341no—
426C22×C4—341no—
427C22×C4—341no—
428C22×C4—341no—
429C22×C4C2×C22⋊D1711yescharacteristic
430C22×C4C2×C22⋊D1711yescharacteristic
431C22×C4C2×C22⋊D1711yescharacteristic
432C22×C4C2×C22⋊D1711yescharacteristic
433C22×C4C2×C22⋊D1711yescharacteristic
434C22×C4C2×C22⋊D1711yescharacteristic
435C22×C4C23×D1711yescharacteristic
436C2×C8—172no—
437C2×C8—172no—
438C2×C8—341no—
439C2×C8—341no—
440C2×C8—341no—
441C2×C8—341no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8—341no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—172no—
456C4⋊C4—172no—
457C4⋊C4—341no—
458C4⋊C4—341no—
459C4⋊C4—344no—
460C4⋊C4—344no—
461C4⋊C4—341no—
462C4⋊C4—341no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—1361no—
466C4⋊C4—1361no—
467C4⋊C4—1361no—
468C4⋊C4—1361no—
469C4⋊C4C2×C4×D1712yes—
470C22⋊C4—24no—
471C22⋊C4—81no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—81no—
480C42—1361no—
481C42—1361no—
482C42C2×C4×D1712yes—
483C42#6325212yes—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C22×C22⋊D1711yescharacteristic
492D4—22no—
493D4—22no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
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—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
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—84no—
522C2×C4—342no—
523C2×C4—342no—
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—681no—
534C2×C4C4×C22⋊D1712yes—
535C2×C4C4×C22⋊D1712yes—
536C2×C4C22×C4×D1711yescharacteristic
537C2×C4C2×#6325211yescharacteristic
538C2×C4#1058769611yescharacteristic
539C2×C4#1058769711yescharacteristic
540C2×C4#1058769711yescharacteristic
541C2×C4#1058769711yescharacteristic
542C8—342no—
543C8—342no—
544C8—681no—
545C8—681no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×C4×C22⋊D1711yescharacteristic
554C22#1176338011yescharacteristic
555C22#1176338711yescharacteristic
556C22#1176361211yescharacteristic
557C22#1176361211yescharacteristic
558C22#1176362011yescharacteristic
559C22#1176362411yescharacteristic
560C4—22no—
561C4—22no—
562C4—41no—
563C4—42no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—682no—
570C4—682no—
571C4—1361no—
572C4—1361no—
573C4#1176328611yescharacteristic
574C4#1176328911yescharacteristic
575C4#1176342111yescharacteristic
576C4#1176343011yescharacteristic
577C2—82no—
578C2#1453146611yescharacteristic
579C2#1453761611yescharacteristic
580C2#1453765711yescharacteristic
581C2#1454003111yescharacteristic
582C2#1454006111yescharacteristic
583C2#1454021011yescharacteristic
584C2#1454021911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2319C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#2319D17
hypocenterC17#22126
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
2. #14531525C2semidirect
3. #14529741C2semidirect
4. #14529542C2semidirect
5. #14529524C2semidirect
6. #14529514C2semidirect
11. #14529373C2semidirect
12. #14529359C2semidirect
14. #14528641C2semidirect
17. #11763109C4semidirect
411. C17#22126semidirect

Maximal subgroups

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

subgroupindex
C2×#117631092
#145315252
#145297412
#145295422
#145295242
#145295142
#145294842
#145294702
#145294012
#145293732
#145293592
#145287842
#145286412
C17×#23192
#2212617