Finite Group Databasebeta

← #17042097

Subgroups — #17042097

Order 4352.

876 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763096C211yesmaximal, characteristic
2#14531533C211yesmaximal, characteristic, complemented
3#14531144C211yesmaximal, characteristic
4#14529552C211yesmaximal, characteristic, complemented
5#14529372C211yesmaximal, characteristic, complemented
6#14529365C211yesmaximal, characteristic, complemented
7#14529046C211yesmaximal, characteristic
8#14529032C211yesmaximal, characteristic
9#14528795C211yesmaximal, characteristic
10#14528680C211yesmaximal, characteristic, complemented
11#14528672C211yesmaximal, characteristic, complemented
12#14528643C211yesmaximal, characteristic, complemented
13#14528608C211yesmaximal, characteristic
14#14528598C211yesmaximal, characteristic
15C17×#1988C211yesmaximal, characteristic, complemented
16C2×C4×D68C2211yescharacteristic
17C2×#10587579C2211yescharacteristic
18C2×#10587579C2211yescharacteristic
19#11763096C418yescomplemented
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×#10587492C2211yescharacteristic
26C2×#10587492C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×#10587486C2211yescharacteristic
30C2×C8×Dic17C2211yescharacteristic
31#11762716—21no—
32#11762716—21no—
33#11762708C2211yescharacteristic
34#11762708C2211yescharacteristic
35#11762695—21no—
36#11762695—21no—
37#11762675C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42C4×#63227—21no—
43C4×#63227—21no—
44#11762656—21no—
45#11762655C2211yescharacteristic
46#95×D17—21no—
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53C2×C17×C4⋊Q8C2211yescharacteristic
54C2×C17×C4⋊C8C2211yescharacteristic
55C4×C4⋊C4×C17C2211yescharacteristic
56C17×#340C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C22×Q8×C17—21no—
61C22×D68C2311yescharacteristic
62C22×C4×D17—21no—
63C22×C4×D17—21no—
64C22×C4×D17—21no—
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C17×C4⋊Q8—22no—
68C17×C4⋊Q8C2×C414yescomplemented
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C42×C17—21no—
75C2×C42×C17C2311yescharacteristic
76C2×#63227—21no—
77C2×#63227—21no—
78C2×#63227—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4×Dic17—21no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17C2311yescharacteristic
89C2×C4×Dic17C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92#10587579—22no—
93#10587579—22no—
94#10587579—22no—
95#10587579Q814yescomplemented
96C4×D68—22no—
97C4×D68C2×C414yes—
98C17×C4⋊C8—22no—
99C17×#95—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#10587502D411yescharacteristic
109#10587502D411yescharacteristic
110#10587502D411yescharacteristic
111#10587502D411yescharacteristic
112#10587492—22no—
113#10587492—22no—
114#10587486—22no—
115C8×Dic17—22no—
116#10587480C2×C414yes—
117#10587480C2×C414yes—
118#10587479—22no—
119#10587479C2×C414yescomplemented
120#10587479C2×C414yescomplemented
121#10587478—22no—
122C4⋊#3622C2×C414yescomplemented
123C23×D17—21no—
124C2×Q8×C17—24no—
125C2×Q8×C17—81no—
126C2×Q8×C17—81no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2411yescharacteristic
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17C2×D411yescharacteristic
138C22×Dic17C2×D411yescharacteristic
139C22×Dic17C2×D411yescharacteristic
140C22×Dic17C2×D411yescharacteristic
141C2×D68—21no—
142C2×D68—21no—
143C2×D68—21no—
144C2×D68C22×C412yes—
145C2×D68C2×Q812yes—
146C2×D68#4112yes—
147C2×C4×D17—24no—
148C2×C4×D17—24no—
149C2×C4×D17—24no—
150C2×C4×D17—81no—
151C2×C4×D17—81no—
152C2×C4×D17—81no—
153C2×C4×D17—81no—
154C2×C4×D17—81no—
155C2×C4×D17—81no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C4⋊C4×C17—24no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17C22×C412yes—
168C4⋊C4×C17C22×C412yes—
169C4⋊C4×C17C2×Q812yes—
170C4⋊C4×C17#4112yes—
171C42×C17—81no—
172C42×C17C22×C412yes—
173C42×C17#4112yes—
174#63227—24no—
175#63227—24no—
176#63227—24no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—81no—
183C4⋊Dic17—81no—
184C4⋊Dic17C22×C412yes—
185C4×Dic17—21no—
186C4×Dic17—21no—
187C4×Dic17—81no—
188C4×Dic17—81no—
189C4×Dic17—81no—
190C4×Dic17—81no—
191C4×Dic17C2×Q812yes—
192C4×Dic17#4112yes—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622C22×C412yes—
198C2×#3622C22×C412yes—
199#21623—171nomaximal, Sylow
200C23×C17C22×D411yescharacteristic
201C22×D17—22no—
202C22×D17—22no—
203C22×D17—22no—
204C22×D17—41no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17—41no—
208Q8×C17—84no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—22no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17C4×D412yes—
226C2×C4×C17C4×D412yes—
227C2×C4×C17C23×C411yescharacteristic
228C2×C4×C17C22×Q811yescharacteristic
229C2×C4×C17C2×#4111yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—22no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
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×Dic17C22⋊Q812yes—
253C2×Dic17C22⋊Q812yes—
254D68—22no—
255D68—22no—
256D68—22no—
257D68C4×Q814yes—
258C4×D17—84no—
259C4×D17—84no—
260C4×D17—84no—
261C8×C17—41no—
262C8×C17—41no—
263#3622—22no—
264#3622—22no—
265C2×#476—171no—
266#2331—171no—
267#2275—171no—
268#1988D1711yescharacteristic, complemented
269#1949—171no—
270#1920—171no—
271#1919—171no—
272#1864—171no—
273#1862—171no—
274#1804—171no—
275#1776—171no—
276C4×#106—171no—
277#1768—171no—
278#1762—171no—
279C4×#102—171no—
280C22×C17C2×C4×D411yescharacteristic
281C22×C17C2×C22⋊Q811yescharacteristic
282C22×C17#52511yescharacteristic
283C22×C17#57611yescharacteristic
284C22×C17#57711yescharacteristic
285C22×C17#57711yescharacteristic
286C22×C17#57711yescharacteristic
287C2×D17—24no—
288C2×D17—42no—
289C2×D17—42no—
290C2×D17—42no—
291C2×D17—42no—
292C2×D17—42no—
293C2×D17—42no—
294C4×C17—22no—
295C4×C17—22no—
296C4×C17—42no—
297C4×C17—42no—
298C4×C17—42no—
299C4×C17—42no—
300C4×C17C2×C4×Q811yescharacteristic
301C4×C17C4×#4111yescharacteristic
302C4×C17#52911yescharacteristic
303C4×C17#53311yescharacteristic
304Dic17—22no—
305Dic17—22no—
306Dic17—41no—
307Dic17—41no—
308Dic17—42no—
309Dic17—81no—
310Dic17—81no—
311C2×C4⋊Q8C2×D1711yescharacteristic
312C2×C4×D4—171no—
313C2×C4×D4—171no—
314C2×C4×D4—171no—
315#476—178no—
316C2×#107—171no—
317C2×#106—171no—
318C2×#106—171no—
319C2×#106—171no—
320C2×C4⋊C8—171no—
321C2×C4⋊C8C2×D1711yescharacteristic
322C2×#102—171no—
323C2×#102—171no—
324C2×#102—171no—
325C2×#102—171no—
326C2×#97—171no—
327C2×C4×C8—171no—
328#391—341no—
329#391—341no—
330#389—171no—
331#389—171no—
332#387—341no—
333#387—341no—
334#382—171no—
335C4×C4⋊C4—171no—
336C4×C4⋊C4—171no—
337C4×C4⋊C4—171no—
338C4×C4⋊C4—171no—
339C4×C4⋊C4C2×D1711yescharacteristic
340C4×C22⋊C4—341no—
341C4×C22⋊C4—341no—
342C4×C22⋊C4—341no—
343#376—171no—
344C2×#95—341no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#340C2×D1711yescharacteristic
350#340C2×D1711yescharacteristic
351#340C2×D1711yescharacteristic
352#340C2×D1711yescharacteristic
353#336—171no—
354#336—171no—
355C2×C17C4×C22⋊Q811yescharacteristic
356C2×C17#294611yescharacteristic
357C2×C17#294911yescharacteristic
358C2×C17#308111yescharacteristic
359C2×C17#308411yescharacteristic
360C2×C17#311811yescharacteristic
361C2×C17#312511yescharacteristic
362D17—44no—
363C22×Q8—21no—
364C22×D4—171no—
365C23×C4—341no—
366C23×C4—341no—
367C23×C4—341no—
368C22×C8—171no—
369C22×C8—171no—
370C22×C8C22×D1711yescharacteristic
371C22×C8C22×D1711yescharacteristic
372C4⋊Q8—22no—
373C4⋊Q8C4×D1714yescomplemented
374C4×D4—174no—
375C4×D4—174no—
376C4×D4—342no—
377C4×D4—342no—
378C4×D4—342no—
379C4×D4—342no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C4⋊C4C22×D1711yescharacteristic
388C2×C4⋊C4C22×D1711yescharacteristic
389C2×C4⋊C4C22×D1711yescharacteristic
390C2×C22⋊C4—341no—
391C2×C22⋊C4—341no—
392C2×C22⋊C4—341no—
393C2×C42—21no—
394C2×C42—171no—
395C2×C42—171no—
396C2×C42—341no—
397C2×C42—341no—
398C2×C42—341no—
399C2×C42—341no—
400C2×C42C22×D1711yescharacteristic
401#107—342no—
402#106—174no—
403#106—174no—
404#106—342no—
405C4⋊C8—22no—
406C4⋊C8—174no—
407#102—174no—
408#102—174no—
409#102—342no—
410#102—342no—
411#97—342no—
412C4×C8—342no—
413#95—21no—
414#95—171no—
415#95—171no—
416#95—171no—
417#95—171no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426C17#2162311yescharacteristic, complemented, Sylow
427C24—341no—
428C2×Q8—24no—
429C2×Q8—81no—
430C2×Q8—81no—
431C2×D4—172no—
432C2×D4—172no—
433C2×D4—172no—
434C2×D4—341no—
435C2×D4—341no—
436C2×D4—341no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—171no—
442C22×C4—171no—
443C22×C4—171no—
444C22×C4—171no—
445C22×C4—344no—
446C22×C4—344no—
447C22×C4—341no—
448C22×C4—344no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—1361no—
452C22×C4—1361no—
453C22×C4—1361no—
454C22×C4—1361no—
455C22×C4—1361no—
456C22×C4—1361no—
457C22×C4D4×D1711yescharacteristic
458C22×C4D4×D1711yescharacteristic
459C22×C4C23×D1711yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—172no—
467C2×C8—172no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—24no—
477C4⋊C4—172no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—1361no—
484C4⋊C4—1361no—
485C4⋊C4C2×C4×D1712yes—
486C4⋊C4C2×C4×D1712yes—
487C4⋊C4Q8×D1712yes—
488C4⋊C4#6325612yes—
489C22⋊C4—344no—
490C22⋊C4—344no—
491C22⋊C4—344no—
492C42—81no—
493C42—172no—
494C42—172no—
495C42—341no—
496C42—341no—
497C42—1361no—
498C42—1361no—
499C42—1361no—
500C42—1361no—
501C42C2×C4×D1712yes—
502C42#6325412yes—
503C23—342no—
504C23—342no—
505C23—342no—
506C23—681no—
507C23—681no—
508C23—681no—
509C23—681no—
510C23C2×D4×D1711yescharacteristic
511Q8—84no—
512D4—174no—
513D4—342no—
514D4—342no—
515D4—342no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—172no—
533C2×C4—172no—
534C2×C4—341no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—341no—
538C2×C4—342no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—1364no—
554C2×C4—1364no—
555C2×C4—1364no—
556C2×C4#1058756712yes—
557C2×C4#1058756712yes—
558C2×C4C22×C4×D1711yescharacteristic
559C2×C4C2×#6325411yescharacteristic
560C2×C4C2×Q8×D1711yescharacteristic
561C2×C4C2×#6325611yescharacteristic
562C2×C4#41×D1711yescharacteristic
563C2×C4#41×D1711yescharacteristic
564C8—41no—
565C8—41no—
566C8—342no—
567C8—342no—
568C22—344no—
569C22—682no—
570C22—682no—
571C22—682no—
572C22—682no—
573C22—682no—
574C22—682no—
575C22C2×#1058756711yescharacteristic
576C22C22⋊Q8×D1711yescharacteristic
577C22#1176332211yescharacteristic
578C22#1176361011yescharacteristic
579C22#1176361511yescharacteristic
580C22#1176361811yescharacteristic
581C22#1176361811yescharacteristic
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—342no—
589C4—342no—
590C4—681no—
591C4—682no—
592C4—681no—
593C4—1361no—
594C4—1361no—
595C4C4×Q8×D1711yescharacteristic
596C4C4×#6325611yescharacteristic
597C4#1176333811yescharacteristic
598C4#1176336411yescharacteristic
599C2—684no—
600C2#1453132211yescharacteristic
601C2#1453761011yescharacteristic
602C2#1453763911yescharacteristic
603C2#1453860811yescharacteristic
604C2#1453864311yescharacteristic
605C2#1453897511yescharacteristic
606C2#1453906811yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1988C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1988D17
hypocenterC17#21623
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
2. #14531533C2semidirect
4. #14529552C2semidirect
5. #14529372C2semidirect
6. #14529365C2semidirect
10. #14528680C2semidirect
11. #14528672C2semidirect
12. #14528643C2semidirect
15. C17×#1988C2semidirect
19. #11763096C4semidirect
68. C17×C4⋊Q8C2×C4semidirect
95. #10587579Q8semidirect
119. #10587479C2×C4semidirect
120. #10587479C2×C4semidirect
122. C4⋊#3622C2×C4semidirect
268. #1988D17semidirect
373. C4⋊Q8C4×D17semidirect
426. C17#21623semidirect

Maximal subgroups

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

subgroupindex
C2×#117630962
#145315332
#145311442
#145295522
#145293722
#145293652
#145290462
#145290322
#145287952
#145286802
#145286722
#145286432
#145286082
#145285982
C17×#19882
#2162317