Finite Group Databasebeta

← #17042088

Subgroups — #17042088

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#14531146C211yesmaximal, characteristic
3#14529737C211yesmaximal, characteristic, complemented
4#14529550C211yesmaximal, characteristic, complemented
5#14529366C211yesmaximal, characteristic, complemented
6#14529359C211yesmaximal, characteristic, complemented
7#14529044C211yesmaximal, characteristic
8#14529030C211yesmaximal, characteristic
9#14528791C211yesmaximal, characteristic
10#14528682C211yesmaximal, characteristic, complemented
11C4×#10587479C211yesmaximal, characteristic, complemented
12#14528641C211yesmaximal, characteristic, complemented
13#14528612C211yesmaximal, characteristic
14C4×#10587480C211yesmaximal, characteristic
15C17×#2331C211yesmaximal, characteristic, complemented
16C2×C4×D68C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19#11763096C418yescomplemented
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×C4⋊#3622C2211yescharacteristic
26C2×#10587480C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×#10587480C2211yescharacteristic
30C2×#10587474C2211yescharacteristic
31C2×C4×#3622C2211yescharacteristic
32C4×C4⋊Dic17C2211yescharacteristic
33C4×C4⋊Dic17C2211yescharacteristic
34C4×C4⋊Dic17C2211yescharacteristic
35#11762656—21no—
36#11762656—21no—
37#11762656—21no—
38#95×D17—21no—
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C2×C17×C4⋊Q8C2211yescharacteristic
50C17×#391—21no—
51C17×#391—21no—
52C17×#389C2211yescharacteristic
53C17×#389C2211yescharacteristic
54C17×#387—21no—
55C17×#387—21no—
56C17×#382C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C4⋊C4×C17C2211yescharacteristic
59C17×#376C2211yescharacteristic
60C22×Q8×C17—21no—
61C22×D68C2311yescharacteristic
62C22×C4×D17—21no—
63C22×C4×D17—21no—
64C22×C4×D17—21no—
65C17×C4⋊Q8—22no—
66C17×C4⋊Q8C2×C414yescomplemented
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—21no—
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×C17—21no—
76C2×C42×C17C2311yescharacteristic
77C2×C42×C17C2311yescharacteristic
78C2×C42×C17C2311yescharacteristic
79C2×#63227—21no—
80C2×#63227—21no—
81C2×#63227—21no—
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4×Dic17—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C22×#3622C2311yescharacteristic
89C22×#3622C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C4×D68—22no—
93C4×D68—22no—
94C4×D68—22no—
95C4×D68—22no—
96C4×D68C2×C414yes—
97C4×D68Q814yescomplemented
98C17×#95—21no—
99C17×#95—21no—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104C17×#95D411yescharacteristic
105C17×#95D411yescharacteristic
106C17×#95D411yescharacteristic
107C17×#95D411yescharacteristic
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587480—22no—
112#10587480—22no—
113#10587480C2×C414yes—
114#10587480C2×C414yes—
115#10587479—22no—
116#10587479C2×C414yescomplemented
117#10587479C2×C414yescomplemented
118#10587478—22no—
119C4⋊#3622—22no—
120C4⋊#3622C2×C414yescomplemented
121#10587474—22no—
122C4×#3622—22no—
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×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2411yescharacteristic
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—21no—
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—
156C4⋊C4×C17—24no—
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—81no—
164C4⋊C4×C17—81no—
165C4⋊C4×C17C22×C412yes—
166C4⋊C4×C17C22×C412yes—
167C42×C17—21no—
168C42×C17—21no—
169C42×C17—81no—
170C42×C17—81no—
171C42×C17C22×C412yes—
172C42×C17C2×Q812yes—
173C42×C17#4112yes—
174C42×C17#4112yes—
175#63227—24no—
176#63227—24no—
177#63227—24no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17C22×C412yes—
182C4⋊Dic17C2×Q812yes—
183C4⋊Dic17#4112yes—
184C4×Dic17—81no—
185C4×Dic17—81no—
186C4×Dic17—81no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
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—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17C4×D412yes—
236C2×C4×C17C4×D412yes—
237C2×C4×C17C22⋊Q812yes—
238C2×C4×C17C22⋊Q812yes—
239C2×C4×C17C23×C411yescharacteristic
240C2×C4×C17C22×Q811yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×C4×C17C2×#4111yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×Dic17—22no—
246C2×Dic17—22no—
247C2×Dic17—22no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254D68—22no—
255D68—22no—
256D68—22no—
257D68C4×Q814yes—
258C4×D17—84no—
259C4×D17—84no—
260C4×D17—84no—
261#3622—22no—
262#3622—22no—
263#3622—41no—
264#3622—41no—
265C2×#476—171no—
266#2331D1711yescharacteristic, complemented
267#2275—171no—
268#1988—171no—
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—22no—
297C4×C17—22no—
298C4×C17—41no—
299C4×C17—41no—
300C4×C17—42no—
301C4×C17—42no—
302C4×C17—81no—
303C4×C17—81no—
304C4×C17C2×C4×Q811yescharacteristic
305C4×C17C4×#4111yescharacteristic
306C4×C17#52911yescharacteristic
307C4×C17#53311yescharacteristic
308Dic17—42no—
309Dic17—42no—
310Dic17—42no—
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⋊C8—171no—
322C2×#102—171no—
323C2×#102—171no—
324C2×#102—171no—
325C2×#102—171no—
326C2×#97—171no—
327C2×C4×C8—171no—
328#391—21no—
329#391—21no—
330#389C2×D1711yescharacteristic
331#389C2×D1711yescharacteristic
332#387—21no—
333#387—21no—
334#382C2×D1711yescharacteristic
335C4×C4⋊C4—171no—
336C4×C4⋊C4—171no—
337C4×C4⋊C4—171no—
338C4×C4⋊C4C2×D1711yescharacteristic
339C4×C4⋊C4C2×D1711yescharacteristic
340C4×C22⋊C4—341no—
341C4×C22⋊C4—341no—
342C4×C22⋊C4—341no—
343#376C2×D1711yescharacteristic
344C2×#95—341no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340—171no—
351#340—171no—
352#340—171no—
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×C8—171no—
371C22×C8—171no—
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—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—171no—
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—21no—
395C2×C42—341no—
396C2×C42—341no—
397C2×C42—341no—
398C2×C42C22×D1711yescharacteristic
399C2×C42C22×D1711yescharacteristic
400C2×C42C22×D1711yescharacteristic
401#107—342no—
402#106—174no—
403#106—174no—
404#106—342no—
405C4⋊C8—174no—
406C4⋊C8—342no—
407#102—174no—
408#102—174no—
409#102—342no—
410#102—342no—
411#97—342no—
412C4×C8—342no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—341no—
420#95—341no—
421#95—341no—
422#95C22⋊D1711yescharacteristic
423#95C22⋊D1711yescharacteristic
424#95C22⋊D1711yescharacteristic
425#95C22⋊D1711yescharacteristic
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—344no—
442C22×C4—341no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—344no—
446C22×C4—344no—
447C22×C4—1361no—
448C22×C4—1361no—
449C22×C4—1361no—
450C22×C4—1361no—
451C22×C4—1361no—
452C22×C4—1361no—
453C22×C4C2×C22⋊D1711yescharacteristic
454C22×C4C2×C22⋊D1711yescharacteristic
455C22×C4C2×C22⋊D1711yescharacteristic
456C22×C4C2×C22⋊D1711yescharacteristic
457C22×C4C2×C22⋊D1711yescharacteristic
458C22×C4C2×C22⋊D1711yescharacteristic
459C22×C4C23×D1711yescharacteristic
460C2×C8—172no—
461C2×C8—172no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—172no—
482C4⋊C4—172no—
483C4⋊C4—172no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4C2×C4×D1712yes—
488C4⋊C4C2×C4×D1712yes—
489C22⋊C4—344no—
490C22⋊C4—344no—
491C22⋊C4—344no—
492C42—21no—
493C42—21no—
494C42—81no—
495C42—81no—
496C42—1361no—
497C42—1361no—
498C42—1361no—
499C42C2×C4×D1712yes—
500C42#6325212yes—
501C42Q8×D1712yes—
502C42#6325612yes—
503C23—342no—
504C23—342no—
505C23—342no—
506C23—681no—
507C23—681no—
508C23—681no—
509C23—681no—
510C23C22×C22⋊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—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—342no—
543C2×C4—342no—
544C2×C4—342no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—1364no—
552C2×C4—1364no—
553C2×C4—1364no—
554C2×C4C4×C22⋊D1712yes—
555C2×C4C4×C22⋊D1712yes—
556C2×C4#1058763212yes—
557C2×C4#1058763212yes—
558C2×C4C22×C4×D1711yescharacteristic
559C2×C4C2×#6325211yescharacteristic
560C2×C4C2×Q8×D1711yescharacteristic
561C2×C4C2×#6325611yescharacteristic
562C2×C4#41×D1711yescharacteristic
563C2×C4#41×D1711yescharacteristic
564C8—342no—
565C8—342no—
566C8—681no—
567C8—681no—
568C22—344no—
569C22—682no—
570C22—682no—
571C22—682no—
572C22—682no—
573C22—682no—
574C22—682no—
575C22C2×C4×C22⋊D1711yescharacteristic
576C22C2×#1058763211yescharacteristic
577C22#1176331611yescharacteristic
578C22#1176361211yescharacteristic
579C22#1176361711yescharacteristic
580C22#1176361711yescharacteristic
581C22#1176362011yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—41no—
587C4—42no—
588C4—42no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—682no—
593C4—682no—
594C4—682no—
595C4C4×Q8×D1711yescharacteristic
596C4C4×#6325611yescharacteristic
597C4#1176334611yescharacteristic
598C4#1176336311yescharacteristic
599C2—684no—
600C2C4×#1058763211yescharacteristic
601C2#1453761611yescharacteristic
602C2#1453764311yescharacteristic
603C2#1453860911yescharacteristic
604C2#1453864211yescharacteristic
605C2#1453896211yescharacteristic
606C2#1453905511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2331C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#2331D17
hypocenterC17#21623
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ #376 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #14529737C2semidirect
4. #14529550C2semidirect
5. #14529366C2semidirect
6. #14529359C2semidirect
10. #14528682C2semidirect
11. C4×#10587479C2semidirect
12. #14528641C2semidirect
15. C17×#2331C2semidirect
19. #11763096C4semidirect
66. C17×C4⋊Q8C2×C4semidirect
97. C4×D68Q8semidirect
116. #10587479C2×C4semidirect
117. #10587479C2×C4semidirect
120. C4⋊#3622C2×C4semidirect
266. #2331D17semidirect
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
#145311462
#145297372
#145295502
#145293662
#145293592
#145290442
#145290302
#145287912
#145286822
C4×#105874792
#145286412
#145286122
C4×#105874802
C17×#23312
#2162317