Finite Group Databasebeta

← #17053499

Subgroups — #17053499

Order 4352.

858 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic17C211yesmaximal, characteristic
2#14531748C211yesmaximal, characteristic, complemented
3#14531689C211yesmaximal, characteristic
4#14529477C211yesmaximal, characteristic, complemented
5#14529471C211yesmaximal, characteristic, complemented
6#14529363C211yesmaximal, characteristic, complemented
7#14529361C211yesmaximal, characteristic, complemented
8#14529120C211yesmaximal, characteristic
9#14528913C211yesmaximal, characteristic, complemented
10#14528797C211yesmaximal, characteristic, complemented
11#14528792C211yesmaximal, characteristic
12#14528650C211yesmaximal, characteristic, complemented
13#14528611C211yesmaximal, characteristic
14#14528609C211yesmaximal, characteristic
15C17×#1877C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C2×D4×Dic17C2211yescharacteristic
18C2×D4×Dic17C2211yescharacteristic
19SD16×Dic17C418yescomplemented
20C2×#10587488C2211yescharacteristic
21C2×#10587485C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587474C2211yescharacteristic
27C2×#10587486C2211yescharacteristic
28C2×C8×Dic17C2211yescharacteristic
29#11762716C2211yescharacteristic
30#11762707C2211yescharacteristic
31#11762705C2211yescharacteristic
32#11762694—21no—
33#11762692—21no—
34#11762682—21no—
35#11762681—21no—
36#11762676—21no—
37#11762674C2211yescharacteristic
38#11762674C2211yescharacteristic
39#11762667—21no—
40C4⋊C4×Dic17C2211yescharacteristic
41C4×#63225C2211yescharacteristic
42C4×C4⋊Dic17C2211yescharacteristic
43#11762654—21no—
44C2×#10587502—21no—
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762563C2211yescharacteristic
53C22×SD16×C17C2211yescharacteristic
54C2×C4×D4×C17C2211yescharacteristic
55C2×C17×#103C2211yescharacteristic
56C2×C17×#102C2211yescharacteristic
57C17×#391C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×C4×C17—21no—
63C23×Dic17—21no—
64C23×Dic17—21no—
65C2×SD16×C17—22no—
66C2×SD16×C17—22no—
67C2×SD16×C17C2×C414yes—
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C4×D4×C17—22no—
71C4×D4×C17—41no—
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C22⋊C4×C17—21no—
75C2×C42×C17C2311yescharacteristic
76C2×C22⋊Dic17—21no—
77C2×C22⋊Dic17—21no—
78Q8×Dic17—22no—
79Q8×Dic17C2×C414yescomplemented
80D4×Dic17—22no—
81D4×Dic17—22no—
82D4×Dic17—41no—
83D4×Dic17C2×C414yes—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×#63225—21no—
90C2×#63225—21no—
91C2×#63225D411yescharacteristic
92C2×#63225D411yescharacteristic
93C2×C4×Dic17—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17C2311yescharacteristic
97C2×C4×Dic17C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C17×#103—41no—
101C17×#102—22no—
102C17×#95—21no—
103C17×#95—21no—
104#10587506—41no—
105#10587506C2×C414yescomplemented
106#10587503—22no—
107#10587503C2×C414yes—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502D411yescharacteristic
119#10587502D411yescharacteristic
120#10587488C2×C414yescomplemented
121#10587486—41no—
122#10587485—22no—
123C8×Dic17C2×C414yescomplemented
124#10587474—41no—
125C24×C17—21no—
126C2×Q8×C17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17C22×C412yes—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17C22×C412yes—
136C2×D4×C17#4112yes—
137C2×D4×C17#4112yes—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—42no—
142C22×C4×C17—42no—
143C22×C4×C17—81no—
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2411yescharacteristic
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—24no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—42no—
153C22×Dic17—42no—
154C22×Dic17—81no—
155C22×Dic17—81no—
156C22×Dic17—81no—
157C22×Dic17C2×D411yescharacteristic
158C22×Dic17C2×D411yescharacteristic
159C22×Dic17C2×D411yescharacteristic
160C22×Dic17C2×D411yescharacteristic
161SD16×C17—28no—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17C22×C412yes—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C22⋊C4×C17—42no—
173C42×C17—21no—
174C42×C17—21no—
175C22⋊Dic17—24no—
176C22⋊Dic17—42no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—24no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17C22×C412yes—
185C4⋊Dic17C22×C412yes—
186#63225—41no—
187#63225—41no—
188#63225—81no—
189#63225—81no—
190C4×Dic17—21no—
191C4×Dic17—21no—
192C4×Dic17—24no—
193C4×Dic17—42no—
194C4×Dic17—42no—
195C4×Dic17C22×C412yes—
196C4×Dic17#4112yes—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622C22×C412yes—
203#22760—171nomaximal, Sylow
204C23×C17—22no—
205C23×C17—22no—
206C23×C17—22no—
207C23×C17—41no—
208C23×C17—41no—
209C23×C17—41no—
210C23×C17—41no—
211C23×C17C22×D411yescharacteristic
212Q8×C17—22no—
213Q8×C17—22no—
214Q8×C17—41no—
215Q8×C17—41no—
216D4×C17—22no—
217D4×C17—22no—
218D4×C17—22no—
219D4×C17#11714yes—
220C2×C4×C17—22no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
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×C17—84no—
236C2×C4×C17C23×C411yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17H(5,𝔽2)11yescharacteristic
241C2×C4×C172−1+411yescharacteristic
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—22no—
247C2×Dic17—22no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—84no—
265C2×Dic17—84no—
266C2×Dic17C4×D412yes—
267C2×Dic17C4×D412yes—
268C8×C17—22no—
269C8×C17—41no—
270#3622—22no—
271#3622—41no—
272C2×C4×SD16—171no—
273#2356—171no—
274#2349—171no—
275#1935—171no—
276#1935—171no—
277#1919—171no—
278#1919—171no—
279#1877—171no—
280#1877D1711yescharacteristic
281#1835—171no—
282#1805—171no—
283#1804—171no—
284#1770—171no—
285#1762—171no—
286#1762—171no—
287C22×C17—24no—
288C22×C17—42no—
289C22×C17—42no—
290C22×C17—42no—
291C22×C17—42no—
292C22×C17—42no—
293C22×C17—42no—
294C22×C17C2×C4×D411yescharacteristic
295C22×C17#54611yescharacteristic
296C22×C17#54811yescharacteristic
297C22×C17#57511yescharacteristic
298C22×C17#57511yescharacteristic
299C22×C17#57711yescharacteristic
300C22×C17#57711yescharacteristic
301C4×C17—42no—
302C4×C17—41no—
303C4×C17—41no—
304C4×C17—81no—
305C4×C17—81no—
306C4×C17C2×#11711yescharacteristic
307C4×C17#52011yescharacteristic
308C4×C17#55211yescharacteristic
309C4×C17#55311yescharacteristic
310Dic17—22no—
311Dic17—22no—
312Dic17—42no—
313Dic17—42no—
314Dic17—41no—
315Dic17—41no—
316Dic17—81no—
317Dic17—81no—
318C22×SD16C2×D1711yescharacteristic
319C2×C4×Q8—171no—
320C2×C4×D4—171no—
321C2×C4×D4—171no—
322C2×C4×D4C2×D1711yescharacteristic
323C4×SD16—178no—
324C2×#106—171no—
325C2×C4⋊C8—171no—
326C2×#103—171no—
327C2×#103—171no—
328C2×#103C2×D1711yescharacteristic
329C2×#102—171no—
330C2×#102—171no—
331C2×#102C2×D1711yescharacteristic
332C2×#97—171no—
333C2×#97—171no—
334C2×C4×C8—171no—
335#391—171no—
336#391C2×D1711yescharacteristic
337#389—171no—
338#389—171no—
339#387—341no—
340#387—341no—
341#385—341no—
342#385—341no—
343#383—341no—
344#382—171no—
345#382—171no—
346#380—341no—
347C4×C4⋊C4—171no—
348C4×C4⋊C4—171no—
349C4×C4⋊C4—171no—
350#376—341no—
351C2×#95—341no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#340—171no—
357#340—171no—
358#340—171no—
359#340C2×D1711yescharacteristic
360#336—171no—
361#336C2×D1711yescharacteristic
362C2×C17—44no—
363C2×C17#235211yescharacteristic
364C2×C17#294311yescharacteristic
365C2×C17#294911yescharacteristic
366C2×C17#332311yescharacteristic
367C2×C17#332311yescharacteristic
368C2×C17#337611yescharacteristic
369C2×C17#337611yescharacteristic
370C22×Q8C22×D1711yescharacteristic
371C22×D4C22×D1711yescharacteristic
372C23×C4—21no—
373C23×C4—341no—
374C23×C4—341no—
375C2×SD16—22no—
376C2×SD16—22no—
377C2×SD16C4×D1714yes—
378C22×C8—171no—
379C22×C8—171no—
380C22×C8C22×D1711yescharacteristic
381C22×C8C22×D1711yescharacteristic
382C4×Q8—174no—
383C4×Q8—342no—
384C4×D4—22no—
385C4×D4—41no—
386C4×D4—174no—
387C4×D4—342no—
388C4×D4—342no—
389C4×D4—681no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—171no—
393C2×C4⋊C4—171no—
394C2×C4⋊C4—171no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—341no—
397C2×C4⋊C4—341no—
398C2×C4⋊C4—341no—
399C2×C4⋊C4C22×D1711yescharacteristic
400C2×C4⋊C4C22×D1711yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—341no—
403C2×C22⋊C4—341no—
404C2×C42—171no—
405C2×C42—171no—
406C2×C42—341no—
407C2×C42—341no—
408C2×C42—341no—
409C2×C42C22×D1711yescharacteristic
410#106—174no—
411C4⋊C8—342no—
412#103—41no—
413#103—174no—
414#103—681no—
415#102—22no—
416#102—174no—
417#102—342no—
418#97—681no—
419#97—681no—
420C4×C8—174no—
421#95—21no—
422#95—21no—
423#95—171no—
424#95—171no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429#95—341no—
430#95—341no—
431#95—341no—
432#95—341no—
433#95—341no—
434#95—341no—
435C17#2276011yescharacteristic, complemented, Sylow
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D1712yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4C2×C4×D1712yes—
447C2×D4#6325412yes—
448C2×D4#6325412yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—171no—
456C22×C4—171no—
457C22×C4—171no—
458C22×C4—171no—
459C22×C4—344no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—341no—
464C22×C4—682no—
465C22×C4—682no—
466C22×C4—1361no—
467C22×C4—1361no—
468C22×C4—1361no—
469C22×C4D4×D1711yescharacteristic
470C22×C4D4×D1711yescharacteristic
471C22×C4C23×D1711yescharacteristic
472SD16—28no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—172no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8C2×C4×D1712yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—172no—
490C4⋊C4—172no—
491C4⋊C4—344no—
492C4⋊C4—341no—
493C4⋊C4—341no—
494C4⋊C4—341no—
495C4⋊C4—341no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—681no—
499C4⋊C4—681no—
500C4⋊C4—1361no—
501C4⋊C4—1361no—
502C22⋊C4—42no—
503C22⋊C4—344no—
504C22⋊C4—682no—
505C42—21no—
506C42—21no—
507C42—172no—
508C42—172no—
509C42—344no—
510C42—341no—
511C42—341no—
512C42—682no—
513C42—682no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D1711yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4#1058756314yes—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—84no—
546C2×C4—172no—
547C2×C4—172no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—342no—
553C2×C4—342no—
554C2×C4—341no—
555C2×C4—341no—
556C2×C4—341no—
557C2×C4—341no—
558C2×C4—341no—
559C2×C4—341no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—681no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—1364no—
571C2×C4—1364no—
572C2×C4C22×C4×D1711yescharacteristic
573C2×C4C2×#6325411yescharacteristic
574C2×C4C2×#6325411yescharacteristic
575C2×C4#1058769111yescharacteristic
576C2×C4#41×D1711yescharacteristic
577C2×C4#1058769711yescharacteristic
578C8—22no—
579C8—41no—
580C8—342no—
581C8—681no—
582C22—24no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22C4×D4×D1711yescharacteristic
590C22#1176344611yescharacteristic
591C22#1176346511yescharacteristic
592C22#1176360511yescharacteristic
593C22#1176360811yescharacteristic
594C22#1176361511yescharacteristic
595C22#1176361611yescharacteristic
596C4—42no—
597C4—41no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—342no—
602C4—342no—
603C4—682no—
604C4—682no—
605C4—681no—
606C4—681no—
607C4—1361no—
608C4—1361no—
609C4C2×#1058756311yescharacteristic
610C4#1176328811yescharacteristic
611C4#1176349611yescharacteristic
612C4#1176350211yescharacteristic
613C2—44no—
614C2#1453171111yescharacteristic
615C2#1453757911yescharacteristic
616C2#1453763811yescharacteristic
617C2#1454099311yescharacteristic
618C2#1454100411yescharacteristic
619C2#1454160411yescharacteristic
620C2#1454160811yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1877C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1877D17
hypocenterC17#22760
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
2. #14531748C2semidirect
4. #14529477C2semidirect
5. #14529471C2semidirect
6. #14529363C2semidirect
7. #14529361C2semidirect
9. #14528913C2semidirect
10. #14528797C2semidirect
12. #14528650C2semidirect
19. SD16×Dic17C4semidirect
79. Q8×Dic17C2×C4semidirect
105. #10587506C2×C4semidirect
120. #10587488C2×C4semidirect
123. C8×Dic17C2×C4semidirect
435. C17#22760semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic172
#145317482
#145316892
#145294772
#145294712
#145293632
#145293612
#145291202
#145289132
#145287972
#145287922
#145286502
#145286112
#145286092
C17×#18772
#2276017