Finite Group Databasebeta

← #17035328

Subgroups — #17035328

Order 4352.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762849C211yesmaximal, characteristic, complemented
2C2×#11762810C211yesmaximal, characteristic
3C2×#11762805C211yesmaximal, characteristic
4#14531400C211yesmaximal, characteristic, complemented
5#14529052C212yesmaximal, complemented
6#14529045C212yesmaximal, complemented
7#14529038C212yesmaximal, complemented
8#14529031C212yesmaximal, complemented
9C2×#11762571C211yesmaximal, characteristic, complemented
10#14528481C211yesmaximal, characteristic, complemented
11C2×C17×#520C211yesmaximal, characteristic
12C22×#63223C2211yescharacteristic
13C23×#3622C2211yescharacteristic
14C2×#10587612C2211yescharacteristic
15C22×C4⋊Dic17C2211yescharacteristic
16#11762849—24no—
17C2×#10587478C2212yes—
18C2×#10587479C2212yes—
19#11762810C418yescomplemented
20#11762805—24no—
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2212yes—
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2212yes—
27#11762707C2212yes—
28#11762707C2212yes—
29#11762705—21no—
30#11762705—21no—
31#11762667—22no—
32#11762571C2212yes—
33#11762571C2212yes—
34#11762564C2212yes—
35#11762564C2212yes—
36C22×#41×C17C2211yescharacteristic
37C17×#520—24no—
38C17×#520—24no—
39C2×C4×Q8×C17C2212yes—
40C2×C4×D4×C17—22no—
41C2×C4×D4×C17C2212yes—
42C2×C17×#117—21no—
43C2×C17×#117C2211yescharacteristic
44C22×C4⋊C4×C17—21no—
45C22×C4⋊C4×C17C2211yescharacteristic
46C2×#41×C17—22no—
47C2×#41×C17—22no—
48C2×#41×C17—22no—
49C2×#41×C17C2×C414yes—
50C2×#41×C17D414yes—
51C2×#41×C17Q814yes—
52C22×Q8×C17C2311yescharacteristic
53C22×D4×C17—21no—
54C22×D4×C17C2311yescharacteristic
55C23×C4×C17—21no—
56C23×C4×C17—21no—
57C23×C4×C17—22no—
58C23×C4×C17C2311yescharacteristic
59C23×Dic17—21no—
60C4×Q8×C17—24no—
61C4×Q8×C17—24no—
62C4×D4×C17—216no—
63C4×D4×C17—24no—
64C4×D4×C17—24no—
65C17×#117—28no—
66C17×#117—22no—
67C17×#117—22no—
68C2×C4⋊C4×C17—28no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—22no—
71C2×C4⋊C4×C17—22no—
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17D414yes—
75C2×C4⋊C4×C17C2312yes—
76C2×C22⋊C4×C17—22no—
77C2×C22⋊C4×C17D412yes—
78C2×C42×C17—22no—
79C2×C42×C17C2312yes—
80C2×C22⋊Dic17—21no—
81#10587612—22no—
82#10587612—22no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—22no—
85C2×C4⋊Dic17C2×C414yescomplemented
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2312yes—
89C2×#63225—22no—
90C2×#63225—22no—
91C2×C4×Dic17—21no—
92C2×#63223—22no—
93C2×#63223—22no—
94C2×#63223C2×C414yescomplemented
95C22×#3622—22no—
96C22×#3622—22no—
97C22×#3622—22no—
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2312yes—
101#10587506—22no—
102#10587506—22no—
103#10587506C2×C418yescomplemented
104#10587503—22no—
105#10587503—22no—
106#10587503C2×C418yescomplemented
107#10587502—22no—
108#10587502—22no—
109#10587479—24no—
110#10587478—24no—
111C24×C17—21no—
112C24×C17C2×D411yescharacteristic
113#41×C17—28no—
114#41×C17—28no—
115#41×C17—28no—
116#41×C17C4⋊C4116yescomplemented
117C2×Q8×C17—21no—
118C2×Q8×C17—21no—
119C2×Q8×C17—21no—
120C2×Q8×C17C22×C412yes—
121C2×Q8×C17C2×D412yes—
122C2×Q8×C17C2×Q812yes—
123C2×D4×C17—24no—
124C2×D4×C17—24no—
125C2×D4×C17—24no—
126C2×D4×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17C22×C412yes—
130C2×D4×C17C2×D412yes—
131C2×D4×C17C2×Q812yes—
132C22×C4×C17—21no—
133C22×C4×C17—24no—
134C22×C4×C17—24no—
135C22×C4×C17—24no—
136C22×C4×C17—22no—
137C22×C4×C17—28no—
138C22×C4×C17—24no—
139C22×C4×C17—82no—
140C22×C4×C17—82no—
141C22×C4×C17—81no—
142C22×C4×C17—81no—
143C22×C4×C17C22×C412yes—
144C22×C4×C17C2×D412yes—
145C22×C4×C17C2×D412yes—
146C22×C4×C17C2×D412yes—
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D412yes—
149C22×C4×C17C2×Q812yes—
150C22×C4×C17#4112yes—
151C22×C4×C17#4112yes—
152C22×C4×C17C2411yescharacteristic
153C22×Dic17—22no—
154C22×Dic17—24no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—81no—
158C22×Dic17—81no—
159C4⋊C4×C17—28no—
160C4⋊C4×C17—22no—
161C4⋊C4×C17—24no—
162C4⋊C4×C17—22no—
163C4⋊C4×C17—22no—
164C4⋊C4×C17C2×D414yes—
165C22⋊C4×C17—28no—
166C22⋊C4×C17—24no—
167C42×C17—28no—
168C42×C17—22no—
169C42×C17—22no—
170C22⋊Dic17—42no—
171C4⋊Dic17—24no—
172C4⋊Dic17—22no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17C22×C414yes—
178#63225—82no—
179#63225—82no—
180C4×Dic17—42no—
181#63223—28no—
182C2×#3622—22no—
183C2×#3622—22no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—44no—
191C2×#3622C22×C414yes—
192#20878—171nomaximal, Sylow
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—22no—
198C23×C17—22no—
199C23×C17—22no—
200C23×C17—41no—
201C23×C17—41no—
202C23×C17—41no—
203C23×C17—41no—
204C23×C17C4×D412yes—
205C23×C17#12012yes—
206C23×C17C22⋊Q812yes—
207C23×C17C22×D411yescharacteristic
208Q8×C17—22no—
209Q8×C17—22no—
210Q8×C17—22no—
211Q8×C17C2×C4⋊C414yes—
212D4×C17—216no—
213D4×C17—22no—
214D4×C17—22no—
215D4×C17—22no—
216D4×C17C2×C4⋊C414yes—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—216no—
221C2×C4×C17—22no—
222C2×C4×C17—22no—
223C2×C4×C17—24no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—22no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—84no—
231C2×C4×C17—88no—
232C2×C4×C17C2×C4⋊C414yes—
233C2×C4×C17C4×D414yes—
234C2×C4×C17C4×D412yes—
235C2×C4×C17#12012yes—
236C2×C4×C17C22⋊Q814yes—
237C2×C4×C17C22⋊Q812yes—
238C2×C4×C17#12314yes—
239C2×C4×C17C23×C411yescharacteristic
240C2×C4×C17C22×D411yescharacteristic
241C2×C4×C17C22×D411yescharacteristic
242C2×C4×C17C22×Q811yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×Dic17—24no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—84no—
255#3622—24no—
256#3622—41no—
257#3622—41no—
258C2×#520D1711yescharacteristic
259C2×#428—171no—
260C2×#417—171no—
261C2×#416—171no—
262#2316—171no—
263#1865—172no—
264#1864—172no—
265#1863—172no—
266#1862—172no—
267C2×#340—171no—
268#1729—171no—
269C22×C17—22no—
270C22×C17—22no—
271C22×C17—22no—
272C22×C17—22no—
273C22×C17—22no—
274C22×C17—22no—
275C22×C17—24no—
276C22×C17—42no—
277C22×C17—42no—
278C22×C17—42no—
279C22×C17—42no—
280C22×C17—42no—
281C22×C17—42no—
282C22×C17#38514yes—
283C22×C17C2×C4×D411yescharacteristic
284C22×C17C2×#12011yescharacteristic
285C22×C17C2×C22⋊Q811yescharacteristic
286C22×C17#57611yescharacteristic
287C22×C17#57711yescharacteristic
288C22×C17#57711yescharacteristic
289C22×C17#57811yescharacteristic
290C4×C17—24no—
291C4×C17—42no—
292C4×C17—44no—
293C4×C17—42no—
294C4×C17#38514yes—
295C4×C17C22×C4⋊C411yescharacteristic
296C4×C17C2×C4×D411yescharacteristic
297C4×C17C2×C22⋊Q811yescharacteristic
298C4×C17C2×#12311yescharacteristic
299Dic17—44no—
300Dic17—81no—
301Dic17—81no—
302C22×#41C2×D1711yescharacteristic
303C22×M16—171no—
304C23×C8—171no—
305#520—24no—
306#520—24no—
307C2×C4×Q8C2×D1712yes—
308C2×C4×D4—22no—
309C2×C4×D4C2×D1712yes—
310C2×#117—21no—
311C2×#117—171no—
312C2×#117C2×D1711yescharacteristic
313C22×C4⋊C4—21no—
314C22×C4⋊C4—171no—
315C22×C4⋊C4C2×D1711yescharacteristic
316#428—344no—
317C2×#107—172no—
318C2×#106—172no—
319#417—178no—
320#416—344no—
321C2×#103—172no—
322C2×#103—171no—
323C2×#103—171no—
324C2×#102—172no—
325C2×#102—171no—
326C2×#102—171no—
327#389—172no—
328#389—172no—
329#389—341no—
330#389—341no—
331#380—342no—
332#340—172no—
333#340—172no—
334#336—172no—
335#336—172no—
336C2×C17—24no—
337C2×C17—44no—
338C2×C17C2×#38511yescharacteristic
339C2×C17#294611yescharacteristic
340C2×C17#294711yescharacteristic
341C2×C17#301211yescharacteristic
342C2×C17#301311yescharacteristic
343C2×C17#308311yescharacteristic
344C2×C17#308411yescharacteristic
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41Dic3414yes—
349C2×#41C4×D1714yes—
350C2×#41C22⋊D1714yes—
351C22×Q8C22×D1711yescharacteristic
352C22×D4—21no—
353C22×D4C22×D1711yescharacteristic
354C23×C4—21no—
355C23×C4—21no—
356C23×C4—22no—
357C23×C4—341no—
358C23×C4C22×D1711yescharacteristic
359C2×M16—174no—
360C2×M16—342no—
361C2×M16—342no—
362C22×C8—172no—
363C22×C8—171no—
364C22×C8—171no—
365C22×C8—342no—
366C22×C8—342no—
367C22×C8—342no—
368C4×Q8—24no—
369C4×Q8—24no—
370C4×D4—216no—
371C4×D4—24no—
372C4×D4—24no—
373#117—28no—
374#117—22no—
375#117—22no—
376#117—342no—
377#117—342no—
378C2×C4⋊C4—28no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—174no—
383C2×C4⋊C4—172no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—342no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—342no—
389C2×C4⋊C4—342no—
390C2×C4⋊C4C22⋊D1711yescharacteristic
391C2×C4⋊C4C22⋊D1711yescharacteristic
392C2×C4⋊C4C22⋊D1714yes—
393C2×C4⋊C4C22×D1712yes—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—341no—
396C2×C22⋊C4C22⋊D1712yes—
397C2×C42—22no—
398C2×C42—341no—
399C2×C42C22×D1712yes—
400#107—344no—
401#106—344no—
402#103—178no—
403#103—342no—
404#103—342no—
405#102—178no—
406#102—342no—
407#102—342no—
408#95—342no—
409#95—342no—
410C17#2087811yescharacteristic, complemented, Sylow
411C24—21no—
412C24C2×C22⋊D1711yescharacteristic
413#41—28no—
414#41—28no—
415#41—28no—
416#41#63225116yescomplemented
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8C2×Dic3412yes—
421C2×Q8C2×C4×D1712yes—
422C2×Q8C2×C22⋊D1712yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4C2×Dic3412yes—
430C2×D4C2×C4×D1712yes—
431C2×D4C2×C22⋊D1712yes—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—342no—
444C22×C4—344no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4—1361no—
448C22×C4—1361no—
449C22×C4C2×Dic3412yes—
450C22×C4C2×C4×D1712yes—
451C22×C4#6325212yes—
452C22×C4#6325212yes—
453C22×C4C2×C22⋊D1712yes—
454C22×C4C2×C22⋊D1712yes—
455C22×C4C2×C22⋊D1712yes—
456C22×C4C2×C22⋊D1712yes—
457C22×C4C2×C22⋊D1711yescharacteristic
458C22×C4C23×D1711yescharacteristic
459M16—348no—
460C2×C8—174no—
461C2×C8—342no—
462C2×C8—342no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—684no—
470C4⋊C4—28no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—174no—
476C4⋊C4—342no—
477C4⋊C4—344no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—1362no—
483C4⋊C4—1362no—
484C4⋊C4C2×C22⋊D1714yes—
485C22⋊C4—28no—
486C22⋊C4—24no—
487C22⋊C4—682no—
488C42—28no—
489C42—22no—
490C42—22no—
491C42—682no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23#1058761012yes—
504C23C4×C22⋊D1712yes—
505C23#1058763912yes—
506C23C22×C22⋊D1711yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8C2×#6322514yes—
511D4—216no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4C2×#6322514yes—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—216no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—84no—
530C2×C4—88no—
531C2×C4—344no—
532C2×C4—682no—
533C2×C4—682no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4—1364no—
541C2×C4C2×#6322514yes—
542C2×C4#1058761014yes—
543C2×C4#1058761012yes—
544C2×C4C4×C22⋊D1714yes—
545C2×C4C4×C22⋊D1712yes—
546C2×C4#1058761514yes—
547C2×C4#1058763912yes—
548C2×C4C22×Dic3411yescharacteristic
549C2×C4C22×C4×D1711yescharacteristic
550C2×C4C2×#6325211yescharacteristic
551C2×C4C2×#6325211yescharacteristic
552C2×C4C22×C22⋊D1711yescharacteristic
553C2×C4C22×C22⋊D1711yescharacteristic
554C8—344no—
555C8—681no—
556C8—681no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22#1176268214yes—
571C22C2×C4×C22⋊D1711yescharacteristic
572C22C2×#1058763911yescharacteristic
573C22C2×#1058761011yescharacteristic
574C22#1176361211yescharacteristic
575C22#1176361711yescharacteristic
576C22#1176362011yescharacteristic
577C22#1176362411yescharacteristic
578C4—24no—
579C4—42no—
580C4—44no—
581C4—42no—
582C4—684no—
583C4—1361no—
584C4—1361no—
585C4#1176268214yes—
586C4C22×#6322511yescharacteristic
587C4C2×C4×C22⋊D1711yescharacteristic
588C4C2×#1058761011yescharacteristic
589C4C2×#1058761511yescharacteristic
590C2—24no—
591C2—44no—
592C2C2×#1176268211yescharacteristic
593C2#1453761211yescharacteristic
594C2#1453762311yescharacteristic
595C2#1453808811yescharacteristic
596C2#1453809911yescharacteristic
597C2#1453863511yescharacteristic
598C2#1453864611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#520C2
socleC23×C17C22×D4
prime radicalC22×#41×C17C22
cosocleC22×C4×C17C24
hypercenterC2×#520D17
hypocenterC17#20878
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#11762849C2semidirect
4. #14531400C2semidirect
5. #14529052C2semidirect
6. #14529045C2semidirect
7. #14529038C2semidirect
8. #14529031C2semidirect
9. C2×#11762571C2semidirect
10. #14528481C2semidirect
19. #11762810C4semidirect
85. C2×C4⋊Dic17C2×C4semidirect
94. C2×#63223C2×C4semidirect
103. #10587506C2×C4semidirect
106. #10587503C2×C4semidirect
116. #41×C17C4⋊C4semidirect
410. C17#20878semidirect
416. #41#63225semidirect

Maximal subgroups

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

subgroupindex
C2×#117628492
C2×#117628102
C2×#117628052
#145314002
#145290522
#145290452
#145290382
#145290312
C2×#117625712
#145284812
C2×C17×#5202
#2087817