Finite Group Databasebeta

← #17035098

Subgroups — #17035098

Order 4352.

1462 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762849C211yesmaximal, characteristic, complemented
2C2×#11762845C211yesmaximal, characteristic, complemented
3C2×#11762776C211yesmaximal, characteristic
4#14531376C211yesmaximal, characteristic
5#14528983C212yesmaximal, complemented
6#14528983C212yesmaximal, complemented
7#14528977C212yesmaximal, complemented
8#14528972C212yesmaximal, complemented
9#14528505C211yesmaximal, characteristic, complemented
10C2×#11762571C211yesmaximal, characteristic, complemented
11C2×C17×#520C211yesmaximal, characteristic
12C22×#63223C2211yescharacteristic
13C23×#3622C2211yescharacteristic
14C2×#10587612C2211yescharacteristic
15C22×C4⋊Dic17C2211yescharacteristic
16#11762849—24no—
17#11762845—24no—
18C2×#10587478C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2212yes—
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2212yes—
24#11762776C418yescomplemented
25C2×C22⋊#3622C2212yes—
26C2×C22⋊#3622C2212yes—
27#11762716—21no—
28#11762712—21no—
29#11762689—22no—
30#11762669C2212yes—
31#11762669C2212yes—
32#11762571C2212yes—
33#11762571C2212yes—
34#11762571C2212yes—
35#11762571C2212yes—
36C22×#41×C17C2211yescharacteristic
37C17×#520—24no—
38C17×#520—24no—
39C2×C4×Q8×C17—21no—
40C2×C4×D4×C17—22no—
41C2×C4×D4×C17—21no—
42C2×C17×#117C2212yes—
43C2×C17×#117C2211yescharacteristic
44C22×C4⋊C4×C17C2212yes—
45C22×C4⋊C4×C17C2211yescharacteristic
46C2×#41×C17—22no—
47C2×#41×C17—22no—
48C2×#41×C17—22no—
49C2×#41×C17—22no—
50C2×#41×C17C2×C414yes—
51C2×#41×C17Q814yes—
52C22×Q8×C17D411yescharacteristic
53C22×D4×C17D412yes—
54C22×D4×C17D411yescharacteristic
55C23×C4×C17—22no—
56C23×C4×C17—21no—
57C23×C4×C17C2312yes—
58C23×C4×C17C2311yescharacteristic
59C23×Dic17—21no—
60C4×Q8×C17—28no—
61C4×D4×C17—216no—
62C4×D4×C17—28no—
63C17×#117—22no—
64C17×#117—24no—
65C17×#117—24no—
66C17×#117D414yes—
67C2×C4⋊C4×C17—22no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—22no—
70C2×C4⋊C4×C17—24no—
71C2×C4⋊C4×C17—24no—
72C2×C4⋊C4×C17D414yes—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2312yes—
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C22⋊C4×C17—22no—
77C2×C22⋊C4×C17—21no—
78C2×C42×C17—22no—
79C2×C42×C17—21no—
80C2×C22⋊Dic17—21no—
81C2×C22⋊Dic17—22no—
82C2×C22⋊Dic17—22no—
83#10587612—22no—
84#10587612—22no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17—22no—
87C2×C4⋊Dic17—22no—
88C2×C4⋊Dic17C2312yes—
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4×Dic17—21no—
92C2×#63223—22no—
93C2×#63223—22no—
94C2×#63223C2×C414yescomplemented
95C22×#3622—22no—
96C22×#3622—22no—
97C22×#3622C2×C414yescomplemented
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2312yes—
101#10587502—22no—
102#10587502—22no—
103C22⋊#3622C2×C418yescomplemented
104C22⋊#3622C2×C418yescomplemented
105#10587479—22no—
106#10587479—22no—
107#10587479—24no—
108#10587478—22no—
109#10587478—22no—
110#10587478—24no—
111C24×C17C2×D412yes—
112C24×C17C2×D411yescharacteristic
113#41×C17—28no—
114#41×C17—28no—
115#41×C17—28no—
116#41×C17—28no—
117C2×Q8×C17—22no—
118C2×Q8×C17—22no—
119C2×Q8×C17C4⋊C414yes—
120C2×D4×C17—24no—
121C2×D4×C17—24no—
122C2×D4×C17—22no—
123C2×D4×C17—22no—
124C2×D4×C17C4⋊C418yes—
125C2×D4×C17C4⋊C414yes—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—28no—
134C22×C4×C17—24no—
135C22×C4×C17—81no—
136C22×C4×C17—82no—
137C22×C4×C17—81no—
138C22×C4×C17—82no—
139C22×C4×C17C22×C414yes—
140C22×C4×C17C22×C412yes—
141C22×C4×C17C2×D412yes—
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D412yes—
144C22×C4×C17C2×D412yes—
145C22×C4×C17C2×Q814yes—
146C22×C4×C17C2×Q812yes—
147C22×C4×C17#4112yes—
148C22×C4×C17#4112yes—
149C22×C4×C17C2411yescharacteristic
150C22×Dic17—22no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—42no—
154C22×Dic17—81no—
155C22×Dic17—81no—
156C4⋊C4×C17—28no—
157C4⋊C4×C17—24no—
158C4⋊C4×C17—22no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17C2×D412yes—
162C4⋊C4×C17C2×D414yes—
163C4⋊C4×C17C2×D412yes—
164C22⋊C4×C17—28no—
165C22⋊C4×C17—24no—
166C42×C17—28no—
167C42×C17—24no—
168C22⋊Dic17—42no—
169C22⋊Dic17—82no—
170C22⋊Dic17—82no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—22no—
176C4⋊Dic17—22no—
177C4⋊Dic17—42no—
178C4×Dic17—42no—
179#63223—28no—
180C2×#3622—28no—
181C2×#3622—21no—
182C2×#3622—22no—
183C2×#3622—21no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—22no—
187C2×#3622C22×C412yes—
188C2×#3622C22×C412yes—
189C2×#3622C22×C414yes—
190#20849—171nomaximal, Sylow
191C23×C17—21no—
192C23×C17—21no—
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—22no—
196C23×C17—22no—
197C23×C17—22no—
198C23×C17—22no—
199C23×C17—22no—
200C23×C17—22no—
201C23×C17C2×C4⋊C414yes—
202C23×C17C2×C4⋊C412yes—
203C23×C17#12012yes—
204C23×C17#12312yes—
205C23×C17C22×D411yescharacteristic
206Q8×C17—28no—
207D4×C17—216no—
208D4×C17—28no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—24no—
212C2×C4×C17—24no—
213C2×C4×C17—24no—
214C2×C4×C17—24no—
215C2×C4×C17—22no—
216C2×C4×C17—24no—
217C2×C4×C17—22no—
218C2×C4×C17—42no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—42no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—84no—
225C2×C4×C17—88no—
226C2×C4×C17C2×C4⋊C414yes—
227C2×C4×C17C2×C4⋊C412yes—
228C2×C4×C17C4×D414yes—
229C2×C4×C17C4×D414yes—
230C2×C4×C17#12012yes—
231C2×C4×C17C22⋊Q814yes—
232C2×C4×C17C22⋊Q814yes—
233C2×C4×C17#12312yes—
234C2×C4×C17C23×C411yescharacteristic
235C2×C4×C17C22×D411yescharacteristic
236C2×C4×C17C22×D411yescharacteristic
237C2×C4×C17C22×Q811yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—42no—
243C2×Dic17—41no—
244C2×Dic17—42no—
245C2×Dic17—41no—
246C2×Dic17—42no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—84no—
250#3622—22no—
251#3622—22no—
252#3622—24no—
253C2×#520D1711yescharacteristic
254C2×#428—171no—
255C2×#427—171no—
256C2×#408—171no—
257#2311—171no—
258#1854—172no—
259#1854—172no—
260#1853—172no—
261#1852—172no—
262#1736—171no—
263C2×#340—171no—
264C22×C17—22no—
265C22×C17—22no—
266C22×C17—22no—
267C22×C17—22no—
268C22×C17—22no—
269C22×C17—22no—
270C22×C17—24no—
271C22×C17—24no—
272C22×C17—24no—
273C22×C17—42no—
274C22×C17—42no—
275C22×C17—42no—
276C22×C17—42no—
277C22×C17#38514yes—
278C22×C17C22×C4⋊C411yescharacteristic
279C22×C17C2×#12011yescharacteristic
280C22×C17C2×#12311yescharacteristic
281C22×C17#57611yescharacteristic
282C22×C17#57711yescharacteristic
283C22×C17#57711yescharacteristic
284C22×C17#57811yescharacteristic
285C4×C17—24no—
286C4×C17—44no—
287C4×C17—42no—
288C4×C17—42no—
289C4×C17#38514yes—
290C4×C17C2×C4×D411yescharacteristic
291C4×C17C2×C4×D411yescharacteristic
292C4×C17C2×C22⋊Q811yescharacteristic
293C4×C17C2×C22⋊Q811yescharacteristic
294Dic17—82no—
295Dic17—81no—
296Dic17—81no—
297C22×#41C2×D1711yescharacteristic
298C22×M16—171no—
299C23×C8—171no—
300#520—24no—
301#520—24no—
302C2×C4×Q8—21no—
303C2×C4×D4—22no—
304C2×C4×D4—21no—
305C2×#117—171no—
306C2×#117C2×D1712yes—
307C2×#117C2×D1711yescharacteristic
308C22×C4⋊C4—171no—
309C22×C4⋊C4C2×D1712yes—
310C22×C4⋊C4C2×D1711yescharacteristic
311#428—344no—
312#427—344no—
313C2×#107—171no—
314C2×#107—172no—
315C2×#107—171no—
316C2×#106—171no—
317C2×#106—172no—
318C2×#106—171no—
319#408—178no—
320C2×C22⋊C8—172no—
321C2×C22⋊C8—172no—
322#391—341no—
323#390—341no—
324#386—342no—
325#380—172no—
326#380—172no—
327#340—172no—
328#340—172no—
329#340—172no—
330#340—172no—
331C2×C17—24no—
332C2×C17—44no—
333C2×C17C2×#38511yescharacteristic
334C2×C17#291211yescharacteristic
335C2×C17#291311yescharacteristic
336C2×C17#301211yescharacteristic
337C2×C17#301311yescharacteristic
338C2×C17#309611yescharacteristic
339C2×C17#309711yescharacteristic
340C2×#41—22no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41—22no—
344C2×#41Dic3414yes—
345C2×#41C4×D1714yes—
346C22×Q8C22⋊D1711yescharacteristic
347C22×D4C22⋊D1712yes—
348C22×D4C22⋊D1711yescharacteristic
349C23×C4—22no—
350C23×C4—21no—
351C23×C4—341no—
352C23×C4C22×D1712yes—
353C23×C4C22×D1711yescharacteristic
354C2×M16—174no—
355C2×M16—342no—
356C2×M16—342no—
357C22×C8—174no—
358C22×C8—171no—
359C22×C8—171no—
360C22×C8—172no—
361C22×C8—342no—
362C22×C8—342no—
363C4×Q8—28no—
364C4×D4—216no—
365C4×D4—28no—
366#117—22no—
367#117—24no—
368#117—24no—
369#117—342no—
370#117—342no—
371#117C22⋊D1714yes—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—24no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—172no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—171no—
380C2×C4⋊C4—342no—
381C2×C4⋊C4—342no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4C22⋊D1714yes—
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C4⋊C4C22×D1712yes—
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C22⋊C4—22no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—341no—
390C2×C22⋊C4—342no—
391C2×C22⋊C4—342no—
392C2×C42—22no—
393C2×C42—21no—
394C2×C42—341no—
395#107—342no—
396#107—342no—
397#107—344no—
398#106—342no—
399#106—342no—
400#106—344no—
401C22⋊C8—178no—
402C22⋊C8—178no—
403#95—342no—
404#95—342no—
405C17#2084911yescharacteristic, complemented, Sylow
406C24C2×C22⋊D1712yes—
407C24C2×C22⋊D1711yescharacteristic
408#41—28no—
409#41—28no—
410#41—28no—
411#41—28no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8#6322514yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4#6322518yes—
420C2×D4#6322514yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—81no—
431C22×C4—82no—
432C22×C4—81no—
433C22×C4—82no—
434C22×C4—341no—
435C22×C4—341no—
436C22×C4—342no—
437C22×C4—682no—
438C22×C4—1361no—
439C22×C4—1361no—
440C22×C4C2×Dic3414yes—
441C22×C4C2×Dic3412yes—
442C22×C4C2×C4×D1714yes—
443C22×C4C2×C4×D1712yes—
444C22×C4#6325212yes—
445C22×C4#6325212yes—
446C22×C4C2×C22⋊D1712yes—
447C22×C4C2×C22⋊D1712yes—
448C22×C4C2×C22⋊D1712yes—
449C22×C4C2×C22⋊D1711yescharacteristic
450C22×C4C23×D1711yescharacteristic
451M16—348no—
452C2×C8—172no—
453C2×C8—172no—
454C2×C8—174no—
455C2×C8—341no—
456C2×C8—341no—
457C2×C8—342no—
458C2×C8—341no—
459C2×C8—342no—
460C2×C8—341no—
461C2×C8—348no—
462C4⋊C4—28no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—24no—
467C4⋊C4—342no—
468C4⋊C4—342no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—682no—
474C4⋊C4C2×C22⋊D1712yes—
475C4⋊C4C2×C22⋊D1714yes—
476C4⋊C4C2×C22⋊D1712yes—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—682no—
480C22⋊C4—1362no—
481C22⋊C4—1362no—
482C42—28no—
483C42—24no—
484C42—682no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23C2×#6322514yes—
496C23C2×#6322512yes—
497C23#1058761512yes—
498C23#1058763912yes—
499C23C22×C22⋊D1711yescharacteristic
500Q8—28no—
501D4—216no—
502D4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—84no—
519C2×C4—88no—
520C2×C4—681no—
521C2×C4—681no—
522C2×C4—682no—
523C2×C4—681no—
524C2×C4—682no—
525C2×C4—681no—
526C2×C4—682no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—1364no—
530C2×C4C2×#6322514yes—
531C2×C4C2×#6322512yes—
532C2×C4#1058761014yes—
533C2×C4#1058761014yes—
534C2×C4C4×C22⋊D1714yes—
535C2×C4C4×C22⋊D1714yes—
536C2×C4#1058761512yes—
537C2×C4#1058763912yes—
538C2×C4C22×Dic3411yescharacteristic
539C2×C4C22×C4×D1711yescharacteristic
540C2×C4C2×#6325211yescharacteristic
541C2×C4C2×#6325211yescharacteristic
542C2×C4C22×C22⋊D1711yescharacteristic
543C2×C4C22×C22⋊D1711yescharacteristic
544C8—342no—
545C8—342no—
546C8—344no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22#1176268214yes—
561C22C22×#6322511yescharacteristic
562C22C2×#1058763911yescharacteristic
563C22C2×#1058761511yescharacteristic
564C22#1176361211yescharacteristic
565C22#1176361711yescharacteristic
566C22#1176362011yescharacteristic
567C22#1176362411yescharacteristic
568C4—24no—
569C4—44no—
570C4—42no—
571C4—42no—
572C4—1361no—
573C4—1361no—
574C4—1362no—
575C4#1176268214yes—
576C4C2×C4×C22⋊D1711yescharacteristic
577C4C2×C4×C22⋊D1711yescharacteristic
578C4C2×#1058761011yescharacteristic
579C4C2×#1058761011yescharacteristic
580C2—24no—
581C2—44no—
582C2C2×#1176268211yescharacteristic
583C2#1453737711yescharacteristic
584C2#1453738211yescharacteristic
585C2#1453808811yescharacteristic
586C2#1453809911yescharacteristic
587C2#1453874211yescharacteristic
588C2#1453875311yescharacteristic
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#20849
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#11762849C2semidirect
2. C2×#11762845C2semidirect
5. #14528983C2semidirect
6. #14528983C2semidirect
7. #14528977C2semidirect
8. #14528972C2semidirect
9. #14528505C2semidirect
10. C2×#11762571C2semidirect
24. #11762776C4semidirect
94. C2×#63223C2×C4semidirect
97. C22×#3622C2×C4semidirect
103. C22⋊#3622C2×C4semidirect
104. C22⋊#3622C2×C4semidirect
405. C17#20849semidirect

Maximal subgroups

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

subgroupindex
C2×#117628492
C2×#117628452
C2×#117627762
#145313762
#145289832
#145289772
#145289722
#145285052
C2×#117625712
C2×C17×#5202
#2084917