Finite Group Databasebeta

← #17054189

Subgroups — #17054189

Order 4352.

810 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762909C211yesmaximal, characteristic
2#14531739C211yesmaximal, characteristic, complemented
3#14529664C211yesmaximal, characteristic
4#14529637C211yesmaximal, characteristic
5#14529425C211yesmaximal, characteristic, complemented
6#14529418C211yesmaximal, characteristic, complemented
7#14529408C211yesmaximal, characteristic
8#14529401C211yesmaximal, characteristic
9#14528953C211yesmaximal, characteristic, complemented
10#14528796C211yesmaximal, characteristic, complemented
11#14528784C211yesmaximal, characteristic
12#14528682C211yesmaximal, characteristic, complemented
13C4×#10587478C211yesmaximal, characteristic, complemented
14#14528659C211yesmaximal, characteristic, complemented
15C17×#2357C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18#11762909C418yescomplemented
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×#10587481C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×#10587503C2211yescharacteristic
30C2×#10587503C2211yescharacteristic
31C2×#10587474C2211yescharacteristic
32C2×#10587474C2211yescharacteristic
33C2×C4×#3622C2211yescharacteristic
34#11762715C2211yescharacteristic
35#11762715C2211yescharacteristic
36#11762696—21no—
37#11762696—21no—
38#11762679C2211yescharacteristic
39#11762674—21no—
40C4⋊C4×Dic17—21no—
41C4×C4⋊Dic17C2211yescharacteristic
42C4×C4⋊Dic17C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C2×C4×D4×C17C2211yescharacteristic
50C17×#390C2211yescharacteristic
51C17×#390C2211yescharacteristic
52C17×#388—21no—
53C17×#388—21no—
54C17×#382C2211yescharacteristic
55C17×#381—21no—
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C4⋊C4×C17C2211yescharacteristic
59C4×C22⋊C4×C17—21no—
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C22×Dic34C2311yescharacteristic
63C4×D4×C17—22no—
64C4×D4×C17C2×C414yes—
65C2×C4⋊C4×C17D411yescharacteristic
66C2×C4⋊C4×C17D411yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C22⋊C4×C17—21no—
71C2×C22⋊C4×C17—21no—
72C2×C22⋊C4×C17—21no—
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17—21no—
75C2×C42×C17—21no—
76C2×C42×C17—21no—
77C2×C42×C17C2311yescharacteristic
78C2×C42×C17C2311yescharacteristic
79C2×C42×C17C2311yescharacteristic
80C2×#63229—22no—
81C2×#63229—22no—
82C2×#63229C2×C414yes—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×#63225—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C4×Dic34—22no—
97C4×Dic34C2×C414yes—
98C17×#95—21no—
99C17×#95—21no—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95D411yescharacteristic
103C17×#95D411yescharacteristic
104#10587503—41no—
105#10587503—41no—
106#10587503C2×C414yes—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587481—41no—
114#10587481—41no—
115#10587481C2×C414yes—
116#10587479—22no—
117#10587479—41no—
118#10587478—22no—
119#10587478—41no—
120#10587478C2×C414yes—
121C4⋊#3622—22no—
122#10587474—22no—
123#10587474C2×C414yes—
124C4×#3622—22no—
125C24×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17C22×C412yes—
132C22×C4×C17—21no—
133C22×C4×C17—24no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—81no—
137C22×C4×C17—81no—
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2411yescharacteristic
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C2×Dic34—21no—
150C2×Dic34—21no—
151C2×Dic34—21no—
152C2×Dic34—21no—
153C2×Dic34—21no—
154C2×Dic34C22×C412yes—
155C4⋊C4×C17—22no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—41no—
162C4⋊C4×C17C22×C412yes—
163C22⋊C4×C17—24no—
164C22⋊C4×C17—42no—
165C22⋊C4×C17—42no—
166C22⋊C4×C17—81no—
167C22⋊C4×C17—81no—
168C42×C17—21no—
169C42×C17—21no—
170C42×C17—21no—
171C42×C17—21no—
172C42×C17—81no—
173C42×C17—81no—
174C42×C17C22×C412yes—
175C42×C17#4112yes—
176#63229—28no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—42no—
183C4⋊Dic17—81no—
184C4⋊Dic17C22×C412yes—
185#63225—24no—
186C4×Dic17—24no—
187C4×Dic17—81no—
188C4×Dic17—81no—
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—
199C2×#3622#4112yes—
200C2×#3622#4112yes—
201#22819—171nomaximal, Sylow
202C23×C17—22no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17—41no—
208C23×C17—41no—
209C23×C17C22×D411yescharacteristic
210D4×C17—22no—
211D4×C17—22no—
212D4×C17—41no—
213D4×C17—41no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
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—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—84no—
239C2×C4×C17C4×D412yes—
240C2×C4×C17C4×D412yes—
241C2×C4×C17#13712yes—
242C2×C4×C17C23×C411yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17H(5,𝔽2)11yescharacteristic
247C2×C4×C172−1+411yescharacteristic
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260Dic34—22no—
261Dic34—22no—
262Dic34—41no—
263Dic34—41no—
264#3622—22no—
265#3622—41no—
266#3622—41no—
267#3622#11714yes—
268C2×#440—171no—
269#2357D1711yescharacteristic
270#2355—171no—
271#1973—171no—
272#1968—171no—
273#1928—171no—
274#1927—171no—
275#1926—171no—
276#1925—171no—
277#1847—171no—
278#1805—171no—
279#1802—171no—
280#1776—171no—
281C4×#107—171no—
282#1771—171no—
283C22×C17—42no—
284C22×C17—42no—
285C22×C17—42no—
286C22×C17—81no—
287C22×C17—81no—
288C22×C17—81no—
289C22×C17—81no—
290C22×C17C2×C4×D411yescharacteristic
291C22×C17#54711yescharacteristic
292C22×C17#55011yescharacteristic
293C22×C17C2×#13711yescharacteristic
294C22×C17#57511yescharacteristic
295C22×C17#57611yescharacteristic
296C22×C17#57711yescharacteristic
297C4×C17—22no—
298C4×C17—22no—
299C4×C17—22no—
300C4×C17—41no—
301C4×C17—41no—
302C4×C17—41no—
303C4×C17—42no—
304C4×C17—81no—
305C4×C17—81no—
306C4×C17C2×#11711yescharacteristic
307C4×C17#52011yescharacteristic
308C4×C17#55511yescharacteristic
309C4×C17#55611yescharacteristic
310Dic17—42no—
311Dic17—42no—
312Dic17—81no—
313Dic17—81no—
314C22×SD16—171no—
315C2×C4×Q8—171no—
316C2×C4×D4C2×D1711yescharacteristic
317#440—178no—
318C2×#107—171no—
319C2×#107—171no—
320C2×#107—171no—
321C2×#106—171no—
322C2×#106—171no—
323C2×C4⋊C8—171no—
324C2×#103—171no—
325C2×#103—171no—
326C2×#103—171no—
327C2×#102—171no—
328C2×#102—171no—
329C2×#102—171no—
330C2×#97—171no—
331C2×#97—171no—
332C2×C4×C8—171no—
333#391—171no—
334#391—171no—
335#390C2×D1711yescharacteristic
336#390C2×D1711yescharacteristic
337#388—21no—
338#388—21no—
339#387—341no—
340#387—341no—
341#384—171no—
342#382—341no—
343#382C2×D1711yescharacteristic
344#381—21no—
345C4×C4⋊C4—171no—
346C4×C4⋊C4—171no—
347C4×C4⋊C4—341no—
348C4×C4⋊C4C2×D1711yescharacteristic
349C4×C4⋊C4C2×D1711yescharacteristic
350C4×C4⋊C4C2×D1711yescharacteristic
351C4×C22⋊C4—21no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#336—171no—
357#336—171no—
358C2×C17—82no—
359C2×C17#235411yescharacteristic
360C2×C17C4×#13711yescharacteristic
361C2×C17#294611yescharacteristic
362C2×C17#334611yescharacteristic
363C2×C17#335011yescharacteristic
364C2×C17#340011yescharacteristic
365C2×C17#340311yescharacteristic
366C22×Q8—171no—
367C22×D4C22×D1711yescharacteristic
368C23×C4—21no—
369C2×SD16—174no—
370C2×SD16—342no—
371C2×SD16—342no—
372C22×C8—171no—
373C22×C8—171no—
374C22×C8—171no—
375C22×C8—171no—
376C4×Q8—174no—
377C4×Q8—342no—
378C4×D4—22no—
379C4×D4C4×D1714yes—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—341no—
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×C4⋊C4C22×D1711yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—341no—
399C2×C42—341no—
400C2×C42—341no—
401C2×C42C22×D1711yescharacteristic
402C2×C42C22×D1711yescharacteristic
403C2×C42C22×D1711yescharacteristic
404#107—174no—
405#107—342no—
406#107—681no—
407#106—342no—
408#106—681no—
409C4⋊C8—342no—
410#103—174no—
411#103—681no—
412#103—681no—
413#102—174no—
414#102—681no—
415#102—681no—
416#97—174no—
417#97—342no—
418C4×C8—342no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429#95C22⋊D1711yescharacteristic
430#95C22⋊D1711yescharacteristic
431C17#2281911yescharacteristic, complemented, Sylow
432C24—21no—
433C2×Q8—172no—
434C2×Q8—341no—
435C2×Q8—341no—
436C2×Q8—341no—
437C2×Q8—341no—
438C2×Q8—341no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C2×C4×D1712yes—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—341no—
452C22×C4—341no—
453C22×C4—341no—
454C22×C4—341no—
455C22×C4C2×C22⋊D1711yescharacteristic
456C22×C4C2×C22⋊D1711yescharacteristic
457C22×C4C2×C22⋊D1711yescharacteristic
458C22×C4C2×C22⋊D1711yescharacteristic
459C22×C4C2×C22⋊D1711yescharacteristic
460C22×C4C2×C22⋊D1711yescharacteristic
461C22×C4C23×D1711yescharacteristic
462SD16—348no—
463C2×C8—172no—
464C2×C8—172no—
465C2×C8—172no—
466C2×C8—172no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C2×C8—341no—
473C2×C8—341no—
474C2×C8—341no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—172no—
483C4⋊C4—341no—
484C4⋊C4—344no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—341no—
488C4⋊C4—341no—
489C4⋊C4—682no—
490C4⋊C4—1361no—
491C4⋊C4C2×C4×D1712yes—
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—81no—
496C22⋊C4—81no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—81no—
502C42—81no—
503C42—344no—
504C42—1361no—
505C42—1361no—
506C42C2×C4×D1712yes—
507C42#6325212yes—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C22×C22⋊D1711yescharacteristic
516Q8—342no—
517Q8—342no—
518Q8—681no—
519Q8—681no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—84no—
549C2×C4—342no—
550C2×C4—342no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4C4×C22⋊D1712yes—
562C2×C4C4×C22⋊D1712yes—
563C2×C4#1058763712yes—
564C2×C4C22×C4×D1711yescharacteristic
565C2×C4C2×#6325211yescharacteristic
566C2×C4#41×D1711yescharacteristic
567C2×C4#41×D1711yescharacteristic
568C2×C4#1058769611yescharacteristic
569C2×C4#1058769711yescharacteristic
570C8—174no—
571C8—342no—
572C8—681no—
573C8—681no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×C4×C22⋊D1711yescharacteristic
582C22#1176345811yescharacteristic
583C22#1176347811yescharacteristic
584C22C2×#1058763711yescharacteristic
585C22#1176360711yescharacteristic
586C22#1176361211yescharacteristic
587C22#1176362011yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—41no—
592C4—41no—
593C4—42no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—682no—
598C4—682no—
599C4—1361no—
600C4—1361no—
601C4#117×D1711yescharacteristic
602C4#1176328911yescharacteristic
603C4#1176352211yescharacteristic
604C4#1176352711yescharacteristic
605C2—82no—
606C2#1453173411yescharacteristic
607C2C4×#1058763711yescharacteristic
608C2#1453761611yescharacteristic
609C2#1454126911yescharacteristic
610C2#1454132311yescharacteristic
611C2#1454183711yescharacteristic
612C2#1454186811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2357C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#2357D17
hypocenterC17#22819
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. #14531739C2semidirect
5. #14529425C2semidirect
6. #14529418C2semidirect
9. #14528953C2semidirect
10. #14528796C2semidirect
12. #14528682C2semidirect
13. C4×#10587478C2semidirect
14. #14528659C2semidirect
18. #11762909C4semidirect
431. C17#22819semidirect

Maximal subgroups

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

subgroupindex
C2×#117629092
#145317392
#145296642
#145296372
#145294252
#145294182
#145294082
#145294012
#145289532
#145287962
#145287842
#145286822
C4×#105874782
#145286592
C17×#23572
#2281917