Finite Group Databasebeta

← #17093196

Subgroups — #17093196

Order 4352.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763218C211yesmaximal, characteristic, complemented
2C2×#11762851C211yesmaximal, characteristic
3#14532508C211yesmaximal, characteristic
4#14532475C211yesmaximal, characteristic
5#14530267C211yesmaximal, characteristic, complemented
6#14530267C211yesmaximal, characteristic, complemented
7#14530123C211yesmaximal, characteristic, complemented
8#14530113C211yesmaximal, characteristic, complemented
9#14529522C211yesmaximal, characteristic, complemented
10#14529517C211yesmaximal, characteristic, complemented
11#14528985C211yesmaximal, characteristic
12#14528974C211yesmaximal, characteristic
13#14528948C211yesmaximal, characteristic, complemented
14#14528817C211yesmaximal, characteristic
15C17×#1753C211yesmaximal, characteristic
16C2×#10587555C2211yescharacteristic
17C2×#10587556C2211yescharacteristic
18C2×#10587612C2211yescharacteristic
19C22×C4⋊Dic17C2211yescharacteristic
20#11763218—24no—
21#11762851—24no—
22C2×#10587489C2211yescharacteristic
23C2×#10587489C2211yescharacteristic
24C2×#10587489C2211yescharacteristic
25C2×#10587489C2211yescharacteristic
26C2×#10587489C2211yescharacteristic
27C2×#10587489C2211yescharacteristic
28C2×#10587488C2211yescharacteristic
29C2×#10587488C2211yescharacteristic
30C2×#10587488C2211yescharacteristic
31C2×#10587488C2211yescharacteristic
32#11762751—21no—
33#11762743—21no—
34#11762732—21no—
35#11762721—21no—
36#11762713—21no—
37#11762707C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762707C2211yescharacteristic
40#11762707C2211yescharacteristic
41#11762696—21no—
42#11762681—21no—
43#11762675C2211yescharacteristic
44#11762675C2211yescharacteristic
45#11762669C2211yescharacteristic
46#11762669C2211yescharacteristic
47#11762667—21no—
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53#11762572C2211yescharacteristic
54C22×M16×C17C2211yescharacteristic
55C2×C17×C22⋊C8C2211yescharacteristic
56C2×C17×#97C2211yescharacteristic
57C2×C4×C8×C17C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C17×#336C2211yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C22×Dic34—21no—
64C2×M16×C17—22no—
65C2×M16×C17—22no—
66C2×M16×C17Q814yescomplemented
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C22×C8×C17C2311yescharacteristic
71C2×C22⋊C4×C17D411yescharacteristic
72C2×C22⋊C4×C17D411yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C22⋊Dic17—21no—
76C2×C22⋊Dic17—21no—
77C2×C22⋊Dic17—21no—
78#10587612—22no—
79#10587612—41no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17—22no—
83C2×C4⋊Dic17—41no—
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×#63225—21no—
97C2×#63225—21no—
98C2×C4×Dic17—21no—
99#10587556—22no—
100#10587556—41no—
101#10587555—22no—
102#10587555—41no—
103C17×C22⋊C8Q814yescomplemented
104C17×#97—22no—
105C4×C8×C17—22no—
106C17×#95D411yescharacteristic
107C17×#95D411yescharacteristic
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#10587489—22no—
119#10587489—22no—
120#10587489—22no—
121#10587489—41no—
122#10587489—41no—
123#10587489D414yes—
124#10587488—22no—
125#10587488—22no—
126#10587488—22no—
127#10587488—22no—
128C24×C17C2×D411yescharacteristic
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×Q812yes—
139C22×C4×C17C2×Q812yes—
140C22×C4×C17C2411yescharacteristic
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—42no—
150C22×Dic17—42no—
151C22×Dic17—81no—
152C2×Dic34—42no—
153C2×Dic34—42no—
154C2×Dic34—81no—
155M16×C17—28no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17C2×D412yes—
163C2×C8×C17C2×Q812yes—
164C2×C8×C17C2×Q812yes—
165C2×C8×C17C2×Q812yes—
166C2×C8×C17C2×Q812yes—
167C2×C8×C17#4112yes—
168C22⋊C4×C17—41no—
169C22⋊C4×C17—41no—
170C42×C17—21no—
171C42×C17—21no—
172C42×C17—21no—
173C42×C17—21no—
174C22⋊Dic17—42no—
175C22⋊Dic17—42no—
176C22⋊Dic17—81no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—42no—
192C4⋊Dic17—42no—
193C4⋊Dic17C2×D412yes—
194C4⋊Dic17C2×D412yes—
195#63225—42no—
196#63225—42no—
197#63225—42no—
198#63225—81no—
199#63225—81no—
200#63225—81no—
201C4×Dic17—42no—
202#26079—171nomaximal, Sylow
203C23×C17—21no—
204C23×C17—21no—
205C23×C17—21no—
206C23×C17—21no—
207C23×C17—21no—
208C23×C17—21no—
209C23×C17C4⋊Q812yes—
210C23×C17C22×D411yescharacteristic
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—22no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
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—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17C4⋊Q812yes—
230C2×C4×C17C22×D411yescharacteristic
231C2×C4×C17C22×Q811yescharacteristic
232C2×C4×C17C22×Q811yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17H(5,𝔽2)11yescharacteristic
235C2×C4×C172−1+411yescharacteristic
236C2×Dic17—22no—
237C2×Dic17—22no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
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—
260C2×Dic17—84no—
261Dic34—84no—
262C8×C17—22no—
263C8×C17—22no—
264C8×C17—22no—
265C8×C17C22⋊Q814yes—
266C2×#501—171no—
267C2×#428—171no—
268#2442—171no—
269#2439—171no—
270#2085—171no—
271#2085—171no—
272#2059—171no—
273#2057—171no—
274#1943—171no—
275#1942—171no—
276#1854—171no—
277#1852—171no—
278#1846—171no—
279#1809—171no—
280#1753D1711yescharacteristic
281C22×C17—22no—
282C22×C17—22no—
283C22×C17—22no—
284C22×C17—41no—
285C22×C17—41no—
286C22×C17—41no—
287C22×C17—41no—
288C22×C17C2×C4⋊Q811yescharacteristic
289C22×C17D4211yescharacteristic
290C22×C17#54711yescharacteristic
291C22×C17#57511yescharacteristic
292C22×C17#57511yescharacteristic
293C22×C17#57611yescharacteristic
294C22×C17#57811yescharacteristic
295C4×C17—22no—
296C4×C17—41no—
297C4×C17—41no—
298C4×C17—41no—
299C4×C17—41no—
300C4×C17C2×C22⋊Q811yescharacteristic
301C4×C17#54411yescharacteristic
302C4×C17D4×Q811yescharacteristic
303C4×C17#55411yescharacteristic
304Dic17—42no—
305Dic17—42no—
306Dic17—81no—
307Dic17—81no—
308Dic17—81no—
309Dic17—81no—
310Dic17—81no—
311Dic17—81no—
312C22×M16C2×D1711yescharacteristic
313C2×C4⋊Q8—171no—
314C2×#125—171no—
315C2×#117—171no—
316C22×C4⋊C4—171no—
317#501—344no—
318#428—344no—
319C2×#107—171no—
320C2×#107—171no—
321C2×#107—171no—
322C2×#107—171no—
323C2×#107—171no—
324C2×#107—171no—
325C2×#106—171no—
326C2×#106—171no—
327C2×#106—171no—
328C2×#106—171no—
329C2×C22⋊C8C2×D1711yescharacteristic
330C2×#97C2×D1711yescharacteristic
331C2×C4×C8C2×D1711yescharacteristic
332#400—341no—
333#398—341no—
334#396—341no—
335#393—341no—
336#391—341no—
337#389—171no—
338#389—171no—
339#389—171no—
340#389—171no—
341#387—341no—
342#385—341no—
343#382—171no—
344#382—171no—
345#380—171no—
346#380—171no—
347#380—341no—
348C4×C22⋊C4C2×D1711yescharacteristic
349#340—171no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#336C2×D1711yescharacteristic
356#336C2×D1711yescharacteristic
357C2×C17—42no—
358C2×C17#244011yescharacteristic
359C2×C17#316411yescharacteristic
360C2×C17#316611yescharacteristic
361C2×C17#327311yescharacteristic
362C2×C17#327711yescharacteristic
363C2×C17#334711yescharacteristic
364C2×C17#334911yescharacteristic
365C22×Q8—341no—
366C23×C4—341no—
367C23×C4C22×D1711yescharacteristic
368C2×M16—22no—
369C2×M16—22no—
370C2×M16Dic3414yescomplemented
371C22×C8C22×D1711yescharacteristic
372C22×C8C22×D1711yescharacteristic
373C22×C8C22×D1711yescharacteristic
374C22×C8C22×D1711yescharacteristic
375C4⋊Q8—342no—
376C4⋊Q8—681no—
377#125—342no—
378#125—681no—
379#117—342no—
380#117—681no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—342no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—341no—
394C2×C4⋊C4—341no—
395C2×C4⋊C4—341no—
396C2×C4⋊C4—341no—
397C2×C4⋊C4—341no—
398C2×C4⋊C4—681no—
399C2×C22⋊C4—341no—
400C2×C22⋊C4—341no—
401C2×C22⋊C4—341no—
402C2×C22⋊C4D6811yescharacteristic
403C2×C22⋊C4D6811yescharacteristic
404C2×C42—341no—
405C2×C42C22×D1711yescharacteristic
406C2×C42C22×D1711yescharacteristic
407#107—174no—
408#107—342no—
409#107—342no—
410#107—342no—
411#107—681no—
412#107—681no—
413#106—342no—
414#106—342no—
415#106—342no—
416#106—342no—
417C22⋊C8Dic3414yescomplemented
418#97—22no—
419C4×C8—22no—
420#95—341no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429#95—341no—
430#95D6811yescharacteristic
431#95D6811yescharacteristic
432C17#2607911yescharacteristic, complemented, Sylow
433C24C2×D6811yescharacteristic
434C2×Q8—682no—
435C2×Q8—682no—
436C2×Q8—1361no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—341no—
442C22×C4—341no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4—341no—
448C22×C4—341no—
449C22×C4—682no—
450C22×C4—682no—
451C22×C4—1361no—
452C22×C4C2×Dic3412yes—
453C22×C4C2×Dic3412yes—
454C22×C4C2×D6811yescharacteristic
455C22×C4C2×D6811yescharacteristic
456C22×C4C2×D6811yescharacteristic
457C22×C4C2×D6811yescharacteristic
458C22×C4C2×D6811yescharacteristic
459C22×C4C23×D1711yescharacteristic
460M16—28no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8C2×Dic3412yes—
468C2×C8C2×Dic3412yes—
469C2×C8C2×Dic3412yes—
470C2×C8C2×Dic3412yes—
471C2×C8D4×D1712yes—
472C2×C8#6325412yes—
473C4⋊C4—172no—
474C4⋊C4—172no—
475C4⋊C4—341no—
476C4⋊C4—341no—
477C4⋊C4—341no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—341no—
488C4⋊C4—341no—
489C4⋊C4—682no—
490C4⋊C4—682no—
491C4⋊C4—682no—
492C4⋊C4—682no—
493C4⋊C4—682no—
494C4⋊C4—1361no—
495C4⋊C4—1361no—
496C4⋊C4—1361no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—682no—
500C22⋊C4—682no—
501C22⋊C4—1361no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—682no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#1058755512yes—
514C23C22×D6811yescharacteristic
515Q8—1364no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
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—21no—
534C2×C4—342no—
535C2×C4—342no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4—681no—
541C2×C4—681no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—1364no—
559C2×C4#1058755512yes—
560C2×C4C22×Dic3411yescharacteristic
561C2×C4C22×Dic3411yescharacteristic
562C2×C4C2×D4×D1711yescharacteristic
563C2×C4C2×#6325411yescharacteristic
564C2×C4#1058769611yescharacteristic
565C2×C4#1058769711yescharacteristic
566C8—22no—
567C8—22no—
568C8—22no—
569C8#1058756414yes—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#1058755511yescharacteristic
578C22D4×D6811yescharacteristic
579C22#1176346311yescharacteristic
580C22#1176360911yescharacteristic
581C22#1176360911yescharacteristic
582C22#1176361411yescharacteristic
583C22#1176362611yescharacteristic
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—682no—
590C4—682no—
591C4—1361no—
592C4—1361no—
593C4—1361no—
594C4—1361no—
595C4—1361no—
596C4—1361no—
597C4C2×#1058756411yescharacteristic
598C4#1176343711yescharacteristic
599C4D4×Dic3411yescharacteristic
600C4#1176351611yescharacteristic
601C2—42no—
602C2#1453248611yescharacteristic
603C2#1453932311yescharacteristic
604C2#1453933611yescharacteristic
605C2#1454039611yescharacteristic
606C2#1454044211yescharacteristic
607C2#1454128611yescharacteristic
608C2#1454131611yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1753C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1753D17
hypocenterC17#26079
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×#11763218C2semidirect
5. #14530267C2semidirect
6. #14530267C2semidirect
7. #14530123C2semidirect
8. #14530113C2semidirect
9. #14529522C2semidirect
10. #14529517C2semidirect
13. #14528948C2semidirect
66. C2×M16×C17Q8semidirect
103. C17×C22⋊C8Q8semidirect
370. C2×M16Dic34semidirect
417. C22⋊C8Dic34semidirect
432. C17#26079semidirect

Maximal subgroups

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

subgroupindex
C2×#117632182
C2×#117628512
#145325082
#145324752
#145302672
#145301232
#145301132
#145295222
#145295172
#145289852
#145289742
#145289482
#145288172
C17×#17532
#2607917