Finite Group Databasebeta

← #17093203

Subgroups — #17093203

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×#11763217C211yesmaximal, characteristic, complemented
2C2×#11762849C211yesmaximal, characteristic
3#14532502C211yesmaximal, characteristic
4#14532476C211yesmaximal, characteristic
5#14530270C211yesmaximal, characteristic, complemented
6#14530268C211yesmaximal, characteristic, complemented
7#14530125C211yesmaximal, characteristic, complemented
8#14530110C211yesmaximal, characteristic, complemented
9#14529521C211yesmaximal, characteristic, complemented
10#14529516C211yesmaximal, characteristic, complemented
11#14528985C211yesmaximal, characteristic
12#14528947C211yesmaximal, characteristic, complemented
13#14528815C211yesmaximal, characteristic
14#14528570C211yesmaximal, characteristic
15C17×#1852C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×#10587574C2211yescharacteristic
18C2×#10587576C2211yescharacteristic
19C2×#10587612C2211yescharacteristic
20#11763217—24no—
21#11762849—24no—
22C2×#10587489C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587478C2211yescharacteristic
26C2×#10587488C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×#10587479C2211yescharacteristic
29C2×#10587479C2211yescharacteristic
30C2×#10587486C2211yescharacteristic
31C2×C8×Dic17C2211yescharacteristic
32#11762748—21no—
33#11762742—21no—
34#11762732—21no—
35#11762721—21no—
36#11762713—21no—
37#11762708C2211yescharacteristic
38#11762708C2211yescharacteristic
39#11762705C2211yescharacteristic
40#11762705C2211yescharacteristic
41#11762692—21no—
42#11762682—21no—
43#11762674C2211yescharacteristic
44#11762674C2211yescharacteristic
45#11762669—21no—
46#11762669C2211yescharacteristic
47C4×C22⋊Dic17C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762563C2211yescharacteristic
53#11762563C2211yescharacteristic
54C22×C4⋊C4×C17C2211yescharacteristic
55C2×C17×#107C2211yescharacteristic
56C2×C17×#107C2211yescharacteristic
57C2×C17×C22⋊C8C2211yescharacteristic
58C17×#380C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17C2311yescharacteristic
63C22×Dic34—21no—
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C2×C4⋊C4×C17—22no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—41no—
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C22⋊C4×C17—21no—
74C2×C22⋊Dic17—21no—
75C2×C22⋊Dic17—21no—
76C2×C22⋊Dic17D411yescharacteristic
77C2×C22⋊Dic17D411yescharacteristic
78#10587612—22no—
79#10587612—41no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17—21no—
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×#63225—21no—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×#63225—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17C2311yescharacteristic
93C2×C4×Dic17C2311yescharacteristic
94C2×#63223—22no—
95C2×#63223—22no—
96C2×#63223Q814yescomplemented
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99#10587576—22no—
100#10587576—41no—
101#10587574—22no—
102#10587574—41no—
103C17×#107—22no—
104C17×#107—41no—
105C17×C22⋊C8Q814yescomplemented
106C17×#95—21no—
107#10587502—21no—
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#10587502D411yescharacteristic
117#10587502D411yescharacteristic
118#10587489—41no—
119#10587488—22no—
120#10587486—22no—
121C8×Dic17—22no—
122#10587479—22no—
123#10587479—22no—
124#10587479—22no—
125#10587478—22no—
126#10587478—22no—
127#10587478D414yes—
128C24×C17C2×D411yescharacteristic
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—42no—
138C22×C4×C17—42no—
139C22×C4×C17—81no—
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×Q812yes—
142C22×C4×C17C2×Q812yes—
143C22×C4×C17C2411yescharacteristic
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17C2×D411yescharacteristic
149C22×Dic17C2×D411yescharacteristic
150C22×Dic17C2×D411yescharacteristic
151C22×Dic17C2×D411yescharacteristic
152C2×Dic34—42no—
153C2×Dic34—42no—
154C2×Dic34—81no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17C2×Q812yes—
160C2×C8×C17C2×Q812yes—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—42no—
169C4⋊C4×C17C2×D412yes—
170C22⋊C4×C17—81no—
171C22⋊Dic17—41no—
172C22⋊Dic17—41no—
173C22⋊Dic17—42no—
174C22⋊Dic17—42no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—42no—
183C4⋊Dic17—42no—
184C4⋊Dic17—81no—
185C4⋊Dic17C2×D412yes—
186#63225—42no—
187#63225—42no—
188#63225—81no—
189#63225—81no—
190C4×Dic17—21no—
191C4×Dic17—21no—
192C4×Dic17—21no—
193C4×Dic17—21no—
194C4×Dic17—42no—
195#63223—28no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622C2×D412yes—
199C2×#3622C2×Q812yes—
200C2×#3622C2×Q812yes—
201C2×#3622#4112yes—
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—22no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—84no—
230C2×C4×C17C4⋊Q812yes—
231C2×C4×C17C22×D411yescharacteristic
232C2×C4×C17C22×Q811yescharacteristic
233C2×C4×C17C22×Q811yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17H(5,𝔽2)11yescharacteristic
236C2×C4×C172−1+411yescharacteristic
237C2×Dic17—22no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
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—41no—
261Dic34—84no—
262C8×C17—22no—
263C8×C17—22no—
264#3622—22no—
265#3622C22⋊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#1852D1711yescharacteristic
278#1846—171no—
279#1809—171no—
280#1753—171no—
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—42no—
297C4×C17—81no—
298C4×C17—81no—
299C4×C17—81no—
300C4×C17C2×C22⋊Q811yescharacteristic
301C4×C17#54411yescharacteristic
302C4×C17D4×Q811yescharacteristic
303C4×C17#55411yescharacteristic
304Dic17—41no—
305Dic17—41no—
306Dic17—41no—
307Dic17—42no—
308Dic17—41no—
309Dic17—81no—
310Dic17—81no—
311Dic17—81no—
312C22×M16—171no—
313C2×C4⋊Q8—171no—
314C2×#125—171no—
315C2×#117—171no—
316C22×C4⋊C4C2×D1711yescharacteristic
317#501—344no—
318#428—344no—
319C2×#107—171no—
320C2×#107—171no—
321C2×#107—171no—
322C2×#107—171no—
323C2×#107C2×D1711yescharacteristic
324C2×#107C2×D1711yescharacteristic
325C2×#106—171no—
326C2×#106—171no—
327C2×#106—171no—
328C2×#106—171no—
329C2×C22⋊C8C2×D1711yescharacteristic
330C2×#97—171no—
331C2×C4×C8—171no—
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—341no—
347#380C2×D1711yescharacteristic
348C4×C22⋊C4—171no—
349#340—171no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340C2×D1711yescharacteristic
354#340C2×D1711yescharacteristic
355#336—171no—
356#336—171no—
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—21no—
367C23×C4C22×D1711yescharacteristic
368C2×M16—174no—
369C2×M16—342no—
370C2×M16—342no—
371C22×C8—171no—
372C22×C8—171no—
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—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—41no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—341no—
394C2×C4⋊C4—341no—
395C2×C4⋊C4C22×D1711yescharacteristic
396C2×C4⋊C4C22×D1711yescharacteristic
397C2×C4⋊C4C22×D1711yescharacteristic
398C2×C4⋊C4C22×D1711yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—171no—
401C2×C22⋊C4—171no—
402C2×C22⋊C4—341no—
403C2×C22⋊C4—341no—
404C2×C42—171no—
405C2×C42—171no—
406C2×C42—341no—
407#107—22no—
408#107—41no—
409#107—174no—
410#107—342no—
411#107—342no—
412#107—681no—
413#106—342no—
414#106—342no—
415#106—342no—
416#106—342no—
417C22⋊C8Dic3414yescomplemented
418#97—342no—
419C4×C8—342no—
420#95—21no—
421#95—171no—
422#95—171no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429#95—341no—
430#95—341no—
431#95—341no—
432C17#2607911yescharacteristic, complemented, Sylow
433C24D4×D1711yescharacteristic
434C2×Q8—682no—
435C2×Q8—682no—
436C2×Q8—1361no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—171no—
449C22×C4—171no—
450C22×C4—171no—
451C22×C4—171no—
452C22×C4—341no—
453C22×C4—341no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4C2×Dic3412yes—
457C22×C4D4×D1711yescharacteristic
458C22×C4Q8×D1712yes—
459C22×C4C23×D1711yescharacteristic
460M16—348no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—172no—
466C2×C8—172no—
467C2×C8—172no—
468C2×C8—172no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8C2×Dic3412yes—
472C2×C8C2×Dic3412yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—172no—
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—1361no—
494C4⋊C4—1361no—
495C4⋊C4—1361no—
496C4⋊C4D4×D1712yes—
497C22⋊C4—81no—
498C22⋊C4—681no—
499C22⋊C4—681no—
500C22⋊C4—682no—
501C22⋊C4—682no—
502C42—341no—
503C42—341no—
504C42—341no—
505C42—341no—
506C42—682no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#1058757412yes—
514C23C2×D4×D1711yescharacteristic
515Q8—1364no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—84no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—342no—
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—681no—
559C2×C4#1058757412yes—
560C2×C4C22×Dic3411yescharacteristic
561C2×C4C2×D4×D1711yescharacteristic
562C2×C4#1058769111yescharacteristic
563C2×C4C2×Q8×D1711yescharacteristic
564C2×C4#41×D1711yescharacteristic
565C2×C4#1058769711yescharacteristic
566C8—22no—
567C8—22no—
568C8—174no—
569C8—342no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#1058757411yescharacteristic
578C22#1176344111yescharacteristic
579C22#1176345711yescharacteristic
580C22#1176360511yescharacteristic
581C22#1176360811yescharacteristic
582C22#1176361011yescharacteristic
583C22#1176362211yescharacteristic
584C4—22no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—681no—
590C4—682no—
591C4—681no—
592C4—681no—
593C4—681no—
594C4—1361no—
595C4—1361no—
596C4—1361no—
597C4C22⋊Q8×D1711yescharacteristic
598C4#1176343811yescharacteristic
599C4D4×Dic3411yescharacteristic
600C4#1176351811yescharacteristic
601C2—42no—
602C2#1453248511yescharacteristic
603C2#1453932011yescharacteristic
604C2#1453933411yescharacteristic
605C2#1454039511yescharacteristic
606C2#1454044111yescharacteristic
607C2#1454128711yescharacteristic
608C2#1454130711yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1852C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1852D17
hypocenterC17#26079
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
1. C2×#11763217C2semidirect
5. #14530270C2semidirect
6. #14530268C2semidirect
7. #14530125C2semidirect
8. #14530110C2semidirect
9. #14529521C2semidirect
10. #14529516C2semidirect
12. #14528947C2semidirect
96. C2×#63223Q8semidirect
105. C17×C22⋊C8Q8semidirect
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×#117632172
C2×#117628492
#145325022
#145324762
#145302702
#145302682
#145301252
#145301102
#145295212
#145295162
#145289852
#145289472
#145288152
#145285702
C17×#18522
#2607917