Finite Group Databasebeta

← #17068261

Subgroups — #17068261

Order 4352.

1350 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763464C211yesmaximal, characteristic
2C2×#11763094C211yesmaximal, characteristic, complemented
3#14533702C211yesmaximal, characteristic, complemented
4#14529971C212yesmaximal, complemented
5#14529784C212yesmaximal, complemented
6#14529745C211yesmaximal, characteristic, complemented
7#14529578C211yesmaximal, characteristic, complemented
8#14529088C212yesmaximal, complemented
9#14528975C212yesmaximal, complemented
10#14528784C211yesmaximal, characteristic, complemented
11C2×C4×SD16×C17C211yesmaximal, characteristic, complemented
12C22×#63252—21no—
13#11763464—24no—
14#11763464—24no—
15C2×#10587555C2211yescharacteristic
16C2×#10587569—22no—
17C2×#10587610—21no—
18C2×#10587632C2212yes—
19C2×#10587624C2212yes—
20C2×C4×D68C2211yescharacteristic
21C2×C4⋊C4×D17C2212yes—
22#11763094—24no—
23C2×#10587479C2212yes—
24C2×C4⋊#3622C2211yescharacteristic
25C2×#10587506C2212yes—
26C2×#10587503C2212yes—
27C2×#10587491C2212yes—
28C2×#10587491C2212yes—
29#11762731—22no—
30#11762711—21no—
31#11762707—21no—
32#11762679C2211yescharacteristic
33#11762666C2212yes—
34#11762572C2212yes—
35#11762572C2212yes—
36#11762571C2212yes—
37C22×SD16×C17C2211yescharacteristic
38C2×C4×Q8×C17C2211yescharacteristic
39C2×C4×D4×C17C2211yescharacteristic
40C4×SD16×C17—24no—
41C2×C17×#106C2211yescharacteristic
42C2×C17×#103C2211yescharacteristic
43C2×C17×#102C2211yescharacteristic
44C2×C4×C8×C17C2211yescharacteristic
45C22×Q8×C17C2311yescharacteristic
46C22×D4×C17C2311yescharacteristic
47C23×C4×C17—21no—
48C22×C22⋊D17—22no—
49C2×#63252—28no—
50C2×#63252—44no—
51C2×#63252—44no—
52C22×D68D411yescharacteristic
53C22×C4×D17—22no—
54C22×C4×D17C2312yes—
55C22×Dic34—21no—
56C2×SD16×C17—22no—
57C2×SD16×C17—22no—
58C2×SD16×C17—22no—
59C22×C8×C17C2311yescharacteristic
60C22×C8×C17C2311yescharacteristic
61C4×Q8×C17—22no—
62C4×Q8×C17—22no—
63C4×D4×C17—22no—
64C4×D4×C17D414yes—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C22⋊C4×C17—21no—
69C2×C42×C17—21no—
70C2×C42×C17C2311yescharacteristic
71C2×C22⋊Dic17—22no—
72#10587632—24no—
73#10587632—42no—
74#10587624—24no—
75#10587624—24no—
76C2×#63227—22no—
77C2×#63227—22no—
78C2×#63227D412yes—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17D411yescharacteristic
83C2×C4⋊Dic17C2312yes—
84C2×C4⋊Dic17C2312yes—
85#10587610—28no—
86C2×#63225—22no—
87C2×#63225—22no—
88C22×#3622C2312yes—
89C4⋊C4×D17—24no—
90C4⋊C4×D17—24no—
91#10587569—216no—
92C4×D68—22no—
93C4×D68—22no—
94#10587555—22no—
95#10587555D414yes—
96C17×#106—22no—
97C17×#103—22no—
98C17×#102—22no—
99C4×C8×C17—22no—
100#10587506—24no—
101#10587503—42no—
102#10587502—22no—
103#10587491—24no—
104#10587491—42no—
105#10587479—24no—
106C4⋊#3622—22no—
107C24×C17—21no—
108C23×D17C2×D412yes—
109C2×Q8×C17—21no—
110C2×Q8×C17—21no—
111C2×Q8×C17—21no—
112C2×Q8×C17—21no—
113C2×Q8×C17—21no—
114C2×Q8×C17#4112yescomplemented
115C2×D4×C17—21no—
116C2×D4×C17—21no—
117C2×D4×C17—21no—
118C2×D4×C17—21no—
119C2×D4×C17—21no—
120C2×D4×C17C2×D412yes—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—24no—
124C22×C4×C17—21no—
125C22×C4×C17—42no—
126C22×C4×C17—42no—
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2411yescharacteristic
130C2×C22⋊D17—28no—
131C2×C22⋊D17—44no—
132C2×C22⋊D17—44no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17C2×D412yes—
136#63252—432no—
137C2×D68—22no—
138C2×D68—22no—
139C2×D68SD1614yescomplemented
140C2×C4×D17—28no—
141C2×C4×D17—22no—
142C2×C4×D17—22no—
143C2×C4×D17—22no—
144C2×C4×D17—22no—
145C2×C4×D17—22no—
146C2×C4×D17—22no—
147C2×C4×D17—44no—
148C2×C4×D17—44no—
149C2×Dic34—24no—
150C2×Dic34—42no—
151C2×Dic34—42no—
152SD16×C17—28no—
153C2×C8×C17—21no—
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17#4112yescomplemented
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—42no—
163C4⋊C4×C17C2×D412yes—
164C22⋊C4×C17—24no—
165C42×C17—42no—
166C42×C17C2×D412yes—
167C42×C17C2×D412yes—
168C22⋊Dic17—28no—
169#63227—24no—
170#63227—44no—
171#63227—82no—
172C4⋊Dic17—24no—
173C4⋊Dic17—22no—
174C4⋊Dic17—22no—
175C4⋊Dic17—22no—
176C4⋊Dic17—22no—
177C4⋊Dic17—42no—
178C4⋊Dic17—81no—
179C4⋊Dic17C2×D414yes—
180#63225—28no—
181#63225—44no—
182C2×#3622—22no—
183C2×#3622—22no—
184C2×#3622—22no—
185#24120—171nomaximal, Sylow
186C23×C17—22no—
187C23×C17—41no—
188C23×C17—41no—
189C23×C17—41no—
190C23×C17—41no—
191C23×C17—41no—
192C23×C17—41no—
193C23×C17C22×D411yescharacteristic
194C22×D17—22no—
195C22×D17—22no—
196C22×D17—22no—
197C22×D17—22no—
198C22×D17—22no—
199C22×D17—22no—
200C22×D17C2×SD1614yes—
201Q8×C17—22no—
202Q8×C17—22no—
203Q8×C17—41no—
204Q8×C17—41no—
205D4×C17—22no—
206D4×C17—22no—
207D4×C17—41no—
208D4×C17—41no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—22no—
212C2×C4×C17—22no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—48no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17#12012yes—
222C2×C4×C17#12012yes—
223C2×C4×C17C2×SD1612yes—
224C2×C4×C17AGL(1,ℤ/8)12yes—
225C2×C4×C17C22×D411yescharacteristic
226C2×C4×C17C22×D411yescharacteristic
227C2×C4×C17C2×#4111yescharacteristic
228C2×C4×C17C2×#4111yescharacteristic
229C2×C4×C17H(5,𝔽2)11yescharacteristic
230C2×C4×C172−1+411yescharacteristic
231C22⋊D17—416no—
232C2×Dic17—22no—
233C2×Dic17—24no—
234C2×Dic17—22no—
235C2×Dic17—22no—
236C2×Dic17—42no—
237C2×Dic17—42no—
238C2×Dic17—42no—
239C2×Dic17—42no—
240C2×Dic17—42no—
241D68—28no—
242C4×D17—416no—
243C4×D17—42no—
244C4×D17—42no—
245C4×D17—42no—
246C4×D17—42no—
247Dic34—48no—
248C8×C17—22no—
249C8×C17—41no—
250#3622—42no—
251C2×#547—171no—
252C2×#475—171no—
253C2×C4×SD16D1711yescharacteristic, complemented
254#2538—171no—
255#2034—172no—
256#2000—172no—
257#1990—171no—
258#1954—171no—
259#1872—172no—
260#1853—172no—
261#1802—171no—
262C22×C17—42no—
263C22×C17—42no—
264C22×C17—42no—
265C22×C17—81no—
266C22×C17—81no—
267C22×C17—81no—
268C22×C17—81no—
269C22×C17C2×#12011yescharacteristic
270C22×C17#54611yescharacteristic
271C22×C17#54711yescharacteristic
272C22×C17C22×SD1611yescharacteristic
273C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C17#57711yescharacteristic
275C22×C17#57811yescharacteristic
276C2×D17—24no—
277C2×D17—24no—
278C2×D17—24no—
279C2×D17—42no—
280C2×D17—42no—
281C2×D17—42no—
282C2×D17—42no—
283C4×C17—22no—
284C4×C17—42no—
285C4×C17—42no—
286C4×C17—81no—
287C4×C17#45014yes—
288C4×C17#52511yescharacteristic
289C4×C17#53711yescharacteristic
290C4×C17#54811yescharacteristic
291C4×C17#55011yescharacteristic
292Dic17—42no—
293Dic17—44no—
294Dic17—82no—
295C22×#41—341no—
296C22×SD16C2×D1711yescharacteristic
297#547—344no—
298#547—344no—
299C2×C4⋊Q8—171no—
300C2×#123—342no—
301C2×C22⋊Q8—172no—
302C2×C22⋊Q8—341no—
303C2×#121—172no—
304C2×C4×Q8C2×D1711yescharacteristic
305C2×C4×D4—171no—
306C2×C4×D4C2×D1711yescharacteristic
307C22×C4⋊C4—172no—
308#475—344no—
309C4×SD16—24no—
310C2×#106—172no—
311C2×#106C2×D1711yescharacteristic
312C2×C4⋊C8—171no—
313C2×#103—172no—
314C2×#103C2×D1711yescharacteristic
315C2×#102—172no—
316C2×#102C2×D1711yescharacteristic
317C2×C22⋊C8—172no—
318C2×C22⋊C8—172no—
319C2×C4×C8C2×D1711yescharacteristic
320#396—342no—
321#390—341no—
322#389—341no—
323#384—171no—
324#380—172no—
325#340—172no—
326#340—172no—
327#340—172no—
328C2×C17—82no—
329C2×C17#240711yescharacteristic
330C2×C17C2×#45011yescharacteristic
331C2×C17#301311yescharacteristic
332C2×C17#329811yescharacteristic
333C2×C17#333211yescharacteristic
334C2×C17#333511yescharacteristic
335C2×C17#335311yescharacteristic
336D17—44no—
337C2×#41—348no—
338C2×#41—684no—
339C2×#41—684no—
340C22×Q8—341no—
341C22×Q8C22×D1711yescharacteristic
342C22×D4—171no—
343C22×D4—342no—
344C22×D4C22×D1711yescharacteristic
345C23×C4—21no—
346C23×C4—172no—
347C23×C4—342no—
348C2×SD16—22no—
349C2×SD16—22no—
350C2×SD16—22no—
351C22×C8—172no—
352C22×C8C22×D1711yescharacteristic
353C22×C8C22×D1711yescharacteristic
354C4⋊Q8—174no—
355C4⋊Q8—342no—
356#123—3416no—
357C22⋊Q8—348no—
358C22⋊Q8—344no—
359C22⋊Q8—682no—
360#121—344no—
361#121—344no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×D4—22no—
365C4×D4—342no—
366C4×D4—342no—
367C4×D4C22⋊D1714yes—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—172no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—172no—
372C2×C4⋊C4—344no—
373C2×C4⋊C4—342no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—344no—
376C2×C4⋊C4—342no—
377C2×C4⋊C4—341no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C4⋊C4C22×D1711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—172no—
383C2×C22⋊C4—342no—
384C2×C22⋊C4—342no—
385C2×C22⋊C4—342no—
386C2×C42—21no—
387C2×C42C22×D1711yescharacteristic
388#106—22no—
389#106—344no—
390C4⋊C8—342no—
391#103—22no—
392#103—344no—
393#102—22no—
394#102—682no—
395C22⋊C8—344no—
396C22⋊C8—682no—
397C4×C8—22no—
398#95—342no—
399C17#2412011yescharacteristic, complemented, Sylow
400C24—21no—
401C24—172no—
402#41—6832no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—344no—
409C2×Q8—682no—
410C2×Q8—682no—
411C2×Q8#6325212yescomplemented
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—174no—
418C2×D4—348no—
419C2×D4—342no—
420C2×D4—342no—
421C2×D4—684no—
422C2×D4—684no—
423C2×D4C2×C22⋊D1712yes—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—172no—
431C22×C4—348no—
432C22×C4—342no—
433C22×C4—342no—
434C22×C4—342no—
435C22×C4—342no—
436C22×C4—342no—
437C22×C4—342no—
438C22×C4—342no—
439C22×C4—342no—
440C22×C4—684no—
441C22×C4—684no—
442C22×C4D4×D1711yescharacteristic
443C22×C4D4×D1711yescharacteristic
444C22×C4C23×D1711yescharacteristic
445SD16—28no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—342no—
452C2×C8—342no—
453C2×C8—342no—
454C2×C8#6325212yescomplemented
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—174no—
460C4⋊C4—348no—
461C4⋊C4—342no—
462C4⋊C4—342no—
463C4⋊C4—344no—
464C4⋊C4—342no—
465C4⋊C4—342no—
466C4⋊C4—684no—
467C4⋊C4—682no—
468C4⋊C4—1361no—
469C4⋊C4C2×C22⋊D1712yes—
470C22⋊C4—24no—
471C22⋊C4—348no—
472C22⋊C4—344no—
473C22⋊C4—684no—
474C22⋊C4—1362no—
475C42—42no—
476C42D4×D1712yes—
477C42C2×C22⋊D1712yes—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—174no—
486C23—342no—
487C23—342no—
488C23—342no—
489C23—342no—
490C23—342no—
491C23—342no—
492C23C2×D4×D1711yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497Q8—688no—
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502D4—348no—
503D4—6816no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—48no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—342no—
517C2×C4—342no—
518C2×C4—344no—
519C2×C4—342no—
520C2×C4—6816no—
521C2×C4—682no—
522C2×C4—682no—
523C2×C4—682no—
524C2×C4—682no—
525C2×C4—682no—
526C2×C4—682no—
527C2×C4—682no—
528C2×C4—682no—
529C2×C4—682no—
530C2×C4SD16×D1712yes—
531C2×C4#1058760012yes—
532C2×C4#1058762312yes—
533C2×C4#1058762312yes—
534C2×C4C2×#6325211yescharacteristic
535C2×C4C2×#6325211yescharacteristic
536C2×C4C2×D4×D1711yescharacteristic
537C2×C4#1058769611yescharacteristic
538C2×C4#1058769711yescharacteristic
539C2×C4C22×C22⋊D1711yescharacteristic
540C8—22no—
541C8—41no—
542C8—682no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—344no—
551C22—344no—
552C22—344no—
553C22—682no—
554C22—682no—
555C22—682no—
556C22—682no—
557C22C2×#1058762311yescharacteristic
558C22#1176345511yescharacteristic
559C22#1176346411yescharacteristic
560C22C2×SD16×D1711yescharacteristic
561C22C2×#1058760011yescharacteristic
562C22#1176361611yescharacteristic
563C22#1176362311yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—81no—
568C4—682no—
569C4—684no—
570C4—1362no—
571C4#1176295814yes—
572C4#1176332011yescharacteristic
573C4#1176338711yescharacteristic
574C4#1176346811yescharacteristic
575C4#1176348011yescharacteristic
576C2—82no—
577C2—684no—
578C2#1453212911yescharacteristic
579C2C2×#1176295811yescharacteristic
580C2#1453809511yescharacteristic
581C2#1454065111yescharacteristic
582C2#1454111311yescharacteristic
583C2#1454114311yescharacteristic
584C2#1454137011yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C4×SD16×C17C2
socleC23×C17C22×D4
prime radicalC2×#11763464C2
hypercenterC2×C4×SD16D17
hypocenterC17#24120
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#11763094C2semidirect
3. #14533702C2semidirect
4. #14529971C2semidirect
5. #14529784C2semidirect
6. #14529745C2semidirect
7. #14529578C2semidirect
8. #14529088C2semidirect
9. #14528975C2semidirect
10. #14528784C2semidirect
11. C2×C4×SD16×C17C2semidirect
114. C2×Q8×C17#41semidirect
139. C2×D68SD16semidirect
158. C2×C8×C17#41semidirect
253. C2×C4×SD16D17semidirect
399. C17#24120semidirect
411. C2×Q8#63252semidirect
454. C2×C8#63252semidirect

Maximal subgroups

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

subgroupindex
C2×#117634642
C2×#117630942
#145337022
#145299712
#145297842
#145297452
#145295782
#145290882
#145289752
#145287842
C2×C4×SD16×C172
#2412017