Finite Group Databasebeta

← #17068235

Subgroups — #17068235

Order 4352.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763516C211yesmaximal, characteristic, complemented
2C2×#11763099C211yesmaximal, characteristic, complemented
3C2×C4×#63229C211yesmaximal, characteristic, complemented
4#14529980C212yesmaximal, complemented
5#14529787C212yesmaximal, complemented
6#14529744C211yesmaximal, characteristic, complemented
7#14529579C211yesmaximal, characteristic, complemented
8#14529102C212yesmaximal, complemented
9#14528976C212yesmaximal, complemented
10#14528786C211yesmaximal, characteristic, complemented
11C17×#2537C211yesmaximal, characteristic
12C22×#63229C2211yescharacteristic
13#11763516—24no—
14#11763516—24no—
15C2×#10587555C2211yescharacteristic
16C2×#10587576—21no—
17C2×#10587610C2212yes—
18C2×#10587564—22no—
19C2×C4×Dic34C2211yescharacteristic
20C2×D4×Dic17—21no—
21C22×C4⋊Dic17C2212yes—
22#11763099—24no—
23C4×#63229—24no—
24C2×#10587479C2212yes—
25C2×#10587479C2211yescharacteristic
26C2×C4⋊#3622C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×#10587481C2212yes—
29C2×#10587503C2212yes—
30C2×#10587503C2211yescharacteristic
31C2×C22⋊#3622C2212yes—
32C2×C22⋊#3622C2212yes—
33C2×C4×#3622C2211yescharacteristic
34#11762571C2212yes—
35#11762571C2212yes—
36#11762571C2212yes—
37C2×C17×#121—21no—
38C2×C17×#121C2212yes—
39C2×C4×D4×C17C2211yescharacteristic
40C2×C4×D4×C17C2211yescharacteristic
41C17×#394—22no—
42C17×#384C2211yescharacteristic
43C17×#380—21no—
44C17×#380C2212yes—
45C22×D4×C17—22no—
46C22×D4×C17D411yescharacteristic
47C22×D4×C17C2311yescharacteristic
48C23×C4×C17—21no—
49C23×C4×C17C2312yes—
50C23×Dic17—22no—
51C22×Dic34—21no—
52C22×Dic34C2311yescharacteristic
53C17×#121—24no—
54C17×#121—42no—
55C17×#121—44no—
56C4×D4×C17—22no—
57C4×D4×C17—22no—
58C4×D4×C17—22no—
59C4×D4×C17—22no—
60C2×C4⋊C4×C17—21no—
61C2×C4⋊C4×C17—21no—
62C2×C4⋊C4×C17D411yescharacteristic
63C2×C4⋊C4×C17C2311yescharacteristic
64C2×C4⋊C4×C17C2312yes—
65C2×C22⋊C4×C17—22no—
66C2×C22⋊C4×C17—21no—
67C2×C22⋊C4×C17—22no—
68C2×C22⋊C4×C17D412yes—
69C2×C42×C17C2311yescharacteristic
70C2×C22⋊Dic17—22no—
71D4×Dic17—28no—
72C2×#63229—22no—
73C2×#63229—22no—
74C2×#63229—22no—
75C2×C4⋊Dic17—24no—
76C2×C4⋊Dic17—24no—
77C2×C4⋊Dic17—22no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2312yes—
81#10587610—24no—
82#10587610—24no—
83C2×#63225—21no—
84C2×#63225—22no—
85C2×C4×Dic17—21no—
86C22×#3622C2311yescharacteristic
87C22×#3622C2312yes—
88C22×#3622C2311yescharacteristic
89#10587576—28no—
90#10587564—216no—
91#10587555—22no—
92#10587555D414yescomplemented
93C4×Dic34—22no—
94C4×Dic34D414yescomplemented
95C17×#95—22no—
96#10587503—24no—
97#10587503—22no—
98C22⋊#3622—24no—
99C22⋊#3622—42no—
100#10587481—22no—
101#10587481—42no—
102#10587479—24no—
103#10587479—22no—
104C4⋊#3622—22no—
105C4×#3622—22no—
106C24×C17—21no—
107C24×C17C2×D412yes—
108C2×D4×C17—22no—
109C2×D4×C17—22no—
110C2×D4×C17—21no—
111C2×D4×C17—21no—
112C2×D4×C17—21no—
113C2×D4×C17—21no—
114C2×D4×C17—21no—
115C2×D4×C17—44no—
116C2×D4×C17—44no—
117C2×D4×C17—82no—
118C2×D4×C17SD1614yescomplemented
119C2×D4×C17#4112yes—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—21no—
127C22×C4×C17—22no—
128C22×C4×C17—42no—
129C22×C4×C17—42no—
130C22×C4×C17—42no—
131C22×C4×C17C2×D412yes—
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2411yescharacteristic
135C22×Dic17—22no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—28no—
139C22×Dic17—82no—
140C22×Dic17—82no—
141C2×Dic34—21no—
142C2×Dic34—21no—
143C2×Dic34—21no—
144C2×Dic34—21no—
145C2×Dic34—21no—
146C2×Dic34—24no—
147C2×Dic34—81no—
148C2×Dic34—81no—
149C2×Dic34C2×D412yes—
150C4⋊C4×C17—22no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—22no—
154C4⋊C4×C17—22no—
155C4⋊C4×C17—42no—
156C4⋊C4×C17—81no—
157C22⋊C4×C17—24no—
158C22⋊C4×C17—44no—
159C22⋊C4×C17—44no—
160C22⋊C4×C17—42no—
161C42×C17C2×D412yes—
162C42×C17C2×D412yes—
163C22⋊Dic17—28no—
164#63229—28no—
165C4⋊Dic17—28no—
166C4⋊Dic17—24no—
167C4⋊Dic17—21no—
168C4⋊Dic17—22no—
169C4⋊Dic17C2×D412yes—
170C4⋊Dic17C2×D414yes—
171#63225—28no—
172#63225—24no—
173C4×Dic17—24no—
174C2×#3622—21no—
175C2×#3622—21no—
176C2×#3622—21no—
177C2×#3622—22no—
178C2×#3622—22no—
179C2×#3622—22no—
180C2×#3622—21no—
181C2×#3622—21no—
182C2×#3622#4112yes—
183#24118—171nomaximal, Sylow
184C23×C17—22no—
185C23×C17—22no—
186C23×C17—22no—
187C23×C17—22no—
188C23×C17—22no—
189C23×C17—22no—
190C23×C17—22no—
191C23×C17—41no—
192C23×C17—41no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17C2×SD1614yes—
198C23×C17C22×D411yescharacteristic
199D4×C17—22no—
200D4×C17—22no—
201D4×C17—28no—
202D4×C17—41no—
203D4×C17—41no—
204D4×C17—88no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—42no—
211C2×C4×C17—42no—
212C2×C4×C17—42no—
213C2×C4×C17—42no—
214C2×C4×C17—41no—
215C2×C4×C17—42no—
216C2×C4×C17—41no—
217C2×C4×C17—42no—
218C2×C4×C17—41no—
219C2×C4×C17—42no—
220C2×C4×C17—84no—
221C2×C4×C17#12012yes—
222C2×C4×C17#12012yes—
223C2×C4×C17C2×SD1612yes—
224C2×C4×C17#13712yes—
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
231C2×Dic17—22no—
232C2×Dic17—24no—
233C2×Dic17—22no—
234C2×Dic17—41no—
235C2×Dic17—41no—
236C2×Dic17—42no—
237C2×Dic17—41no—
238C2×Dic17—42no—
239C2×Dic17—41no—
240C2×Dic17—88no—
241Dic34—22no—
242Dic34—22no—
243Dic34—41no—
244Dic34—41no—
245Dic34—84no—
246#3622—22no—
247#3622—41no—
248#3622—42no—
249C2×#554—171no—
250C2×#476—171no—
251C2×C4×SD16—171no—
252#2537D1711yescharacteristic
253#2035—172no—
254#2000—172no—
255#1990—171no—
256#1954—171no—
257#1874—172no—
258#1853—172no—
259#1803—171no—
260C22×C17—24no—
261C22×C17—24no—
262C22×C17—24no—
263C22×C17—42no—
264C22×C17—42no—
265C22×C17—42no—
266C22×C17—42no—
267C22×C17—42no—
268C22×C17—42no—
269C22×C17—42no—
270C22×C17—81no—
271C22×C17—81no—
272C22×C17—81no—
273C22×C17—81no—
274C22×C17C2×#12011yescharacteristic
275C22×C17#54611yescharacteristic
276C22×C17#54711yescharacteristic
277C22×C17C22×SD1611yescharacteristic
278C22×C17C2×#13711yescharacteristic
279C22×C17#57611yescharacteristic
280C22×C17#57711yescharacteristic
281C4×C17—22no—
282C4×C17—42no—
283C4×C17—81no—
284C4×C17—82no—
285C4×C17#44814yes—
286C4×C17#52511yescharacteristic
287C4×C17#53711yescharacteristic
288C4×C17#54811yescharacteristic
289C4×C17#55011yescharacteristic
290Dic17—42no—
291Dic17—44no—
292Dic17—42no—
293C22×SD16—171no—
294#554—344no—
295#554—344no—
296C2×C4⋊Q8—171no—
297C2×#125—341no—
298C2×C22⋊Q8—172no—
299C2×C22⋊Q8—342no—
300C2×#121—21no—
301C2×#121C2×D1712yes—
302C2×C4×Q8—171no—
303C2×C4×D4—341no—
304C2×C4×D4C2×D1711yescharacteristic
305C2×C4×D4C2×D1711yescharacteristic
306C22×C4⋊C4—172no—
307#476—344no—
308C4×SD16—344no—
309C2×#106—171no—
310C2×#106—172no—
311C2×C4⋊C8—171no—
312C2×#103—171no—
313C2×#103—172no—
314C2×#102—171no—
315C2×#102—172no—
316C2×C22⋊C8—172no—
317C2×C22⋊C8—172no—
318C2×C4×C8—171no—
319#394—22no—
320#384C2×D1711yescharacteristic
321#380—21no—
322#380C2×D1712yes—
323#340—172no—
324#340—172no—
325#340—172no—
326C2×C17—44no—
327C2×C17—82no—
328C2×C17#240711yescharacteristic
329C2×C17C2×#44811yescharacteristic
330C2×C17#301211yescharacteristic
331C2×C17#329511yescharacteristic
332C2×C17#333111yescharacteristic
333C2×C17#333511yescharacteristic
334C2×C17#335711yescharacteristic
335C22×Q8—171no—
336C22×Q8—341no—
337C22×D4—22no—
338C22×D4C22⋊D1711yescharacteristic
339C22×D4C22×D1711yescharacteristic
340C23×C4—21no—
341C23×C4—342no—
342C23×C4C22×D1712yes—
343C2×SD16—342no—
344C2×SD16—342no—
345C2×SD16—342no—
346C22×C8—171no—
347C22×C8—171no—
348C22×C8—172no—
349C4⋊Q8—174no—
350C4⋊Q8—342no—
351#125—348no—
352C22⋊Q8—3416no—
353C22⋊Q8—344no—
354C22⋊Q8—344no—
355#121—24no—
356#121—42no—
357#121—44no—
358C4×Q8—174no—
359C4×Q8—342no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—348no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—172no—
369C2×C4⋊C4—344no—
370C2×C4⋊C4—344no—
371C2×C4⋊C4—342no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—342no—
375C2×C4⋊C4C22⋊D1711yescharacteristic
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C4⋊C4C22×D1712yes—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—342no—
382C2×C22⋊C4C22⋊D1712yes—
383C2×C42—341no—
384C2×C42C22×D1711yescharacteristic
385#106—342no—
386#106—344no—
387C4⋊C8—342no—
388#103—342no—
389#103—682no—
390#102—342no—
391#102—344no—
392C22⋊C8—344no—
393C22⋊C8—682no—
394C4×C8—342no—
395#95—22no—
396C17#2411811yescharacteristic, complemented, Sylow
397C24—21no—
398C24C2×C22⋊D1712yes—
399C2×Q8—172no—
400C2×Q8—341no—
401C2×Q8—344no—
402C2×Q8—341no—
403C2×Q8—341no—
404C2×Q8—341no—
405C2×Q8—341no—
406C2×Q8—1361no—
407C2×Q8—1361no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—82no—
418C2×D4#6322914yescomplemented
419C2×D4#6325412yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—341no—
432C22×C4—341no—
433C22×C4—342no—
434C22×C4—348no—
435C22×C4—1362no—
436C22×C4—1362no—
437C22×C4C2×C22⋊D1712yes—
438C22×C4C2×C22⋊D1711yescharacteristic
439C22×C4C2×C22⋊D1711yescharacteristic
440C22×C4C23×D1711yescharacteristic
441SD16—348no—
442C2×C8—172no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—342no—
447C2×C8—341no—
448C2×C8—342no—
449C2×C8—342no—
450C2×C8—341no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—81no—
458C4⋊C4—172no—
459C4⋊C4—174no—
460C4⋊C4—348no—
461C4⋊C4—348no—
462C4⋊C4—341no—
463C4⋊C4—342no—
464C4⋊C4—344no—
465C4⋊C4—344no—
466C22⋊C4—24no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—42no—
470C22⋊C4—348no—
471C42—344no—
472C42D4×D1712yes—
473C42D4×D1712yes—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23C2×#6322914yes—
488C23C22×C22⋊D1711yescharacteristic
489Q8—342no—
490Q8—342no—
491Q8—681no—
492Q8—681no—
493Q8—1364no—
494D4—28no—
495D4—22no—
496D4—22no—
497D4—41no—
498D4—41no—
499D4—88no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—84no—
516C2×C4—342no—
517C2×C4—342no—
518C2×C4—344no—
519C2×C4—681no—
520C2×C4—681no—
521C2×C4—682no—
522C2×C4—681no—
523C2×C4—682no—
524C2×C4—681no—
525C2×C4—1368no—
526C2×C4C2×#6322912yes—
527C2×C4#1058762312yes—
528C2×C4#1058762312yes—
529C2×C4#1058763712yes—
530C2×C4C2×D4×D1711yescharacteristic
531C2×C4C2×D4×D1711yescharacteristic
532C2×C4C2×#6325411yescharacteristic
533C2×C4#1058769111yescharacteristic
534C2×C4#41×D1711yescharacteristic
535C2×C4#1058769711yescharacteristic
536C8—342no—
537C8—682no—
538C8—681no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×#1058762311yescharacteristic
554C22#1176345311yescharacteristic
555C22#1176345811yescharacteristic
556C22C22×#6322911yescharacteristic
557C22C2×#1058763711yescharacteristic
558C22#1176361211yescharacteristic
559C22#1176361711yescharacteristic
560C4—22no—
561C4—42no—
562C4—82no—
563C4—81no—
564C4—682no—
565C4—682no—
566C4—684no—
567C4#1176294714yes—
568C4#1176331911yescharacteristic
569C4#1176338811yescharacteristic
570C4#1176346511yescharacteristic
571C4#1176347711yescharacteristic
572C2—44no—
573C2—82no—
574C2#1453212511yescharacteristic
575C2C2×#1176294711yescharacteristic
576C2#1453808511yescharacteristic
577C2#1454060811yescharacteristic
578C2#1454110011yescharacteristic
579C2#1454114111yescharacteristic
580C2#1454140411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC17×#2537C2
socleC23×C17C22×D4
hypercenter#2537D17
hypocenterC17#24118
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. C2×#11763516C2semidirect
2. C2×#11763099C2semidirect
3. C2×C4×#63229C2semidirect
4. #14529980C2semidirect
5. #14529787C2semidirect
6. #14529744C2semidirect
7. #14529579C2semidirect
8. #14529102C2semidirect
9. #14528976C2semidirect
10. #14528786C2semidirect
92. #10587555D4semidirect
94. C4×Dic34D4semidirect
118. C2×D4×C17SD16semidirect
396. C17#24118semidirect
418. C2×D4#63229semidirect

Maximal subgroups

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

subgroupindex
C2×#117635162
C2×#117630992
C2×C4×#632292
#145299802
#145297872
#145297442
#145295792
#145291022
#145289762
#145287862
C17×#25372
#2411817