Finite Group Databasebeta

← #17066317

Subgroups — #17066317

Order 4352.

1222 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762964C211yesmaximal, characteristic, complemented
2C2×#11762947C211yesmaximal, characteristic
3#14531368C211yesmaximal, characteristic, complemented
4#14529922C212yesmaximal, complemented
5#14529878C212yesmaximal, complemented
6#14529125C212yesmaximal, complemented
7#14529118C212yesmaximal, complemented
8#14529107C211yesmaximal, characteristic
9#14529097C211yesmaximal, characteristic, complemented
10#14528855C211yesmaximal, characteristic, complemented
11C2×C17×#542C211yesmaximal, characteristic
12C23×Dic34C2211yescharacteristic
13C22×#63231C2212yes—
14C22×#63229C2212yes—
15C2×#10587612C2211yescharacteristic
16#11762964—24no—
17#11762947—24no—
18C2×#10587506C2212yes—
19C2×#10587481C2212yes—
20C2×#10587481C2212yes—
21C2×#10587503C2212yes—
22C2×C22⋊#3622C2211yescharacteristic
23C2×C22⋊#3622C2211yescharacteristic
24#11762715C2212yes—
25#11762715C2212yes—
26#11762714—21no—
27#11762714—21no—
28#11762670—21no—
29#11762670—21no—
30#11762571C2212yes—
31#11762571C2212yes—
32#11762564C2212yes—
33#11762564C2212yes—
34C17×#542—24no—
35C17×#542—24no—
36C2×C17×C4⋊Q8C2211yescharacteristic
37C2×C17×#126—21no—
38C2×C17×#124—21no—
39C2×C17×#124C2211yescharacteristic
40C2×C17×#123—21no—
41C2×C17×C22⋊Q8—21no—
42C2×C17×C22⋊Q8C2211yescharacteristic
43C2×C17×#121C2211yescharacteristic
44C2×C4×Q8×C17C2211yescharacteristic
45C2×C4×D4×C17C2211yescharacteristic
46C2×C17×#117C2211yescharacteristic
47C22×Q8×C17—21no—
48C22×Q8×C17C2311yescharacteristic
49C22×D4×C17—21no—
50C22×D4×C17C2311yescharacteristic
51C23×C4×C17—21no—
52C23×C4×C17C2311yescharacteristic
53C23×Dic17—21no—
54C22×Dic34—21no—
55C22×Dic34—24no—
56C22×Dic34—22no—
57C22×Dic34—22no—
58C22×Dic34—42no—
59C22×Dic34C2312yes—
60C17×C4⋊Q8—22no—
61C17×C4⋊Q8—22no—
62C17×#126—28no—
63C17×#124—28no—
64C17×#124—22no—
65C17×#124—22no—
66C17×#123—28no—
67C17×C22⋊Q8—28no—
68C17×C22⋊Q8—22no—
69C17×C22⋊Q8D414yes—
70C17×#121—22no—
71C17×#121D414yes—
72C4×Q8×C17—22no—
73C4×Q8×C17—22no—
74C4×D4×C17—22no—
75C4×D4×C17—22no—
76C17×#117—22no—
77C17×#117—22no—
78C2×C4⋊C4×C17—21no—
79C2×C4⋊C4×C17—21no—
80C2×C4⋊C4×C17—21no—
81C2×C4⋊C4×C17D411yescharacteristic
82C2×C4⋊C4×C17D411yescharacteristic
83C2×C4⋊C4×C17C2311yescharacteristic
84C2×C4⋊C4×C17C2311yescharacteristic
85C2×C22⋊C4×C17—21no—
86C2×C22⋊C4×C17—21no—
87C2×C22⋊C4×C17—21no—
88C2×C22⋊C4×C17—21no—
89C2×C22⋊C4×C17—21no—
90C2×C22⋊C4×C17D411yescharacteristic
91C2×C22⋊C4×C17D411yescharacteristic
92C2×C42×C17—21no—
93C2×C42×C17C2311yescharacteristic
94C2×C42×C17C2311yescharacteristic
95C2×C22⋊Dic17—21no—
96C2×#63231—24no—
97C2×#63231—24no—
98C2×#63231—42no—
99C2×#63229—24no—
100C2×#63229—24no—
101C2×#63229—42no—
102#10587612—42no—
103C2×C4⋊Dic17C2312yes—
104C2×C4×Dic17—21no—
105C22×#3622C2312yes—
106C22×#3622C2312yes—
107#10587506—42no—
108#10587503—42no—
109#10587502—22no—
110#10587502—22no—
111#10587502—22no—
112#10587502—22no—
113C22⋊#3622—22no—
114C22⋊#3622—22no—
115#10587481—24no—
116#10587481—24no—
117C24×C17—21no—
118C24×C17C2×D411yescharacteristic
119C2×Q8×C17—24no—
120C2×Q8×C17—21no—
121C2×Q8×C17—22no—
122C2×Q8×C17—22no—
123C2×Q8×C17—82no—
124C2×Q8×C17C2×D412yes—
125C2×D4×C17—24no—
126C2×D4×C17—21no—
127C2×D4×C17—22no—
128C2×D4×C17—22no—
129C2×D4×C17—82no—
130C2×D4×C17C2×D412yes—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—24no—
137C22×C4×C17—82no—
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D412yes—
144C22×C4×C17C2×D412yes—
145C22×C4×C17C2411yescharacteristic
146C22×Dic17—24no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—42no—
150C22×Dic17—42no—
151C2×Dic34—22no—
152C2×Dic34—22no—
153C2×Dic34—22no—
154C2×Dic34—22no—
155C2×Dic34—24no—
156C2×Dic34—22no—
157C2×Dic34—48no—
158C2×Dic34—48no—
159C2×Dic34—42no—
160C2×Dic34—42no—
161C2×Dic34#4114yes—
162C4⋊C4×C17—24no—
163C4⋊C4×C17—24no—
164C4⋊C4×C17—22no—
165C4⋊C4×C17—24no—
166C4⋊C4×C17—22no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17C2×D412yes—
170C4⋊C4×C17C2×D412yes—
171C22⋊C4×C17—24no—
172C22⋊C4×C17—24no—
173C22⋊C4×C17—24no—
174C22⋊C4×C17—24no—
175C22⋊C4×C17—22no—
176C22⋊C4×C17—22no—
177C22⋊C4×C17—24no—
178C42×C17—24no—
179C42×C17—21no—
180C42×C17—21no—
181C42×C17—21no—
182C42×C17—21no—
183C22⋊Dic17—42no—
184#63231—48no—
185#63229—48no—
186C4⋊Dic17—22no—
187C4⋊Dic17—22no—
188C4×Dic17—42no—
189C2×#3622—22no—
190C2×#3622—22no—
191C2×#3622—22no—
192C2×#3622—22no—
193C2×#3622—22no—
194C2×#3622—22no—
195#23946—171nomaximal, Sylow
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—22no—
199C23×C17—22no—
200C23×C17—22no—
201C23×C17—41no—
202C23×C17—41no—
203C23×C17—42no—
204C23×C17—42no—
205C23×C17#12012yes—
206C23×C17C22×D411yescharacteristic
207Q8×C17—24no—
208Q8×C17—42no—
209Q8×C17—84no—
210D4×C17—24no—
211D4×C17—42no—
212D4×C17—84no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—24no—
216C2×C4×C17—24no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—22no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—22no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—84no—
233C2×C4×C17#12012yes—
234C2×C4×C17C22×D411yescharacteristic
235C2×C4×C17C22×D411yescharacteristic
236C2×C4×C17C2×#4112yes—
237C2×C4×C17H(5,𝔽2)12yes—
238C2×Dic17—24no—
239C2×Dic17—48no—
240C2×Dic17—42no—
241C2×Dic17—42no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245Dic34—24no—
246Dic34—24no—
247Dic34—48no—
248Dic34—42no—
249Dic34—42no—
250#3622—42no—
251#3622—42no—
252C2×#542D1711yescharacteristic
253C2×#451—171no—
254C2×#448—171no—
255#2310—171no—
256#2024—172no—
257#2015—172no—
258#1878—172no—
259#1877—172no—
260#1875—171no—
261#1873—171no—
262#1818—171no—
263C22×C17—22no—
264C22×C17—22no—
265C22×C17—22no—
266C22×C17—42no—
267C22×C17—42no—
268C22×C17—42no—
269C22×C17—42no—
270C22×C17—42no—
271C22×C17—82no—
272C22×C17—82no—
273C22×C17C2×#12011yescharacteristic
274C22×C17#52511yescharacteristic
275C22×C17#53511yescharacteristic
276C22×C17#57712yes—
277C22×C17#57812yes—
278C4×C17—22no—
279C4×C17—42no—
280C4×C17—41no—
281C4×C17—41no—
282C4×C17—42no—
283C4×C17—42no—
284C4×C17—41no—
285C4×C17—41no—
286C4×C17#54612yes—
287C4×C17#54612yes—
288Dic17—44no—
289Dic17—82no—
290C23×Q8—171no—
291C22×Q16—172no—
292C22×SD16—172no—
293#542—24no—
294#542—24no—
295C2×C4⋊Q8C2×D1711yescharacteristic
296C2×#126—21no—
297C2×#124—21no—
298C2×#124C2×D1711yescharacteristic
299C2×#123—21no—
300C2×C22⋊Q8—21no—
301C2×C22⋊Q8C2×D1711yescharacteristic
302C2×#121C2×D1711yescharacteristic
303C2×C4×Q8C2×D1711yescharacteristic
304C2×C4×D4C2×D1711yescharacteristic
305C2×#117—171no—
306C2×#117C2×D1711yescharacteristic
307#451—344no—
308#448—344no—
309C2×#103—172no—
310C2×#103—172no—
311C2×#103—172no—
312C2×#102—172no—
313C2×C22⋊C8—171no—
314C2×C22⋊C8—171no—
315#391—172no—
316#391—172no—
317#391—341no—
318#391—341no—
319#381—341no—
320#381—341no—
321#340—172no—
322#340—172no—
323#336—172no—
324#336—172no—
325C2×C17—42no—
326C2×C17—82no—
327C2×C17#240411yescharacteristic
328C2×C17#300911yescharacteristic
329C2×C17#301111yescharacteristic
330C2×C17#312112yes—
331C2×C17#322512yes—
332C22×Q8—21no—
333C22×Q8—172no—
334C22×Q8—344no—
335C22×Q8—341no—
336C22×Q8—342no—
337C22×Q8—342no—
338C22×Q8—682no—
339C22×Q8C22×D1711yescharacteristic
340C22×D4—21no—
341C22×D4C22×D1711yescharacteristic
342C23×C4—21no—
343C23×C4—341no—
344C23×C4C22×D1711yescharacteristic
345C2×Q16—344no—
346C2×Q16—344no—
347C2×Q16—682no—
348C2×SD16—344no—
349C2×SD16—344no—
350C2×SD16—682no—
351C22×C8—172no—
352C22×C8—172no—
353C4⋊Q8—22no—
354C4⋊Q8—22no—
355#126—28no—
356#124—28no—
357#124—22no—
358#124—22no—
359#123—28no—
360C22⋊Q8—28no—
361C22⋊Q8—22no—
362C22⋊Q8C22⋊D1714yes—
363#121—22no—
364#121C22⋊D1714yes—
365C4×Q8—22no—
366C4×Q8—22no—
367C4×D4—22no—
368C4×D4—22no—
369#117—22no—
370#117—22no—
371#117—682no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—172no—
376C2×C4⋊C4C22⋊D1711yescharacteristic
377C2×C4⋊C4C22⋊D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—341no—
386C2×C22⋊C4C22⋊D1711yescharacteristic
387C2×C22⋊C4C22⋊D1711yescharacteristic
388C2×C42—21no—
389C2×C42—341no—
390C2×C42C22×D1711yescharacteristic
391C2×C42C22×D1711yescharacteristic
392#103—344no—
393#103—344no—
394#103—682no—
395#102—682no—
396C22⋊C8—342no—
397C22⋊C8—342no—
398#95—342no—
399#95—342no—
400#95—342no—
401#95—342no—
402C17#2394611yescharacteristic, complemented, Sylow
403C24—21no—
404C24C2×C22⋊D1711yescharacteristic
405C2×Q8—24no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—82no—
410C2×Q8—174no—
411C2×Q8—344no—
412C2×Q8—342no—
413C2×Q8—342no—
414C2×Q8—342no—
415C2×Q8—342no—
416C2×Q8—342no—
417C2×Q8—688no—
418C2×Q8—688no—
419C2×Q8—682no—
420C2×Q8—682no—
421C2×Q8C2×C22⋊D1712yes—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—82no—
427C2×D4C2×C22⋊D1712yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—82no—
435C22×C4—344no—
436C22×C4—342no—
437C22×C4—342no—
438C22×C4—682no—
439C22×C4—682no—
440C22×C4C2×C22⋊D1711yescharacteristic
441C22×C4C2×C22⋊D1711yescharacteristic
442C22×C4C2×C22⋊D1711yescharacteristic
443C22×C4C2×C22⋊D1711yescharacteristic
444C22×C4C2×C22⋊D1712yes—
445C22×C4C2×C22⋊D1712yes—
446C22×C4C2×C22⋊D1711yescharacteristic
447C22×C4C23×D1711yescharacteristic
448Q16—688no—
449SD16—688no—
450C2×C8—342no—
451C2×C8—342no—
452C2×C8—342no—
453C2×C8—342no—
454C2×C8—342no—
455C2×C8—342no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—342no—
464C4⋊C4—342no—
465C4⋊C4C2×C22⋊D1712yes—
466C4⋊C4C2×C22⋊D1712yes—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—24no—
474C22⋊C4—682no—
475C42—24no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—682no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23—21no—
485C23—22no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23#1058763912yes—
491C23C22×C22⋊D1711yescharacteristic
492Q8—24no—
493Q8—42no—
494Q8—84no—
495Q8—344no—
496Q8—344no—
497Q8—688no—
498Q8—682no—
499Q8—682no—
500D4—24no—
501D4—42no—
502D4—84no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—84no—
523C2×C4—344no—
524C2×C4—688no—
525C2×C4—682no—
526C2×C4—682no—
527C2×C4—682no—
528C2×C4—682no—
529C2×C4—682no—
530C2×C4#1058763912yes—
531C2×C4#41×D1712yes—
532C2×C4#1058769612yes—
533C2×C4C22×C22⋊D1711yescharacteristic
534C2×C4C22×C22⋊D1711yescharacteristic
535C8—682no—
536C8—682no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—82no—
546C22—82no—
547C22C2×#1058763911yescharacteristic
548C22#1176331611yescharacteristic
549C22#1176337311yescharacteristic
550C22#1176362012yes—
551C22#1176362412yes—
552C4—22no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—684no—
561C4—1362no—
562C4#1176344712yes—
563C4#1176344712yes—
564C2—42no—
565C2—82no—
566C2#1453209911yescharacteristic
567C2#1453806111yescharacteristic
568C2#1453807911yescharacteristic
569C2#1453900712yes—
570C2#1453988712yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#542C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenterC2×#542D17
hypocenterC17#23946
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. C2×#11762964C2semidirect
3. #14531368C2semidirect
4. #14529922C2semidirect
5. #14529878C2semidirect
6. #14529125C2semidirect
7. #14529118C2semidirect
9. #14529097C2semidirect
10. #14528855C2semidirect
402. C17#23946semidirect

Maximal subgroups

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

subgroupindex
C2×#117629642
C2×#117629472
#145313682
#145299222
#145298782
#145291252
#145291182
#145291072
#145290972
#145288552
C2×C17×#5422
#2394617