Finite Group Databasebeta

← #17068168

Subgroups — #17068168

Order 4352.

1068 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763137C211yesmaximal, characteristic, complemented
2C2×#11762852C211yesmaximal, characteristic, complemented
3#14533101C211yesmaximal, characteristic
4#14530119C212yesmaximal, complemented
5#14529984C212yesmaximal, complemented
6#14529218C211yesmaximal, characteristic, complemented
7#14529082C211yesmaximal, characteristic
8#14529052C212yesmaximal, complemented
9#14528977C212yesmaximal, complemented
10#14528879C211yesmaximal, characteristic, complemented
11C2×C17×#552C211yesmaximal, characteristic
12C22×#63223C2211yescharacteristic
13C22×C4⋊Dic17C2211yescharacteristic
14#11763137—24no—
15#11762852—24no—
16C2×#10587478C2212yes—
17C2×#10587479C2212yes—
18C2×#10587479C2212yes—
19C2×#10587506C2212yes—
20C2×#10587506C2212yes—
21C2×C22⋊#3622C2211yescharacteristic
22C2×C22⋊#3622C2212yes—
23#11762748—21no—
24#11762745—21no—
25#11762707C2212yes—
26#11762689—21no—
27#11762681—21no—
28#11762669C2211yescharacteristic
29#11762669C2212yes—
30#11762571C2212yes—
31#11762571C2212yes—
32#11762571C2212yes—
33#11762564C2212yes—
34C17×#552—24no—
35C17×#552—24no—
36C2×C17×#126—21no—
37C2×C17×#125—21no—
38C2×C17×#125C2211yescharacteristic
39C2×C17×#123—21no—
40C2×C17×C22⋊Q8C2211yescharacteristic
41C2×C17×C22⋊Q8C2211yescharacteristic
42C2×C4×Q8×C17C2211yescharacteristic
43C2×C4×D4×C17C2211yescharacteristic
44C2×C17×#117—21no—
45C2×C17×#117C2211yescharacteristic
46C22×C4⋊C4×C17C2211yescharacteristic
47C22×Q8×C17C2311yescharacteristic
48C22×D4×C17D411yescharacteristic
49C23×C4×C17—21no—
50C23×C4×C17C2311yescharacteristic
51C23×C4×C17C2311yescharacteristic
52C23×Dic17—21no—
53C17×#126—28no—
54C17×#125—28no—
55C17×#125—22no—
56C17×#125—22no—
57C17×#123—28no—
58C17×C22⋊Q8—22no—
59C17×C22⋊Q8—22no—
60C17×C22⋊Q8—22no—
61C17×C22⋊Q8D414yes—
62C4×Q8×C17—22no—
63C4×Q8×C17—22no—
64C4×D4×C17—22no—
65C4×D4×C17—22no—
66C17×#117—28no—
67C17×#117—22no—
68C17×#117D414yes—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—22no—
76C2×C4⋊C4×C17—22no—
77C2×C4⋊C4×C17D411yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17—21no—
83C2×C22⋊C4×C17—21no—
84C2×C22⋊C4×C17D411yescharacteristic
85C2×C22⋊C4×C17D411yescharacteristic
86C2×C42×C17—21no—
87C2×C42×C17—21no—
88C2×C42×C17C2311yescharacteristic
89C2×C22⋊Dic17—21no—
90C2×C22⋊Dic17—22no—
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17—22no—
93C2×C4⋊Dic17—22no—
94C2×C4⋊Dic17C2312yes—
95C2×C4⋊Dic17C2312yes—
96C2×#63225—22no—
97C2×#63223—24no—
98C2×#63223—22no—
99C22×#3622C2312yes—
100C22×#3622C2312yes—
101#10587506—24no—
102#10587506—24no—
103#10587502—22no—
104#10587502—21no—
105#10587502—22no—
106C22⋊#3622—24no—
107C22⋊#3622—22no—
108#10587479—24no—
109#10587479—24no—
110#10587478—24no—
111C24×C17C2×D411yescharacteristic
112C24×C17C2×D411yescharacteristic
113C2×Q8×C17—21no—
114C2×Q8×C17—22no—
115C2×Q8×C17—22no—
116C2×Q8×C17C2×D412yes—
117C2×D4×C17—22no—
118C2×D4×C17—22no—
119C2×D4×C17—41no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—24no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17—42no—
132C22×C4×C17—42no—
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D412yes—
138C22×C4×C17C2×D412yes—
139C22×C4×C17C2411yescharacteristic
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—42no—
143C22×Dic17—82no—
144C4⋊C4×C17—24no—
145C4⋊C4×C17—24no—
146C4⋊C4×C17—24no—
147C4⋊C4×C17—24no—
148C4⋊C4×C17—24no—
149C4⋊C4×C17—24no—
150C4⋊C4×C17—22no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17C2×D412yes—
155C4⋊C4×C17C2×D412yes—
156C4⋊C4×C17C2×D412yes—
157C22⋊C4×C17—24no—
158C22⋊C4×C17—24no—
159C22⋊C4×C17—24no—
160C22⋊C4×C17—22no—
161C22⋊C4×C17—22no—
162C42×C17—24no—
163C42×C17—24no—
164C42×C17—21no—
165C42×C17—21no—
166C22⋊Dic17—42no—
167C22⋊Dic17—82no—
168C4⋊Dic17—22no—
169C4⋊Dic17—22no—
170C4⋊Dic17—22no—
171C4⋊Dic17—22no—
172C4⋊Dic17—42no—
173#63225—82no—
174#63223—28no—
175C2×#3622—22no—
176C2×#3622—22no—
177C2×#3622—22no—
178C2×#3622—22no—
179C2×#3622—22no—
180C2×#3622#4114yes—
181#24111—171nomaximal, Sylow
182C23×C17—21no—
183C23×C17—22no—
184C23×C17—22no—
185C23×C17—21no—
186C23×C17—21no—
187C23×C17—21no—
188C23×C17—22no—
189C23×C17—22no—
190C23×C17#12012yes—
191C23×C17#13712yes—
192C23×C17C22×D411yescharacteristic
193Q8×C17—24no—
194Q8×C17—41no—
195Q8×C17—41no—
196D4×C17—44no—
197C2×C4×C17—22no—
198C2×C4×C17—22no—
199C2×C4×C17—22no—
200C2×C4×C17—24no—
201C2×C4×C17—21no—
202C2×C4×C17—22no—
203C2×C4×C17—22no—
204C2×C4×C17—22no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—22no—
212C2×C4×C17—48no—
213C2×C4×C17—42no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—42no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—42no—
222C2×C4×C17#12012yes—
223C2×C4×C17C22×D411yescharacteristic
224C2×C4×C17C22×D411yescharacteristic
225C2×C4×C17C2×#4112yes—
226C2×C4×C172−1+412yes—
227C2×Dic17—42no—
228C2×Dic17—42no—
229C2×Dic17—42no—
230C2×Dic17—42no—
231C2×Dic17—42no—
232C2×Dic17—42no—
233C2×Dic17—84no—
234#3622—24no—
235#3622—42no—
236C2×#552D1711yescharacteristic
237C2×#483—171no—
238C2×#428—171no—
239#2487—171no—
240#2059—172no—
241#2036—172no—
242#1894—171no—
243#1870—171no—
244#1865—172no—
245#1853—172no—
246#1825—171no—
247C22×C17—24no—
248C22×C17—22no—
249C22×C17—22no—
250C22×C17—22no—
251C22×C17—22no—
252C22×C17—42no—
253C22×C17—42no—
254C22×C17—42no—
255C22×C17—41no—
256C22×C17—41no—
257C22×C17C2×#12011yescharacteristic
258C22×C17#54712yes—
259C22×C17C2×#13711yescharacteristic
260C22×C17#57511yescharacteristic
261C22×C17#57712yes—
262C4×C17—22no—
263C4×C17—41no—
264C4×C17—42no—
265C4×C17—41no—
266C4×C17—42no—
267C4×C17—42no—
268C4×C17—41no—
269C4×C17—42no—
270C4×C17#52511yescharacteristic
271C4×C17#53611yescharacteristic
272C4×C17#54712yes—
273Dic17—82no—
274Dic17—82no—
275C22×M16—171no—
276#552—24no—
277#552—24no—
278C2×#126—21no—
279C2×#125—21no—
280C2×#125C2×D1711yescharacteristic
281C2×#123—21no—
282C2×C22⋊Q8C2×D1711yescharacteristic
283C2×C22⋊Q8C2×D1711yescharacteristic
284C2×C4×Q8C2×D1711yescharacteristic
285C2×C4×D4C2×D1711yescharacteristic
286C2×#117—21no—
287C2×#117C2×D1711yescharacteristic
288C22×C4⋊C4—171no—
289C22×C4⋊C4C2×D1711yescharacteristic
290#483—344no—
291#428—344no—
292C2×#107—172no—
293C2×#106—172no—
294C2×#106—172no—
295C2×#103—172no—
296C2×#103—172no—
297C2×C22⋊C8—171no—
298C2×C22⋊C8—172no—
299#400—341no—
300#399—341no—
301#389—172no—
302#386—341no—
303#385—341no—
304#380—171no—
305#380—172no—
306#340—172no—
307#340—172no—
308#340—172no—
309#336—172no—
310C2×C17—42no—
311C2×C17—42no—
312C2×C17#240611yescharacteristic
313C2×C17#300711yescharacteristic
314C2×C17#301011yescharacteristic
315C2×C17#334412yes—
316C2×C17#335112yes—
317C22×Q8C22×D1711yescharacteristic
318C22×D4C22⋊D1711yescharacteristic
319C23×C4—21no—
320C23×C4—341no—
321C23×C4C22×D1711yescharacteristic
322C23×C4C22×D1711yescharacteristic
323C2×M16—344no—
324C2×M16—342no—
325C22×C8—172no—
326C22×C8—172no—
327#126—28no—
328#125—28no—
329#125—22no—
330#125—22no—
331#123—28no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8C22⋊D1714yes—
336C4×Q8—22no—
337C4×Q8—22no—
338C4×D4—22no—
339C4×D4—22no—
340#117—28no—
341#117—22no—
342#117C22⋊D1714yes—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—172no—
352C2×C4⋊C4—172no—
353C2×C4⋊C4—342no—
354C2×C4⋊C4—342no—
355C2×C4⋊C4—341no—
356C2×C4⋊C4—342no—
357C2×C4⋊C4C22⋊D1711yescharacteristic
358C2×C4⋊C4C22×D1711yescharacteristic
359C2×C4⋊C4C22×D1711yescharacteristic
360C2×C4⋊C4C22×D1711yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—341no—
365C2×C22⋊C4—342no—
366C2×C22⋊C4C22⋊D1711yescharacteristic
367C2×C22⋊C4C22⋊D1711yescharacteristic
368C2×C42—21no—
369C2×C42—21no—
370C2×C42C22×D1711yescharacteristic
371#107—344no—
372#106—344no—
373#106—344no—
374#103—344no—
375#103—344no—
376C22⋊C8—344no—
377C22⋊C8—342no—
378#95—342no—
379#95—341no—
380#95—342no—
381C17#2411111yescharacteristic, complemented, Sylow
382C24C2×C22⋊D1711yescharacteristic
383C24C2×C22⋊D1711yescharacteristic
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8C2×C22⋊D1712yes—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—41no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—342no—
405C22×C4—342no—
406C22×C4—682no—
407C22×C4—1362no—
408C22×C4C2×C22⋊D1711yescharacteristic
409C22×C4C2×C22⋊D1711yescharacteristic
410C22×C4C2×C22⋊D1711yescharacteristic
411C22×C4C2×C22⋊D1712yes—
412C22×C4C2×C22⋊D1712yes—
413C22×C4C2×C22⋊D1711yescharacteristic
414C22×C4C23×D1711yescharacteristic
415M16—348no—
416C2×C8—174no—
417C2×C8—342no—
418C2×C8—342no—
419C2×C8—342no—
420C2×C8—342no—
421C2×C8—342no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—342no—
433C4⋊C4—342no—
434C4⋊C4—342no—
435C4⋊C4—342no—
436C4⋊C4—682no—
437C4⋊C4—1362no—
438C4⋊C4C2×C22⋊D1712yes—
439C4⋊C4C2×C22⋊D1712yes—
440C4⋊C4C2×C22⋊D1712yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—682no—
447C22⋊C4—1362no—
448C42—24no—
449C42—21no—
450C42—21no—
451C42—24no—
452C23—22no—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—22no—
460C23#1058763712yes—
461C23#1058763912yes—
462C23C22×C22⋊D1711yescharacteristic
463Q8—24no—
464Q8—41no—
465Q8—41no—
466D4—44no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—682no—
493C2×C4—682no—
494C2×C4—682no—
495C2×C4—682no—
496C2×C4—682no—
497C2×C4—682no—
498C2×C4—1364no—
499C2×C4#1058763912yes—
500C2×C4#41×D1712yes—
501C2×C4#1058769712yes—
502C2×C4C22×C22⋊D1711yescharacteristic
503C2×C4C22×C22⋊D1711yescharacteristic
504C8—344no—
505C8—682no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—24no—
511C22—41no—
512C22—42no—
513C22—42no—
514C22—41no—
515C22—42no—
516C22C2×#1058763911yescharacteristic
517C22#1176345812yes—
518C22C2×#1058763711yescharacteristic
519C22#1176360711yescharacteristic
520C22#1176361712yes—
521C4—22no—
522C4—41no—
523C4—42no—
524C4—41no—
525C4—42no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—1362no—
530C4—1362no—
531C4#1176331611yescharacteristic
532C4#1176338011yescharacteristic
533C4#1176345812yes—
534C2—42no—
535C2—42no—
536C2#1453211511yescharacteristic
537C2#1453804411yescharacteristic
538C2#1453807011yescharacteristic
539C2#1454123912yes—
540C2#1454134412yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#552C2
socleC23×C17C22×D4
prime radicalC22×D4×C17D4
hypercenterC2×#552D17
hypocenterC17#24111
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#11763137C2semidirect
2. C2×#11762852C2semidirect
4. #14530119C2semidirect
5. #14529984C2semidirect
6. #14529218C2semidirect
8. #14529052C2semidirect
9. #14528977C2semidirect
10. #14528879C2semidirect
381. C17#24111semidirect

Maximal subgroups

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

subgroupindex
C2×#117631372
C2×#117628522
#145331012
#145301192
#145299842
#145292182
#145290822
#145290522
#145289772
#145288792
C2×C17×#5522
#2411117