Finite Group Databasebeta

← #17068131

Subgroups — #17068131

Order 4352.

1062 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#14530114C212yesmaximal, complemented
5#14529977C212yesmaximal, complemented
6#14529218C211yesmaximal, characteristic, complemented
7#14529082C211yesmaximal, characteristic
8#14529038C212yesmaximal, complemented
9#14528983C212yesmaximal, complemented
10#14528879C211yesmaximal, characteristic, complemented
11C2×C17×#556C211yesmaximal, 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×#556—24no—
35C17×#556—24no—
36C2×C17×C4⋊Q8C2211yescharacteristic
37C2×C17×#126—21no—
38C2×C17×#126—21no—
39C2×C17×#124—21no—
40C2×C17×C22⋊Q8C2211yescharacteristic
41C2×C17×C22⋊Q8C2211yescharacteristic
42C2×C4×Q8×C17—21no—
43C2×C4×Q8×C17C2211yescharacteristic
44C2×C4×D4×C17C2211yescharacteristic
45C2×C17×#117C2211yescharacteristic
46C2×C17×#117C2211yescharacteristic
47C22×Q8×C17—21no—
48C22×Q8×C17C2311yescharacteristic
49C22×D4×C17D411yescharacteristic
50C23×C4×C17C2311yescharacteristic
51C23×C4×C17C2311yescharacteristic
52C23×Dic17—21no—
53C17×C4⋊Q8—22no—
54C17×C4⋊Q8—22no—
55C17×#126—28no—
56C17×#126—28no—
57C17×#124—28no—
58C17×C22⋊Q8—22no—
59C17×C22⋊Q8—22no—
60C17×C22⋊Q8—22no—
61C17×C22⋊Q8D414yes—
62C4×Q8×C17—28no—
63C4×Q8×C17—22no—
64C4×Q8×C17—22no—
65C4×D4×C17—22no—
66C4×D4×C17—22no—
67C17×#117—22no—
68C17×#117—22no—
69C17×#117—22no—
70C17×#117D414yes—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17D411yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17—21no—
83C2×C22⋊C4×C17D411yescharacteristic
84C2×C22⋊C4×C17D411yescharacteristic
85C2×C42×C17—21no—
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—24no—
114C2×Q8×C17—21no—
115C2×Q8×C17—22no—
116C2×Q8×C17—22no—
117C2×Q8×C17—42no—
118C2×Q8×C17—42no—
119C2×Q8×C17C2×D412yes—
120C2×D4×C17—22no—
121C2×D4×C17—22no—
122C2×D4×C17—41no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D412yes—
137C22×C4×C17C2×D412yes—
138C22×C4×C17C2×D411yescharacteristic
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—22no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17C2×D412yes—
153C4⋊C4×C17C2×D412yes—
154C4⋊C4×C17C2×D412yes—
155C22⋊C4×C17—24no—
156C22⋊C4×C17—24no—
157C22⋊C4×C17—24no—
158C22⋊C4×C17—24no—
159C22⋊C4×C17—22no—
160C22⋊C4×C17—22no—
161C42×C17—24no—
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#24108—171nomaximal, Sylow
182C23×C17—22no—
183C23×C17—21no—
184C23×C17—21no—
185C23×C17—22no—
186C23×C17—21no—
187C23×C17—22no—
188C23×C17—21no—
189C23×C17—21no—
190C23×C17—22no—
191C23×C17#12012yes—
192C23×C17C22×D411yescharacteristic
193Q8×C17—24no—
194Q8×C17—48no—
195Q8×C17—41no—
196Q8×C17—41no—
197D4×C17—44no—
198C2×C4×C17—21no—
199C2×C4×C17—22no—
200C2×C4×C17—21no—
201C2×C4×C17—22no—
202C2×C4×C17—22no—
203C2×C4×C17—24no—
204C2×C4×C17—22no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—42no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—42no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—42no—
221C2×C4×C17#12012yes—
222C2×C4×C17#13712yes—
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×#556D1711yescharacteristic
237C2×#483—171no—
238C2×#428—171no—
239#2487—171no—
240#2058—172no—
241#2035—172no—
242#1894—171no—
243#1870—171no—
244#1863—172no—
245#1854—172no—
246#1825—171no—
247C22×C17—22no—
248C22×C17—22no—
249C22×C17—22no—
250C22×C17—42no—
251C22×C17—42no—
252C22×C17—42no—
253C22×C17—42no—
254C22×C17—41no—
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—22no—
264C4×C17—41no—
265C4×C17—42no—
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#556—24no—
277#556—24no—
278C2×C4⋊Q8C2×D1711yescharacteristic
279C2×#126—21no—
280C2×#126—21no—
281C2×#124—21no—
282C2×C22⋊Q8C2×D1711yescharacteristic
283C2×C22⋊Q8C2×D1711yescharacteristic
284C2×C4×Q8—21no—
285C2×C4×Q8C2×D1711yescharacteristic
286C2×C4×D4C2×D1711yescharacteristic
287C2×#117C2×D1711yescharacteristic
288C2×#117C2×D1711yescharacteristic
289C22×C4⋊C4—171no—
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—81no—
312C2×C17#240611yescharacteristic
313C2×C17#300711yescharacteristic
314C2×C17#301011yescharacteristic
315C2×C17#335012yes—
316C2×C17#335612yes—
317C22×Q8—21no—
318C22×Q8C22×D1711yescharacteristic
319C22×D4C22⋊D1711yescharacteristic
320C23×C4—341no—
321C23×C4C22×D1711yescharacteristic
322C23×C4C22×D1711yescharacteristic
323C2×M16—344no—
324C2×M16—342no—
325C22×C8—172no—
326C22×C8—172no—
327C4⋊Q8—22no—
328C4⋊Q8—22no—
329#126—28no—
330#126—28no—
331#124—28no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8C22⋊D1714yes—
336C4×Q8—28no—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×D4—22no—
340C4×D4—22no—
341#117—22no—
342#117—22no—
343#117—22no—
344#117C22⋊D1714yes—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—172no—
350C2×C4⋊C4—172no—
351C2×C4⋊C4—342no—
352C2×C4⋊C4—342no—
353C2×C4⋊C4—341no—
354C2×C4⋊C4—342no—
355C2×C4⋊C4C22⋊D1711yescharacteristic
356C2×C4⋊C4C22×D1711yescharacteristic
357C2×C4⋊C4C22×D1711yescharacteristic
358C2×C4⋊C4C22×D1711yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—341no—
364C2×C22⋊C4—342no—
365C2×C22⋊C4C22⋊D1711yescharacteristic
366C2×C22⋊C4C22⋊D1711yescharacteristic
367C2×C42—21no—
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#2410811yescharacteristic, complemented, Sylow
382C24C2×C22⋊D1711yescharacteristic
383C24C2×C22⋊D1711yescharacteristic
384C2×Q8—24no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8C2×C22⋊D1712yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—41no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—342no—
405C22×C4—342no—
406C22×C4—682no—
407C22×C4—1362no—
408C22×C4C2×C22⋊D1711yescharacteristic
409C22×C4C2×C22⋊D1711yescharacteristic
410C22×C4C2×C22⋊D1712yes—
411C22×C4C2×C22⋊D1712yes—
412C22×C4C2×C22⋊D1711yescharacteristic
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—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—342no—
431C4⋊C4—342no—
432C4⋊C4—342no—
433C4⋊C4—342no—
434C4⋊C4—682no—
435C4⋊C4—1362no—
436C4⋊C4C2×C22⋊D1712yes—
437C4⋊C4C2×C22⋊D1712yes—
438C4⋊C4C2×C22⋊D1712yes—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—682no—
446C22⋊C4—1362no—
447C42—24no—
448C42—24no—
449C42—21no—
450C42—21no—
451C42—24no—
452C23—22no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23#1058763912yes—
462C23C22×C22⋊D1711yescharacteristic
463Q8—24no—
464Q8—48no—
465Q8—41no—
466Q8—41no—
467D4—44no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—682no—
492C2×C4—682no—
493C2×C4—682no—
494C2×C4—682no—
495C2×C4—682no—
496C2×C4—682no—
497C2×C4—1364no—
498C2×C4#1058763712yes—
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—42no—
510C22—42no—
511C22—42no—
512C22—41no—
513C22—41no—
514C22—42no—
515C22—41no—
516C22C2×#1058763911yescharacteristic
517C22#1176345812yes—
518C22C2×#1058763711yescharacteristic
519C22#1176360711yescharacteristic
520C22#1176361712yes—
521C4—22no—
522C4—22no—
523C4—41no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—1362no—
530C4—1362no—
531C4#1176331611yescharacteristic
532C4#1176338011yescharacteristic
533C4#1176345812yes—
534C2—42no—
535C2—81no—
536C2#1453211511yescharacteristic
537C2#1453804411yescharacteristic
538C2#1453807011yescharacteristic
539C2#1454132912yes—
540C2#1454139312yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#556C2
socleC23×C17C22×D4
prime radicalC22×D4×C17D4
hypercenterC2×#556D17
hypocenterC17#24108
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×#11763137C2semidirect
2. C2×#11762852C2semidirect
4. #14530114C2semidirect
5. #14529977C2semidirect
6. #14529218C2semidirect
8. #14529038C2semidirect
9. #14528983C2semidirect
10. #14528879C2semidirect
381. C17#24108semidirect

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
#145301142
#145299772
#145292182
#145290822
#145290382
#145289832
#145288792
C2×C17×#5562
#2410817