Finite Group Databasebeta

← #17068157

Subgroups — #17068157

Order 4352.

1088 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763142C211yesmaximal, characteristic, complemented
2C2×#11762852C211yesmaximal, characteristic, complemented
3#14533092C211yesmaximal, characteristic
4#14530119C212yesmaximal, complemented
5#14529977C212yesmaximal, complemented
6#14529218C211yesmaximal, characteristic, complemented
7#14529082C211yesmaximal, characteristic
8#14529038C212yesmaximal, complemented
9#14528972C212yesmaximal, complemented
10#14528879C211yesmaximal, characteristic, complemented
11C2×C17×#552C211yesmaximal, characteristic
12C22×#63223C2211yescharacteristic
13C2×#10587621—21no—
14C22×C4⋊Dic17C2211yescharacteristic
15#11763142—24no—
16#11762852—24no—
17C2×#10587478C2212yes—
18C2×#10587478C2212yes—
19C2×#10587479C2212yes—
20C2×#10587506C2212yes—
21C2×#10587506C2212yes—
22C2×C22⋊#3622C2211yescharacteristic
23C2×C22⋊#3622C2212yes—
24#11762739—21no—
25#11762734—21no—
26#11762707C2212yes—
27#11762670—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—22no—
90C2×C22⋊Dic17—21no—
91#10587621—44no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17—22no—
94C2×C4⋊Dic17—22no—
95C2×C4⋊Dic17C2312yes—
96C2×C4⋊Dic17C2312yes—
97C2×#63225—22no—
98C2×#63223—24no—
99C2×#63223—22no—
100C22×#3622C2312yes—
101C22×#3622C2312yes—
102#10587506—24no—
103#10587506—24no—
104#10587502—22no—
105#10587502—21no—
106#10587502—22no—
107C22⋊#3622—24no—
108C22⋊#3622—22no—
109#10587479—24no—
110#10587478—24no—
111#10587478—24no—
112C24×C17C2×D411yescharacteristic
113C24×C17C2×D411yescharacteristic
114C2×Q8×C17—21no—
115C2×Q8×C17—22no—
116C2×Q8×C17—22no—
117C2×Q8×C17C2×D412yes—
118C2×D4×C17—22no—
119C2×D4×C17—22no—
120C2×D4×C17—41no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—24no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—42no—
133C22×C4×C17—42no—
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D412yes—
139C22×C4×C17C2×D412yes—
140C22×C4×C17C2411yescharacteristic
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—42no—
144C22×Dic17—42no—
145C22×Dic17—42no—
146C4⋊C4×C17—24no—
147C4⋊C4×C17—24no—
148C4⋊C4×C17—24no—
149C4⋊C4×C17—24no—
150C4⋊C4×C17—24no—
151C4⋊C4×C17—24no—
152C4⋊C4×C17—22no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17C2×D412yes—
157C4⋊C4×C17C2×D412yes—
158C4⋊C4×C17C2×D412yes—
159C22⋊C4×C17—24no—
160C22⋊C4×C17—24no—
161C22⋊C4×C17—24no—
162C22⋊C4×C17—22no—
163C22⋊C4×C17—22no—
164C42×C17—24no—
165C42×C17—24no—
166C42×C17—21no—
167C42×C17—21no—
168C22⋊Dic17—42no—
169C22⋊Dic17—44no—
170C4⋊Dic17—22no—
171C4⋊Dic17—22no—
172C4⋊Dic17—22no—
173C4⋊Dic17—22no—
174C4⋊Dic17—42no—
175#63225—44no—
176#63223—28no—
177C2×#3622—22no—
178C2×#3622—22no—
179C2×#3622—22no—
180C2×#3622—22no—
181C2×#3622—22no—
182C2×#3622#4114yes—
183#24110—171nomaximal, Sylow
184C23×C17—21no—
185C23×C17—22no—
186C23×C17—22no—
187C23×C17—21no—
188C23×C17—21no—
189C23×C17—21no—
190C23×C17—22no—
191C23×C17—22no—
192C23×C17#12012yes—
193C23×C17AGL(1,ℤ/8)12yes—
194C23×C17C22×D411yescharacteristic
195Q8×C17—24no—
196Q8×C17—41no—
197Q8×C17—41no—
198D4×C17—44no—
199C2×C4×C17—22no—
200C2×C4×C17—22no—
201C2×C4×C17—22no—
202C2×C4×C17—24no—
203C2×C4×C17—21no—
204C2×C4×C17—22no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—22no—
214C2×C4×C17—48no—
215C2×C4×C17—42no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—42no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—42no—
224C2×C4×C17#12012yes—
225C2×C4×C17C22×D411yescharacteristic
226C2×C4×C17C22×D411yescharacteristic
227C2×C4×C17C2×#4112yes—
228C2×C4×C172−1+412yes—
229C2×Dic17—42no—
230C2×Dic17—42no—
231C2×Dic17—42no—
232C2×Dic17—42no—
233C2×Dic17—42no—
234C2×Dic17—42no—
235C2×Dic17—84no—
236#3622—24no—
237#3622—42no—
238C2×#552D1711yescharacteristic
239C2×#484—171no—
240C2×#428—171no—
241#2486—171no—
242#2059—172no—
243#2035—172no—
244#1894—171no—
245#1870—171no—
246#1863—172no—
247#1852—172no—
248#1825—171no—
249C22×C17—24no—
250C22×C17—22no—
251C22×C17—22no—
252C22×C17—22no—
253C22×C17—22no—
254C22×C17—42no—
255C22×C17—42no—
256C22×C17—42no—
257C22×C17—41no—
258C22×C17—41no—
259C22×C17C2×#12011yescharacteristic
260C22×C17#54712yes—
261C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C17#57511yescharacteristic
263C22×C17#57812yes—
264C4×C17—22no—
265C4×C17—41no—
266C4×C17—42no—
267C4×C17—41no—
268C4×C17—42no—
269C4×C17—42no—
270C4×C17—41no—
271C4×C17—42no—
272C4×C17#52511yescharacteristic
273C4×C17#53611yescharacteristic
274C4×C17#54712yes—
275Dic17—82no—
276Dic17—82no—
277C22×M16—171no—
278#552—24no—
279#552—24no—
280C2×#126—21no—
281C2×#125—21no—
282C2×#125C2×D1711yescharacteristic
283C2×#123—21no—
284C2×#123—341no—
285C2×C22⋊Q8C2×D1711yescharacteristic
286C2×C22⋊Q8C2×D1711yescharacteristic
287C2×C4×Q8C2×D1711yescharacteristic
288C2×C4×D4C2×D1711yescharacteristic
289C2×#117—21no—
290C2×#117C2×D1711yescharacteristic
291C22×C4⋊C4—171no—
292C22×C4⋊C4C2×D1711yescharacteristic
293#484—344no—
294#428—344no—
295C2×#107—172no—
296C2×#107—172no—
297C2×#106—172no—
298C2×#103—172no—
299C2×#103—172no—
300C2×C22⋊C8—171no—
301C2×C22⋊C8—172no—
302#398—341no—
303#397—341no—
304#389—172no—
305#381—341no—
306#380—171no—
307#380—172no—
308#340—172no—
309#340—172no—
310#340—172no—
311#336—172no—
312C2×C17—42no—
313C2×C17—42no—
314C2×C17#240611yescharacteristic
315C2×C17#300611yescharacteristic
316C2×C17#301111yescharacteristic
317C2×C17#334312yes—
318C2×C17#335212yes—
319C22×Q8C22×D1711yescharacteristic
320C22×D4C22⋊D1711yescharacteristic
321C23×C4—21no—
322C23×C4—341no—
323C23×C4C22×D1711yescharacteristic
324C23×C4C22×D1711yescharacteristic
325C2×M16—344no—
326C2×M16—342no—
327C22×C8—172no—
328C22×C8—172no—
329#126—28no—
330#125—28no—
331#125—22no—
332#125—22no—
333#123—28no—
334#123—684no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8C22⋊D1714yes—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×D4—22no—
342C4×D4—22no—
343#117—28no—
344#117—22no—
345#117C22⋊D1714yes—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—172no—
355C2×C4⋊C4—172no—
356C2×C4⋊C4—342no—
357C2×C4⋊C4—342no—
358C2×C4⋊C4—341no—
359C2×C4⋊C4—342no—
360C2×C4⋊C4C22⋊D1711yescharacteristic
361C2×C4⋊C4C22×D1711yescharacteristic
362C2×C4⋊C4C22×D1711yescharacteristic
363C2×C4⋊C4C22×D1711yescharacteristic
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—341no—
368C2×C22⋊C4—342no—
369C2×C22⋊C4C22⋊D1711yescharacteristic
370C2×C22⋊C4C22⋊D1711yescharacteristic
371C2×C42—21no—
372C2×C42—21no—
373C2×C42C22×D1711yescharacteristic
374#107—344no—
375#107—344no—
376#106—344no—
377#103—344no—
378#103—344no—
379C22⋊C8—344no—
380C22⋊C8—342no—
381#95—342no—
382#95—341no—
383#95—342no—
384C17#2411011yescharacteristic, complemented, Sylow
385C24C2×C22⋊D1711yescharacteristic
386C24C2×C22⋊D1711yescharacteristic
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8C2×C22⋊D1712yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—41no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—342no—
408C22×C4—342no—
409C22×C4—682no—
410C22×C4—682no—
411C22×C4—682no—
412C22×C4C2×C22⋊D1711yescharacteristic
413C22×C4C2×C22⋊D1711yescharacteristic
414C22×C4C2×C22⋊D1711yescharacteristic
415C22×C4C2×C22⋊D1712yes—
416C22×C4C2×C22⋊D1712yes—
417C22×C4C2×C22⋊D1711yescharacteristic
418C22×C4C23×D1711yescharacteristic
419M16—348no—
420C2×C8—174no—
421C2×C8—342no—
422C2×C8—342no—
423C2×C8—342no—
424C2×C8—342no—
425C2×C8—342no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—342no—
437C4⋊C4—342no—
438C4⋊C4—342no—
439C4⋊C4—342no—
440C4⋊C4—684no—
441C4⋊C4—682no—
442C4⋊C4C2×C22⋊D1712yes—
443C4⋊C4C2×C22⋊D1712yes—
444C4⋊C4C2×C22⋊D1712yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—682no—
451C22⋊C4—684no—
452C42—24no—
453C42—21no—
454C42—21no—
455C42—24no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—22no—
463C23—22no—
464C23#1058763512yes—
465C23#1058763912yes—
466C23C22×C22⋊D1711yescharacteristic
467Q8—24no—
468Q8—41no—
469Q8—41no—
470D4—44no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—48no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—682no—
497C2×C4—682no—
498C2×C4—682no—
499C2×C4—682no—
500C2×C4—682no—
501C2×C4—682no—
502C2×C4—1364no—
503C2×C4#1058763912yes—
504C2×C4#41×D1712yes—
505C2×C4#1058769712yes—
506C2×C4C22×C22⋊D1711yescharacteristic
507C2×C4C22×C22⋊D1711yescharacteristic
508C8—344no—
509C8—682no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—24no—
515C22—41no—
516C22—42no—
517C22—42no—
518C22—41no—
519C22—42no—
520C22C2×#1058763911yescharacteristic
521C22#1176345812yes—
522C22C2×#1058763511yescharacteristic
523C22#1176360711yescharacteristic
524C22#1176362412yes—
525C4—22no—
526C4—41no—
527C4—42no—
528C4—41no—
529C4—42no—
530C4—42no—
531C4—41no—
532C4—42no—
533C4—1362no—
534C4—1362no—
535C4#1176331611yescharacteristic
536C4#1176338011yescharacteristic
537C4#1176345812yes—
538C2—42no—
539C2—42no—
540C2#1453211511yescharacteristic
541C2#1453803611yescharacteristic
542C2#1453807911yescharacteristic
543C2#1454122412yes—
544C2#1454135312yes—
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#24110
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×#11763142C2semidirect
2. C2×#11762852C2semidirect
4. #14530119C2semidirect
5. #14529977C2semidirect
6. #14529218C2semidirect
8. #14529038C2semidirect
9. #14528972C2semidirect
10. #14528879C2semidirect
384. C17#24110semidirect

Maximal subgroups

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

subgroupindex
C2×#117631422
C2×#117628522
#145330922
#145301192
#145299772
#145292182
#145290822
#145290382
#145289722
#145288792
C2×C17×#5522
#2411017