Finite Group Databasebeta

← #17036735

Subgroups — #17036735

Order 4352.

1218 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762849C211yesmaximal, characteristic, complemented
2C2×#11762810C211yesmaximal, characteristic
3#14531275C211yesmaximal, characteristic
4C2×C4⋊C4×Dic17C211yesmaximal, characteristic, complemented
5#14529428C212yesmaximal, complemented
6#14529421C212yesmaximal, complemented
7#14529414C212yesmaximal, complemented
8#14529407C212yesmaximal, complemented
9#14528504C211yesmaximal, characteristic, complemented
10#14528477C211yesmaximal, characteristic, complemented
11C2×C17×#417C211yesmaximal, characteristic
12C22×#63223C2211yescharacteristic
13C2×#10587612C2211yescharacteristic
14C22×C4⋊Dic17C2211yescharacteristic
15C22×C4×Dic17—21no—
16C22×C4×Dic17C2211yescharacteristic
17#11762849—24no—
18C2×#10587478C2212yes—
19C2×#10587479C2212yes—
20#11762810C418yescomplemented
21C2×#10587506C2212yes—
22C2×#10587503C2212yes—
23#11762716C2211yescharacteristic
24#11762716C2211yescharacteristic
25#11762712C2211yescharacteristic
26#11762712C2211yescharacteristic
27#11762689—21no—
28#11762689—21no—
29C4⋊C4×Dic17—21no—
30C4⋊C4×Dic17—21no—
31C4⋊C4×Dic17C2212yes—
32C4⋊C4×Dic17C2212yes—
33C4×C22⋊Dic17—21no—
34C4×C22⋊Dic17—21no—
35C2×#10587502—21no—
36#11762572C2212yes—
37#11762572C2212yes—
38#11762563C2212yes—
39#11762563C2212yes—
40C22×#41×C17C2211yescharacteristic
41C22×M16×C17C2211yescharacteristic
42C22×C4⋊C4×C17C2211yescharacteristic
43C17×#417—24no—
44C2×C17×#103C2212yes—
45C2×C17×#102C2212yes—
46C2×#41×C17—22no—
47C2×#41×C17—22no—
48C2×#41×C17—22no—
49C2×#41×C17—22no—
50C2×#41×C17—22no—
51C2×#41×C17C2×C414yes—
52C22×Q8×C17C2311yescharacteristic
53C22×D4×C17—21no—
54C22×D4×C17C2311yescharacteristic
55C23×C4×C17—21no—
56C23×C4×C17—21no—
57C23×C4×C17C2311yescharacteristic
58C23×Dic17—21no—
59C23×Dic17D411yescharacteristic
60C23×Dic17D411yescharacteristic
61C2×M16×C17—22no—
62C2×M16×C17—22no—
63C2×M16×C17—22no—
64C22×C8×C17C2312yes—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—22no—
67C2×C4⋊C4×C17D414yes—
68C2×C4⋊C4×C17C2312yes—
69C2×C22⋊Dic17—21no—
70C2×C22⋊Dic17—21no—
71C2×C22⋊Dic17—21no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74#10587612—22no—
75#10587612—22no—
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17—22no—
78C2×C4⋊Dic17C2×C414yes—
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2312yes—
82C2×C4×Dic17—21no—
83C2×C4×Dic17—22no—
84C2×C4×Dic17—22no—
85C2×C4×Dic17—22no—
86C2×C4×Dic17—22no—
87C2×C4×Dic17—44no—
88C2×C4×Dic17D411yescharacteristic
89C2×C4×Dic17D411yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C2×#63223—22no—
93C2×#63223—22no—
94C2×#63223C2×C414yes—
95C22×#3622C2312yes—
96C17×#103—24no—
97C17×#102—24no—
98#10587506C2×C418yescomplemented
99#10587503C2×C418yescomplemented
100#10587502—21no—
101#10587502—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—22no—
105#10587502—22no—
106#10587479—24no—
107#10587478—24no—
108C24×C17—21no—
109C24×C17C2×D411yescharacteristic
110#41×C17—28no—
111#41×C17—28no—
112#41×C17—44no—
113#41×C17—44no—
114C2×Q8×C17—21no—
115C2×Q8×C17—21no—
116C2×Q8×C17—21no—
117C2×Q8×C17—21no—
118C2×Q8×C17—21no—
119C2×Q8×C17C22×C412yes—
120C2×D4×C17—24no—
121C2×D4×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17—42no—
127C2×D4×C17—42no—
128C2×D4×C17C22×C412yes—
129C22×C4×C17—24no—
130C22×C4×C17—21no—
131C22×C4×C17—22no—
132C22×C4×C17—24no—
133C22×C4×C17—42no—
134C22×C4×C17—42no—
135C22×C4×C17—42no—
136C22×C4×C17—42no—
137C22×C4×C17C22×C412yes—
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D412yes—
140C22×C4×C17#4112yes—
141C22×C4×C17#4112yes—
142C22×C4×C17#4112yes—
143C22×C4×C17#4112yes—
144C22×C4×C17C2411yescharacteristic
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—24no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—41no—
154C22×Dic17—41no—
155C22×Dic17—42no—
156C22×Dic17—42no—
157C22×Dic17C2×D411yescharacteristic
158C22×Dic17C2×D411yescharacteristic
159C22×Dic17C2×D411yescharacteristic
160C22×Dic17C2×D411yescharacteristic
161M16×C17—44no—
162C2×C8×C17—22no—
163C2×C8×C17—22no—
164C2×C8×C17—22no—
165C4⋊C4×C17—24no—
166C4⋊C4×C17—22no—
167C4⋊C4×C17C2×D414yes—
168C22⋊Dic17—42no—
169C22⋊Dic17—42no—
170C22⋊Dic17—42no—
171C22⋊Dic17—42no—
172C22⋊Dic17—42no—
173C4⋊Dic17—24no—
174C4⋊Dic17—22no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17C22×C414yes—
180C4×Dic17—22no—
181C4×Dic17—22no—
182C4×Dic17—21no—
183C4×Dic17—21no—
184C4×Dic17—21no—
185C4×Dic17—21no—
186C4×Dic17—44no—
187C4×Dic17—42no—
188C4×Dic17—44no—
189#63223—28no—
190C2×#3622—22no—
191C2×#3622—22no—
192C2×#3622C22×C414yes—
193#21059—171nomaximal, Sylow
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—22no—
199C23×C17—41no—
200C23×C17—41no—
201C23×C17—41no—
202C23×C17—41no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17C4×D412yes—
206C23×C17#12112yes—
207C23×C17#12312yes—
208C23×C17C22×D411yescharacteristic
209Q8×C17—22no—
210Q8×C17—22no—
211Q8×C17—41no—
212Q8×C17—41no—
213D4×C17—22no—
214D4×C17—22no—
215D4×C17—48no—
216D4×C17—41no—
217D4×C17—41no—
218C2×C4×C17—22no—
219C2×C4×C17—24no—
220C2×C4×C17—48no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17—48no—
226C2×C4×C17#11714yes—
227C2×C4×C17C4×D412yes—
228C2×C4×C17C4×D414yes—
229C2×C4×C17#12112yes—
230C2×C4×C17#12312yes—
231C2×C4×C17#12414yes—
232C2×C4×C17#12614yes—
233C2×C4×C17C23×C411yescharacteristic
234C2×C4×C17C22×D411yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×Dic17—28no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—24no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—44no—
252C2×Dic17—48no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17AGL(1,ℤ/8)12yes—
262C2×Dic17#13712yes—
263C8×C17—42no—
264#3622—24no—
265C2×#428—171no—
266C2×#417—171no—
267C2×#417D1711yescharacteristic
268#2297—171no—
269C2×C4×C4⋊C4—171no—
270#1928—172no—
271#1927—172no—
272#1926—172no—
273#1925—172no—
274#1736—171no—
275#1728—171no—
276C22×C17—22no—
277C22×C17—22no—
278C22×C17—22no—
279C22×C17—22no—
280C22×C17—22no—
281C22×C17—22no—
282C22×C17—42no—
283C22×C17—42no—
284C22×C17—42no—
285C22×C17—81no—
286C22×C17—81no—
287C22×C17—81no—
288C22×C17—81no—
289C22×C17#38814yes—
290C22×C17C2×C4×D411yescharacteristic
291C22×C17C2×#12111yescharacteristic
292C22×C17C2×#12311yescharacteristic
293C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C17C2×#13711yescharacteristic
295C22×C17#57511yescharacteristic
296C22×C17#57511yescharacteristic
297C4×C17—44no—
298C4×C17—42no—
299C4×C17#38814yes—
300C4×C17C2×#11711yescharacteristic
301C4×C17C2×C4×D411yescharacteristic
302C4×C17C2×#12411yescharacteristic
303C4×C17C2×#12611yescharacteristic
304Dic17—22no—
305Dic17—22no—
306Dic17—41no—
307Dic17—41no—
308Dic17—44no—
309Dic17—81no—
310Dic17—81no—
311C22×#41C2×D1711yescharacteristic
312C22×M16—171no—
313C22×M16C2×D1711yescharacteristic
314C2×#117—171no—
315C22×C4⋊C4—171no—
316C22×C4⋊C4C2×D1711yescharacteristic
317C22×C42—171no—
318C22×C42—341no—
319#428—344no—
320C2×#107—172no—
321C2×#106—172no—
322#417—24no—
323#417—178no—
324C2×#103—172no—
325C2×#103C2×D1712yes—
326C2×#102—172no—
327C2×#102C2×D1712yes—
328#391—171no—
329#391—171no—
330#390—171no—
331#390—171no—
332#386—341no—
333#386—341no—
334C4×C4⋊C4—172no—
335C4×C4⋊C4—172no—
336C4×C4⋊C4—341no—
337C4×C4⋊C4—341no—
338C4×C22⋊C4—341no—
339C4×C22⋊C4—341no—
340C2×#95—341no—
341#340—172no—
342#340—172no—
343#336—172no—
344#336—172no—
345C2×C17—24no—
346C2×C17—82no—
347C2×C17C2×#38811yescharacteristic
348C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
349C2×C17C4×#13711yescharacteristic
350C2×C17#303611yescharacteristic
351C2×C17#303711yescharacteristic
352C2×C17#309211yescharacteristic
353C2×C17#309311yescharacteristic
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41C4×D1714yes—
360C22×Q8C22×D1711yescharacteristic
361C22×D4—21no—
362C22×D4C22×D1711yescharacteristic
363C23×C4—21no—
364C23×C4—21no—
365C23×C4—171no—
366C23×C4—171no—
367C23×C4—341no—
368C23×C4C22×D1711yescharacteristic
369C2×M16—22no—
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—174no—
373C2×M16—342no—
374C2×M16—342no—
375C22×C8—172no—
376C22×C8C22×D1712yes—
377#117—342no—
378#117—342no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—174no—
382C2×C4⋊C4—172no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—342no—
387C2×C4⋊C4C22⋊D1714yes—
388C2×C4⋊C4C22×D1712yes—
389C2×C22⋊C4—341no—
390C2×C22⋊C4—341no—
391C2×C22⋊C4—341no—
392C2×C22⋊C4—341no—
393C2×C22⋊C4—341no—
394C2×C42—171no—
395C2×C42—171no—
396C2×C42—171no—
397C2×C42—171no—
398C2×C42—342no—
399C2×C42—342no—
400C2×C42—342no—
401C2×C42—342no—
402C2×C42—341no—
403C2×C42—684no—
404#107—344no—
405#106—344no—
406#103—24no—
407#103—178no—
408#102—24no—
409#102—178no—
410#95—342no—
411#95—342no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416C17#2105911yescharacteristic, complemented, Sylow
417C24—21no—
418C24D4×D1711yescharacteristic
419#41—28no—
420#41—28no—
421#41—44no—
422#41—44no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8C2×C4×D1712yes—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4C2×C4×D1712yes—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—171no—
447C22×C4—171no—
448C22×C4—171no—
449C22×C4—171no—
450C22×C4—342no—
451C22×C4—342no—
452C22×C4—342no—
453C22×C4—342no—
454C22×C4—344no—
455C22×C4—342no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—681no—
459C22×C4—681no—
460C22×C4—682no—
461C22×C4—682no—
462C22×C4C2×C4×D1712yes—
463C22×C4#6325212yes—
464C22×C4D4×D1711yescharacteristic
465C22×C4#6325412yes—
466C22×C4#6325412yes—
467C22×C4#6325412yes—
468C22×C4C2×C22⋊D1712yes—
469C22×C4C23×D1711yescharacteristic
470M16—44no—
471M16—348no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—174no—
476C2×C8—342no—
477C2×C8—342no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—174no—
481C4⋊C4—342no—
482C4⋊C4—344no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4C2×C22⋊D1714yes—
488C22⋊C4—682no—
489C22⋊C4—682no—
490C22⋊C4—682no—
491C22⋊C4—682no—
492C22⋊C4—682no—
493C42—341no—
494C42—341no—
495C42—341no—
496C42—341no—
497C42—342no—
498C42—342no—
499C42—684no—
500C42—684no—
501C42—682no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23#1058756712yes—
514C23#1058756912yes—
515C23#1058762512yes—
516C23C2×D4×D1711yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—48no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—48no—
534C2×C4—172no—
535C2×C4—172no—
536C2×C4—348no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—344no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—688no—
549C2×C4—684no—
550C2×C4—682no—
551C2×C4—682no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4#1058756314yes—
559C2×C4#1058756514yes—
560C2×C4#1058756712yes—
561C2×C4#1058756912yes—
562C2×C4C4×C22⋊D1714yes—
563C2×C4#1058762214yes—
564C2×C4#1058762512yes—
565C2×C4C22×C4×D1711yescharacteristic
566C2×C4C2×#6325211yescharacteristic
567C2×C4C2×#6325411yescharacteristic
568C2×C4C2×#6325411yescharacteristic
569C2×C4C2×#6325411yescharacteristic
570C2×C4C22×C22⋊D1711yescharacteristic
571C8—42no—
572C8—344no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22#1176270314yes—
587C22C2×#1058756711yescharacteristic
588C22C2×#1058762511yescharacteristic
589C22C2×#1058756911yescharacteristic
590C22AGL(1,ℤ/8)×D1711yescharacteristic
591C22#137×D1711yescharacteristic
592C22#1176360511yescharacteristic
593C22#1176360811yescharacteristic
594C4—44no—
595C4—42no—
596C4—342no—
597C4—342no—
598C4—684no—
599C4—681no—
600C4—681no—
601C4—1361no—
602C4—1361no—
603C4#1176270314yes—
604C4C2×#1058756311yescharacteristic
605C4C2×C4×C22⋊D1711yescharacteristic
606C4C2×#1058762211yescharacteristic
607C4C2×#1058756511yescharacteristic
608C2—24no—
609C2—82no—
610C2C2×#1176270311yescharacteristic
611C2#1453759711yescharacteristic
612C2#1453760811yescharacteristic
613C2#1453824511yescharacteristic
614C2#1453825311yescharacteristic
615C2#1453870511yescharacteristic
616C2#1453871411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#417C2
socleC23×C17C22×D4
prime radicalC22×#41×C17C22
cosocleC22×C4×C17C24
hypercenterC2×#417D17
hypocenterC17#21059
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
1. C2×#11762849C2semidirect
4. C2×C4⋊C4×Dic17C2semidirect
5. #14529428C2semidirect
6. #14529421C2semidirect
7. #14529414C2semidirect
8. #14529407C2semidirect
9. #14528504C2semidirect
10. #14528477C2semidirect
20. #11762810C4semidirect
98. #10587506C2×C4semidirect
99. #10587503C2×C4semidirect
416. C17#21059semidirect

Maximal subgroups

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

subgroupindex
C2×#117628492
C2×#117628102
#145312752
C2×C4⋊C4×Dic172
#145294282
#145294212
#145294142
#145294072
#145285042
#145284772
C2×C17×#4172
#2105917