Finite Group Databasebeta

← #17036726

Subgroups — #17036726

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×#11762852C211yesmaximal, characteristic, complemented
2C2×#11762810C211yesmaximal, characteristic
3C2×#11762810C211yesmaximal, characteristic
4C2×C4×C4⋊Dic17C211yesmaximal, characteristic, complemented
5#14529422C212yesmaximal, complemented
6#14529415C212yesmaximal, complemented
7#14529408C212yesmaximal, complemented
8#14529401C212yesmaximal, complemented
9#14528505C211yesmaximal, characteristic, complemented
10#14528478C211yesmaximal, characteristic, complemented
11C17×#2297C211yesmaximal, characteristic
12C22×#63223C2211yescharacteristic
13C22×#63223C2211yescharacteristic
14C22×C4⋊Dic17C2211yescharacteristic
15C22×C4⋊Dic17C2211yescharacteristic
16C22×C4×Dic17—21no—
17#11762852—24no—
18C2×#10587478C2212yes—
19C2×#10587479C2212yes—
20#11762810—24no—
21#11762810C418yescomplemented
22C2×#10587506C2212yes—
23C2×#10587506C2212yes—
24C2×#10587503C2212yes—
25C2×#10587503C2212yes—
26C4×C4⋊Dic17—21no—
27C4×C4⋊Dic17—21no—
28C4×C4⋊Dic17C2212yes—
29C4×C4⋊Dic17C2212yes—
30C2×#10587502—21no—
31#11762571C2212yes—
32#11762571C2212yes—
33#11762564C2212yes—
34#11762564C2212yes—
35C22×#41×C17C2211yescharacteristic
36C2×C17×#117C2211yescharacteristic
37C22×C42×C17C2211yescharacteristic
38C17×#391C2211yescharacteristic
39C17×#391C2211yescharacteristic
40C17×#390C2211yescharacteristic
41C17×#390C2211yescharacteristic
42C17×#386—21no—
43C17×#386—21no—
44C4×C22⋊C4×C17—21no—
45C4×C22⋊C4×C17—21no—
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×C17D411yescharacteristic
57C23×C4×C17D411yescharacteristic
58C23×C4×C17C2311yescharacteristic
59C23×Dic17—21no—
60C23×Dic17—21no—
61C17×#117—22no—
62C17×#117—22no—
63C2×C4⋊C4×C17C2311yescharacteristic
64C2×C4⋊C4×C17C2311yescharacteristic
65C2×C22⋊C4×C17—21no—
66C2×C22⋊C4×C17—21no—
67C2×C22⋊C4×C17—21no—
68C2×C22⋊C4×C17—21no—
69C2×C22⋊C4×C17—21no—
70C2×C42×C17—22no—
71C2×C42×C17—22no—
72C2×C42×C17—21no—
73C2×C42×C17D411yescharacteristic
74C2×C42×C17D411yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×C42×C17C2311yescharacteristic
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—22no—
80C2×C4⋊Dic17—22no—
81C2×C4⋊Dic17C2×C414yes—
82C2×C4⋊Dic17D414yes—
83C2×C4⋊Dic17C2312yes—
84C2×C4⋊Dic17C2312yes—
85C2×C4×Dic17—22no—
86C2×C4×Dic17—22no—
87C2×C4×Dic17—44no—
88C2×#63223—22no—
89C2×#63223—22no—
90C2×#63223—22no—
91C2×#63223—22no—
92C2×#63223—22no—
93C2×#63223C2×C414yes—
94C22×#3622C2312yes—
95C22×#3622C2312yes—
96C17×#95—21no—
97C17×#95—21no—
98C17×#95—21no—
99C17×#95—21no—
100#10587506—24no—
101#10587506C2×C418yescomplemented
102#10587503—24no—
103#10587503C2×C418yescomplemented
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—22no—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—21no—
134C22×C4×C17—24no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—41no—
138C22×C4×C17—41no—
139C22×C4×C17—42no—
140C22×C4×C17—42no—
141C22×C4×C17C22×C412yes—
142C22×C4×C17C2×D412yes—
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17#4112yes—
149C22×C4×C17#4112yes—
150C22×C4×C17#4112yes—
151C22×C4×C17#4112yes—
152C22×C4×C17C2411yescharacteristic
153C22×Dic17—24no—
154C22×Dic17—24no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—42no—
158C22×Dic17—42no—
159C22×Dic17—42no—
160C22×Dic17—42no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C22⋊C4×C17—42no—
166C22⋊C4×C17—42no—
167C22⋊C4×C17—42no—
168C22⋊C4×C17—42no—
169C22⋊C4×C17—42no—
170C42×C17—22no—
171C42×C17—22no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—42no—
177C4⋊Dic17—24no—
178C4⋊Dic17—24no—
179C4⋊Dic17—22no—
180C4⋊Dic17—22no—
181C4⋊Dic17C22×C414yes—
182C4⋊Dic17C2×D414yes—
183C4×Dic17—44no—
184C4×Dic17—44no—
185#63223—28no—
186#63223—44no—
187C2×#3622—22no—
188C2×#3622—22no—
189C2×#3622—22no—
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—28no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—22no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—44no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—48no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17#11714yes—
241C2×C4×C17C4×D414yes—
242C2×C4×C17C4×D412yes—
243C2×C4×C17#12112yes—
244C2×C4×C17#12312yes—
245C2×C4×C17#12414yes—
246C2×C4×C17#12614yes—
247C2×C4×C17AGL(1,ℤ/8)12yes—
248C2×C4×C17#13712yes—
249C2×C4×C17C23×C411yescharacteristic
250C2×C4×C17C22×D411yescharacteristic
251C2×C4×C17C2×#4111yescharacteristic
252C2×C4×C17C2×#4111yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×Dic17—24no—
256C2×Dic17—24no—
257C2×Dic17—48no—
258C2×Dic17—48no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263#3622—24no—
264#3622—42no—
265C2×#428—171no—
266C2×#417—171no—
267C2×#417—171no—
268#2297D1711yescharacteristic
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—22no—
298C4×C17—22no—
299C4×C17—41no—
300C4×C17—41no—
301C4×C17—42no—
302C4×C17—81no—
303C4×C17—81no—
304C4×C17#38814yes—
305C4×C17C2×#11711yescharacteristic
306C4×C17C2×C4×D411yescharacteristic
307C4×C17C2×#12411yescharacteristic
308C4×C17C2×#12611yescharacteristic
309Dic17—44no—
310Dic17—44no—
311C22×#41C2×D1711yescharacteristic
312C22×M16—171no—
313C22×M16—171no—
314C2×#117C2×D1711yescharacteristic
315C22×C4⋊C4—171no—
316C22×C4⋊C4—171no—
317C22×C42—341no—
318C22×C42C2×D1711yescharacteristic
319#428—344no—
320C2×#107—172no—
321C2×#106—172no—
322#417—178no—
323#417—344no—
324C2×#103—172no—
325C2×#103—172no—
326C2×#102—172no—
327C2×#102—172no—
328#391C2×D1711yescharacteristic
329#391C2×D1711yescharacteristic
330#390C2×D1711yescharacteristic
331#390C2×D1711yescharacteristic
332#386—21no—
333#386—21no—
334C4×C4⋊C4—172no—
335C4×C4⋊C4—172no—
336C4×C4⋊C4—341no—
337C4×C4⋊C4—341no—
338C4×C22⋊C4—21no—
339C4×C22⋊C4—21no—
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—341no—
365C23×C4—341no—
366C23×C4C22⋊D1711yescharacteristic
367C23×C4C22⋊D1711yescharacteristic
368C23×C4C22×D1711yescharacteristic
369C2×M16—174no—
370C2×M16—342no—
371C2×M16—342no—
372C2×M16—342no—
373C2×M16—342no—
374C2×M16—342no—
375C22×C8—172no—
376C22×C8—172no—
377#117—22no—
378#117—22no—
379C2×C4⋊C4—174no—
380C2×C4⋊C4—174no—
381C2×C4⋊C4—172no—
382C2×C4⋊C4—172no—
383C2×C4⋊C4—342no—
384C2×C4⋊C4—342no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4C22×D1711yescharacteristic
388C2×C4⋊C4C22×D1711yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—342no—
398C2×C42—342no—
399C2×C42—684no—
400C2×C42C22⋊D1711yescharacteristic
401C2×C42C22⋊D1711yescharacteristic
402C2×C42C22×D1711yescharacteristic
403C2×C42C22×D1711yescharacteristic
404#107—344no—
405#106—344no—
406#103—178no—
407#103—344no—
408#102—178no—
409#102—344no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—342no—
415#95—342no—
416C17#2105911yescharacteristic, complemented, Sylow
417C24—21no—
418C24C2×C22⋊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—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—342no—
451C22×C4—342no—
452C22×C4—344no—
453C22×C4—344no—
454C22×C4—682no—
455C22×C4—682no—
456C22×C4—682no—
457C22×C4—682no—
458C22×C4C2×C4×D1712yes—
459C22×C4#6325212yes—
460C22×C4D4×D1712yes—
461C22×C4#6325412yes—
462C22×C4#6325412yes—
463C22×C4#6325412yes—
464C22×C4C2×C22⋊D1711yescharacteristic
465C22×C4C2×C22⋊D1711yescharacteristic
466C22×C4C2×C22⋊D1711yescharacteristic
467C22×C4C2×C22⋊D1711yescharacteristic
468C22×C4C2×C22⋊D1711yescharacteristic
469C22×C4C23×D1711yescharacteristic
470M16—348no—
471M16—684no—
472C2×C8—174no—
473C2×C8—342no—
474C2×C8—342no—
475C2×C8—342no—
476C2×C8—342no—
477C2×C8—342no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—174no—
483C4⋊C4—174no—
484C4⋊C4—342no—
485C4⋊C4—342no—
486C4⋊C4—344no—
487C4⋊C4—344no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—22no—
498C42—22no—
499C42—42no—
500C42—684no—
501C42—684no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C4×C22⋊D1712yes—
514C23#1058762112yes—
515C23#1058762412yes—
516C23C22×C22⋊D1711yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—28no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—48no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—344no—
549C2×C4—344no—
550C2×C4—688no—
551C2×C4—688no—
552C2×C4—682no—
553C2×C4—682no—
554C2×C4—682no—
555C2×C4—682no—
556C2×C4#1058756314yes—
557C2×C4#1058756514yes—
558C2×C4#1058756714yes—
559C2×C4#1058757114yes—
560C2×C4C4×C22⋊D1712yes—
561C2×C4#1058762112yes—
562C2×C4#1058762412yes—
563C2×C4#1058763512yes—
564C2×C4#1058763712yes—
565C2×C4C22×C4×D1711yescharacteristic
566C2×C4C2×#6325211yescharacteristic
567C2×C4C2×D4×D1711yescharacteristic
568C2×C4C2×#6325411yescharacteristic
569C2×C4C2×#6325411yescharacteristic
570C2×C4C2×#6325411yescharacteristic
571C8—344no—
572C8—682no—
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#1176269714yes—
587C22C2×C4×C22⋊D1711yescharacteristic
588C22C2×#1058762411yescharacteristic
589C22C2×#1058762111yescharacteristic
590C22C2×#1058763511yescharacteristic
591C22C2×#1058763711yescharacteristic
592C22#1176360711yescharacteristic
593C22#1176360711yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—684no—
602C4—684no—
603C4#1176269714yes—
604C4C2×#1058756311yescharacteristic
605C4C2×#1058756711yescharacteristic
606C4C2×#1058757111yescharacteristic
607C4C2×#1058756511yescharacteristic
608C2—24no—
609C2—82no—
610C2C2×#1176269711yescharacteristic
611C2C4×#1058763511yescharacteristic
612C2C4×#1058763711yescharacteristic
613C2#1453824411yescharacteristic
614C2#1453825211yescharacteristic
615C2#1453870411yescharacteristic
616C2#1453871311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2297C2
socleC23×C17C22×D4
prime radicalC22×#41×C17C22
cosocleC22×C4×C17C24
hypercenter#2297D17
hypocenterC17#21059
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
1. C2×#11762852C2semidirect
4. C2×C4×C4⋊Dic17C2semidirect
5. #14529422C2semidirect
6. #14529415C2semidirect
7. #14529408C2semidirect
8. #14529401C2semidirect
9. #14528505C2semidirect
10. #14528478C2semidirect
21. #11762810C4semidirect
101. #10587506C2×C4semidirect
103. #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×#117628522
C2×#117628102
C2×C4×C4⋊Dic172
#145294222
#145294152
#145294082
#145294012
#145285052
#145284782
C17×#22972
#2105917