Finite Group Databasebeta

← #16956892

Subgroups — #16956892

Order 4352.

1846 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762702C212yesmaximal
2#14527037C212yesmaximal, complemented
3#14527033C212yesmaximal
4#14527016C212yesmaximal, complemented
5#14527014C211yesmaximal, characteristic, complemented
6#14527011C211yesmaximal, characteristic, complemented
7#14527011C211yesmaximal, characteristic, complemented
8#14527007C211yesmaximal, characteristic
9#14527007C211yesmaximal, characteristic
10C4×#10587502C211yesmaximal, characteristic, complemented
11C22×C17×#117C211yesmaximal, characteristic
12C22×C4⋊Dic17C2212yes—
13C22×C22⋊Dic17C2212yes—
14C22×C22⋊Dic17C2212yes—
15C22×C4×Dic17C2212yes—
16#11762702—28no—
17C2×#10587502C2211yescharacteristic
18C2×#10587502C2211yescharacteristic
19C2×#10587502C2212yes—
20C2×#10587502C2211yescharacteristic
21C2×#10587502C2211yescharacteristic
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C24×C4×C17C2211yescharacteristic
30C2×C17×#117—28no—
31C2×C17×#117C418yescomplemented
32C22×C4⋊C4×C17C2211yescharacteristic
33C22×C4⋊C4×C17C2211yescharacteristic
34C22×C22⋊C4×C17C2211yescharacteristic
35C22×C22⋊C4×C17C2211yescharacteristic
36C22×C42×C17C2211yescharacteristic
37C22×C42×C17C2211yescharacteristic
38C25×C17C2311yescharacteristic
39C23×C4×C17—24no—
40C23×C4×C17—22no—
41C23×C4×C17—22no—
42C23×C4×C17—24no—
43C23×C4×C17C2×C414yes—
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×C4×C17C2311yescharacteristic
47C23×C4×C17C2311yescharacteristic
48C23×C4×C17C2311yescharacteristic
49C23×C4×C17C2311yescharacteristic
50C23×Dic17C2312yes—
51C23×Dic17C2312yes—
52C23×Dic17C2312yes—
53C23×Dic17C2312yes—
54C17×#117—232no—
55C2×C4⋊C4×C17—24no—
56C2×C4⋊C4×C17—24no—
57C2×C4⋊C4×C17C2×C414yes—
58C2×C4⋊C4×C17C2×C414yes—
59C2×C22⋊C4×C17—24no—
60C2×C22⋊C4×C17C2×C414yes—
61C2×C22⋊C4×C17C2×C414yes—
62C2×C22⋊C4×C17D418yes—
63C2×C42×C17—24no—
64C2×C42×C17—24no—
65C2×C42×C17C2×C414yes—
66C2×C42×C17C2×C414yes—
67C2×C22⋊Dic17—24no—
68C2×C22⋊Dic17—24no—
69C2×C22⋊Dic17—24no—
70C2×C22⋊Dic17—24no—
71C2×C22⋊Dic17—42no—
72C2×C22⋊Dic17—42no—
73C2×C4⋊Dic17—24no—
74C2×C4⋊Dic17—24no—
75C2×C4⋊Dic17—42no—
76C2×C4×Dic17—24no—
77C2×C4×Dic17—24no—
78C2×C4×Dic17—42no—
79#10587502—24no—
80#10587502—24no—
81#10587502—22no—
82#10587502—22no—
83#10587502—24no—
84#10587502—24no—
85#10587502—22no—
86#10587502—22no—
87#10587502—24no—
88#10587502—24no—
89#10587502—42no—
90#10587502—42no—
91C24×C17—21no—
92C24×C17—21no—
93C24×C17—22no—
94C24×C17—22no—
95C24×C17—22no—
96C24×C17—21no—
97C24×C17—21no—
98C24×C17—21no—
99C24×C17—21no—
100C24×C17C22×C412yes—
101C24×C17C2×D414yes—
102C24×C17C2411yescharacteristic
103C22×C4×C17—216no—
104C22×C4×C17—216no—
105C22×C4×C17—28no—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—21no—
109C22×C4×C17—21no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—21no—
114C22×C4×C17—21no—
115C22×C4×C17—21no—
116C22×C4×C17—21no—
117C22×C4×C17—28no—
118C22×C4×C17—28no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17C22×C412yes—
128C22×C4×C17C22×C412yes—
129C22×C4×C17C22×C412yes—
130C22×C4×C17C22×C412yes—
131C22×C4×C17C22×C412yes—
132C22×C4×C17C22×C412yes—
133C22×C4×C17C2×D414yes—
134C22×C4×C17C2×D414yes—
135C22×C4×C17#4112yes—
136C22×C4×C17#4112yes—
137C22×C4×C17#4114yes—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17#4114yes—
165C22×Dic17#4114yes—
166C4⋊C4×C17—28no—
167C4⋊C4×C17—28no—
168C22⋊C4×C17—28no—
169C22⋊C4×C17C22⋊C4116yes—
170C42×C17—28no—
171C42×C17—28no—
172C22⋊Dic17—48no—
173C22⋊Dic17—48no—
174C4⋊Dic17—48no—
175C4×Dic17—48no—
176#13943—171nomaximal, Sylow
177C23×C17—22no—
178C23×C17—22no—
179C23×C17—22no—
180C23×C17—22no—
181C23×C17—22no—
182C23×C17—22no—
183C23×C17—24no—
184C23×C17—24no—
185C23×C17—24no—
186C23×C17—24no—
187C23×C17—24no—
188C23×C17—24no—
189C23×C17—42no—
190C23×C17—42no—
191C23×C17—41no—
192C23×C17—41no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—42no—
196C23×C17—42no—
197C23×C17—42no—
198C23×C17—42no—
199C23×C17C2×C22⋊C414yes—
200C23×C17C23×C411yescharacteristic
201C23×C17C22×D412yes—
202C23×C17C2×#4111yescharacteristic
203C23×C17C2×#4111yescharacteristic
204C23×C17C2×#4112yes—
205C23×C17C2×#4112yes—
206C23×C17C2×#4112yes—
207C23×C17H(5,𝔽2)11yescharacteristic
208C23×C17H(5,𝔽2)11yescharacteristic
209C23×C172−1+411yescharacteristic
210C23×C172−1+411yescharacteristic
211C2×C4×C17—232no—
212C2×C4×C17—22no—
213C2×C4×C17—22no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
222C2×C4×C17—24no—
223C2×C4×C17—24no—
224C2×C4×C17—24no—
225C2×C4×C17—24no—
226C2×C4×C17—24no—
227C2×C4×C17—24no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—22no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17C2×C22⋊C414yes—
239C2×C4×C17C2×C22⋊C414yes—
240C2×C4×C17#11714yes—
241C2×C4×C17#11714yes—
242C2×Dic17—24no—
243C2×Dic17—24no—
244C2×Dic17—24no—
245C2×Dic17—24no—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—42no—
249C2×Dic17—42no—
250C2×Dic17—42no—
251C2×Dic17—42no—
252C2×Dic17—42no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—42no—
266C2×Dic17—42no—
267C2×Dic17—42no—
268C2×Dic17—42no—
269C2×Dic17—42no—
270C22×#117D1711yescharacteristic
271C2×#388—172no—
272#1441—172no—
273#1440—172no—
274#1436—172no—
275C4⋊#95—171no—
276#1434—171no—
277#1434—171no—
278C22⋊#95—171no—
279C22⋊#95—171no—
280C4×#95—171no—
281C22×C17—24no—
282C22×C17—24no—
283C22×C17—24no—
284C22×C17—42no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17—44no—
288C22×C17—44no—
289C22×C17—44no—
290C22×C17—44no—
291C22×C17—42no—
292C22×C17C22×C22⋊C411yescharacteristic
293C22×C17C2×#11711yescharacteristic
294C22×C17C2×#11711yescharacteristic
295C22×C17C4×#4111yescharacteristic
296C22×C17C4×#4111yescharacteristic
297C22×C17#52011yescharacteristic
298C22×C17#52011yescharacteristic
299C22×C17#52512yes—
300C22×C17#52512yes—
301C22×C17#52912yes—
302C22×C17#52912yes—
303C22×C17#54011yescharacteristic
304C22×C17#54111yescharacteristic
305C22×C17#54211yescharacteristic
306C22×C17#54211yescharacteristic
307C22×C17#54612yes—
308C22×C17#54612yes—
309C22×C17#54712yes—
310C22×C17#54712yes—
311C22×C17#55212yes—
312C22×C17#55312yes—
313C22×C17#55512yes—
314C22×C17#55612yes—
315C4×C17—24no—
316C4×C17—24no—
317C4×C17—24no—
318C4×C17—24no—
319C4×C17—42no—
320C4×C17—42no—
321Dic17—44no—
322Dic17—44no—
323Dic17—82no—
324Dic17—82no—
325C24×C4C2×D1711yescharacteristic
326C2×#117—28no—
327C2×#117Dic1718yescomplemented
328C22×C4⋊C4—172no—
329C22×C4⋊C4C2×D1711yescharacteristic
330C22×C4⋊C4C2×D1711yescharacteristic
331C22×C22⋊C4—172no—
332C22×C22⋊C4—172no—
333C22×C22⋊C4C2×D1711yescharacteristic
334C22×C22⋊C4C2×D1711yescharacteristic
335C22×C42—172no—
336C22×C42C2×D1711yescharacteristic
337C22×C42C2×D1711yescharacteristic
338#388—348no—
339C2×#95—171no—
340C2×#95—171no—
341C2×#95—172no—
342C2×#95—171no—
343C2×#95—171no—
344C2×#95—172no—
345C2×#95—172no—
346C2×#95—172no—
347C2×#95—172no—
348C2×#95—172no—
349C2×#95—172no—
350C2×#95—172no—
351C2×C17—44no—
352C2×C17#228011yescharacteristic
353C2×C17#229811yescharacteristic
354C2×C17#231411yescharacteristic
355C2×C17#233911yescharacteristic
356C2×C17#233911yescharacteristic
357C2×C17#234611yescharacteristic
358C2×C17#234711yescharacteristic
359C2×C17#247012yes—
360C2×C17#247412yes—
361C2×C17#247612yes—
362C2×C17#247812yes—
363C25C22×D1711yescharacteristic
364C23×C4—24no—
365C23×C4—24no—
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—172no—
369C23×C4—172no—
370C23×C4—172no—
371C23×C4—172no—
372C23×C4C2×Dic1714yes—
373C23×C4C22×D1711yescharacteristic
374C23×C4C22×D1711yescharacteristic
375C23×C4C22×D1711yescharacteristic
376C23×C4C22×D1711yescharacteristic
377C23×C4C22×D1711yescharacteristic
378C23×C4C22×D1711yescharacteristic
379#117—232no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—344no—
383C2×C4⋊C4—344no—
384C2×C4⋊C4—682no—
385C2×C4⋊C4C2×Dic1714yes—
386C2×C4⋊C4C2×Dic1714yes—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—344no—
389C2×C22⋊C4—344no—
390C2×C22⋊C4—344no—
391C2×C22⋊C4—344no—
392C2×C22⋊C4—682no—
393C2×C22⋊C4—682no—
394C2×C22⋊C4C2×Dic1714yes—
395C2×C22⋊C4C2×Dic1714yes—
396C2×C22⋊C4C22⋊D1718yes—
397C2×C42—24no—
398C2×C42—24no—
399C2×C42—344no—
400C2×C42—344no—
401C2×C42—682no—
402C2×C42C2×Dic1714yes—
403C2×C42C2×Dic1714yes—
404#95—344no—
405#95—344no—
406#95—342no—
407#95—342no—
408#95—344no—
409#95—344no—
410#95—342no—
411#95—342no—
412#95—344no—
413#95—344no—
414#95—682no—
415#95—682no—
416C17#1394311yescharacteristic, complemented, Sylow
417C24—22no—
418C24—21no—
419C24—21no—
420C24—22no—
421C24—22no—
422C24—21no—
423C24—21no—
424C24—21no—
425C24—21no—
426C24C22×Dic1712yes—
427C24C2×C22⋊D1714yes—
428C24C23×D1711yescharacteristic
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—174no—
454C22×C4—174no—
455C22×C4—342no—
456C22×C4—342no—
457C22×C4—342no—
458C22×C4—342no—
459C22×C4—342no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—342no—
464C22×C4—342no—
465C22×C4—342no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4#6325214yes—
482C22×C4#6325212yes—
483C22×C4#6325212yes—
484C22×C4C22×Dic1712yes—
485C22×C4C22×Dic1712yes—
486C22×C4C22×Dic1712yes—
487C22×C4C22×Dic1712yes—
488C22×C4C22×Dic1712yes—
489C22×C4C22×Dic1712yes—
490C22×C4C2×C22⋊D1714yes—
491C22×C4C2×C22⋊D1714yes—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—688no—
495C22⋊C4—28no—
496C22⋊C4—688no—
497C22⋊C4—688no—
498C22⋊C4C22⋊Dic17116yes—
499C42—28no—
500C42—28no—
501C42—688no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23C2×C22⋊Dic1714yes—
525C23C2×#6325212yes—
526C23C2×#6325211yescharacteristic
527C23C2×#6325211yescharacteristic
528C23#41×D1712yes—
529C23#41×D1712yes—
530C23#1058769611yescharacteristic
531C23#1058769611yescharacteristic
532C23#1058769711yescharacteristic
533C23#1058769711yescharacteristic
534C23C23×Dic1711yescharacteristic
535C23C22×C22⋊D1712yes—
536C2×C4—232no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—344no—
564C2×C4—344no—
565C2×C4—344no—
566C2×C4—344no—
567C2×C4—344no—
568C2×C4—344no—
569C2×C4—682no—
570C2×C4—682no—
571C2×C4—682no—
572C2×C4—682no—
573C2×C4—682no—
574C2×C4—682no—
575C2×C4—682no—
576C2×C4—682no—
577C2×C4—682no—
578C2×C4—682no—
579C2×C4—682no—
580C2×C4—682no—
581C2×C4—682no—
582C2×C4—682no—
583C2×C4—682no—
584C2×C4—682no—
585C2×C4—682no—
586C2×C4—682no—
587C2×C4—682no—
588C2×C4—682no—
589C2×C4—682no—
590C2×C4—682no—
591C2×C4#1058761214yes—
592C2×C4#1058761214yes—
593C2×C4C2×C22⋊Dic1714yes—
594C2×C4C2×C22⋊Dic1714yes—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—44no—
599C22—42no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22C22×C22⋊Dic1711yescharacteristic
607C22C2×#1058761211yescharacteristic
608C22C2×#1058761211yescharacteristic
609C22#41×Dic1711yescharacteristic
610C22#41×Dic1711yescharacteristic
611C22#1176329211yescharacteristic
612C22#1176329211yescharacteristic
613C22#1176332012yes—
614C22#1176332012yes—
615C22#1176333712yes—
616C22#1176333712yes—
617C22#1176341411yescharacteristic
618C22#1176342111yescharacteristic
619C22#1176343011yescharacteristic
620C22#1176343011yescharacteristic
621C22#1176344712yes—
622C22#1176344712yes—
623C22#1176345812yes—
624C22#1176345812yes—
625C22#1176349812yes—
626C22#1176350312yes—
627C22#1176352212yes—
628C22#1176352712yes—
629C4—24no—
630C4—24no—
631C4—24no—
632C4—24no—
633C4—42no—
634C4—42no—
635C4—684no—
636C4—684no—
637C4—1362no—
638C4—1362no—
639C2—44no—
640C2#1453116711yescharacteristic
641C2#1453128211yescharacteristic
642C2#1453139311yescharacteristic
643C2#1453161411yescharacteristic
644C2#1453161411yescharacteristic
645C2#1453166911yescharacteristic
646C2#1453167711yescharacteristic
647C2#1453287412yes—
648C2#1453293412yes—
649C2#1453296412yes—
650C2#1453299412yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC22×C17×#117C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC22×#117D17
hypocenterC17#13943
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. #14527037C2semidirect
4. #14527016C2semidirect
5. #14527014C2semidirect
6. #14527011C2semidirect
7. #14527011C2semidirect
10. C4×#10587502C2semidirect
31. C2×C17×#117C4semidirect
327. C2×#117Dic17semidirect
416. C17#13943semidirect

Maximal subgroups

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

subgroupindex
C2×#117627022
#145270372
#145270332
#145270162
#145270142
#145270112
#145270072
C4×#105875022
C22×C17×#1172
#1394317