Finite Group Databasebeta

← #16956210

Subgroups — #16956210

Order 4352.

2438 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587578C211yesmaximal, characteristic
2C22×C4⋊C4×D17C211yesmaximal, characteristic
3C2×#11762671C211yesmaximal, characteristic
4C2×#11762671C212yesmaximal
5#14527038C211yesmaximal, characteristic, complemented
6#14527037C211yesmaximal, characteristic, complemented
7#14527034C211yesmaximal, characteristic, complemented
8#14527018C212yesmaximal, complemented
9#14527015C212yesmaximal, complemented
10#14527005C212yesmaximal
11C17×#1441C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C23×C4×D17C2211yescharacteristic
14C23×C4×D17C2211yescharacteristic
15C2×#10587578—28no—
16C2×#10587578—24no—
17C22×C4⋊Dic17C2211yescharacteristic
18C22×C4⋊Dic17C2211yescharacteristic
19C22×#63225C2211yescharacteristic
20C22×#63225C2211yescharacteristic
21C2×C4⋊C4×D17—24no—
22C2×C4⋊C4×D17—24no—
23C2×C4⋊C4×D17C418yescomplemented
24C22×#63227C2212yes—
25C22×#63227C2212yes—
26C22×C4×Dic17C2212yes—
27#11762671—28no—
28#11762671—24no—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C2×#10587502C2212yes—
36C2×#10587502C2211yescharacteristic
37C22×C4⋊C4×C17C2211yescharacteristic
38C22×C4⋊C4×C17C2211yescharacteristic
39C2×C17×#95C2212yes—
40C2×C17×#95C2212yes—
41C2×C17×#95C2211yescharacteristic
42C24×D17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2312yes—
46C23×C4×C17C2311yescharacteristic
47C23×C4×C17C2311yescharacteristic
48C23×Dic17C2312yes—
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2312yes—
51C23×Dic17C2311yescharacteristic
52C23×Dic17C2311yescharacteristic
53C22×C4×D17—22no—
54C22×C4×D17—22no—
55C22×C4×D17—24no—
56C22×C4×D17—24no—
57C22×C4×D17—22no—
58C22×C4×D17—22no—
59C22×C4×D17—22no—
60C22×C4×D17—22no—
61C22×C4×D17—22no—
62C22×C4×D17—22no—
63C22×C4×D17—24no—
64C22×C4×D17—24no—
65C22×C4×D17C2×C414yes—
66C22×C4×D17C2×C414yes—
67C22×C4×D17C2×C414yes—
68C22×C4×D17D414yes—
69C22×C4×D17D414yes—
70C2×C4⋊C4×C17—22no—
71C2×C4⋊C4×C17—22no—
72C2×C4⋊C4×C17—24no—
73C2×C4⋊C4×C17—22no—
74C2×C4⋊C4×C17C2×C414yes—
75C2×#63227—24no—
76C2×#63227—24no—
77C2×#63227—24no—
78C2×#63227—24no—
79C2×#63227—24no—
80C2×#63227D418yes—
81C2×C4⋊Dic17—22no—
82C2×C4⋊Dic17—22no—
83C2×C4⋊Dic17—22no—
84C2×C4⋊Dic17—24no—
85C2×C4⋊Dic17C2×C414yes—
86C2×#63225—22no—
87C2×#63225—22no—
88C2×#63225—22no—
89C2×#63225—22no—
90C2×#63225C2×C414yes—
91C2×#63225C2×C414yes—
92C2×C4×Dic17—24no—
93C2×C4×Dic17—24no—
94C2×C4×Dic17—24no—
95#10587578—232no—
96C4⋊C4×D17—232no—
97C17×#95—24no—
98C17×#95—22no—
99C17×#95—24no—
100#10587502—24no—
101#10587502—22no—
102#10587502—24no—
103#10587502—24no—
104#10587502—24no—
105#10587502—24no—
106#10587502—24no—
107#10587502—24no—
108C24×C17C2411yescharacteristic
109C23×D17—21no—
110C23×D17—21no—
111C23×D17—21no—
112C23×D17—21no—
113C23×D17—22no—
114C23×D17—22no—
115C23×D17C22×C412yes—
116C23×D17C2×D412yes—
117C23×D17C2×D412yes—
118C23×D17C2×D414yes—
119C23×D17#4114yes—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—22no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—22no—
138C22×C4×C17—22no—
139C22×C4×C17—22no—
140C22×C4×C17—22no—
141C22×C4×C17—22no—
142C22×C4×C17—22no—
143C22×C4×C17C22×C412yes—
144C22×C4×C17C22×C412yes—
145C22×C4×C17C22×C412yes—
146C22×C4×C17C2×D414yes—
147C22×C4×C17C2×D412yes—
148C22×C4×C17C2×D412yes—
149C22×C4×C17#4112yes—
150C22×C4×C17#4112yes—
151C22×Dic17—22no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—22no—
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17—22no—
168C22×Dic17—22no—
169C22×Dic17—22no—
170C22×Dic17—22no—
171C22×Dic17—22no—
172C22×Dic17—22no—
173C22×Dic17—22no—
174C22×Dic17C22×C412yes—
175C22×Dic17C22×C412yes—
176C22×Dic17C22×C412yes—
177C22×Dic17C2×D414yes—
178C22×Dic17C2×D412yes—
179C22×Dic17C2×D412yes—
180C22×Dic17#4112yes—
181C22×Dic17#4112yes—
182C2×C4×D17—28no—
183C2×C4×D17—28no—
184C2×C4×D17—28no—
185C2×C4×D17—28no—
186C2×C4×D17—28no—
187C2×C4×D17—216no—
188C2×C4×D17—28no—
189C2×C4×D17—216no—
190C2×C4×D17—28no—
191C2×C4×D17—28no—
192C2×C4×D17—28no—
193C2×C4×D17—28no—
194C2×C4×D17—216no—
195C2×C4×D17—44no—
196C2×C4×D17—48no—
197C2×C4×D17—44no—
198C2×C4×D17C22⋊C4116yes—
199C4⋊C4×C17—28no—
200C4⋊C4×C17—28no—
201#63227—216no—
202#63227—48no—
203C4⋊Dic17—28no—
204C4⋊Dic17—28no—
205#63225—28no—
206#63225—28no—
207C4×Dic17—216no—
208#13870—171nomaximal, Sylow
209C23×C17C23×C411yescharacteristic
210C23×C17C22×D411yescharacteristic
211C23×C17C22×D411yescharacteristic
212C23×C17C22×D412yes—
213C23×C17C2×#4111yescharacteristic
214C23×C17C2×#4111yescharacteristic
215C23×C17C2×#4111yescharacteristic
216C23×C17C2×#4111yescharacteristic
217C23×C17C2×#4112yes—
218C23×C172−1+412yes—
219C23×C172−1+412yes—
220C22×D17—22no—
221C22×D17—22no—
222C22×D17—24no—
223C22×D17—22no—
224C22×D17—22no—
225C22×D17—22no—
226C22×D17—22no—
227C22×D17—22no—
228C22×D17—24no—
229C22×D17—22no—
230C22×D17—24no—
231C22×D17—24no—
232C22×D17—24no—
233C22×D17—24no—
234C22×D17—24no—
235C22×D17—24no—
236C22×D17—24no—
237C22×D17—24no—
238C22×D17C2×C22⋊C414yes—
239C22×D17C4×D418yes—
240C22×D17#12014yes—
241C22×D17#12118yes—
242C22×D17#12314yes—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—24no—
246C2×C4×C17—22no—
247C2×C4×C17—24no—
248C2×C4×C17—22no—
249C2×C4×C17—22no—
250C2×C4×C17—22no—
251C2×C4×C17—22no—
252C2×C4×C17—22no—
253C2×C4×C17—22no—
254C2×C4×C17—24no—
255C2×C4×C17—24no—
256C2×C4×C17—24no—
257C2×C4×C17—24no—
258C2×C4×C17—41no—
259C2×C4×C17—42no—
260C2×C4×C17—41no—
261C2×C4×C17—42no—
262C2×C4×C17—42no—
263C2×C4×C17—42no—
264C2×C4×C17—42no—
265C2×C4×C17—42no—
266C2×C4×C17—42no—
267C2×C4×C17—42no—
268C2×C4×C17—42no—
269C2×C4×C17—42no—
270C2×C4×C17—42no—
271C2×C4×C17—42no—
272C2×C4×C17C2×C22⋊C414yes—
273C2×C4×C17#11714yescomplemented
274C2×Dic17—22no—
275C2×Dic17—22no—
276C2×Dic17—24no—
277C2×Dic17—22no—
278C2×Dic17—24no—
279C2×Dic17—22no—
280C2×Dic17—22no—
281C2×Dic17—22no—
282C2×Dic17—22no—
283C2×Dic17—22no—
284C2×Dic17—22no—
285C2×Dic17—24no—
286C2×Dic17—24no—
287C2×Dic17—24no—
288C2×Dic17—24no—
289C2×Dic17—41no—
290C2×Dic17—42no—
291C2×Dic17—41no—
292C2×Dic17—42no—
293C2×Dic17—42no—
294C2×Dic17—42no—
295C2×Dic17—42no—
296C2×Dic17—42no—
297C2×Dic17—42no—
298C2×Dic17—42no—
299C2×Dic17—42no—
300C2×Dic17—42no—
301C2×Dic17—42no—
302C2×Dic17—42no—
303C2×Dic17C2×C22⋊C414yes—
304C2×Dic17#11714yescomplemented
305C4×D17—232no—
306C4×D17—232no—
307C4×D17—416no—
308C22×#117—171no—
309C23×C4⋊C4—171no—
310C2×#381—172no—
311C2×#381—171no—
312#1441—171no—
313#1441—171no—
314#1441—171no—
315#1441D1711yescharacteristic, complemented
316#1436—172no—
317#1436—172no—
318C22⋊#95—172no—
319C22×C17C22×C22⋊C411yescharacteristic
320C22×C17C2×#11711yescharacteristic
321C22×C17C2×#11711yescharacteristic
322C22×C17C2×C4×D412yes—
323C22×C17#51912yes—
324C22×C17C2×#12011yescharacteristic
325C22×C17C2×#12112yes—
326C22×C17C2×#12311yescharacteristic
327C22×C17#52511yescharacteristic
328C22×C17#52511yescharacteristic
329C22×C17#52511yescharacteristic
330C22×C17#52511yescharacteristic
331C22×C17#53611yescharacteristic
332C22×C17#53612yes—
333C22×C17#53611yescharacteristic
334C22×C17#54712yes—
335C22×C17#54712yes—
336C22×C17#54712yes—
337C22×C17#54712yes—
338C22×C17#55212yes—
339C22×C17#55212yes—
340C22×C17#55212yes—
341C22×C17#55212yes—
342C2×D17—24no—
343C2×D17—28no—
344C2×D17—28no—
345C2×D17—24no—
346C2×D17—24no—
347C2×D17—28no—
348C2×D17—28no—
349C2×D17—24no—
350C2×D17—24no—
351C2×D17—24no—
352C2×D17#38618yes—
353C4×C17—24no—
354C4×C17—24no—
355C4×C17—44no—
356C4×C17—42no—
357C4×C17—82no—
358Dic17—24no—
359Dic17—24no—
360Dic17—44no—
361Dic17—42no—
362Dic17—82no—
363C24×C4—171no—
364C24×C4—171no—
365C24×C4—171no—
366C2×#117—348no—
367C2×#117—344no—
368C22×C4⋊C4—171no—
369C22×C4⋊C4—171no—
370C22×C4⋊C4—171no—
371C22×C4⋊C4—178no—
372C22×C4⋊C4—171no—
373C22×C4⋊C4—344no—
374C22×C4⋊C4—344no—
375C22×C4⋊C4C2×D1711yescharacteristic
376C22×C4⋊C4C2×D1711yescharacteristic
377C22×C22⋊C4—172no—
378C22×C22⋊C4—172no—
379C22×C42—172no—
380#381—348no—
381#381—344no—
382C2×#95—172no—
383C2×#95—172no—
384C2×#95—172no—
385C2×#95—172no—
386C2×#95—172no—
387C2×#95—172no—
388C2×#95—172no—
389C2×#95—171no—
390C2×#95C2×D1712yes—
391C2×#95C2×D1712yes—
392C2×#95C2×D1711yescharacteristic
393C2×C17#228011yescharacteristic
394C2×C17C2×#38611yescharacteristic
395C2×C17#231011yescharacteristic
396C2×C17#234412yes—
397C2×C17#234412yes—
398C2×C17#240611yescharacteristic
399C2×C17#240611yescharacteristic
400C2×C17#241312yes—
401C2×C17#241312yes—
402C2×C17#248411yescharacteristic
403C2×C17#248411yescharacteristic
404D17—28no—
405C25—171no—
406C23×C4—171no—
407C23×C4—171no—
408C23×C4—174no—
409C23×C4—174no—
410C23×C4—174no—
411C23×C4—172no—
412C23×C4—171no—
413C23×C4—174no—
414C23×C4—174no—
415C23×C4—172no—
416C23×C4—342no—
417C23×C4—342no—
418C23×C4—342no—
419C23×C4—342no—
420C23×C4—344no—
421C23×C4—344no—
422C23×C4—342no—
423C23×C4—342no—
424C23×C4—344no—
425C23×C4—344no—
426C23×C4—342no—
427C23×C4—342no—
428C23×C4C22×D1712yes—
429C23×C4C22×D1711yescharacteristic
430C23×C4C22×D1712yes—
431C23×C4C22×D1711yescharacteristic
432C23×C4C22×D1711yescharacteristic
433#117—3432no—
434C2×C4⋊C4—22no—
435C2×C4⋊C4—22no—
436C2×C4⋊C4—24no—
437C2×C4⋊C4—22no—
438C2×C4⋊C4—174no—
439C2×C4⋊C4—174no—
440C2×C4⋊C4—174no—
441C2×C4⋊C4—3432no—
442C2×C4⋊C4—342no—
443C2×C4⋊C4—342no—
444C2×C4⋊C4—342no—
445C2×C4⋊C4—342no—
446C2×C4⋊C4—344no—
447C2×C4⋊C4—342no—
448C2×C4⋊C4—342no—
449C2×C4⋊C4—342no—
450C2×C4⋊C4C4×D1714yes—
451C2×C22⋊C4—178no—
452C2×C22⋊C4—344no—
453C2×C22⋊C4—344no—
454C2×C22⋊C4—344no—
455C2×C22⋊C4—344no—
456C2×C22⋊C4—344no—
457C2×C42—344no—
458C2×C42—344no—
459C2×C42—344no—
460#95—24no—
461#95—22no—
462#95—24no—
463#95—344no—
464#95—344no—
465#95—344no—
466#95—344no—
467#95—344no—
468#95—344no—
469#95—342no—
470#95—344no—
471C17#1387011yescharacteristic, complemented, Sylow
472C24—172no—
473C24—174no—
474C24—172no—
475C24—172no—
476C24—174no—
477C24—341no—
478C24—341no—
479C24—342no—
480C24—341no—
481C24—341no—
482C24—342no—
483C24C23×D1711yescharacteristic
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—172no—
508C22×C4—172no—
509C22×C4—172no—
510C22×C4—172no—
511C22×C4—172no—
512C22×C4—172no—
513C22×C4—1716no—
514C22×C4—172no—
515C22×C4—174no—
516C22×C4—348no—
517C22×C4—348no—
518C22×C4—348no—
519C22×C4—3416no—
520C22×C4—3416no—
521C22×C4—348no—
522C22×C4—348no—
523C22×C4—348no—
524C22×C4—348no—
525C22×C4—348no—
526C22×C4—348no—
527C22×C4—3416no—
528C22×C4—342no—
529C22×C4—341no—
530C22×C4—341no—
531C22×C4—341no—
532C22×C4—341no—
533C22×C4—342no—
534C22×C4—348no—
535C22×C4—342no—
536C22×C4—342no—
537C22×C4—342no—
538C22×C4—341no—
539C22×C4—341no—
540C22×C4—342no—
541C22×C4—342no—
542C22×C4—342no—
543C22×C4—342no—
544C22×C4—342no—
545C22×C4—342no—
546C22×C4—342no—
547C22×C4—342no—
548C22×C4—342no—
549C22×C4—342no—
550C22×C4—342no—
551C22×C4—342no—
552C22×C4—684no—
553C22×C4—688no—
554C22×C4—684no—
555C22×C4C2×C4×D1712yes—
556C22×C4C2×C4×D1712yes—
557C22×C4C2×C4×D1712yes—
558C22×C4D4×D1714yes—
559C22×C4D4×D1712yes—
560C22×C4D4×D1712yes—
561C22×C4#6325412yes—
562C22×C4#6325612yes—
563C4⋊C4—28no—
564C4⋊C4—28no—
565C4⋊C4—348no—
566C4⋊C4—348no—
567C4⋊C4—348no—
568C4⋊C4—348no—
569C22⋊C4—3416no—
570C22⋊C4—688no—
571C42—3416no—
572C23—174no—
573C23—178no—
574C23—174no—
575C23—178no—
576C23—174no—
577C23—344no—
578C23—344no—
579C23—344no—
580C23—344no—
581C23—344no—
582C23—344no—
583C23—344no—
584C23—344no—
585C23—342no—
586C23—342no—
587C23—344no—
588C23—342no—
589C23—342no—
590C23—342no—
591C23—342no—
592C23—342no—
593C23—344no—
594C23—342no—
595C23C22×C4×D1711yescharacteristic
596C23C2×D4×D1712yes—
597C23C2×D4×D1711yescharacteristic
598C23C2×D4×D1711yescharacteristic
599C23C2×#6325411yescharacteristic
600C23C2×#6325611yescharacteristic
601C23#1058769412yes—
602C23#41×D1711yescharacteristic
603C23#41×D1711yescharacteristic
604C23#41×D1712yes—
605C23#1058769712yes—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—174no—
636C2×C4—174no—
637C2×C4—3432no—
638C2×C4—3432no—
639C2×C4—342no—
640C2×C4—342no—
641C2×C4—344no—
642C2×C4—342no—
643C2×C4—344no—
644C2×C4—342no—
645C2×C4—342no—
646C2×C4—342no—
647C2×C4—342no—
648C2×C4—342no—
649C2×C4—344no—
650C2×C4—344no—
651C2×C4—342no—
652C2×C4—344no—
653C2×C4—344no—
654C2×C4—6816no—
655C2×C4—682no—
656C2×C4—682no—
657C2×C4—681no—
658C2×C4—682no—
659C2×C4—681no—
660C2×C4—682no—
661C2×C4—682no—
662C2×C4—682no—
663C2×C4—682no—
664C2×C4—682no—
665C2×C4—682no—
666C2×C4—682no—
667C2×C4—682no—
668C2×C4—682no—
669C2×C4C22⋊C4×D1714yes—
670C2×C4#1058757814yescomplemented
671C22—178no—
672C22—344no—
673C22—344no—
674C22—344no—
675C22—348no—
676C22—348no—
677C22—348no—
678C22—348no—
679C22—344no—
680C22—344no—
681C22—344no—
682C22C2×C22⋊C4×D1711yescharacteristic
683C22#117×D1711yescharacteristic
684C22C2×#1058757811yescharacteristic
685C22C4×D4×D1712yes—
686C22#1176328512yes—
687C22#120×D1711yescharacteristic
688C22#121×D1712yes—
689C22#123×D1711yescharacteristic
690C22#1176331911yescharacteristic
691C22#1176331911yescharacteristic
692C22#1176332211yescharacteristic
693C22#1176332211yescharacteristic
694C22#1176337911yescharacteristic
695C22#1176338212yes—
696C22#1176338411yescharacteristic
697C22#1176345712yes—
698C22#1176345912yes—
699C22#1176346112yes—
700C22#1176346212yes—
701C22#1176349112yes—
702C22#1176349312yes—
703C22#1176349512yes—
704C22#1176349712yes—
705C4—24no—
706C4—24no—
707C4—44no—
708C4—42no—
709C4—82no—
710C4—344no—
711C4—344no—
712C4—684no—
713C4—682no—
714C4—1362no—
715C2—348no—
716C2#1453116611yescharacteristic
717C2D17×#38611yescharacteristic
718C2#1453136711yescharacteristic
719C2#1453165012yes—
720C2#1453165212yes—
721C2#1453211411yescharacteristic
722C2#1453212011yescharacteristic
723C2#1453218412yes—
724C2#1453218612yes—
725C2#1453306611yescharacteristic
726C2#1453307211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1441C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1441D17
hypocenterC17#13870
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
5. #14527038C2semidirect
6. #14527037C2semidirect
7. #14527034C2semidirect
8. #14527018C2semidirect
9. #14527015C2semidirect
11. C17×#1441C2semidirect
23. C2×C4⋊C4×D17C4semidirect
273. C2×C4×C17#117semidirect
304. C2×Dic17#117semidirect
315. #1441D17semidirect
471. C17#13870semidirect
670. C2×C4#10587578semidirect

Maximal subgroups

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

subgroupindex
C22×#105875782
C22×C4⋊C4×D172
C2×#117626712
#145270382
#145270372
#145270342
#145270182
#145270152
#145270052
C17×#14412
#1387017