Finite Group Databasebeta

← #16956213

Subgroups — #16956213

Order 4352.

2438 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587558C211yesmaximal, characteristic
2C22×C4⋊C4×D17C211yesmaximal, characteristic
3C2×#11762671C211yesmaximal, characteristic
4C2×#11762671C212yesmaximal
5#14527036C212yesmaximal, complemented
6#14527035C212yesmaximal, complemented
7#14527019C211yesmaximal, characteristic, complemented
8#14527017C211yesmaximal, characteristic, complemented
9#14527016C211yesmaximal, characteristic, complemented
10#14527005C212yesmaximal
11C17×#1436C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C23×C4×D17C2211yescharacteristic
14C23×C4×D17C2211yescharacteristic
15C2×#10587558—28no—
16C2×#10587558—24no—
17C22×C4⋊Dic17C2211yescharacteristic
18C22×#63225C2212yes—
19C22×#63225C2211yescharacteristic
20C22×#63225C2211yescharacteristic
21C2×C4⋊C4×D17—24no—
22C2×C4⋊C4×D17—24no—
23C2×C4⋊C4×D17C418yescomplemented
24C22×#63227C2212yes—
25C22×#63227C2212yes—
26C22×C4×Dic17C2211yescharacteristic
27#11762671—28no—
28#11762671—24no—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2211yescharacteristic
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C2×#10587502C2212yes—
36C2×#10587502C2212yes—
37C22×C4⋊C4×C17C2211yescharacteristic
38C22×C42×C17C2211yescharacteristic
39C2×C17×#95C2212yes—
40C2×C17×#95C2211yescharacteristic
41C2×C17×#95C2212yes—
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—24no—
58C22×C4×D17—24no—
59C22×C4×D17—22no—
60C22×C4×D17—22no—
61C22×C4×D17—24no—
62C22×C4×D17—24no—
63C22×C4×D17C2×C414yes—
64C22×C4×D17C2×C414yes—
65C22×C4×D17C2×C414yes—
66C22×C4×D17D414yes—
67C22×C4×D17D414yes—
68C2×C4⋊C4×C17—22no—
69C2×C4⋊C4×C17—22no—
70C2×C4⋊C4×C17C2×C414yes—
71C2×C42×C17—24no—
72C2×C42×C17—22no—
73C2×#63227—24no—
74C2×#63227—24no—
75C2×#63227—24no—
76C2×#63227—24no—
77C2×#63227—24no—
78C2×#63227D418yes—
79C2×C4⋊Dic17—22no—
80C2×C4⋊Dic17—22no—
81C2×C4⋊Dic17C2×C414yes—
82C2×#63225—22no—
83C2×#63225—22no—
84C2×#63225—24no—
85C2×#63225—22no—
86C2×#63225—22no—
87C2×#63225—24no—
88C2×#63225—24no—
89C2×#63225C2×C414yes—
90C2×#63225C2×C414yes—
91C2×C4×Dic17—22no—
92C2×C4×Dic17—24no—
93C4⋊C4×D17—232no—
94#10587558—232no—
95C17×#95—24no—
96C17×#95—22no—
97C17×#95—24no—
98#10587502—24no—
99#10587502—22no—
100#10587502—24no—
101#10587502—24no—
102#10587502—24no—
103#10587502—24no—
104#10587502—24no—
105#10587502—24no—
106C24×C17C2411yescharacteristic
107C23×D17—22no—
108C23×D17—22no—
109C23×D17—22no—
110C23×D17—22no—
111C23×D17C22×C412yes—
112C23×D17C2×D412yes—
113C23×D17C2×D412yes—
114C23×D17C2×D414yes—
115C23×D17#4114yes—
116C22×C4×C17—22no—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
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×C17C22×C412yes—
139C22×C4×C17C22×C412yes—
140C22×C4×C17C22×C412yes—
141C22×C4×C17C2×D414yes—
142C22×C4×C17C2×D412yes—
143C22×C4×C17C2×D412yes—
144C22×C4×C17#4114yes—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—21no—
152C22×Dic17—21no—
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—22no—
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17C22×C412yes—
168C22×Dic17C22×C412yes—
169C22×Dic17C22×C412yes—
170C22×Dic17C2×D414yes—
171C22×Dic17C2×D412yes—
172C22×Dic17C2×D412yes—
173C22×Dic17#4114yes—
174C2×C4×D17—216no—
175C2×C4×D17—216no—
176C2×C4×D17—28no—
177C2×C4×D17—216no—
178C2×C4×D17—28no—
179C2×C4×D17—216no—
180C2×C4×D17—28no—
181C2×C4×D17—28no—
182C2×C4×D17—28no—
183C2×C4×D17—28no—
184C2×C4×D17—216no—
185C2×C4×D17—44no—
186C2×C4×D17—48no—
187C2×C4×D17—44no—
188C2×C4×D17C22⋊C4116yes—
189C4⋊C4×C17—28no—
190C42×C17—28no—
191#63227—216no—
192#63227—48no—
193C4⋊Dic17—28no—
194#63225—216no—
195#63225—28no—
196#63225—28no—
197C4×Dic17—28no—
198#13870—171nomaximal, Sylow
199C23×C17C23×C411yescharacteristic
200C23×C17C22×D411yescharacteristic
201C23×C17C22×D411yescharacteristic
202C23×C17C22×D412yes—
203C23×C17C2×#4112yes—
204C23×C17C2×#4112yes—
205C23×C17C2×#4112yes—
206C23×C172−1+412yes—
207C23×C172−1+412yes—
208C22×D17—22no—
209C22×D17—22no—
210C22×D17—24no—
211C22×D17—24no—
212C22×D17—24no—
213C22×D17—22no—
214C22×D17—24no—
215C22×D17—22no—
216C22×D17—24no—
217C22×D17—24no—
218C22×D17—24no—
219C22×D17—24no—
220C22×D17—24no—
221C22×D17—24no—
222C22×D17—24no—
223C22×D17—24no—
224C22×D17C2×C22⋊C414yes—
225C22×D17C4×D418yes—
226C22×D17#12014yes—
227C22×D17#12118yes—
228C22×D17#12314yes—
229C2×C4×C17—22no—
230C2×C4×C17—22no—
231C2×C4×C17—24no—
232C2×C4×C17—22no—
233C2×C4×C17—24no—
234C2×C4×C17—22no—
235C2×C4×C17—24no—
236C2×C4×C17—24no—
237C2×C4×C17—22no—
238C2×C4×C17—24no—
239C2×C4×C17—24no—
240C2×C4×C17—24no—
241C2×C4×C17—24no—
242C2×C4×C17—41no—
243C2×C4×C17—42no—
244C2×C4×C17—41no—
245C2×C4×C17—42no—
246C2×C4×C17—42no—
247C2×C4×C17—42no—
248C2×C4×C17—42no—
249C2×C4×C17—42no—
250C2×C4×C17—42no—
251C2×C4×C17—42no—
252C2×C4×C17—42no—
253C2×C4×C17—42no—
254C2×C4×C17—42no—
255C2×C4×C17—42no—
256C2×C4×C17C2×C22⋊C414yes—
257C2×C4×C17#11714yescomplemented
258C2×Dic17—22no—
259C2×Dic17—22no—
260C2×Dic17—24no—
261C2×Dic17—22no—
262C2×Dic17—24no—
263C2×Dic17—22no—
264C2×Dic17—24no—
265C2×Dic17—24no—
266C2×Dic17—22no—
267C2×Dic17—24no—
268C2×Dic17—24no—
269C2×Dic17—24no—
270C2×Dic17—24no—
271C2×Dic17—41no—
272C2×Dic17—42no—
273C2×Dic17—41no—
274C2×Dic17—42no—
275C2×Dic17—42no—
276C2×Dic17—42no—
277C2×Dic17—42no—
278C2×Dic17—42no—
279C2×Dic17—42no—
280C2×Dic17—42no—
281C2×Dic17—42no—
282C2×Dic17—42no—
283C2×Dic17—42no—
284C2×Dic17—42no—
285C2×Dic17C2×C22⋊C414yes—
286C2×Dic17#11714yescomplemented
287C4×D17—232no—
288C4×D17—232no—
289C4×D17—416no—
290C22×#117—171no—
291C23×C4⋊C4—171no—
292C2×#381—172no—
293C2×#381—171no—
294#1441—172no—
295#1441—172no—
296#1436—171no—
297#1436—171no—
298#1436—171no—
299#1436D1711yescharacteristic, complemented
300C22⋊#95—172no—
301C22×C17C22×C22⋊C411yescharacteristic
302C22×C17C2×#11711yescharacteristic
303C22×C17C2×#11711yescharacteristic
304C22×C17C2×C4×D412yes—
305C22×C17#51912yes—
306C22×C17C2×#12011yescharacteristic
307C22×C17C2×#12112yes—
308C22×C17C2×#12311yescharacteristic
309C22×C17#52512yes—
310C22×C17#52512yes—
311C22×C17#53611yescharacteristic
312C22×C17#53612yes—
313C22×C17#53611yescharacteristic
314C22×C17#54712yes—
315C22×C17#54712yes—
316C22×C17#54712yes—
317C22×C17#54712yes—
318C22×C17#55212yes—
319C22×C17#55212yes—
320C22×C17#55212yes—
321C22×C17#55212yes—
322C2×D17—24no—
323C2×D17—28no—
324C2×D17—28no—
325C2×D17—28no—
326C2×D17—28no—
327C2×D17—28no—
328C2×D17—28no—
329C2×D17—24no—
330C2×D17#38618yes—
331C4×C17—24no—
332C4×C17—24no—
333C4×C17—44no—
334C4×C17—42no—
335C4×C17—82no—
336Dic17—24no—
337Dic17—24no—
338Dic17—44no—
339Dic17—42no—
340Dic17—82no—
341C24×C4—171no—
342C24×C4—171no—
343C24×C4—171no—
344C2×#117—348no—
345C2×#117—344no—
346C22×C4⋊C4—171no—
347C22×C4⋊C4—171no—
348C22×C4⋊C4—171no—
349C22×C4⋊C4—178no—
350C22×C4⋊C4—172no—
351C22×C4⋊C4—344no—
352C22×C4⋊C4—344no—
353C22×C4⋊C4C2×D1711yescharacteristic
354C22×C22⋊C4—172no—
355C22×C22⋊C4—172no—
356C22×C42—171no—
357C22×C42C2×D1711yescharacteristic
358#381—348no—
359#381—344no—
360C2×#95—172no—
361C2×#95—172no—
362C2×#95—172no—
363C2×#95—171no—
364C2×#95—172no—
365C2×#95—172no—
366C2×#95—172no—
367C2×#95—172no—
368C2×#95C2×D1712yes—
369C2×#95C2×D1711yescharacteristic
370C2×#95C2×D1712yes—
371C2×C17#228011yescharacteristic
372C2×C17C2×#38611yescharacteristic
373C2×C17#231011yescharacteristic
374C2×C17#234412yes—
375C2×C17#234412yes—
376C2×C17#240612yes—
377C2×C17#241312yes—
378C2×C17#241312yes—
379C2×C17#248412yes—
380D17—28no—
381C25—171no—
382C23×C4—171no—
383C23×C4—171no—
384C23×C4—174no—
385C23×C4—174no—
386C23×C4—174no—
387C23×C4—172no—
388C23×C4—171no—
389C23×C4—174no—
390C23×C4—174no—
391C23×C4—172no—
392C23×C4—344no—
393C23×C4—342no—
394C23×C4—342no—
395C23×C4—344no—
396C23×C4—344no—
397C23×C4—342no—
398C23×C4—342no—
399C23×C4—344no—
400C23×C4—344no—
401C23×C4—344no—
402C23×C4C22×D1712yes—
403C23×C4C22×D1711yescharacteristic
404C23×C4C22×D1712yes—
405C23×C4C22×D1711yescharacteristic
406C23×C4C22×D1711yescharacteristic
407#117—3432no—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—174no—
411C2×C4⋊C4—174no—
412C2×C4⋊C4—174no—
413C2×C4⋊C4—3432no—
414C2×C4⋊C4—342no—
415C2×C4⋊C4—342no—
416C2×C4⋊C4—342no—
417C2×C4⋊C4—342no—
418C2×C4⋊C4—344no—
419C2×C4⋊C4—344no—
420C2×C4⋊C4—342no—
421C2×C4⋊C4—342no—
422C2×C4⋊C4—344no—
423C2×C4⋊C4C4×D1714yes—
424C2×C22⋊C4—178no—
425C2×C22⋊C4—344no—
426C2×C22⋊C4—344no—
427C2×C22⋊C4—344no—
428C2×C22⋊C4—344no—
429C2×C22⋊C4—344no—
430C2×C42—24no—
431C2×C42—22no—
432C2×C42—344no—
433C2×C42—342no—
434#95—24no—
435#95—24no—
436#95—22no—
437#95—344no—
438#95—344no—
439#95—344no—
440#95—344no—
441#95—344no—
442#95—344no—
443#95—344no—
444#95—342no—
445C17#1387011yescharacteristic, complemented, Sylow
446C24—172no—
447C24—174no—
448C24—172no—
449C24—172no—
450C24—174no—
451C24—342no—
452C24—342no—
453C24—342no—
454C24—342no—
455C24C23×D1711yescharacteristic
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—172no—
479C22×C4—172no—
480C22×C4—172no—
481C22×C4—172no—
482C22×C4—174no—
483C22×C4—1716no—
484C22×C4—172no—
485C22×C4—174no—
486C22×C4—348no—
487C22×C4—348no—
488C22×C4—348no—
489C22×C4—3416no—
490C22×C4—3416no—
491C22×C4—348no—
492C22×C4—3416no—
493C22×C4—3416no—
494C22×C4—348no—
495C22×C4—3416no—
496C22×C4—342no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4—348no—
502C22×C4—342no—
503C22×C4—342no—
504C22×C4—342no—
505C22×C4—342no—
506C22×C4—341no—
507C22×C4—341no—
508C22×C4—342no—
509C22×C4—342no—
510C22×C4—342no—
511C22×C4—342no—
512C22×C4—342no—
513C22×C4—342no—
514C22×C4—342no—
515C22×C4—342no—
516C22×C4—342no—
517C22×C4—342no—
518C22×C4—342no—
519C22×C4—684no—
520C22×C4—688no—
521C22×C4—684no—
522C22×C4C2×C4×D1712yes—
523C22×C4C2×C4×D1712yes—
524C22×C4C2×C4×D1712yes—
525C22×C4#6325214yes—
526C22×C4D4×D1714yes—
527C22×C4D4×D1712yes—
528C22×C4D4×D1712yes—
529C4⋊C4—28no—
530C4⋊C4—348no—
531C4⋊C4—348no—
532C4⋊C4—3416no—
533C4⋊C4—348no—
534C22⋊C4—3416no—
535C22⋊C4—688no—
536C42—28no—
537C42—348no—
538C23—174no—
539C23—178no—
540C23—174no—
541C23—178no—
542C23—174no—
543C23—344no—
544C23—344no—
545C23—344no—
546C23—344no—
547C23—344no—
548C23—344no—
549C23—344no—
550C23—344no—
551C23—342no—
552C23—342no—
553C23—344no—
554C23—344no—
555C23—344no—
556C23—342no—
557C23—344no—
558C23—342no—
559C23C22×C4×D1711yescharacteristic
560C23C2×#6325212yes—
561C23C2×D4×D1712yes—
562C23C2×D4×D1711yescharacteristic
563C23C2×D4×D1711yescharacteristic
564C23#1058769412yes—
565C23#41×D1712yes—
566C23#41×D1712yes—
567C23#1058769712yes—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—174no—
596C2×C4—174no—
597C2×C4—3432no—
598C2×C4—3432no—
599C2×C4—342no—
600C2×C4—342no—
601C2×C4—344no—
602C2×C4—342no—
603C2×C4—344no—
604C2×C4—342no—
605C2×C4—344no—
606C2×C4—344no—
607C2×C4—344no—
608C2×C4—344no—
609C2×C4—342no—
610C2×C4—344no—
611C2×C4—344no—
612C2×C4—6816no—
613C2×C4—682no—
614C2×C4—682no—
615C2×C4—681no—
616C2×C4—682no—
617C2×C4—681no—
618C2×C4—682no—
619C2×C4—682no—
620C2×C4—682no—
621C2×C4—682no—
622C2×C4—682no—
623C2×C4—682no—
624C2×C4—682no—
625C2×C4—682no—
626C2×C4—682no—
627C2×C4#1058755814yescomplemented
628C2×C4C22⋊C4×D1714yes—
629C22—178no—
630C22—348no—
631C22—344no—
632C22—348no—
633C22—348no—
634C22—348no—
635C22—344no—
636C22—348no—
637C22—348no—
638C22C2×C22⋊C4×D1711yescharacteristic
639C22#117×D1711yescharacteristic
640C22C2×#1058755811yescharacteristic
641C22C4×D4×D1712yes—
642C22#1176328512yes—
643C22#120×D1711yescharacteristic
644C22#121×D1712yes—
645C22#123×D1711yescharacteristic
646C22#1176331812yes—
647C22#1176331812yes—
648C22#1176337911yescharacteristic
649C22#1176338212yes—
650C22#1176338411yescharacteristic
651C22#1176345712yes—
652C22#1176346012yes—
653C22#1176346112yes—
654C22#1176346412yes—
655C22#1176349112yes—
656C22#1176349512yes—
657C22#1176349812yes—
658C22#1176350012yes—
659C4—24no—
660C4—24no—
661C4—44no—
662C4—42no—
663C4—82no—
664C4—344no—
665C4—344no—
666C4—684no—
667C4—682no—
668C4—1362no—
669C2—348no—
670C2#1453116511yescharacteristic
671C2D17×#38611yescharacteristic
672C2#1453136711yescharacteristic
673C2#1453164912yes—
674C2#1453165012yes—
675C2#1453211712yes—
676C2#1453218812yes—
677C2#1453219012yes—
678C2#1453306912yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1436C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1436D17
hypocenterC17#13870
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
5. #14527036C2semidirect
6. #14527035C2semidirect
7. #14527019C2semidirect
8. #14527017C2semidirect
9. #14527016C2semidirect
11. C17×#1436C2semidirect
23. C2×C4⋊C4×D17C4semidirect
257. C2×C4×C17#117semidirect
286. C2×Dic17#117semidirect
299. #1436D17semidirect
445. C17#13870semidirect
627. C2×C4#10587558semidirect

Maximal subgroups

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

subgroupindex
C22×#105875582
C22×C4⋊C4×D172
C2×#117626712
#145270362
#145270352
#145270192
#145270172
#145270162
#145270052
C17×#14362
#1387017