Finite Group Databasebeta

← #16965354

Subgroups — #16965354

Order 4352.

1702 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587564C211yesmaximal, characteristic
2C2×#11762670C211yesmaximal, characteristic
3C2×#11762667C211yesmaximal, characteristic
4#14527054C212yesmaximal, complemented
5#14527050C212yesmaximal, complemented
6#14527036C212yesmaximal, complemented
7#14527035C212yesmaximal, complemented
8#14527033C211yesmaximal, characteristic
9#14527031C211yesmaximal, characteristic
10#14527031C211yesmaximal, characteristic
11C17×#1440C211yesmaximal, characteristic
12C23×Dic34C2211yescharacteristic
13C24×Dic17C2211yescharacteristic
14C2×#10587564—24no—
15C2×#10587564—24no—
16C2×#10587564C418yescomplemented
17C22×C4⋊Dic17C2211yescharacteristic
18C22×#63225C2211yescharacteristic
19C22×#63225C2211yescharacteristic
20C22×#63225C2212yes—
21C22×C22⋊Dic17C2211yescharacteristic
22C22×C22⋊Dic17C2211yescharacteristic
23C22×C22⋊Dic17C2211yescharacteristic
24#11762670—24no—
25#11762667—24no—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C22×C22⋊C4×C17C2211yescharacteristic
36C22×C22⋊C4×C17C2211yescharacteristic
37C22×C22⋊C4×C17C2211yescharacteristic
38C2×C17×#95C2212yes—
39C2×C17×#95C2212yes—
40C25×C17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C23×Dic17—42no—
46C23×Dic17—42no—
47C23×Dic17C2×C414yes—
48C23×Dic17D414yes—
49C23×Dic17D414yes—
50C23×Dic17C2312yes—
51C23×Dic17C2312yes—
52C23×Dic17C2311yescharacteristic
53C23×Dic17C2311yescharacteristic
54C23×Dic17C2311yescharacteristic
55C23×Dic17C2311yescharacteristic
56C22×Dic34—22no—
57C22×Dic34—22no—
58C22×Dic34—42no—
59C22×Dic34—42no—
60C22×Dic34C2×C414yes—
61C2×C22⋊C4×C17—22no—
62C2×C22⋊C4×C17—22no—
63C2×C22⋊C4×C17—22no—
64C2×C22⋊C4×C17—22no—
65C2×C22⋊C4×C17—42no—
66C2×C22⋊C4×C17—42no—
67C2×C22⋊C4×C17C2×C414yes—
68C2×C22⋊Dic17—22no—
69C2×C22⋊Dic17—22no—
70C2×C22⋊Dic17—22no—
71C2×C22⋊Dic17—22no—
72C2×C22⋊Dic17—42no—
73C2×C22⋊Dic17—42no—
74C2×C22⋊Dic17C2×C414yes—
75C2×C4⋊Dic17—22no—
76C2×C4⋊Dic17—22no—
77C2×C4⋊Dic17C2×C414yes—
78C2×#63225—22no—
79C2×#63225—22no—
80C2×#63225—22no—
81C2×#63225—22no—
82C2×#63225—24no—
83C2×#63225—42no—
84C2×#63225—42no—
85C2×#63225C2×C414yes—
86C2×#63225C2×C414yes—
87#10587564—232no—
88C17×#95—42no—
89C17×#95—42no—
90#10587502—24no—
91#10587502—24no—
92#10587502—24no—
93#10587502—42no—
94#10587502—42no—
95#10587502—42no—
96#10587502—42no—
97#10587502—42no—
98#10587502—42no—
99C24×C17—22no—
100C24×C17—22no—
101C24×C17—22no—
102C24×C17—22no—
103C24×C17—22no—
104C24×C17—22no—
105C24×C17C22×C412yes—
106C24×C17C2×D412yes—
107C24×C17C2×D412yes—
108C24×C17C2411yescharacteristic
109C22×C4×C17—21no—
110C22×C4×C17—21no—
111C22×C4×C17—21no—
112C22×C4×C17—21no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
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×C17C22×C412yes—
132C22×C4×C17C22×C412yes—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—21no—
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—48no—
160C22×Dic17—44no—
161C22×Dic17—44no—
162C22×Dic17—48no—
163C22×Dic17C22⋊C4116yes—
164C22×Dic17C22×C412yes—
165C22×Dic17C22×C412yes—
166C22×Dic17C22×C412yes—
167C22×Dic17C22×C412yes—
168C22×Dic17C2×D412yes—
169C22×Dic17C2×D412yes—
170C22×Dic17C2×D412yes—
171C22×Dic17C2×D412yes—
172C2×Dic34—28no—
173C2×Dic34—48no—
174C2×Dic34—48no—
175C2×Dic34—44no—
176C2×Dic34—44no—
177C22⋊C4×C17—28no—
178C22⋊C4×C17—44no—
179C22⋊C4×C17—44no—
180C22⋊Dic17—28no—
181C22⋊Dic17—44no—
182C22⋊Dic17—44no—
183C4⋊Dic17—28no—
184#63225—28no—
185#63225—28no—
186#63225—48no—
187#14660—171nomaximal, Sylow
188C23×C17—24no—
189C23×C17—22no—
190C23×C17—22no—
191C23×C17—24no—
192C23×C17—22no—
193C23×C17—22no—
194C23×C17—24no—
195C23×C17—22no—
196C23×C17—24no—
197C23×C17—24no—
198C23×C17—22no—
199C23×C17—24no—
200C23×C17—42no—
201C23×C17—42no—
202C23×C17—42no—
203C23×C17—42no—
204C23×C17—42no—
205C23×C17—42no—
206C23×C17—42no—
207C23×C17—42no—
208C23×C17C2×C22⋊C414yes—
209C23×C17C23×C411yescharacteristic
210C23×C17C22×D411yescharacteristic
211C23×C17C22×D411yescharacteristic
212C23×C17H(5,𝔽2)12yes—
213C23×C17H(5,𝔽2)12yes—
214C23×C17H(5,𝔽2)12yes—
215C23×C172−1+412yes—
216C23×C172−1+412yes—
217C23×C172−1+412yes—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
222C2×C4×C17—42no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—42no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—42no—
233C2×C4×C17—42no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—42no—
241C2×C4×C17—42no—
242C2×Dic17—24no—
243C2×Dic17—24no—
244C2×Dic17—24no—
245C2×Dic17—22no—
246C2×Dic17—22no—
247C2×Dic17—22no—
248C2×Dic17—22no—
249C2×Dic17—24no—
250C2×Dic17—22no—
251C2×Dic17—22no—
252C2×Dic17—24no—
253C2×Dic17—24no—
254C2×Dic17—416no—
255C2×Dic17—42no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
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—
270C2×Dic17—42no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C2×Dic17—42no—
275C2×Dic17C2×C22⋊C414yes—
276C2×Dic17C2×C22⋊C414yes—
277Dic34—416no—
278C22×C22⋊Q8—171no—
279C2×#381—171no—
280C2×#380—171no—
281#1446—172no—
282#1445—172no—
283#1441—172no—
284#1441—172no—
285#1440—171no—
286#1440—171no—
287#1440—171no—
288#1440D1711yescharacteristic
289C22×C17—24no—
290C22×C17—24no—
291C22×C17—24no—
292C22×C17—44no—
293C22×C17—44no—
294C22×C17—44no—
295C22×C17—44no—
296C22×C17—44no—
297C22×C17—44no—
298C22×C17C22×C22⋊C411yescharacteristic
299C22×C17#51812yes—
300C22×C17#51911yescharacteristic
301C22×C17#51911yescharacteristic
302C22×C17#52012yes—
303C22×C17#53412yes—
304C22×C17#53411yescharacteristic
305C22×C17#53411yescharacteristic
306C22×C17#53511yescharacteristic
307C22×C17#53612yes—
308C22×C17#53611yescharacteristic
309C22×C17#53612yes—
310C22×C17#53712yes—
311C22×C17#56212yes—
312C22×C17#56212yes—
313C22×C17#56212yes—
314C22×C17#56212yes—
315C22×C17#56312yes—
316C22×C17#56312yes—
317C22×C17#56312yes—
318C22×C17#56312yes—
319C4×C17—42no—
320C4×C17—42no—
321C4×C17—82no—
322C4×C17—82no—
323Dic17—24no—
324Dic17—24no—
325Dic17—42no—
326Dic17—42no—
327Dic17—82no—
328Dic17—82no—
329C23×Q8—171no—
330C24×C4—171no—
331C2×C22⋊Q8—178no—
332C2×C22⋊Q8—344no—
333C2×C22⋊Q8—344no—
334C22×C4⋊C4—171no—
335C22×C4⋊C4—171no—
336C22×C4⋊C4—171no—
337C22×C4⋊C4—172no—
338C22×C22⋊C4—171no—
339C22×C22⋊C4—171no—
340C22×C22⋊C4—171no—
341C22×C22⋊C4C2×D1711yescharacteristic
342C22×C22⋊C4C2×D1711yescharacteristic
343C22×C22⋊C4C2×D1711yescharacteristic
344#381—344no—
345#380—344no—
346C2×#95—172no—
347C2×#95—172no—
348C2×#95—172no—
349C2×#95—172no—
350C2×#95—172no—
351C2×#95—172no—
352C2×#95—172no—
353C2×#95—172no—
354C2×#95—172no—
355C2×#95C2×D1712yes—
356C2×#95C2×D1712yes—
357C2×C17—44no—
358C2×C17#230811yescharacteristic
359C2×C17#231011yescharacteristic
360C2×C17#231111yescharacteristic
361C2×C17#237812yes—
362C2×C17#237812yes—
363C2×C17#281412yes—
364C2×C17#281412yes—
365C2×C17#281612yes—
366C2×C17#281712yes—
367C25C22×D1711yescharacteristic
368C22×Q8—174no—
369C22×Q8—342no—
370C22×Q8—342no—
371C22×Q8—682no—
372C22×Q8—682no—
373C23×C4—171no—
374C23×C4—171no—
375C23×C4—174no—
376C23×C4—174no—
377C23×C4—174no—
378C23×C4—171no—
379C23×C4—171no—
380C23×C4—172no—
381C23×C4—172no—
382C23×C4—682no—
383C23×C4—682no—
384C23×C4C22×D1712yes—
385C23×C4C22×D1712yes—
386C23×C4C22×D1711yescharacteristic
387C23×C4C22×D1711yescharacteristic
388C22⋊Q8—3432no—
389C2×C4⋊C4—174no—
390C2×C4⋊C4—174no—
391C2×C4⋊C4—174no—
392C2×C4⋊C4—342no—
393C2×C4⋊C4—342no—
394C2×C4⋊C4—342no—
395C2×C4⋊C4—342no—
396C2×C4⋊C4—342no—
397C2×C4⋊C4—342no—
398C2×C4⋊C4—344no—
399C2×C4⋊C4—682no—
400C2×C4⋊C4—682no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—174no—
408C2×C22⋊C4—342no—
409C2×C22⋊C4—342no—
410C2×C22⋊C4—342no—
411C2×C22⋊C4—342no—
412C2×C22⋊C4—682no—
413C2×C22⋊C4—682no—
414C2×C22⋊C4C4×D1714yes—
415#95—42no—
416#95—42no—
417#95—344no—
418#95—344no—
419#95—344no—
420#95—682no—
421#95—682no—
422#95—682no—
423#95—682no—
424#95—682no—
425#95—682no—
426C17#1466011yescharacteristic, complemented, Sylow
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24C2×C4×D1712yes—
434C24D4×D1712yes—
435C24D4×D1712yes—
436C24C23×D1711yescharacteristic
437C2×Q8—348no—
438C2×Q8—688no—
439C2×Q8—688no—
440C2×Q8—684no—
441C2×Q8—684no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—172no—
465C22×C4—172no—
466C22×C4—172no—
467C22×C4—172no—
468C22×C4—172no—
469C22×C4—172no—
470C22×C4—1716no—
471C22×C4—172no—
472C22×C4—172no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—341no—
478C22×C4—341no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4—342no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4—342no—
486C22×C4—342no—
487C22×C4—342no—
488C22×C4—342no—
489C22×C4—342no—
490C22×C4—342no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4—342no—
494C22×C4—342no—
495C22×C4—342no—
496C22×C4—342no—
497C22×C4—342no—
498C22×C4—342no—
499C22×C4—688no—
500C22×C4—688no—
501C22×C4—684no—
502C22×C4—684no—
503C22×C4C2×C4×D1712yes—
504C22×C4C2×C4×D1712yes—
505C4⋊C4—348no—
506C4⋊C4—348no—
507C4⋊C4—348no—
508C4⋊C4—688no—
509C22⋊C4—28no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—348no—
513C22⋊C4—684no—
514C22⋊C4—684no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23C22⋊C4×D1714yes—
536C23C22×C4×D1711yescharacteristic
537C23C2×D4×D1711yescharacteristic
538C23C2×D4×D1711yescharacteristic
539C23#1058769112yes—
540C23#1058769112yes—
541C23#1058769612yes—
542C23#1058769712yes—
543C23#1058769712yes—
544C23#1058769712yes—
545Q8—6816no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—174no—
571C2×C4—174no—
572C2×C4—344no—
573C2×C4—344no—
574C2×C4—344no—
575C2×C4—344no—
576C2×C4—342no—
577C2×C4—342no—
578C2×C4—342no—
579C2×C4—342no—
580C2×C4—342no—
581C2×C4—342no—
582C2×C4—344no—
583C2×C4—344no—
584C2×C4—6816no—
585C2×C4—682no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—682no—
591C2×C4—682no—
592C2×C4—682no—
593C2×C4—682no—
594C2×C4—682no—
595C2×C4—682no—
596C2×C4—682no—
597C2×C4—682no—
598C2×C4—682no—
599C2×C4—682no—
600C2×C4—682no—
601C2×C4—682no—
602C2×C4—682no—
603C2×C4—682no—
604C2×C4—682no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22C2×C22⋊C4×D1711yescharacteristic
615C22#1176328112yes—
616C22#1176328611yescharacteristic
617C22#1176328611yescharacteristic
618C22#1176328812yes—
619C22#1176336712yes—
620C22#1176336911yescharacteristic
621C22#1176337011yescharacteristic
622C22#1176337411yescharacteristic
623C22#1176338112yes—
624C22#1176338112yes—
625C22#1176338111yescharacteristic
626C22#1176338612yes—
627C22#1176355312yes—
628C22#1176355412yes—
629C22#1176355412yes—
630C22#1176355612yes—
631C22#1176355912yes—
632C22#1176356112yes—
633C22#1176356112yes—
634C22#1176356212yes—
635C4—42no—
636C4—42no—
637C4—82no—
638C4—82no—
639C4—344no—
640C4—344no—
641C4—682no—
642C4—682no—
643C4—1362no—
644C4—1362no—
645C2—44no—
646C2#1453135811yescharacteristic
647C2#1453136911yescharacteristic
648C2#1453137111yescharacteristic
649C2#1453194412yes—
650C2#1453195112yes—
651C2#1453680112yes—
652C2#1453680312yes—
653C2#1453681612yes—
654C2#1453682212yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1440C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenter#1440D17
hypocenterC17#14660
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
4. #14527054C2semidirect
5. #14527050C2semidirect
6. #14527036C2semidirect
7. #14527035C2semidirect
16. C2×#10587564C4semidirect
426. C17#14660semidirect

Maximal subgroups

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

subgroupindex
C22×#105875642
C2×#117626702
C2×#117626672
#145270542
#145270502
#145270362
#145270352
#145270332
#145270312
C17×#14402
#1466017