Finite Group Databasebeta

← #16960967

Subgroups — #16960967

Order 4352.

1978 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587572C211yesmaximal, characteristic
2C2×#11762667C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×Dic17C211yesmaximal, characteristic, complemented
4#14527036C212yesmaximal, complemented
5#14527033C211yesmaximal, characteristic, complemented
6#14527028C212yesmaximal, complemented
7#14527023C211yesmaximal, characteristic
8#14527023C211yesmaximal, characteristic
9#14527020C212yesmaximal, complemented
10#14527018C212yesmaximal, complemented
11C17×#1440C211yesmaximal, characteristic, complemented
12C23×C22⋊D17C2211yescharacteristic
13C24×Dic17C2211yescharacteristic
14C2×#10587572—24no—
15C2×#10587572—24no—
16C2×#10587572C418yescomplemented
17C22×C4⋊Dic17C2211yescharacteristic
18C22×C4⋊Dic17C2211yescharacteristic
19C22×#63225C2212yes—
20C22×C22⋊Dic17C2211yescharacteristic
21C22×C22⋊Dic17C2211yescharacteristic
22C22×#63227C2211yescharacteristic
23C22×#63227C2211yescharacteristic
24C22×#63227C2212yes—
25C22×#63227C2212yes—
26C22×C4×Dic17C2212yes—
27#11762667—24no—
28C22⋊C4×Dic17—24no—
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
41C24×D17C2311yescharacteristic
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2312yes—
46C22×C22⋊D17—22no—
47C22×C22⋊D17—22no—
48C22×C22⋊D17—42no—
49C22×C22⋊D17—42no—
50C22×C22⋊D17C2×C414yes—
51C23×Dic17—42no—
52C23×Dic17—42no—
53C23×Dic17C2×C414yes—
54C23×Dic17D414yes—
55C23×Dic17D414yes—
56C23×Dic17C2312yes—
57C23×Dic17C2311yescharacteristic
58C23×Dic17C2312yes—
59C23×Dic17C2311yescharacteristic
60C23×Dic17C2311yescharacteristic
61C2×C22⋊C4×C17—22no—
62C2×C22⋊C4×C17—22no—
63C2×C22⋊C4×C17—22no—
64C2×C22⋊C4×C17—24no—
65C2×C22⋊C4×C17—22no—
66C2×C22⋊C4×C17—42no—
67C2×C22⋊C4×C17C2×C414yes—
68C2×C22⋊Dic17—22no—
69C2×C22⋊Dic17—24no—
70C2×C22⋊Dic17—22no—
71C2×C22⋊Dic17—42no—
72C2×#63227—24no—
73C2×#63227—22no—
74C2×#63227—22no—
75C2×#63227—22no—
76C2×#63227—22no—
77C2×#63227—24no—
78C2×#63227—42no—
79C2×#63227—42no—
80C2×#63227—42no—
81C2×#63227—42no—
82C2×#63227C2×C414yes—
83C2×#63227C2×C414yes—
84C2×C4⋊Dic17—22no—
85C2×C4⋊Dic17—22no—
86C2×C4⋊Dic17—22no—
87C2×C4⋊Dic17—22no—
88C2×C4⋊Dic17C2×C414yes—
89C2×C4⋊Dic17C2×C414yes—
90C2×#63225—24no—
91C2×#63225—42no—
92C2×#63225—42no—
93C2×C4×Dic17—24no—
94C2×C4×Dic17—42no—
95C2×C4×Dic17—42no—
96#10587572—232no—
97C17×#95—24no—
98C17×#95—42no—
99#10587502—24no—
100#10587502—24no—
101#10587502—24no—
102#10587502—42no—
103#10587502—42no—
104#10587502—42no—
105C24×C17—22no—
106C24×C17—22no—
107C24×C17—22no—
108C24×C17—22no—
109C24×C17—22no—
110C24×C17—22no—
111C24×C17C22×C412yes—
112C24×C17C2×D412yes—
113C24×C17C2×D412yes—
114C24×C17C2411yescharacteristic
115C23×D17—21no—
116C23×D17—21no—
117C23×D17—22no—
118C23×D17—22no—
119C23×D17—22no—
120C23×D17—22no—
121C23×D17—22no—
122C23×D17—22no—
123C23×D17C22×C412yes—
124C22×C4×C17—22no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
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×C17—22no—
144C22×C4×C17—22no—
145C22×C4×C17C22×C412yes—
146C22×C4×C17C22×C412yes—
147C22×C4×C17#4114yes—
148C2×C22⋊D17—28no—
149C2×C22⋊D17—44no—
150C2×C22⋊D17—44no—
151C2×C22⋊D17—48no—
152C2×C22⋊D17—48no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—21no—
158C22×Dic17—21no—
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×Dic17—22no—
168C22×Dic17—22no—
169C22×Dic17—22no—
170C22×Dic17—22no—
171C22×Dic17—22no—
172C22×Dic17—22no—
173C22×Dic17—22no—
174C22×Dic17—44no—
175C22×Dic17—44no—
176C22×Dic17—48no—
177C22×Dic17—48no—
178C22×Dic17C22⋊C4116yescomplemented
179C22×Dic17C22×C412yes—
180C22×Dic17C22×C412yes—
181C22×Dic17C22×C412yes—
182C22×Dic17C2×D412yes—
183C22×Dic17C2×D412yes—
184C22×Dic17C2×D412yes—
185C22×Dic17C2×D412yes—
186C22×Dic17#4114yes—
187C22⋊C4×C17—28no—
188C22⋊C4×C17—28no—
189C22⋊C4×C17—44no—
190C22⋊Dic17—28no—
191C22⋊Dic17—44no—
192#63227—28no—
193#63227—28no—
194#63227—48no—
195#63227—48no—
196C4⋊Dic17—28no—
197C4⋊Dic17—28no—
198#63225—48no—
199C4×Dic17—48no—
200#14287—171nomaximal, Sylow
201C23×C17—24no—
202C23×C17—22no—
203C23×C17—22no—
204C23×C17—24no—
205C23×C17—22no—
206C23×C17—22no—
207C23×C17—24no—
208C23×C17—24no—
209C23×C17—24no—
210C23×C17—22no—
211C23×C17—22no—
212C23×C17—24no—
213C23×C17—42no—
214C23×C17—42no—
215C23×C17—42no—
216C23×C17—42no—
217C23×C17—42no—
218C23×C17—42no—
219C23×C17—42no—
220C23×C17—42no—
221C23×C17C2×C22⋊C414yes—
222C23×C17C23×C411yescharacteristic
223C23×C17C22×D411yescharacteristic
224C23×C17C22×D411yescharacteristic
225C23×C17C2×#4112yes—
226C23×C17C2×#4112yes—
227C23×C17H(5,𝔽2)12yes—
228C23×C17H(5,𝔽2)12yes—
229C23×C17H(5,𝔽2)12yes—
230C23×C172−1+412yes—
231C22×D17—22no—
232C22×D17—24no—
233C22×D17—22no—
234C22×D17—22no—
235C22×D17—24no—
236C22×D17—41no—
237C22×D17—41no—
238C22×D17—42no—
239C22×D17—42no—
240C22×D17—42no—
241C22×D17—41no—
242C22×D17—41no—
243C22×D17—42no—
244C22×D17—42no—
245C22×D17—42no—
246C22×D17—42no—
247C22×D17—42no—
248C22×D17—42no—
249C22×D17—42no—
250C22×D17—42no—
251C22×D17—42no—
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—24no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17—42no—
262C2×C4×C17—42no—
263C2×C4×C17—41no—
264C2×C4×C17—41no—
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×C17—42no—
273C2×C4×C17—42no—
274C2×C4×C17—42no—
275C2×C4×C17—42no—
276C22⋊D17—416no—
277C2×Dic17—22no—
278C2×Dic17—22no—
279C2×Dic17—22no—
280C2×Dic17—22no—
281C2×Dic17—24no—
282C2×Dic17—24no—
283C2×Dic17—24no—
284C2×Dic17—24no—
285C2×Dic17—22no—
286C2×Dic17—22no—
287C2×Dic17—22no—
288C2×Dic17—24no—
289C2×Dic17—24no—
290C2×Dic17—24no—
291C2×Dic17—416no—
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×Dic17—42no—
304C2×Dic17—42no—
305C2×Dic17C2×C22⋊C414yes—
306C2×Dic17C2×C22⋊C414yes—
307C22×#123—171no—
308C2×#380—171no—
309C2×C4×C22⋊C4—171no—
310#1441—172no—
311#1440—171no—
312#1440D1711yescharacteristic, complemented
313#1439—172no—
314#1438—171no—
315#1438—171no—
316#1437—172no—
317#1436—172no—
318C22×C17—24no—
319C22×C17—24no—
320C22×C17—24no—
321C22×C17—44no—
322C22×C17—44no—
323C22×C17—44no—
324C22×C17—44no—
325C22×C17—44no—
326C22×C17—44no—
327C22×C17C22×C22⋊C411yescharacteristic
328C22×C17C4×#4112yes—
329C22×C17#51811yescharacteristic
330C22×C17#51811yescharacteristic
331C22×C17#52012yes—
332C22×C17#52511yescharacteristic
333C22×C17#52511yescharacteristic
334C22×C17#53212yes—
335C22×C17#53412yes—
336C22×C17#53412yes—
337C22×C17#53511yescharacteristic
338C22×C17#53611yescharacteristic
339C22×C17#53712yes—
340C22×C17#53812yes—
341C22×C17#53812yes—
342C22×C17#53912yes—
343C22×C17#53912yes—
344C22×C17#54012yes—
345C22×C17#54012yes—
346C22×C17#54212yes—
347C22×C17#54212yes—
348C2×D17—42no—
349C2×D17—42no—
350C2×D17—44no—
351C2×D17—44no—
352C2×D17—42no—
353C2×D17—82no—
354C2×D17—82no—
355C2×D17—82no—
356C2×D17—82no—
357C4×C17—42no—
358C4×C17—44no—
359C4×C17—42no—
360C4×C17—82no—
361Dic17—24no—
362Dic17—24no—
363Dic17—42no—
364Dic17—44no—
365Dic17—82no—
366C23×D4—171no—
367C24×C4—171no—
368C2×#123—178no—
369C2×#123—344no—
370C2×#123—344no—
371C22×C4⋊C4—171no—
372C22×C4⋊C4—171no—
373C22×C4⋊C4—172no—
374C22×C22⋊C4—171no—
375C22×C22⋊C4—171no—
376C22×C22⋊C4—171no—
377C22×C22⋊C4—171no—
378C22×C22⋊C4—172no—
379C22×C22⋊C4—172no—
380C22×C22⋊C4C2×D1711yescharacteristic
381C22×C22⋊C4C2×D1711yescharacteristic
382C22×C22⋊C4C2×D1711yescharacteristic
383C22×C42—172no—
384#380—344no—
385C4×C22⋊C4—344no—
386C2×#95—172no—
387C2×#95—172no—
388C2×#95—172no—
389C2×#95—172no—
390C2×#95—172no—
391C2×#95—172no—
392C2×#95C2×D1712yes—
393C2×#95C2×D1712yes—
394C2×C17—44no—
395C2×C17#229711yescharacteristic
396C2×C17#230811yescharacteristic
397C2×C17#231111yescharacteristic
398C2×C17#233512yes—
399C2×C17#233512yes—
400C2×C17#263012yes—
401C2×C17#263012yes—
402C2×C17#266112yes—
403C2×C17#266312yes—
404D17—82no—
405C25—171no—
406C25C22×D1711yescharacteristic
407C22×D4—174no—
408C22×D4—342no—
409C22×D4—342no—
410C22×D4—682no—
411C22×D4—682no—
412C23×C4—171no—
413C23×C4—171no—
414C23×C4—174no—
415C23×C4—174no—
416C23×C4—174no—
417C23×C4—172no—
418C23×C4—171no—
419C23×C4—172no—
420C23×C4—682no—
421C23×C4—682no—
422C23×C4C22×D1712yes—
423C23×C4C22×D1711yescharacteristic
424C23×C4C22×D1711yescharacteristic
425C23×C4C22×D1712yes—
426#123—3432no—
427C2×C4⋊C4—174no—
428C2×C4⋊C4—174no—
429C2×C4⋊C4—342no—
430C2×C4⋊C4—342no—
431C2×C4⋊C4—342no—
432C2×C4⋊C4—342no—
433C2×C4⋊C4—344no—
434C2×C4⋊C4—682no—
435C2×C4⋊C4—682no—
436C2×C22⋊C4—22no—
437C2×C22⋊C4—22no—
438C2×C22⋊C4—22no—
439C2×C22⋊C4—24no—
440C2×C22⋊C4—22no—
441C2×C22⋊C4—42no—
442C2×C22⋊C4—174no—
443C2×C22⋊C4—174no—
444C2×C22⋊C4—342no—
445C2×C22⋊C4—342no—
446C2×C22⋊C4—342no—
447C2×C22⋊C4—342no—
448C2×C22⋊C4—344no—
449C2×C22⋊C4—342no—
450C2×C22⋊C4—344no—
451C2×C22⋊C4—342no—
452C2×C22⋊C4—344no—
453C2×C22⋊C4—682no—
454C2×C22⋊C4—682no—
455C2×C22⋊C4—682no—
456C2×C22⋊C4—682no—
457C2×C22⋊C4—682no—
458C2×C22⋊C4C4×D1714yes—
459C2×C42—344no—
460C2×C42—682no—
461C2×C42—682no—
462#95—24no—
463#95—42no—
464#95—344no—
465#95—344no—
466#95—344no—
467#95—682no—
468#95—682no—
469#95—682no—
470C17#1428711yescharacteristic, complemented, Sylow
471C24—22no—
472C24—22no—
473C24—22no—
474C24—22no—
475C24—22no—
476C24—22no—
477C24—172no—
478C24—341no—
479C24—341no—
480C24—342no—
481C24—342no—
482C24—342no—
483C24—342no—
484C24—342no—
485C24—342no—
486C24C2×C4×D1712yes—
487C24D4×D1712yes—
488C24D4×D1712yes—
489C24C23×D1711yescharacteristic
490C2×D4—348no—
491C2×D4—684no—
492C2×D4—684no—
493C2×D4—688no—
494C2×D4—688no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—172no—
517C22×C4—172no—
518C22×C4—172no—
519C22×C4—172no—
520C22×C4—172no—
521C22×C4—172no—
522C22×C4—1716no—
523C22×C4—172no—
524C22×C4—174no—
525C22×C4—342no—
526C22×C4—342no—
527C22×C4—342no—
528C22×C4—342no—
529C22×C4—342no—
530C22×C4—341no—
531C22×C4—341no—
532C22×C4—342no—
533C22×C4—342no—
534C22×C4—342no—
535C22×C4—342no—
536C22×C4—342no—
537C22×C4—342no—
538C22×C4—342no—
539C22×C4—342no—
540C22×C4—342no—
541C22×C4—342no—
542C22×C4—342no—
543C22×C4—342no—
544C22×C4—342no—
545C22×C4—342no—
546C22×C4—684no—
547C22×C4—684no—
548C22×C4—688no—
549C22×C4—688no—
550C22×C4C2×C4×D1712yes—
551C22×C4C2×C4×D1712yes—
552C22×C4#6325414yes—
553C4⋊C4—348no—
554C4⋊C4—348no—
555C4⋊C4—688no—
556C22⋊C4—28no—
557C22⋊C4—28no—
558C22⋊C4—44no—
559C22⋊C4—348no—
560C22⋊C4—348no—
561C22⋊C4—348no—
562C22⋊C4—688no—
563C22⋊C4—688no—
564C22⋊C4—684no—
565C42—688no—
566C23—24no—
567C23—22no—
568C23—22no—
569C23—24no—
570C23—22no—
571C23—22no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—22no—
576C23—22no—
577C23—24no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—342no—
587C23—344no—
588C23—342no—
589C23—342no—
590C23—344no—
591C23—681no—
592C23—681no—
593C23—682no—
594C23—682no—
595C23—682no—
596C23—681no—
597C23—681no—
598C23—682no—
599C23—682no—
600C23—682no—
601C23—682no—
602C23—682no—
603C23—682no—
604C23—682no—
605C23—682no—
606C23—682no—
607C23C22⋊C4×D1714yes—
608C23C22×C4×D1711yescharacteristic
609C23C2×D4×D1711yescharacteristic
610C23C2×D4×D1711yescharacteristic
611C23C2×#6325412yes—
612C23#1058769112yes—
613C23#41×D1712yes—
614C23#1058769612yes—
615C23#1058769612yes—
616C23#1058769712yes—
617D4—6816no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—174no—
643C2×C4—174no—
644C2×C4—342no—
645C2×C4—342no—
646C2×C4—342no—
647C2×C4—342no—
648C2×C4—344no—
649C2×C4—344no—
650C2×C4—344no—
651C2×C4—344no—
652C2×C4—342no—
653C2×C4—342no—
654C2×C4—342no—
655C2×C4—344no—
656C2×C4—344no—
657C2×C4—344no—
658C2×C4—6816no—
659C2×C4—682no—
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×C4—682no—
670C2×C4—682no—
671C2×C4—682no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—682no—
682C22—682no—
683C22—682no—
684C22—684no—
685C22—684no—
686C22—1362no—
687C22—1362no—
688C22—1362no—
689C22—1362no—
690C22C2×C22⋊C4×D1711yescharacteristic
691C22C4×#6325412yes—
692C22#1176328311yescharacteristic
693C22#1176328311yescharacteristic
694C22#1176328812yes—
695C22#1176331511yescharacteristic
696C22#1176331711yescharacteristic
697C22#1176335412yes—
698C22#1176337012yes—
699C22#1176337012yes—
700C22#1176337411yescharacteristic
701C22#1176338111yescharacteristic
702C22#1176338612yes—
703C22#1176339312yes—
704C22#1176339612yes—
705C22#1176340012yes—
706C22#1176340312yes—
707C22#1176341112yes—
708C22#1176341212yes—
709C22#1176342512yes—
710C22#1176342812yes—
711C4—42no—
712C4—42no—
713C4—44no—
714C4—82no—
715C4—344no—
716C4—344no—
717C4—682no—
718C4—684no—
719C4—1362no—
720C2—44no—
721C2—1362no—
722C2#1453126911yescharacteristic
723C2#1453136111yescharacteristic
724C2#1453137111yescharacteristic
725C2#1453156712yes—
726C2#1453157912yes—
727C2#1453462212yes—
728C2#1453462612yes—
729C2#1453491112yes—
730C2#1453493312yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1440C2
socle, cosocleC24×C17C24
prime radicalC23×C22⋊D17C22
hypercenter#1440D17
hypocenterC17#14287
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
2. C2×#11762667C2semidirect
3. C2×C22⋊C4×Dic17C2semidirect
4. #14527036C2semidirect
5. #14527033C2semidirect
6. #14527028C2semidirect
9. #14527020C2semidirect
10. #14527018C2semidirect
11. C17×#1440C2semidirect
16. C2×#10587572C4semidirect
178. C22×Dic17C22⋊C4semidirect
312. #1440D17semidirect
470. C17#14287semidirect

Maximal subgroups

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

subgroupindex
C22×#105875722
C2×#117626672
C2×C22⋊C4×Dic172
#145270362
#145270332
#145270282
#145270232
#145270202
#145270182
C17×#14402
#1428717