Finite Group Databasebeta

← #16963504

Subgroups — #16963504

Order 4352.

1962 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762702C212yesmaximal, complemented
2#14527052C212yesmaximal, complemented
3#14527029C212yesmaximal, complemented
4#14527025C211yesmaximal, characteristic
5#14527025C211yesmaximal, characteristic
6#14527021C212yesmaximal, complemented
7#14527013C212yesmaximal, complemented
8#14527007C212yesmaximal, complemented
9C22×C17×#123C211yesmaximal, characteristic
10C22×C4⋊Dic17C2212yes—
11C22×C22⋊Dic17C2212yes—
12C22×C22⋊Dic17C2212yes—
13C22×C22⋊Dic17C2211yescharacteristic
14C22×C22⋊Dic17C2212yes—
15C22×C22⋊Dic17C2211yescharacteristic
16C22×C4×Dic17C2212yes—
17#11762702—28no—
18C2×#10587502C2212yes—
19C2×#10587502C2212yes—
20C2×#10587502C2212yes—
21C2×#10587502C2212yes—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2211yescharacteristic
25C2×#10587502C2212yes—
26C2×#10587502C2211yescharacteristic
27C23×D4×C17C2211yescharacteristic
28C24×C4×C17C2211yescharacteristic
29C2×C17×#123—28no—
30C2×C17×#123C418yescomplemented
31C22×C4⋊C4×C17C2212yes—
32C22×C22⋊C4×C17C2212yes—
33C22×C22⋊C4×C17C2211yescharacteristic
34C25×C17C2311yescharacteristic
35C25×C17C2311yescharacteristic
36C22×D4×C17—24no—
37C22×D4×C17—42no—
38C22×D4×C17—41no—
39C22×D4×C17—41no—
40C22×D4×C17C2×C414yes—
41C23×C4×C17—24no—
42C23×C4×C17—42no—
43C23×C4×C17—41no—
44C23×C4×C17—41no—
45C23×C4×C17C2×C414yes—
46C23×C4×C17C2312yes—
47C23×C4×C17C2311yescharacteristic
48C23×C4×C17C2312yes—
49C23×Dic17C2312yes—
50C23×Dic17C2312yes—
51C23×Dic17C2312yes—
52C23×Dic17C2312yes—
53C17×#123—232no—
54C2×C4⋊C4×C17—24no—
55C2×C4⋊C4×C17—24no—
56C2×C4⋊C4×C17C2×C418yes—
57C2×C22⋊C4×C17—24no—
58C2×C22⋊C4×C17—24no—
59C2×C22⋊C4×C17C2×C418yes—
60C2×C22⋊C4×C17C2×C414yes—
61C2×C22⋊C4×C17D418yes—
62C2×C22⋊Dic17—24no—
63C2×C22⋊Dic17—24no—
64C2×C22⋊Dic17—24no—
65C2×C22⋊Dic17—24no—
66C2×C22⋊Dic17—24no—
67C2×C22⋊Dic17—24no—
68C2×C22⋊Dic17—41no—
69C2×C22⋊Dic17—42no—
70C2×C22⋊Dic17—42no—
71C2×C22⋊Dic17—42no—
72C2×C22⋊Dic17—42no—
73C2×C22⋊Dic17—41no—
74C2×C22⋊Dic17—42no—
75C2×C4⋊Dic17—24no—
76C2×C4⋊Dic17—24no—
77C2×C4⋊Dic17—42no—
78C2×C4×Dic17—24no—
79C2×C4×Dic17—24no—
80C2×C4×Dic17—42no—
81#10587502—24no—
82#10587502—24no—
83#10587502—41no—
84#10587502—42no—
85#10587502—42no—
86#10587502—42no—
87#10587502—42no—
88#10587502—41no—
89#10587502—42no—
90C24×C17—22no—
91C24×C17—22no—
92C24×C17—22no—
93C24×C17—21no—
94C24×C17—21no—
95C24×C17—22no—
96C24×C17—22no—
97C24×C17—22no—
98C24×C17—22no—
99C24×C17—22no—
100C24×C17—22no—
101C24×C17—21no—
102C24×C17—21no—
103C24×C17—22no—
104C24×C17—22no—
105C24×C17C22×C412yes—
106C24×C17C22×C412yes—
107C24×C17C2×D414yes—
108C24×C17C2411yescharacteristic
109C2×D4×C17—28no—
110C2×D4×C17—44no—
111C2×D4×C17—44no—
112C2×D4×C17—48no—
113C2×D4×C17—48no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—28no—
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—21no—
128C22×C4×C17—21no—
129C22×C4×C17—44no—
130C22×C4×C17—44no—
131C22×C4×C17—48no—
132C22×C4×C17—48no—
133C22×C4×C17C22×C414yes—
134C22×C4×C17C22×C412yes—
135C22×C4×C17C22×C414yes—
136C22×C4×C17C2×D414yes—
137C22×C4×C17C2×D414yes—
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—216no—
167C22⋊C4×C17—216no—
168C22⋊C4×C17C22⋊C4116yescomplemented
169C22⋊Dic17—44no—
170C22⋊Dic17—48no—
171C22⋊Dic17—48no—
172C22⋊Dic17—48no—
173C22⋊Dic17—44no—
174C4⋊Dic17—48no—
175C4×Dic17—48no—
176#14506—171nomaximal, Sylow
177C23×C17—24no—
178C23×C17—22no—
179C23×C17—22no—
180C23×C17—24no—
181C23×C17—24no—
182C23×C17—24no—
183C23×C17—24no—
184C23×C17—24no—
185C23×C17—24no—
186C23×C17—24no—
187C23×C17—22no—
188C23×C17—24no—
189C23×C17—22no—
190C23×C17—22no—
191C23×C17—24no—
192C23×C17—42no—
193C23×C17—42no—
194C23×C17—42no—
195C23×C17—42no—
196C23×C17—41no—
197C23×C17—42no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17—42no—
201C23×C17—41no—
202C23×C17—42no—
203C23×C17—42no—
204C23×C17—42no—
205C23×C17—42no—
206C23×C17—42no—
207C23×C17—42no—
208C23×C17—42no—
209C23×C17—42no—
210C23×C17—42no—
211C23×C17—42no—
212C23×C17—41no—
213C23×C17—42no—
214C23×C17—41no—
215C23×C17—41no—
216C23×C17—42no—
217C23×C17—41no—
218C23×C17C2×C22⋊C414yes—
219C23×C17C23×C411yescharacteristic
220C23×C17C22×D412yes—
221C23×C17C2×#4112yes—
222C23×C17C2×#4112yes—
223C23×C17H(5,𝔽2)12yes—
224C23×C17H(5,𝔽2)11yescharacteristic
225C23×C17H(5,𝔽2)11yescharacteristic
226C23×C17H(5,𝔽2)12yes—
227C23×C172−1+412yes—
228D4×C17—416no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—24no—
233C2×C4×C17—24no—
234C2×C4×C17—24no—
235C2×C4×C17—24no—
236C2×C4×C17—22no—
237C2×C4×C17—24no—
238C2×C4×C17—24no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—416no—
242C2×C4×C17—42no—
243C2×C4×C17—42no—
244C2×C4×C17—42no—
245C2×C4×C17—42no—
246C2×C4×C17—42no—
247C2×C4×C17—42no—
248C2×C4×C17C2×C22⋊C418yes—
249C2×Dic17—24no—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—24no—
254C2×Dic17—24no—
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—
270C2×Dic17—42no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C2×Dic17—42no—
275C2×Dic17—42no—
276C2×Dic17—42no—
277C22×#123D1711yescharacteristic
278C2×#388—172no—
279#1445—172no—
280#1439—172no—
281#1438—171no—
282#1438—171no—
283#1437—172no—
284C4⋊#95—172no—
285C22⋊#95—172no—
286C22×C17—24no—
287C22×C17—28no—
288C22×C17—44no—
289C22×C17—42no—
290C22×C17—42no—
291C22×C17—44no—
292C22×C17—44no—
293C22×C17—44no—
294C22×C17—44no—
295C22×C17—42no—
296C22×C17—44no—
297C22×C17—44no—
298C22×C17—42no—
299C22×C17—42no—
300C22×C17—82no—
301C22×C17—82no—
302C22×C17—82no—
303C22×C17—82no—
304C22×C17C22×C22⋊C411yescharacteristic
305C22×C17C4×#4112yes—
306C22×C17#51811yescharacteristic
307C22×C17#51811yescharacteristic
308C22×C17#52012yes—
309C22×C17#53512yes—
310C22×C17#53512yes—
311C22×C17#53812yes—
312C22×C17#53912yes—
313C22×C17#54012yes—
314C22×C17#54212yes—
315C22×C17#54612yes—
316C22×C17#54612yes—
317C22×C17#54712yes—
318C22×C17#54712yes—
319C22×C17#55111yescharacteristic
320C22×C17#55312yes—
321C22×C17#55911yescharacteristic
322C22×C17#56011yescharacteristic
323C22×C17#56212yes—
324C22×C17#56311yescharacteristic
325C4×C17—28no—
326C4×C17—44no—
327C4×C17—42no—
328Dic17—44no—
329Dic17—44no—
330Dic17—82no—
331Dic17—82no—
332C23×D4C2×D1711yescharacteristic
333C24×C4C2×D1711yescharacteristic
334C2×#123—28no—
335C2×#123Dic1718yescomplemented
336C22×C4⋊C4—172no—
337C22×C4⋊C4C2×D1712yes—
338C22×C22⋊C4—172no—
339C22×C22⋊C4—172no—
340C22×C22⋊C4—171no—
341C22×C22⋊C4—172no—
342C22×C22⋊C4—171no—
343C22×C22⋊C4C2×D1712yes—
344C22×C22⋊C4C2×D1711yescharacteristic
345C22×C42—172no—
346#388—348no—
347C2×#95—172no—
348C2×#95—172no—
349C2×#95—172no—
350C2×#95—172no—
351C2×#95—172no—
352C2×#95—172no—
353C2×#95—171no—
354C2×#95—172no—
355C2×#95—171no—
356C2×C17—44no—
357C2×C17—82no—
358C2×C17#230911yescharacteristic
359C2×C17#233512yes—
360C2×C17#233912yes—
361C2×C17#237411yescharacteristic
362C2×C17#237611yescharacteristic
363C2×C17#270312yes—
364C2×C17#270612yes—
365C2×C17#271012yes—
366C2×C17#271812yes—
367C25C22×D1711yescharacteristic
368C25C22×D1711yescharacteristic
369C22×D4—24no—
370C22×D4—41no—
371C22×D4—41no—
372C22×D4—42no—
373C22×D4C2×Dic1714yes—
374C23×C4—24no—
375C23×C4—41no—
376C23×C4—41no—
377C23×C4—42no—
378C23×C4—172no—
379C23×C4—172no—
380C23×C4—172no—
381C23×C4—172no—
382C23×C4C2×Dic1714yes—
383C23×C4C22×D1712yes—
384C23×C4C22×D1711yescharacteristic
385C23×C4C22×D1712yes—
386#123—232no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—344no—
390C2×C4⋊C4—344no—
391C2×C4⋊C4—682no—
392C2×C4⋊C4C2×Dic1718yes—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—344no—
396C2×C22⋊C4—344no—
397C2×C22⋊C4—344no—
398C2×C22⋊C4—344no—
399C2×C22⋊C4—344no—
400C2×C22⋊C4—344no—
401C2×C22⋊C4—681no—
402C2×C22⋊C4—682no—
403C2×C22⋊C4—682no—
404C2×C22⋊C4—682no—
405C2×C22⋊C4—682no—
406C2×C22⋊C4—681no—
407C2×C22⋊C4—682no—
408C2×C22⋊C4C2×Dic1718yes—
409C2×C22⋊C4C2×Dic1714yes—
410C2×C22⋊C4C22⋊D1718yes—
411C2×C42—344no—
412C2×C42—344no—
413C2×C42—682no—
414#95—344no—
415#95—344no—
416#95—682no—
417#95—682no—
418#95—682no—
419#95—682no—
420#95—681no—
421#95—682no—
422#95—681no—
423C17#1450611yescharacteristic, complemented, Sylow
424C24—21no—
425C24—21no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—21no—
432C24—21no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24C22×Dic1712yes—
440C24C22×Dic1712yes—
441C24C2×C22⋊D1714yes—
442C24C23×D1711yescharacteristic
443C2×D4—28no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—48no—
447C2×D4—48no—
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—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—28no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—174no—
468C22×C4—174no—
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—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×C4C22×Dic1714yes—
496C22×C4C22×Dic1712yes—
497C22×C4C22×Dic1714yes—
498C22×C4C2×C22⋊D1714yes—
499C22×C4C2×C22⋊D1714yes—
500C4⋊C4—216no—
501C4⋊C4—688no—
502C22⋊C4—216no—
503C22⋊C4—684no—
504C22⋊C4—688no—
505C22⋊C4—688no—
506C22⋊C4—688no—
507C22⋊C4—684no—
508C22⋊C4C22⋊Dic17116yescomplemented
509C42—688no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23C2×C22⋊Dic1714yes—
552C23#1058769111yescharacteristic
553C23#1058769111yescharacteristic
554C23#1058769112yes—
555C23#41×D1712yes—
556C23#41×D1712yes—
557C23#1058769612yes—
558C23#1058769712yes—
559C23C23×Dic1711yescharacteristic
560C23C22×C22⋊D1712yes—
561D4—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—416no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—344no—
582C2×C4—344no—
583C2×C4—344no—
584C2×C4—344no—
585C2×C4—344no—
586C2×C4—344no—
587C2×C4—682no—
588C2×C4—682no—
589C2×C4—682no—
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—
605C2×C4—682no—
606C2×C4—682no—
607C2×C4—682no—
608C2×C4—682no—
609C2×C4C2×C22⋊Dic1718yes—
610C22—28no—
611C22—24no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×C22⋊Dic1711yescharacteristic
629C22#41×Dic1712yes—
630C22#1176328411yescharacteristic
631C22#1176328411yescharacteristic
632C22#1176329212yes—
633C22#1176337612yes—
634C22#1176337612yes—
635C22#1176339512yes—
636C22#1176340212yes—
637C22#1176341112yes—
638C22#1176342712yes—
639C22#1176344712yes—
640C22#1176344712yes—
641C22#1176345812yes—
642C22#1176345812yes—
643C22#1176348711yescharacteristic
644C22#1176350412yes—
645C22#1176354311yescharacteristic
646C22#1176354911yescharacteristic
647C22#1176355612yes—
648C22#1176356211yescharacteristic
649C4—28no—
650C4—44no—
651C4—42no—
652C4—684no—
653C4—684no—
654C4—1362no—
655C4—1362no—
656C2—44no—
657C2—82no—
658C2#1453136611yescharacteristic
659C2#1453157812yes—
660C2#1453161412yes—
661C2#1453192011yescharacteristic
662C2#1453193211yescharacteristic
663C2#1453542112yes—
664C2#1453546612yes—
665C2#1453552212yes—
666C2#1453563012yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC22×C17×#123C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenterC22×#123D17
hypocenterC17#14506
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C2×#11762702C2semidirect
2. #14527052C2semidirect
3. #14527029C2semidirect
6. #14527021C2semidirect
7. #14527013C2semidirect
8. #14527007C2semidirect
30. C2×C17×#123C4semidirect
168. C22⋊C4×C17C22⋊C4semidirect
335. C2×#123Dic17semidirect
423. C17#14506semidirect
508. C22⋊C4C22⋊Dic17semidirect

Maximal subgroups

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

subgroupindex
C2×#117627022
#145270522
#145270292
#145270252
#145270212
#145270132
#145270072
C22×C17×#1232
#1450617