Finite Group Databasebeta

← #16964383

Subgroups — #16964383

Order 4352.

1966 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762726C211yesmaximal, characteristic
2C2×#11762698C212yesmaximal, complemented
3C2×#11762687C212yesmaximal
4#14527039C212yesmaximal, complemented
5#14527028C212yesmaximal, complemented
6#14527016C211yesmaximal, characteristic, complemented
7#14527005C212yesmaximal, complemented
8#14527005C212yesmaximal, complemented
9C17×#1445C211yesmaximal, characteristic, complemented
10C23×D68C2211yescharacteristic
11C23×C4×D17C2212yes—
12C22×#63225C2212yes—
13C22×#63227C2212yes—
14C22×#63227C2212yes—
15C22×#63227C2212yes—
16C22×#63227C2212yes—
17C22×#63227C2212yes—
18C22×C4×Dic17C2211yescharacteristic
19#11762726C418yescomplemented
20#11762698—28no—
21#11762687—28no—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2211yescharacteristic
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2211yescharacteristic
29C22×C4⋊C4×C17C2211yescharacteristic
30C2×C17×#95C2212yes—
31C2×C17×#95C2212yes—
32C2×C17×#95C2211yescharacteristic
33C2×C17×#95C2211yescharacteristic
34C24×D17C2312yes—
35C23×C4×C17C2312yes—
36C23×C4×C17C2311yescharacteristic
37C23×C4×C17C2311yescharacteristic
38C23×C4×C17C2311yescharacteristic
39C23×C4×C17C2312yes—
40C23×Dic17C2312yes—
41C23×Dic17C2312yes—
42C23×Dic17C2312yes—
43C22×D68—24no—
44C22×D68—24no—
45C22×D68—41no—
46C22×D68—41no—
47C22×D68C2×C414yes—
48C22×C4×D17—24no—
49C22×C4×D17—24no—
50C22×C4×D17—24no—
51C22×C4×D17—42no—
52C22×C4×D17—42no—
53C22×C4×D17—42no—
54C22×C4×D17—42no—
55C2×C4⋊C4×C17—41no—
56C2×C4⋊C4×C17—41no—
57C2×C4⋊C4×C17C2×C414yescomplemented
58C2×#63227—24no—
59C2×#63227—24no—
60C2×#63227—24no—
61C2×#63227—24no—
62C2×#63227—24no—
63C2×#63227—24no—
64C2×#63227—42no—
65C2×#63227—42no—
66C2×#63227—42no—
67C2×#63227—42no—
68C2×#63227—42no—
69C2×#63227—42no—
70C2×#63227C2×C418yescomplemented
71C2×#63227C2×C418yescomplemented
72C2×#63227D418yes—
73C2×#63225—24no—
74C2×#63225—42no—
75C2×#63225—42no—
76C2×C4×Dic17—24no—
77C2×C4×Dic17—41no—
78C17×#95—24no—
79C17×#95—24no—
80C17×#95—41no—
81C17×#95—41no—
82#10587502—24no—
83#10587502—24no—
84#10587502—24no—
85#10587502—41no—
86#10587502—42no—
87#10587502—42no—
88#10587502C2×C414yescomplemented
89C24×C17C2411yescharacteristic
90C23×D17—22no—
91C23×D17—22no—
92C23×D17—22no—
93C23×D17—22no—
94C23×D17—22no—
95C23×D17—22no—
96C23×D17—22no—
97C23×D17—22no—
98C23×D17—22no—
99C23×D17—22no—
100C23×D17—22no—
101C23×D17—22no—
102C23×D17C22×C414yes—
103C23×D17C2×D414yes—
104C23×D17#4114yes—
105C22×C4×C17—22no—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—21no—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
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×C17C22×C412yes—
131C22×C4×C17C22×C412yes—
132C22×C4×C17C22×C412yes—
133C22×C4×C17C2×D414yes—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
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×Dic17C22×C414yes—
153C22×Dic17C2×D414yes—
154C22×Dic17#4114yes—
155C2×D68—216no—
156C2×D68—44no—
157C2×D68—44no—
158C2×D68—48no—
159C2×D68—44no—
160C2×C4×D17—48no—
161C2×C4×D17—48no—
162C2×C4×D17—48no—
163C2×C4×D17—48no—
164C2×C4×D17—48no—
165C2×C4×D17—48no—
166C2×C4×D17—48no—
167C4⋊C4×C17—44no—
168#63227—216no—
169#63227—48no—
170#63227—48no—
171#63227—48no—
172#63227—48no—
173#63225—48no—
174C4×Dic17—44no—
175#14571—171nomaximal, Sylow
176C23×C17C23×C411yescharacteristic
177C23×C17C22×D412yes—
178C23×C17C2×#4112yes—
179C23×C17C2×#4112yes—
180C23×C17H(5,𝔽2)11yescharacteristic
181C23×C17H(5,𝔽2)11yescharacteristic
182C23×C17H(5,𝔽2)12yes—
183C23×C17H(5,𝔽2)12yes—
184C23×C172−1+412yes—
185C22×D17—24no—
186C22×D17—24no—
187C22×D17—24no—
188C22×D17—24no—
189C22×D17—24no—
190C22×D17—24no—
191C22×D17—24no—
192C22×D17—24no—
193C22×D17—24no—
194C22×D17—24no—
195C22×D17—24no—
196C22×D17—24no—
197C22×D17—24no—
198C22×D17—24no—
199C22×D17—24no—
200C22×D17—24no—
201C22×D17—24no—
202C22×D17—24no—
203C22×D17—42no—
204C22×D17—42no—
205C22×D17—42no—
206C22×D17—42no—
207C22×D17—42no—
208C22×D17—42no—
209C22×D17—42no—
210C22×D17—42no—
211C22×D17—42no—
212C22×D17—42no—
213C22×D17—42no—
214C22×D17—42no—
215C22×D17—42no—
216C22×D17—42no—
217C22×D17—42no—
218C22×D17—42no—
219C22×D17C4×D418yes—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—24no—
228C2×C4×C17—24no—
229C2×C4×C17—24no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—42no—
233C2×C4×C17—42no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17—42no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—42no—
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×C17—42no—
249C2×Dic17—24no—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—24no—
254C2×Dic17—24no—
255C2×Dic17—24no—
256C2×Dic17—24no—
257C2×Dic17—24no—
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—
270D68—416no—
271C4×D17—432no—
272C2×#394—171no—
273C2×#388—172no—
274C2×#386—172no—
275#1445D1711yescharacteristic, complemented
276#1442—172no—
277#1439—172no—
278#1436—171no—
279C22⋊#95—172no—
280C22⋊#95—172no—
281C22×C17C2×C4×D412yes—
282C22×C17C4×#4111yescharacteristic
283C22×C17#51811yescharacteristic
284C22×C17#51811yescharacteristic
285C22×C17#52012yes—
286C22×C17#53412yes—
287C22×C17#53712yes—
288C22×C17#53812yes—
289C22×C17#53812yes—
290C22×C17#53912yes—
291C22×C17#54012yes—
292C22×C17D4211yescharacteristic
293C22×C17#54612yes—
294C22×C17#54612yes—
295C22×C17#54712yes—
296C22×C17#55112yes—
297C22×C17#55111yescharacteristic
298C22×C17#55212yes—
299C22×C17H(3,𝔽4)11yescharacteristic
300C22×C17#56212yes—
301C22×C17#56311yescharacteristic
302C2×D17—28no—
303C2×D17—28no—
304C2×D17—28no—
305C2×D17—44no—
306C2×D17—44no—
307C2×D17—44no—
308C2×D17—44no—
309C2×D17—44no—
310C2×D17—44no—
311C2×D17—44no—
312C2×D17—44no—
313C2×D17—44no—
314C2×D17—44no—
315C2×D17—44no—
316C2×D17—44no—
317C4×C17—42no—
318C4×C17—42no—
319C4×C17—42no—
320C4×C17—44no—
321C4×C17—82no—
322Dic17—44no—
323Dic17—44no—
324Dic17—44no—
325C23×D4—171no—
326C24×C4—172no—
327C22×C4⋊C4—172no—
328C22×C4⋊C4C2×D1711yescharacteristic
329C22×C22⋊C4—172no—
330C22×C22⋊C4—172no—
331C22×C22⋊C4—172no—
332C22×C22⋊C4—172no—
333C22×C22⋊C4—172no—
334C22×C42—171no—
335#394—178no—
336#388—348no—
337#386—348no—
338C2×#95—171no—
339C2×#95—172no—
340C2×#95—172no—
341C2×#95—172no—
342C2×#95—171no—
343C2×#95—172no—
344C2×#95—172no—
345C2×#95C2×D1712yes—
346C2×#95C2×D1712yes—
347C2×#95C2×D1711yescharacteristic
348C2×#95C2×D1711yescharacteristic
349C2×C17#232512yes—
350C2×C17#233311yescharacteristic
351C2×C17#235811yescharacteristic
352C2×C17#235912yes—
353C2×C17#237911yescharacteristic
354C2×C17#267112yes—
355C2×C17#268412yes—
356C2×C17#269512yes—
357C2×C17#274312yes—
358D17—48no—
359C25—172no—
360C22×D4—174no—
361C22×D4—344no—
362C22×D4—344no—
363C22×D4—681no—
364C22×D4—681no—
365C23×C4—172no—
366C23×C4—172no—
367C23×C4—172no—
368C23×C4—344no—
369C23×C4—344no—
370C23×C4—344no—
371C23×C4—682no—
372C23×C4—682no—
373C23×C4—682no—
374C23×C4—682no—
375C23×C4C22×D1712yes—
376C23×C4C22×D1711yescharacteristic
377C23×C4C22×D1711yescharacteristic
378C23×C4C22×D1711yescharacteristic
379C23×C4C22×D1712yes—
380C2×C4⋊C4—41no—
381C2×C4⋊C4—41no—
382C2×C4⋊C4—344no—
383C2×C4⋊C4—682no—
384C2×C4⋊C4—682no—
385C2×C4⋊C4C4×D1714yescomplemented
386C2×C22⋊C4—178no—
387C2×C22⋊C4—178no—
388C2×C22⋊C4—178no—
389C2×C22⋊C4—344no—
390C2×C22⋊C4—344no—
391C2×C22⋊C4—344no—
392C2×C22⋊C4—344no—
393C2×C22⋊C4—344no—
394C2×C22⋊C4—344no—
395C2×C22⋊C4—682no—
396C2×C22⋊C4—682no—
397C2×C22⋊C4—682no—
398C2×C22⋊C4—682no—
399C2×C22⋊C4—682no—
400C2×C22⋊C4—682no—
401C2×C42—344no—
402C2×C42—681no—
403#95—24no—
404#95—24no—
405#95—41no—
406#95—41no—
407#95—174no—
408#95—344no—
409#95—344no—
410#95—344no—
411#95—681no—
412#95—682no—
413#95—682no—
414C17#1457111yescharacteristic, complemented, Sylow
415C24—174no—
416C24—174no—
417C24—174no—
418C24—342no—
419C24—342no—
420C24—342no—
421C24—342no—
422C24—342no—
423C24—342no—
424C24—342no—
425C24—342no—
426C24—342no—
427C24—342no—
428C24—342no—
429C24—342no—
430C24C23×D1711yescharacteristic
431C2×D4—3416no—
432C2×D4—684no—
433C2×D4—684no—
434C2×D4—684no—
435C2×D4—688no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
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—174no—
462C22×C4—174no—
463C22×C4—174no—
464C22×C4—342no—
465C22×C4—342no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—342no—
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—688no—
483C22×C4—688no—
484C22×C4—688no—
485C22×C4—688no—
486C22×C4—688no—
487C22×C4—688no—
488C22×C4—688no—
489C22×C4C2×C4×D1712yes—
490C22×C4C2×C4×D1712yes—
491C22×C4C2×C4×D1712yes—
492C22×C4D4×D1714yes—
493C4⋊C4—44no—
494C4⋊C4—688no—
495C22⋊C4—3416no—
496C22⋊C4—688no—
497C22⋊C4—688no—
498C22⋊C4—688no—
499C22⋊C4—688no—
500C42—684no—
501C23—178no—
502C23—344no—
503C23—344no—
504C23—344no—
505C23—344no—
506C23—344no—
507C23—344no—
508C23—344no—
509C23—344no—
510C23—344no—
511C23—344no—
512C23—344no—
513C23—344no—
514C23—344no—
515C23—344no—
516C23—344no—
517C23—344no—
518C23—344no—
519C23—344no—
520C23—682no—
521C23—682no—
522C23—682no—
523C23—682no—
524C23—682no—
525C23—682no—
526C23—682no—
527C23—682no—
528C23—682no—
529C23—682no—
530C23—682no—
531C23—682no—
532C23—682no—
533C23—682no—
534C23—682no—
535C23—682no—
536C23C22×C4×D1711yescharacteristic
537C23C2×D4×D1712yes—
538C23#1058769111yescharacteristic
539C23#1058769111yescharacteristic
540C23#1058769412yes—
541C23#41×D1712yes—
542C23#41×D1712yes—
543C23#1058769612yes—
544C23#1058769612yes—
545D4—6816no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—344no—
576C2×C4—344no—
577C2×C4—344no—
578C2×C4—344no—
579C2×C4—344no—
580C2×C4—344no—
581C2×C4—344no—
582C2×C4—344no—
583C2×C4—344no—
584C2×C4—6832no—
585C2×C4—682no—
586C2×C4—682no—
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—
597C22—348no—
598C22—348no—
599C22—348no—
600C22—684no—
601C22—684no—
602C22—684no—
603C22—684no—
604C22—684no—
605C22—684no—
606C22—684no—
607C22—684no—
608C22—684no—
609C22—684no—
610C22—684no—
611C22—684no—
612C22C4×D4×D1712yes—
613C22#1176327711yescharacteristic
614C22#1176328211yescharacteristic
615C22#1176328311yescharacteristic
616C22#1176329112yes—
617C22#1176337012yes—
618C22#1176338912yes—
619C22#1176339512yes—
620C22#1176339812yes—
621C22#1176340712yes—
622C22#1176341112yes—
623C22#1176344411yescharacteristic
624C22#1176344812yes—
625C22#1176345112yes—
626C22#1176346112yes—
627C22#1176348412yes—
628C22#1176348911yescharacteristic
629C22#1176349112yes—
630C22#1176355111yescharacteristic
631C22#1176355512yes—
632C22#1176355811yescharacteristic
633C4—42no—
634C4—42no—
635C4—42no—
636C4—44no—
637C4—82no—
638C4—684no—
639C4—684no—
640C4—684no—
641C2—688no—
642C2#1453148312yes—
643C2#1453154611yescharacteristic
644C2#1453176511yescharacteristic
645C2#1453176912yes—
646C2#1453195611yescharacteristic
647C2#1453502512yes—
648C2#1453516612yes—
649C2#1453532512yes—
650C2#1453591612yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1445C2
socle, cosocleC24×C17C24
prime radicalC23×D68C22
hypercenter#1445D17
hypocenterC17#14571
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
2. C2×#11762698C2semidirect
4. #14527039C2semidirect
5. #14527028C2semidirect
6. #14527016C2semidirect
7. #14527005C2semidirect
8. #14527005C2semidirect
9. C17×#1445C2semidirect
19. #11762726C4semidirect
57. C2×C4⋊C4×C17C2×C4semidirect
70. C2×#63227C2×C4semidirect
71. C2×#63227C2×C4semidirect
88. #10587502C2×C4semidirect
275. #1445D17semidirect
385. C2×C4⋊C4C4×D17semidirect
414. C17#14571semidirect

Maximal subgroups

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

subgroupindex
C2×#117627262
C2×#117626982
C2×#117626872
#145270392
#145270282
#145270162
#145270052
C17×#14452
#1457117