Finite Group Databasebeta

← #16964382

Subgroups — #16964382

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×#11762702C212yesmaximal, complemented
2C2×#11762689C212yesmaximal
3#14527052C211yesmaximal, characteristic, complemented
4#14527041C212yesmaximal, complemented
5#14527029C212yesmaximal, complemented
6#14527016C211yesmaximal, characteristic, complemented
7#14527006C212yesmaximal, complemented
8#14527006C212yesmaximal, complemented
9C2×C17×#394C211yesmaximal, characteristic
10C24×Dic17C2212yes—
11C22×C4⋊Dic17C2212yes—
12C22×C22⋊Dic17C2212yes—
13C22×C22⋊Dic17C2212yes—
14C22×C22⋊Dic17C2212yes—
15C22×C4×Dic17C2211yescharacteristic
16#11762702—28no—
17#11762689—28no—
18C2×#10587502C2212yes—
19C2×#10587502C2212yes—
20C2×#10587502C2212yes—
21C2×#10587502C2212yes—
22C2×#10587502C2212yes—
23C2×#10587502C2211yescharacteristic
24C2×#10587502C2212yes—
25C2×#10587502C2211yescharacteristic
26C2×#10587502C2212yes—
27C2×#10587502C2211yescharacteristic
28C23×D4×C17C2211yescharacteristic
29C22×C4⋊C4×C17C2211yescharacteristic
30C22×C22⋊C4×C17C2212yes—
31C22×C22⋊C4×C17C2212yes—
32C17×#394C418yescomplemented
33C2×C17×#95C2211yescharacteristic
34C25×C17C2312yes—
35C22×D4×C17—24no—
36C22×D4×C17—24no—
37C22×D4×C17—41no—
38C22×D4×C17—41no—
39C22×D4×C17C2×C414yes—
40C23×C4×C17C2311yescharacteristic
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×Dic17—24no—
45C23×Dic17—24no—
46C23×Dic17—24no—
47C23×Dic17—42no—
48C23×Dic17—42no—
49C23×Dic17—42no—
50C23×Dic17—42no—
51C23×Dic17C2312yes—
52C23×Dic17C2312yes—
53C23×Dic17C2312yes—
54C23×Dic17C2312yes—
55C2×C4⋊C4×C17—41no—
56C2×C4⋊C4×C17—41no—
57C2×C4⋊C4×C17C2×C414yescomplemented
58C2×C22⋊C4×C17—24no—
59C2×C22⋊C4×C17—24no—
60C2×C22⋊C4×C17—42no—
61C2×C22⋊C4×C17—42no—
62C2×C22⋊C4×C17C2×C418yescomplemented
63C2×C22⋊C4×C17C2×C418yescomplemented
64C2×C22⋊Dic17—24no—
65C2×C22⋊Dic17—24no—
66C2×C22⋊Dic17—24no—
67C2×C22⋊Dic17—24no—
68C2×C22⋊Dic17—42no—
69C2×C22⋊Dic17—42no—
70C2×C22⋊Dic17—42no—
71C2×C22⋊Dic17—42no—
72C2×C22⋊Dic17D418yes—
73C2×C4⋊Dic17—24no—
74C2×C4⋊Dic17—42no—
75C2×C4⋊Dic17—42no—
76C2×C4×Dic17—24no—
77C2×C4×Dic17—41no—
78C17×#95C2×C414yescomplemented
79#10587502—24no—
80#10587502—24no—
81#10587502—24no—
82#10587502—24no—
83#10587502—24no—
84#10587502—42no—
85#10587502—41no—
86#10587502—41no—
87#10587502—42no—
88#10587502—41no—
89C24×C17—22no—
90C24×C17—22no—
91C24×C17—22no—
92C24×C17—22no—
93C24×C17—22no—
94C24×C17—22no—
95C24×C17—22no—
96C24×C17—22no—
97C24×C17—22no—
98C24×C17—22no—
99C24×C17—22no—
100C24×C17—22no—
101C24×C17C22×C414yes—
102C24×C17C2×D414yes—
103C24×C17#4114yes—
104C24×C17C2411yescharacteristic
105C2×D4×C17—216no—
106C2×D4×C17—44no—
107C2×D4×C17—44no—
108C2×D4×C17—48no—
109C2×D4×C17—44no—
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—22no—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17C22×C412yes—
129C22×C4×C17C22×C414yes—
130C22×C4×C17C22×C412yes—
131C22×C4×C17C22×C412yes—
132C22×Dic17—22no—
133C22×Dic17—22no—
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×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—48no—
158C22×Dic17—48no—
159C22×Dic17—48no—
160C22×Dic17—48no—
161C22×Dic17—48no—
162C22×Dic17—48no—
163C22×Dic17—48no—
164C22×Dic17C2×D414yes—
165C22×Dic17C2×D414yes—
166C22×Dic17#4114yes—
167C4⋊C4×C17—44no—
168C22⋊C4×C17—216no—
169C22⋊C4×C17—48no—
170C22⋊Dic17—48no—
171C22⋊Dic17—48no—
172C22⋊Dic17—48no—
173C4⋊Dic17—48no—
174C4×Dic17—44no—
175#14571—171nomaximal, Sylow
176C23×C17—24no—
177C23×C17—24no—
178C23×C17—24no—
179C23×C17—24no—
180C23×C17—24no—
181C23×C17—24no—
182C23×C17—24no—
183C23×C17—24no—
184C23×C17—24no—
185C23×C17—24no—
186C23×C17—24no—
187C23×C17—24no—
188C23×C17—24no—
189C23×C17—24no—
190C23×C17—24no—
191C23×C17—24no—
192C23×C17—24no—
193C23×C17—24no—
194C23×C17—42no—
195C23×C17—42no—
196C23×C17—42no—
197C23×C17—42no—
198C23×C17—42no—
199C23×C17—42no—
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×C17—42no—
209C23×C17—42no—
210C23×C17C4×D418yes—
211C23×C17C23×C411yescharacteristic
212C23×C17C22×D412yes—
213C23×C17C2×#4112yes—
214C23×C17C2×#4112yes—
215C23×C17H(5,𝔽2)12yes—
216C23×C17H(5,𝔽2)11yescharacteristic
217C23×C17H(5,𝔽2)11yescharacteristic
218C23×C17H(5,𝔽2)12yes—
219C23×C172−1+412yes—
220D4×C17—416no—
221C2×C4×C17—24no—
222C2×C4×C17—24no—
223C2×C4×C17—24no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—24no—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—24no—
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—41no—
238C2×C4×C17—41no—
239C2×C4×C17—42no—
240C2×C4×C17—42no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×Dic17—24no—
244C2×Dic17—24no—
245C2×Dic17—24no—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—24no—
249C2×Dic17—24no—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—432no—
253C2×Dic17—42no—
254C2×Dic17—42no—
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×#394D1711yescharacteristic
273C2×#388—172no—
274C2×#386—172no—
275#1445—171no—
276#1442—172no—
277#1439—172no—
278#1436—171no—
279C22⋊#95—172no—
280C22⋊#95—172no—
281C22×C17—28no—
282C22×C17—28no—
283C22×C17—28no—
284C22×C17—44no—
285C22×C17—44no—
286C22×C17—44no—
287C22×C17—44no—
288C22×C17—44no—
289C22×C17—44no—
290C22×C17—44no—
291C22×C17—44no—
292C22×C17—44no—
293C22×C17—44no—
294C22×C17—44no—
295C22×C17—44no—
296C22×C17C2×C4×D412yes—
297C22×C17C4×#4111yescharacteristic
298C22×C17#51811yescharacteristic
299C22×C17#51811yescharacteristic
300C22×C17#52012yes—
301C22×C17#53412yes—
302C22×C17#53712yes—
303C22×C17#53812yes—
304C22×C17#53812yes—
305C22×C17#53912yes—
306C22×C17#54012yes—
307C22×C17D4211yescharacteristic
308C22×C17#54612yes—
309C22×C17#54612yes—
310C22×C17#54712yes—
311C22×C17#55112yes—
312C22×C17#55111yescharacteristic
313C22×C17#55212yes—
314C22×C17H(3,𝔽4)11yescharacteristic
315C22×C17#56212yes—
316C22×C17#56311yescharacteristic
317C4×C17—44no—
318C4×C17—42no—
319C4×C17—42no—
320C4×C17—42no—
321Dic17—44no—
322Dic17—44no—
323Dic17—44no—
324Dic17—82no—
325C23×D4C2×D1711yescharacteristic
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⋊C4C2×D1712yes—
333C22×C22⋊C4C2×D1712yes—
334C22×C42—171no—
335#394Dic1718yescomplemented
336#388—348no—
337#386—348no—
338C2×#95—172no—
339C2×#95—172no—
340C2×#95—172no—
341C2×#95—172no—
342C2×#95—172no—
343C2×#95—171no—
344C2×#95—172no—
345C2×#95—171no—
346C2×#95—172no—
347C2×#95—171no—
348C2×#95C2×D1711yescharacteristic
349C2×C17—48no—
350C2×C17#232512yes—
351C2×C17#233311yescharacteristic
352C2×C17#235811yescharacteristic
353C2×C17#235912yes—
354C2×C17#237911yescharacteristic
355C2×C17#267112yes—
356C2×C17#268412yes—
357C2×C17#269512yes—
358C2×C17#274312yes—
359C25C22×D1712yes—
360C22×D4—24no—
361C22×D4—24no—
362C22×D4—41no—
363C22×D4—41no—
364C22×D4C2×Dic1714yes—
365C23×C4—172no—
366C23×C4—172no—
367C23×C4—172no—
368C23×C4—172no—
369C23×C4—344no—
370C23×C4—344no—
371C23×C4—344no—
372C23×C4—682no—
373C23×C4—682no—
374C23×C4—682no—
375C23×C4—682no—
376C23×C4C22×D1712yes—
377C23×C4C22×D1711yescharacteristic
378C23×C4C22×D1711yescharacteristic
379C23×C4C22×D1711yescharacteristic
380C2×C4⋊C4—41no—
381C2×C4⋊C4—41no—
382C2×C4⋊C4—344no—
383C2×C4⋊C4—682no—
384C2×C4⋊C4—682no—
385C2×C4⋊C4C2×Dic1714yescomplemented
386C2×C22⋊C4—24no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—42no—
389C2×C22⋊C4—42no—
390C2×C22⋊C4—178no—
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⋊C4C2×Dic1718yescomplemented
400C2×C22⋊C4C2×Dic1718yescomplemented
401C2×C42—344no—
402C2×C42—681no—
403#95—344no—
404#95—344no—
405#95—344no—
406#95—344no—
407#95—344no—
408#95—682no—
409#95—681no—
410#95—681no—
411#95—681no—
412#95—682no—
413#95C2×Dic1714yescomplemented
414C17#1457111yescharacteristic, complemented, Sylow
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24D4×D1714yes—
428C24#6325414yes—
429C24C22×Dic1714yes—
430C24C23×D1711yescharacteristic
431C2×D4—216no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—48no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—174no—
455C22×C4—174no—
456C22×C4—174no—
457C22×C4—342no—
458C22×C4—342no—
459C22×C4—342no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—342no—
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×C4C22×Dic1714yes—
490C22×C4C22×Dic1712yes—
491C22×C4C22×Dic1712yes—
492C22×C4C22×Dic1712yes—
493C4⋊C4—44no—
494C4⋊C4—688no—
495C22⋊C4—216no—
496C22⋊C4—48no—
497C22⋊C4—688no—
498C22⋊C4—688no—
499C22⋊C4—688no—
500C42—684no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23D4×Dic1718yes—
536C23C2×D4×D1712yes—
537C23C2×#6325412yes—
538C23#1058769111yescharacteristic
539C23#1058769111yescharacteristic
540C23#1058769112yes—
541C23#41×D1712yes—
542C23#1058769612yes—
543C23#1058769712yes—
544C23C23×Dic1711yescharacteristic
545D4—416no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
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—344no—
569C2×C4—344no—
570C2×C4—344no—
571C2×C4—344no—
572C2×C4—344no—
573C2×C4—344no—
574C2×C4—344no—
575C2×C4—344no—
576C2×C4—344no—
577C2×C4—6832no—
578C2×C4—682no—
579C2×C4—682no—
580C2×C4—682no—
581C2×C4—682no—
582C2×C4—682no—
583C2×C4—682no—
584C2×C4—682no—
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—28no—
598C22—28no—
599C22—28no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22C2×D4×Dic1712yes—
613C22#41×Dic1711yescharacteristic
614C22#1176328411yescharacteristic
615C22#1176328411yescharacteristic
616C22#1176329212yes—
617C22#1176336712yes—
618C22#1176338612yes—
619C22#1176339112yes—
620C22#1176339612yes—
621C22#1176340412yes—
622C22#1176340912yes—
623C22#1176344111yescharacteristic
624C22#1176344612yes—
625C22#1176345112yes—
626C22#1176345712yes—
627C22#1176348212yes—
628C22#1176349011yescharacteristic
629C22#1176349612yes—
630C22#1176355111yescharacteristic
631C22#1176355312yes—
632C22#1176355911yescharacteristic
633C4—44no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—684no—
638C4—684no—
639C4—684no—
640C4—1362no—
641C2—48no—
642C2#1453148912yes—
643C2#1453155111yescharacteristic
644C2#1453176811yescharacteristic
645C2#1453178012yes—
646C2#1453195811yescharacteristic
647C2#1453501312yes—
648C2#1453515912yes—
649C2#1453531812yes—
650C2#1453592512yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#394C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenterC2×#394D17
hypocenterC17#14571
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C2×#11762702C2semidirect
3. #14527052C2semidirect
4. #14527041C2semidirect
5. #14527029C2semidirect
6. #14527016C2semidirect
7. #14527006C2semidirect
8. #14527006C2semidirect
32. C17×#394C4semidirect
57. C2×C4⋊C4×C17C2×C4semidirect
62. C2×C22⋊C4×C17C2×C4semidirect
63. C2×C22⋊C4×C17C2×C4semidirect
78. C17×#95C2×C4semidirect
335. #394Dic17semidirect
385. C2×C4⋊C4C2×Dic17semidirect
399. C2×C22⋊C4C2×Dic17semidirect
400. C2×C22⋊C4C2×Dic17semidirect
413. #95C2×Dic17semidirect
414. C17#14571semidirect

Maximal subgroups

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

subgroupindex
C2×#117627022
C2×#117626892
#145270522
#145270412
#145270292
#145270162
#145270062
C2×C17×#3942
#1457117