Finite Group Databasebeta

← #17061812

Subgroups — #17061812

Order 4352.

1902 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762849C211yesmaximal, characteristic, complemented
2C2×#11762814C211yesmaximal, characteristic, complemented
3C2×#11762814C211yesmaximal, characteristic, complemented
4C2×#11762773C211yesmaximal, characteristic, complemented
5#14531167C211yesmaximal, characteristic, complemented
6#14529970C214yesmaximal, complemented
7#14529963C214yesmaximal, complemented
8#14528503C211yesmaximal, characteristic, complemented
9C22×C17×#121C211yesmaximal, characteristic
10C22×#63223C2211yescharacteristic
11C22×#63223C2211yescharacteristic
12C2×#10587612C2211yescharacteristic
13C2×#10587612C2211yescharacteristic
14#11762849—24no—
15C2×#10587478C2212yescomplemented
16C2×#10587479C2212yescomplemented
17#11762814—24no—
18#11762814—24no—
19C2×#10587503C2212yescomplemented
20C2×#10587503C2214yescomplemented
21C2×#10587503C2212yescomplemented
22#11762773—24no—
23C2×C22⋊#3622C2214yescomplemented
24#11762672—21no—
25#11762669C2212yescomplemented
26#11762669C2212yescomplemented
27#11762665—21no—
28C4×C22⋊Dic17—22no—
29#11762571C2214yescomplemented
30C23×D4×C17—21no—
31C23×D4×C17C2211yescharacteristic
32C24×C4×C17C2211yescharacteristic
33C2×C17×#121—24no—
34C2×C17×#121—24no—
35C2×C17×#121—22no—
36C2×C17×#121C2214yes—
37C22×C4⋊C4×C17C2211yescharacteristic
38C22×C22⋊C4×C17—21no—
39C25×C17—21no—
40C25×C17D411yescharacteristic
41C22×D4×C17—28no—
42C22×D4×C17—24no—
43C22×D4×C17—22no—
44C22×D4×C17—21no—
45C22×D4×C17—22no—
46C22×D4×C17—22no—
47C22×D4×C17—22no—
48C22×D4×C17—22no—
49C22×D4×C17—48no—
50C22×D4×C17D414yes—
51C22×D4×C17C2312yes—
52C23×C4×C17—21no—
53C23×C4×C17—22no—
54C23×C4×C17—22no—
55C23×C4×C17—22no—
56C23×C4×C17—22no—
57C23×C4×C17—22no—
58C23×C4×C17—22no—
59C23×C4×C17D411yescharacteristic
60C23×C4×C17D412yes—
61C23×C4×C17C2312yes—
62C23×C4×C17C2311yescharacteristic
63C17×#121—216no—
64C17×#121—28no—
65C17×#121—28no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17—22no—
68C2×C4⋊C4×C17D414yes—
69C2×C4⋊C4×C17C2312yes—
70C2×C22⋊C4×C17—28no—
71C2×C22⋊C4×C17—24no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—22no—
75C2×C22⋊Dic17—22no—
76#10587612—22no—
77#10587612—22no—
78#10587612—22no—
79#10587612—22no—
80C2×C4⋊Dic17C2312yes—
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4×Dic17—21no—
84C2×C4×Dic17—21no—
85C2×#63223—22no—
86C2×#63223—22no—
87C2×#63223—22no—
88C2×#63223—22no—
89C2×#63223—22no—
90C2×#63223D414yescomplemented
91C22×#3622C2312yes—
92C22×#3622C2312yes—
93#10587503—28no—
94#10587503—24no—
95#10587503—24no—
96#10587502—22no—
97#10587502—22no—
98C22⋊#3622—28no—
99#10587479—24no—
100#10587478—24no—
101C24×C17—22no—
102C24×C17—24no—
103C24×C17—24no—
104C24×C17—22no—
105C24×C17—22no—
106C24×C17—22no—
107C24×C17—42no—
108C24×C17—44no—
109C24×C17—44no—
110C24×C17—42no—
111C24×C17—42no—
112C24×C17C2×D412yes—
113C24×C17C2×D411yescharacteristic
114C2×D4×C17—216no—
115C2×D4×C17—24no—
116C2×D4×C17—28no—
117C2×D4×C17—28no—
118C2×D4×C17—22no—
119C2×D4×C17—22no—
120C2×D4×C17—22no—
121C2×D4×C17—22no—
122C2×D4×C17—22no—
123C2×D4×C17—432no—
124C2×D4×C17—48no—
125C2×D4×C17—48no—
126C2×D4×C17—44no—
127C2×D4×C17—42no—
128C2×D4×C17—42no—
129C2×D4×C17—44no—
130C2×D4×C17C2×D414yes—
131C22×C4×C17—22no—
132C22×C4×C17—24no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—24no—
137C22×C4×C17—24no—
138C22×C4×C17—24no—
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×C17—44no—
146C22×C4×C17—44no—
147C22×C4×C17—44no—
148C22×C4×C17—44no—
149C22×C4×C17—44no—
150C22×C4×C17C2×D412yes—
151C22×C4×C17C2×D412yes—
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D412yes—
154C22×C4×C17C2×D412yes—
155C22×C4×C17#4112yes—
156C22×C4×C17#4112yes—
157C22×C4×C17#4112yes—
158C22×C4×C17C2411yescharacteristic
159C22×Dic17—21no—
160C22×Dic17—21no—
161C22×Dic17—22no—
162C4⋊C4×C17—24no—
163C4⋊C4×C17—22no—
164C4⋊C4×C17C2×D414yes—
165C22⋊C4×C17—216no—
166C22⋊Dic17—44no—
167C22⋊Dic17—42no—
168C22⋊Dic17—42no—
169C22⋊Dic17—44no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—22no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—22no—
176C4×Dic17—42no—
177C4×Dic17—42no—
178#63223—28no—
179#63223—44no—
180C2×#3622—22no—
181C2×#3622—22no—
182C2×#3622—22no—
183C2×#3622—22no—
184C2×#3622—22no—
185C2×#3622C2×D414yescomplemented
186#23480—171nomaximal, Sylow
187C23×C17—22no—
188C23×C17—22no—
189C23×C17—28no—
190C23×C17—22no—
191C23×C17—22no—
192C23×C17—22no—
193C23×C17—22no—
194C23×C17—22no—
195C23×C17—22no—
196C23×C17—24no—
197C23×C17—48no—
198C23×C17—48no—
199C23×C17—44no—
200C23×C17—48no—
201C23×C17—48no—
202C23×C17—44no—
203C23×C17—42no—
204C23×C17—42no—
205C23×C17—44no—
206C23×C17—44no—
207C23×C17—48no—
208C23×C17—88no—
209C23×C17—88no—
210C23×C17#12012yes—
211C23×C17#12112yes—
212C23×C17#12312yes—
213C23×C17AGL(1,ℤ/8)14yes—
214C23×C17C22×D411yescharacteristic
215D4×C17—24no—
216D4×C17—24no—
217D4×C17—432no—
218D4×C17—42no—
219D4×C17—42no—
220D4×C17—48no—
221C2×C4×C17—22no—
222C2×C4×C17—24no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—48no—
226C2×C4×C17—44no—
227C2×C4×C17—44no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—48no—
233C2×C4×C17#12012yes—
234C2×C4×C17#12114yes—
235C2×C4×C17#12114yes—
236C2×C4×C17#12112yes—
237C2×C4×C17#12314yes—
238C2×C4×C17#12312yes—
239C2×C4×C17#12414yes—
240C2×C4×C17C22×D411yescharacteristic
241C2×C4×C17C22×D411yescharacteristic
242C2×C4×C17C22×D411yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—42no—
249C2×Dic17—42no—
250C2×Dic17—42no—
251C2×Dic17—42no—
252C2×Dic17—42no—
253#3622—24no—
254#3622—42no—
255C22×#121D1711yescharacteristic
256C2×#428—171no—
257C2×#418—171no—
258C2×#418—171no—
259C2×#407—171no—
260#2280—171no—
261#2034—174no—
262#2033—174no—
263#1736—171no—
264C22×C17—24no—
265C22×C17—22no—
266C22×C17—22no—
267C22×C17—24no—
268C22×C17—24no—
269C22×C17—24no—
270C22×C17—24no—
271C22×C17—416no—
272C22×C17—42no—
273C22×C17—42no—
274C22×C17—42no—
275C22×C17—42no—
276C22×C17—44no—
277C22×C17—44no—
278C22×C17—44no—
279C22×C17—816no—
280C22×C17—84no—
281C22×C17—84no—
282C22×C17#39414yes—
283C22×C17C2×#12011yescharacteristic
284C22×C17C2×#12111yescharacteristic
285C22×C17C2×#12311yescharacteristic
286C22×C17C2×AGL(1,ℤ/8)12yes—
287C22×C17#57512yes—
288C4×C17—42no—
289C4×C17—44no—
290C4×C17#39414yes—
291C4×C17C2×#12111yescharacteristic
292C4×C17C2×#12111yescharacteristic
293C4×C17C2×#12311yescharacteristic
294C4×C17C2×#12411yescharacteristic
295Dic17—81no—
296Dic17—81no—
297Dic17—82no—
298C23×D4—21no—
299C23×D4C2×D1711yescharacteristic
300C24×C4C2×D1711yescharacteristic
301C22×M16—171no—
302C22×M16—171no—
303C2×#121—24no—
304C2×#121—22no—
305C2×#121—24no—
306C2×#121C2×D1714yes—
307C2×#117—171no—
308C2×#117—171no—
309C22×C4⋊C4C2×D1711yescharacteristic
310C22×C22⋊C4—21no—
311#428—344no—
312C2×#107—172no—
313C2×#106—172no—
314#418—344no—
315#418—344no—
316C2×#102—172no—
317C2×#102—174no—
318C2×#102—172no—
319#407—344no—
320C2×C22⋊C8—174no—
321#381—341no—
322#380—172no—
323#380—172no—
324#379—341no—
325C4×C22⋊C4—342no—
326#340—174no—
327C2×C17—24no—
328C2×C17—44no—
329C2×C17—84no—
330C2×C17C2×#39411yescharacteristic
331C2×C17#300311yescharacteristic
332C2×C17#300511yescharacteristic
333C2×C17#305512yes—
334C2×C17#309412yes—
335C25—21no—
336C25C22⋊D1711yescharacteristic
337C22×D4—28no—
338C22×D4—24no—
339C22×D4—22no—
340C22×D4—22no—
341C22×D4—22no—
342C22×D4—22no—
343C22×D4—22no—
344C22×D4—21no—
345C22×D4—48no—
346C22×D4C22⋊D1714yes—
347C22×D4C22×D1712yes—
348C23×C4—21no—
349C23×C4—22no—
350C23×C4—22no—
351C23×C4—22no—
352C23×C4—22no—
353C23×C4—22no—
354C23×C4—22no—
355C23×C4C22⋊D1711yescharacteristic
356C23×C4C22⋊D1712yes—
357C23×C4C22×D1712yes—
358C23×C4C22×D1711yescharacteristic
359C2×M16—174no—
360C2×M16—342no—
361C2×M16—342no—
362C2×M16—342no—
363C2×M16—342no—
364C2×M16—342no—
365C22×C8—172no—
366C22×C8—172no—
367#121—216no—
368#121—28no—
369#121—28no—
370#117—342no—
371#117—342no—
372#117—342no—
373#117—342no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—172no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4C22⋊D1714yes—
380C2×C4⋊C4C22×D1712yes—
381C2×C22⋊C4—28no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—341no—
384C2×C22⋊C4—341no—
385C2×C22⋊C4—342no—
386C2×C22⋊C4—342no—
387C2×C42—341no—
388C2×C42—341no—
389#107—344no—
390#106—344no—
391#102—348no—
392#102—344no—
393#102—344no—
394C22⋊C8—348no—
395#95—342no—
396#95—342no—
397C17#2348011yescharacteristic, complemented, Sylow
398C24—24no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—24no—
404C24—42no—
405C24—44no—
406C24—44no—
407C24—42no—
408C24—42no—
409C24C2×C22⋊D1712yes—
410C24C2×C22⋊D1711yescharacteristic
411C2×D4—216no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—24no—
420C2×D4—432no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4C2×C22⋊D1714yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—341no—
448C22×C4—341no—
449C22×C4—342no—
450C22×C4D4×D1712yes—
451C22×C4#6325412yes—
452C22×C4#6325412yes—
453C22×C4#6325412yes—
454C22×C4C2×C22⋊D1712yes—
455C22×C4C2×C22⋊D1712yes—
456C22×C4C2×C22⋊D1712yes—
457C22×C4C2×C22⋊D1711yescharacteristic
458C22×C4C23×D1711yescharacteristic
459M16—348no—
460M16—684no—
461C2×C8—174no—
462C2×C8—342no—
463C2×C8—342no—
464C2×C8—342no—
465C2×C8—342no—
466C2×C8—342no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—342no—
470C4⋊C4—342no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4C2×C22⋊D1714yes—
476C22⋊C4—216no—
477C22⋊C4—684no—
478C22⋊C4—684no—
479C22⋊C4—682no—
480C22⋊C4—682no—
481C42—682no—
482C42—682no—
483C23—22no—
484C23—22no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—48no—
494C23—48no—
495C23—44no—
496C23—48no—
497C23—48no—
498C23—44no—
499C23—48no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—88no—
505C23—88no—
506C23#1058761814yes—
507C23#1058762112yes—
508C23#1058762512yes—
509C23#1058763912yes—
510C23C22×C22⋊D1711yescharacteristic
511D4—24no—
512D4—24no—
513D4—432no—
514D4—48no—
515D4—42no—
516D4—42no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—48no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—682no—
532C2×C4—682no—
533C2×C4—682no—
534C2×C4—682no—
535C2×C4—682no—
536C2×C4#1058762112yes—
537C2×C4#1058762114yes—
538C2×C4#1058762214yes—
539C2×C4#1058762414yes—
540C2×C4#1058762512yes—
541C2×C4#1058762514yes—
542C2×C4#1058763912yes—
543C2×C4C2×D4×D1711yescharacteristic
544C2×C4C2×#6325411yescharacteristic
545C2×C4C2×#6325411yescharacteristic
546C2×C4C2×#6325411yescharacteristic
547C2×C4C22×C22⋊D1711yescharacteristic
548C2×C4C22×C22⋊D1711yescharacteristic
549C8—344no—
550C8—682no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—416no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—816no—
567C22—84no—
568C22—84no—
569C22#1176272314yes—
570C22C2×#1058763911yescharacteristic
571C22C2×#1058762511yescharacteristic
572C22C2×#1058762111yescharacteristic
573C22C2×#1058761812yes—
574C22#1176360712yes—
575C4—44no—
576C4—42no—
577C4—1361no—
578C4—1361no—
579C4—1362no—
580C4#1176272314yes—
581C4C2×#1058762411yescharacteristic
582C4C2×#1058762511yescharacteristic
583C4C2×#1058762111yescharacteristic
584C4C2×#1058762211yescharacteristic
585C2—24no—
586C2—44no—
587C2—84no—
588C2C2×#1176272311yescharacteristic
589C2#1453801111yescharacteristic
590C2#1453802611yescharacteristic
591C2#1453839912yes—
592C2#1453872012yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC22×C17×#121C2
socleC23×C17C22×D4
hypercenterC22×#121D17
hypocenterC17#23480
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. C2×#11762849C2semidirect
2. C2×#11762814C2semidirect
3. C2×#11762814C2semidirect
4. C2×#11762773C2semidirect
5. #14531167C2semidirect
6. #14529970C2semidirect
7. #14529963C2semidirect
8. #14528503C2semidirect
15. C2×#10587478C22semidirect
16. C2×#10587479C22semidirect
19. C2×#10587503C22semidirect
20. C2×#10587503C22semidirect
21. C2×#10587503C22semidirect
23. C2×C22⋊#3622C22semidirect
25. #11762669C22semidirect
26. #11762669C22semidirect
29. #11762571C22semidirect
90. C2×#63223D4semidirect
185. C2×#3622C2×D4semidirect
397. C17#23480semidirect

Maximal subgroups

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

subgroupindex
C2×#117628492
C2×#117628142
C2×#117627732
#145311672
#145299702
#145299632
#145285032
C22×C17×#1212
#2348017