Finite Group Databasebeta

← #17070401

Subgroups — #17070401

Order 4352.

1150 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763364C211yesmaximal, characteristic, complemented
2C2×#11763099C211yesmaximal, characteristic, complemented
3C4×#10587624C211yesmaximal, characteristic
4#14530120C212yesmaximal, complemented
5#14529928C212yesmaximal, complemented
6#14529737C211yesmaximal, characteristic, complemented
7#14529731C211yesmaximal, characteristic, complemented
8#14529103C212yesmaximal
9#14528980C212yesmaximal, complemented
10#14528761C211yesmaximal, characteristic, complemented
11C2×C17×#474C211yesmaximal, characteristic, complemented
12#11763364—24no—
13#11763364—24no—
14C2×#10587555C2211yescharacteristic
15C2×#10587576C2212yes—
16C2×#10587583—21no—
17C2×#10587632—21no—
18C2×#10587624C2211yescharacteristic
19C2×Q8×Dic17—21no—
20C2×#10587573—21no—
21C2×C4×C22⋊D17—21no—
22C2×#10587578C2212yes—
23C2×C42×D17C2211yescharacteristic
24#11763099—24no—
25C2×#10587479C2212yes—
26C2×C4⋊#3622C2211yescharacteristic
27C2×#10587503C2212yes—
28C2×#10587491C2212yes—
29#11762712C2211yescharacteristic
30#11762705C2211yescharacteristic
31#11762697—21no—
32#11762685—21no—
33#11762666C2211yescharacteristic
34C4×C4⋊Dic17C2211yescharacteristic
35C4×C22⋊Dic17—21no—
36#11762572C2212yes—
37#11762572C2212yes—
38#11762571C2212yes—
39#11762571C2212yes—
40#11762563C2212yes—
41C2×C17×C4⋊Q8C2211yescharacteristic
42C2×C4×D4×C17C2211yescharacteristic
43C17×#474—24no—
44C2×C17×#107C2212yes—
45C2×C17×C4⋊C8C2211yescharacteristic
46C2×C17×#102C2212yes—
47C22×Q8×C17—21no—
48C22×D4×C17C2311yescharacteristic
49C23×C4×C17—21no—
50C22×C22⋊D17—21no—
51C22×C4×D17D411yescharacteristic
52C22×C4×D17D411yescharacteristic
53C22×C4×D17C2311yescharacteristic
54C22×Dic34—21no—
55C22×C8×C17C2312yes—
56C17×C4⋊Q8—22no—
57C17×C4⋊Q8D414yescomplemented
58C4×D4×C17—22no—
59C4×D4×C17—22no—
60C2×C4⋊C4×C17—21no—
61C2×C4⋊C4×C17C2311yescharacteristic
62C2×C4⋊C4×C17C2312yes—
63C2×C22⋊C4×C17—21no—
64C2×C42×C17C2311yescharacteristic
65C2×C22⋊Dic17—21no—
66C2×C22⋊Dic17—21no—
67C2×C22⋊Dic17—21no—
68#10587632—28no—
69Q8×Dic17—28no—
70#10587624—22no—
71#10587624Q814yescomplemented
72C4×C22⋊D17—44no—
73C2×#63227—22no—
74C2×#63227—21no—
75C2×C4⋊Dic17—21no—
76C2×C4⋊Dic17—22no—
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2312yes—
79C2×C4⋊Dic17C2311yescharacteristic
80C2×#63225—22no—
81C2×#63225—21no—
82C2×C4×Dic17—21no—
83C2×C4×Dic17—22no—
84C2×C4×Dic17D411yescharacteristic
85C2×C4×Dic17D411yescharacteristic
86C2×C4×Dic17C2311yescharacteristic
87C22×#3622C2312yes—
88#10587583—28no—
89#10587578—24no—
90#10587578—24no—
91#10587576—24no—
92#10587576—24no—
93#10587573—28no—
94C42×D17—22no—
95C42×D17—22no—
96#10587555—22no—
97#10587555D414yescomplemented
98C17×#107—24no—
99C17×C4⋊C8—22no—
100C17×#102—24no—
101#10587503—24no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587491—24no—
106#10587479—24no—
107C4⋊#3622—22no—
108C24×C17—21no—
109C23×D17C2×D411yescharacteristic
110C2×Q8×C17—24no—
111C2×Q8×C17—42no—
112C2×Q8×C17—42no—
113C2×D4×C17—21no—
114C2×D4×C17—21no—
115C2×D4×C17—21no—
116C2×D4×C17—21no—
117C2×D4×C17—21no—
118C2×D4×C17C2×Q812yes—
119C22×C4×C17—22no—
120C22×C4×C17—21no—
121C22×C4×C17—42no—
122C22×C4×C17—42no—
123C22×C4×C17—42no—
124C22×C4×C17C2×D411yescharacteristic
125C22×C4×C17C2×D411yescharacteristic
126C22×C4×C17C2411yescharacteristic
127C2×C22⋊D17—24no—
128C2×C22⋊D17—42no—
129C2×C22⋊D17—42no—
130C22×Dic17—22no—
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17C2×D411yescharacteristic
134C22×Dic17C2×D411yescharacteristic
135C22×Dic17C2×D411yescharacteristic
136C2×C4×D17—21no—
137C2×C4×D17—21no—
138C2×C4×D17—21no—
139C2×C4×D17—21no—
140C2×C4×D17—22no—
141C2×C4×D17—22no—
142C2×C4×D17—22no—
143C2×C4×D17—22no—
144C2×C4×D17—41no—
145C2×C4×D17—41no—
146C2×C4×D17C2×Q812yes—
147C2×C4×D17#4112yes—
148C2×Dic34—24no—
149C2×Dic34—42no—
150C2×Dic34—42no—
151C2×C8×C17—22no—
152C2×C8×C17—22no—
153C2×C8×C17—22no—
154C4⋊C4×C17—24no—
155C4⋊C4×C17—22no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17C2×D414yes—
159C22⋊C4×C17—42no—
160C42×C17C2×D412yes—
161C42×C17C2×D412yes—
162C22⋊Dic17—24no—
163C22⋊Dic17—42no—
164C22⋊Dic17—42no—
165#63227—28no—
166#63227—42no—
167C4⋊Dic17—28no—
168C4⋊Dic17—24no—
169C4⋊Dic17—22no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17C2×D414yes—
174C4⋊Dic17C2×Q812yes—
175#63225—28no—
176#63225—42no—
177C4×Dic17—28no—
178C4×Dic17—22no—
179C4×Dic17—22no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—42no—
183C2×#3622—22no—
184C2×#3622—22no—
185C2×#3622—22no—
186#24300—171nomaximal, Sylow
187C23×C17—22no—
188C23×C17—41no—
189C23×C17—41no—
190C23×C17—41no—
191C23×C17—41no—
192C23×C17—41no—
193C23×C17—41no—
194C23×C17C22×D411yescharacteristic
195C22×D17—21no—
196C22×D17—21no—
197C22×D17—21no—
198C22×D17—21no—
199C22×D17—21no—
200C22×D17—21no—
201C22×D17C22⋊Q812yes—
202Q8×C17—48no—
203D4×C17—22no—
204D4×C17—22no—
205D4×C17—41no—
206D4×C17—41no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—24no—
210C2×C4×C17—42no—
211C2×C4×C17—42no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—84no—
216C2×C4×C17#12012yes—
217C2×C4×C17#12012yes—
218C2×C4×C17#13512yes—
219C2×C4×C17AGL(1,ℤ/8)12yes—
220C2×C4×C17C22×D411yescharacteristic
221C2×C4×C17C22×D411yescharacteristic
222C2×C4×C17C22×Q811yescharacteristic
223C2×C4×C17C2×#4111yescharacteristic
224C2×C4×C172−1+411yescharacteristic
225C2×C4×C172−1+411yescharacteristic
226C22⋊D17—48no—
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—24no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—22no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—41no—
238C2×Dic17—42no—
239C2×Dic17—42no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17C22⋊Q812yes—
245C4×D17—22no—
246C4×D17—22no—
247C4×D17—22no—
248C4×D17—22no—
249C4×D17—44no—
250C4×D17—44no—
251Dic34—48no—
252C8×C17—42no—
253#3622—42no—
254C2×#533—171no—
255C2×#476—171no—
256C2×#474D1711yescharacteristic, complemented
257C4×#121—171no—
258#2059—172no—
259#2025—172no—
260#1988—171no—
261#1987—171no—
262#1875—172no—
263#1854—172no—
264#1797—171no—
265C22×C17—42no—
266C22×C17—42no—
267C22×C17—42no—
268C22×C17—81no—
269C22×C17—81no—
270C22×C17—81no—
271C22×C17—81no—
272C22×C17C2×#12011yescharacteristic
273C22×C17C2×C22⋊Q811yescharacteristic
274C22×C17#53611yescharacteristic
275C22×C17C2×#13511yescharacteristic
276C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C17#57511yescharacteristic
278C22×C17#57511yescharacteristic
279C2×D17—22no—
280C2×D17—22no—
281C2×D17—22no—
282C2×D17—41no—
283C2×D17—41no—
284C2×D17—41no—
285C2×D17—41no—
286C4×C17—22no—
287C4×C17—44no—
288C4×C17—81no—
289C4×C17#44914yes—
290C4×C17#54811yescharacteristic
291C4×C17#54811yescharacteristic
292C4×C17D4×Q811yescharacteristic
293C4×C17D4×Q811yescharacteristic
294Dic17—22no—
295Dic17—42no—
296Dic17—41no—
297Dic17—44no—
298Dic17—41no—
299Dic17—81no—
300#533—344no—
301#533—344no—
302C2×C4⋊Q8—171no—
303C2×C4⋊Q8C2×D1711yescharacteristic
304C2×#125—172no—
305C2×C22⋊Q8—341no—
306C2×C22⋊Q8—341no—
307C2×#121—171no—
308C2×C4×Q8—341no—
309C2×C4×Q8—341no—
310C2×C4×D4—341no—
311C2×C4×D4C2×D1711yescharacteristic
312C2×#117—172no—
313C22×C42—171no—
314#476—344no—
315#474—24no—
316C2×#107C2×D1712yes—
317C2×#106—172no—
318C2×C4⋊C8—171no—
319C2×C4⋊C8C2×D1711yescharacteristic
320C2×#102—172no—
321C2×#102C2×D1712yes—
322C2×C22⋊C8—172no—
323#390—171no—
324#389—171no—
325#388—341no—
326#386—341no—
327#380—171no—
328C4×C4⋊C4—171no—
329C4×C22⋊C4—341no—
330#340—172no—
331#340—172no—
332#340—172no—
333#340—172no—
334#336—172no—
335C2×C17—82no—
336C2×C17#240911yescharacteristic
337C2×C17C2×#44911yescharacteristic
338C2×C17#300511yescharacteristic
339C2×C17#307611yescharacteristic
340C2×C17#307711yescharacteristic
341C2×C17#323311yescharacteristic
342C2×C17#323511yescharacteristic
343D17—42no—
344C22×Q8—21no—
345C22×Q8—341no—
346C22×D4—341no—
347C22×D4C22×D1711yescharacteristic
348C23×C4—21no—
349C23×C4—171no—
350C23×C4—171no—
351C23×C4—171no—
352C22×C8—172no—
353C22×C8C22×D1712yes—
354C4⋊Q8—22no—
355C4⋊Q8—174no—
356C4⋊Q8—342no—
357C4⋊Q8C22⋊D1714yescomplemented
358#125—344no—
359#125—344no—
360C22⋊Q8—348no—
361C22⋊Q8—348no—
362#121—174no—
363#121—342no—
364C4×Q8—348no—
365C4×Q8—348no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—684no—
369#117—344no—
370#117—344no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—172no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—342no—
376C2×C4⋊C4—342no—
377C2×C4⋊C4—341no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C4⋊C4C22×D1712yes—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—342no—
383C2×C22⋊C4—341no—
384C2×C22⋊C4—341no—
385C2×C22⋊C4—341no—
386C2×C22⋊C4—341no—
387C2×C42—171no—
388C2×C42—171no—
389C2×C42—171no—
390C2×C42—342no—
391C2×C42—342no—
392C2×C42—342no—
393C2×C42—341no—
394C2×C42C22×D1711yescharacteristic
395#107—24no—
396#106—344no—
397C4⋊C8—22no—
398C4⋊C8—342no—
399#102—24no—
400#102—344no—
401C22⋊C8—344no—
402#95—341no—
403#95—341no—
404#95—341no—
405C17#2430011yescharacteristic, complemented, Sylow
406C24—21no—
407C24—171no—
408C2×Q8—24no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—344no—
412C2×Q8—682no—
413C2×Q8—682no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—344no—
420C2×D4—682no—
421C2×D4—682no—
422C2×D4Q8×D1712yes—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—171no—
429C22×C4—172no—
430C22×C4—172no—
431C22×C4—171no—
432C22×C4—171no—
433C22×C4—341no—
434C22×C4—341no—
435C22×C4—341no—
436C22×C4—341no—
437C22×C4—341no—
438C22×C4—342no—
439C22×C4—342no—
440C22×C4—342no—
441C22×C4—342no—
442C22×C4—342no—
443C22×C4—341no—
444C22×C4—681no—
445C22×C4—681no—
446C22×C4D4×D1711yescharacteristic
447C22×C4D4×D1711yescharacteristic
448C22×C4C23×D1711yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—342no—
453C2×C8—342no—
454C2×C8—342no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—174no—
460C4⋊C4—172no—
461C4⋊C4—348no—
462C4⋊C4—348no—
463C4⋊C4—341no—
464C4⋊C4—342no—
465C4⋊C4—344no—
466C4⋊C4—341no—
467C4⋊C4—341no—
468C4⋊C4—682no—
469C4⋊C4C2×C22⋊D1714yes—
470C22⋊C4—42no—
471C22⋊C4—348no—
472C22⋊C4—344no—
473C22⋊C4—682no—
474C22⋊C4—682no—
475C22⋊C4—682no—
476C42—348no—
477C42—341no—
478C42—341no—
479C42—342no—
480C42—342no—
481C42—682no—
482C42D4×D1712yes—
483C42C2×C22⋊D1712yes—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—172no—
492C23—341no—
493C23—341no—
494C23—341no—
495C23—341no—
496C23—341no—
497C23—341no—
498C23C2×D4×D1711yescharacteristic
499Q8—48no—
500Q8—688no—
501D4—22no—
502D4—22no—
503D4—41no—
504D4—41no—
505D4—688no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—84no—
515C2×C4—172no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—342no—
519C2×C4—342no—
520C2×C4—342no—
521C2×C4—342no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—344no—
525C2×C4—342no—
526C2×C4—341no—
527C2×C4—341no—
528C2×C4—341no—
529C2×C4—341no—
530C2×C4—684no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—682no—
534C2×C4—682no—
535C2×C4—684no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4#1058759712yes—
540C2×C4#1058759812yes—
541C2×C4#1058762312yes—
542C2×C4#1058762312yes—
543C2×C4C2×D4×D1711yescharacteristic
544C2×C4C2×Q8×D1711yescharacteristic
545C2×C4#1058769411yescharacteristic
546C2×C4#41×D1711yescharacteristic
547C2×C4#1058769711yescharacteristic
548C2×C4C22×C22⋊D1711yescharacteristic
549C8—42no—
550C8—682no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—342no—
559C22—342no—
560C22—342no—
561C22—681no—
562C22—681no—
563C22—681no—
564C22—681no—
565C22C2×#1058762311yescharacteristic
566C22C22⋊Q8×D1711yescharacteristic
567C22#1176338211yescharacteristic
568C22C2×#1058759811yescharacteristic
569C22C2×#1058759711yescharacteristic
570C22#1176360811yescharacteristic
571C22#1176360811yescharacteristic
572C4—22no—
573C4—44no—
574C4—81no—
575C4—342no—
576C4—681no—
577C4—684no—
578C4—682no—
579C4—681no—
580C4—1361no—
581C4#1176295114yes—
582C4#1176346611yescharacteristic
583C4#1176346911yescharacteristic
584C4#1176347211yescharacteristic
585C4Q8×C22⋊D1711yescharacteristic
586C2—82no—
587C2—682no—
588C2#1453214511yescharacteristic
589C2C2×#1176295111yescharacteristic
590C2#1453802511yescharacteristic
591C2#1453856511yescharacteristic
592C2#1453857611yescharacteristic
593C2#1453996111yescharacteristic
594C2#1453998111yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#474C2
socleC23×C17C22×D4
prime radicalC2×#10587624C22
hypercenterC2×#474D17
hypocenterC17#24300
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#11763364C2semidirect
2. C2×#11763099C2semidirect
4. #14530120C2semidirect
5. #14529928C2semidirect
6. #14529737C2semidirect
7. #14529731C2semidirect
9. #14528980C2semidirect
10. #14528761C2semidirect
11. C2×C17×#474C2semidirect
57. C17×C4⋊Q8D4semidirect
71. #10587624Q8semidirect
97. #10587555D4semidirect
256. C2×#474D17semidirect
357. C4⋊Q8C22⋊D17semidirect
405. C17#24300semidirect

Maximal subgroups

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

subgroupindex
C2×#117633642
C2×#117630992
C4×#105876242
#145301202
#145299282
#145297372
#145297312
#145291032
#145289802
#145287612
C2×C17×#4742
#2430017