Finite Group Databasebeta

← #17005312

Subgroups — #17005312

Order 4352.

610 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538804C211yesmaximal, characteristic
2#14537799C211yesmaximal, characteristic, complemented
3#14537788C211yesmaximal, characteristic, complemented
4#14527694C211yesmaximal, characteristic, complemented
5#14527688C211yesmaximal, characteristic, complemented
6#14527657C211yesmaximal, characteristic, complemented
7#14527650C211yesmaximal, characteristic, complemented
8#14527643C211yesmaximal, characteristic, complemented
9#14527643C211yesmaximal, characteristic, complemented
10#14527609C211yesmaximal, characteristic, complemented
11#14527598C211yesmaximal, characteristic, complemented
12#14527564C211yesmaximal, characteristic, complemented
13#14527245C211yesmaximal, characteristic, complemented
14#14527167C211yesmaximal, characteristic, complemented
15C17×#1451C211yesmaximal, characteristic, complemented
16#11763343C2211yescharacteristic
17C4×#63252C2211yescharacteristic
18#11763151C411yescharacteristic
19#11763134C411yescharacteristic
20#11763117C411yescharacteristic
21#11763110C411yescharacteristic
22#11763032C411yescharacteristic
23#11763025C411yescharacteristic
24#11762973C411yescharacteristic
25#11762956C411yescharacteristic
26#11762939C2211yescharacteristic
27#11762939C2211yescharacteristic
28#11762939C2211yescharacteristic
29C8×Dic34C2211yescharacteristic
30C4×Dic68C2211yescharacteristic
31C4×#63219C2211yescharacteristic
32C4×#63219C2211yescharacteristic
33C4×D136C2211yescharacteristic
34#11762888C2211yescharacteristic
35#11762885—21no—
36#11762878—21no—
37#11762878—21no—
38#11762878—21no—
39C8×D68C2211yescharacteristic
40C8×D68C2211yescharacteristic
41C8×D68C2211yescharacteristic
42#11762871—21no—
43#11762866C2211yescharacteristic
44#11762864—21no—
45#11762862C2211yescharacteristic
46#11762766—21no—
47#97×D17—21no—
48#11762562C2211yescharacteristic
49#11762562C2211yescharacteristic
50#11762560C2211yescharacteristic
51#11762560C2211yescharacteristic
52#11762541C2211yescharacteristic
53#11762541C2211yescharacteristic
54#11762541C2211yescharacteristic
55#11762541C2211yescharacteristic
56#11762538C2211yescharacteristic
57#11762538C2211yescharacteristic
58#11762538C2211yescharacteristic
59#11762538C2211yescharacteristic
60C17×#431C2211yescharacteristic
61C4×M16×C17C2211yescharacteristic
62C2×C4×C8×C17C2211yescharacteristic
63C17×#322C2211yescharacteristic
64C17×#322C2211yescharacteristic
65C17×#322C2211yescharacteristic
66C17×#322C2211yescharacteristic
67C2×#63252C2×C411yescharacteristic
68C2×M16×C17D411yescharacteristic
69C2×M16×C17D411yescharacteristic
70C22×C8×C17—21no—
71C2×C42×C17C2311yescharacteristic
72#10587616C2×C411yescharacteristic
73#10587615—21no—
74C4×C22⋊D17—21no—
75C4×C22⋊D17—21no—
76C4×C22⋊D17—21no—
77#10587612C2×C411yescharacteristic
78#10587610C2×C411yescharacteristic
79C2×Dic68C2×C411yescharacteristic
80C2×D136C2×C411yescharacteristic
81C2×#63219C2×C411yescharacteristic
82C2×#63219C2×C411yescharacteristic
83C2×#63218—21no—
84C2×C8×D17—21no—
85C2×C8×D17—21no—
86C2×C8×D17—21no—
87#10587562—21no—
88#10587561C2×C411yescharacteristic
89C4×D68C2×C411yescharacteristic
90C4×D68C2311yescharacteristic
91C4×D68C2311yescharacteristic
92#10587558—21no—
93#10587558—21no—
94C42×D17—21no—
95#10587556C2×C411yescharacteristic
96C4×Dic34C2×C411yescharacteristic
97C4×Dic34C2311yescharacteristic
98C4×Dic34C2311yescharacteristic
99C17×C4⋊C8C2×C411yescharacteristic
100C17×C4⋊C8C2×C411yescharacteristic
101C17×C22⋊C8C2×C411yescharacteristic
102C17×C22⋊C8C2×C411yescharacteristic
103C17×#97D411yescharacteristic
104C17×#97D411yescharacteristic
105C4×C8×C17—21no—
106C4×C8×C17C2311yescharacteristic
107C4×C8×C17C2311yescharacteristic
108C4×C8×C17C2311yescharacteristic
109C4×C8×C17C2311yescharacteristic
110C4×C8×C17C2311yescharacteristic
111C4×C8×C17C2311yescharacteristic
112C22⋊#3622—21no—
113C22⋊#3622—21no—
114#10587492C2×C411yescharacteristic
115#10587492C2×C411yescharacteristic
116#10587492C2×C411yescharacteristic
117#10587492C2×C411yescharacteristic
118#10587491—21no—
119#10587491—21no—
120#10587491—21no—
121#10587491—21no—
122#10587489C2×C411yescharacteristic
123#10587489C2×C411yescharacteristic
124#10587488C2×C411yescharacteristic
125#10587488C2×C411yescharacteristic
126#10587487C2×C411yescharacteristic
127#10587487C2×C411yescharacteristic
128#10587487C2×C411yescharacteristic
129#10587487C2×C411yescharacteristic
130#10587486—21no—
131#10587486—21no—
132#10587486—21no—
133#10587485—21no—
134#10587485—21no—
135#10587485—21no—
136#10587485—21no—
137C8×Dic17—21no—
138C4⋊#3622C2311yescharacteristic
139C4⋊#3622C2311yescharacteristic
140C4⋊#3622C2311yescharacteristic
141C4⋊#3622C2311yescharacteristic
142#10587474—21no—
143C4×#3622—21no—
144C22×C4×C17C22×C411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17#4111yescharacteristic
147C2×C22⋊D17—21no—
148C2×C22⋊D17—21no—
149#63252—22no—
150#63252—41no—
151C2×D68C22×C411yescharacteristic
152C2×D68C22×C411yescharacteristic
153C2×C4×D17—21no—
154C2×C4×D17—21no—
155C2×C4×D17—21no—
156C2×C4×D17—21no—
157C2×Dic34C22×C411yescharacteristic
158C2×Dic34C22×C411yescharacteristic
159M16×C17—41no—
160M16×C17—41no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—42no—
164C2×C8×C17C22×C411yescharacteristic
165C2×C8×C17C22×C411yescharacteristic
166C2×C8×C17C22×C411yescharacteristic
167C2×C8×C17C22×C411yescharacteristic
168C2×C8×C17C2×D411yescharacteristic
169C2×C8×C17C2×D411yescharacteristic
170C2×C8×C17C2×D411yescharacteristic
171C2×C8×C17C2×D411yescharacteristic
172C42×C17C22×C411yescharacteristic
173C42×C17C2×D411yescharacteristic
174C42×C17#4111yescharacteristic
175C42×C17C2411yescharacteristic
176C22⋊Dic17—21no—
177C22⋊Dic17—21no—
178#63227—21no—
179#63227—21no—
180#63227—21no—
181#63227—21no—
182#63227—21no—
183#63227—21no—
184C4⋊Dic17C22×C411yescharacteristic
185C4⋊Dic17C22×C411yescharacteristic
186C4⋊Dic17C22×C411yescharacteristic
187C4⋊Dic17C22×C411yescharacteristic
188#63225—21no—
189#63225—21no—
190#63225—21no—
191#63225—21no—
192#63225—21no—
193#63225—21no—
194C4×Dic17—21no—
195C4×Dic17—21no—
196C4×Dic17—21no—
197C4×Dic17—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202Dic68—41no—
203D136—22no—
204#63219—22no—
205#63219—41no—
206#63218—42no—
207C8×D17—42no—
208C8×D17—42no—
209C8×D17—42no—
210#18002—171nomaximal, Sylow
211C23×C17C4×D411yescharacteristic
212C22×D17—21no—
213C22×D17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—41no—
219C2×C4×C17C4×D411yescharacteristic
220C2×C4×C17C4×D411yescharacteristic
221C2×C4×C17C4×D411yescharacteristic
222C2×C4×C17C23×C411yescharacteristic
223C2×C4×C17C22×D411yescharacteristic
224C2×C4×C17C2×#4111yescharacteristic
225C22⋊D17—42no—
226C22⋊D17—81no—
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233D68—21no—
234D68—21no—
235D68—21no—
236D68#13112yes—
237C4×D17—42no—
238C4×D17—41no—
239C4×D17—41no—
240C4×D17—41no—
241C4×D17—41no—
242C4×D17—81no—
243Dic34—21no—
244Dic34—21no—
245Dic34—21no—
246Dic34—21no—
247C8×C17—22no—
248C8×C17—21no—
249C8×C17—21no—
250C8×C17—21no—
251C8×C17—21no—
252C8×C17—21no—
253C8×C17—21no—
254C8×C17—21no—
255C8×C17—21no—
256C8×C17—41no—
257#3622—41no—
258#3622—41no—
259#3622—41no—
260#3622—41no—
261#3102—171no—
262#2970—171no—
263#2969—171no—
264#1593—171no—
265#1592—171no—
266#1587—171no—
267#1586—171no—
268#1585—171no—
269#1585—171no—
270#1580—171no—
271SD16⋊C8—171no—
272#1572—171no—
273#1494—171no—
274#1476—171no—
275#1451D1711yescharacteristic, complemented
276C22×C17—21no—
277C22×C17—21no—
278C22×C17—21no—
279C22×C17C2×C4×D411yescharacteristic
280C2×D17—22no—
281C2×D17—41no—
282C2×D17—41no—
283C2×D17—41no—
284C2×D17—41no—
285C2×D17—41no—
286C4×C17—21no—
287C4×C17—21no—
288C4×C17—21no—
289C4×C17—21no—
290C4×C17C2×#13111yescharacteristic
291C4×C17#56811yescharacteristic
292C4×C17#57511yescharacteristic
293C4×C17#57511yescharacteristic
294Dic17—41no—
295Dic17—41no—
296Dic17—41no—
297Dic17—41no—
298Dic17—41no—
299Dic17—41no—
300#529—171no—
301C4×#41—171no—
302#485—171no—
303#482—171no—
304#479—171no—
305#478—171no—
306#464—171no—
307#463—171no—
308#452—171no—
309#449—171no—
310#446—171no—
311#446—171no—
312#446—171no—
313C8×Q8—171no—
314C4×Q16—171no—
315C4×SD16—171no—
316C4×SD16—171no—
317C4×D8—171no—
318#436—171no—
319#436—341no—
320#435—341no—
321#435—341no—
322#435—341no—
323C8×D4—171no—
324C8×D4—171no—
325C8×D4—171no—
326#433—341no—
327#432—171no—
328#432—341no—
329#431—171no—
330#431C2×D1711yescharacteristic
331#405—341no—
332C4×M16C2×D1711yescharacteristic
333C2×#97—341no—
334C2×C4×C8C2×D1711yescharacteristic
335#335—171no—
336#335—171no—
337#334—171no—
338#334—171no—
339Q8⋊C8—171no—
340Q8⋊C8—171no—
341Q8⋊C8—171no—
342Q8⋊C8—171no—
343D4⋊C8—171no—
344D4⋊C8—171no—
345D4⋊C8—171no—
346D4⋊C8—171no—
347#322C2×D1711yescharacteristic
348#322C2×D1711yescharacteristic
349#322C2×D1711yescharacteristic
350#322C2×D1711yescharacteristic
351C2×C17—41no—
352C2×C17#293311yescharacteristic
353C2×C17#294311yescharacteristic
354C2×C17#295611yescharacteristic
355D17—42no—
356D17—81no—
357C2×#41—171no—
358C2×Q16—171no—
359C2×SD16—171no—
360C2×SD16—171no—
361C2×D8—171no—
362C2×M16—341no—
363C2×M16D6811yescharacteristic
364C2×M16D6811yescharacteristic
365C22×C8—21no—
366C22×C8—341no—
367C22×C8—341no—
368C22×C8—341no—
369#126—341no—
370#125—171no—
371#124—171no—
372#123—341no—
373C22⋊Q8—171no—
374#121—171no—
375C4×Q8—171no—
376C4×Q8—171no—
377C4×Q8—171no—
378C4×D4—171no—
379C4×D4—171no—
380C4×D4—171no—
381C4×D4—341no—
382C4×D4—341no—
383C4×D4—341no—
384#117—171no—
385#117—341no—
386#117—341no—
387C2×C42—341no—
388C2×C42C22×D1711yescharacteristic
389#107—171no—
390#107—171no—
391#106—171no—
392#106—171no—
393C4⋊C8—171no—
394C4⋊C8—171no—
395C4⋊C8—171no—
396C4⋊C8—171no—
397C4⋊C8—341no—
398C4⋊C8—341no—
399C4⋊C8—341no—
400C4⋊C8—341no—
401C4⋊C8C4×D1711yescharacteristic
402C4⋊C8C4×D1711yescharacteristic
403#103—171no—
404#103—171no—
405#103—171no—
406#103—171no—
407#102—171no—
408#102—171no—
409#102—171no—
410#102—171no—
411C22⋊C8—341no—
412C22⋊C8—341no—
413C22⋊C8—341no—
414C22⋊C8—341no—
415C22⋊C8—341no—
416C22⋊C8—341no—
417C22⋊C8C4×D1711yescharacteristic
418C22⋊C8C4×D1711yescharacteristic
419#97—341no—
420#97—341no—
421#97—341no—
422#97—341no—
423#97D6811yescharacteristic
424#97D6811yescharacteristic
425C4×C8—21no—
426C4×C8—341no—
427C4×C8—341no—
428C4×C8C22×D1711yescharacteristic
429C4×C8C22×D1711yescharacteristic
430C4×C8C22×D1711yescharacteristic
431C4×C8C22×D1711yescharacteristic
432C4×C8C22×D1711yescharacteristic
433C4×C8C22×D1711yescharacteristic
434C17#1800211yescharacteristic, complemented, Sylow
435#41—342no—
436#41—681no—
437C2×Q8—171no—
438C2×Q8—171no—
439C2×D4—171no—
440C2×D4—171no—
441C2×D4—341no—
442C2×D4—341no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4C2×C4×D1711yescharacteristic
448C22×C4C2×D6811yescharacteristic
449C22×C4#6325211yescharacteristic
450Q16—681no—
451SD16—342no—
452SD16—681no—
453D8—342no—
454M16—41no—
455M16—41no—
456M16—682no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—42no—
460C2×C8—341no—
461C2×C8—341no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—682no—
465C2×C8—682no—
466C2×C8—682no—
467C2×C8C2×C4×D1711yescharacteristic
468C2×C8C2×C4×D1711yescharacteristic
469C2×C8C2×C4×D1711yescharacteristic
470C2×C8C2×C4×D1711yescharacteristic
471C2×C8C2×D6811yescharacteristic
472C2×C8C2×D6811yescharacteristic
473C2×C8C2×D6811yescharacteristic
474C2×C8C2×D6811yescharacteristic
475C4⋊C4—171no—
476C4⋊C4—171no—
477C4⋊C4—171no—
478C4⋊C4—171no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C22⋊C4—341no—
486C22⋊C4—341no—
487C22⋊C4—341no—
488C22⋊C4—341no—
489C22⋊C4—341no—
490C22⋊C4—341no—
491C22⋊C4—341no—
492C22⋊C4—341no—
493C42—341no—
494C42—341no—
495C42—341no—
496C42—341no—
497C42C2×C4×D1711yescharacteristic
498C42C2×D6811yescharacteristic
499C42#6325211yescharacteristic
500C42C23×D1711yescharacteristic
501C23—341no—
502C23—341no—
503C23C4×D6811yescharacteristic
504Q8—341no—
505Q8—341no—
506Q8—341no—
507Q8—341no—
508D4—172no—
509D4—341no—
510D4—341no—
511D4—341no—
512D4—682no—
513D4—1361no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—341no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—682no—
526C2×C4—681no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—1361no—
531C2×C4C4×D6811yescharacteristic
532C2×C4C4×D6811yescharacteristic
533C2×C4C4×D6811yescharacteristic
534C2×C4C22×C4×D1711yescharacteristic
535C2×C4C22×D6811yescharacteristic
536C2×C4C2×#6325211yescharacteristic
537C8—22no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—21no—
542C8—21no—
543C8—21no—
544C8—21no—
545C8—21no—
546C8—41no—
547C8—681no—
548C8—681no—
549C8—681no—
550C8—681no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—342no—
555C22—681no—
556C22—681no—
557C22—681no—
558C22—681no—
559C22—681no—
560C22C2×C4×D6811yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—681no—
566C4—681no—
567C4—681no—
568C4—681no—
569C4—681no—
570C4—681no—
571C4#131×D1711yescharacteristic
572C4#1176357411yescharacteristic
573C4#1176360911yescharacteristic
574C4#1176360911yescharacteristic
575C2—41no—
576C2—682no—
577C2—1361no—
578C2#1453751211yescharacteristic
579C2#1453758611yescharacteristic
580C2#1453769511yescharacteristic
subgroupG/H
centerC2×C4C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1451C2
socleC22×C17C2×C4×D4
prime radical#11762956C4
cosocleC42×C17C24
hypercenter#1451D17
hypocenterC17#18002
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1451

Complements

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

NH ≅ G/Ntype
2. #14537799C2semidirect
3. #14537788C2semidirect
4. #14527694C2semidirect
5. #14527688C2semidirect
6. #14527657C2semidirect
7. #14527650C2semidirect
8. #14527643C2semidirect
9. #14527643C2semidirect
10. #14527609C2semidirect
11. #14527598C2semidirect
12. #14527564C2semidirect
13. #14527245C2semidirect
14. #14527167C2semidirect
15. C17×#1451C2semidirect
275. #1451D17semidirect
434. C17#18002semidirect

Maximal subgroups

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

subgroupindex
#145388042
#145377992
#145377882
#145276942
#145276882
#145276572
#145276502
#145276432
#145276092
#145275982
#145275642
#145272452
#145271672
C17×#14512
#1800217